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Problem-Based and Inquiry-Based Learning: What’s the difference?

Sep 30th, 2019 by Kathryn Mulholland

“If your goal is to engage students in critical thinking… you need to present interesting challenges to solve, rather than simply explaining how other smart people have already solved those challenges.” – Therese Huston

Problem-Based Learning (PBL) and Inquiry-Based Learning (IBL) are both student-centered teaching pedagogies that encourage active learning and critical thinking through investigation. Both methods offer students interesting problems to consider. And research shows that both PBL and IBL are effective models of learning. 

So, what’s the difference between the two?

According to Banchi and Bell [4], there are four different levels of inquiry.

• Confirmation Inquiry: Students confirm a principle through an activity when the results are known in advance.
• Structured Inquiry: Students investigate a teacher-presented question through a prescribed procedure. 
• Guided Inquiry: Students investigate a teacher-presented question using student designed or selected procedures.
• Open Inquiry: Students investigate questions that are student formulated through student designed or selected procedures.

Most academics define Inquiry-Based-Learning as a pedagogy that is based on one of these levels. So IBL can be as methodical as guiding students through a procedure to discover a known result or as free-form as encouraging students to formulate original questions. For example, in a Physics laboratory, suppose the topic is Newton’s Second Law of Motion. The lab instructions could define a procedure to record the mass and impact force of various objects. Multiplying the mass by the acceleration due to gravity, the students should recover the force they recorded, thus confirming Newton’s Second Law.

Problem-Based-Learning can be classified as guided inquiry where the teacher-presented question is an unsolved, real-world problem. For example, in a Middle Eastern Studies course, the main problem posed by the instructor could be “Propose a solution to the Israeli–Palestinian conflict.” This question will motivate the study of the history of the region, the theological differences between Judaism and Islam, and current events. At the end of the semester, students would be expected to present and justify their solution. 

Therefore, using the definition above, PBL is a type of IBL .

PBL is great because it motivates course content and maximizes learning via investigation, explanation, and resolution of real and meaningful problems. At any level, inquiry can be an effective method of learning because it is student-centered and encourages the development of practical skills and higher-level thinking. 

As you plan for your next class, I invite you to reflect on your method of content delivery. Is it motivated? How? Would your students benefit from a day based on inquiry?

References.

• Inquiry Based Learning. University of Notre Dame Notes on Teaching and Learning. https://sites.nd.edu/kaneb/2014/11/10/inquiry-based-learning/ .
• Problem-Based Learning. Cornell University Center for Teaching Innovation . https://teaching.cornell.edu/teaching-resources/engaging-students/problem-based-learning .
• Hmelo-Silver, Cindy E.; Duncan, Ravit Golan; Chinn, Clark A. (2007). “Scaffolding and Achievement in Problem-Based and Inquiry Learning: A Response to Kirschner, Sweller, and Clark (2006)”. Educational Psychologi st. 42 (2): 99–107. doi : 10.1080/00461520701263368 .
• Banchi, H., & Bell R. (2008). The many levels of inquiry. Science and Children.

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What Is “Inquiry-Based Learning”?: Types, Benefits, Examples

What Is Inquiry-Based Learning?

The 4 types of inquiry-based learning, 7 benefits of inquiry-based learning, 5 inquiry-based learning examples, 5 strategies and tips for implementing inquiry-based learning, 4 models to use in the classroom, let’s wrap, frequently asked questions (faqs).

Are you looking for a teaching strategy that will engage your students in the learning process? Do you want them to be able to ask questions and investigate real-world problems? If so, you should consider using inquiry-based learning in your classroom.

Inquiry-based learning is a teaching method that encourages students to ask questions and investigate real-world problems. This type of learning has many benefits and can be used in various subject areas.

This blog will discuss the benefits of inquiry-based learning and provide some strategies, tips, and models that you can use in your classroom. But first, let’s take a closer look at what inquiry-based learning is.

• What is inquiry-based learning
• Types of inquiry-based learning
• Benefits of inquiry-based learning
• Inquiry-based learning examples
• Strategies for implementing inquiry-based learning in the classroom
• Four models to use in the classroom

Inquiry-based learning is a student-centered teaching method that encourages students to ask questions and investigate real-world problems. In this type of learning environment, students are actively engaged in the learning process and are given the opportunity to explore their natural curiosities.

This type of learning is often hands-on and allows students to connect what they learn in the classroom and the real world. Inquiry-based learning has been shown to improve critical thinking skills, problem-solving skills, and creativity.

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There are four types of inquiry-based learning:

1. The Structured Inquiry Approach

The structured inquiry approach is a sequential process that helps students learn how to ask questions and investigate real-world problems. This type of inquiry-based learning is often used in science classes, where students are given a problem to investigate and are taught how to use the scientific process to find a solution.

2. The Open-Ended Inquiry Approach

The open-ended inquiry approach is a more free-form approach to inquiry-based learning. In this type of learning environment, students are given the freedom to explore their interests and ask questions about the topic they are studying. This type of inquiry-based learning is often used in humanities classes, where students are asked to explore a topic in-depth and debate different viewpoints.

3. The Problem-Based Inquiry Approach

A problem-based inquiry approach is a problem-solving approach to inquiry-based learning. In this type of approach, students are given a real-world problem to solve. This type of inquiry-based learning is often used in mathematics and engineering classes, where students are asked to apply what they have learned to solve a real-world problem.

4. The Guided Inquiry Approach

The guided inquiry approach is a teacher-led approach to inquiry-based learning. In this type of approach, the teacher guides the students through the inquiry process and helps them to ask questions and find solutions to real-world problems. This type of inquiry-based learning is often used in elementary and middle school classrooms.

Now that we have a better understanding of the different types of inquiry-based learning, let’s take a look at the benefits.

With so many benefits, it is no wonder that inquiry-based learning has become a popular teaching method . Some of the benefits of inquiry-based learning include:

1. Encourages critical thinking

Inquiry-based learning encourages students to think critically about the information they are presented with. They are asked to question the information and develop their own solutions. This type of learning helps students develop problem-solving skills and critical-thinking skills.

2. Improves problem-solving skills

Inquiry-based learning helps students develop problem-solving skills. When they are given the opportunity to explore real-world problems, they are forced to think outside the box and come up with their own solutions. This is an important skill that will help them in their future careers.

3. Encourages creativity

This concept of learning encourages creativity. When students are given the opportunity to explore a problem independently, they often come up with creative solutions. This is due to the fact that any particular way of thinking does not restrict them.

4. Improves communication skills

It also helps students improve their communication skills. When working on a problem, they often have to explain their thoughts and ideas to others. This helps them learn how to communicate effectively with others.

5. Connects learning to the real world

Inquiry-based learning helps connect learning to the real world. When students are allowed to explore problems that exist in the real world, they can see how what they are learning in the classroom is relevant. This also helps them develop a better understanding of the material.

6. Helps students understand complex topics

Inquiry-based learning can also help students understand complex topics. When they are allowed to explore these topics in a hands-on environment, they can learn about them more meaningfully.

7. Encourages engaged learning

Finally, this type of learning encourages engaged learning. When students are actively involved in the learning process, they are more likely to retain the information. This is due to the fact that they are invested in what they are doing.

Now that we have looked at the benefits of inquiry-based learning, let’s take a look at some examples.

1. Science Experiments

One way to incorporate inquiry-based learning into your classroom is to allow students to conduct experiments. This will encourage them to ask questions and think critically about the results.

2. Field Trips

Another way to encourage inquiry-based learning is to take students on field trips. This will allow them to explore real-world problems and see how what they are learning in the classroom is relevant.

3. Classroom Debates

Classroom debates are another great way to encourage this type of learning. When students debate a topic, they are forced to think critically about both sides of the argument.

4. Projects

Projects are another great way to encourage inquiry-based learning. When students are given the opportunity to work on a project that is related to the topic they are studying, they will be more likely to learn and remember the information.

5. Group Work

When students work in groups, they are able to share their ideas and thoughts with others. This helps them to understand the material better.

Now that we have looked at the benefits of inquiry-based learning and some examples, let’s look at some inquiry-based strategies and tips that you can use in your classroom.

1. Start with a Question

The best way to start an inquiry-based lesson is by asking a question. This will get students thinking about the topic and will encourage them to ask their own questions.

2. Allow for Exploration

Once you have asked a question, allow students to explore the topic on their own. This will help them to understand the material better.

3. Encourage Discussion

Encourage students to discuss their ideas with each other. This will help them to develop a better understanding of the material.

4. Provide Resources

Be sure to provide students with resources that they can use to explore the topic. This will help them develop a better understanding. Teachers can also give access to online learning platforms like SplashLearn , which further help enhance the knowledge of the concepts.

5. Summarize What Was Learned

At the end of the lesson, be sure to summarize what was learned. This will help students to remember the information.

You can use different models to encourage inquiry-based learning in your classroom. The important thing is that you allow students to be actively involved in the learning process. Let’s have a look at a few models that you can use.

Now that we have looked at the benefits of inquiry-based learning and some strategies for implementing it in your classroom , let’s take a look at four models you can use.

1. The Question Model

The question model is one of the most basic models for inquiry-based learning. It involves asking students questions about the topic you are teaching. This will encourage them to think critically about the material.

2. The Problem-Based Learning Model

The problem-based learning model is another excellent option for inquiry-based learning. This model involves giving students a problem to solve. They will need to think critically about the problem and find a solution.

3. The Project-Based Learning Model

Project-based learning is a great way for students to explore a topic in depth. This model involves giving students a project to work on that is related to the topic you are teaching.

4. The Inquiry Cycle Model

With the inquiry cycle model, students are given the opportunity to ask questions, investigate a topic, and then share their findings. This model allows students to explore a topic in-depth and share their discoveries with others.

Inquiry-based learning is a teaching method that encourages students to ask questions and explore their answers. This type of learning has many benefits, both for students and teachers. In this article, we’ve looked at some of the critical benefits of inquiry-based learning as well as strategies you can use to get started in your own classroom. We hope you’re inspired to give it a try!

What is the importance of inquiry-based learning?

Inquiry-based learning is important because it allows students to explore and ask questions about the world around them. This type of learning helps students develop critical thinking and problem-solving skills.

What is the definition of inquiry-based learning?

Inquiry-based learning is a type of active learning that encourages students to ask questions, conduct research, and explore new ideas. This approach to learning helps students develop critical thinking, problem-solving, and research skills.

What are the roles of students in inquiry-based learning?

In inquiry-based learning, students take on the role of researcher. They are encouraged to ask questions and explore new ideas. Students also have the opportunity to share their findings with their classmates and learn from each other.

How do you plan an inquiry-based lesson?

Inquiry-based lessons are typically designed around a central question or problem. From there, teachers can provide resources and scaffolding to help students investigate the topic. It is important to leave room for student exploration and allow them to ask their own questions.

What are the five guiding questions of inquiry?

The 5 guiding questions of inquiry are:

Do inquiry-based and project-based learning have to be the same thing?

No, inquiry-based and project-based learning are two different approaches. Inquiry-based learning is focused on student-driven research and exploration. Project-based learning is focused on students working together to complete a real-world project. However, both approaches can include elements of inquiry and problem-solving.

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Learn about inquiry-based learning, a teaching approach that encourages students to ask questions, find answers, and make meaningful connections. Discover key strategies for implementing inquiry-based learning in your classroom.

Inquiry-based learning requires teachers to shift the focus away from delivering content and instead emphasize the process of learning. To help students gain a deeper understanding of the world around them, teachers can use Profs online neuroscience tutors to provide guidance and support during the inquiry-based learning process. Teachers should create an environment where students feel comfortable asking questions, exploring ideas, and taking risks through inquiry-based methods. To ensure that students are engaged in their learning, teachers should incorporate opportunities for inquiry-based collaboration and discussion among students.

Moreover, teachers should design meaningful assignments that require students to think critically and make connections between concepts using inquiry-based approaches. To ensure that students are actively involved in their learning, teachers should also provide timely feedback on student progress through inquiry-based methods. This can include providing written or verbal feedback on assignments as well as offering guidance on how to improve work. Teachers can also encourage students to reflect on their own learning process by asking them to think about what worked well and what could be improved upon. Finally, teachers should also consider how they can assess student learning in an inquiry-based classroom. Rather than relying solely on traditional tests or quizzes, teachers should look for ways to assess student understanding through more creative methods such as projects or presentations.

Providing Meaningful Assignments

Additionally, assigning group projects or activities that allow students to collaborate on solving problems is an effective way to further engage students in the inquiry process. In order to create meaningful assignments, teachers should consider providing students with a range of materials to work with. For example, providing students with primary sources such as historical documents or scientific experiments can help them develop a more comprehensive understanding of the material. Additionally, providing students with a range of options for how they can present their research and solutions can help to engage them in the inquiry process.

Offering Timely Feedback

Using technology to provide real-time feedback, giving oral feedback, providing written feedback, using visual aids, creating an inquiry-based learning environment, encourage collaboration:, encourage risk-taking:, provide resources:, be patient:, assessing student learning, observation, performance-based assessments.

This means creating a classroom atmosphere that is conducive to exploration, experimentation, and critical thinking. One way to create an inquiry-based learning environment is to encourage collaboration among students. Group activities and projects can be a great way to foster a collaborative atmosphere in the classroom, as it allows students to work together to find answers to their questions. Additionally, providing students with the freedom to explore and ask questions without fear of being wrong can help create an environment that encourages inquiry.

It is also important for teachers to provide their students with the necessary resources for inquiry-based learning. This includes access to textbooks, reference materials, and other resources such as computers and technology. When students have access to these materials, they are able to research and discover answers on their own. Finally, it is important for teachers to provide guidance and support when necessary.

2.Dialogue:

3.online discussion forums:, 4.peer feedback:, observation:, class discussions:, tests and quizzes:.

By providing meaningful assignments, offering timely feedback, and assessing student understanding through creative methods, teachers can help create an environment where students feel comfortable asking questions and engaging in meaningful discussions. Inquiry-based learning can be used to help students develop critical thinking skills and make meaningful connections that will serve them in their academic and professional lives. Through inquiry-based learning, teachers can help students explore their own curiosities and develop their problem-solving abilities. By encouraging students to ask questions and search for answers, teachers can help create a classroom environment that fosters creativity and collaboration. With the right strategies in place, inquiry-based learning can be a powerful tool for engaging and motivating students.

Shahid Lakha

Shahid Lakha is a seasoned educational consultant with a rich history in the independent education sector and EdTech. With a solid background in Physics, Shahid has cultivated a career that spans tutoring, consulting, and entrepreneurship. As an Educational Consultant at Spires Online Tutoring since October 2016, he has been instrumental in fostering educational excellence in the online tutoring space. Shahid is also the founder and director of Specialist Science Tutors, a tutoring agency based in West London, where he has successfully managed various facets of the business, including marketing, web design, and client relationships. His dedication to education is further evidenced by his role as a self-employed tutor, where he has been teaching Maths, Physics, and Engineering to students up to university level since September 2011. Shahid holds a Master of Science in Photon Science from the University of Manchester and a Bachelor of Science in Physics from the University of Bath.

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Inquiry-based learning

On this page:, what is it.

Inquiry-based learning is an education approach that focuses on investigation and problem-solving. Inquiry-based learning is different from traditional approaches because it reverses the order of learning. Instead of presenting information, or ‘the answer’, up-front, teachers start with a range of scenarios, questions and problems for students to navigate.

Inquiry-based learning prioritises problems that require critical and creative thinking so students can develop their abilities to ask questions, design investigations, interpret evidence, form explanations and arguments, and communicate findings.

How does it help?

Students learn key STEM and life skills through inquiry-based learning. Inquiry-based learning also promotes:

• Social interaction. This helps attention span and develops reasoning skills. Social interaction encourages students to generate their own ideas and critique in group discussions. It develops agency, ownership and engagement with student learning .
• Exploration. This allows students to investigate, design, imagine and explore, therefore developing curiosity, resilience and optimism.
• Argumentation and reasoning. This creates a safe and supportive environment for students to engage in discussion and debate. It promotes engagement in scientific discussion and improves learning of scientific concepts. It encourages students to generate questions , formulate positions and make decisions.
• Positive attitudes to failure. The iterative and evaluative nature of many STEM problems means failure is an important part of the problem-solving process. A healthy attitude to failure encourages reflection, resilience and continual improvement.

How do you do it?

• set a challenge for students
• encourage active student investigations
• make generalisations
• For more information on inquiry-based learning and examples of classroom strategies. Griffith University has prepared a useful resource.

Want to know more?

Research reports.

• STEM Education: A review of the contribution of the disciplines of science, technology, engineering and mathematics - Science Education International Vol. 27, Issue 4, 2016, 530-569
• Opening up pathways : Engagement in STEM across the Primary-Secondary school transition. A review of the literature concerning supports and barriers to Science, Technology, Engineering and Mathematics engagement at Primary- Secondary transition. Commissioned by the Australian Department of Education, Employment and Workplace Relations. June, 2008
• Studies in Science Education - Volume 44, 2008 - Issue 1 - Students' questions: a potential resource for teaching and learning science
• From concept to classroom Translating STEM education research into practice - Australian Council for Educational Research - June 2016

Case study: reSolve: Mathematics by Inquiry

reSolve: Mathematics by Inquiry is a national program to help teachers adopt inquiry-based methods when teaching primary and secondary mathematics. The reSolve approach encourages students to ask questions, test ideas, seek meaning and explain reasons. reSolve provides classroom resources, professional learning modules and a protocol that underpins excellent inquiry-based teaching and learning. It also trains ‘reSolve Champions’: teachers and leaders who take the messages and resources of reSolve into the wider mathematics teaching community. Approximately 300 teachers and leaders have either completed or are undertaking a 12-month professional learning program to become reSolve Champions. The reSolve program is funded by the Australian Government Department of Education Skills and Employment.

How Students Learn: History, Mathematics, and Science in the Classroom (2005)

Chapter: 12 developing understanding through model-based inquiry, 12 developing understanding through model-based inquiry.

James Stewart, Jennifer L. Cartier, and Cynthia M. Passmore

A classroom of students need only look at each other to see remarkable variation in height, hair color and texture, skin tone, and eye color, as well as in behaviors. Some differences, such as gender, are discrete: students are male or female. Others, such as hair color or height, vary continuously within a certain range. Some characteristics—10 fingers, 10 toes, and one head—do not vary at all except in the rarest of cases. There are easily observed similarities between children and their parents or among siblings, yet there are many differences as well. How can we understand the patterns we observe?

Students need only look through the classroom window to take these questions a next step. Birds have feathers and wings—characteristics on which they vary somewhat from each other but on which they are completely distinct from humans. Dogs, cats, and squirrels have four legs. Why do we have only two? As with much of science, students can begin the study of genetics and evolution by questioning the familiar. The questions mark a port of entry into more than a century of fascinating discovery that has changed our understanding of our similarities, our differences, and our diseases and how to cure them. That inquiry has never been more vital than it is today.

It is likely that people observed and wondered about similarities of offspring and their parents, and about how species of animals are similar and distinct, long before the tools to record those musings were available. But major progress in understanding these phenomena has come only relatively recently through scientific inquiry. At the heart of that inquiry is the careful collection of data, the observation of patterns in the data, and the generation of causal models to construct and test explanations for those

patterns. Our goal in teaching genetics and evolution is to introduce students to the conceptual models and the wealth of knowledge that have been generated by that scientific enterprise. Equally important, however, we want to build students’ understanding of scientific modeling processes more generally—how scientific knowledge is generated and justified. We want to foster students’ abilities not only to understand, but also to use such understandings to engage in inquiry.

For nearly two decades, we have developed science curricula in which the student learning outcomes comprise both disciplinary knowledge and knowledge about the nature of science. Such learning outcomes are realized in classrooms where students learn by “doing science” in ways that are similar to the work scientists do in their intellectual communities. We have created classrooms in which students are engaged in discipline-specific inquiry as they learn and employ the causal models and reasoning patterns of the discipline. The topics of genetics and evolution illustrate two different discipline-specific approaches to inquiry. While causal models are central in both disciplines, different reasoning patterns are involved in the use or construction of such models. The major difference is that the reconstruction of past events, a primary activity in the practice of evolutionary biology, is not common in the practice of genetics. The first section of this chapter focuses on genetics and the second on evolution. The third describes our approach to designing classroom environments, with reference to both units.

Our approach to curriculum development emerged as a result of collaborative work with high school teachers and their students (our collaborative group is known as MUSE, or Modeling for Understanding in Science Education). 1 As part of that collaboration, we have conducted research on student learning, problem solving, and reasoning. This research has led to refinements to the instruction, which in turn have led to improved student understanding.

An important step in course design is to clarify what we want students to know and be able to do. 2 Our goal for the course in genetics is for students to come away with a meaningful understanding of the concepts introduced above—that they will become adept at identifying patterns in the variations and similarities in observable traits (phenotypes) found within family lines. We expect students will do this using realistic data that they generate themselves or, in some cases, that is provided. However, while simply being familiar with data patterns may allow students to predict the outcomes of future genetic crosses, it provides a very incomplete understanding of genetics because it does not have explanatory power. Explanatory power comes from understanding that there is a physical basis for those

patterns in the transmission of genetic material (i.e., that there are genes, and those genes are “carried” on chromosomes from mother and father to offspring as a result of the highly specialized process of cell division known as meiosis) and as a result of fertilization.

To achieve this understanding, students must learn to explain the patterns they see in their data using several models in a consistent fashion. Genetics models (or inheritance pattern models) explain how genes interact to produce variations in traits. These models include Mendel’s simple dominance model, codominance, and multiple alleles. But to understand how the observed pairings of genes (the genotype) came about in the first place, students must also understand models of chromosome behavior, particularly the process of segregation and independent assortment during meiosis (the meiotic model).

We have one additional learning outcome for students—that they will couple their understanding of the transmission of the genetic material and their rudimentary understanding of how alleles interact to influence phenotype with an understanding of the relationship of DNA to genes and the role played by DNA products (proteins) in the formation of an organism’s phenotype (biomolecular models). DNA provides the key to understanding why there are different models of gene interaction and introduces students to the frontier of genetic inquiry today.

These three models (genetic, meiotic, and biomolecular) and the relationships among them form the basic conceptual framework for understanding genetics. We have designed our instruction to support students in putting this complex framework in place.

Attending to Students’ Existing Knowledge

While knowledge of the discipline of genetics has shaped our instructional goals, students’ knowledge—the preconceptions they bring to the classroom and the difficulties they encounter in understanding the new material—have played a major role in our instructional design as well.

The genetics course is centered around a set of scientific models. However, in our study of student learning we have found, as have others, 3 that students have misunderstandings about the origin, the function, and the very nature of causal models (see Box 12-1 ). They view models in a “naïve realistic” manner rather than as conceptual structures that scientists use to explain data and ask questions about the natural world. 4

Following our study of student thinking about models, we altered the instruction in the genetics unit to take into consideration students’ prior knowledge about models and particular vocabulary for describing model attributes. Most important, we recognized the powerful prior ideas students had brought with them about models as representational entities and explic-

itly addressed these ideas at the outset of the unit. In the genetics unit, teachers employ tasks early on that solicit students’ ideas about scientific models and explicitly define the term “model” as it will be used in the science unit. Frequently, teachers present sample models that purport to explain the phenomena at hand and ask students to evaluate these models. Teachers create models that have particular shortcomings in order to prompt discussion by students. Most commonly, students will describe the need for a model to explain all the data, predict new experimental outcomes, and be realistic (their term for conceptual consistency). Throughout the course, teachers return to these assessment criteria in each discussion about students’ own inheritance models.

A subsequent study has shown that these instructional modifications (along with other curricular changes in the genetics unit) help students understand the conceptual nature of scientific models and learn how to evaluate them for consistency with other ideas. 6 We now provide an example of an initial instructional activity—the black box—designed to focus students’ attention on scientific modeling.

As Chapter 1 suggests, children begin at a very young age to develop informal models of how things work in the world around them. Scientific modeling, however, is more demanding. Students must articulate their model as a set of propositions and consider how those propositions can be confirmed or disconfirmed. Because this more disciplined modeling is different from what students do in their daily lives, we begin the course with an activity that focuses only on the process of modeling. No new scientific content is introduced. The complexity of the task itself is controlled to focus students on the “modeling game” and introduce them to scientific norms of argumentation concerning data, explanations, causal models, and their relationships. This initial activity prepares students for similar modeling pursuits in the context of sophisticated disciplinary content.

During the first few days of the genetics course, the teacher presents the students with a black box—either an actual box or a diagram and description of a hypothetical box—and demonstrates or describes the phenomenon associated with it. For example, one box is a cardboard detergent container that dispenses a set amount of detergent each time it is tipped, while another is a large wooden box with a funnel on top and an outlet tube at the bottom that dispenses water in varying amounts, shown in Figure 12-1 . Once the students have had an opportunity to establish the data pattern associated with the particular box in question, the teacher explains that the students’ task is to determine what mechanisms might give rise to this observable pattern. During this activity (which can take anywhere from 3 to 11 class periods, depending on the black box that is used and the extent to which students can collect their own data), the students work in small teams. At the conclusion of the task, each team creates a poster representing its explana-

FIGURE 12-1 One black box used in the MUSE science curriculum and typical data patterns associated with the box.

tion for the box mechanism and presents it to the class. Classmates offer criticism and seek clarification during these presentations.

As the dialogue below suggests, the exercise begins with students engaged in a central activity of scientists—making observations.

The students respond with a variety of observations:

After several minutes of listening to the students, the teacher stops them and invites them to take a closer look at the carton, prompting them to identify patterns associated with their observations. Their reflection on these patterns leads the students to propose manipulations of the container, which in turn produce more observations. The teacher now interrupts them to guide their attention, saying:

She goes on to challenge them:

Over the next two class periods, the students work in animated groups to develop models that can be used to explain their observations. They describe, draw, and create three-dimensional representations of what they think is in the carton. They argue. They negotiate. They revise. Then they share drawings of their models with one another.

The students also propose tests of their models:

A visitor to the classroom would notice that Mrs. S. listens attentively to the descriptions that each group gives of its model and the observations the model is designed to explain. She pays special attention to the group’s interactions with other groups and is skillful in how she converses with the students during their presentations. Through her comments she demonstrates how to question the models of others and how to present a scientific argument. To one group she says, “I think I follow your model, but I am not sure how it explains why you get 90 milliliters of liquid each time you tip the box.” To another she comments, “You say that you have used something similar to a toilet bowl valve. But I don’t understand how your valve allows soap to flow in both directions.” And to a third group she asks, “Do you think that Ian’s model explains the data? What question would you ask his group at this point?” By the end of the multiday activity, the students are explicit about how their prior knowledge and experiences influence their observations and their models. They also ask others to explain how a proposed model is consistent with the data and challenge them when a component of a model, designed to explain patterns in observations, does not appear to work as described.

This activity creates many opportunities to introduce and reinforce foundational ideas about the nature of scientific inquiry and how one judges scientific models and related explanations. As the class shares early ideas, the teacher leads discussion about the criteria they are using to decide whether and how to modify these initial explanations. Together, the class establishes that causal models must be able to explain the data at hand, accurately predict the results of future experiments, and be consistent with prior knowledge (or be “realistic”) (see the example in Box 12-2 ). Through discussion and a short reading about scientific inquiry and model assessment, the teacher helps students connect their own work on the black boxes with that of scientists attempting to understand how the natural world works. This framework for thinking about scientific inquiry and determining the validity of knowledge claims is revisited repeatedly throughout the genetics unit.

Other modeling problems might serve just as well as the one we introduce here. What is key is for the problem to be complex enough so that students have experiences that allow them to understand the rigors of scien-

tific modeling. In particular, the activity is designed to give students an opportunity to do the following:

Use prior knowledge to pose problems and generate data . When science teaching emphasizes results rather than the process of scientific inquiry, students can easily think about science as truths to be memorized, rather than as understandings that grow out of a creative process of observing, imagining, and reasoning by making connections with what one already knows. This latter view is critical not only because it offers a view of science that is more engaging and inviting, but also because it allows students to

grasp that what we understand today can be changed, sometimes radically, by tomorrow’s new observations, insights, and tools. By carrying out a modeling activity they see as separate from the academic content they are studying in the unit, students are more likely to engage in understanding how models are generated rather than in learning about a particular model.

Search for patterns in data . Often the point of departure between science and everyday observation and reasoning is the collection of data and close attention to its patterns. To appreciate this, students must take part in a modeling activity that produces data showing an interesting pattern in need of explanation.

Develop causal models to account for patterns . 7 The data produced by the activity need to be difficult enough so that the students see the modeling activity as posing a challenge. If an obvious model is apparent, the desired discourse regarding model testing and consideration of the features of alternative models will not be realized.

Use patterns in data and models to make predictions . A model that is adequate to explain a pattern in data provides relatively little power if it cannot also be used for predictive purposes. The activity is used to call students’ attention to predictive power as a critical feature of a model.

Make ideas public, and revise initial models in light of anomalous data and in response to critiques of others . Much of the schoolwork in which students engage ends with a completed assignment that is graded by a teacher. Progress in science is supported by a culture in which even the best work is scrutinized by others, in which one’s observations are complemented by those of others, and in which one’s reasoning is continually critiqued. For some students, making ideas public and open to critique is highly uncomfortable. A low-stakes activity like this introductory modeling exercise can create a relatively comfortable setting for familiarizing students with the culture of science and its expectations. A teacher might both acknowledge the discomfort of public exposure and the benefits of the discussion and the revised thinking that results in progress in the modeling effort. Students have ample opportunity to see that scientific ideas, even those that are at the root of our most profound advances, are initially critiqued harshly and often rejected for a period before they are embraced.

Learning Genetics Content

Having provided this initial exposure to a modeling exercise, we turn to instruction focused specifically on genetics. While the core set of causal models, assumptions, and argument structures generated the content and learning outcomes for our genetics unit, our study of student understanding and reasoning influenced both the design and the sequencing of instructional activities. For example, many high school students do not understand

the interrelationships among genetic, meiotic, and biomolecular models, relationships that are key to a deep understanding of inheritance phenomena. 8 To deal with this problem, we identified learning outcomes that address the conceptual connections among these families of models, and the models are introduced in a sequence that emphasizes their relatedness. Initially, for example, we introduced genetic models, beginning with Mendel’s model of simple dominance, first. This is typical of many genetics courses. In our early studies (as well as in similar studies on problem solving in genetics 9 ), students often did not examine their inheritance models to see whether they were consistent with meiosis. In fact, students proposed models whereby offspring received unequal amounts of genetic information from their two parents or had fewer alleles at a particular locus than did their parents. 10 Because of their struggles and the fact that meiosis is central to any model of inheritance, we placed this model first in the revised curricular sequence. Students now begin their exploration of Mendelian inheritance with a firm understanding of a basic meiotic model and continue to refer to this model as they examine increasingly complex inheritance patterns.

A solid integration of the models does not come easily, however. In early versions of the course, it became apparent that students were solving problems, even sophisticated ones, without adequately drawing on an integrated understanding of meiotic and genetic models. 11 In response, we designed a set of data analysis activities and related homework that required students to integrate across models (cytology, genetics, and molecular biology) when conducting their genetic investigations and when presenting model-based explanations to account for patterns in their data. By providing tasks that require students to attend to knowledge across domains and by structuring classrooms so that students must make their thinking about such integration public, we have seen improvements in their understanding of genetics. 12

We then focus on inheritance models, beginning with Mendel’s model of simple dominance. Mendel, a nineteenth-century monk, grew generation after generation of pea plants in an attempt to understand how traits were passed from parent plants to their offspring. As Chapter 9 indicates, Mendel’s work represented a major breakthrough in understanding inheritance, achieved in large part by selecting a subject for study—peas—that had discontinuous trait variations. The peas were yellow or green, smooth or wrinkled. Peas can be self-fertilized, allowing Mendel to observe that some offspring from a single genetic source have the same phenotype as the parent plants and some have a different phenotype. Mendel’s work confirmed that individuals can carry alleles that are recessive—not expressed in the phenotype. By performing many such crosses, Mendel was able to deduce that the distribution of alleles follows the laws of probability when the pairing of alleles is random. These insights are fundamental to all the work

in classical transmission genetics since Mendel. Students need ample opportunity to work with Mendel’s model if they are to make these fundamental insights their own.

The development of modern genetic theory from its classical Mendelian origins has been the subject of much historical and philosophical analysis. Darden 13 draws on historical evidence to identify a set of strategies used by scientists to generate and test ideas while conducting early inquiries into the phenomenon of inheritance. She traces the development of a number of inheritance models that were seen at least originally to be at odds with those underlying a Mendelian (i.e., simple dominance) explanation of inheritance. Among these models are those based on the notions of linkage and multiple forms (alleles) of a single gene. In short, Darden provides a philosophical analysis of the history of model-based inquiry into the phenomenon of inheritance from a classical genetics perspective. Drawing on Darden’s work and our own experiences as teachers and researchers, we made a primary feature of the course engaging students in building and revising Mendel’s simple dominance model. Students thereby have rich opportunities to learn important genetics concepts, as well as key ideas about the practice of genetics.

Inheritance is considerably more complex than Mendel’s simple dominance model suggests. Mendel was not wrong. However, simple dominance applies to only a subset of heritable traits. Just as geneticists have done, students need opportunities to observe cases that cannot be explained by a simple dominance model. We provide such opportunities and thus allow students to conclude that Mendel’s model is not adequate to explain the data. Students propose alternatives, such as the codominance model, to explain these more complex patterns.

Once students have come to understand that there are multiple models of allele interaction, they are primed for an explanation of why we observe these different inheritance patterns. How can a recessive allele sometimes have an influence and sometimes not? With that question in mind, we introduce DNA and its role in protein production. What drives the instructional experience throughout is students’ active engagement in inquiry, which we turn to in the next section.

Student Inquiry in Genetics

Early instruction in the genetics class includes a few days during which students learn about the meiotic model 14 and the phenomena this model can explain. In an introductory activity, students look at sets of pictures and are asked to determine which individuals are members of the same families. The bases for their decisions include physical similarities between parents and children and between siblings. Thus, instruction about meiosis focuses

on how the meiotic model can account for these patterns: children resemble their parents because they receive information from both of them, and siblings resemble each other but are not exactly alike because of the random assortment of parental information during meiosis.

After students have developed some understanding of meiosis, they create, with guidance from the teacher, a representation of Mendel’s model of simple dominance (see Figures 12-2a and 12-b ) in an attempt to further explain why offspring look like parents. First, “Mendel” (a teacher dressed in a monk’s habit) pays the class a visit and tells them he would like to share some phenomena and one important model from his own research with them. In character, “Mendel” passes out three packets of peas representing a parental generation and the F1 and F2 generations (the first and second filial generations, respectively). He asks the students to characterize the peas according to color and shape. For example, the parental generation includes round green peas and wrinkled yellow peas. The F1 generation contains only round yellow peas. Finally, the F2 generation contains a mix of round yellow, wrinkled yellow, round green, and wrinkled green peas in a ratio of approximately 9:3:3:1. Using what they already know about meiosis—particularly the fact that offspring receive information from both parents—the students reconstruct Mendel’s model of simple dominance to explain these patterns (see Figures 12-2a and 12-b ).

While Darden’s work (discussed above) aides in the identification of important inheritance models and strategies used by scientists to judge those models, it is the work of Kitcher 15 that places the simple dominance model developed by students into context with comparable models of geneticists. According to Kitcher, 16 genetic models provide the following information:

(a) Specification of the number of relevant loci and the number of alleles at each locus; (b) Specification of the relationships between genotypes and phenotypes; (c) Specification of the relations between genes and chromosomes, of facts about the transmission of chromosomes to gametes (for example, resolution of the question whether there is disruption of normal segregation) and about the details of zygote formation; (d) Assignment of genotypes to individuals in the pedigree.

Moreover, Kitcher 17 describes how such models might be used in inquiry:

… after showing that the genetic hypothesis is consistent with the data and constraints of the problem, the principles of cytology and the laws of probability are used to compute expected distributions of phenotypes from crosses. The expected distributions are then compared with those assigned in part (d) of the genetic hypothesis.

FIGURE 12-2 Mendel’s model of simple dominance.

(a) Students’ representation of Mendel’s simple dominance model. This model accounts for the inheritance of discrete traits for which there are two variants (designated A and B). Each individual in the population possesses two alleles (designated 1 and 2) for the trait; one allele (here, allele 1) is completely dominant over the other. For plant height, for example, there are two phenotypic variants: short and tall. There are only two different alleles in the population. Plants with a genetic makeup of (1,1) or (1,2) will be tall, whereas plants with a genetic makeup of (2,2) will be short.

(b) Meiotic processes governing inheritance. The underlying processes governing simple dominance are Mendel’s law of segregation (the meiotic process of sex cell formation during which half of all parental genetic information is packaged into sperm or egg cells) and fertilization (during which genetic information from both parents combines in the offspring).

With their teacher’s guidance, students represent Mendel’s simple dominance model in a manner consistent with Kitcher’s description of the models of geneticists. They pay particular attention to (b) and (d) above: specifying the relationships between genotypes and phenotypes and identifying the genotypes of individuals in their experimental populations. Because our unit does not address multigene traits, one locus per trait is assumed (thus part of criterion (a) above is not applicable in this case), and students focus on determining the number of alleles at that locus. Finally, the students’ prior understanding of meiosis—developed earlier in the unit—enables them to specify chromosomal transmission of genes for each particular case (item (c) above). The vignette below portrays students engaged in this type of inquiry.

Genetic Inquiry in the Classroom: A Vignette

Nineteen students are sitting at lab tables in a small and cluttered high school biology classroom. The demonstration desk at the front of the room is barely visible under the stacks of papers and replicas of mitotic cells. A human skeleton wearing a lab coat and a sign reading “Mr. Stempe” stands in a corner at the front of the room, and the countertops are stacked with books, dissecting trays, and cages holding snakes and gerbils.

During the previous few days, the students in this class have studied the work of Mendel. Years of work resulted in his publication of Experiments on Plant Hybridization , 18 a paper in which he presented his model explaining the inheritance of discontinuous traits in plants. 19 The students have read an edited version of this paper and refer to Mendel’s idea as the “simple dominance model” because it explains the inheritance of traits derived from two alleles (or pieces of genetic information) when one of the alleles is completely dominant over the other (see Figures 12-2a and 12-2b ).

During class on this day, the students’ attention is drawn to the cabinet doors along the length of the room. These doors are covered with students’ drawings of family pedigrees labeled “Summers: Marfan” (see Figure 12-3a ), “Healey: Blood Types,” “Jacques: Osteogenesis Imperfecta,” and “Cohen: Achondroplasia.” The teacher is standing at the side of the room facilitating a discussion about these family pedigrees.

Kelly walks to one of the cabinets at the side of the room and begins to label each of the circles and squares on the pedigree with two alleles: some are assigned the genotype 1,2 (heterozygous or possessing two different alleles) and others 2,2 (homozygous recessive or possessing two recessive alleles) (see Figures 12-3a and 12-3b , respectively).

Curtis proceeds to label the same pedigree consistently with his idea that the Marfan allele is actually recessive (see Figure 12-3c ).

FIGURE 12-3 Pedigrees representing inheritance of Marfan symdrome in the Summers family. (a) The original pedigree, representing the inheritance pattern within the Summers family without specifying individual genotypes.

FIGURE 12-3 (b) Kelly’s genotype assignments, assuming that Marfan syndrome is inherited as a dominant trait.

FIGURE 12-3 (c) Curtis’ genotype assignments, assuming that Marfan syndrome is inherited as a recessive trait.

The students in this high school biology class are engaged in genetic inquiry: they are examining data and identifying patterns of inheritance for various traits. They are also attempting to use a powerful causal model, Mendel’s model of simple dominance, to explain the patterns they see. And just as scientists do, they recognize the limitations of their model when it simply cannot explain certain data patterns. These students are poised to continue their inquiry in genetics by revising Mendel’s model such that the resulting models will be able to explain a variety of inheritance patterns, including the multiple allele/codominance pattern within the Healey pedigree.

Multiple Examples in Different Contexts

Chapter 1 argues that learning new concepts with understanding requires multiple opportunities to use those concepts in different contexts. Our course is designed to provide those opportunities. Once the students have represented and used the simple dominance model to explain phenomena such as the inheritance of characteristics in peas and disease traits in humans, they use the model to explain data they generate using the software program Genetics Construction Kit or GCK. 20 This program enables students to manipulate populations of virtual organisms (usually fruit flies) by performing matings (or crosses) on any individuals selected. Each cross produces a new generation of organisms whose variations for particular traits (e.g., eye color, wing shape) are described. Thus, the students develop expertise using the simple dominance model to explain new data, and they also design and perform crosses to test their initial genotype-to-phenotype mappings within these populations.

The beginning of this process is illustrated in Figure 12-4 , which shows an excerpt from one student’s work with GCK and the simple dominance model. After the student’s model is discussed, the teacher presents or revisits phenomena that the simple dominance model cannot explain. For example, students realize when applying the model to explain their human pedigrees that it is inadequate in some cases: it cannot account for the inheritance of human blood types or achondroplasia. The next step for the class is to study these “anomalous” inheritance patterns using GCK. They begin with achondroplasia, a trait for which there are three variations rather than two. Students revise the simple dominance model to account for the codominant

FIGURE 12-4 Example of student work on a GCK homework assignment. Students were asked to infer as much as possible from each successive cross within this population. The student’s work is shown to the right of each cross.

inheritance pattern observed for this trait. While solving GCK problems such as this, students propose models that specify some or all of the information (a through d) noted above and then test their models for fit with existing data, as well as for the ability to predict the results of new experiments accurately.

Since most students ultimately explain the inheritance of achondroplasia using a codominant model (whereby each possible genotype maps to a distinct phenotype), they must also revise their understanding of dominance and recessiveness. Up to this point, most students tend to associate recessiveness with either (1) a phenotype, (2) any genotype that contains a recessive allele (designated with the number 2), or (3) both. It is quite common for students to conclude that the phenotypes associated with (1,1) and (1,2) genotypes are both “recessive.” 21 However, this conclusion is inconsistent with the students’ prior concept of recessiveness as it was developed under the simple dominance model. Thus, it is at this point in the unit that we emphasize the need for models to be consistent with other knowledge in a scientific discipline. In other words, geneticists must assess a new inheritance model in part on the basis of how well it fits within an existing family of related models, such as those for meiosis (including cytological data) and molecular biology (which specifies the relationships between DNA and proteins, as well as protein actions in cells). After explicit instruction about DNA transcription, translation, and protein function, students attempt to reconcile their codominance models with this new model of protein action in cells. In the case of codominance, doing so requires them to conceptualize recessiveness at the level of alleles and their relationships to one another, rather than at the level of phenotypes or genotypes. 22 In the process, students construct meanings for dominance and recessiveness that are consistent across various inheritance models (e.g., simple dominance, codominance, multiple alleles, etc.), as well as models of meiosis and molecular biology.

For the final GCK inquiry, the students are organized into two research teams, each of which consists of four small research groups. Each team is assigned a population of virtual fruit flies and told to explain the inheritance of four traits within this population (see Figure 12-5 ). The work is divided such that each research group studies two of the traits. Consequently, there is some overlap of trait assignments among the groups within a team. The teams hold research meetings periodically, and a minimal structure for those meetings is imposed: two groups present some data and tentative explanations of the data, one group moderates the meeting, and one group records the proceedings. The roles of individual groups alternate in successive meetings.

Each of the fly populations in this last problem contains traits that exhibit the following inheritance patterns: (1) Mendelian simple dominance; (2) codominance; (3) multiple alleles (specifically, three different alleles with varying dominant/codominant relationships between pairs of alleles); and

FIGURE 12-5 Initial GCK population for the final GCK inquiry.

(4) x-linkage. After about a week of data collection, model testing, and team meetings, each small research group is usually able to describe a model of inheritance for at least one of the traits in its population, and most groups can describe inheritance models for both of the traits on which they chose to focus. The entire class then gathers for a final conference during which students create posters that summarize their research findings, take turns

making formal presentations of their models, and critique their classmates’ models.

This high school biology curriculum is designed to give students opportunities to learn about genetic inquiry in part by providing them with realistic experiences in conducting inquiry in the discipline. As a primary goal of practicing scientists is to construct causal models to explain natural phenomena, involving students in the construction of their own models is given major emphasis in the classroom. The students work in groups structured like scientific communities to develop, revise, and defend models for inheritance phenomena. The overall instructional goals include helping students to understand mechanistic explanations for inheritance patterns in fruit flies and humans, and to appreciate the degree to which scientists rely on empirical data as well as broader conceptual knowledge to assess models.

Metacognition: Engaging Students in Reflective Scientific Practice

Ultimately, students need to learn to reflect on and judge their own work rather than relying solely on assessments from others. Several early studies of students’ GCK work in our genetics unit revealed that students assessed their tentative models primarily on the basis of empirical rather than conceptual criteria. 23 Even when conceptual inconsistencies occurred between the students’ proposed models and other models or biological knowledge, their primary focus was usually on how well a given model could explain the data at hand. They frequently had difficulty recognizing specific inconsistencies between their models and meiosis or other biological knowledge, such as the method of sex determination in humans. In some instances, students recognized that their models were inconsistent with other knowledge but were willing to overlook such inconsistencies when they judged their models to have adequate explanatory power. (For example, students sometimes proposed models to account for x-linkage inheritance patterns wherein a male organism simply could never be heterozygous. They gave no explanation consistent with independent assortment in meiosis for this model.) Thus, students paid more attention to empirical than conceptual issues and tended to value empirical power over conceptual consistency in models when both criteria were brought to bear.

White and Frederiksen 24 describe a middle school science curriculum designed to teach students about the nature of inquiry generally and the role of modeling in specific scientific inquiries. One aspect of the curriculum that had a measurable effect on its success was the emphasis on students’ reflective (metacognitive) assessment. Following modeling activities, students were asked to rate themselves and others in various categories, including “understanding the science,” “understanding the processes of inquiry,” “being systematic,” and “writing and communicating well.” Involving the students in

such an explicit evaluation task helped emphasize the importance of learning about inquiry and modeling in addition to learning how to do inquiry.

Our work in developing tasks for students is also predicated on the importance of metacognitive reflection on the students’ part. Influenced by our research in the genetics unit, we built into the curriculum more tasks that require students to reflect upon, write about, and discuss conceptual aspects of genetic modeling. These tasks include journal writing, written self-assessments, homework assignments that require students to explain their reasoning (see Box 12-4 ), and class presentations (both formal and

informal). Most important, we created a complex problem involving several different inheritance patterns and asked the students to account for these new data while working in cooperative laboratory teams. As described above, the regular team interactions required students to be critical of their own thinking and that of others. Moreover, situating the study of these inheritance patterns within the context of a single population of organisms helped emphasize the need for each inheritance model to be basically consistent with other models within genetics. We have found that in this new context, students are more successful at proposing causal models and have a better understanding of the conceptual nature of such scientific models. 25

The structure of the genetics class that we have described reflects important aspects of scientific practice: students are engaged in an extended inquiry into inheritance in which they collect data, seek patterns, and attempt to explain those patterns using causal models. The models proposed by students are also highly similar to those of practicing geneticists in that they specify allelic relationships and genotype-to-phenotype mappings for particular traits. In the next section, we describe a course in evolutionary biology that provides opportunities for students to participate in realistic inquiry within another subdiscipline of biology.

DEVELOPING DARWIN’S MODEL OF NATURAL SELECTION IN HIGH SCHOOL EVOLUTION

Hillary and Jerome are sitting next to each other in their sixth-hour science class waiting for the bell to ring.

The bell rings, and their teacher announces that the class will start work on the last of three case studies designed to allow the students to continue to develop and use Darwin’s model of natural selection. She tells the students that there are two parts to this third case. First, they will need to use their knowledge of the natural selection model to develop an explanation for the bright coloration of the male ring-necked pheasant. Second, they will have to write a research proposal that will then be considered by the rest of the students in a research grant competition.

As the teacher passes out the eight pages of case materials, she asks the students to get to work. Each group receives a file folder containing the task description and information about the ring-necked pheasant. There are color pictures that show adult males, adult females, and young. Some of the pages contain information about predators, mating behavior, and mating success. Hillary, Jerome, and their third group member, Grace, begin to shuffle through the pages in the folder.

The three students spend the remainder of the period looking over and discussing various aspects of the case. By the middle of the period on Tuesday, this group is just finalizing their explanation when Casey, a member of another group, asks if she can talk to them.

These two groups have very different explanations. Hillary’s group is thinking that the males’ bright coloration distracts predators from the nest, while Casey’s group has decided that the bright coloration confers an advantage on the males by helping them attract more mates. A lively discussion ensues.

The groups agree to disagree on their interpretation of this piece of data and continue to compare their explanations on other points.

The students in the above vignette are using Darwin’s model of natural selection and realistic data to create arguments about evolution in a population of organisms. In doing so, they attend to and discuss such ideas as selective advantage and reproductive success that are core components of the Darwinian model. Early in the course, students have opportunities to learn about natural selection, but as the course progresses, they are required to use their understanding to develop explanations (as illustrated in the vignette). As was true in teaching genetics, our goals for student learning include both deep understanding of evolution and an understanding of how knowledge in evolution is generated and justified. And once again we want students to be able to use their understanding to engage in scientific inquiry—to construct their own Darwinian explanations.

There is an important difference between the two units, however, that motivated the decision to include both in this chapter. The nature of the scientific inquiry involved in the study of evolution is different from that involved in the study of genetics—or in some other scientific disciplines for that matter. The difference arises because of the important role that history plays in evolution and the inability of biologists to “replay the tape of the earth’s history.” Engaging students in authentic inquiry therefore presents a new set of challenges. Mayr 26 suggests that “there is probably no more original, more complex, and bolder concept in the history of ideas than Darwin’s mechanistic explanation of adaptation.” Our teacher/researcher collaborative took on the challenge of designing a course that would allow students to master this powerful concept and to use it in ways that are analogous to those of evolutionary biologists.

Attending to Significant Disciplinary Knowledge

The choices we make when designing curricula are determined in part by an examination of the discipline under study. In the case of evolution, it is clear that a solid understanding of natural selection provides a foundation upon which further knowledge depends—the knowledge-centered conceptual framework referred to in the principles of How People Learn (see Chapter 1 ). But that foundation is hard won and takes time to develop because the concepts that make up the natural selection model are difficult for students to understand and apply. To understand natural selection, students must understand the concept of random variation. They must understand that while some differences are insignificant, others confer an advantage or a disadvantage under certain conditions. The length of a finch’s beak, for example, may give it access to a type of food that allows it to survive a drought. Survivors produce offspring, passing their genes along to the next generation. In this way, nature “selects” for particular characteristics within species.

Equally important in our instructional approach is that students understand how Darwinian explanations are generated and justified. Kitcher 27 describes a Darwinian history as a “narrative which traces the successive modifications of a lineage of organisms from generation to generation in terms of various factors, most notably that of natural selection.” The use of narrative explanation is a key means of distinguishing evolutionary biology from other scientific disciplines. “Narratives fix events along a temporal dimension, so that prior events are understood to have given rise to subsequent events and thereby explain them.” 28 Thus, our concept of a Darwinian explanation draws together the components of the natural selection model and a narrative structure that demands attention to historical contingency. Textbook examples of explanations for particular traits frequently take the

form of “state explanations”—that is, they explain the present function of particular character states without reference to their history. 29 In contrast, what we call a Darwinian explanation attempts to explain an event or how a trait might have come into being. This type of explanation is summarized by Mayr: 30

When a biologist tries to answer a question about a unique occurrence such as “Why are there no hummingbirds in the Old World?” or “Where did the species Homo sapiens originate?” he cannot rely on universal laws. The biologist has to study all the known facts relating to the particular problem, infer all sorts of consequences from the reconstructed constellation of factors, and then attempt to construct a scenario that would explain the observed facts in this particular case. In other words, he constructs a historical narrative.

Providing opportunities for students to use the natural selection model to develop narrative explanations that are consistent with the view described above is a central goal of the course.

Attending to Student Knowledge

Anyone who has ever taught evolution can attest to the fact that students bring a wide range of conceptions and attitudes to the classroom. During the past two decades, researchers have documented student ideas both before and after instruction. 31 These studies have confirmed what teachers already know: students have very tenacious misconceptions about the mechanism of evolution and its assumptions.

As Mayr suggests, the scientific method employed by evolutionary biologists in some respects resembles history more than it does other natural sciences. This resemblance can be problematic. In disciplines such as history, for example, we look for motivations. While students may struggle to understand that in different times and under different circumstances, the motivations of others may be different from our motivations today, motivation itself is a legitimate subject for inquiry. But in the Darwinian model, naturally occurring, random variation within species allows some individuals to survive the forces of nature in larger numbers. The random nature of the variation, the role of natural phenomena in selecting who flourishes and who withers, and the absence of motivation or intent make Darwinian narrative antithetical to much of the literary or historical narrative that students encounter outside the science classroom.

We have found that replacing this familiar approach to constructing a narrative with the scientific approach used in evolutionary biology requires

a significant period of time and multiple opportunities to try out the Darwinian model in different contexts. Many courses or units in evolutionary biology at the high school level require far shorter periods of time than the 9 weeks described here and also include additional sophisticated concepts, such as genetic drift and speciation. With a large number of concepts being covered in a short period of time, however, the likelihood that students will develop a deep understanding of any concept is diminished; a survey of content is not sufficient to support the required conceptual change.

In the next section, we highlight key instructional activities that we have developed over time to support students in acquiring an understanding of evolution and an ability to engage in evolutionary inquiry.

Instruction

The three principles of How People Learn are interwoven in the design of the instructional activities that make up the course in evolutionary biology. For example, the related set of concepts that we consider to be central to students’ understanding (Principle 2) was expanded when we realized that students’ preconceptions (about variation, for example) or weak foundational knowledge (about drawing inferences and developing arguments) served as barriers to learning. Instructional activities designed to support students’ ability to draw inferences and make arguments at the same time strengthen their metacognitive abilities. All three principles are tightly woven in the instruction described below.

Laying the Groundwork

Constructing and defending Darwinian explanations involves drawing inferences and developing arguments from observed patterns in data. In early versions of the course, we found that students’ ability to draw inferences was relatively weak, as was their ability to critique particular arguments. Our course has since been modified to provide opportunities for students to develop a common framework for making and critiquing arguments. As with the black box activity at the beginning of the genetics course, we use a cartoon sequencing activity that does not introduce course content, thus allowing students to focus more fully on drawing inferences and developing arguments.

Students are given a set of 13 cartoon frames (see Box 12-5 ) that have been placed in random order. Their task is to work with their group to reconstruct a story using the information they can glean from the images. Students are enthusiastic about this task as they imagine how the images relate to one another and how they can all be tied together in a coherent story. The whole class then assembles to compare stories and discuss how

decisions were made. The sequences presented by different groups usually vary quite a bit (see Box 12-5 for two examples). This variation provides a context for discussing how inferences are drawn.

The initial discussion centers on students’ observations about the images. However, it quickly becomes apparent that each person does not place the same importance on specific observations and that even though groups may have observed the same thing, they may not have made the same decisions about the order of the cards. What ensues is a conversation about considerations that entered into the students’ decision making. Students realize that they are all examining the same images (the data), but that each also brings a lifetime of experience with cartoons and stories to the table. Together the students establish that the process of drawing inferences about the order of the cards is influenced by both what they observe (the data) and their own prior knowledge and beliefs. This notion is then generalized, and students see that all inferences can be thought of as having these two bases. They discuss how scientific arguments are usually a collection of several inferences, all of which are dependent on data and prior knowledge and beliefs. The teacher supports this discussion by pointing out examples of fruitful questioning and encouraging the students to think about what it means to foster a community in which communication about important ideas is expected.

In addition to introducing general norms for classrooms in which scientific argumentation is central, the cartoon activity serves to orient students to a framework for critiquing arguments in evolution. At one level, this framework is common to all science disciplines. In this capacity, the emphasis is on the importance of being explicit about how prior knowledge and beliefs influence the inferences drawn from particular data. At this general level, the activity is linked to the common MUSE framework of models and modeling as the teacher connects the ideas concerning inferences to those concerning models. The teacher does this by explaining that a causal model is an idea that is used to create explanations for some set of phenomena and that models are based on several inferences. Students then read some material on models and as a class discuss the ways in which models can be assessed. Through examples in the reading and from their own experience, the group settles on criteria for judging models: explanatory power and consistency with other knowledge. Note that, in contrast with the genetics course, there is no mention of predictive adequacy here as a major assessment criterion because explanation is much more central than prediction in the evolution course. This is one example of the assertion we have made previously: disciplines do rely on differing methods for making and evaluating claims. The demonstrative inference that is common in the genetics course gives way to a greater reliance on nondemonstrative inference in the evolution course. This occurs as students create Darwinian explanations. Such expla-

nations, with their characteristic narrative structure, are developed to make sense out of the diverse data (structural and behavioral characteristics of organisms, patterns in their molecular biology, patterns of distribution in both time and geography, and so on) that are characteristic of evolutionary argumentation.

A second evolution-specific function served by the cartoon activity is to introduce students to one of the more important undertakings of evolutionary biologists—the reconstruction of past events (the development of a trait, such as the vertebrate eye, or the speciation events that led to the “tree of life”). Such historical reconstructions do not have close analogues in genetic inquiry.

A second instructional component was added to the course when we observed students’ difficulties in understanding the concept of variation. These difficulties have been documented in the literature, 32 and we have encountered them in our own classrooms. Because of the experiences students have with variability in most genetics instruction—in which they usually examine traits with discrete variations—the concept of continuous variation can be a significant challenge for them. We have seen that an incomplete understanding of variation in populations promotes students’ ideas that adaptations are a result of a single dramatic mutation and that selection is an all-or-none event operating on one of two to three possible phenotypes. Recognition of these problems has led us to incorporate explicit instruction on variability in populations and, perhaps more important, to provide opportunities for students to examine and characterize the variability present in real organisms before they begin using the concept in constructing Darwinian explanations.

One of the activities used for this purpose is a relatively simple one, but it provides a powerful visual representation on which students can draw later when thinking about variation in populations. Typically, students do not recognize the wide range of variation that is present even in familiar organisms. To give them experience in thinking about and characterizing variation, we have them examine sunflower seeds. Their task is to count the stripes on a small sample of seeds (but even this simple direction is less than straightforward since the class must then negotiate such matters as what counts as a stripe and whether to count one side or two).

Once they have come up with common criteria and have sorted their sample into small piles, the teacher has them place their seeds into correspondingly numbered test tubes. The result, once the test tubes have been lined up in a row, is a clear visual representation of a normal distribution. The subsequent discussion centers on ways to describe distributions using such concepts as mean, median, and mode. This activity takes place before students need to draw on their understanding of variation to construct explanations using the natural selection model.

Understanding the Darwinian Model

The second major section of the course engages students in examining three historical models that account for species’ adaptation and diversity. The students must draw on the framework established during the cartoon activity to accomplish this comparison. This means that as they examine each argument, they also identify the major inferences drawn and the data and prior knowledge and beliefs that formed the basis for those inferences. The three models are (1) William Paley’s model of intelligent design, which asserts that all organisms were made perfectly for their function by an intelligent creator; (2) Jean Baptiste de Lamarck’s model of acquired characteristics, which is based on a view that adaptations can result from the use or disuse of body parts and that changes accumulated during an organism’s lifetime will be passed on to offspring; and (3) Darwin’s model of natural selection. The models of Paley and Lamarck were chosen because each represents some of the common ideas students bring with them to the classroom. Specifically, it is clear that many students attribute evolutionary change to the needs of an organism and believe that extended exposure to particular environments will result in lasting morphological change. Many students are also confused about the role of supernatural forces in evolution. Darwin’s model is included in the analysis so students can see how the underlying assumptions of his model compare with those of the Paley and Lamarck models.

For students to compare the prior knowledge and beliefs of the authors, however, they must first become familiar with the models. To this end, each model is examined in turn, and students are discouraged from making comparisons until each model has been fully explored. All three models are presented in the same way. Students read edited selections of the author’s original writing, answer questions about the reading, and participate in a class discussion in which the proposed explanation for species diversity and adaptation is clarified and elaborated. In the following example, Claire and Casey are working with Hillary in a group during class. They are trying to analyze and understand an excerpt of original writing by Lamarck. Hillary is looking over the discussion questions:

Students are also given an opportunity to explore the natural phenomena or data that served as an inspiration for each author: they examine fossils as discussed by Lamarck, dissect an eye to examine the structure/ function relationships that so fascinated Paley, and are visited by a pigeon breeder who brings several of the pigeon varieties that Darwin described in his Origin of Species . Once students have developed an understanding of the explanation that each author proposed and some familiarity with the observations on which it was based, they examine the readings again to identify the prior knowledge and beliefs that each author may have held.

Following this discussion, the students compare the three models. First, they assess the explanatory power of the models, using each to explain phenomena other than those described in the original writings. For example, they attempt to use Paley’s model to explain the presence of fossils and Lamarck’s model to explain the structure of the eye. Sometimes the model can easily account for new phenomena; Lamarck’s model of use inheritance, for example, is easily adapted to explaining the diversity of pigeon varieties. In other instances, the students recognize the limitations of the model; Paley’s model, for instance, cannot easily account for the presence of fossils or extinct organisms. The students then compare the underlying assumptions or beliefs of the authors. Even if a model can account for diverse phenomena on its own terms, it is still necessary to examine and critique the underlying assumptions. Many students question the necessity of the supernatural force underlying Paley’s model, and still more find the role of need to be a questionable assumption in Lamarck’s model.

These explicit discussions of some of the major views students bring to the study of evolution lay the groundwork for the future use and extension of Darwin’s model. Comparing the assumptions of the three models enables students to distinguish between those beliefs that underlie the model of natural selection and those that do not. Unlike some classroom contexts, however, in which it is the students’ ideas that are laid bare and examined for inconsistencies, here we have developed a situation in which students’ ideas are represented by the models of Paley and Lamarck. We have found that through this approach, students are willing to attend to the differences between ideas rather than spending their time and energy being defensive; because they do not feel that their own ideas are being criticized, the discussions are fruitful.

These two activities foster a classroom community that operates from a common set of commitments. For our purposes, the most important of these is that Darwin proposed a naturalistic mechanism of species change that acts on variation among individuals within a species and that assumptions of supernatural influence and individual need are not a part of his model. Keeping this distinction in mind while using the natural selection model later in the course enables students to avoid some common misconceptions, or at least makes identification of those misconceptions more straightforward. For example, when students use the natural selection model to explain the bright coloration of the monarch butterfly, they often challenge each other when need-based or Lamarckian language is used.

Using the Darwinian Model

During the final weeks of the course, students are engaged in creating Darwinian explanations using the components of the natural selection model to make sense of realistic data they have been given. Each scenario is presented to the students as a case study, and they are given materials that describe the natural history of the organism. Photographs, habitat and predator information, mating behavior and success, and phylogenetic data are examples of the types of information that may be included in a given case. Students then weave the information into a narrative that must take into account all of the components of a natural selection model and describe the change over time that may have occurred (see Box 12-6 for one group’s Darwinian explanation). As students hone their abilities to develop and assess evolutionary arguments over three successive case studies, they are able to participate in realistic evolutionary inquiry.

In the first case study, students develop a Darwinian explanation for differences in seed coat characteristics among populations of a hypothetical plant species. The second case study involves explaining the bright, and similar, coloration of monarch and viceroy butterflies. The final case requires that students develop an explanation for how the sexual dimorphism exhibited by ring-necked pheasants might have arisen.

During each case study, the time is structured so that a group will consult with at least one other group as they develop their explanations. This task organization reinforces the nature of argumentation in evolutionary biology, as it includes the expectation that students will attend to the central feature of any Darwinian explanation—that it have a historical component. But it is not enough to just have a history. In tracing the possible historical development of a trait, students must weave a complex story that draws on available data, as well as their understanding of an array of biological models (e.g., genetic models), to explain the role of heritable variation, superfecundity, competition, and agents of selection. Within their research

groups, meetings between research groups, and whole-class discussions, students question one another using a variety of sophisticated stances. These include ensuring that there is consistency among the data, the natural selection model, and claims; that the history of the shift in a trait is feasible (i.e., consistent with genetics); and that the proposed selection agent could have brought about the change in the trait between times 1 and 2. The students question one another to ensure that their explanations are both internally

and externally consistent. In so doing, they normally propose more than a single explanation, thus recognizing that, in evolution at least, it is important to consider multiple interpretations. As they examine competing Darwinian explanations for the same phenomena, they invoke an evolution-specific argument-analysis norm—that the explanation of the history of a trait has to be consistent with the natural selection model. For example, the second case requires students to provide a Darwinian explanation for the similarity in color between the monarch and viceroy butterflies. Frequently students will say such things as “the viceroy needs to look like the monarch so that the birds won’t eat it.” When statements such as these are made, other students will often challenge the speaker to use Darwinian rather than Lamarckian language. The work on the cases allows students to practice using the Darwinian model in appropriate ways, and the interactive nature of all of the work in class affords them opportunities to think explicitly about and defend their own ideas.

The culminating activities for each of the three cases require public sharing of ideas in a forum where the expectation is that the presenting groups and audience members will consider thoughtfully the ideas before them. Each case has a different type of final presentation. The first case ends with a poster session, the second with a roundtable discussion, and the last with a research proposal and an oral presentation.

One particularly powerful experience students have occurs during the final case study. For the first two case studies, students use their understanding of the Darwinian model to account for the changes that may have occurred in particular populations and to explicitly tie data from the case materials to their claims. For the final case study, they must construct a Darwinian explanation for the sexual dimorphism observed between male and female ring-necked pheasants, and in addition, they must produce a research proposal to shed light on their explanation. Typically, students choose to focus their research proposal on a single aspect of their explanation. This activity requires that they think carefully about the components of their explanation and the confidence they place in each of those components. Thus in this instance they are not evaluating the entire explanation as a single entity, but are considering each part in relation to the others. Once they have decided on a research proposal, they must determine how their proposed research would strengthen their argument. Being able to examine an argument as a whole and according to its parts is an important skill that this task helps develop. This case also stimulates interesting conversations among groups. The nonpresenting groups act as a proposal review panel and interact with the presenting groups in an attempt to understand the proposal. Once all groups have presented, the students discuss the merits and shortcomings of each proposal and then decide individually which proposal should be funded.

CLASSROOM ENVIRONMENTS THAT SUPPORT LEARNING WITH UNDERSTANDING

We have found that much of what students learn in genetics and evolutionary biology units grounded in model-based inquiry depends on their active and thoughtful participation in the classroom community. 33 To learn about the process of modeling and about discipline-specific patterns of argumentation, students must be critically aware of the elements that influence their own knowledge generation and justification. The MUSE curricula are designed to facilitate this type of student thinking through explicit discussion of students’ expectations for engaging in argumentation, the design of student tasks, and the use of various tools for interacting with and representing abstract concepts.

Knowledge-Centered

By the end of our courses, students are able to reason in sophisticated ways about inheritance patterns and about evolutionary phenomena. Realizing that goal, we believe, is due in large measure to careful attention to the core disciplinary knowledge, as well as persistent attention to students’ preconceptions and the supports required for effective conceptual change. The instructional activities we have described highlight a classroom environment that is knowledge-centered in putting both the core concepts and scientific approaches to generating and justifying those concepts at the center of instruction.

Learner-Centered

The classrooms are also learner-centered in several respects. The curriculum was designed to address existing conceptions that we had observed were creating problems for students as they tried to master new material. We also identified weaknesses in students’ knowledge base—such as their understanding of models and their ability to draw inferences and develop arguments—and designed activities to strengthen those competencies. The use of frequent dialogue in our courses allows an attentive teacher to continuously monitor students’ developing thinking.

Assessment-Centered

We have attempted to embed formative and authentic assessments throughout our courses. Assessment of student understanding needs to be undertaken with an eye to the various types of prior knowledge described above (misconceptions of science concepts, ideas about what science is,

and the extent to which students’ knowledge is integrated). We have seen, time and again, teachers becoming aware of students’ common struggles and beginning to “hear” their own students differently. Thus, an important feature of instructional activities that give students opportunities to make their thinking and knowledge public and therefore visible to teachers is that they make assessment and instruction seamless. This becomes possible when students articulate the process of arriving at a solution and not simply the solution itself.

Because students struggle with conceptual problems in the genetics unit, for example, we incorporate a number of assessments that require them to describe the relationships between models or ideas that they have learned (see Box 12-7 ). Whenever possible, we design formal assessments as well as written classroom tasks that reflect the structure of students’ work in the classroom. Our students spend a great deal of their class time working in groups, pouring over data, and talking with one another about their ideas. Thus, assessments also require them to look at data, propose explanations, and describe the thinking that led to particular conclusions.

In the evolution course, students are required during instruction to use the natural selection model to develop Darwinian explanations that account for rich data sets. To then ask them about data or the components of natural selection in a multiple-choice format that would require them to draw on only bits and pieces of knowledge for any one question appears incomplete at best. Instead, we provide them with novel data and ask them to describe their reasoning about those data using the natural selection model—a task analogous to what they have been doing in class. An instance of this type of assessment on the final exam asks students to write a Darwinian explanation for the color of polar bear fur using information about ancestral populations. In this way, during assessment we draw on students’ ideas and skills as they were developed in class rather than asking students to simply recall bits of information in contrived testing situations.

While assessments provide teachers with information about student understanding, students also benefit from assessments that give them opportunities to see how their understanding has changed during a unit of study. One method we have used is to require each student to critique her or his own early work based on what she or he knows at the conclusion of a course. Not only does this approach give teachers insights into students’ knowledge, but it also allows students to glimpse how much their knowledge and their ability to critique arguments have changed. Students’ consideration of their own ideas has been incorporated into the assessment tasks in both units. On several occasions and in different ways, students examine their own ideas and explicitly discuss how those ideas have changed. For example, one of the questions on the final exam in evolution requires students to read and critique a Darwinian explanation they created on the first

day of class (see Box 12-8 ). We have found this to be one of the most powerful moments for many students, as they recognize how much their own ideas have changed. Many students are critical of the need-based language that was present in their original explanation, or they find that they described evolutionary change as having happened at the individual rather than the population level.

Community-Centered

As Chapter 1 suggests, the knowledge-centered, learner-centered, and assessment-centered classrooms come together in the context of a classroom community. The culture of successful scientific communities includes both collaboration and questioning among colleagues. It involves norms for making and justifying claims. At the source of the productivity of such a community is an understanding of central causal models, the ability to use such models to conduct inquiry, and the ability to engage in the assessment of causal models and related explanations. We have found that these outcomes can be realized in classrooms where students are full participants in a scientific community. 34 Interestingly, one unexpected outcome of structuring classrooms so that students are expected to participate in the intellectual work of science has been increased involvement and achievement by students not previously identified as successful in science.

In addition to establishing expectations for class participation and a shared framework for knowledge assessment, MUSE curricula promote metacognitive reflection on the part of students by incorporating tasks that require discourse (formal and informal) at all stages of student work. While working in groups and presenting results to the class as a whole, students are required to share their ideas even when those ideas may not be fully formed. Moreover, recall that the context for idea sharing is one in which discipline-specific criteria for assessment of ideas have been established. Thus, discourse is anchored in norms of argumentation that reflect scientific practice to the extent possible.

Learning with Understanding

While the four features of classroom environments can be described individually, in practice they must interact if students are to deeply engage in learning for understanding. High school students have had more than 9 years of practice at playing the “game of school.” Most have become quite adept at memorizing and reiterating information, seeking answers to questions or problems, and moving quickly from one topic to another. Typically during the game of school, students win when they present the correct answer. The process by which one determines the answer is irrelevant or, at best, undervalued. The students described here are quite typical in this regard: they enter our genetics and evolution classes anticipating that they will be called upon to provide answers and are prepared to do so. In fact, seeking an end product is so ingrained that even when we design tasks that involve multiple iterations of modeling and testing ideas, such as within the genetics course, students frequently reduce the work to seeking algorithms that have predictive power instead of engaging in the much more difficult

task of evaluating models on the basis of their conceptual consistency within a family of related ideas. 35

After studying how people solved problems in a variety of situations, Klayman and Ha 36 noted the frequent use of what they call a “positive test strategy.” That is, solvers would propose a model (or solution) and test it by attempting to apply it to the situation most likely to fit the model in the first place. If the idea had explanatory or predictive power, the solver remained satisfied with it; if not, the solver would quickly test another idea. The positive test strategy was frequently applied by students in early versions of our genetics course. 37 This method of problem solving does not map well to scientific practice in most cases, however: it is the absence of disproving evidence, and not the presence of confirming evidence that is more commonly persuasive to scientists. Moreover, testing a model in limited situations in which one expects a data–model match would be considered “confirmation bias” within scientific communities. Nevertheless, Klayman and Ha point out that this positive test strategy is often quite useful in real-life situations.

Given our students’ facility with the game of school and the general tendency to apply less scientific model-testing strategies when problem solving, we were forced to create tasks that not only afford the opportunity for reflection, but actually require students to think more deeply about the ways in which they have come to understand science concepts, as well as what is involved in scientific argumentation. We want students to realize that the models and explanations they propose are likely to be challenged and that the conflicts surrounding such challenges are the lifeblood of science. Thus, we explicitly discuss with our students the expectations for their participation in the course. Teachers state that the students’ task is not simply to produce an “answer” (a model in genetics or a Darwinian explanation in evolutionary biology), but also to be able to defend and critique ideas according to the norms of a particular scientific discipline. In other words, we ask the students to abandon the game of school and begin to play the game of science.

Examination of ideas requires more than simply providing space for reflection to occur; it also involves working with students to develop systematic ways of critiquing their own ideas and those of others. This is why we begin each course with an activity whose focus is the introduction of discipline-specific ways of generating and critiquing knowledge claims. These activities do not require that students will come to understand any particular scientific concepts upon their completion. Rather, they will have learned about the process of constructing and evaluating arguments in genetics or evolutionary biology. Specific criteria for weighing scientific explanations are revisited throughout each course as students engage in extended inquiries within these biological disciplines.

For students to develop understanding in any scientific discipline, teachers and curriculum developers must attend to a set of complex and interrelated components, including the nature of practice in particular scientific disciplines, students’ prior knowledge, and the establishment of a collaborative environment that engages students in reflective scientific practice. These design components allow educators to create curricula and instructional materials that help students learn about science both as and by inquiry.

The students in the biology classrooms described in this chapter have developed sophisticated understandings of some of the most central explanatory frameworks in genetics and evolutionary biology. In addition, they have, unlike many high school students, shown great maturity in their abilities to reason about realistic biological data and phenomena using these models. Moreover, they have accomplished this in classrooms that are structured along the lines of scientific communities. This has all been made possible by a concerted collaboration involving high school teachers and their students, university science educators, and university biologists. That MUSE combined this collaboration with a research program on student learning and reasoning was essential. With the knowledge thus gained, we believe it is possible to help others realize the expectations for improving science education that are set forth in reform documents such as the National Science Education Standards . 38 In particular, there has been a call for curricular reforms that allow students to be “engaged in inquiry” that involves “combin[ing] processes and scientific knowledge as they use scientific reasoning and critical thinking to develop their understanding of science.” 39 Recommendations for improved teaching of science are solidly rooted in a commitment to teaching both through and about inquiry. Furthermore, the National Science Education Standards do not simply suggest that science teachers incorporate inquiry in classrooms; rather, they demand that teachers embrace inquiry in order to:

Plan an inquiry-based science program for their students.

Focus and support inquiries while interacting with students.

Create a setting for student work that is flexible and supportive of science inquiry.

Model and emphasize the skills, attitudes, and values of scientific inquiry.

It is just these opportunities that have been described in this chapter.

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How do you get a fourth-grader excited about history? How do you even begin to persuade high school students that mathematical functions are relevant to their everyday lives? In this volume, practical questions that confront every classroom teacher are addressed using the latest exciting research on cognition, teaching, and learning.

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The book explores the importance of balancing students' knowledge of historical fact against their understanding of concepts, such as change and cause, and their skills in assessing historical accounts. It discusses how to build straightforward science experiments into true understanding of scientific principles. And it shows how to overcome the difficulties in teaching math to generate real insight and reasoning in math students. It also features illustrated suggestions for classroom activities.

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Steps for Inquiry and Problem-Solving

Although problem-solving and inquiry are technically different, they require many of the same skills. At a minimum, students need to recognize the language that they will need and the processes involved. Critical thinking skills are also crucial for successful inquiry projects. Teachers can model some critical thinking skills, such as

• analyzing arguments, claims, or evidence
• making inferences using inductive or deductive reasoning
• judging or evaluating
• making decisions or solving problems
• asking and answering questions for clarification
• defining terms
• identifying assumptions
• interpreting and explaining
• reasoning verbally
• seeing both sides of an issue (Lai, 2011, pp. 9-10)

Critical thinking skills do not necessarily come naturally with second language learning, and they are culturally situated, so students need to learn and practice them before, during, and after each step in the process. According to Molnar, Boninger, and Fogarty (2011), an environment that “encourages students to ask questions, to think about their thought processes, and thus to develop habits of mind that enable them to transfer the critical thinking skills they learn in class to other, unrelated, situations” is where critical thinking is cultivated (p. i).

Although sources describe the steps in the inquiry process differently, most sources include the same five basic steps of establishing orientation , conceptualization , investigation , conclusion , and discussion (Pedaste, et al., 2015). These steps are addressed to the learner and include:

• What are you interested in? Ask a question that has meaning, define the problem, and figure out what you need to do to answer it.
• Investigate by researching. Plan, gather resources and information, and record what you have found.
• Create new ideas, thoughts, and directions for action. Make sense of the information you have gathered by summarizing, synthesizing, and interpreting.
• Discuss with others. Interaction can shed new light on the question, the investigation, and the process. Share what you have learned and then use the feedback to return to the process.
• Reflect on the inquiry process. Did the process lead to unexpected conclusions? Is there something else that needs researching? Has the problem been solved?

For younger or less proficient learners, Freeman and Freeman (1998) present six steps that follow these same basic guidelines. They call this the “Wonderfilled Way of Learning,” and the steps are addressed to the teacher:

• Ask the students: What do we know about ______ ?
• Ask the students: What do we wonder about _____ ?
• Ask the students: How can we find  out about _____ ?
• With the students, work out a plan of action, and, at the same time, work school district curriculum requirements into the unit.
• Plan an event to celebrate what you have learned together.
• Learning is continuous. From any unit, more topics and questions come up. Begin the cycle again. (pp. 138–139)

Regardless of which set of guidelines you and your students follow, inquiry projects can be used to support language and content learning.

In the chapter’s opening scenario, Ms. Petrie guides the students through learning experiences. She has planned that throughout the project, they will not only learn techniques for inquiry such as planning, brainstorming, reflecting, and evaluating, but through their interactions, they will also acquire a variety of language content and structures. These activities facilitate many of the CALL principles—for example, students have many opportunities for language input and output, they have many choices (structure/autonomy), they are motivated to learn because they are answering meaningful questions (authenticity and connection), and they interact with peers and community members (social interaction and feedback).

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STEM Problem Solving: Inquiry, Concepts, and Reasoning

• Published: 29 January 2022
• Volume 32 , pages 381–397, ( 2023 )

Cite this article

• Aik-Ling Tan   ORCID: orcid.org/0000-0002-4627-4977 1 ,
• Yann Shiou Ong   ORCID: orcid.org/0000-0002-6092-2803 1 ,
• Yong Sim Ng   ORCID: orcid.org/0000-0002-8400-2040 1 &
• Jared Hong Jie Tan 1  

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Balancing disciplinary knowledge and practical reasoning in problem solving is needed for meaningful learning. In STEM problem solving, science subject matter with associated practices often appears distant to learners due to its abstract nature. Consequently, learners experience difficulties making meaningful connections between science and their daily experiences. Applying Dewey’s idea of practical and science inquiry and Bereiter’s idea of referent-centred and problem-centred knowledge, we examine how integrated STEM problem solving offers opportunities for learners to shuttle between practical and science inquiry and the kinds of knowledge that result from each form of inquiry. We hypothesize that connecting science inquiry with practical inquiry narrows the gap between science and everyday experiences to overcome isolation and fragmentation of science learning. In this study, we examine classroom talk as students engage in problem solving to increase crop yield. Qualitative content analysis of the utterances of six classes of 113 eighth graders and their teachers were conducted for 3 hours of video recordings. Analysis showed an almost equal amount of science and practical inquiry talk. Teachers and students applied their everyday experiences to generate solutions. Science talk was at the basic level of facts and was used to explain reasons for specific design considerations. There was little evidence of higher-level scientific conceptual knowledge being applied. Our observations suggest opportunities for more intentional connections of science to practical problem solving, if we intend to apply higher-order scientific knowledge in problem solving. Deliberate application and reference to scientific knowledge could improve the quality of solutions generated.

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1 Introduction

As we enter to second quarter of the twenty-first century, it is timely to take stock of both the changes and demands that continue to weigh on our education system. A recent report by World Economic Forum highlighted the need to continuously re-position and re-invent education to meet the challenges presented by the disruptions brought upon by the fourth industrial revolution (World Economic Forum, 2020 ). There is increasing pressure for education to equip children with the necessary, relevant, and meaningful knowledge, skills, and attitudes to create a “more inclusive, cohesive and productive world” (World Economic Forum, 2020 , p. 4). Further, the shift in emphasis towards twenty-first century competencies over mere acquisition of disciplinary content knowledge is more urgent since we are preparing students for “jobs that do not yet exist, technology that has not yet been invented, and problems that has yet exist” (OECD, 2018 , p. 2). Tan ( 2020 ) concurred with the urgent need to extend the focus of education, particularly in science education, such that learners can learn to think differently about possibilities in this world. Amidst this rhetoric for change, the questions that remained to be answered include how can science education transform itself to be more relevant; what is the role that science education play in integrated STEM learning; how can scientific knowledge, skills and epistemic practices of science be infused in integrated STEM learning; what kinds of STEM problems should we expose students to for them to learn disciplinary knowledge and skills; and what is the relationship between learning disciplinary content knowledge and problem solving skills?

In seeking to understand the extent of science learning that took place within integrated STEM learning, we dissected the STEM problems that were presented to students and examined in detail the sense making processes that students utilized when they worked on the problems. We adopted Dewey’s ( 1938 ) theoretical idea of scientific and practical/common-sense inquiry and Bereiter’s ideas of referent-centred and problem-centred knowledge building process to interpret teacher-students’ interactions during problem solving. There are two primary reasons for choosing these two theoretical frameworks. Firstly, Dewey’s ideas about the relationship between science inquiry and every day practical problem-solving is important in helping us understand the role of science subject matter knowledge and science inquiry in solving practical real-world problems that are commonly used in STEM learning. Secondly, Bereiter’s ideas of referent-centred and problem-centred knowledge augment our understanding of the types of knowledge that students can learn when they engage in solving practical real-world problems.

Taken together, Dewey’s and Bereiter’s ideas enable us to better understand the types of problems used in STEM learning and their corresponding knowledge that is privileged during the problem-solving process. As such, the two theoretical lenses offered an alternative and convincing way to understand the actual types of knowledge that are used within the context of integrated STEM and help to move our understanding of STEM learning beyond current focus on examining how engineering can be used as an integrative mechanism (Bryan et al., 2016 ) or applying the argument of the strengths of trans-, multi-, or inter-disciplinary activities (Bybee, 2013 ; Park et al., 2020 ) or mapping problems by the content and context as pure STEM problems, STEM-related problems or non-STEM problems (Pleasants, 2020 ). Further, existing research (for example, Gale et al., 2000 ) around STEM education focussed largely on description of students’ learning experiences with insufficient attention given to the connections between disciplinary conceptual knowledge and inquiry processes that students use to arrive at solutions to problems. Clarity in the role of disciplinary knowledge and the related inquiry will allow for more intentional design of STEM problems for students to learn higher-order knowledge. Applying Dewey’s idea of practical and scientific inquiry and Bereiter’s ideas of referent-centred and problem-centred knowledge, we analysed six lessons where students engaged with integrated STEM problem solving to propose answers to the following research questions: What is the extent of practical and scientific inquiry in integrated STEM problem solving? and What conceptual knowledge and problem-solving skills are learnt through practical and science inquiry during integrated STEM problem solving?

2 Inquiry in Problem Solving

Inquiry, according to Dewey ( 1938 ), involves the direct control of unknown situations to change them into a coherent and unified one. Inquiry usually encompasses two interrelated activities—(1) thinking about ideas related to conceptual subject-matter and (2) engaging in activities involving our senses or using specific observational techniques. The National Science Education Standards released by the National Research Council in the US in 1996 defined inquiry as “…a multifaceted activity that involves making observations; posing questions; examining books and other sources of information to see what is already known; planning investigations; reviewing what is already known in light of experimental evidence; using tools to gather, analyze, and interpret data; proposing answers, explanations, and predictions; and communicating the results. Inquiry requires identification of assumptions, use of critical and logical thinking, and consideration of alternative explanations” (p. 23). Planning investigation; collecting empirical evidence; using tools to gather, analyse and interpret data; and reasoning are common processes shared in the field of science and engineering and hence are highly relevant to apply to integrated STEM education.

In STEM education, establishing the connection between general inquiry and its application helps to link disciplinary understanding to epistemic knowledge. For instance, methods of science inquiry are popular in STEM education due to the familiarity that teachers have with scientific methods. Science inquiry, a specific form of inquiry, has appeared in many science curriculum (e.g. NRC, 2000 ) since Dewey proposed in 1910 that learning of science should be perceived as both subject-matter and a method of learning science (Dewey, 1910a , 1910b ). Science inquiry which involved ways of doing science should also encompass the ways in which students learn the scientific knowledge and investigative methods that enable scientific knowledge to be constructed. Asking scientifically orientated questions, collecting empirical evidence, crafting explanations, proposing models and reasoning based on available evidence are affordances of scientific inquiry. As such, science should be pursued as a way of knowing rather than merely acquisition of scientific knowledge.

Building on these affordances of science inquiry, Duschl and Bybee ( 2014 ) advocated the 5D model that focused on the practice of planning and carrying out investigations in science and engineering, representing two of the four disciplines in STEM. The 5D model includes science inquiry aspects such as (1) deciding on what and how to measure, observe and sample; (2) developing and selecting appropriate tools to measure and collect data; (3) recording the results and observations in a systematic manner; (4) creating ways to represent the data and patterns that are observed; and (5) determining the validity and the representativeness of the data collected. The focus on planning and carrying out investigations in the 5D model is used to help teachers bridge the gap between the practices of building and refining models and explanation in science and engineering. Indeed, a common approach to incorporating science inquiry in integrated STEM curriculum involves student planning and carrying out scientific investigations and making sense of the data collected to inform engineering design solution (Cunningham & Lachapelle, 2016 ; Roehrig et al., 2021 ). Duschl and Bybee ( 2014 ) argued that it is needful to design experiences for learners to appreciate that struggles are part of problem solving in science and engineering. They argued that “when the struggles of doing science is eliminated or simplified, learners get the wrong perceptions of what is involved when obtaining scientific knowledge and evidence” (Duschl & Bybee, 2014 , p. 2). While we concur with Duschl and Bybee about the need for struggles, in STEM learning, these struggles must be purposeful and grade appropriate so that students will also be able to experience success amidst failure.

The peculiar nature of science inquiry was scrutinized by Dewey ( 1938 ) when he cross-examined the relationship between science inquiry and other forms of inquiry, particularly common-sense inquiry. He positioned science inquiry along a continuum with general or common-sense inquiry that he termed as “logic”. Dewey argued that common-sense inquiry serves a practical purpose and exhibits features of science inquiry such as asking questions and a reliance on evidence although the focus of common-sense inquiry tends to be different. Common-sense inquiry deals with issues or problems that are in the immediate environment where people live, whereas the objects of science inquiry are more likely to be distant (e.g. spintronics) from familiar experiences in people’s daily lives. While we acknowledge the fundamental differences (such as novel discovery compared with re-discovering science, ‘messy’ science compared with ‘sanitised’ science) between school science and science that is practiced by scientists, the subject of interest in science (understanding the world around us) remains the same.

The unfamiliarity between the functionality and purpose of science inquiry to improve the daily lives of learners does little to motivate learners to learn science (Aikenhead, 2006 ; Lee & Luykx, 2006 ) since learners may not appreciate the connections of science inquiry in their day-to-day needs and wants. Bereiter ( 1992 ) has also distinguished knowledge into two forms—referent-centred and problem-centred. Referent-centred knowledge refers to subject-matter that is organised around topics such as that in textbooks. Problem-centred knowledge is knowledge that is organised around problems, whether they are transient problems, practical problems or problems of explanations. Bereiter argued that referent-centred knowledge that is commonly taught in schools is limited in their applications and meaningfulness to the lives of students. This lack of familiarity and affinity to referent-centred knowledge is likened to the science subject-matter knowledge that was mentioned by Dewey. Rather, it is problem-centred knowledge that would be useful when students encounter problems. Learning problem-centred knowledge will allow learners to readily harness the relevant knowledge base that is useful to understand and solve specific problems. This suggests a need to help learners make the meaningful connections between science and their daily lives.

Further, Dewey opined that while the contexts in which scientific knowledge arise could be different from our daily common-sense world, careful consideration of scientific activities and applying the resultant knowledge to daily situations for use and enjoyment is possible. Similarly, in arguing for problem-centred knowledge, Bereiter ( 1992 ) questioned the value of inert knowledge that plays no role in helping us understand or deal with the world around us. Referent-centred knowledge has a higher tendency to be inert due to the way that the knowledge is organised and the way that the knowledge is encountered by learners. For instance, learning about the equation and conditions for photosynthesis is not going to help learners appreciate how plants are adapted for photosynthesis and how these adaptations can allow plants to survive changes in climate and for farmers to grow plants better by creating the best growing conditions. Rather, students could be exposed to problems of explanations where they are asked to unravel the possible reasons for low crop yield and suggest possible ways to overcome the problem. Hence, we argue here that the value of the referent knowledge is that they form the basis and foundation for the students to be able to discuss or suggest ways to overcome real life problems. Referent-centred knowledge serves as part of the relevant knowledge base that can be harnessed to solve specific problems or as foundational knowledge students need to progress to learn higher-order conceptual knowledge that typically forms the foundations or pillars within a discipline. This notion of referent-centred knowledge serving as foundational knowledge that can be and should be activated for application in problem-solving situation is shown by Delahunty et al. ( 2020 ). They found that students show high reliance on memory when they are conceptualising convergent problem-solving tasks.

While Bereiter argues for problem-centred knowledge, he cautioned that engagement should be with problems of explanation rather than transient or practical problems. He opined that if learners only engage in transient or practical problem alone, they will only learn basic-category types of knowledge and fail to understand higher-order conceptual knowledge. For example, for photosynthesis, basic-level types of knowledge included facts about the conditions required for photosynthesis, listing the products formed from the process of photosynthesis and knowing that green leaves reflect green light. These basic-level knowledges should intentionally help learners learn higher-level conceptual knowledge that include learners being able to draw on the conditions for photosynthesis when they encounter that a plant is not growing well or is exhibiting discoloration of leaves.

Transient problems disappear once a solution becomes available and there is a high likelihood that we will not remember the problem after that. Practical problems, according to Bereiter are “stuck-door” problems that could be solved with or without basic-level knowledge and often have solutions that lacks precise definition. There are usually a handful of practical strategies, such as pulling or pushing the door harder, kicking the door, etc. that will work for the problems. All these solutions lack a well-defined approach related to general scientific principles that are reproducible. Problems of explanations are the most desirable types of problems for learners since these are problems that persist and recur such that they can become organising points for knowledge. Problems of explanations consist of the conceptual representations of (1) a text base that serves to represent the text content and (2) a situation model that shows the portion of the world in which the text is relevant. The idea of text base to represent text content in solving problems of explanations is like the idea of domain knowledge and structural knowledge (refers to knowledge of how concepts within a domain are connected) proposed by Jonassen ( 2000 ). He argued that both types of knowledges are required to solve a range of problems from well-structured problems to ill-structured problems with a simulated context, to simple ill-structured problems and to complex ill-structured problems.

Jonassen indicated that complex ill-structured problems are typically design problems and are likely to be the most useful forms of problems for learners to be engaged in inquiry. Complex ill-structured design problems are the “wicked” problems that Buchanan ( 1992 ) discussed. Buchanan’s idea is that design aims to incorporate knowledge from different fields of specialised inquiry to become whole. Complex or wicked problems are akin to the work of scientists who navigate multiple factors and evidence to offer models that are typically oversimplified, but they apply them to propose possible first approximation explanations or solutions and iteratively relax constraints or assumptions to refine the model. The connections between the subject matter of science and the design process to engineer a solution are delicate. While it is important to ensure that practical concerns and questions are taken into consideration in designing solutions (particularly a material artefact) to a practical problem, the challenge here lies in ensuring that creativity in design is encouraged even if students initially lack or neglect the scientific conceptual understanding to explain/justify their design. In his articulation of wicked problems and the role of design thinking, Buchanan ( 1992 ) highlighted the need to pay attention to category and placement. Categories “have fixed meanings that are accepted within the framework of a theory or a philosophy and serve as the basis for analyzing what already exist” (Buchanan, 1992 , p. 12). Placements, on the other hand, “have boundaries to shape and constrain meaning, but are not rigidly fixed and determinate” (p. 12).

The difference in the ideas presented by Dewey and Bereiter lies in the problem design. For Dewey, scientific knowledge could be learnt from inquiring into practical problems that learners are familiar with. After all, Dewey viewed “modern science as continuous with, and to some degree an outgrowth and refinement of, practical or ‘common-sense’ inquiry” (Brown, 2012 ). For Bereiter, he acknowledged the importance of familiar experiences, but instead of using them as starting points for learning science, he argued that practical problems are limiting in helping learners acquire higher-order knowledge. Instead, he advocated for learners to organize their knowledge around problems that are complex, persistent and extended and requiring explanations to better understand the problems. Learners are to have a sense of the kinds of problems to which the specific concept is relevant before they can be said to have grasp the concept in a functionally useful way.

To connect between problem solving, scientific knowledge and everyday experiences, we need to examine ways to re-negotiate the disciplinary boundaries (such as epistemic understanding, object of inquiry, degree of precision) of science and make relevant connections to common-sense inquiry and to the problem at hand. Integrated STEM appears to be one way in which the disciplinary boundaries of science can be re-negotiated to include practices from the fields of technology, engineering and mathematics. In integrated STEM learning, inquiry is seen more holistically as a fluid process in which the outcomes are not absolute but are tentative. The fluidity of the inquiry process is reflected in the non-deterministic inquiry approach. This means that students can use science inquiry, engineering design, design process or any other inquiry approaches that fit to arrive at the solution. This hybridity of inquiry between science, common-sense and problems allows for some familiar aspects of the science inquiry process to be applied to understand and generate solutions to familiar everyday problems. In attempting to infuse elements of common-sense inquiry with science inquiry in problem-solving, logic plays an important role to help learners make connections. Hypothetically, we argue that with increasing exposure to less familiar ways of thinking such as those associated with science inquiry, students’ familiarity with scientific reasoning increases, and hence such ways of thinking gradually become part of their common-sense, which students could employ to solve future relevant problems. The theoretical ideas related to complexities of problems, the different forms of inquiry afforded by different problems and the arguments for engaging in problem solving motivated us to examine empirically how learners engage with ill-structured problems to generate problem-centred knowledge. Of particular interest to us is how learners and teachers weave between practical and scientific reasoning as they inquire to integrate the components in the original problem into a unified whole.

3.1 Context

The integrated STEM activity in our study was planned using the S-T-E-M quartet instructional framework (Tan et al., 2019 ). The S-T-E-M quartet instructional framework positions complex, persistent and extended problems at its core and focusses on the vertical disciplinary knowledge and understanding of the horizontal connections between the disciplines that could be gained by learners through solving the problem (Tan et al., 2019 ). Figure  1 depicts the disciplinary aspects of the problem that was presented to the students. The activity has science and engineering as the two lead disciplines. It spanned three 1-h lessons and required students to both learn and apply relevant scientific conceptual knowledge to solve a complex, real-world problem through processes that resemble the engineering design process (Wheeler et al., 2019 ).

Connections across disciplines in integrate STEM activity

Frequency of different types of reasoning

In the first session (1 h), students were introduced to the problem and its context. The problem pertains to the issue of limited farmland in a land scarce country that imports 90% of food (Singapore Food Agency [SFA], 2020 ). The students were required to devise a solution by applying knowledge of the conditions required for photosynthesis and plant growth to design and build a vertical farming system to help farmers increase crop yield with limited farmland. This context was motivated by the government’s effort to generate interests and knowledge in farming to achieve the 30 by 30 goal—supplying 30% of country’s nutritional needs by 2030. The scenario was a fictitious one where they were asked to produce 120 tonnes of Kailan (a type of leafy vegetable) with two hectares of land instead of the usual six hectares over a specific period. In addition to the abovementioned constraints, the teacher also discussed relevant success criteria for evaluating the solution with the students. Students then researched about existing urban farming approaches. They were given reading materials pertaining to urban farming to help them understand the affordances and constraints of existing solutions. In the second session (6 h), students engaged in ideation to generate potential solutions. They then designed, built and tested their solution and had opportunities to iteratively refine their solution. Students were given a list of materials (e.g. mounting board, straws, ice-cream stick, glue, etc.) that they could use to design their solutions. In the final session (1 h), students presented their solution and reflected on how well their solution met the success criteria. The prior scientific conceptual knowledge that students require to make sense of the problem include knowledge related to plant nutrition, namely, conditions for photosynthesis, nutritional requirements of Kailin and growth cycle of Kailin. The problem resembles a real-world problem that requires students to engage in some level of explanation of their design solution.

A total of 113 eighth graders (62 boys and 51 girls), 14-year-olds, from six classes and their teachers participated in the study. The students and their teachers were recruited as part of a larger study that examined the learning experiences of students when they work on integrated STEM activities that either begin with a problem, a solution or are focused on the content. Invitations were sent to schools across the country and interested schools opted in for the study. For the study reported here, all students and teachers were from six classes within a school. The teachers had all undergone 3 h of professional development with one of the authors on ways of implementing the integrated STEM activity used in this study. During the professional development session, the teachers learnt about the rationale of the activity, familiarize themselves with the materials and clarified the intentions and goals of the activity. The students were mostly grouped in groups of three, although a handful of students chose to work independently. The group size of students was not critical for the analysis of talk in this study as the analytic focus was on the kinds of knowledge applied rather than collaborative or group think. We assumed that the types of inquiry adopted by teachers and students were largely dependent on the nature of problem. Eighth graders were chosen for this study since lower secondary science offered at this grade level is thematic and integrated across biology, chemistry and physics. Furthermore, the topic of photosynthesis is taught under the theme of Interactions at eighth grade (CPDD, 2021 ). This thematic and integrated nature of science at eighth grade offered an ideal context and platform for integrated STEM activities to be trialled.

The final lessons in a series of three lessons in each of the six classes was analysed and reported in this study. Lessons where students worked on their solutions were not analysed because the recordings had poor audibility due to masking and physical distancing requirements as per COVID-19 regulations. At the start of the first lesson, the instructions given by the teacher were:

You are going to present your models. Remember the scenario that you were given at the beginning that you were tasked to solve using your model. …. In your presentation, you have to present your prototype and its features, what is so good about your prototype, how it addresses the problem and how it saves costs and space. So, this is what you can talk about during your presentation. ….. pay attention to the presentation and write down questions you like to ask the groups after the presentation… you can also critique their model, you can evaluate, critique and ask questions…. Some examples of questions you can ask the groups are? Do you think your prototype can achieve optimal plant growth? You can also ask questions specific to their models.

3.2 Data collection

Parental consent was sought a month before the start of data collection. The informed consent adhered to confidentiality and ethics guidelines as described by the Institutional Review Board. The data collection took place over a period of one month with weekly video recording. Two video cameras, one at the front and one at the back of the science laboratory were set up. The front camera captured the students seated at the front while the back video camera recorded the teacher as well as the groups of students at the back of the laboratory. The video recordings were synchronized so that the events captured from each camera can be interpreted from different angles. After transcription of the raw video files, the identities of students were substituted with pseudonyms.

3.3 Data analysis

The video recordings were analysed using the qualitative content analysis approach. Qualitative content analysis allows for patterns or themes and meanings to emerge from the process of systematic classification (Hsieh & Shannon, 2005 ). Qualitative content analysis is an appropriate analytic method for this study as it allows us to systematically identify episodes of practical inquiry and science inquiry to map them to the purposes and outcomes of these episodes as each lesson unfolds.

In total, six h of video recordings where students presented their ideas while the teachers served as facilitator and mentor were analysed. The video recordings were transcribed, and the transcripts were analysed using the NVivo software. Our unit of analysis is a single turn of talk (one utterance). We have chosen to use utterances as proxy indicators of reasoning practices based on the assumption that an utterance relates to both grammar and context. An utterance is a speech act that reveals both meaning and intentions of the speaker within specific contexts (Li, 2008 ).

Our research analytical lens is also interpretative in nature and the validity of our interpretation is through inter-rater discussion and agreement. Each utterance at the speaker level in transcripts was examined and coded either as relevant to practical reasoning or scientific reasoning based on the content. The utterances could be a comment by the teacher, a question by a student or a response by another student. Deductive coding is deployed with the two codes, practical reasoning and scientific reasoning derived from the theoretical ideas of Dewey and Bereiter as described earlier. Practical reasoning refers to utterances that reflect commonsensical knowledge or application of everyday understanding. Scientific reasoning refers to utterances that consist of scientifically oriented questions, scientific terms, or the use of empirical evidence to explain. Examples of each type of reasoning are highlighted in the following section. Each coded utterance is then reviewed for detailed description of the events that took place that led to that specific utterance. The description of the context leading to the utterance is considered an episode. The episodes and codes were discussed and agreed upon by two of the authors. Two coders simultaneously watched the videos to identify and code the episodes. The coders interpreted the content of each utterance, examine the context where the utterance was made and deduced the purpose of the utterance. Once each coder has established the sense-making aspect of the utterance in relation to the context, a code of either practical reasoning or scientific reasoning is assigned. Once that was completed, the two coders compared their coding for similarities and differences. They discussed the differences until an agreement was reached. Through this process, an agreement of 85% was reached between the coders. Where disagreement persisted, codes of the more experienced coder were adopted.

4 Results and Discussion

The specific STEM lessons analysed were taken from the lessons whereby students presented the model of their solutions to the class for peer evaluation. Every group of students stood in front of the class and placed their model on the bench as they presented. There was also a board where they could sketch or write their explanations should they want to. The instructions given by the teacher to the students were to explain their models and state reasons for their design.

4.1 Prevalence of Reasoning

The 6h of videos consists of 1422 turns of talk. Three hundred four turns of talk (21%) were identified as talk related to reasoning, either practical reasoning or scientific reasoning. Practical reasoning made up 62% of the reasoning turns while 38% were scientific reasoning (Fig. 2 ).

The two types of reasoning differ in the justifications that are used to substantiate the claims or decisions made. Table 1 describes the differences between the two categories of reasoning.

4.2 Applications of Scientific Reasoning

Instances of engagement with scientific reasoning (for instance, using scientific concepts to justify, raising scientifically oriented questions, or providing scientific explanations) revolved around the conditions for photosynthesis and the concept of energy conversion when students were presenting their ideas or when they were questioned by their peers. For example, in explaining the reason for including fish in their plant system, one group of students made connection to cyclical energy transfer: “…so as the roots of the plants submerged in the water, faeces from the fish will be used as fertilizers so that the plant can grow”. The students considered how organic matter that is still trapped within waste materials can be released and taken up by plants to enhance the growth. The application of scientific reasoning made their design one that is innovative and sustainable as evaluated by the teacher. Some students attempted more ecofriendly designs by considering energy efficiencies through incorporating water turbines in their farming systems. They applied the concept of different forms of energy and energy conversion when their peers inquired about their design. The same scientific concepts were explained at different levels of details by different students. At one level, the students explained in a purely descriptive manner of what happens to the different entities in their prototypes, with implied changes to the forms of energy─ “…spins then generates electricity. So right, when the water falls down, then it will spin. The water will fall on the fan blade thing, then it will spin and then it generates electricity. So, it saves electricity, and also saves water”. At another level, students defended their design through an explanation of energy conversion─ “…because when the water flows right, it will convert gravitational potential energy so, when it reaches the bottom, there is not really much gravitational potential energy”. While these instances of applying scientific reasoning indicated that students have knowledge about the scientific phenomena and can apply them to assist in the problem-solving process, we are not able to establish if students understood the science behind how the dynamo works to generate electricity. Students in eighth grade only need to know how a generator works at a descriptive level and the specialized understanding how a dynamo works is beyond the intended learning outcomes at this grade level.

The application of scientific concepts for justification may not always be accurate. For instance, the naïve conception that students have about plants only respiring at night and not in the day surfaced when one group of students tried to justify the growth rates of Kailan─ “…I mean, they cannot be making food 24/7 and growing 24/7. They have nighttime for a reason. They need to respire”. These students do not appreciate that plants respire in the day as well, and hence respiration occurs 24/7. This naïve conception that plants only respire at night is one that is common among learners of biology (e.g. Svandova, 2014 ) since students learn that plant gives off oxygen in the day and takes in oxygen at night. The hasty conclusion to that observation is that plants carry out photosynthesis in the day and respire at night. The relative rates of photosynthesis and respiration were not considered by many students.

Besides naïve conceptions, engagement with scientific ideas to solve a practical problem offers opportunities for unusual and alternative ideas about science to surface. For instance, another group of students explained that they lined up their plants so that “they can take turns to absorb sunlight for photosynthesis”. These students appear to be explaining that the sun will move and depending on the position of the sun, some plants may be under shade, and hence rates of photosynthesis are dependent on the position of the sun. However, this idea could also be interpreted as (1) the students failed to appreciate that sunlight is everywhere, and (2) plants, unlike animals, particularly humans, do not have the concept of turn-taking. These diverse ideas held by students surfaced when students were given opportunities to apply their knowledge of photosynthesis to solve a problem.

4.3 Applications of Practical Reasoning

Teachers and students used more practical reasoning during an integrated STEM activity requiring both science and engineering practices as seen from 62% occurrence of practical reasoning compared with 38% for scientific reasoning. The intention of the activity to integrate students’ scientific knowledge related to plant nutrition to engineering practice of building a model of vertical farming system could be the reason for the prevalence of practical reasoning. The practical reasoning used related to structural design considerations of the farming system such as how water, light and harvesting can be carried out in the most efficient manner. Students defended the strengths of designs using logic based on their everyday experiences. In the excerpt below (transcribed verbatim), we see students applied their everyday experiences when something is “thinner” (likely to mean narrower), logically it would save space. Further, to reach a higher level, you use a machine to climb up.

Excerpt 1. “Thinner, more space” Because it is more thinner, so like in terms of space, it’s very convenient. So right, because there is – because it rotates right, so there is this button where you can stop it. Then I also installed steps, so that – because there are certain places you can’t reach even if you stop the – if you stop the machine, so when you stop it and you climb up, and then you see the condition of the plants, even though it costs a lot of labour, there is a need to have an experienced person who can grow plants. Then also, when like – when water reach the plants, cos the plants I want to use is soil-based, so as the water reach the soil, the soil will xxx, so like the water will be used, and then we got like – and then there’s like this filter that will filter like the dirt.

In the examples of practical reasoning, we were not able to identify instances where students and teachers engaged with discussion around trade-off and optimisation. Understanding constraints, trade-offs and optimisations are important ideas in informed design matrix for engineering as suggested by Crismond and Adams ( 2012 ). For instance, utterances such as “everything will be reused”, “we will be saving space”, “it looks very flimsy” or “so that it can contains [sic] the plants” were used. These utterances were made both by students while justifying their own prototypes and also by peers who challenged the design of others. Longer responses involving practical reasoning were made based on common-sense, everyday logic─ “…the product does not require much manpower, so other than one or two supervisors like I said just now, to harvest the Kailan, hence, not too many people need to be used, need to be hired to help supervise the equipment and to supervise the growth”. We infer that the higher instances of utterances related to practical reasoning could be due to the presence of more concrete artefacts that is shown, and the students and teachers were more focused on questioning the structure at hand. This inference was made as instructions given by the teacher at the start of students’ presentation focus largely on the model rather than the scientific concepts or reasoning behind the model.

4.4 Intersection Between Scientific and Practical Reasoning

Comparing science subject matter knowledge and problem-solving to the idea of categories and placement (Buchanan, 1992 ), subject matter is analogous to categories where meanings are fixed with well-established epistemic practices and norms. The problem-solving process and design of solutions are likened to placements where boundaries are less rigid, hence opening opportunities for students’ personal experiences and ideas to be presented. Placements allow students to apply their knowledge from daily experiences and common-sense logic to justify decisions. Common-sense knowledge and logic are more accessible, and hence we observe higher frequency of usage. Comparatively, while science subject matter (categories) is also used, it is observed less frequently. This could possibly be due either to less familiarity with the subject matter or lack of appropriate opportunity to apply in practical problem solving. The challenge for teachers during implementation of a STEM problem-solving activity, therefore, lies in the balance of the application of scientific and practical reasoning to deepen understanding of disciplinary knowledge in the context of solving a problem in a meaningful manner.

Our observations suggest that engaging students with practical inquiry tasks with some engineering demands such as the design of modern farm systems offers opportunities for them to convert their personal lived experiences into feasible concrete ideas that they can share in a public space for critique. The peer critique following the sharing of their practical ideas allows for both practical and scientific questions to be asked and for students to defend their ideas. For instance, after one group of students presented their prototype that has silvered surfaces, a student asked a question: “what is the function of the silver panels?”, to which his peers replied : “Makes the light bounce. Bounce the sunlight away and then to other parts of the tray.” This question indicated that students applied their knowledge that shiny silvered surfaces reflect light, and they used this knowledge to disperse the light to other trays where the crops were growing. An example of a practical question asked was “what is the purpose of the ladder?”, to which the students replied: “To take the plants – to refill the plants, the workers must climb up”. While the process of presentation and peer critique mimic peer review in the science inquiry process, the conceptual knowledge of science may not always be evident as students paid more attention to the design constraints such as lighting, watering, and space that was set in the activity. Given the context of growing plants, engagement with the science behind nutritional requirements of plants, the process of photosynthesis, and the adaptations of plants could be more deliberately explored.

5 Conclusion

The goal of our work lies in applying the theoretical ideas of Dewey and Bereiter to better understand reasoning practices in integrate STEM problem solving. We argue that this is a worthy pursue to better understand the roles of scientific reasoning in practical problem solving. One of the goals of integrated STEM education in schools is to enculture students into the practices of science, engineering and mathematics that include disciplinary conceptual knowledge, epistemic practices, and social norms (Kelly & Licona, 2018 ). In the integrated form, the boundaries and approaches to STEM learning are more diverse compared with monodisciplinary ways of problem solving. For instance, in integrated STEM problem solving, besides scientific investigations and explanations, students are also required to understand constraints, design optimal solutions within specific parameters and even to construct prototypes. For students to learn the ways of speaking, doing and being as they participate in integrated STEM problem solving in schools in a meaningful manner, students could benefit from these experiences.

With reference to the first research question of What is the extent of practical and scientific reasoning in integrated STEM problem solving, our analysis suggests that there are fewer instances of scientific reasoning compared with practical reasoning. Considering the intention of integrated STEM learning and adopting Bereiter’s idea that students should learn higher-order conceptual knowledge through engagement with problem solving, we argue for a need for scientific reasoning to be featured more strongly in integrated STEM lessons so that students can gain higher order scientific conceptual knowledge. While the lessons observed were strong in design and building, what was missing in generating solutions was the engagement in investigations, where learners collected or are presented with data and make decisions about the data to allow them to assess how viable the solutions are. Integrated STEM problems can be designed so that science inquiry can be infused, such as carrying out investigations to figure out relationships between variables. Duschl and Bybee ( 2014 ) have argued for the need to engage students in problematising science inquiry and making choices about what works and what does not.

With reference to the second research question , What is achieved through practical and scientific reasoning during integrated STEM problem solving? , our analyses suggest that utterance for practical reasoning are typically used to justify the physical design of the prototype. These utterances rely largely on what is observable and are associated with basic-level knowledge and experiences. The higher frequency of utterances related to practical reasoning and the nature of the utterances suggests that engagement with practical reasoning is more accessible since they relate more to students’ lived experiences and common-sense. Bereiter ( 1992 ) has urged educators to engage learners in learning that is beyond basic-level knowledge since accumulation of basic-level knowledge does not lead to higher-level conceptual learning. Students should be encouraged to use scientific knowledge also to justify their prototype design and to apply scientific evidence and logic to support their ideas. Engagement with scientific reasoning is preferred as conceptual knowledge, epistemic practices and social norms of science are more widely recognised compared with practical reasoning that are likely to be more varied since they rely on personal experiences and common-sense. This leads us to assert that both context and content are important in integrated STEM learning. Understanding the context or the solution without understanding the scientific principles that makes it work makes the learning less meaningful since we “…cannot strip learning of its context, nor study it in a ‘neutral’ context. It is always situated, always relayed to some ongoing enterprise”. (Bruner, 2004 , p. 20).

To further this discussion on how integrated STEM learning experiences harness the ideas of practical and scientific reasoning to move learners from basic-level knowledge to higher-order conceptual knowledge, we propose the need for further studies that involve working with teachers to identify and create relevant problems-of-explanations that focuses on feasible, worthy inquiry ideas such as those related to specific aspects of transportation, alternative energy sources and clean water that have impact on the local community. The design of these problems can incorporate opportunities for systematic scientific investigations and scaffolded such that there are opportunities to engage in epistemic practices of the constitute disciplines of STEM. Researchers could then examine the impact of problems-of-explanations on students’ learning of higher order scientific concepts. During the problem-solving process, more attention can be given to elicit students’ initial and unfolding ideas (practical) and use them as a basis to start the science inquiry process. Researchers can examine how to encourage discussions that focus on making meaning of scientific phenomena that are embedded within specific problems. This will help students to appreciate how data can be used as evidence to support scientific explanations as well as justifications for the solutions to problems. With evidence, learners can be guided to work on reasoning the phenomena with explanatory models. These aspects should move engagement in integrated STEM problem solving from being purely practice to one that is explanatory.

6 Limitations

There are four key limitations of our study. Firstly, the degree of generalisation of our observations is limited. This study sets out to illustrate what how Dewey and Bereiter’s ideas can be used as lens to examine knowledge used in problem-solving. As such, the findings that we report here is limited in its ability to generalise across different contexts and problems. Secondly, the lessons that were analysed came from teacher-frontal teaching and group presentation of solution and excluded students’ group discussions. We acknowledge that there could potentially be talk that could involve practical and scientific reasonings within group work. There are two practical consideration for choosing to analyse the first and presentation segments of the suite of lesson. Firstly, these two lessons involved participation from everyone in class and we wanted to survey the use of practical and scientific reasoning by the students as a class. Secondly, methodologically, clarity of utterances is important for accurate analysis and as students were wearing face masks during the data collection, their utterances during group discussions lack the clarity for accurate transcription and analysis. Thirdly, insights from this study were gleaned from a small sample of six classes of students. Further work could involve more classes of students although that could require more resources devoted to analysis of the videos. Finally, the number of students varied across groups and this could potentially affect the reasoning practices during discussions.

Data Availability

The datasets used and analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to acknowledge the contributions of the other members of the research team who gave their comment and feedback in the conceptualization stage.

This study is funded by Office of Education Research grant OER 24/19 TAL.

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Tan, AL., Ong, Y.S., Ng, Y.S. et al. STEM Problem Solving: Inquiry, Concepts, and Reasoning. Sci & Educ 32 , 381–397 (2023). https://doi.org/10.1007/s11191-021-00310-2

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An Inquiry-Led Science Classroom

Teachers can guide students to be in control of their learning by asking questions that promote reflection and engaging class discussions.

Teachers agree that student questions are important during classroom discussions. Yet in practice, students ask remarkably few questions, and even fewer in search of knowledge. In particular, the three-part exchange structure known as triadic dialogue is widely used in middle and high school science classrooms.

Triadic dialogue consists of three stages: initiation, response, and evaluation, commonly referred to as the IRE model. Typically, the teacher asks a knowledge-based question that requires a predetermined short answer at the recall or lower-order cognitive level. Students respond to the question. Responses are evaluated, whereby correct answers are praised and wrong answers are corrected.

Modify the IRE Model to Encourage Dialogue and Reflection

While the IRE model might appear to restrict students’ thinking, it has its merits in the classroom, usually in the form of a process called revoicing . By revoicing and paraphrasing the students’ correct answers in the evaluation stage, teachers affirm the students’ responses, make their ideas available to the class, and assist them in verbalizing their thoughts as they co-construct a response. Through revoicing, teachers can scaffold students’ answers by encouraging further dialogue and questioning to support their construction of meaning and to reciprocally build on their contributions.

One modification to the IRE model is the “reflective toss,” where the teacher throws the responsibility for thinking back to the whole class by asking a question in response to a student’s answer, thereby shifting the classroom discussion toward more reflective discourse. These questions are usually open-ended and require one- or two-sentence answers. The teacher engages students in higher-order thinking to appraise and extend students’ ideas and encourage deeper thinking. While the dialogic discourse allows students to argue and justify their ideas, the triadic dialogue aids in emphasizing the constructed shared knowledge.

Alternating between these two types of discourse is ideal for developing conceptual thinking. Flexibility in questioning allows teachers to adjust the questions to accommodate their students’ contributions and to respond to their thinking in a neutral rather than evaluative manner.

The subsequent questions that we ask challenge students and encourage them to elaborate on their ideas by hypothesizing, predicting outcomes, generating explanations, inferring, concluding, and self-evaluating their thinking. Essentially, by redirecting the evaluation stage of the IRE model back to the whole class, teachers establish a classroom environment that values discussion, justification, exploration, and the co-construction of knowledge. This approach shifts authority for evaluating answers from the teacher to the entire class so that students can make an effort to understand what their peers are saying.

Welcome All Student Questions

Several factors may affect the frequency and quality of questions that your students ask—such as the grade level, their prior knowledge, the nature of the topic, your attitude, your teaching style, and the overall classroom environment. Students might initially ask closed-ended, knowledge-based questions because they’re easier to formulate compared with questions that involve analysis or extension of ideas. These questions are usually not investigable and can be answered with yes or no. Additionally, if the classroom environment is unwelcoming and if students are fearful that they might be bullied or labeled as the “student who doesn’t understand,” they will tend to raise questions in their minds but rarely out loud in the classroom.

Closed-ended, knowledge-based student questions:

• Does volume increase when water freezes?
• Does a glass bottle crack in low temperatures?
• Why does a mealworm become a beetle?

By welcoming all questions, you increase the probability that “better” questions will eventually be asked. Each question your students ask has the potential to address the conceptual goals of the lesson. Even the seemingly lower-level questions can serve as means to formulate more sophisticated questions. You can guide students to shift from asking typical questions to questions that have the potential to focus their learning on explanation instead of recalling facts.

These guiding questions are referred to as wonderment questions . Wonderment questions stimulate students to generate explanations, propose solutions, hypothesize, predict, and enrich the classroom discussion—leading to a cascade of knowledge construction. This isn’t to say that comprehension-level questions are less important. On the contrary, knowledge-based questions are imperative to review learning, identify misconceptions, and facilitate the connection of preexisting concepts with new ones.

Teacher-guided wonderment questions:

• Why did the bottle break?
• Why does the bottle break when it freezes?
• Do all worms become beetles?
• When do mealworms look like beetles?

The questions you ask can serve as models for the students. It’s important for us as teachers to demonstrate to students that their questions are valued, model asking investigable or exploratory questions by providing examples, and provide explicit practice in refining questions.

The questions that we ask can do the following:

• Guide students’ participation in class discussions
• Increase students’ interest
• Identify misconceptions
• Stimulate students’ prior knowledge
• Allow students to develop an opinion about a topic

In inquiry-based learning, questions include the problem context, are testable, and begin with “What,” “How,” or “Why”: “How is…?” “How did … happen?” “What does … mean?” “What can you say about…?” “What would happen if…?” “How would you test…?” “Can you predict the outcome if…?”

In inquiry-based learning, problem-finding (questioning) is equally important to problem-solving. Encourage students to seek knowledge so that their questions become a critical part of their self-directed learning process. We want classrooms where teachers elicit and use student questions effectively in instruction to extend their thinking and reasoning.

Discussing science is more than using scientific terminology and answering questions. Formulating a question is a cognitive skill that lies at the heart of inquiry-based learning. Questions posed at critical junctures of a lesson can focus our students’ attention on the main aspects of the concept and create the space for inquiry and learning.

Activity Ideas / Math / Planning & Teaching

How to Teach Math Using Inquiry-Based Learning

Teaching math has slowly been evolving from a subject that, for many students, feels useless and trivial, to one that helps provide relevance and build important learning skills inside and outside of the classroom. Inquiry learning helps provide students with opportunities to develop critical thinking, teamwork, and resilience – skills that are absolutely essential to be “good” at math.

Teaching math using inquiry-based learning is a powerful way to reinforce the skill of problem-solving for students. In the inquiry math classroom, students are not passive recipients of formulas and facts, but are instead encouraged to ask questions, investigate rules and patterns, and persevere when confronted with challenges. Using an inquiry approach in math helps build important skills such as resilience, problem-solving, and perseverance.

The post below outlines some ways to incorporate an inquiry-based approach into your math classroom (or within your lessons). It also provides some examples of inquiry and PBL projects to help your students gain confidence, build good learning habits, and apply their skills.

Traditional vs. Inquiry-Based Approaches

There’s been a lot of debate about using a “back to basics” traditional approach or an inquiry-based approach when teaching math. Traditional math teaching typically follows a structured curriculum with a focus on rote memorization, guided procedures, and step-by-step instructions, which many would describe as providing a solid foundation for students to build mathematical skills. On the other hand, inquiry-based math emphasizes exploration, critical thinking, and problem-solving through open-ended tasks. Neither of these approaches are wrong.

In my own classrooms, I’ve often opted for a combination of approaches. Recognizing the strengths of each method and using them complementary to each other has resulted in the best outcomes for my students in the past. Traditional math works very well for students who need the reliability of steps and formulas to grasp numerical concepts and skills. Inquiry-based methods work well for students when they need to apply these concepts and skills.

Encouraging students to apply their skills and understanding in the “real world” and helping them learn to make connections between concepts allows them to develop a deeper understanding of math. Combining the two approaches can create a balanced and well-rounded math education, and equip students with not only the procedural knowledge they need, but also the necessary problem-solving skills to apply this knowledge in other contexts.

Why should you use an inquiry approach in math?

For most of us, we’re used to math being presented as a set of rules and procedures we follow to arrive at a correct answer. However, this is usually far from the way math is applied in the real world. A good example is when scientists and climatologists are studying weather patterns – they might ask something like, “how much rain can we expect from this hurricane?”. From there, meteorologists and atmospheric scientists combine data to create models to help approach the question. They apply their understanding of mathematical principles to situations to determine the correct answer, then see if their solution applies in the real world. If not, it’s back to the drawing board.

Framing math in this kind of way, as opposed to the traditional “memorize the formula, practice some questions, then repeat”, can be far more effective. Some of the benefits include:

• Emphasizing the process as opposed to the final product
• Finding several different approaches and choosing the one(s) that work for you
• Seeing math from a more holistic viewpoint
• Teaching students that mistakes are where we learn the most
• Reducing the stigma that math is “hard”, “useless”, or “irrelevant”
• Building critical thinking habits and resilience to persevere through difficulties
• Having the ability to transfer skills to all types of mathematical problems
• Learning more deeply and in a more problem-based way

Below are some unique ways to incorporate an inquiry focus into your math lessons. These ideas can be used as stand-alone projects or as complements to your traditional lessons. They also work well as projects that call for the application of new learning.

Math Inquiry Idea #1 – Urban Planning and Sustainable Cities

For this inquiry idea, students play the role of an urban planner tasked with designing a new layout for a sustainable city. To start, discuss the concepts of urban planning and sustainability. 

Ask students how the two are related and ask them what characteristics a sustainable city has. Some characteristics include:

• Reliance on renewable energy (such as wind, solar, and smart grid technologies)
• Plenty of green spaces and biodiversity
• Examples of urban agriculture, such as community gardens
• Climate change adaptation
• Efficient transportation and waste management services
• Mixed-use development
• Community engagement and participation

Spend some time discussing and envisioning what a sustainable city would look like. Allow them to utilize videos, books, and maps to explore their ideas.

Suggested resources:

• Green City: How One Community Survived a Tornado and Rebuilt for a Sustainable Future (hardcover, ages 5-8)
• Sustainable Development Goals (PDF from the United Nations for your learning wall)
• How Cities Work (Lonely Planet hardcover/interactive book)
• Climate Change and Sustainable Cities (NRDC)

Provide students with resources that highlight real-world examples of sustainable urban planning projects. Guide them to explore how these cities are laid out and where the major roads and buildings are. Are there green spaces? Where are they? Where could they go? Are there clear public transportation links? Could unused space be utilized in a more efficient way?

Once students have an idea of what features they’d like their city to have, it’s time to draw. Using paper or computers, have students create a map that lays out their city and all of its features. Utilize geography concepts like scale, elevation, and topography so that students understand the connection between math and other subjects. Encourage students to use materials like cardboard, wood, foam, or recyclable materials to bring their maps to life.

Provide students opportunities to reflect on their learning and the challenges they encountered during the design process. Ask questions about the application of mathematical principles and how they helped students with things like scale, measurement, and spacing. Facilitate discussions that encourage problem-solving and allow other students to respectfully ask questions about the sustainability features they see. Prompt students to consider the trade-offs and compromises they had to make in their decisions.

Incorporating Mathematical Concepts

There are a few different ways that math can be applied to this inquiry project:

1. Creating scale models

Determine the scale at which you want to create the model city. Common scales include 1:100, 1:200, or 1:500. Encourage students to plan a layout for the city, including streets, buildings, parks, landmarks, and other features of a sustainable city. Consider the use of grids and geometric shapes for simpler planning. or use pattern blocks to map out the features of the city.

2. Simple calculations

Planning a sustainable city requires the calculation of things like distance, areas, and volume in order to optimize for land use, transportation routes, and building placements. For example, calculating the distance between a major airport and the city centre, or calculating the amount of rainwater that a barrel can hold.

3. Statistical analysis

Through examining articles of pre-existing cities, students gather and analyze demographic data, population trends, and socioeconomic indicators to identify patterns. These can then be used to inform decisions along the way. Students could also conduct surveys to find out what features their classmates would want in a sustainable city and analyze that data too.

4. Graphing and equations

If students conduct surveys or collect forms of data from their peers, they can create formulas for tracking things like future population growth, resource demand, and infrastructure needs. They could also predict greenhouse gas emissions or energy consumption trends too. Encourage students to graph their results so the data can be understood simply.

5. Financial costs

Building a sustainable city isn’t cheap. Discuss the cost of materials, paying workers, and other variables that go into the construction of a city. Older students may choose to calculate the cost-effectiveness of sustainable initiatives and projects by using financial metrics, or conduct a risk assessment to evaluate the probability of things like natural disasters and climate-related events.

Related Reading :

Solving the Problems of Vacant Spaces and Empty Rooftops (ideas #2 and 3) Blueprints for a Greener Footprint PDF (WEF) The City in 2050: Creating Blueprints for Change (ULI) SmartAfrica Blueprint (PDF)

Math Inquiry Idea #2 – Applying Mathematics to Cryptography

It’s no surprise that coding and math share a lot of similarities. This inquiry idea is suitable for older students studying computer science and other high-school level mathematics or coding courses.

Cryptography is the science of using math to hide data behind encryption. This helps store, secure, and protect communication. Math and cryptography share a deep connection because math is what provides the foundation for creating secure cryptographic systems.

Most cryptographic systems combine two things:

1. A set of rules that specify the mathematical steps needed to encipher or decipher data (also known as a process or algorithm) 2. A cryptographic key (a string of numbers of characters), or keys

In simpler terms, imagine you had a recipe for baking a cake. In the same way the recipe tells you the steps to follow to bake the cake, a cryptographic algorithm is a set of rules that tells a computer how to turn secret information (plaintext) into scrambled information (ciphertext).

Imagine you put that recipe in a locked box. Having the key to unlock that secret box is like having a cryptographic key, which is a string of numbers or characters that the algorithm uses to scramble and unscramble your data. Without the right keys, you can’t get into the box.

So, when you combine the algorithm with the cryptographic key, it’s like following a secret recipe with a special key to unlock and lock your secret message. The key is what keeps your recipe safe, and the algorithm is the set of rules that keeps everything secure.

Inquiry Projects Involving Cryptography

1. self-guided inquiry.

Students can explore the various theories involved in encryption schemes, including:

• Diffie-Hellman key exchange scheme
• Chinese remainder theorem
• Probability theory
• The RSA encryption algorithm
• Complexity theory
• Elliptic curve cyptography
• EIGamal encryption system
• Information theory

Challenge students to not only understand one of the above theories, but to simplify it so that younger students can understand. Similar to the cake example, students use their knowledge and experiment with different analogies to help others understand in an easy-to-understand way. Students can choose to share their information as a presentation, workshop, or through a demonstration.

2. Simple ciphers

Show students simple substitution ciphers, such as:

Caesar cipher – each plaintext letter is replaced with a different one a fixed number of places down the alphabet. For example, a left shift of three places, as seen below:

Pigpen cipher – each plaintext letter is replaced with a simple geometric picture symbol (created using grids and crosses so that each letter is represented by fragments of a grid or cross with or without a dot)

Provide them with decoding charts and encourage them to encode and decode their messages using these ciphers. For example, provide math equations where each answer corresponds to a letter in the alphabet, and challenge students to decode the message by solving the equations. These types of problems utilize deductive reasoning and problem-solving skills as well as helping to solidify basic concepts such as addition, subtraction, multiplication, and division.

Related Reading:

Guide to Cryptography Mathematics Explorer Academy: Code-Breaking Activity Adventure (National Geographic) Code Cracking for Kids (Codes and Ciphers) Cryptography as a Teaching Tool

Math Inquiry Idea #3 – Sports and Statistics

While not uncommon in the math classroom, sports and statistics can help foster critical thinking skills in a way that appeals to a lot of students. By encouraging students to explore the connection between math and sports, they get the chance to see how mathematical concepts apply to familiar, real-life situations.

Of course, not every student is interested in sports. In these cases, it helps to point out the fact that sports aren’t just about the rules, plays, and competition. Instead, merging sports and mathematics offers students new and interesting ways to apply their learning in strands such as geometry, statistics, algebra, and measurement.

To get students thinking about the intersection of math and sports, consider starting off by discussing the importance of statistics and how they are used in sports such as basketball, soccer, baseball, or football.

Students may suggest the following points:

• Statistics help track player performance and evaluation (helpful for managers and scouts to assess a players’ strengths and weaknesses)
• They allow coaches and managers to notice trends and patterns in team dynamics (which helps identify areas for improvement or adjustment)
• The use of statistics helps coaches and team members make strategic and informed decisions (helpful in making lineup changes, substitutions, and set goals)
• Statistics also help enhance fan engagement by providing valuable insight into the dynamics of the game (which fuels discussions and debates within the sports community)

Encourage students to generate questions and inquiries related to sports statistics. Prompt them to think about what types of data are collected in sports, and what they’re curious about in general. Provide them with time and opportunities to research statistics for players or sports they’re interested in. Statistics can come from a variety of courses, such as online databases, sports websites, and historical data archives. Guide students in collecting and organizing data on player and team stats, game scores, and other relevant metrics.

From here, students can take their learning in several directions. Some suggestions are below:

1. Drawing comparisons

Students love making comparisons, whether between each other, celebrities, sports stars, or their favourite sports team. In a statistics inquiry, encourage students to make comparisons between different players, teams, or seasons using measures such as averages, percentages, ratios, and rates. For example, students can compare the shooting percentages of basketball players, betting averages of baseball players, or the win-loss records of sports teams.

Guide students in exploring mathematical concepts and principles embedded in these statistics. Prompt them to ask inquiry questions to guide their own learning. One example might be, “how did Tom Brady’s passing yards change from when he played for New England vs. when he played for Tampa Bay?” Here are some other examples:

• How does a player’s shooting percentage affect their team’s overall performance?
• What factors influence a team’s success in a particular sport?
• How do player statistics change over the course of a season?

Facilitate investigations by helping students analyze their data, generate hypotheses, conduct experiments, and draw conclusions based on their findings. Encourage students to share their learning and insights through various inquiry-based presentation formats, including traditional presentations, discussions, the use of charts or graphs, and multimedia presentations. Ensure they are explaining their mathematical reasoning and encourage peer feedback.

2. Creating a “Statistics Palette”

Have you ever heard of “movie palettes”? Basically, it’s a set of specific colours used throughout a particular movie or film. They’re carefully curated to evoke specific moods and feelings, convey thematic elements, and enhance storytelling. Movie palettes are relatively recent and can be purchased as a unique piece to showcase your favourite movie. During the process, the dominant colour from each scene of your favourite movie is turned into a vertical strip, and arranged chronologically, side-by-side, onto a canvas. This results in a unique piece of artwork that represents your favourite movie as a colour palette.

Students can apply this idea to a sports statistics inquiry in math by assigning colours to specific data categories. For example, examine Tom Brady’s passing yards when he played for the New England Patriots and compare them to his passing yards when he played for the Tampa Bay Buccaneers. Group his statistics by number of yards (for example <1000 per season, 1000-1500 per season, etc.) and assign each group a colour, shown below:

The United States of Sports (Sports Illustrated for Kids) It’s a Numbers Game! (Co-authored by Patrick Mahomes) Sports Reference (website full of up-to-date sports statistics)

Math Inquiry Idea #4 – Geometry in Architecture

This inquiry idea is great because it allows teachers to facilitate learning in a cross-curricular way , often incorporating several subjects. To start, give students the opportunity to explore the history and significance of geometry in architecture, from ancient civilizations to the contemporary world. Students can examine great architectural wonders of the past, including:

• The Great Pyramid of Giza (Egypt, 2560 BCE)
• The Hanging Gardens of Babylon (Iraq, 600 BCE)
• The Colossus of Rhodes (Greece, 280 BCE)
• Chichen Itza (Mexico, 700 AD)
• The Taj Mahal (India, 1648 AD)

By examining architectural masterpieces from the past, students will gain an appreciation for the role of geometry in creating these structures. As facilitator, help students to identify geometric shapes and principles found within these structures and explore how proportions, patterns, and culture influence architectural design and construction processes.

Recognizing Historical Successes

A great example is the use of geometry by the Inca at Machu Picchu, where they built intricate terraces and staircases, both for agricultural purposes and in order to navigate the steep Andean terrain. In this case, geometry helped them design and construct these structures with precision. Show students a photo of the staircases and terraces, and ask students to identify their characteristics; for example, the terraces follow the contours of the mountainside, and the staircases were engineered to conform to the natural topography of the landscape.

Once students feel comfortable identifying these characteristics, have them explore the use of water irrigation systems and aqueducts. Encourage them to make inferences about how the architectural design of these systems resulted in optimization water distribution, minimal erosion, and the conservation of natural resources.

As students continue to gain confidence in exploring these architectural features, encourage them to use more precise mathematical language. For example, instead of saying that the Inca “made steep paths”, they can say that the Inca “constructed steep slopes”.

Solving Modern-Day Problems

Once students understand the link between geometry and construction, and have had some time to explore the impact of mathematics on construction, challenge them to apply their understanding to real-world design challenges and problems. This is where a wonderful intersection of subjects – math, design, engineering, physics, and environmental science – occurs. Students may be tasked with designing and construction scale models of popular landmarks, exploring principles of stability and load-bearing, or analyzing the benefits of using specific shapes such as arches, domes, and columns.

A good application of these skills is to tackle the challenge of how to create safe and stable wildlife crossings over busy roads and highways. This is a problem that a lot of students show interest in because they want to see animals be able to coexist safely in our modern world. The use of bridges, tunnels, overpasses to allow animals to safely cross roads and highways requires problem-solving skills and spatial thinking in addition to mathematical thinking. A good place to start is by getting familiar with the Banff Wildlife Crossings Project , which has reduced animal-vehicle collisions in the province of Alberta by more than 80% since its construction.

The use of manipulatives such as pattern tiles, tile boards, and geoboards ( available in a set of 6 here ) help students to connect the building process with geometric concepts. Encourage them to explore how shape, symmetry, and proportion work together to create stunning buildings, homes, and other architectural wonders. Challenge students to sketch their ideas, incorporating geometric elements such as a variety of shapes, angles, patterns, tessellations, and symmetry in their designs.

Teachers can prompt students to consider other elements, such as:

• Proportions and measurements of windows, doors, and other features
• Sustainable and eco-friendly features, such as solar panels, wind turbines, rainwater harvesting, living walls, green roods, and natural ventilation
• Aesthetics (for example, creating building that mesh nicely with the culture and traditions of a particular place)

Once students have visualized their ideas, they are encouraged to display their learning in a way that is most meaningful to them. For example, if they sketched a scale-drawing of a wildlife bridge, the next natural step would be to create a physical model or prototype to demonstrate how it would look in real-life. Some may choose to conduct a gallery walk, showing their peers and others their idea progress through rough sketches and blueprints. Regardless of their presentation method, students should be encourage to provide feedback on both their own work and their peers’ work. Review sessions can be conducted to facilitate the exchange of constructive feedback and suggestions for improvement, which are a natural part of the inquiry process.

Related reading: 

Wildlife Crossings (National Geographic article) The Wallis Annenberg Wildlife Crossing Wildlife Crossings and Fish Passages (AIL)

Final Thoughts

Fusing math and inquiry learning in the classroom is important in all 21st century classrooms. An inquiry-based approach encourages students to ask questions, explore mathematical principles, and see how math relates to many different aspects of their lives. By allowing students to make connections and participate in authentic learning, they will hopefully develop an appreciation for the relevance and applicability of mathematics in everyday life, even if they are not keen to pursue a job that requires extensive math knowledge. 

Key Takeaways:

(1) Teaching math using inquiry-based learning is a powerful way to reinforce the skill of problem-solving for students

(2) Applying inquiry-based learning in math doesn’t have to be a “one or the other” approach. Rather, you can combine elements of traditional math teaching and incorporate tenets of inquiry along the way to solidify concepts and apply learning in new contexts

(3) Using scenario-based learning or delving into real-life problems that involve math helps build problem-solving and communication skills, as well as perseverance in students of all ages

(4) Employing a cross-curricular approach allows students to appreciate the applicability of math in everyday life

(5) Inquiry-based learning in math empowers students to take ownership of their learning and draw conclusions independently or collaboratively

Cover photo by Alena Darmel

Have you used any of these math inquiry project ideas in your classroom? Do you have any other ideas you’d like to share? Feel free to comment down below, or browse some more ideas on  Pinterest !

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Inquiry and Problem Solving

Inquiry and problem solving refer to an array of learner-centered processes that facilitate deep engagement with a question or problem and strategies to develop subsequent solutions and explanations.

With any approach to inquiry and problem-solving, students follow a series of phases or specific discipline-based practices to take them through an intentional process. These phases or practices are not necessarily completed in a linear or lock-step manner, but provide some structure for students to address a question, issue, problem, or need.

Common elements of inquiry and problem-solving include:

Asking questions or defining problems

Exploring solutions or explanations

Analyzing or testing solutions or explanations

Communicating or taking action on solutions or explanations

The elements are most effective when applied in an iterative cycle, so that students have the opportunity to revise their solutions and explanations based on utility, analysis, and feedback. Here's how different sets of discipline-based practices and inquiry models can align to the four common elements of inquiry.

Inquiry Elements

Scientific and Engineering Practices

Standards for Mathematical Practice

Inquiry Arc

Design Thinking

Asking questions and defining problems

Make sense of problems and persevere in solving them

Developing questions and planning inquiries

Empathize with users

Define the problem

Planning and carrying out investigations

Ideate potential solutions

Prototype solutions

Developing and using models

Analyzing and interpreting data

Using mathematics and computational thinking

Constructing explanations and designing solutions

Model with mathematics

Reason abstractly and quantitatively

Use appropriate tools strategically

Attend to precision

Look for and make sense of structure

Look for and express regularity in repeated reasoning

Applying disciplinary concepts and tools

Evaluating sources and using evidence

Test solutions

Engaging in argument from evidence

Obtaining, evaluating and communicating information

Construct viable arguments and critique the reasoning of others

Communicating conclusions and taking informed action

Implementation of solution

Inquiry and problem-solving opportunities encourage students to engage with relevant concepts and skills within authentic, real-world contexts. Research suggests that combining real-world application with sustained participation in inquiry and problem solving experiences can lead to increased student engagement and support deeper learning of concepts and skills. Additionally, opportunities for inquiry and problem solving can help students develop:

• Professional skills - e.g. communication, professionalism, collaboration, and empathy;
• Innovative mindsets - e.g. creative, ‘out of the box’ thinking; and
• Metacognition - e.g. self-regulation and self-monitoring of their learning progress

Developing these skills prepares students for college, career, and community success, and help students meet the HIDOE general learner outcomes . To further examine the benefits of inquiry-based learning, see the following resources:

Center for Inspired Teaching

Review of Research on Inquiry-Based Learning

Inquiry-Based Teaching in World Languages

Inquiry-Based Models

Students can engage in deep learning of concepts and skills through a variety of inquiry-based models, including: instructional approaches, instructional sequences, and/or design processes.

• Instructional approaches - models of inquiry that are designated by essential features more so than one consensus sequence of events
• Instructional sequences - models of inquiry that have a specific sequence of events based on learning theory or practices of a discipline
• Design processes - models of design based on an iterative process of prototyping and testing to develop a final product

Examples of inquiry-based models, their essential features and ideas for getting started with them are presented below:

Instructional Approaches

• Project-Based Learning: Essential Features | Getting Started
• Problem-Based Learning: Essential Features | Getting Started
• Place-Based Learning: Essential Features | Getting Started

Instructional Sequences

• 5E Instructional Model: Essential Features | Getting Started
• Inquiry Design Model: Essential Features | Getting Started
• Gather - Reason - Communicate: Essential Features | Getting Started

Design Processes

• Engineering Design Processes: Essential Features | Getting Started
• Design Thinking: Essential Features | Getting Started

Considerations for Implementing Inquiry-Based Learning

Whatever process or model is chosen to engage students in inquiry and problem-solving, it is important to consider different approaches to structured vs open inquiry . For many students and at earlier grade levels, a scaffolded approach is ideal when students first encounter an inquiry investigation. Considering a variety of scaffolding and differentiation approaches will ensure that students’ different learning needs are addressed. Over time, with more practice, the teacher can gradually release decision-making and control to the students. It is also helpful for the teacher to consider the types of questions they will ask students to support their inquiry process.

Lesson and Activity Resources

The following sites showcase examples of learning experiences that are inquiry- and problem-based.

All Content Areas

Visible Thinking : A site with a variety of thinking routines that can be used to scaffold students’ inquiry and problem solving skills.

Design Thinking : Resources for facilitating design thinking with students curated by Hawai‘i educators.

Mathematics

Making Sense of Problems : A site with examples of classroom practice of the standards for mathematical practice, including, “Making sense of problems and persevere in solving them.”

Hawai‘i educators developed inquiry-based STEM Units that align to the FAIR features of STEM learning experiences .

Social Studies

C3 Hawai‘i Hub : Provides inquiry lessons for social studies developed by Hawai‘i educators using the inquiry design model.

Science: GRC Lessons

Provides inquiry lessons for science developed by Hawai‘i and other U.S. educators using the Gather - Reason - Communicate Approach and 5E instructional sequence.

Back to School Design

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Inquiry Learning Model

Inquiry Learning Model is a student-centered, active learning approach that places students at the center of the learning process. Instead of traditional lecture-style teaching, where information is delivered to students passively, inquiry learning encourages students to ask questions, explore topics, and actively engage in problem-solving and critical thinking.

Here are the key elements and principles of an inquiry learning model:

• Questioning : Students are encouraged to ask questions and express their curiosity about a particular topic. These questions can be open-ended and drive the direction of their learning.
• Exploration : Students engage in hands-on activities, research, and investigations to gather information and data related to their questions. This may involve experiments, surveys, reading, or other forms of data collection.
• Critical Thinking : Through inquiry, students develop critical thinking skills by analyzing information, evaluating evidence, and making informed decisions or drawing conclusions based on their findings.
• Problem-Solving : Inquiry learning often presents students with real-world problems or challenges that require creative problem-solving. Students work independently or collaboratively to develop solutions.
• Active Participation : Students actively participate in their learning process. They are responsible for setting goals, planning their activities, and monitoring their progress.
• Teacher Facilitation : While students take a more active role in inquiry learning, teachers play a crucial role as facilitators. They guide students, provide resources, offer support, and help frame questions and problems.
• Reflection : Students are encouraged to reflect on their learning experiences. This reflection helps them consolidate their knowledge and develop metacognitive skills, allowing them to become more effective learners.
• Interdisciplinary Approach : Inquiry learning often blurs the boundaries between subjects. It allows students to explore topics from various angles, incorporating elements from multiple disciplines.
• Student Autonomy : Students have a degree of autonomy in choosing the topics they want to explore and the methods they want to use for investigation. This autonomy fosters a sense of ownership over their learning.
• Assessment : Assessment in inquiry learning focuses on the process as well as the product. It may involve self-assessment, peer assessment, and teacher evaluation of the skills and knowledge gained through the inquiry process.
• Continuous Learning : Inquiry learning promotes a culture of continuous learning. Students learn how to learn, develop a growth mindset, and become more self-directed in their pursuit of knowledge.

Inquiry Learning Model can be applied at various educational levels, from primary school through higher education. It encourages students to become active, engaged, and curious learners while developing critical thinking and problem-solving skills that are valuable in both academic and real-world contexts.

Also Read : Use of ICT in Education

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STEM Problem Solving: Inquiry, Concepts, and Reasoning

Aik-ling tan.

Natural Sciences and Science Education, meriSTEM@NIE, National Institute of Education, Nanyang Technological University, Singapore, Singapore

Yann Shiou Ong

Yong sim ng, jared hong jie tan, associated data.

The datasets used and analysed during the current study are available from the corresponding author on reasonable request.

Balancing disciplinary knowledge and practical reasoning in problem solving is needed for meaningful learning. In STEM problem solving, science subject matter with associated practices often appears distant to learners due to its abstract nature. Consequently, learners experience difficulties making meaningful connections between science and their daily experiences. Applying Dewey’s idea of practical and science inquiry and Bereiter’s idea of referent-centred and problem-centred knowledge, we examine how integrated STEM problem solving offers opportunities for learners to shuttle between practical and science inquiry and the kinds of knowledge that result from each form of inquiry. We hypothesize that connecting science inquiry with practical inquiry narrows the gap between science and everyday experiences to overcome isolation and fragmentation of science learning. In this study, we examine classroom talk as students engage in problem solving to increase crop yield. Qualitative content analysis of the utterances of six classes of 113 eighth graders and their teachers were conducted for 3 hours of video recordings. Analysis showed an almost equal amount of science and practical inquiry talk. Teachers and students applied their everyday experiences to generate solutions. Science talk was at the basic level of facts and was used to explain reasons for specific design considerations. There was little evidence of higher-level scientific conceptual knowledge being applied. Our observations suggest opportunities for more intentional connections of science to practical problem solving, if we intend to apply higher-order scientific knowledge in problem solving. Deliberate application and reference to scientific knowledge could improve the quality of solutions generated.

Introduction

As we enter to second quarter of the twenty-first century, it is timely to take stock of both the changes and demands that continue to weigh on our education system. A recent report by World Economic Forum highlighted the need to continuously re-position and re-invent education to meet the challenges presented by the disruptions brought upon by the fourth industrial revolution (World Economic Forum, 2020 ). There is increasing pressure for education to equip children with the necessary, relevant, and meaningful knowledge, skills, and attitudes to create a “more inclusive, cohesive and productive world” (World Economic Forum, 2020 , p. 4). Further, the shift in emphasis towards twenty-first century competencies over mere acquisition of disciplinary content knowledge is more urgent since we are preparing students for “jobs that do not yet exist, technology that has not yet been invented, and problems that has yet exist” (OECD, 2018 , p. 2). Tan ( 2020 ) concurred with the urgent need to extend the focus of education, particularly in science education, such that learners can learn to think differently about possibilities in this world. Amidst this rhetoric for change, the questions that remained to be answered include how can science education transform itself to be more relevant; what is the role that science education play in integrated STEM learning; how can scientific knowledge, skills and epistemic practices of science be infused in integrated STEM learning; what kinds of STEM problems should we expose students to for them to learn disciplinary knowledge and skills; and what is the relationship between learning disciplinary content knowledge and problem solving skills?

In seeking to understand the extent of science learning that took place within integrated STEM learning, we dissected the STEM problems that were presented to students and examined in detail the sense making processes that students utilized when they worked on the problems. We adopted Dewey’s ( 1938 ) theoretical idea of scientific and practical/common-sense inquiry and Bereiter’s ideas of referent-centred and problem-centred knowledge building process to interpret teacher-students’ interactions during problem solving. There are two primary reasons for choosing these two theoretical frameworks. Firstly, Dewey’s ideas about the relationship between science inquiry and every day practical problem-solving is important in helping us understand the role of science subject matter knowledge and science inquiry in solving practical real-world problems that are commonly used in STEM learning. Secondly, Bereiter’s ideas of referent-centred and problem-centred knowledge augment our understanding of the types of knowledge that students can learn when they engage in solving practical real-world problems.

Taken together, Dewey’s and Bereiter’s ideas enable us to better understand the types of problems used in STEM learning and their corresponding knowledge that is privileged during the problem-solving process. As such, the two theoretical lenses offered an alternative and convincing way to understand the actual types of knowledge that are used within the context of integrated STEM and help to move our understanding of STEM learning beyond current focus on examining how engineering can be used as an integrative mechanism (Bryan et al., 2016 ) or applying the argument of the strengths of trans-, multi-, or inter-disciplinary activities (Bybee, 2013 ; Park et al., 2020 ) or mapping problems by the content and context as pure STEM problems, STEM-related problems or non-STEM problems (Pleasants, 2020 ). Further, existing research (for example, Gale et al., 2000 ) around STEM education focussed largely on description of students’ learning experiences with insufficient attention given to the connections between disciplinary conceptual knowledge and inquiry processes that students use to arrive at solutions to problems. Clarity in the role of disciplinary knowledge and the related inquiry will allow for more intentional design of STEM problems for students to learn higher-order knowledge. Applying Dewey’s idea of practical and scientific inquiry and Bereiter’s ideas of referent-centred and problem-centred knowledge, we analysed six lessons where students engaged with integrated STEM problem solving to propose answers to the following research questions: What is the extent of practical and scientific inquiry in integrated STEM problem solving? and What conceptual knowledge and problem-solving skills are learnt through practical and science inquiry during integrated STEM problem solving?

Inquiry in Problem Solving

Inquiry, according to Dewey ( 1938 ), involves the direct control of unknown situations to change them into a coherent and unified one. Inquiry usually encompasses two interrelated activities—(1) thinking about ideas related to conceptual subject-matter and (2) engaging in activities involving our senses or using specific observational techniques. The National Science Education Standards released by the National Research Council in the US in 1996 defined inquiry as “…a multifaceted activity that involves making observations; posing questions; examining books and other sources of information to see what is already known; planning investigations; reviewing what is already known in light of experimental evidence; using tools to gather, analyze, and interpret data; proposing answers, explanations, and predictions; and communicating the results. Inquiry requires identification of assumptions, use of critical and logical thinking, and consideration of alternative explanations” (p. 23). Planning investigation; collecting empirical evidence; using tools to gather, analyse and interpret data; and reasoning are common processes shared in the field of science and engineering and hence are highly relevant to apply to integrated STEM education.

In STEM education, establishing the connection between general inquiry and its application helps to link disciplinary understanding to epistemic knowledge. For instance, methods of science inquiry are popular in STEM education due to the familiarity that teachers have with scientific methods. Science inquiry, a specific form of inquiry, has appeared in many science curriculum (e.g. NRC, 2000 ) since Dewey proposed in 1910 that learning of science should be perceived as both subject-matter and a method of learning science (Dewey, 1910a , 1910b ). Science inquiry which involved ways of doing science should also encompass the ways in which students learn the scientific knowledge and investigative methods that enable scientific knowledge to be constructed. Asking scientifically orientated questions, collecting empirical evidence, crafting explanations, proposing models and reasoning based on available evidence are affordances of scientific inquiry. As such, science should be pursued as a way of knowing rather than merely acquisition of scientific knowledge.

Building on these affordances of science inquiry, Duschl and Bybee ( 2014 ) advocated the 5D model that focused on the practice of planning and carrying out investigations in science and engineering, representing two of the four disciplines in STEM. The 5D model includes science inquiry aspects such as (1) deciding on what and how to measure, observe and sample; (2) developing and selecting appropriate tools to measure and collect data; (3) recording the results and observations in a systematic manner; (4) creating ways to represent the data and patterns that are observed; and (5) determining the validity and the representativeness of the data collected. The focus on planning and carrying out investigations in the 5D model is used to help teachers bridge the gap between the practices of building and refining models and explanation in science and engineering. Indeed, a common approach to incorporating science inquiry in integrated STEM curriculum involves student planning and carrying out scientific investigations and making sense of the data collected to inform engineering design solution (Cunningham & Lachapelle, 2016 ; Roehrig et al., 2021 ). Duschl and Bybee ( 2014 ) argued that it is needful to design experiences for learners to appreciate that struggles are part of problem solving in science and engineering. They argued that “when the struggles of doing science is eliminated or simplified, learners get the wrong perceptions of what is involved when obtaining scientific knowledge and evidence” (Duschl & Bybee, 2014 , p. 2). While we concur with Duschl and Bybee about the need for struggles, in STEM learning, these struggles must be purposeful and grade appropriate so that students will also be able to experience success amidst failure.

The peculiar nature of science inquiry was scrutinized by Dewey ( 1938 ) when he cross-examined the relationship between science inquiry and other forms of inquiry, particularly common-sense inquiry. He positioned science inquiry along a continuum with general or common-sense inquiry that he termed as “logic”. Dewey argued that common-sense inquiry serves a practical purpose and exhibits features of science inquiry such as asking questions and a reliance on evidence although the focus of common-sense inquiry tends to be different. Common-sense inquiry deals with issues or problems that are in the immediate environment where people live, whereas the objects of science inquiry are more likely to be distant (e.g. spintronics) from familiar experiences in people’s daily lives. While we acknowledge the fundamental differences (such as novel discovery compared with re-discovering science, ‘messy’ science compared with ‘sanitised’ science) between school science and science that is practiced by scientists, the subject of interest in science (understanding the world around us) remains the same.

The unfamiliarity between the functionality and purpose of science inquiry to improve the daily lives of learners does little to motivate learners to learn science (Aikenhead, 2006 ; Lee & Luykx, 2006 ) since learners may not appreciate the connections of science inquiry in their day-to-day needs and wants. Bereiter ( 1992 ) has also distinguished knowledge into two forms—referent-centred and problem-centred. Referent-centred knowledge refers to subject-matter that is organised around topics such as that in textbooks. Problem-centred knowledge is knowledge that is organised around problems, whether they are transient problems, practical problems or problems of explanations. Bereiter argued that referent-centred knowledge that is commonly taught in schools is limited in their applications and meaningfulness to the lives of students. This lack of familiarity and affinity to referent-centred knowledge is likened to the science subject-matter knowledge that was mentioned by Dewey. Rather, it is problem-centred knowledge that would be useful when students encounter problems. Learning problem-centred knowledge will allow learners to readily harness the relevant knowledge base that is useful to understand and solve specific problems. This suggests a need to help learners make the meaningful connections between science and their daily lives.

Further, Dewey opined that while the contexts in which scientific knowledge arise could be different from our daily common-sense world, careful consideration of scientific activities and applying the resultant knowledge to daily situations for use and enjoyment is possible. Similarly, in arguing for problem-centred knowledge, Bereiter ( 1992 ) questioned the value of inert knowledge that plays no role in helping us understand or deal with the world around us. Referent-centred knowledge has a higher tendency to be inert due to the way that the knowledge is organised and the way that the knowledge is encountered by learners. For instance, learning about the equation and conditions for photosynthesis is not going to help learners appreciate how plants are adapted for photosynthesis and how these adaptations can allow plants to survive changes in climate and for farmers to grow plants better by creating the best growing conditions. Rather, students could be exposed to problems of explanations where they are asked to unravel the possible reasons for low crop yield and suggest possible ways to overcome the problem. Hence, we argue here that the value of the referent knowledge is that they form the basis and foundation for the students to be able to discuss or suggest ways to overcome real life problems. Referent-centred knowledge serves as part of the relevant knowledge base that can be harnessed to solve specific problems or as foundational knowledge students need to progress to learn higher-order conceptual knowledge that typically forms the foundations or pillars within a discipline. This notion of referent-centred knowledge serving as foundational knowledge that can be and should be activated for application in problem-solving situation is shown by Delahunty et al. ( 2020 ). They found that students show high reliance on memory when they are conceptualising convergent problem-solving tasks.

While Bereiter argues for problem-centred knowledge, he cautioned that engagement should be with problems of explanation rather than transient or practical problems. He opined that if learners only engage in transient or practical problem alone, they will only learn basic-category types of knowledge and fail to understand higher-order conceptual knowledge. For example, for photosynthesis, basic-level types of knowledge included facts about the conditions required for photosynthesis, listing the products formed from the process of photosynthesis and knowing that green leaves reflect green light. These basic-level knowledges should intentionally help learners learn higher-level conceptual knowledge that include learners being able to draw on the conditions for photosynthesis when they encounter that a plant is not growing well or is exhibiting discoloration of leaves.

Transient problems disappear once a solution becomes available and there is a high likelihood that we will not remember the problem after that. Practical problems, according to Bereiter are “stuck-door” problems that could be solved with or without basic-level knowledge and often have solutions that lacks precise definition. There are usually a handful of practical strategies, such as pulling or pushing the door harder, kicking the door, etc. that will work for the problems. All these solutions lack a well-defined approach related to general scientific principles that are reproducible. Problems of explanations are the most desirable types of problems for learners since these are problems that persist and recur such that they can become organising points for knowledge. Problems of explanations consist of the conceptual representations of (1) a text base that serves to represent the text content and (2) a situation model that shows the portion of the world in which the text is relevant. The idea of text base to represent text content in solving problems of explanations is like the idea of domain knowledge and structural knowledge (refers to knowledge of how concepts within a domain are connected) proposed by Jonassen ( 2000 ). He argued that both types of knowledges are required to solve a range of problems from well-structured problems to ill-structured problems with a simulated context, to simple ill-structured problems and to complex ill-structured problems.

Jonassen indicated that complex ill-structured problems are typically design problems and are likely to be the most useful forms of problems for learners to be engaged in inquiry. Complex ill-structured design problems are the “wicked” problems that Buchanan ( 1992 ) discussed. Buchanan’s idea is that design aims to incorporate knowledge from different fields of specialised inquiry to become whole. Complex or wicked problems are akin to the work of scientists who navigate multiple factors and evidence to offer models that are typically oversimplified, but they apply them to propose possible first approximation explanations or solutions and iteratively relax constraints or assumptions to refine the model. The connections between the subject matter of science and the design process to engineer a solution are delicate. While it is important to ensure that practical concerns and questions are taken into consideration in designing solutions (particularly a material artefact) to a practical problem, the challenge here lies in ensuring that creativity in design is encouraged even if students initially lack or neglect the scientific conceptual understanding to explain/justify their design. In his articulation of wicked problems and the role of design thinking, Buchanan ( 1992 ) highlighted the need to pay attention to category and placement. Categories “have fixed meanings that are accepted within the framework of a theory or a philosophy and serve as the basis for analyzing what already exist” (Buchanan, 1992 , p. 12). Placements, on the other hand, “have boundaries to shape and constrain meaning, but are not rigidly fixed and determinate” (p. 12).

The difference in the ideas presented by Dewey and Bereiter lies in the problem design. For Dewey, scientific knowledge could be learnt from inquiring into practical problems that learners are familiar with. After all, Dewey viewed “modern science as continuous with, and to some degree an outgrowth and refinement of, practical or ‘common-sense’ inquiry” (Brown, 2012 ). For Bereiter, he acknowledged the importance of familiar experiences, but instead of using them as starting points for learning science, he argued that practical problems are limiting in helping learners acquire higher-order knowledge. Instead, he advocated for learners to organize their knowledge around problems that are complex, persistent and extended and requiring explanations to better understand the problems. Learners are to have a sense of the kinds of problems to which the specific concept is relevant before they can be said to have grasp the concept in a functionally useful way.

To connect between problem solving, scientific knowledge and everyday experiences, we need to examine ways to re-negotiate the disciplinary boundaries (such as epistemic understanding, object of inquiry, degree of precision) of science and make relevant connections to common-sense inquiry and to the problem at hand. Integrated STEM appears to be one way in which the disciplinary boundaries of science can be re-negotiated to include practices from the fields of technology, engineering and mathematics. In integrated STEM learning, inquiry is seen more holistically as a fluid process in which the outcomes are not absolute but are tentative. The fluidity of the inquiry process is reflected in the non-deterministic inquiry approach. This means that students can use science inquiry, engineering design, design process or any other inquiry approaches that fit to arrive at the solution. This hybridity of inquiry between science, common-sense and problems allows for some familiar aspects of the science inquiry process to be applied to understand and generate solutions to familiar everyday problems. In attempting to infuse elements of common-sense inquiry with science inquiry in problem-solving, logic plays an important role to help learners make connections. Hypothetically, we argue that with increasing exposure to less familiar ways of thinking such as those associated with science inquiry, students’ familiarity with scientific reasoning increases, and hence such ways of thinking gradually become part of their common-sense, which students could employ to solve future relevant problems. The theoretical ideas related to complexities of problems, the different forms of inquiry afforded by different problems and the arguments for engaging in problem solving motivated us to examine empirically how learners engage with ill-structured problems to generate problem-centred knowledge. Of particular interest to us is how learners and teachers weave between practical and scientific reasoning as they inquire to integrate the components in the original problem into a unified whole.

The integrated STEM activity in our study was planned using the S-T-E-M quartet instructional framework (Tan et al., 2019 ). The S-T-E-M quartet instructional framework positions complex, persistent and extended problems at its core and focusses on the vertical disciplinary knowledge and understanding of the horizontal connections between the disciplines that could be gained by learners through solving the problem (Tan et al., 2019 ). Figure  1 depicts the disciplinary aspects of the problem that was presented to the students. The activity has science and engineering as the two lead disciplines. It spanned three 1-h lessons and required students to both learn and apply relevant scientific conceptual knowledge to solve a complex, real-world problem through processes that resemble the engineering design process (Wheeler et al., 2019 ).

Connections across disciplines in integrate STEM activity

In the first session (1 h), students were introduced to the problem and its context. The problem pertains to the issue of limited farmland in a land scarce country that imports 90% of food (Singapore Food Agency [SFA], 2020 ). The students were required to devise a solution by applying knowledge of the conditions required for photosynthesis and plant growth to design and build a vertical farming system to help farmers increase crop yield with limited farmland. This context was motivated by the government’s effort to generate interests and knowledge in farming to achieve the 30 by 30 goal—supplying 30% of country’s nutritional needs by 2030. The scenario was a fictitious one where they were asked to produce 120 tonnes of Kailan (a type of leafy vegetable) with two hectares of land instead of the usual six hectares over a specific period. In addition to the abovementioned constraints, the teacher also discussed relevant success criteria for evaluating the solution with the students. Students then researched about existing urban farming approaches. They were given reading materials pertaining to urban farming to help them understand the affordances and constraints of existing solutions. In the second session (6 h), students engaged in ideation to generate potential solutions. They then designed, built and tested their solution and had opportunities to iteratively refine their solution. Students were given a list of materials (e.g. mounting board, straws, ice-cream stick, glue, etc.) that they could use to design their solutions. In the final session (1 h), students presented their solution and reflected on how well their solution met the success criteria. The prior scientific conceptual knowledge that students require to make sense of the problem include knowledge related to plant nutrition, namely, conditions for photosynthesis, nutritional requirements of Kailin and growth cycle of Kailin. The problem resembles a real-world problem that requires students to engage in some level of explanation of their design solution.

A total of 113 eighth graders (62 boys and 51 girls), 14-year-olds, from six classes and their teachers participated in the study. The students and their teachers were recruited as part of a larger study that examined the learning experiences of students when they work on integrated STEM activities that either begin with a problem, a solution or are focused on the content. Invitations were sent to schools across the country and interested schools opted in for the study. For the study reported here, all students and teachers were from six classes within a school. The teachers had all undergone 3 h of professional development with one of the authors on ways of implementing the integrated STEM activity used in this study. During the professional development session, the teachers learnt about the rationale of the activity, familiarize themselves with the materials and clarified the intentions and goals of the activity. The students were mostly grouped in groups of three, although a handful of students chose to work independently. The group size of students was not critical for the analysis of talk in this study as the analytic focus was on the kinds of knowledge applied rather than collaborative or group think. We assumed that the types of inquiry adopted by teachers and students were largely dependent on the nature of problem. Eighth graders were chosen for this study since lower secondary science offered at this grade level is thematic and integrated across biology, chemistry and physics. Furthermore, the topic of photosynthesis is taught under the theme of Interactions at eighth grade (CPDD, 2021 ). This thematic and integrated nature of science at eighth grade offered an ideal context and platform for integrated STEM activities to be trialled.

The final lessons in a series of three lessons in each of the six classes was analysed and reported in this study. Lessons where students worked on their solutions were not analysed because the recordings had poor audibility due to masking and physical distancing requirements as per COVID-19 regulations. At the start of the first lesson, the instructions given by the teacher were:

You are going to present your models. Remember the scenario that you were given at the beginning that you were tasked to solve using your model. …. In your presentation, you have to present your prototype and its features, what is so good about your prototype, how it addresses the problem and how it saves costs and space. So, this is what you can talk about during your presentation. ….. pay attention to the presentation and write down questions you like to ask the groups after the presentation… you can also critique their model, you can evaluate, critique and ask questions…. Some examples of questions you can ask the groups are? Do you think your prototype can achieve optimal plant growth? You can also ask questions specific to their models.

Data collection

Parental consent was sought a month before the start of data collection. The informed consent adhered to confidentiality and ethics guidelines as described by the Institutional Review Board. The data collection took place over a period of one month with weekly video recording. Two video cameras, one at the front and one at the back of the science laboratory were set up. The front camera captured the students seated at the front while the back video camera recorded the teacher as well as the groups of students at the back of the laboratory. The video recordings were synchronized so that the events captured from each camera can be interpreted from different angles. After transcription of the raw video files, the identities of students were substituted with pseudonyms.

Data analysis

The video recordings were analysed using the qualitative content analysis approach. Qualitative content analysis allows for patterns or themes and meanings to emerge from the process of systematic classification (Hsieh & Shannon, 2005 ). Qualitative content analysis is an appropriate analytic method for this study as it allows us to systematically identify episodes of practical inquiry and science inquiry to map them to the purposes and outcomes of these episodes as each lesson unfolds.

In total, six h of video recordings where students presented their ideas while the teachers served as facilitator and mentor were analysed. The video recordings were transcribed, and the transcripts were analysed using the NVivo software. Our unit of analysis is a single turn of talk (one utterance). We have chosen to use utterances as proxy indicators of reasoning practices based on the assumption that an utterance relates to both grammar and context. An utterance is a speech act that reveals both meaning and intentions of the speaker within specific contexts (Li, 2008 ).

Our research analytical lens is also interpretative in nature and the validity of our interpretation is through inter-rater discussion and agreement. Each utterance at the speaker level in transcripts was examined and coded either as relevant to practical reasoning or scientific reasoning based on the content. The utterances could be a comment by the teacher, a question by a student or a response by another student. Deductive coding is deployed with the two codes, practical reasoning and scientific reasoning derived from the theoretical ideas of Dewey and Bereiter as described earlier. Practical reasoning refers to utterances that reflect commonsensical knowledge or application of everyday understanding. Scientific reasoning refers to utterances that consist of scientifically oriented questions, scientific terms, or the use of empirical evidence to explain. Examples of each type of reasoning are highlighted in the following section. Each coded utterance is then reviewed for detailed description of the events that took place that led to that specific utterance. The description of the context leading to the utterance is considered an episode. The episodes and codes were discussed and agreed upon by two of the authors. Two coders simultaneously watched the videos to identify and code the episodes. The coders interpreted the content of each utterance, examine the context where the utterance was made and deduced the purpose of the utterance. Once each coder has established the sense-making aspect of the utterance in relation to the context, a code of either practical reasoning or scientific reasoning is assigned. Once that was completed, the two coders compared their coding for similarities and differences. They discussed the differences until an agreement was reached. Through this process, an agreement of 85% was reached between the coders. Where disagreement persisted, codes of the more experienced coder were adopted.

Results and Discussion

The specific STEM lessons analysed were taken from the lessons whereby students presented the model of their solutions to the class for peer evaluation. Every group of students stood in front of the class and placed their model on the bench as they presented. There was also a board where they could sketch or write their explanations should they want to. The instructions given by the teacher to the students were to explain their models and state reasons for their design.

Prevalence of Reasoning

The 6h of videos consists of 1422 turns of talk. Three hundred four turns of talk (21%) were identified as talk related to reasoning, either practical reasoning or scientific reasoning. Practical reasoning made up 62% of the reasoning turns while 38% were scientific reasoning (Fig. ​ (Fig.2 2 ).

Frequency of different types of reasoning

The two types of reasoning differ in the justifications that are used to substantiate the claims or decisions made. Table ​ Table1 1 describes the differences between the two categories of reasoning.

Types of reasoning used in the integrated STEM activity

Applications of Scientific Reasoning

Instances of engagement with scientific reasoning (for instance, using scientific concepts to justify, raising scientifically oriented questions, or providing scientific explanations) revolved around the conditions for photosynthesis and the concept of energy conversion when students were presenting their ideas or when they were questioned by their peers. For example, in explaining the reason for including fish in their plant system, one group of students made connection to cyclical energy transfer: “…so as the roots of the plants submerged in the water, faeces from the fish will be used as fertilizers so that the plant can grow”. The students considered how organic matter that is still trapped within waste materials can be released and taken up by plants to enhance the growth. The application of scientific reasoning made their design one that is innovative and sustainable as evaluated by the teacher. Some students attempted more ecofriendly designs by considering energy efficiencies through incorporating water turbines in their farming systems. They applied the concept of different forms of energy and energy conversion when their peers inquired about their design. The same scientific concepts were explained at different levels of details by different students. At one level, the students explained in a purely descriptive manner of what happens to the different entities in their prototypes, with implied changes to the forms of energy─ “…spins then generates electricity. So right, when the water falls down, then it will spin. The water will fall on the fan blade thing, then it will spin and then it generates electricity. So, it saves electricity, and also saves water”. At another level, students defended their design through an explanation of energy conversion─ “…because when the water flows right, it will convert gravitational potential energy so, when it reaches the bottom, there is not really much gravitational potential energy”. While these instances of applying scientific reasoning indicated that students have knowledge about the scientific phenomena and can apply them to assist in the problem-solving process, we are not able to establish if students understood the science behind how the dynamo works to generate electricity. Students in eighth grade only need to know how a generator works at a descriptive level and the specialized understanding how a dynamo works is beyond the intended learning outcomes at this grade level.

The application of scientific concepts for justification may not always be accurate. For instance, the naïve conception that students have about plants only respiring at night and not in the day surfaced when one group of students tried to justify the growth rates of Kailan─ “…I mean, they cannot be making food 24/7 and growing 24/7. They have nighttime for a reason. They need to respire”. These students do not appreciate that plants respire in the day as well, and hence respiration occurs 24/7. This naïve conception that plants only respire at night is one that is common among learners of biology (e.g. Svandova, 2014 ) since students learn that plant gives off oxygen in the day and takes in oxygen at night. The hasty conclusion to that observation is that plants carry out photosynthesis in the day and respire at night. The relative rates of photosynthesis and respiration were not considered by many students.

Besides naïve conceptions, engagement with scientific ideas to solve a practical problem offers opportunities for unusual and alternative ideas about science to surface. For instance, another group of students explained that they lined up their plants so that “they can take turns to absorb sunlight for photosynthesis”. These students appear to be explaining that the sun will move and depending on the position of the sun, some plants may be under shade, and hence rates of photosynthesis are dependent on the position of the sun. However, this idea could also be interpreted as (1) the students failed to appreciate that sunlight is everywhere, and (2) plants, unlike animals, particularly humans, do not have the concept of turn-taking. These diverse ideas held by students surfaced when students were given opportunities to apply their knowledge of photosynthesis to solve a problem.

Applications of Practical Reasoning

Teachers and students used more practical reasoning during an integrated STEM activity requiring both science and engineering practices as seen from 62% occurrence of practical reasoning compared with 38% for scientific reasoning. The intention of the activity to integrate students’ scientific knowledge related to plant nutrition to engineering practice of building a model of vertical farming system could be the reason for the prevalence of practical reasoning. The practical reasoning used related to structural design considerations of the farming system such as how water, light and harvesting can be carried out in the most efficient manner. Students defended the strengths of designs using logic based on their everyday experiences. In the excerpt below (transcribed verbatim), we see students applied their everyday experiences when something is “thinner” (likely to mean narrower), logically it would save space. Further, to reach a higher level, you use a machine to climb up.

Excerpt 1. “Thinner, more space” Because it is more thinner, so like in terms of space, it’s very convenient. So right, because there is – because it rotates right, so there is this button where you can stop it. Then I also installed steps, so that – because there are certain places you can’t reach even if you stop the – if you stop the machine, so when you stop it and you climb up, and then you see the condition of the plants, even though it costs a lot of labour, there is a need to have an experienced person who can grow plants. Then also, when like – when water reach the plants, cos the plants I want to use is soil-based, so as the water reach the soil, the soil will xxx, so like the water will be used, and then we got like – and then there’s like this filter that will filter like the dirt.

In the examples of practical reasoning, we were not able to identify instances where students and teachers engaged with discussion around trade-off and optimisation. Understanding constraints, trade-offs and optimisations are important ideas in informed design matrix for engineering as suggested by Crismond and Adams ( 2012 ). For instance, utterances such as “everything will be reused”, “we will be saving space”, “it looks very flimsy” or “so that it can contains [sic] the plants” were used. These utterances were made both by students while justifying their own prototypes and also by peers who challenged the design of others. Longer responses involving practical reasoning were made based on common-sense, everyday logic─ “…the product does not require much manpower, so other than one or two supervisors like I said just now, to harvest the Kailan, hence, not too many people need to be used, need to be hired to help supervise the equipment and to supervise the growth”. We infer that the higher instances of utterances related to practical reasoning could be due to the presence of more concrete artefacts that is shown, and the students and teachers were more focused on questioning the structure at hand. This inference was made as instructions given by the teacher at the start of students’ presentation focus largely on the model rather than the scientific concepts or reasoning behind the model.

Intersection Between Scientific and Practical Reasoning

Comparing science subject matter knowledge and problem-solving to the idea of categories and placement (Buchanan, 1992 ), subject matter is analogous to categories where meanings are fixed with well-established epistemic practices and norms. The problem-solving process and design of solutions are likened to placements where boundaries are less rigid, hence opening opportunities for students’ personal experiences and ideas to be presented. Placements allow students to apply their knowledge from daily experiences and common-sense logic to justify decisions. Common-sense knowledge and logic are more accessible, and hence we observe higher frequency of usage. Comparatively, while science subject matter (categories) is also used, it is observed less frequently. This could possibly be due either to less familiarity with the subject matter or lack of appropriate opportunity to apply in practical problem solving. The challenge for teachers during implementation of a STEM problem-solving activity, therefore, lies in the balance of the application of scientific and practical reasoning to deepen understanding of disciplinary knowledge in the context of solving a problem in a meaningful manner.

Our observations suggest that engaging students with practical inquiry tasks with some engineering demands such as the design of modern farm systems offers opportunities for them to convert their personal lived experiences into feasible concrete ideas that they can share in a public space for critique. The peer critique following the sharing of their practical ideas allows for both practical and scientific questions to be asked and for students to defend their ideas. For instance, after one group of students presented their prototype that has silvered surfaces, a student asked a question: “what is the function of the silver panels?”, to which his peers replied : “Makes the light bounce. Bounce the sunlight away and then to other parts of the tray.” This question indicated that students applied their knowledge that shiny silvered surfaces reflect light, and they used this knowledge to disperse the light to other trays where the crops were growing. An example of a practical question asked was “what is the purpose of the ladder?”, to which the students replied: “To take the plants – to refill the plants, the workers must climb up”. While the process of presentation and peer critique mimic peer review in the science inquiry process, the conceptual knowledge of science may not always be evident as students paid more attention to the design constraints such as lighting, watering, and space that was set in the activity. Given the context of growing plants, engagement with the science behind nutritional requirements of plants, the process of photosynthesis, and the adaptations of plants could be more deliberately explored.

The goal of our work lies in applying the theoretical ideas of Dewey and Bereiter to better understand reasoning practices in integrate STEM problem solving. We argue that this is a worthy pursue to better understand the roles of scientific reasoning in practical problem solving. One of the goals of integrated STEM education in schools is to enculture students into the practices of science, engineering and mathematics that include disciplinary conceptual knowledge, epistemic practices, and social norms (Kelly & Licona, 2018 ). In the integrated form, the boundaries and approaches to STEM learning are more diverse compared with monodisciplinary ways of problem solving. For instance, in integrated STEM problem solving, besides scientific investigations and explanations, students are also required to understand constraints, design optimal solutions within specific parameters and even to construct prototypes. For students to learn the ways of speaking, doing and being as they participate in integrated STEM problem solving in schools in a meaningful manner, students could benefit from these experiences.

With reference to the first research question of What is the extent of practical and scientific reasoning in integrated STEM problem solving, our analysis suggests that there are fewer instances of scientific reasoning compared with practical reasoning. Considering the intention of integrated STEM learning and adopting Bereiter’s idea that students should learn higher-order conceptual knowledge through engagement with problem solving, we argue for a need for scientific reasoning to be featured more strongly in integrated STEM lessons so that students can gain higher order scientific conceptual knowledge. While the lessons observed were strong in design and building, what was missing in generating solutions was the engagement in investigations, where learners collected or are presented with data and make decisions about the data to allow them to assess how viable the solutions are. Integrated STEM problems can be designed so that science inquiry can be infused, such as carrying out investigations to figure out relationships between variables. Duschl and Bybee ( 2014 ) have argued for the need to engage students in problematising science inquiry and making choices about what works and what does not.

With reference to the second research question , What is achieved through practical and scientific reasoning during integrated STEM problem solving? , our analyses suggest that utterance for practical reasoning are typically used to justify the physical design of the prototype. These utterances rely largely on what is observable and are associated with basic-level knowledge and experiences. The higher frequency of utterances related to practical reasoning and the nature of the utterances suggests that engagement with practical reasoning is more accessible since they relate more to students’ lived experiences and common-sense. Bereiter ( 1992 ) has urged educators to engage learners in learning that is beyond basic-level knowledge since accumulation of basic-level knowledge does not lead to higher-level conceptual learning. Students should be encouraged to use scientific knowledge also to justify their prototype design and to apply scientific evidence and logic to support their ideas. Engagement with scientific reasoning is preferred as conceptual knowledge, epistemic practices and social norms of science are more widely recognised compared with practical reasoning that are likely to be more varied since they rely on personal experiences and common-sense. This leads us to assert that both context and content are important in integrated STEM learning. Understanding the context or the solution without understanding the scientific principles that makes it work makes the learning less meaningful since we “…cannot strip learning of its context, nor study it in a ‘neutral’ context. It is always situated, always relayed to some ongoing enterprise”. (Bruner, 2004 , p. 20).

To further this discussion on how integrated STEM learning experiences harness the ideas of practical and scientific reasoning to move learners from basic-level knowledge to higher-order conceptual knowledge, we propose the need for further studies that involve working with teachers to identify and create relevant problems-of-explanations that focuses on feasible, worthy inquiry ideas such as those related to specific aspects of transportation, alternative energy sources and clean water that have impact on the local community. The design of these problems can incorporate opportunities for systematic scientific investigations and scaffolded such that there are opportunities to engage in epistemic practices of the constitute disciplines of STEM. Researchers could then examine the impact of problems-of-explanations on students’ learning of higher order scientific concepts. During the problem-solving process, more attention can be given to elicit students’ initial and unfolding ideas (practical) and use them as a basis to start the science inquiry process. Researchers can examine how to encourage discussions that focus on making meaning of scientific phenomena that are embedded within specific problems. This will help students to appreciate how data can be used as evidence to support scientific explanations as well as justifications for the solutions to problems. With evidence, learners can be guided to work on reasoning the phenomena with explanatory models. These aspects should move engagement in integrated STEM problem solving from being purely practice to one that is explanatory.

Limitations

There are four key limitations of our study. Firstly, the degree of generalisation of our observations is limited. This study sets out to illustrate what how Dewey and Bereiter’s ideas can be used as lens to examine knowledge used in problem-solving. As such, the findings that we report here is limited in its ability to generalise across different contexts and problems. Secondly, the lessons that were analysed came from teacher-frontal teaching and group presentation of solution and excluded students’ group discussions. We acknowledge that there could potentially be talk that could involve practical and scientific reasonings within group work. There are two practical consideration for choosing to analyse the first and presentation segments of the suite of lesson. Firstly, these two lessons involved participation from everyone in class and we wanted to survey the use of practical and scientific reasoning by the students as a class. Secondly, methodologically, clarity of utterances is important for accurate analysis and as students were wearing face masks during the data collection, their utterances during group discussions lack the clarity for accurate transcription and analysis. Thirdly, insights from this study were gleaned from a small sample of six classes of students. Further work could involve more classes of students although that could require more resources devoted to analysis of the videos. Finally, the number of students varied across groups and this could potentially affect the reasoning practices during discussions.

Acknowledgements

The authors would like to acknowledge the contributions of the other members of the research team who gave their comment and feedback in the conceptualization stage.

Authors’ Contribution

The first author conceptualized, researched, read, analysed and wrote the article.

The second author worked on compiling the essential features and the variations tables.

The third and fourth authors worked with the first author on the ideas and refinements of the idea.

This study is funded by Office of Education Research grant OER 24/19 TAL.

Data Availability

Declarations.

The authors declare that they have no competing interests.

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Contributor Information

Aik-Ling Tan, Email: [email protected] .

Yann Shiou Ong, Email: [email protected] .

Yong Sim Ng, Email: [email protected] .

Jared Hong Jie Tan, Email: moc.liamg@derajeijgnohnat .

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Physical Review B

Covering condensed matter and materials physics.

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Beyond the Tamura model of phonon-isotope scattering

Nakib h. protik and claudia draxl, phys. rev. b 109 , 165201 – published 1 april 2024.

• No Citing Articles
• INTRODUCTION
• COMPUTATIONAL METHODS
• NUMERICAL RESULTS AND DISCUSSION
• SUMMARY AND OUTLOOK
• ACKNOWLEDGMENTS

The Tamura model is a particular type of first Born approximation of the phonon-isotope scattering problem. The expression for the mode-resolved phonon-isotope scattering rates in this model, derived in 1983, is still widely used in ab initio transport calculations. While the original work emphasized its applicability to low-energy acoustic phonons only, nevertheless, it has also been applied to optical phonons in the field of phonon transport. The model has the salient feature of being a perturbation theory on top of a virtual-crystal background. As such, this approach does not correspond to the proper methodology for solving the phonon-substitution defect problem in the respective limit. Here we explore three avenues to go beyond the Tamura model and carry out calculations on a set of common materials to compare the different approaches. This work allows a systematic improvement of the treatment of phonon-isotope scattering in ab initio phonon transport while unifying it with the general phonon-substitution scattering problem.

• Received 22 January 2024
• Revised 6 March 2024
• Accepted 12 March 2024

DOI: https://doi.org/10.1103/PhysRevB.109.165201

©2024 American Physical Society

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• Research Areas

Authors & Affiliations

• Department of Physics and CSMB, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
• Department of Physics and CSMB, Humboldt-Universität zu Berlin, 12489 Berlin, Germany and Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany
• * [email protected]
• † [email protected]

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Vol. 109, Iss. 16 — 15 April 2024

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Cartoon illustrating the difference between the Tamura (top) and the DIB (bottom) approaches to the isotope substitution problem in a 1D monatomic chain. The hatched circles represent VCA atoms created from the average of the dominant (solid circles) and the minority (empty circles) isotopes. Arrows symbolize substitution.

Phonon density of states (top panel; yellow squares for VCA and red dots for DIB) and phonon-isotope scattering rates (bottom panel; four different models) for Si.

Same as Fig.  2 but for LiF. Note that in the lower panel inset, the blue (yellow) symbols are hardly visible, since they mostly overlap with the black (red) symbols.

Same as Fig.  2 but for BAs.

Same as Fig.  2 but for wurtzite GaN.

Isotropic average of the phonon thermal conductivity κ as a function of temperature for wurtzite GaN. For comparison, experimental data from Ref. [ 23 ] (blue triangles) and Ref. [ 24 ] are shown (red circles). To highlight that the DIB-1B shows excellent agreement with the experimentally observed temperature dependence, we repeat this result with a − 20 Wm − 1 K − 1 offset (dashed line).

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