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Unpacking learner's growth in geometric understanding when solving problems in a dynamic geometry environment: Coordinating two frames

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The emergence of dynamic geometry environments challenges researchers in mathematics education to develop theories that capture learner's growth in geometric understanding in this particular environment. This study coordinated the Pirie-Kieren theory and instrumental genesis to examine learner's growth in geometric understanding when solving problems in a dynamic geometry environment. Data analysis suggested that coordinating the two theoretical approaches provided a productive means to capture the dynamic interaction between the growth in mathematical understanding and the formation/application of utilization scheme during a learner's mathematical exploration with dynamic geometry software. The analysis of one episode on inscribing a square in a triangle was shared to illustrate this approach. This study contributes to the continuing conversation of “networking theories” in the mathematics education research community. By networking the two theoretical approaches, this paper presents a model for studying learner's growth in mathematical understanding in a dynamic learning environment while accounting for interaction with digital tools.

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• Applied Mathematics

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• Dynamic geometry Mathematics 100%
• Mathematics Education Mathematics 48%
• Networking Mathematics 47%
• mathematics Social Sciences 44%
• Mathematics Medicine & Life Sciences 42%
• Geometry Engineering & Materials Science 40%
• Growth Medicine & Life Sciences 33%
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T1 - Unpacking learner's growth in geometric understanding when solving problems in a dynamic geometry environment

T2 - Coordinating two frames

AU - Yao, Xiangquan

N1 - Publisher Copyright: © 2020 Elsevier Inc.

PY - 2020/12

Y1 - 2020/12

N2 - The emergence of dynamic geometry environments challenges researchers in mathematics education to develop theories that capture learner's growth in geometric understanding in this particular environment. This study coordinated the Pirie-Kieren theory and instrumental genesis to examine learner's growth in geometric understanding when solving problems in a dynamic geometry environment. Data analysis suggested that coordinating the two theoretical approaches provided a productive means to capture the dynamic interaction between the growth in mathematical understanding and the formation/application of utilization scheme during a learner's mathematical exploration with dynamic geometry software. The analysis of one episode on inscribing a square in a triangle was shared to illustrate this approach. This study contributes to the continuing conversation of “networking theories” in the mathematics education research community. By networking the two theoretical approaches, this paper presents a model for studying learner's growth in mathematical understanding in a dynamic learning environment while accounting for interaction with digital tools.

AB - The emergence of dynamic geometry environments challenges researchers in mathematics education to develop theories that capture learner's growth in geometric understanding in this particular environment. This study coordinated the Pirie-Kieren theory and instrumental genesis to examine learner's growth in geometric understanding when solving problems in a dynamic geometry environment. Data analysis suggested that coordinating the two theoretical approaches provided a productive means to capture the dynamic interaction between the growth in mathematical understanding and the formation/application of utilization scheme during a learner's mathematical exploration with dynamic geometry software. The analysis of one episode on inscribing a square in a triangle was shared to illustrate this approach. This study contributes to the continuing conversation of “networking theories” in the mathematics education research community. By networking the two theoretical approaches, this paper presents a model for studying learner's growth in mathematical understanding in a dynamic learning environment while accounting for interaction with digital tools.

UR - http://www.scopus.com/inward/record.url?scp=85089740233&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85089740233&partnerID=8YFLogxK

U2 - 10.1016/j.jmathb.2020.100803

DO - 10.1016/j.jmathb.2020.100803

M3 - Article

AN - SCOPUS:85089740233

SN - 0732-3123

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Dynamic Programming

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• Karleigh Moore
• Norbert Madarász

Dynamic programming refers to a problem-solving approach, in which we precompute and store simpler, similar subproblems, in order to build up the solution to a complex problem. It is similar to recursion , in which calculating the base cases allows us to inductively determine the final value. This bottom-up approach works well when the new value depends only on previously calculated values.

An important property of a problem that is being solved through dynamic programming is that it should have overlapping subproblems. This is what distinguishes DP from divide and conquer in which storing the simpler values isn't necessary.

To show how powerful the technique can be, here are some of the most famous problems commonly approached through dynamic programming:

• Backpack Problem : Given a set of treasures with known values and weights, which of them should you pick to maximize your profit whilst not damaging your backpack which has a fixed capacity?
• Egg Dropping : What is the best way to drop \(n\) eggs from an \(m\)-floored building to figure out the lowest height from which the eggs when dropped crack?
• Longest Common Subsequence : Given two sequences, which is the longest subsequence common to both of them?
• Subset Sum Problem : Given a set and a value \(n,\) is there a subset the sum of whose elements is \(n?\)
• Fibonacci Numbers : Is there a better way to compute Fibonacci numbers than plain recursion?

In a contest environment, dynamic programming almost always comes up (and often in a surprising way, no matter how familiar the contestant is with it).

Motivational Example: Change of Coins

Recursion with memoization, bidimensional dynamic programming: example, example: maximum paths.

What is the minimum number of coins of values \(v_1,v_2, v_3, \ldots, v_n\) required to amount a total of \(V?\) You may use a denomination more than once.

Optimal Substructure

The most important aspect of this problem that encourages us to solve this through dynamic programming is that it can be simplified to smaller subproblems.

Let \(f(N)\) represent the minimum number of coins required for a value of \(N\).

Visualize \(f(N)\) as a stack of coins. What is the coin at the top of the stack? It could be any of \(v_1,v_2, v_3, \ldots, v_n\). In case it were \(v_1\), the rest of the stack would amount to \(N-v_1;\) or if it were \(v_2\), the rest of the stack would amount to \(N-v_2\), and so on.

How do we decide which is it? Sure enough, we do not know yet. We need to see which of them minimizes the number of coins required.

Going by the above argument, we could state the problem as follows:

\[f(V) = \min \Big( \big\{ 1 + f(V - v_1), 1 + f(V-v_2), \ldots, 1 + f(V-v_n) \big \} \Big). \]

Because the coin at the top of the stack also counts as one coin, and then we can look at the rest.

Overlapping Subproblems

It is easy to see that the subproblems could be overlapping.

For example, if we are trying to make a stack of $11 using $1, $2, and $5, our look-up pattern would be like this: \[\begin{align} f(11) &= \min \Big( \big\{ 1+f(10),\ 1+ f(9),\ 1 + f(6) \big\} \Big) \\ &= \min \Big ( \big \{ 1+ \min {\small \left ( \{ 1 + f(9), 1+ f(8), 1+ f(5) \} \right )},\ 1+ f(9),\ 1 + f(6) \big \} \Big ). \end{align} \] Clearly enough, we'll need to use the value of \(f(9)\) several times.

One of the most important aspects of optimizing our algorithms is that we do not recompute these values. To do this, we compute and store all the values of \(f\) from 1 onwards for potential future use.

The recursion has to bottom out somewhere, in other words, at a known value from which it can start.

For this problem, we need to take care of two things:

Zero : It is clear enough that \(f(0) = 0\) since we do not require any coins at all to make a stack amounting to 0.

Negative and Unreachable Values : One way of dealing with such values is to mark them with a sentinel value so that our code deals with them in a special way. A good choice of a sentinel is \(\infty\), since the minimum value between a reachable value and \(\infty\) could never be infinity.

The Algorithm

Let's sum up the ideas and see how we could implement this as an actual algorithm:

We have claimed that naive recursion is a bad way to solve problems with overlapping subproblems. Why is that? Mainly because of all the recomputations involved.

Another way to avoid this problem is to compute the data first time and store it as we go, in a top-down fashion.

Let's look at how one could potentially solve the previous coin change problem in the memoization way. 1 2 3 4 5 6 7 8 9 10 11 12 def coinsChange ( V , v ): memo = {} def Change ( V ): if V in memo : return memo [ V ] if V == 0 : return 0 if V < 0 : return float ( "inf" ) memo [ V ] = min ([ 1 + Change ( V - vi ) for vi in v ]) return memo [ V ] return Change ( V )

Dynamic Programming vs Recursion with Caching

There are \(k\) types of brackets each with its own opening bracket and closing bracket. We assume that the first pair is denoted by the numbers 1 and \(k+1,\) the second by 2 and \(k+2,\) and so on. Thus the opening brackets are denoted by \(1, 2, \ldots, k,\) and the corresponding closing brackets are denoted by \(k+1, k+2, \ldots, 2k,\) respectively.

Some sequences with elements from \(1, 2, \ldots, 2k\) form well-bracketed sequences while others don't. A sequence is well-bracketed if we can match or pair up opening brackets of the same type in such a way that the following holds:

• Every bracket is paired up.
• In each matched pair, the opening bracket occurs before the closing bracket.
• For a matched pair, any other matched pair lies either completely between them or outside them.

In this problem, you are given a sequence of brackets of length \(N\): \(B[1], \ldots, B[N]\), where each \(B[i]\) is one of the brackets. You are also given an array of Values: \(V[1],\ldots, V[N] \).

Among all the subsequences in the Values array, such that the corresponding bracket subsequence in the B Array is a well-bracketed sequence, you need to find the maximum sum.

Task: Solve the above problem for this input.

Input Format

One line, which contains \((2\times N + 2)\) space separate integers. The first integer denotes \(N.\) The next integer is \(k.\) The next \(N\) integers are \(V[1],..., V[N].\) The last \(N\) integers are \(B[1],..., B[N].\)

Constraints

• \(1 \leq k \leq 7\)
• \(-10^6 \leq V[i] \leq 10^6\), for all \(i\)
• \(1 \leq B[i] \leq 2k\), for all \(i\)

Illustrated Examples

For the examples discussed here, let us assume that \(k = 2\). The sequence 1, 1, 3 is not well-bracketed as one of the two 1's cannot be paired. The sequence 3, 1, 3, 1 is not well-bracketed as there is no way to match the second 1 to a closing bracket occurring after it. The sequence 1, 2, 3, 4 is not well-bracketed as the matched pair 2, 4 is neither completely between the matched pair 1, 3 nor completely outside of it. That is, the matched pairs cannot overlap. The sequence 1, 2, 4, 3, 1, 3 is well-bracketed. We match the first 1 with the first 3, the 2 with the 4, and the second 1 with the second 3, satisfying all the 3 conditions. If you rewrite these sequences using [, {, ], } instead of 1, 2, 3, 4 respectively, this will be quite clear.

Suppose \(N = 6, k = 3,\) and the values of \(V\) and \(B\) are as follows: Then, the brackets in positions 1, 3 form a well-bracketed sequence (1, 4) and the sum of the values in these positions is 2 (4 + (-2) =2). The brackets in positions 1, 3, 4, 5 form a well-bracketed sequence (1, 4, 2, 5) and the sum of the values in these positions is 4. Finally, the brackets in positions 2, 4, 5, 6 form a well-bracketed sequence (3, 2, 5, 6) and the sum of the values in these positions is 13. The sum of the values in positions 1, 2, 5, 6 is 16 but the brackets in these positions (1, 3, 5, 6) do not form a well-bracketed sequence. You can check the best sum from positions whose brackets form a well-bracketed sequence is 13.

We'll try to solve this problem with the help of a dynamic program, in which the state , or the parameters that describe the problem, consist of two variables.

First, we set up a two-dimensional array dp[start][end] where each entry solves the indicated problem for the part of the sequence between start and end inclusive.

We'll try to think what happens when we run across a new end value, and need to solve the new problem in terms of the previously solved subproblems. Here are all the possibilities:

• When end <= start , there are no valid subsequences.
• When b[end] <= k , i.e, the last entry is an open bracket, no valid subsequence can end with it. Effectively, the result is the same if we hadn't included the last entry at all.
• When b[end] > k , i.e, the last entry is a closing bracket, one has to find the best match for it, or simply ignore it, whichever maximizes the sum.

Can you use these ideas to solve the problem?

Very often, dynamic programming helps solve problems that ask us to find the most profitable (or least costly) path in an implicit graph setting. Let us try to illustrate this with an example.

You are supposed to start at the top of a number triangle and chose your passage all the way down by selecting between the numbers below you to the immediate left or right. Your goal is to maximize the sum of the elements lying in your path. For example, in the triangle below, the red path maximizes the sum.

To see the optimal substructures and the overlapping subproblems , notice that everytime we make a move from the top to the bottom right or the bottom left, we are still left with smaller number triangle, much like this:

We could break each of the sub-problems in a similar way until we reach an edge-case at the bottom:

In this case, the solution is a + max(b,c) .

A bottom-up dynamic programming solution is to allocate a number triangle that stores the maximum reachable sum if we were to start from that position . It is easy to compute the number triangles from the bottom row onward using the fact that the

\[\text{best from this point} = \text{this point} + \max(\text{best from the left, best from the right}).\]

Let me demonstrate this principle through the iterations. Iteration 1: 1 8 5 9 3 Iteration 2: 1 2 10 13 15 8 5 9 3 Iteration 3: 1 2 3 20 19 10 13 15 8 5 9 3 Iteration 4: 1 2 3 4 23 20 19 10 13 15 8 5 9 3 So, the effective best we could do from the top is 23, which is our answer.

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Analyzing group coordination when solving geometry problems with dynamic geometry software

• Published: 25 January 2013
• Volume 8 , pages 13–39, ( 2013 )

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In CSCL research, collaborative activity is conceptualized along various yet intertwined dimensions. When functioning within these multiple dimensions, participants make use of several resources, which can be social or content-related (and sometimes temporal) in nature. It is the effective coordination of these resources that appears to characterize successful collaborative activity. This study proposes a methodological approach for studying coordination of resources when solving geometry problems with dynamic geometry software. The aim is to suggest a methodological lens to capture both the content-related and social discourse within the context of geometry problem solving using dynamic geometry software. As an example, the paper also provides an analysis of a dyad’s face-to-face interaction using the suggested framework.

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The author would like to thank the six anonymous reviewers for their comments on an earlier version of this manuscript.

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Oner, D. Analyzing group coordination when solving geometry problems with dynamic geometry software. Computer Supported Learning 8 , 13–39 (2013). https://doi.org/10.1007/s11412-012-9161-0

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DOI : https://doi.org/10.1007/s11412-012-9161-0

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Abstract: The Unique Games Conjecture (UGC) constitutes a highly dynamic subarea within computational complexity theory, intricately linked to the outstanding P versus NP problem. Despite multiple insightful results in the past few years, a proof for the conjecture remains elusive. In this work, we construct a novel dynamical systems-based approach for studying unique games and, more generally, the field of computational complexity. We propose a family of dynamical systems whose equilibria correspond to solutions of unique games and prove that unsatisfiable instances lead to ergodic dynamics. Moreover, as the instance hardness increases, the weight of the invariant measure in the vicinity of the optimal assignments scales polynomially, sub-exponentially, or exponentially depending on the value gap. We numerically reproduce a previously hypothesized hardness plot associated with the UGC. Our results indicate that the UGC is likely true, subject to our proposed conjectures that link dynamical systems theory with computational complexity.

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Problem Solving and Problem Posing in a Dynamic Geometry Environment

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In the current study, I shall describe characteristic elements of a ‘dynamic’ trajectory, a synthesis of experts of mybprevious research regarding an approximation process for the construction of number pi, enriched, analyzed and reviewed in the light of my recent theoretical considerations (not published as yet). Number pi is a mathematical abstract object but it can also be perceived as a result of a process. Specific examples from my experimental research using dynamic active representations will be analyzed in the methodology section. Moreover, a brief report of students (small groups or individuals) which participated in the process. My aim was the students to conceive the meaning of number pi as a limit using the iteration process of the Geometer’s Sketchpad dynamic geometry software. Finally, the role the active representations play in the learning trajectory made me think of a way to define what a ‘dynamic active learning trajectory’ is.

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Experience the power of Pic Math Pro – AI Math, your ultimate AI homework companion, fueled by cutting-edge AI technology. Simply snap a picture of your math homework with your phone camera, highlight the problem you want to solve and get your AI powered answer instantly. You can use Pic Math Pro with your English assignments as well and also to write Essays for your school assignments. Key Features include: 1) Precision and Clarity: Pic Math Pro outshines other homework helpers with its speed and accuracy. Within seconds, receive precise solutions accompanied by detailed instructions and comprehensive explanations. 2) Seamless User Experience: Pic Math Pro streamlines the homework process. Questions are automatically recognized and cropped, with solutions delivered in seconds. Homework becomes not just manageable, but enjoyable with a user-friendly interface. 3) Comprehensive Homework Support: In addition to Math, Pic Math Pro also helps you analyzing your English homework assignments by creating summaries in a snap. It also helps you create essays for your English school work. 4) Share and forward the answers: Share the answers with friends or forward them to yourself for reference instantly. Free to use: You can try Pic Math Pro up to three times for free, without any advertisements. Pic Math Pro – AI Math Unlimited Subscription: - You can subscribe Pic Math Pro unlimited subscription for free for 3 days. After that you will be charged as follows: Subscription charged at $6.99 a week or $29.99 a year - Each subscription is automatically renewed on iTunes each month unless explicitly canceled using iTunes before the start of the next billing cycle. During the period of your subscription you will be provided with unlimited access to all premium features. - Payment will be charged to iTunes Account at confirmation of purchase - Subscription automatically renews unless auto-renew is turned off at least 24-hours before the end of the current period – Account will be charged for renewal within 24-hours prior to the end of the current period, and identify the cost of the renewal – Subscriptions may be managed by the user and auto-renewal may be turned off by going to the user's Account Settings after purchase  – Any unused portion of a free trial period, if offered, will be forfeited when the user purchases a subscription, where applicable Please visit our Terms and Conditions at the link below: https://picturemathapp.blogspot.com/2024/03/terms-and-conditions.html Please visit our Privacy Policy at the link below: https://picturemathapp.blogspot.com/2024/03/privacy-policy.html

App Privacy

The developer, Payal Seth , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

Data Not Collected

The developer does not collect any data from this app.

Privacy practices may vary, for example, based on the features you use or your age. Learn More

Information

• YEARLY ACCESS $29.99
• WEEKLY ACCESS $6.99
• App Support
• Privacy Policy

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