Students regularly struggle with mathematical tasks, particularly those concerning non-routine problems in geometry. Although educators would like for their learners to transfer their knowledge to non-routine and real-life situations, students run into a number of difficulties. The goal of this exploratory study was to analyze three participants’ problem solving processes in a dynamic geometry software (DGS), and therefore, gain insights about how DGS was used to support solving non-routine geometry problems. Here I viewed the DGS as a cognitive tool that can enhance and reorganize the problem solving process. The three participants were in different phases of their educational career in mathematics and/or mathematics education (bachelor, master, and doctoral student). Only one problem—the Land Boundary Problem—from the TIMSS video study will be discussed here. In this problem, the participants had to straighten a bent fence between two farmers’ land so that each farmer would keep the same amount of land. All three participants solved the problem, but used the same computer-based problem-solving tool differently. While a DGS allowed and supported some participants to discover new methods of thinking, and unanticipated ways of using it, it also inhibited the problem solving processes through development of tool-dependency by some. Its different use was dependent on the presence of managerial decisions, ability to manage different resources, and problem solving experience. Based on these findings, I make recommendations for technology-embedded problem solving with an emphasis on the importance of appropriate tool use in educational settings and offer some teaching methods that may be worthwhile for research.
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On the market many different DGS are available, such as Geometer’s Sketchpad (GSP), GeoGebra, Cinderella and Cabri.
In this study, I used the well-received interactive geometry software, the Geometer’s Sketchpad (GSP). I use the term DGS throughout the paper, because the GSP possesses typical features of a DGS. For that reason, the results are transferable to any DGS.
When the problem did not get solved within 90 min, I stopped the session and went onto the second part of the session. The participants solved the problem in the next session, which was then again followed by a semi-structured interview. This occurred only once.
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Appendix 1: Macroscopic Episodes of the Framework for Analyzing Students’ Problem Solving Processes in DGS with Descriptors
|Episode||Description of episodes with possible problem solving processes|
|Reading episode||Student reads the problem statement silently or aloud (instance of doing). That person can, in addition, engage in monitoring and control strategies to avoid missteps in this early stage of the problem solving process (metacognitive).|
|Understanding episode||The problem solver engages in behaviors attending to clarifying the meaning of the problem or making comments about it. For instance, that person may note conditions of the problem, state the goals of the problem, represent the problem (draw a diagram), and assess the current knowledge relative to the task. In other words, the problem solver engages in various both cognitive and metacognitive processes that reflect attempts to make sense of the problem.|
|Analysis episode||If there is no apparent way to proceed, the problem solver may engage in coherent, well-structured and reasonable actions that are closely related to the conditions or goals of the problem. In it, that person attempts to fully understand the problem, decomposes the problem in its basic elements, or simplifies or reformulates it (cognitive), examines the relationships between the given information, conditions and the goals of the problem, and chooses and evaluates appropriate perspectives to solve the problem using different appropriate strategies to assist that person (metacognitive) (e.g., examine special cases, reduces the number of constraints).|
|Exploration episode||The exploration episode is less well structured and further removed from the given problem (broad tour through the problem space). The problem solver searches for relevant information that can be incorporated into the analysis, planning, and implementation sequence relying on the previous knowledge and experience. In it a problem solver may find, for instance a variety of heuristics, the examination of related problems, analogies. Since exploration is weakly structured, monitoring and regulatory activities are important for keeping the exploration focused and controlled (metacognitive). Otherwise, lengthy unproductive problem solving (so called “wild goose chase”) will occur or even promising alternative(s) can get dismissed (cognitive).|
|Planning episode||During the planning episode, the behaviors pertaining to concretely solving the problem are made. In other words, the student creates a plan, i.e. selects steps and strategies that may lead to problem solution, which then gets implemented. However, such behaviors can be well thought out, evaluated, and therefore focused (metacognitive). But also the steps and strategies can be made without careful attention to their productivity or appropriateness (cognitive).|
|Implementation episode||The plan gets implemented on the basis of the previous episodes. Hence, here the givens get transformed into the goals of the problems (cognitive). However, metacognitive decisions (monitoring the process, evaluating considered decisions, regulating the steps to be implemented) are crucial to avoid faulty steps.|
|Verification episode||The student reviews and tests whether the solution passes specific or general tests in relation to problem requirements (e.g., redoing the steps (cognitive); evaluating the reasonableness of the solution (metacognitive)).|
|Observation-interaction sub-episode||This sub-episode can be embedded within any episode and denotes non-verbal interaction between the user and the technology (here DGS). Here a distinction between static and dynamic behaviors was made. Under static behavior I understand such behaviors that do not involve any tacit use of technology, but those such as observing the screen. On the other hand, dynamic behaviors denote such behaviors in which activities are tacit. Such behaviors include but are not limited to, dragging, using measuring and construction tools, using locus and trace function. Observation-interaction behaviors can subsequently prompt different cognitive and metacognitive behaviors within the episode or a new episode.|
|Transition||Transition is a junction between the other episodes and occurs only when a student assesses the current solution state and makes decisions about pursuing a new direction to solve the problem. According to Schoenfeld (1981), this is the episode where managerial decisions or their absence “will make or break a solution” (p. 26).|
Appendix 2: Non-routine Geometry Problems
Problem 1: The Longest Segment Problem
Given two intersecting circles. Draw a line through one of the intersection points, say, A. That line also intersects circles in exactly two points, say, B and C. What choice of the point B results in the segment BC such that the segment BC is the longest?
Formulate and prove your conjecture.
Find the construction for a point B such that the length of BC in the longest.
Justify your answers as best as you can.
Problem 2: Inscribed Square
Given an acute triangle. Develop a Euclidean construction for inscribing a square in the triangle such that two of its vertices are on the base of the triangle and the other two vertices are on the other two sides of the triangle.
Justify your answers as best as you can.
Problem 3: The Airport Problem
Three towns, Athens, Bogart and Columbus, are equally distant from each other and connected by straight roads. An airport will be constructed such that the sum of its distances to the roads is as small as possible.
What are possible locations for the airport?
What is the best location for the airport?
Give a geometric interpretation for the sum of the distances of the optimal point to the sides of the triangle.
Justify your answers as best as you can.
Problem 4: The Treasure Island Problem
Among his great-grandfather’s papers, Marco found a parchment describing the location of a pirate treasure buried on a deserted island. The island contained a pine three, an oak tree, and a gallows where traitors were hung. The parchment was accompanied by the following directions:
Walk from the gallows to the pine tree, counting the number of steps. At the pine tree, turn 90° to the right. Walk the same distance and put a spike in the ground. Return to the gallows ad walk to the oak tree, counting your steps. At the oak tree, turn 90° to the left, walk the same number of steps, and put another spike in the ground. The treasure is halfway between the spikes.
Marco found the island and the two trees but could not find no trace of the gallows or the spikes, as both had probably rotted. In desperation, he began to dig at random but soon gave up because the island was too large. Your quest is to devise a plan to find the exact location of the treasure.
Justify your answers as best as you can.
Problem 5: The Land Boundary Problem
The boundary between two farmers’ land is bent, and they would both like to straighten it out, but each wants to keep the same amount of land. Solve the problem for them.
Justify your answers as best as you can.
What if the common border has three segments? Justify your answers as best as you can.
Appendix 3: A Priori Analysis of Possible Problem Solving Processes and Technology Use
Here I outline several approaches that could be used to solve the Land Boundary Problem in the light of the outlined framework.
Episode 1: Reading the Problem (Cognitive and Metacognitive)
The participants read the problem (cognitive). However, if one reads the problem too fast or fails to note relevant information, it is unlikely to result in a solution. On the other hand, one may read statements slowly, marking relevant words, such as “keep the same amount” or “straighten out”, and/or making the connections between the problem representation and the text. Thus, different controlling strategies (metacognitive) may be employed, which can be executed also with editing tools of DGS.
Episode 2: Understanding the Problem (Metacognitive)
The participants need to understand the given, the goals, and the conditions that must be met when solving the problem (sense making).
The bended boundary must be straightened.
The areas of the two lands must not change.
Also they may use construction tools of DGS to add different auxiliary lines to better understand the problem.
Episode 3: Analyzing the Problem (Cognitive and Metacognitive)
In this episode the participants focus on establishing connections within the given problem (givens, conditions, goals). In order to do so, the participants need to decompose the problem into its basic structure (cognitive), find relationships between the givens and the goals on the basis of previous knowledge relevant to the problem and consider approaches that might be appropriate for solving the problem (metacognitive). Thus, they evaluate the relevancy of chosen approaches and consequences for the following steps.
The problem can be reformulated that the bounded fence has to be straightened with areas of the two lands staying the same.
The bended fence edges could be connected and with it a triangle is built.
Because area is a function of one side and an altitude on that side, the participants need to think about how construct a triangle with the same area but with one side lying on the land boundary.
Episode 4: Exploring the Problem (Cognitive and Metacognitive)
The participants engage in a search for information relevant to the problem that can be later on used. For instance, the participant, by using DGS, can make a series of random guesses as to where the border can be constructed and then with repeated editing try to find an approach to reconstruct it. Here however, the guess is not based on a thoughtful hypothesis, but on random guesses. Such actions are unmonitored and not evaluated, and will likely lead to unproductive actions and will not result in a solution (cognitive). On the other hand, if the participant is altering the givens of the problem (e.g., moving the borders, or the fence point) to gain more information and making well thought guesses, testing them, and refining them to meet the goals, while also monitoring and regulating thoughts, then the exploration will lead to a productive effort (metacognitive). Thus, through dragging modalities of DGS, the participant can engage in a systematic trial-and-error-analyze strategy. Based on the feedback through visualization of undertaken activities, the student can engage in other behaviors, such as to better understand the problem or to gain more information, or proceed to the planning episode.
Episode 5: Planning the Problem (Cognitive and Metacognitive)
There are many ways that a participant may attempt to solve the problem.
One can decide to work backwards—draw border using technological affordances of the tool such that the goals of the problem are met.
One can decide to work forwards and try to construct the new border on the basis of mathematical knowledge. Here, drawing a parallel to the base side of a triangle will lead to a solution of the problem—any of the two intersections of this parallel line with the land boundaries allows constructing a new straight fence.
Depending on the quality of the thought processes, the behaviors can be either of cognitive or metacognitive nature.
Episode 6: Implementing the problem (Cognitive and Metacognitive)
Once a plan is devised, the participants will try to implement it. When the execution of steps is sequential and monitored, this might lead to a solution or revelation that the plan was not good and because of it, they may devise a new plan (metacognitive). When implementation is not monitored, the participant may get buried in the implementation of a not well-thought out plan that is unlikely to result in a solution (cognitive). These behaviors can be easily executed using construction tools of DGS.
Episode 7: Verifying the Solution (Cognitive and Metacognitive)
The participants may verify the plan by using the measure area function of the software (cognitive) or use mathematical reasoning and ability to monitor the results to check whether the conditions of the problem were met (metacognitive).
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Kuzle, A. Delving into the Nature of Problem Solving Processes in a Dynamic Geometry Environment: Different Technological Effects on Cognitive Processing. Tech Know Learn 22, 37–64 (2017). https://doi.org/10.1007/s10758-016-9284-x
- Cognitive tool
- Dynamic geometry software
- Mathematical problem solving
- Non-routine geometry problems
- Teacher education