The Video Clips - My Observation
After viewing the video clips of Prof. A Takahashi teaching a class of secondary students on a mind reader puzzle, several questions surfaced from my own reflection of the activity. While viewing the video clips, I took notice on how a student begins to confront or to solve a problem mathematically. I couldn’t help but wonder if there is a set of “fixed” strategies that students would adopt when they encounter mathematical problems, maybe a set of modus operandi that students will follow whenever they take on a math problem. Or perhaps what do students do in order to make sense of the problem presented to them, before solving the problem mathematically.
Based on my observation from the video (reference: “Students Exploring Puzzle”), many students began with the heuristic of guess and check. These students were seen writing down (on their paper) a few examples to work out what are the results when the sum of two digits, (that make up a two digit number) is subtracted from the two digit number. The examples that the students chose to guess and check are random (i.e. 99, 54, 23, 45), with no particular order or sequences to the two digit numbers that were chosen. There was also no limitation as to how many examples that were chosen by the students. Some students tried four or five while others did eight to ten.
Based on the examples, students came up with the conjecture that the results they obtained are “usually multiples of 9” (guess). They were not sure if it is always 9 since they did not cover all the two digit numbers, but they were guessing that it seems the results “tends to be” multiples of 9. When Prof. Takahashi probed further (reference: Professor Sharing Students Work) to whether “everything” that the students work out are multiples of 9 (check), or maybe there is “one” possibility that the result is not a multiple of 9”, students were hesitant in their respond.
Another approach that students adopted to make sense of the problem was looking for patterns. Students were seen listing down a set of two digit numbers (reference: Students Attempting the Puzzle given the Hint) and observe that each set of two digit numbers that begins with the same tens digit (i.e. 23, 24, 25, 26 …, etc) will have the same result (i.e. 18). Students who adopted the heuristic of looking for pattern, were organized and systematic in their approach in making sense of the problem. These students organized and categorized their data to look for patterns or trends that may tell them something about the phenomena that they had observed. They would make conjectures based on the patterns or trends that they had discovered and associate their findings with the mathematical concepts they had learned.
The Research Questions and other leading questions
Based on my observation of these two heuristics employed by students in the video clips, I wanted to find out more about the problem solving strategies used by students. I wanted to know if the types of problem/question posed have a direct influence on the heuristic(s) employed by an individual in mathematics problem solving. What makes a student chooses the heuristic when he/she is solving a mathematics problem?
After viewing the video clips of Prof. A Takahashi teaching a class of secondary students on a mind reader puzzle, several questions surfaced from my own reflection of the activity. While viewing the video clips, I took notice on how a student begins to confront or to solve a problem mathematically. I couldn’t help but wonder if there is a set of “fixed” strategies that students would adopt when they encounter mathematical problems, maybe a set of modus operandi that students will follow whenever they take on a math problem. Or perhaps what do students do in order to make sense of the problem presented to them, before solving the problem mathematically.
Based on my observation from the video (reference: “Students Exploring Puzzle”), many students began with the heuristic of guess and check. These students were seen writing down (on their paper) a few examples to work out what are the results when the sum of two digits, (that make up a two digit number) is subtracted from the two digit number. The examples that the students chose to guess and check are random (i.e. 99, 54, 23, 45), with no particular order or sequences to the two digit numbers that were chosen. There was also no limitation as to how many examples that were chosen by the students. Some students tried four or five while others did eight to ten.
Based on the examples, students came up with the conjecture that the results they obtained are “usually multiples of 9” (guess). They were not sure if it is always 9 since they did not cover all the two digit numbers, but they were guessing that it seems the results “tends to be” multiples of 9. When Prof. Takahashi probed further (reference: Professor Sharing Students Work) to whether “everything” that the students work out are multiples of 9 (check), or maybe there is “one” possibility that the result is not a multiple of 9”, students were hesitant in their respond.Another approach that students adopted to make sense of the problem was looking for patterns. Students were seen listing down a set of two digit numbers (reference: Students Attempting the Puzzle given the Hint) and observe that each set of two digit numbers that begins with the same tens digit (i.e. 23, 24, 25, 26 …, etc) will have the same result (i.e. 18). Students who adopted the heuristic of looking for pattern, were organized and systematic in their approach in making sense of the problem. These students organized and categorized their data to look for patterns or trends that may tell them something about the phenomena that they had observed. They would make conjectures based on the patterns or trends that they had discovered and associate their findings with the mathematical concepts they had learned.
The Research Questions and other leading questions Based on my observation of these two heuristics employed by students in the video clips, I wanted to find out more about the problem solving strategies used by students. I wanted to know if the types of problem/question posed have a direct influence on the heuristic(s) employed by an individual in mathematics problem solving. What makes a student chooses the heuristic when he/she is solving a mathematics problem?
In Polya’s Four Steps to Problem Solving, choosing a heuristic is part of Step 2: Devising a Plan. In our Singapore Mathematics Framework, the component of Metacognition encompasses the selection and use of problem solving strategies by students. Hence, I had narrowed down my research question to “How will the different types of problem influence the use of heuristic(s) by students in mathematics problem solving?”
The Experiment
To conduct the research, I noted that in the research question, the types of problem are the independent variable and this can be operationalised into mathematical problems that are (1) numerical, (2) arithmetic, (3) geometry, (4) statistic (5) probabilistic, and (6) analytical in nature. We want to determine how these variables will influence the dependent variable, the choice of use of heuristics in mathematics problem solving.
With these definitions in mind, I can conduct an experiment to collect data by crafting problems according to the six categories (i.e. numeric, arithmetic, geometrical, statistic, probabilistic and analytical). I will then assess students’ choice of heuristics in answering these problems. Data will be classified into these six categories and the heuristics that students use to solve these problems are noted down. An interview will also be carried out to find out more about students’ approaches and strategies in choosing the heuristic(s) of their choice. The collection of qualitative data through interviews will further consolidate and formalise our understanding in the choice made by students when they are confronted with a mathematics problem.
Useful towards teaching and learning of Mathematics Problem Solving
The answer to the research question will enhance our knowledge into the strategies used by individuals in mathematics problem solving. It will help teachers to be more aware of their students’ mathematics problem solving abilities and provide remedies or interventions to help their students to become better problem solvers.
Teachers will also improve their teaching and review their pedagogical approaches towards students’ learning and understanding of mathematics problem solving. They will become more aware of the various mathematical problem solving heuristic(s) that student will tend to use for a particular type of questions, or that an individual is more prone to using a certain heuristic over the others. Such valuable information will allow teachers to plan their lessons and explore how each heuristic(s) may approach the problem from different mathematical perspectives yet solving the problem.
Website References
http://sc-math.com/math/heuristics.php
http://www.scribd.com/doc/13916762/Problem-Solving-Heuristics-Series-1
http://findarticles.com/p/articles/mi_qa3735/is_200710/ai_n21100613/
The Experiment
To conduct the research, I noted that in the research question, the types of problem are the independent variable and this can be operationalised into mathematical problems that are (1) numerical, (2) arithmetic, (3) geometry, (4) statistic (5) probabilistic, and (6) analytical in nature. We want to determine how these variables will influence the dependent variable, the choice of use of heuristics in mathematics problem solving.
With these definitions in mind, I can conduct an experiment to collect data by crafting problems according to the six categories (i.e. numeric, arithmetic, geometrical, statistic, probabilistic and analytical). I will then assess students’ choice of heuristics in answering these problems. Data will be classified into these six categories and the heuristics that students use to solve these problems are noted down. An interview will also be carried out to find out more about students’ approaches and strategies in choosing the heuristic(s) of their choice. The collection of qualitative data through interviews will further consolidate and formalise our understanding in the choice made by students when they are confronted with a mathematics problem.
Useful towards teaching and learning of Mathematics Problem Solving
The answer to the research question will enhance our knowledge into the strategies used by individuals in mathematics problem solving. It will help teachers to be more aware of their students’ mathematics problem solving abilities and provide remedies or interventions to help their students to become better problem solvers.
Teachers will also improve their teaching and review their pedagogical approaches towards students’ learning and understanding of mathematics problem solving. They will become more aware of the various mathematical problem solving heuristic(s) that student will tend to use for a particular type of questions, or that an individual is more prone to using a certain heuristic over the others. Such valuable information will allow teachers to plan their lessons and explore how each heuristic(s) may approach the problem from different mathematical perspectives yet solving the problem.
Website References
http://sc-math.com/math/heuristics.php
http://www.scribd.com/doc/13916762/Problem-Solving-Heuristics-Series-1
http://findarticles.com/p/articles/mi_qa3735/is_200710/ai_n21100613/
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