Tag: math research

Posted in Math research

Analyzing and explaining mathematical thinking and learning

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There are four general perspectives one can analyze and explain mathematical thinking and learning.These views should be seen not as competing but complementing each other. The four are:

  1. Mathematics:  where the focus is on the rules and norms of the mathematical research community.
  2. Mathematics education: where the focus is on the individual and social processes in a community of learners, in and out of the classroom.
  3. Cognitive psychology: where the focus is on the universal characteristics of the human mind and behavior, which are shared across individuals, cultures, and different content areas.
  4. Evolutionary psychology: where the focus is on the evolutionary origins of human cognition and behavior and their expression in “universal human nature.”

 -from Uri Leron and Orit Hazzan in their paper Intuitive and Analytical Thinking: Four Perspectives published in Educational Study of Mathematics (2009) 71:263–278.

Let us differentiate these perspectives in terms of following questions:

What is multiplication?

Mathematically, it is not correct to define multiplication as repeated addition and some mathematicians think we should not teach it that way. You can read about the controversy around this in my post Math War over Multiplication. However, from the cognitive and evolutionary psychology point of view, it is but natural and perhaps to be expected that majority of the young students will make this deduction that multiplication is repeated addition. From the math education perspective of course, teachers are expected to eventually challenge this conception.

Are mathematical errors good or bad? (Errors here actually refers to misconceptions, that is, common errors). The following analysis is from Leron and Hazzan’s paper:

  • The mathematical perspective typically views errors (misconceptions) as bugs, something that went wrong due to faulty knowledge, and needs to be corrected.
  • The mathematics educational perspective typically views errors as partial knowledge, still undesirable, but a necessary intermediate stage on the way towards attaining professional norms, and a base on which new or refined knowledge can be constructed.
  • Cognitive psychologists typically view errors as an undesirable but unavoidable feature of the human mind, analogical to optical illusions, which originate at the interface between intuitive and analytical thinking.
  • Evolutionary psychologists, in contrast, view errors as stemming from useful and adaptive features of human cognition. According to this perspective, people make mistakes (at least of the universal recurring kind) not because of deficiencies in their intelligence or their knowledge but because the requirements of modern mathematics, logic, or statistics clash with their “natural” intelligence.”

For those thinking of doing a research on mathematics thinking and learning, you must be clear about which of these different perspectives you will be analyzing your data and explaining your findings.

Recommended readings for your research:

 

Posted in Math investigations

Exercises, Problems, and Math Investigations

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The quality of mathematics students learn depends on the mathematical tasks or activities we let our students engage in.

Mathematical activities/tasks can be categorized into three types: exercises, problem solving, and math investigations.

Standard exercises

These are activities with clearly defined procedure/strategy and goal. Standard exercises are used for mastery of a newly learned skill – computational, use of an instrument, and even new terms or vocabulary. These are important learning activities but must be used in moderation. If our teaching is dominated by these activities, students will begin to think mathematics is about learning facts and procedures only. This is very dangerous.

Problem solving activity

These are activities involving clearly defined goals but the solutions or strategies are not readily apparent. The student makes decision on the latter. If the students already know how to solve the problem then it is no longer a problem. It is an exercise. Click here for features of good problem solving tasks. It is said that problem solving is at the heart of mathematics. Can you imagine mathematics without problem solving?

Math investigations

These are activities that involve exploration of open-ended mathematical situation. The student is free to choose what aspects of the situation he or she would like to do and how to do it. The students pose their own problem to solve and extend it to a directions they want to pursue. In this activity, students experience how mathematicians work and how to conduct a mathematical research. I know there are some parents and teachers who don’t like math investigation. Here are some few reason why we need to let our students to go through it.

  1. Students develop questions, approaches, and results, that are, at least for them, original products
  2. Students use the same general methods used by research mathematicians. They work through cycles of data-gathering, visualization, abstraction, conjecturing and proof.
  3. It gives students the opportunity communicate mathematically: describing their thinking, writing definitions and conjectures, using symbols, justifying their conclusions, and writing and reading mathematics.
  4. When the research involves a class or group, it becomes a ‘community of mathematicians’ sharing and building on each other’s questions, conjectures and theorems.

Students need to be exposed to all these type of mathematical activities. It is unfortunate that  textbooks and  many mathematics classes are dominated by exercises rather than problem solving and investigations tasks, creating the misconception that mathematics is about mastering skills and following procedures and not a way of thinking and communicating.

Samples of these tasks are shown in the picture below:

Click here and here for a sample teaching using math investigation.