Tag: educational studies

Posted in Mathematics education

Levels of Problem Solving Skills

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Here is one way of describing students levels of problem solving skills in mathematics. I call them levels of problem solving skills rather than process of reflective abstraction as described in the original paper. As math teachers it is important that we are aware of our students learning trajectory in problem solving so we can properly help them move into the next level.problem solving

Level 1 – Recognition

Students at this level have the ability to recognize characteristics of a previously solved problem in a new situation and believe that one can do again what one did before. Solvers operating at this level would not be able to anticipate sources of difficulty and would be surprised by complications that might occur as they attempted their solution. A student operating at this level would not be able to mentally run-through a solution method in order to confirm or reject its usefulness.

Level 2 – Re-presentation

Students at this level are able to run through a problem mentally and are able to anticipate potential sources of difficulty and promise. Solvers who operate at this level are more flexible in their thinking and are not only able to recognize similarities between problems, they are also able to notice the differences that might cause them difficulty if they tried to repeat a previously used method of solution. Such solvers could imagine using the methods and could even imagine some of the problems they might encounter but could not take the results as a given. At this level, the subject would be unable to think about potential methods of solution and the anticipated results of such activity.

Level 3 – Structural abstraction

Students at this level evaluates solution prospects based on mental run-throughs of potential methods as well as methods that have been used before. They are able to discern the characteristics that are necessary to solve the problem and are able to evaluate the merits of a solution method based on these characteristics. This level evidences considerable flexibility of thought.

Level 4 – Structural awareness

A solver operating at this level is able to anticipate the results of potential activity without having to complete a mental run-through of the solution activity. The problem structure created by the solver has become an object of reflection. The student is able to consider such structures as objects and is able to make judgments about them without resorting to physically or mentally representing methods of solution.

The levels of problem solving skills described above indicate that as solvers attain the higher levels they become increasingly flexible in their thinking. This framework is from the dissertation of Cifarelli but I read it from the paper The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra by Tracy Goodson-Espy. Educational Studies in Mathematics 36: 219–245, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

You may also be interested on Levels of understanding of function in equation form based on my own research on understanding function.

Image Credit: vidoons.com/how-it-works

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: