Mathematics for Teaching Curriculum Reform, Teaching mathematics What is reasoning? How can we teach it?

What is reasoning? How can we teach it?

The world does not give us complete information that’s why call on our power of reasoning to complete this information the best we can and produce new knowledge.  Mathematics is one of its most famous product.

What is reasoning? When do we learn it?

Reasoning is defined as  the capacity human beings have to make sense of things, to establish and verify facts, and to change or justify practices, institutions and beliefs. We can make this definition more specific using Ol’eron’s:

“Reasoning is an ordered set of statements, which are purposefully linked, combined or opposed to each other respecting certain constraints that can be made explicit.” – Ol’eron (1977; 9)

Teachers’ knowledge of learning trajectory for reasoning is as important as their knowledge of students’ typical learning trajectory for specific content topics. In this post I will share a framework that I think will be useful for teachers in developing the reasoning skills of learners. I cannot anymore trace where I got this idea but I know it’s from a Japanese lesson study document I was reading last year. Reasoning is a skill highly emphasized in Japanese mathematics lessons. They have developed a framework for analyzing their students ‘reasoning trajectory’. This is applicable even in non-mathematics context. The framework even specifies the grade level to which a particular way of reasoning and arguing it is expected.

Levels of reasoning
  1. At the end of 2nd grade, students begin using expressions such as “because…” to describe their reasons and support their ideas.
  2. In 3rd grade, they begin comparing their own ideas with others and use expressions such as “my idea is similar to so-and-so’s idea…”
  3. In 4th grade, students use expression such as “for example…” and “because…,” more frequently Moreover, they begin to use hypothetical statements such as “if it is… then…”
  4. In 5th grade, they can become more sophisticated and make statements such as, “If it is … then it will be *, but if it is # then I think we can say @” by looking at different conditions.
  5. Finally, in 6th grade, students can start describing in ways such as, “It can be said when it is … but in this situation # is much better,” and starting to make decisions about how to choose a better idea.
In teaching mathematics, reasoning need not always be restricted to that of formal, logical or mathematical forms of reasoning. Words and phrases such as those listed above should be part of the students communication. It is therefore important to listen to the way students make their arguments or reason out in whole class and small-group discussion. If these are not yet part of the everyday communication of mathematics in our classes then its time for us to design the lesson that creates the environment where these kind of thinking and communicating is encouraged. Problem solving and mathematical investigation activities are great context where this can happen.



2 thoughts on “What is reasoning? How can we teach it?”

  1. The framework @delta_dc and I use, of which I’ve grown quite fond, is:
    -making sense
    -making conjectures
    -making arguments

    You’ve got nice examples of sense and argument in your post.

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