Instrumental vs. relational teaching - A reflection on today's class

Instrumental vs. relational teaching: a reflection of today's class discussion

Sept. 16, 2015

Today we looked at the pros and cons of teaching instrumental mathematics versus relational mathematics. Yet teaching doesn't have to incorporate just one or the other method. Both can be integrated. When teaching a subject, I think that both instrumental and relational should be taught together at all times. In philosophy, we would say that every object has both form and matter -- the form being the essence of the thing, and the matter embodying its accidental (secondary) attributes. I would propose that this analogy of form and matter fits very well with our understanding of relational and instrumental ways of teaching math. The structure (instrumental) is necessary as a scaffold in which to set the understanding of the student (relational). We teach the rules, the boundaries, the restrictions, the equations, while explaining how these equations and rules are derived, and how they reflect and can predict reality. To give but one example, consider the Pythagorean theorem. For a student to understand this theorem, he must first have an understanding of what a right triangle is. The Pythagorean theorem provides us with an equation which relates the relationship between the different sides of the triangle. Once the equation is memorized, and the framework is understood, we can then explore the consequences of this equation by noticing that the hypotenuse is always going to be larger than the other two sides. We can consider one or two different triangles, which we can draw out accurately on the board, and test our equation on them. We can consider extreme cases where one of the sides is almost zero in length. We can show how if we double the two right-angled sides of the triangle, then the hypotenuse is also doubled in length. We can also prove the Pythagorean theorem using the simple visual proof of triangles inscribed in a square. The understanding will follow within the framework of the instrumental equations and rules.

I think too that if the teacher has a very relational understanding of mathematics, then he will communicate his way of understanding to the students. I have noticed this whenever I taught high school math -- I would always investigate the why of what I was teaching. And yet a prudent teacher must know where the middle road is, and how much time ought to be spent simply on imparting meaning vs. intrumental teaching.