Skemp article

Reaction to article:  Relational Understanding and Instrumental Understanding, R. Skemp

The concept of faux amis in math teaching and learning is a very valid and powerful one.  Valid because of the high preponderance of instrumental learning even in today's math  education (despite the research) ; powerful because a shift to relational learning requires nothing short of a 360 turn of mental paradigm.

Universities and business have decried the lack of creativity, logical reasoning, problem solving mindset (e.g. correctly defining the problem, conceptual mapping of a problem into its interrelated parts, looking at a problem from different perspectives, persistence) of our graduates.  A major cause would be the instrumental approach of most teachers, a view necessarily adopted by students, and implicitly encouraged by the parents (who enrol children in tutorial classes & demand top marks), by the Education Ministry and some universities, that measure mathematical knowledge with mandatory provincial & entrance exams, the bulk of the questions testing instrumental “understanding.”

In my 12 months tutoring K-12 Math at Sylvan Learning, I would many times encounter tutees who are only interested in the “rules,” apply them automatically, with some success to routine problems, and very little success to non-routine ones.  When discussing their errors, I would launch into the rationale behind the rules, but most of them tune out, as if they don't want their minds burdened by more math.  Hence students are happy to be reminded that “you flip inequality signs when multiplying/dividing by negative numbers” or that “you can cancel same factors in the top & bottom numbers of fractions, but not same addends.”  Not one of them asked why this is so. 

Part of the problem seems to be the approach of some of their previous teachers, who introduce the rules early on the topic, and follow these with drill & kill exercises, for skill mastery.  Such was the case when exponents were “discussed” in my Grade 10 son's class.  He knew that he had to add exponents when multiplying similar bases because of the rule, but couldn't explain why. Moreover, he was at a lost why you now multiply exponents when you raise a power to a power.  The interesting part was that when I tried to explain the raison-de-etat of the rules, he tuned out because “that's not the way the teacher explained it.”  In other words, he accepted the teacher's approach with utmost faith, that no other approach or explanation is possible. (or could it be because I'm just a parent & not an official teacher?)

The same thing happened in his elementary math when he asked me if he correctly calculated the area of an irregular figure where an isosceles trapezoid was nested.  For starters, I asked him why the formula of the trapezoid was such, he replied that he recalled his teacher dissecting the trapezoid & moving bits around, but he did not copy the drawing since  “it is only the formula that matters.”  And you guessed it right, he was not receptive to any explanation on my part.

So even in this instance when the teacher was going for relational understanding, such effort was undermined.  I can offer 2 possible explanations:  first, the follow-up assignments and the all-important tests were just plugging into the formula.  Second, the teacher might in the majority of his past lessons explained instrumentally (with matching tests).