# On Two Types of Learning (in Mathematics) and Implications for Teaching

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Abstract

A preliminary identification is made of two types of understanding--nominal and explanatory. The distinction has a bearing on teaching mathematics. To help illustrate the ideas, a detailed outline for an introduction to the dot product in linear algebra is given. Some contemporary issues in teaching are then discussed, particularly in regard to certain trends in contemporary textbook publication.

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Introduction

I have been studying mathematics for most of my life so far. For me, one of the marked features of learning mathematics has been joy. In that I am not alone. In some fashion or another, colleagues frequently express similar sentiments. Of those who do not do so explicitly, there can be the testimony of their lively presence when they talk about mathematical discoveries.

Basically then, mathematics can be a happy science. Of course, this does mean that mathematics is easy, or that there is any lack of struggle required. Even the most gifted reach greatness only after prolonged effort. But, whether one is a beginner or a professional, growing in mathematics is essentially a happy occupation. In other words, from elementary puzzles to advanced theorems, there is the joy of discovery.

Motivation for writing this paper, however, partly comes from continuing reports to the contrary, from what is evidently an accumulating majority of students of all ages. It is troubling how regular and even acceptable it has become to hear expressions like "Oh no, not mathematics!", "I was never any good at mathematics.", "I hate mathematics.", or "Mathematics is boring." For some individuals, these feelings even can become a more or less permanent fear. Along with this widespread distaste for mathematics, there is the increasing lack of competence in basic mathematics skills.

Consider the familiar situation from College Algebra. A few weeks after an exam, of those students who happen to remember the formula for combining fractions, how many are able to explain the sum 1/3 + 1/5? A particular instance is the following: Recently, a very sincere first year student was telling me that, in his high school mathematics classes, he had had no problem with algebra, but he could not do problems involving particular ratios. We spent some time at the board chatting through some examples. And sure enough, for certain cases at least, he could factor algebraic expressions and "cancel" like-terms in rational expressions. But, where did the terms go once they were "canceled?" When trying to solve "real-world" problems, it became apparent that he did not have a basic grasp of numerical fractions. In particular, he was not aware of the helpful elementary school diagram that partitions a square both vertically and horizontally, and so reveals the rule for multiplication of fractions. I mention this student not because his problem is unusual, but because this type of problem is too usual. At the same time, its very prevalence does not imply a lacking in native ability of students. Instead, this is evidence that, despite positive discoveries made by scholars in mathematics education, certain flaws continue to influence teaching methods brought to the classroom.

Recently, some of these difficulties were the topic of a conference organized by the MAA (Conference to Improve College Algebra, Task Force on the First College-Level Mathematics Course, U.S. Military Academy, February 7-10, 2002; supported by the Historically Black Colleges and Universities (HBCU) Consortium for College Algebra Reform). Results of this conference were discussed in the May/June 2002 issue of the MAA publication Focus, in the article "An Urgent Call to Improve Traditional College Algebra Programs" (Small, 2002).

The first part of the present paper addresses a distinction that relates to the broader significance of the MAA recommendations. The distinction is between two types of insight, and would seem to pertain to all contexts of mathematical understanding. …

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