calvinmath
Comic by Bill Watterson

I am of the opinion that many of the students who profess to hate math do not, in fact, hate math. They just dislike computation (the rote skills of addition, subtraction, etc.). This view of math stems from two main problems: 1) lack of understanding, and 2) lack of support for the subject (e.g., parents telling children who have trouble with school math, “Math is hard,” or, “I’m not good at math”).

From an early grade students relate mathematics to procedure. To solve the area of a trapezoid I have to add the lengths of the bases, divide the sum by two, then multiply by the height. To calculate the missing value in this polynomial I must apply the quadratic formula.

Students learn how the formulas work. They collect them through the years, stashing them away in their mathematics tool kits and, when the need arises, rummaging through the lot to find the appropriate tool. While this method works when pressed for time during a standardized test and faced with a question about 30-60-90 triangles for which two lengths are known and the student has to find the length of side b, it fails when it comes to really understanding mathematics.

Handyman's toolbox
The cluttered and confusing mathematical toolbox (Photo from Real Simple)

Mathematics is not about the how so much as the why. Oftentimes mathematicians consider the proof for a mathematical finding more important than the mathematical finding itself.

Now, I’m not suggesting that teachers introduce elementary school students to the tedious though satisfying process of working through the proofs of 1+1 or the area of a circle, \pi{r}^2 , but rather that students intuitively recognize why the computation works.

Some amount of memorization does help, but that is not the goal. The misconception begun in elementary mathematics instruction – that it is a manner of number and sign manipulation – creates an early atmosphere of dislike. Given a solid grounding in the concepts, one can skip down the halls of mathematics without a head full of disconnected equations, formulas, and other mathematical constructs.

In learning about area, a student doesn’t need the whole kit and caboodle for area formulas. With just the area of a rectangle/square, all other polygonal areas could be derived. For example, take a {4}x{4}-inch square. Drawing a grid of square inches in the square’s borders, as below,

sixteensquares_fig2
Image from NM State

shows that its area is 16 square inches, the product of multiplying the height and the length. Cutting the square in half, you have a triangle. What does that suggest about the area of a triangle? Once you have the area of the triangle, you can determine the area of a trapezoid, which looks like a quadrilateral with a triangle attached on either side.

Eventually, by cutting pentagons, hexagons, and other “greater” polygons into triangles, you could see and explain yourself why the area of any polygon is \frac{1}{2} x perimeter x  apothem.

Of course, at some point in mathematics visual explanations are not sufficient. As students move into the realm of calculus, differential equations, linear algebra, and others, math focuses more on functions, variables, and algebraic constructs. Nevertheless, even at these levels – or, perhaps, especially at these levels – understanding the concepts is more important than knowing the “methods.”

The way that schools teach mathematics puts students at a disadvantage when they enter college and enroll in college-level math courses. They arrive unprepared for what mathematics really embodies – exploration, discovery, understanding – because twelve years of elementary and high school left the impression that mathematics is about manipulation, computation, and memorization.

Mathematics is, dare I say it, creative. Alas, it is also sadly misunderstood.