Learning and Memorizing Math. Lesson 11

There was a time that I thought that understanding math was more useful than memorizing it. A certain amount of memorization is essential, especially for lower-level math. For example, you need to memorize the multiplication table up to 9 x 9 in order to be able to multiply two four-digit numbers.

In elementary and high school, this was not an issue for me, because I did not have to work very hard to understand or memorize it. In college, however, they threw engineering calculus at me, and that was a very different story, not so easy. This was in second semester of my freshman year at the University of Tennessee. I struggled to understand the formulas and going into the final exam, my grade was an F. In desperation, I gave up trying to understand the calculus concepts. Instead, I memorized all the formulas we had covered in the course and the kinds of problems that each could solve. I made a 100 on the final exam, which converted my F to a C. While the memorization served a useful purpose, my lack of understanding would surely have doomed me if I had been in a curriculum that required more advanced classes later.

Now, many decades later, I discover that a similar experience happened to the author of an article published by the Dana Center. The author, Monette McIver, liked math before she went to college, and it was relatively easy. But college math was too challenging, and she too realized that maybe she should just focus on memorizing what was necessary rather than trying to understand. She actually got an undergraduate degree in math that way. She laments having to do this, however, because she really wanted a career as an engineer, and she did not end up with needed level of understanding.

So, what should you do when faced with overwhelming math challenges? I say, memorize it, in order not to be overwhelmed, but as you move along, keep striving for understanding. And seek out the teachers who are good at explaining the math. While the modern focus on understanding math, as in “New Math” and “Common Core,” is commendable, it too often leaves students confused, and they don’t learn the basics either. There is a certain utility in memorizing multiplication tables, even if you understand the principles well enough to build a table. The trick is in discovering what should be memorized to free you to apply that knowledge for higher-level problems. Here are some basic principles that might help:

1.     Use the math you do know to help you figure out the math you do not know. For beginners, work on developing number sense. For example, when asked to solve 7 x 8 question, someone with number sense may have memorized 56, but they would also be able to work out that 7 x 7 is 49 and then add 7 to make 56, or they may work out ten 7’s and subtract two 7’s (70-14). Alternatively, know that 7 x 8 means 7 rows of 8 or 8 rows of 7, to enable you to realize that 7 x 7 = 49 plus 7 makes 56. Mnemonics are very helpful with the memorization: for example: 56 = 7x8 is the numbers 5 6 7 8 in order.

2.     Solve many problems at the level you do understand. This will reinforce the memorization, and seeing the same issues in different contexts will gradually build up understanding.

3.     Try to identify, understand, and memorize the really crucial concepts and definitions that underlie many other math ideas. Examples include words like function, theorem, angle, tangent, sine, cosine, derivative, integral, matrix.

4.     Learn a special case of a math concept first and then generalize to more abstract extensions as your understanding improves.

5.     Whenever feasible, make drawings to illustrate a concept.

6.     Find good learning sources. Many are on the Internet. One of the best sites is Kahn Academy

7.     At the appropriate age, master algebra. It is fundamental to most of higher math.

8.     Develop mnemonic devices, but only for basic ideas and don’t overdo it. The point is to use memorization as an adjunct to mathematical thinking, not as a substitute for it.

In one lesson, I can’t cover much mathematics, but I will choose a fundamental of trigonometry to illustrate my point to identify what is useful to memorize and what you need to think through. To remember the core ideas of a triangle, the features you need to memorize are labeled in the figure below.

First, realize that the lines are measures of distance (a,b,c,). For future reference you could think of lines a and b as y and x axes of a graph, but this is not relevant here. The lines could be labeled anything. Notice also there are three angles (1,2,3). They also could be labeled anything. Mathematicians like to use Greek and for angles, they often use theta (θ). Angle #1 is a special angle, called a “right” angle. I don’t know why it is called “right,” but you might think of it being the right angle to focus on in a triangle: because it is 90^o by definition, the sum of the other two angles is also 90.^o Can you figure out why the other two add up to 90? On a scratch sheet, draw two triangles stacked against each other to form a square and a horizontal line running from top left to bottom right. The box has four 90^o angles if you remove the diagonal line. On the right you see that if you flip line b to form a straight line, you have created a 180^o angle, and if you keep on moving the line around, you create a circle of 360^o. I have no idea why a circle is defined as having 360 degrees. Some mathematician early knew it would be useful to carve a circle up into angular pieces and for what ever reason decided 360 would be a good number (not too small, not too big).

See that there are three boundaries, a, b, and the longest one, c, which is always defined as the hypotenuse. At about this point, you should be asking, “What is the point of all this? What is the ultimate objective?” The answer is that trigonometry allows you to calculate certain dimensions without actually having to memorize or even measure them.

At this point you must memorize that in a right triangle the longest line (c) is called the “hypotenuse.” Each of the two variable angles has a line opposite to it and the other non-hypotenuse line is called the adjacent side. Sine is arbitrarily defined as opposite/hypotenuse, cosine as adjacent/hypotenuse, and tangent as opposite/adjacent. In the diagram above, the sine of angle 3 is a/c, the cosine is b/c, and the tangent is a/b. With these ideas firmly cemented in memory, you are now free to explore the mathematical consequences and uses.

Sine, cosine, and tangent are just names; they could have been called anything. But they are useful because they are a way to label the ratios of the lengths of any two sides of the triangle. If you divide the denominator of the ratio into the numerator, you calculate the length of the third side without having to measure it.

You could, for example, measure on of the angles with a protractor and now instantly know the degree of the other angle that is not the right angle of 90.^o If angle 3, for example is 35^o, then angle 2 is 55.^o

Note also that the two non-right angles are complementary. They sum to 90^o and moreover, the sine of one is equal to the cosine of the other. The equations can be consolidated, as follows:

Sin 35^o = a/c, Cos 55^o = a/c, therefore, the sin of any angle, θ, = cos 90 - (θ).

With these few examples, you can see the usefulness for learning math of combining some memorization with reasoning.

                                 

Next Lesson: #11. Deliberate Practice

Sources:

https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-complementary-angles/v/sine-and-cosine-of-compl

 

McIver, Monette. (2018). Memorization versus understanding. Better approaches to teaching mathematics. May 26. https://www.utdanacenter.org/blog/memorization-versus-understanding-better-approaches-teaching-mathematics