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Understanding Mathematics


Why Should You Try to Understand Math?

 

Many students in America graduate from high school with a limited understanding of the math that they studied in school. If they go on to college or university, these students often have difficulty with the math courses there, for three reasons:

  1. College math courses are founded on basic school-level math.
  2. A lack of understanding of basic math usually leads to faulty memory, confusion, and errors.
  3. A habit of studying math without trying hard to understand it backfires in college courses.

These considerations lead to the following questions:

  1. What constitutes “understanding math,” anyway?
  2. What is the proper emphasis on understanding basic math?
  3. How can students re-learn their school math so that they do understand it?

 

What does Understanding Math Mean?

The phrase “understanding math” refers to the understanding of each specific topic in math. Do you understand quadratic equations? Logarithms? Inverse functions? and so on.

Understanding a specific topic implies:

  1. Knowing exactly what the topic (result, theorem, algorithm) is, including knowing the precise definitions of all relevant terms.
  2. Knowing why the result, theorem, or algorithm is true or valid.
  3. Knowing how to use the topic.
  4. Knowing how the topic is related to other topics in math.

Understanding a certain topic in math is more difficult than just memorizing it. Failure to understand a topic may have been the result of laziness on the part of the student, or the result of the teacher’s lack of insistence on understanding. SAT tests don’t usually test understanding directly, so there may be little incentive to learn understanding.

The Distributive Law

Here’s an example, the distributive law:

a (b + c) = ab + ac

This equation is absolutely fundamental in all of mathematics, from algebra to trigonometry and beyond. It’s also one of the most frequent sources of error made by weak students.

Are you familiar with the above equation? What does it say in words? Can you give some examples? What do the letters a, b, and c represent? And especially, why is the equation true?

Does it matter why the equation is true, in general? If you don’t know why it’s true, how would you know whether the similar-looking equation a/(b + c) = a/b + a/c is true? (It isn’t.) It’s quite a feat of memory to remember that the first equation is true while the second is not, without understanding why in either case. But a student who remembers (even if vaguely) why the first equation is true, is also likely to realize that the same logic does not apply to the second equation, which is probably therefore incorrect. This student could also probably prove that the second equation is wrong, by supplying a counterexample.

Knowing why the formulas, theorems and algorithms of basic math are true is an important aspect of learning the subject.  Without such knowledge, math becomes just a jumble of unrelated, mysterious factoids.

Too Much Emphasis on Understanding?

Is it possible to over-emphasize understanding of basic math? Yes, if this means that problem-solving skills are downplayed. Students need lots of practice (“drills”) to perform routine operations quickly and accurately, without consciously thinking of the underlying theory. But the fact that you do subconsciously remember the basic logic is strongly confidence-building. A sound mathematical education combines theory (understanding) and practice in balanced proportions.

Math Overboard!

My recently published book Math Overboard! (Basic Math for Adults) reviews all of school mathematics, from kindergarten to Grade 12, emphasizing both understanding and skills. To my knowledge no other available book accomplishes this outcome. Please visit Math Overboard!