## Learning Math – The Wrong Way or the Right Way

June 29th, 2014Like everything we do, there is a wrong way and a right way to learn mathematics. My years of teaching math at the university level showed me that many students had developed unhelpful habits for learning math. These habits, unless corrected, can lead to failure in college math courses.

**The Wrong Way to Learn Mathematics**

- Memorize topics, but don’t bother trying to understand the logic behind them.
- If you have difficulty with a certain topic, skip it. Maybe it won’t be important later anyway.
- Use sloppy, messy writing for solutions to problems.
- Don’t do the practice problems.
- Copy someone else’s homework.
- Don’t learn basic definitions, or their consequences.
- Forget about learning and understanding mathematical proofs.

In short, bad habits (learning math the wrong way) are mostly the result of intellectual laziness. Mathematics is *not* an easy subject to learn. To learn it properly you have to understand every detail. The first way to check whether you do understand a topic is to do the exercises, or problems. But you also need to study and learn the *logical basis* of every topic. Ask yourself whether you could explain it to someone else.

To mention but one example, you probably “know” that the sum of the interior angles in any triangle is , but can you explain why this is true? The proof is not difficult, and I believe that every geometry student should learn and be able to recite the proof. Understanding the logic underlying each math topic helps greatly to reinforce your memory of that topic. It is also useful, and in fact necessary, for learning more advanced topics. Students who rely on memorization alone often remember math incorrectly – or forget things entirely.

Another, very important example is the equation a(b+c) = ab + ac, which is frequently used in Algebra and elsewhere. Misuse of this equation often leads to failure on college math exams. But a student who understands why the equation is valid is unlikely to use it incorrectly.

**The Right Way to Learn Mathematics**

- Always spend the effort to fully understand each topic, whether an algorithm (method of calculation), a proof, or (especially) a definition.
- Memorize everything you learn; invent your own mnemonic “tricks.” Understanding the logical basis of a given topic is a powerful aid to remembering it over the long term.
- Understand and memorize definitions. For example, what is the basic definition for adding fractions? How does it imply the standard method?
- Always do practice problems. It is the problems that cause you some difficulty that are the most helpful. Trying to understand and overcome the difficulty is an important aspect of learning math.
- Watch for “mental blocks” – things that you just don’t understand. Everyone encounters mental blocks once in a while. Take the time and effort to resolve such blocks.
- Work hard to understand and remember proofs. Gradually learn how to produce valid – and clearly written – proofs on your own. Figure out how the assumptions were used in the proof.
- Write down the solutions to practice problems clearly and succinctly, using plain English phrases such as “by substitution,” “therefore we have,” “by Pythagoras’s Theorem,” etc.
- Think about how a given topic being studied is related to other familiar topics. Look for instructive special cases.

Many people seem to think that learning math is like learning history. You memorize a bunch of unrelated facts (or problem-solving techniques) for the exam. But learning math in this way is pretty much a waste of time (the same is probably true for learning history). Mathematics is a tightly organized system of knowledge, based on strictly logical arguments which themselves make sense. Ignoring the underlying logic is a serious mistake, which leads to faulty memory, and eventually to increasing confusion and “math anxiety.”

Most people know whether they do or do not understand a particular topic in math. But what can a person who doubts his or her understanding of parts of math do about it? Should you use online resources as learning aids? Maybe, but many sites that I have looked at were pretty close to “wrong way” approaches – sets of routine problems requiring very little basic comprehension.

Two outstanding exceptions: The Khan Academy http://www.khanacademy.org/ and my book Math Overboard . The Khan Academy is mainly for people who are learning the math for the first time. **Math Overboard (Basic Math for Adults)** reviews all of school math, from kindergarten to Grade 12, with self-contained explanations of every topic. It is designed for people who need to re-learn parts of basic math. Frequent problems support learning. Math Overboard, Parts 1 and 2, are now available in printed and eBook versions. See http://www.mathoverboard.com/ for printed versions, and an online book seller for eBooks.

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