Archive for the 'Understanding Math' Category

Learning Math – The Wrong Way or the Right Way

Sunday, June 29th, 2014

Like 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

  1. Memorize topics, but don’t bother trying to understand the logic behind them.
  2. If you have difficulty with a certain topic, skip it. Maybe it won’t be important later anyway.
  3. Use sloppy, messy writing for solutions to problems.
  4. Don’t do the practice problems.
  5. Copy someone else’s homework.
  6. Don’t learn basic definitions, or their consequences.
  7. 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

  1. Always spend the effort to fully understand each topic, whether an algorithm (method of calculation), a proof, or (especially) a definition.
  2. 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.
  3. Understand and memorize definitions. For example, what is the basic definition for adding fractions? How does it imply the standard method?
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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 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 for printed versions, and an online book seller for eBooks.



Understanding Basic Math – Test 2

Saturday, May 4th, 2013


Algebra of Polynomials

This is the second in a series of tests of understanding basic math. You need to be able to answer these questions accurately and confidently if you intend to take college-level math courses in Calculus, or Statistics. In answering the questions, please provide typical examples.


  1. Explain how the expression xn is defined, if n is a positive whole number (n = 1,2,3,…) and x is any real number.
  2. Prove that xn xm  = xn+m.
  3. Prove that (xy)n  = xn yn and that (xn)m  = xnm.
  4. Explain why x0 is defined as x0  = 1 (for x not equal to 0).
  5. How is xn defined if n is a negative integer?
  6. Prove that xn / xm  = xn-m.


  1. Define “polynomial.” Also define “degree” of a polynomial.
  2. Show how to add and multiply two polynomials.
  3. Expand (x + a)2 and explain.
  4. Explain division of polynomials with remainder. Illustrate with an example.
  5. Show how to solve a linear equation ax + b = 0.

Quadratic Polynomials

  1. What is a quadratic polynomial?
  2. Explain how to factor a quadratic polynomial by trial and error.
  3. State the quadratic formula, with examples.
  4. Explain the method called “completing the square.”
  5. Use completing the square to prove the quadratic formula.
  6. Define the discriminant of a quadratic polynomial.

The Binomial Theorem

  1. Expand (a + b)3.
  2. Describe Pascal’s triangle.
  3. What are the binomial coefficients C(n,k)? How are they related to Pascal’s triangle?
  4. State the Binomial Theorem, using ∑ notation.

Roots of Polynomials

  1. Define the terms “root of a polynomial” and “factor of a polynomial.”
  2. Explain why any polynomial of odd degree must have at least one real root.
  3. State and prove the remainder theorem.
  4. State and prove the factor theorem.


Understanding Basic Math – A Test

Wednesday, February 13th, 2013


Understanding Math

This is the first in a series of posts that allows you to test whether you understand basic math, by which I mean the math that you learned in school, from kindergarten to Grade 12. I will assume that you are either a parent, or a student (school or college-level), and that you realize that in fact you don’t fully understand basic math. No doubt you have already made an effort to upgrade your math skills and understanding, but have found the available material (books or the web) not too helpful for some reason. My recently published book Math Overboard! (Basic Math for Adults), which reviews all of basic math (K-12), attempts to explain fully and understandably all of basic math. By skimming through several of the tests in this series you can find out what you need to study, and what Chapters of Math Overboard! are relevant.

But what does “understanding” math actually mean? I once asked a class of Honors Calculus students to write a short essay on this question, and was surprised to find that they didn’t have a clue. I soon found that I wasn’t very good at answering the question myself. Most mathematicians will say that to understand a theorem you have to understand the proof. That is true, but it’s only part of the story.  First, there are many topics in math other than theorems as such. Algorithms are one example. Applications are another. You have to know (memorize) these things, and also to understand why they are valid. For example, you need to know the long-division algorithm, and why it is valid. How are long division with remainders, and long division with decimal expansions, related? And why is it true that the decimal expansion of a fraction of whole numbers either terminates or repeats?

I am not going to provide you with the answers to the test questions, as this would take too much space. You will need to read Math Overboard! or some other book (or use the web). By reading through the tests you can find out what you need to study.

Arithmetic: Base-Ten Numbers

  1. Explain what base-ten notation, as in 2,316 for example, means, in terms of the powers of ten.
  2. Explain how the addition (or sum) of two base-ten numbers is defined, in terms of counting.
  3. Of course you know that 59 + 37 = 37 + 59 (without actually doing the additions), but how do you explain that a + b = b + a in general?
  4. Describe the addition algorithm for base-ten numbers (which are the same as whole numbers). Explain why the algorithm is valid. (An algorithm is a method of calculation that is routine and repetitive.)
  5. Repeat questions 3-5 for the multiplication of base-ten  numbers.
  6. Another basic law of arithmetic is that a + (b + c) = (a + b) + c. What is the meaning of the brackets (or parentheses)? How do we know that this law is true?
  7. Repeat question 6 for multiplication.
  8. Yet another all-important law of arithmetic is that a(b + c) = ab + ac. Give some examples. Then explain why this law is valid.
  9. What does the expression a < b mean? Give examples.
  10. State and explain the algorithm for deciding which of two given whole numbers is the larger.

Decimal numbers

  1. Explain what decimal number notation, as in 17.62, means.
  2. What is meant by the number line for decimal numbers? How is a given decimal number located on the number line?
  3. Explain the addition algorithm for decimal numbers.
  4. Show how to multiply two decimal numbers (Math Overboard! doesn’t explain why this method is correct, so perhaps you can explain that.)
  5. List the Laws of Arithmetic for decimal numbers. How do we know that these laws are valid?


  1. If a < b, how is the difference b – a defined?
  2. State and explain the subtraction algorithm for decimal numbers a < b.
  3. Explain both addition and subtraction in terms of the number line.

Negative Numbers

  1. Describe the system of integers. Show how to subtract two whole numbers b – a, in the event that a > b.
  2. Explain the subtraction algorithm for two decimal numbers b – a, in the event that a > b.
  3. Define multiplication for integers, and for decimal numbers (positive and negative).

Division and Fractions

  1. If a, b, and c are whole numbers, what does the equation a / b = c mean? Discuss examples. (Be careful: it may not mean anything if we insist that all three symbols refer to whole numbers.)
  2. Now explain the concept of division with remainders.
  3. State and explain the algorithm for division with remainders. Discuss examples.
  4. What is meant by a fraction m/n? Explain how to locate a fraction on the number line.
  5. Explain the cancellation law am / an = m/n.
  6. Explain addition of fractions, in the case that both fractions have the same denominator.
  7. How do you add fractions having different denominators? What is the quickest way to check such an addition?
  8. Explain how to obtain the decimal representation of a fraction.
  9. How are two fractions multiplied?
  10. What is the definition of division for fractions? What is the algorithm, and why is it valid?

If you can answer these questions confidently, you already understand the number system, at least as far as rational numbers (fractions, positive or negative) are concerned. If not, you need to brush up on basic arithmetic.

Understanding Mathematics

Monday, December 10th, 2012

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!

Calculus killed me

Wednesday, November 21st, 2012

Why do so many students fail Calculus?


The young lady behind the counter was making change. I said that the next time I was in her bookstore I hoped my own new book would be on display. “Oh,” she said, “what’s it called?”

“Math Overboard,” I replied, “It’s a review of school math, written for adults.” “Hey, that’s just what I need,” she said, writing down the book’s and my name. “Calculus killed me,” she volunteered.

Calculus seems to kill a lot of pretty bright kids. It’s not that calculus is that hard – it isn’t. The problem is that many students come to calculus with an inadequate understanding of school-level math, including algebra, analytic geometry, functions and graphs, logarithms, exponentials, and trigonometry. This lack of understanding is often combined with poor writing skills in mathematics – brackets are misused, equality signs run on endlessly, and so on. These students lack confidence in their mathematical abilities – they suffer from “Math Anxiety.”

But what can be done about it?

Math Overboard!

My recently published book Math Overboard! (Basic Math for Adults) deals head-on with this problem. Covering all of school math, from kindergarten to Grade 12, Math Overboard! stresses the importance of understanding math in detail, as you learn it, or in this case, re-learn it. Frequent Problems test your understanding as well as your skills.

Math Overboard! consists of two volumes:

Part 1: Arithmetic, Algebra, Geometry, Functions and Graphs. (Published November, 2012; 444 pages. Price if ordered from the website (includes 20% discount from retail price), $24.00.)

Part 2: Trigonometry, Exponential and Logarithmic Functions, Complex Numbers, Statistics and Probability, Advanced Topics. (Expected publication date June, 2013.)


For further information, please visit Math Overboard!