Category Archives: math

My Brother’s Face

As my daughters learned about exponents, they waded through a sea of definitions and formulae. I taught exponents in algebra class, and it really surprised me that exponents were such a difficult topic. There are only a few rules to remember; it seemed straightforward to me. Rules about when to add exponents, when to multiply them, and when to change their sign.

But this is after years of being a mathematician. It’s not that the exponent rules are so ingrained in me that I never make a mistake simplifying expressions. It’s that exponents have a big set of associations for me. I recognize them in different guises. Much like my brother’s face.

For a long time, my brother’s face looked like this:


Now he looks rather different.

But the nice thing about knowing someone for a long time, and through many changes, for me anyway, is that all the essentials are still there. To me his gray hair is not his basic hair color. It’s a recent dusting of frost. He’s still the same. It’s the same with all the faces in my family.

What I’m saying is: when you first learn about exponents, your understanding is like my brother’s face above. Your understanding will grow, and change, and become unforgettable and recognizable in many guises. You might still mistake that face once in a while. But with time you can know it, very well.

Math study skills – tests

Something we get asked about often is how to prepare for a math test. While there is some variation, here are some general guidelines:

1. Know exactly when your test takes place. Three days before the test, do the following things.

2. Know exactly what topics will be on the test. Know the names of topics, techniques, rules, theorems. 3 days in advance!

3. Know the definitions of all of the vocabulary words in the test topics. Usually it’s best to use the definition that is in your text book, not a google definition. Why is this important? Because you might know the word “polynomial” in one context, but then see it on the test in a slightly different context. If you do not know the definition well, you might misunderstand the usage on the test.

Make sure you can explain clearly to yourself what each word means, or give an example of what it is. 3 days in advance!

4. The problems in the textbook are arranged from easy to hard. If you want to do well on a test, you have to do at least some problems from the end of the section in the textbook, not the beginning; in other words, it’s not enough to do the first ten problems. Attempt questions that throw you a curve – are not obvious. 3 days in advance!

5. Spend some time puzzling over problems you cannot answer – but not too long. How long to puzzle depends on your level of math. At the highest levels it’s appropriate to think for days and days! In elementary school, at least a few minutes; in middle school, a longer time. In high school, longer still. But after a certain amount of time, ask for help. First look again at your book, ask a classmate, ask your teacher, ask a tutor. 3 days in advance!

To do this successfully 3 days in advance, you have to attempt some hard problems.

6. Mix up your studying: do problems in your head; write problems and solutions down; tell a friend some math; watch a video. In this way, you are using your mind, eyes, ears, speech, and hands to do the math. This helps you to understand the material better and more fully, and access more of your problem solving ability.

Some people learn best while walking or kicking a ball; others while perfectly still; others with music. Find out what works for you – and mix things up. 3 days in advance!

7. Three days before the test is an approximate number. For some subjects, you need more time. You need enough time to attempt hard problems, find out what you don’t know, and get some help if you need it.

8. Two or three days before the test, write a summary of the material you need to know. Give examples of problems you need to know how to solve. Refer to topics, rules, types of problems by name.

Knowing math is a mixture of memorization and how-to. Some people emphasize one or the other. I prefer both. Memorizing the vocabulary can help.

Sequence Photos: Call for entries

What does a sequence mean to you?

Professor Bear is pleased to announce a new fortnightly math magazine, coming out soon via The topic for the first issue is: Sequences.

We’d love to publish your sequence photo in our inaugural issue.

A sequence is an ordered list of terms. For some people, what matters in a sequence is the pattern followed by its terms, or whether there is a pattern:

1, 2, 3, 4, …
0, 1, 0, 2, 0, 3, …
1/2, 1/3, 1/4, …
sunny, cloudy, rainy, sunny, partially cloudy, …
1/2, 1/4, 1/8, …

For some people, what matters is the “destination” or whether there is a destination:

3, 3.1, 3.14, 3.141, 3.1415, 3.14159, …
-1, 1, -1, 1, -1, 1, …
duck, duck, goose
… -2, 0, 2, 4, ….

If you have an interesting photo of a sequence, and would like it included in our magazine, please send it to:

If your photo is chosen, you’ll have a photographer byline. Creativity encouraged! How about a sequence of sequins?