The reason math teachers assign homework is to give students the practice necessary for entrenching new concepts and skills in the brain.

Math homework is necessary for the same reason practicing the piano is necessary: it’s one thing to “get” what the teacher taught during the lesson, but it’s another thing to be able to perform that same skill independently and fluently.

Yet, all too many students practice math incorrectly, and they therefore gain little benefit, or even worse, they solidify misunderstandings and bad habits.

Practice doesn’t necessarily make perfect, but enough practice does make permanent, which is why guitar teachers, ski instructors, and golf pros are all such sticklers for proper form; they know how hard it is to unlearn errors that have become ingrained.

Many students will do a whole page of math and never check their answers. How do they know they were doing the right procedures? (Answer: They don’t.)

Or, students check their answers after completing the entire assignment, and only then discover that their answers don’t match up with those in the back of the book. In both such cases students tend to declare: Oh, well, the teacher will go over it in class tomorrow.

But in each of these scenarios, the student has now thoroughly practiced BEING WRONG.

Here’s the right way to do math (or math-related) homework:

1. Locate the answer key. If it’s in the back of the textbook, insert a bookmark or Post-it note for easy back-and-forth flipping.
2. Do the first problem.
3. Check it on the answer key.
4. If you were correct, move on to the next problem.
5. If you were incorrect, figure out what you did wrong before you move on.
6. Check your arithmetic; did you make a careless mistake?
7. If you realize you don’t understand how to do the problem, go to your notebook or your textbook and page through until you find the topic and carefully follow the explanation.
8. Still stumped? You can get help on www.KhanAcademy.org. Or, ask a friend or parent or sibling for help.
9. Only as a last resort, put a star next to the problem and make sure and ask the teacher for help tomorrow.
10. Proceed to …
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I’m hoping that as math instruction improves and becomes more “brain-friendly,” we’ll see fewer kids struggling in math.

When I was in my doctoral program, I was amazed at some of the research coming out on kids’ understanding of math concepts. We assume that children all learn pretty much the same math at roughly the same ages, and that they learn these concepts in math class.

In fact, there’s a wide natural variation, and not necessarily a lot of correlation between the math kids are taught in school and the math they actually know.

[I'm going to devote my Thursday blog posts to the topic of All Things Academic: reading, writing, 'rithmetic and the other school subjects.]

Last week I said that I see value in having kids (and all learners) memorize a certain amount of factual information.

I also said that I’m not a fan of rote memorization of multiplication “facts.” Kids should also be learning when and how to apply all of the four operations to various situations.

[I'm going to try devoting my Monday blog posts to the topic of Learners with Special Needs, which, I find, describes us all in some way or another.]

I’m so impressed with the Khan Academy videos, and I’ve been experimenting with ways to use them with my students….and with myself!

Do math tutors ever suffer math anxiety?

I do, sometimes, when I know I’m going to have to teach a topic that lies at the outer boundary of my own expertise.

So, yes, I am feeling anxious right now, because this afternoon I’m going to have to help a student with some pretty sophisticated trigonometry (including those dreaded “ferris wheel” problems). It’s stuff I don’t do every day…and it’s hard!

Here’s how I’m coping:

The most common math error I see is the “dropped” negative sign.

The problem calls for a -3, and the kid copies down a 3. Or they calculate an answer of -14 but they only write down the 14.

I expect to see this when kids are first learning algebra; I attribute it to the heightened demands on their attention that all those new algebra rules impose. The negative signs are like cell phones or car keys; little but important things that get left behind in the shuffle.

Test preparation is big business these days, and I’m part of it.

I’ve been prepping kids for the SAT, ACT, PSAT, SSAT, ISEE and other tests, for decades now.

Is test prep a plus? Or does it do more harm than good?

In my opinion, it depends on how the preparation is done.

Is the goal merely to achieve a higher score? One common approach is to teach tricks and shortcuts, which supposedly produces higher scores quickly. JenBee wrote about how harmful this sort of coaching was for her:

Why do they have these timed tests, like 25 problems in 3 minutes?

This is an excellent question.

I currently work and have worked with quite a few students who receive extra time on standardized tests, and I know for a fact that colleges do not factor this into their decision.

Meaning, if we have two absolutely identical students A and B, and A scores a 2100 out of 2400 with regular constraints while B scores a 2200 with double the time, B gets in and A doesn’t.

So, first, does time really matter? And second, if it does, why does time matter?

My sister has argued that students should not freely be given extra time. I think my hypothetical identical students identified this problem. Her basic point is that in the real world (or a college environment), speed and time are factors. Take two engineers applying for a job: it’s obvious that the guy who’s faster at math has a practical advantage.

Yet, I’ve worked with students who just need more time, and for each of them, I’m so glad that they get the opportunity to let their true intellectual power show.

Here’s a great question from Wanda:

When I was in grade school, I could not do well on timed math tests, even the basic add/subtract/multiply/divide tests. If I could do it on my own time, I did well.

Now my grandson has the same problem. When we do the flashcards, he can do them very fast but we make it fun also.

Why do they have these timed tests, like 25 problems in 3 minutes?

How can I help him do better?

There’s a wide variation in how people learn math, and in what math-related skills they are stronger or weaker in.

We usually teach multiplication in second grade, but many kindergartners are able to solve multiplication problems by using repeated addition.

There were five boxes and each box had three chocolate bars, how many chocolate bars in all?

Adults would say 5 x 3 = 15, but many young kids will do 3 + 3 + 3 + 3 + 3 = 15. They’ll get the correct answer and their logic is sound.

But at some point most people’s brains make a transition from repeated addition to “multiplicative thinking.” The brain now treats multiplication as a simultaneous operation and not as a string of additions.