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Wednesday, February 21, 2018

How to Teach Math

Several years ago, I wrote a column explaining why math is an important subject. In it, I pointed out that the actual content, that is, the actual mathematical manipulations, didn't really matter.  What matters in math is the meta-concepts.

To be successful in math, you have to master an approach to the world that is inherently useful. College majors typically require mastery of a math course not because anyone cares about math, but because passing a math course is the easiest way to demonstrate mastery of the meta-concepts that everyone actually does care about.

Once you realize that math is about meta-concepts, and not about math at all, teaching math becomes really very easy.

1) Buy a math notebook.

Every frustrated math student, everyone who is poor at math, has one thing in common: they don't organize their work. They scribble numbers down all over the page without regard to sequence. If students learn ANYTHING in math, they must learn to break that habit.

So, beginning students should be actively discouraged from doing math in their heads. Yes, I know the Math-Bowl encourages this for the advanced students who compete, but it's not a good idea for beginners. Beginners need the external structure. So, buy a THICK, empty math notebook, with lots of empty pages.

2) Throw out the calculator

For basic math, they don't need it.

Calculators interrupt the student's concentration, forcing him to alternate between doing the procedure the problem requires and doing the procedure the machine requires, figuring out the correct sequence of key punches.

Calculators are mostly a distraction. No one needs a calculator until they start doing trigonometry or statistics. If they aren't doing either, then let them learn the multiplication tables.

If you think calculators should be allowed on basic math tests, then why shouldn't cell phones and internet access be allowed for reference on history or English tests? The Internet is the equivalent of a history or grammar calculator. Why bog down the student with memorization of useless dates and grammar rules when they could be doing higher-order stuff?

Now, watch the history and English teachers howl in outrage that you should suggest such a thing. Watch the math teachers smile sadly and say, "Yeah, well, welcome to our world, suckers."

3) Write down each step

The student must write down each problem as follows:
  1. On the first line, the problem itself
  2. On the second and subsequent lines, write each successive step.
  3. No more than one operation (add, subtract, multiply, divide) is permitted in any step.
  4. The answer is written at the bottom of the step sequence. 
No scribbling in side margins allowed at all. Don't allow multiple operations in any one step because beginning students get themselves confused easily. Each step does exactly one thing, that is all.

Ignore their whining. Even if a beginning student gets the answer correct, the problem is wrong if they haven't shown all their steps. Make that clear. Stand over them for a month, enforce it, and they will gain the habit. Ingraining into them this single, solitary little trick solves over half your math problems overnight.

4) A fresh sheet of paper for every problem 

Math is not an exercise in conserving paper. Be profligate. Paper costs less than half a cent a sheet. Splurge. Once they have successfully trained themselves to write out every step, you can alter this rule to allow more than one problem per page, but even then NEVER let them break a problem over two sheets of paper. Ever. No. I mean it, don't do it.

Beginning students get a feeling of accomplishment from seeing all their work laid out neatly at a glance. It feels restful, as the eye glides downhill through the gears of the problem and finally takes up its ease at the bottom of the sequence, peacefully resting upon the (correct) answer.

5) When they get stuck

First, if there is ANY sign of margin scribbling, turn the old sheet of paper face-down, start on a fresh sheet.
Do it.
If you start on the old sheet, all the old scribblings will be a distraction. The student will wander down rabbit-trails trying to figure out what went wrong with the previous procedure. Clear his mind. Start fresh. Give him the gift of new eyes and a clean slate.

Now, math teaches a lot of (seemingly) arbitrary procedures. The student has to know all the procedures and know when to apply which procedure. Both parts of this are hard, but the second part - knowing when to apply which procedure - is the hardest. So, when he gets stuck and isn't sure what to do next, here's what you do:
  • Ask him a question you are sure he can answer.
  • When he answers correctly, affirm it ("That's right."), 
  • Rinse and repeat. Ask a series of questions, each one of which you are confident he can answer.
  • Build that series of questions so as to lead him to or through the correct procedure.
  • In basic math, this question will always come up at some point: "Do you think you would add, subtract, multiply or divide?" Those are the four basic operations, and one of them is almost certainly going to be part of the path to solving the problem. 
  • The student is almost always able to weed out at least a couple of the operations. That instills self-confidence, it shows partial mastery.
  • Don't give the student the answer. 
  • Ever. 
  • Always respond with a question you know s/he can answer.
  • Once s/he has gotten the answer, point out the truth: "I didn't tell you the answer. All I did was ask questions. You KNEW all the answers. You already KNEW how to do it."
  • And the student DID know how to do it. He just needs to internalize how to ask himself the same series of questions you asked.
  • Don't point this out. 
  • Simply keep repeating this sequence with him on every problem he has, week in, week out, making sure he writes down every problem step-by-step
  • He will learn to internalize the question sequence himself. He will start asking and answering his own questions. 
  • At that point, you can go bake brownies.
  • This whole sequence only takes a couple of months to instill. 
Once this basic skill set is instilled, it is now permissible to have the student walk through the steps of a failed problem to see where the mistake was made.
  • If each step has only one operation, it will be relatively easy to see which step failed. 
  • Now the student will see the wisdom of the step-by-step process.
  • He doesn't have to re-do every problem from scratch. 
  • He can find and correct his own mistakes easily.
  • Once he realizes this, math becomes almost bearable.

6) When YOU get stuck

Don't be afraid to say, "I don't know. Let's Google it." You don't have to know everything in math. In fact, you don't have to know ANYTHING about math. Remember, math isn't about math. Math is about learning how to be
  • organized, 
  • good at documenting details, 
  • good at being detail-oriented, and 
  • good at following and trusting arbitrary procedures.
None of those skills require you to know the arbitrary procedures yourself. Even if you are no good at math, you will naturally be better at searching for the correct way to do it. Model how to search for the right way to do things. Have your student watch you as you bumble along, figuring it out.

The student thereby learns:
(1) it is ok to not know something,
(2) this is how you find out what you don't know,
(3) Searching for the right procedure takes time and that's also ok,
(3) Perseverance can be as important, or more important, than possessing knowledge.

That's all there is to teaching math.

Well, that and liberal use of Khan Academy. Yes, I have a degree in computer science, minor in math, and have taught developmental math at the college level for years, but I taught my children almost no math at all. There's no point. Khan Academy teaches the concepts as well or better than I could. I only got involved if a video was opaque (unusual) or a solution sequence unclear (also unusual).

Often-times, I would walk along through the Khan Academy solution to the problem as perplexed about the correct sequence as my child was. It's not like I remember most of the stuff I learned thirty or forty years ago. We would discover the solution together, which was rather fun.

No, the only way I have ever taught math was to follow the sequence I have described above. It works.


A friend reminds me that I have omitted an important step.  Obviously, patience on the part of both teacher and student is a developed skill that is the absolute key to the method, on both parts.

But here's the step I'm missing: constantly remind the student that math requires only one thing - it simply requires you to be as perfect as God. You can't make any mistakes. Easy, eh?

I used to regale the children with stories of mathematicians who made very simple mistakes and destroyed millions of dollars worth of equipment, or entirely killed people. Everyone makes math mistakes, even the most skilled engineers and mathematicians. As I have frequently pointed out, I am a math teacher because I have gotten thousands more problems wrong than any of my students. My students can only become math teachers if they have failed as often as I have.

Fail early and often!
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1 comment:

Inca said...

You practice this concept of being organized,detailed, step-by-step in math in this article!

I like the encouragement to splurge on paper and to have the problem on one page, to see the work unfold to the correct answer! Helpful to know we are instilling good habits, that that is the goal (then less stress all around means we can bake brownies! Nice goal there too 😎)