Memorizing math facts -- long teaching guide

I was asked to get this back on the board, and now can’t find what board the request was on. So here’s a copy where it belongs and I’ll also repost it on the board where it was requested.

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Here’s an approach to memorizing math facts that is tried and true. It is not magic, it is not overnight; this will take time and hard work (three dirty little four-letter words). But if you go for it for a few weeks you should see progress and in several months you should have wrestled the problem to the ground.

Basic pedagogical principles (good in learning ANYTHING)
- anything is easy — once you already know how. To a new learner it is NOT easy. Avoid insulting the student’s hard work.
- focus on one thing at a time
- once one thing is learned, learn a second, then tie them together. And so on.
- keep the big picture and your eternal values in mind; don’t let misguided or “easy” short-term goals lead you into the opposite of your core beliefs.
- go from big picture to breaking down to the finest details to building back up to big picture. This is THE skill in teaching.
- ALWAYS work on connections since individual facts without connections are valueless trivia.
- connect learning to meaning at all times. Learning with meaning is real learning that will be retained; without meaning it may be “quick”, but what’s the point of quick meaningless mystical incantations?
- details add up; one piece of paper is so light and thin you can’t estimate its weight or thickness, but a pack of 500 sheets weighs several pounds and is several inches thick. Don’t ignore and slough off the details — they come back to haunt you.
- accuracy is vital. Of course we all make mistakes especially when learning and it is important to be accepting of a learner;s attempts, but smart people are the ones who learn to correct their mistakes. Accuracy comes BEFORE speed — what exactly is the value of a fast mistake?
- constantly remind the student to take time and think; never tell them to hurry up with an answer. Thinking IS your long-term goal and value, isn’t it?
- teach the simple and direct and logical methods and regularities first. Slow but steady. Shortcuts and exceptions can come later. A package of shortcuts and exceptions and tricks is a dead end, while a set of regular rules can get you 90% correct.
- it takes much LESS time to do things right the first time. What’s better, doing Books 1, 2, and 3 over three years and succeeding, or starting with Book 3 and failing it repeatedly for three years? The first method teaches success and skills and leads to Book 4; the second method teaches failure and leads to more failure, often permanent. Keep the long run in mind.
- don’t expect students to learn anything you don’t teach. Some may guess, but most will invent inefficient dead-end systems.
- we’re teaching the results of three thousand years of civilization and study. Students, bright as they may be, are simply not going to reinvent this all by themselves. Parents and teachers are needed because we do (hopefully) know more than kids.
- model the behaviour you want the students to acquire.
- REAL learning involves accuracy, retention, and transfer. If you can’t get it right, don’t remember it, and can’t use it away from the workbook, it isn’t much good for anything, is it? Take the time for these; they are not distractions but the central core.

To do math tables — starting with addition, then the same approaches for subtraction, multiplication, and division.
Actually, even before addition, you can use a similar approach to teach counting and number symbols.

Get a package of 100 sheets of 8 1/2 x 11 poster board, and some large-tip permanent markers in red, blue, and black. Buy or make a stencil to make a neat 1 inch circle.
It is actually a good thing to have the student watch you and talk to you as you make the poster (in general you make it, as a small kid’s messier efforts will be distracting— he’ll talk about his drawing nor the colours or his mistakes, not the math).
First poster: One red dot (one-inch circle) near the top. One blue dot a few inches underneath it. Near the bottom, you print in black very large (numbers two inches high) in your neatest printing 1 + 1 = 2. Or you can computer print the numbers and cut and paste them on the poster. With a simple graphics program if you have a good printer you could print the dots and numbers too. That is it. No cute pictures, no arrows, no hearts and flowers, no cute frames, no words. The goal is to focus on the addition, so have the addition the only thing to focus on.
Second poster: one red dot, two blue dots, 1 + 2 = 3
Continue up to (and only up to) 1 + 9 = 10
As you are making the posters, numbers over five can be grouped to make them more identifiable. Put the dots neatly in rows, not scattered which is harder to count. Six is two rows of three, seven is a row of three and a row of four, etc.
Numbers over 9 we take care of with our base 10 system and carrying — no need to overload memory or teach two conflicting approaches.
As you make each poster work at the table, then post these on the wall in a place the student can see clearly and where you can easily reach up to cover the writing.
Start with the first poster. Ask: “How many red dots?” “How many blue?” “How many all together?” “Say the addition.” (one plus one is two). Of course the one-plus row is pretty basic and most kids will go through this quickly - if not, take the time it takes.
At the learning stage, the student SHOULD be counting each group separately and then counting the total — the “counting on” strategy is an advanced thinking skill and we are backing up to the basics here. If the student puts a finger on each dot or later points at each dot to count, that’s OK, in fact a good habit of checking counting by matching. Practice and practice until counting is quick and easy; in math you don’t get anywhere by leaping ahead over steps.
As a side note, watch the hand and eye motions and insist on left-to-right directionality. Direction is vital not only to reading but even more to numbers, and if the student has a habit of random circular motions, this is a large part of the problem, so work on it.
**A VERY important point: the addition is NOT “two”, it is “one plus one is two”. Your goal is to teach the connection of the problem to the result, and having the student say only the result is shooting yourself in the foot, omitting the very topic you are trying to teach! A student who has never been expected to do this before will argue and want to do it the “easy” way (If it’s so easy, why are you having to remediate it?) but be steadily insistent.

Second stage — cover the sums with a file card as you move along and have the student say “one plus one is two, one plus two is three, one plus three is four, (etc.)” as you point to each poster with the sum covered but the problem still visible/

Third stage — turn away from the wall and recite the entire table from memory. It is a *good* thing to do this as a chant in a rhythm. You may move a hand up and down to keep the rhythm ONE plus ONE is TWO, etc. However avoid cute songs and tunes and rhymes, more distractors that take the mind away from the subject instead of into it.

Once you’re good on the one-plus row, do the two-pluses: 2 + 1 = 3, 2 + 2 = 4, 2 + 3 = 5, … again up to 2 + 9 = 11
And slowly continuing row by row.
Your wall will get covered — good. Make a neat block of the 1 + up to the 4+ and when you get too far down, start a second block of 5+ up to 9 +
If you arrange the blocks neatly, the first column downwards will be 1 + 1, 2 + 1, 3 + 1, etc.

Note the order I’m suggesting; many people go the other way around, doing 1 + 1, 2 + 1, 3 + 1, etc; nothing wrong but orally and in memory it seems a little easier to change the second number than the first. Later you will do the other order down the columns anyhow so it makes little difference.

Note also I.m not even mentioning zero yet. It’s a more advanced, abstract concept, so save it until the basics are down.

Many kids will rush through the one-plus and two-plus rows insisting this is “easy” and they “know” all this. Starting around 3 + 4 or so, you’ll start to see memory glitches. At this point repeat the basic lesson, count the red dots, count the blue dots (separately, not counting on, you want to get the second number visualized), count the total, say the “whole” question and answer out loud; repeat it out loud three times immediately to get it into memory: “Three plus four is seven. Three plus four is seven. Three plus four is seven.” Then go on as above.
When reciting the tables orally, insist that each word be clearly pronounced. A fast mumble may “get through” quickly, but teaches excatly the wrong thing.

You spend a *small* amount of time, *very often*. This is most important; it’s called “spaced practice” and is the best-known way to get something into memory. You spend ten to fifteen minutes, no more, in the morning, and ten to fifteen minutes in the afternoon or evening. For older students I suggest reciting under their breath while brushing their teeth.
After the student can recite the table without looking or counting — and then you KEEP doing it. This is called “overlearning” and is the best-known way to get something into permanent memory. This overlearning practice of the tables already mastered can be done in various places, in the car or waiting for the but being excellent ways to fill otherwise wasted time.
Sometime during the day you also do a reasonable amount of written math, a page or two in a workbook, and you allow the student to refer to his tables for a backup. Encourage him to try to remember, then try to count, and finally check the table, but try to work it out himself first before checking. When doing school work or tests, show him how to make small dot diagrams in the margin if necessary (and then try to outgrow these with time and practice).
You can get a sheet of 11 x 14 paper and make a smaller version of the table, *with* the visual dots, questions *and* answers, to keep on his desk; plasticizing with clear Mac-Tac is a good idea.

You do all this until the student can repeat the table with his eyes closed, and you keep doing it a bit longer for the overlearning effect.

You add the 0 + __ row and the __ + 0 column (If you thought ahead way back you may have left space for them …)
To illustrate zeros, where there would ordinarily be a group of dots you lightly make an empty loop with a pale yellow pencil, and then you point at this space and ask “How many red/blue dots?” and the student laughs and says “None!”

Then you go down the columns and do the plus-ones, plus-twos, etc.
Then you go down the diagonals and look at all the ways to add up to four (4 + 0, 3 + 1, 2 + 2, 1 + 3, 0 + 4) and five and so on.
Then you gve random problems — five plus eight is …? and *give the student time* to count up to it. Speed will come with comfort and mastery.

Somewhere in here, when the student is conceptually comfortable with what addition means, you show that 3 + 2 and 2 + 3 are always the same, and so forth, that addition is reversible (commutative law).
When the student is conceptually handling tens and units well, you can show the “tens group” method with pennies — to add eight plus five, put down eight pennies and five pennies; now to make a group of ten, shift two pennies over towards the eight so you are left with ten and three, or thirteen. This isn’t necessary but people (like myself) who are better with logic than with rote memory find it a wonderful tool.

Then you can take the table off his desk because he doesn’t need it any more (it is important to graduate sooner or later).

An average six-year-old will take several days or a week to get one row down. SO? So in ten weeks you KNOW your tables and never have to fight with them again. Unlike hoping and praying and guessing for eight years and more. Even if it takes two or even three weeks for one row, you still get it down by the end of the school year and not have to redo and redo and redo every year. What a time-saver, to get it out of the way once!

This approach combines the visual (dot patterns), oral (saying the numbers and the sum), kinesthetic (pointing at the dots as you count them, chanting in rhythm). The combination is a powerful route to memory.

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Much the same approach for subtaction. Start with 1 - 1 = 0, then 2 - 1 = 1, 2 - 2 = 0, 3 - 1 = 2, 3 - 2 - 1, 3 - 3 = 0, and so on, Again it’s a little easier orally to keep the same first number and change the second. Note that when you do it this way the rows get one longer each time, so you end up with a triangle pattern for the table.
On your dot diagrams, show subtraction by drawing blue dots and then crossing out the last ones. For example five minus three would be OOXXX (with the X’s representing dots with a red bar through them.)

PLEASE teach the words “minus” and “subtraction”. I get college students who still talk about “a take-away problem” and besides embarrassing themselves if they only knew what people thought, they are conceptually stuck in kindergarten and have huge trouble with algebra. Young kids are vocabulary sponges — teach them the terms they need.
Please note that “and” means addition. Never let the student (and never teach) use “and” for subtraction or multiplication — this leads to humongous confusion where students have no idea which operation they mean and choose at random, guaranteeing wrong more than half the time.

Once the student is fairly comfortable with subtraction, show that subtraction and additon come in “fact families”. I call this the “four for the price of one” system. After all, the diagram OOOUU can illustrate 3 + 2 = 5, 2 + 3 = 5, 5 - 3 = 2, or 5 - 2 = 3
Even if you do know your fact families and tens groupings, keep working on knowing the table by memory, too; connect, overlearn, and look at everything from as many points of view as possible. Make numbers your friends, useful tools that you can use and see in many different ways.

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For multiplication, your diagram dots will have to be a little smaller (half inch or the size of a penny) because you have to get up to ten by ten on one poster, and you want all the posters to be consistent.

This time show multiplication by rows and columns, for example

O O O O
O O O O
O O O O

3 x 4 = 12

Be consistent — first number is the number of rows, second number is the number of columns.

Have the student count and repeat as above.
In the learning stage, if the student has addition down well, he uses 4 + 4 = 8, 8 + 4 = 12 to work out the result.
If the student isn’t good at addition, he may count the whole thing. In this case do keep working on addition.

Division is learned by “undoing” multiplication and the four-for-one approach; the same diagram shows four facts

O O O O
O O O O
O O O O

3 x 4 = 12
4 x 3 = 12
12/3 = 4
12/4 = 3

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Of course all of this is *combined* with meaningful problem-solving in texts and/or workbooks as you move along. You also should be teaching the base ten system in detail.