Four generations of my family – or more – are very good with math, often testing years ahead of the curve. Why? Is it all genetics? Or are we doing something different?
Many folks imagine flash cards and lessons and worksheets. Too slow, too late, possibly counter-productive and ineffective.
Changing diapers should be a pleasant process. If baby is fussing, cuddle and soothe. Tell them what’s happening. I almost never in my life changed a crying or squirmy child. After everything is done, thank them, and count their toes and fingers. One, two, three, four, five little toes. Do the little piggy routine. They are learning that grownups use number words. They’re learning the idea of counting, even before they can verbalize it.
Play clapping games. Clap twice. One two. Toddler claps twice. Clap three times. One two three. Rinse and repeat. Even pre-verbal children enjoy emulating the number of claps.
The first idea to get across is not the numerals. The first idea is that of number words, of counting things. Two books. Three forks. Four buttons. Children learn to associate numbers with finite, countable sets of things. Soon they understand that “two forks” and “two buttons” and “two pennies” share a “two-ness,” and they learn the sequence – the ordering – of these numbers. They master the ability to count.
Play games such as chutes and ladders early. At first, verbalize the counting and addition. Point to the die; count the pips aloud. Other die: When they reach the point of recognizing small numbers, just say the count. Verbalize the addition: one and five makes six. Count aloud: one two three four five six spaces. When you think they’re ready, point to a die and wait – they’ll say the number. At some point, you’ll be able to say “two and five makes …” and they’ll give you the answer.
Model what you want them to learn. When it comes to math, verbalize early and often. Keep it at their level, and adapt – when they master one step, introduce another.
An even more basic principle: learning should be fun. There is no need to “test” a toddler, nor to force them to do your choice of task.
All of these things develop “number sense.”
When you cook, verbalize the quantities. Half a cup of milk. Two eggs. Cup of pancake mix. At some point, they ask. “What is a cup? Half a cup?” If the recipe calls for half a cup of water and 3/4 cup of milk, can you substitute? What is half and three fourths? A half is two fourths. Two and three makes five fourths. That would be one cup and a quarter. Cooking is full of math opportunities.
Children are hard-wired to learn language – not by formal instruction, but by listening to it in use.
When they’re ready, use money. Count change, explain pennies, nickels, dimes, and so forth.
How do you count fifty pennies for a roll? (I grew up before CoinStar was available – we rolled change and took it to the bank.) You could do it one-by-one, counting up to fifty. What if you make five stacks of ten? five tens makes fifty. How do you make a stack of ten? How about five pairs? Count them: 2 4 6 8 10.
Make five stacks; count them: 10 20 30 40 50.
Relax. Smile. This is a game. It is not scary. If your child doesn’t catch it today, let go, relax, try again at a later time.
At the grocery store, count your change aloud. It’s OK, you got a kid with you, people understand that it’s a learning process. Did you buy something “three for a dollar?” Look at the tape. The items might ring up as .33, .33, .34. Might be that they ring up as .35 -.02, .35 – .02, .35 – .01. Talk about this. 33 and 33 is 66. 66 and 34 is a dollar.
If you keep building, step by step, you will be amazed how far your child can go, while having fun.
Let’s break down the process of learning to add. A child’s first approach to adding two plus three objects is to count all five objects. One two three four five. Let them practice that skill a few times. Then point out that they can recognize “two” and “three” on sight, and can start with one number, and count up to the sum of both. Demonstrate. So the kid points to one set, says “two three four five.” A little time, a little mental effort is saved. Which has more? Two or three? Point to the concrete examples. They learn that three is bigger than two. OK, how about, when we add two numbers, we start with the bigger one and count? three four five. Soon your child learns that two plus three always adds to five – and so on for many single-digit pairs, especially when one of the pair is small.
Next step? Grouping by tens. When adding eight plus five, how many would we add to get ten? (pointing to the eight.) Two more makes ten. Physically take the two from the five. How many are left? What’s this number? Ten and three is thirteen, eh?
Relax. This is fun. This is not a test. Don’t make this into school, pressure, or anything as wasteful as that.
Some of you may be thinking “common core.” You may be thinking “how roundabout; how slow.”
This is how math wizards learn. This is how a boy learns enough math to score at the 18th grade equivalent when he is 9 or 10 years old. This is the fast way.
Research by Geary supports my contention: these number skills form the foundation of strong math skills. Children who do not learn these skills before first grade will be hindered in school. (Those outside of school have more leeway – they’re not being measured and sorted; they’re not struggling to keep up with an assembly line.)
Drill and kill is the slow way. It’s like sending a child onto the freeway before she has any clue what the brakes and steering and accelerator do. After a few crashes, the child is likely to conclude that math is “too hard.”
The defect of “Common Core” isn’t the ideas – it’s the conversion of them into tests and quizzes and stress; it’s the cogs-on-an-assembly-line approach. It’s teaching to a schedule or test, rather than teaching to the child. (It’s also, from what I have seen, a nearly-universal tendency of textbook writers to contrive the most nonsensical and confusing examples.)
I have talked about powers and square roots and binary arithmetic with six year old children. Long division, multiplication, fractions, decimals, negative numbers – none of it is “too advanced” for a bright child. At age 2, 3, 4, 5 is is important to use tangible concrete models whenever possible. I can use verbal instruction alone when they’ve learned enough vocabulary, but when introducing an idea, I do my best to make it concrete, make it tangible.
There is no “right age to learn X” – each child is a little different, even within the same family. Start where your child is. Think about the “next step” – and think small, go back to the fundamentals if you have to. Invent a game with the materials at hand. Play.
With multiplication, I start with a model and build an example. Checkers, pennies, poker chips – whatever is handy. Make a rectangle. Three times four – three rows of four. Count by threes. Count by fours. Play around. Split it into two stacks of six. Split the sixes into fives and ones. Oh, cool. Two fives is ten. Two more is twelve.
What does it mean to “divide by two?” Count off pairs. Two four six eight ten twelve. Look, six pairs. Two goes into twelve six times. Split the task – five twos make ten; one more pair makes twelve.
The time to introduce formal algorithms is when the informal language is well-understood.
Otherwise, why are you teaching this crazy complicated algorithm? When children already have the idea of rearranging numbers, building tens and hundreds, they can see how the algorithms for addition, subtraction, multiplication, and division all make sense. It doesn’t seem so complex, when you can think about what happens during these little math games.