Back in the good old days, when I was teaching Calculus to undergrads, I remember two types of students: those who could calculate, and those who couldn't. Now, certainly there are other skills which are important for the study of higher mathematics. There's creativity, and imagination, both of which feed into visualization and abstraction. But without the ability to calculate, to do simple arithmetic quickly and easily, the other skills are handicapped.
I had students who could grasp difficult concepts, could creatively plan methods of attack on complex problems, or could abstract fundamental truths from seemingly unrelated data. But they couldn't bring those tools to bear in their full strength, because they had a weak foundation in arithmetic. And they were often slow in following my reasoning on the board, because they were still hung up on a simple factoring that occurred two paragraphs earlier ("Dr. Fulton, where did that come from?").
I had students who got Calculus. They truly understood the conceptual picture. But they struggled solving problems, simply because of arithmetic. Why? I don't know exactly. I can only guess that they were allowed to use calculators throughout grade school, and they missed out on the reflexive training for arithmetic. That's a heartbreaking situation, because it's not easily fixable at a later stage. No Calculus student wants to be told to go back and review multiplication tables.
We mentioned one flash card technique that we've used in our "Shoot-the-Number" post, involving Nerf guns. It involved the child saying the answer, and then shooting it on the number chart if he was right. As he improves, make him get 5 in a row before he can take a shot. Then make him get 5 in a row in under 10 seconds. It's like weightlifting. Keep making it harder, tougher, faster. You want the arithmetic to be reflexive.
Another technique we've used is the trampoline. A child gets to keep jumping on the trampoline, as long as they're doing the flash cards and getting them right. Once they've missed, say, three cards, it's time to get off. Choose an activity your kid enjoys, and do likewise.
If you have children of different age levels, you can teach multiple kids at once. We just did this yesterday. One child held up a card, a younger child pronounced the numbers on the card (for number recognition), and an older child answered the multiplication problem. Meanwhile, the child holding up the card was also getting a review.
Sometimes, it is useful to call out the card to your child, without letting them see it. It's different, and it will slow them down at first. But it's good. It's important to be able to do mathematics without necessarily relying upon optical signals. It builds visualization.
There are many, many variations that can be done with flash cards. Be creative. But be consistent. Use them regularly. They're a good, cheap tool in building an arithmetical foundation that will last...
...to Calculus, and beyond!
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Brian and Melanie Fulton both earned doctoral degrees in mathematics at Virginia Tech. They formerly taught math at the university level, and now run a hobby farm while accuracy-checking collegiate mathematics texts. They homeschool their four children, frequently employing the aid of chicken, dairy goat, cat, and dog tutors.