Recently, it was mentioned that the addition problem “7+4” can be reformulated as “7+3 and one more”. This exploits the fact that the 7+3 subproblem may be easier to remember, if you concentrate on memorizing things that add up to 10. The question arose whether it was OK to teach this trick to grade-school students.
That is a really important question, with a really emphatic answer: Yes, it’s OK. Yes yes yes yes!!!!
At the next level of detail, there are quite a number of things that could be said about this:
I like to tell students: You’re not Superman, and I’m not Superman either. I can’t teach you how to leap tall buildings in a single bound. I can however show you where the stairwell is. We can get to the top one step at a time.
Note that sometimes the patterns involve numbers, but not necessarily. Whenever you notice an interesting pattern, and (!) whenever you notice something peculiar that doesn’t fit the expected pattern, you get a glimpse of what mathematicians do.
I mention this because some people who ought to know better insist that it is important to master basic arithmetic before moving on to more sophisticated mathematical topics. They have noticed that mathematicians tend to be good at arithmetic, but they are wrong about the pedagogical policy implications, and they are wrong about the underlying facts twice over:
There is a mountain of evidence in favor of the spiral approach to teaching and learning, as discussed in reference 1. That is, you first introduce a number of topics, touching on them only lightly. On the next turn of the spiral, you revisit the old topics, make connections between them, and introduce a few new topics. And so on. You can’t teach everything at once, and you shouldn’t try. You can’t teach mastery of anything on the first go-round, and you shouldn’t try.
This is called the spiral approach. Other terms for closely-related ideas include building-block approach and stair-step approach. The importantce of connections has been understood since the late 1800s, as discussed in reference 2.
Let’s be clear: Mathematics is interesting. Basic arithmetic isn’t. The example of 7+4=7+3+1 is an example of using math to help with arithmetic, not vice versa. To say the same thing the other way: Requiring mastery^{1} of arithmetic facts as a strict prerequisite for more advanced mathematics is pedagogical malpractice. This is the sort of cruel and counterproductive regimentation that Einstein called Kadavergehorsamkeit.
Some skill is required to keep track of what’s going on when there is a large number of steps. This is a skill that can be learned. Sometimes this skill is called “sequencing”. It comes more easily to some students than others.
Sometimes the steps are so easy and so numerous that it is not worth writing them down. Often students are told to “show the work”, and sometimes that’s important, but sometimes the opposite is important. That is, sometimes it would be ridiculously laborious to show all the steps. Deciding which steps are worth showing and which are not requires a non-obvious judgment call. Here is the verbose version of our 7+4 example:
[note background knowledge] | |||||||||
4 | = | 3+1 | (1) | ||||||
[restate original question] | |||||||||
7+4 | = | 7+4 | (2a) | ||||||
[now substitute (1) into (2a)] | |||||||||
= | 7+(3+1) | (2b) | |||||||
[use the associative property of addition] | |||||||||
= | (7+3)+1 | (2c) | |||||||
[perform a relatively-easy addition] | |||||||||
= | 10+1 | (2d) | |||||||
[perform a super-easy addition] | |||||||||
= | 11 | (2e) |
However, don’t panic about the number of steps. Most of them are so easy and so obvious that you don’t even notice doing them. However, when introducing the idea, you should be prepared to walk the student through the process, step by step.