Copyright © 2019 jsd

Taking an Indirect Approach
John Denker

1  Introduction

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:

  1. This trick gives you a glimpse of how real-world problems get solved. Such tricks are used by mathematicians, scientists, engineers, and everybody else. The general principle is: Given a problem that you can’t easily solve, break it down in to a larger number of easier steps. If you can’t solve it directly, solve it indirectly.

    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.

  2. This illustrates the principle that "There is More Than One Way of Doing It" (often abbreviated as TiMTOWoDI). (a) It is of course OK to memorize 7+4=11. (b) It is also OK to figure it out using reasoning that is based on other known facts.
  3. Rote memory and reasoning are not mutually exclusive. In a situation where 7+4 crops up repeatedly, you can figure it out the first one or two times, and thereafter just remember the result. TiMTOWoDI. (Insert your favorite joke about Professor Tim Towodi.)
  4. As I have often said:

    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:

    1. For one thing, not all mathematicians are good at arithmetic. Some of them are, but some of them aren’t.
    2. Many of them were good at mathematics first, and got good at arithmetic later. A lifetime of playing with numbers will make you good at it.

    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 mastery1 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.

  5. This trick is worth teaching and worth learning, because you get to use it again and again. Here’s slightly fancier example: What is 999,999 times 5? You can figure that out in your head, in less time than it takes to tell about it, if you re-arrange the calculation slightly.
  6. Often when you break a hard problem into a larger number of easier steps, the number of steps becomes quite large. As the 2300-year-old proverb says, a journey of a thousand miles begins with a single step.

    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.

  7. Most of the examples in a typical algebra book are grossly impractical. This is one of many reasons why some adults have a bad attitude toward algebra. They say “I’m 48 years old and I raised three children, and I’ve never needed algebra, so why do we even bother?” However, it’s not that simple. Algebra provides a way of thinking that people use to simplify their lives all the time. In our example, transforming “7+4” into “7+3 and one more” is essentially an algebraic transformation. The equations given above prove that the transformation is correct and 100% reliable, because every step relies on a fundamental rule of algebra. This is not a particularly grand life-altering example, but it is nevertheless a real example of using algebra to make something easier.
  8. In the education literature, taking an indirect approach is sometimes called “compensation”. You may find that terminology helpful if you are doing a literature search, but otherwise it’s better to avoid arcane jargon. Sane people just call it “taking an indirect approach”. It is a simple example of “thinking outside the box”.

2  References

John Denker,
“The Spiral Approach to Thinking and Learning”

John Denker,
“Principles of Teaching and Learning”

... not to mention demanding “automaticity” or learning “beyond mastery”, as some hard-liners do.
Copyright © 2019 jsd