What’s The Formula For Formulas?

Audrey McLaren asks this great question

There were many responses and as usual, the conversations were many and inquisitive. But as usual with Twitter, for me at least, there is only so much depth one can go into. Hence, a blog post!

There are two parts to Audrey’s question.

  1. Why do we discourage students from using formulas?
  2. What to do when they come up with their own formula?

Discourage Formula Use

For clarity’s sake, the discouragement that Audrey and other math teachers point to is one when students are given formulas / procedures / algorithms and are told to “just use it to solve the problem” with no further explanation as to what is going on. Generally, this isn’t a good way to go about teaching mathematics as the student hasn’t really learned anything other than an arbitrary set of rules and use cases. Generally. There are times and cases where it is probably better to simply give the formula and have students “plug and chug”. But those instances are few and far between. One case where you could do this is to help build intuition / comfort with the behavior of a function. In fact, I do encourage students to toss numbers into a function and see what happens. I call this tinkering and I think it’s a healthy way to get a feel for the objects we are to work with. However, when we want to get to modeling and problem solving, it’s a good idea for students to know the meaning of the functions they are using. And thus, I do discourage simply giving students formulas to use without a proper explanation of how to use them.

Student-Discovered Formulas

Nothing warms my heart more than when a student finds his/her own way of solving a problem. It tells me that they’ve been processing the content. There is alertness and awareness. So what’s a teacher to do when the student makes their own discovery?

As is the case with most things educational, there isn’t one correct answer. Here’s what I do.

Encourage Exploration

If a student has made their own discovery, I want them to push the limits of their discovery. More often than not, the student’s discovery is narrow and works only in a certain set of cases. Sometimes this is a result of the types of problems that the student has seen. Consider this contrived case: suppose a student is asked to find the average of the following sets of numbers:

  • 2, 3, 4
  • 4, 12, 20
  • -5, 0, 5
  • 2,4,6,8,10

In all cases, the average happens to be the median. And if this is the student’s first introduction to average, he/she may conceivably conclude that the average is the middle number. After all, the pattern holds. The best thing a teacher can do here is to encourage the student to use their own discovery on as many different cases as possible. These additional ‘breaking’ cases can be supplied by the teacher and / or constructed by the student.

Or more plausible, consider these fraction subtraction problems:

  • \(\frac{1}{5}-\frac{1}{6} = \frac{1}{30}\)
  • \(\frac{1}{7} – \frac{1}{9} = \frac{2}{63}\)
  • \(\frac{1}{3} – \frac{1}{7} = \frac{4}{21}\)

The “formula” is just “subtract the denominators and put the positive difference for the numerator and multiply the denominators and make that the new denominator”. This formula is indeed correct and will always work for problems of the type $$\frac{1}{a} – \frac{1}{b}$$ where \(b > a > 0\). From here, the student can be given different fraction subtraction problems that would break the discovery. But with a modicum of astuteness, it is well within reason that he/she will discover “cross multiplication” or some such technique which will always work.

And this brings us to a different quandary. What do we do if the student has discovered a shortcut, but doesn’t understand any of the reason for why it works, other than it does? In almost all cases that I’ve seen of student discovery of math content, the student has discovered the “how” based on an extrapolation (informal induction) of recurring patterns in problems.

Encourage Exploration, Part 2

Usually, the correct “shortcut” can be defeated by constructing somewhat annoying problems that show that the student’s approach is the long way around the barn. Take for example, “the butterfly method” (Google it, I refuse to link to any such article 😀 ). How efficient is it for adding \(\frac{2}{17} + \frac{13}{51}\)?

And this is part of the point. Is the method efficient even if it is sound? The butterfly method is sound.

The other part of the point is, does the student or will the student be able to take their discovery and push it to further discover the “why” or will they just be content with having discovered the “how”?

So, the second part of encouraging exploration is to get the student to take the how part and extend it to understand the why part so that both the how and why are efficient. With that said, I do give some quarter to utilizing an inefficient method if the understanding of why is sound and complete. Of course, I would draw the line at something like \(5 \times 28\) solved by twenty-eight additions of five [and I would probably argue that understanding multiplication as repeated addition is probably not a strong enough “why”].

TL;DR

  • Generally, discourage “understanding free” usage of formula / processes / algorithms.
  • Help the student continue to develop their method by either providing counterexamples [to show the limited use] or by providing “annoying” cases [to show the inefficiency].
  • Help the student to understand why their method works (or doesn’t always work).

All in all, I would say continue to encourage the student to explore!

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2 thoughts on “What’s The Formula For Formulas?

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