Every (Math) Trick in the Book
She has plenty of company from Tina Cardone, author of Nix the Tricks, a free downloadable book of tricks that Cardone believes should be removed from the curriculum. (It also includes what and how things should be taught instead. Though Cardone is the author, she admits it was a collaborative effort of #MTBoS.)
But Gojak and Cardone are not talking about fun and mysterious math tricks, like this:
- In a bookstore or library, pick out two identical copies of the same book. Keep one, and give one to your friend.
- Ask your friend to pick a number, and both of you turn to that page in the book.
- Then tell your friend to pick a word in the top half of that page, and silently spell that word while moving forward one word for each letter. (For instance, if they picked “math,” they’d move forward four words as they spelled.) Have them do that again with the word they land on; then again; and again; till they reach the end of the page. They should stop at the last word that won’t take them to the next page.
- Do the same thing with your copy of the book — though make it obvious that you’re not copying them and using the same first word.
- Miraculously, you and your friend will land on the same word at the end of the page.
What Gojak and Cardone are talking about and, rightfully, railing against are the mnemonics and shortcuts that many teachers give to students in lieu of developing true understanding, such as:
- Ours is not to reason why; just invert, and multiply.
- Just add a 0 when multiplying by 10.
- The butterfly method of adding and subtracting fractions.
From my perspective, tricks are not inherently bad, as long as students have developed the conceptual understanding necessary to recognize why the trick works. In fact, the standard algorithms for multiplication and long division are, essentially, math tricks. But if students are able to reason their way to an answer — and, even better, if they can discern these tricks on their own from examples — then the ‘tricks’ can be useful in helping them get answers quickly.
The problem is that many teachers (or well-meaning parents) introduce tricks before students understand the underlying mathematics, and that can be to the students’ detriment. Instead of mathematics appearing to be a cohesive whole, students believe that it is a large bag of disconnected procedures to be memorized, remembered for the test, and then promptly forgotten.
My eighth-grade English teacher taught us the following:
You need to learn the rules of grammar, so you can feel comfortable breaking them.
I believe the corollary in mathematics applies here. If you understand the structure and algorithms associated with procedural fluency, then you can then feel comfortable using well-known shortcuts.
The Common Core identifies three pillars of rigor — conceptual understanding, procedural fluency, and application — and asserts that each pillar must be “pursued with equal intensity” (CCSSM Publishers’ Criteria for K-8 and High School). But I’ll take that one step further. I contend that conceptual understanding must happen prior to procedural fluency for learning to be effective. That contention is based on research as well as personal experience, and it’s consistent with the NCTM Procedural Fluency in Mathematics Position Statement, in which the Council states, “Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving,” citing the Common Core State Standards for Mathematics, Principles and Standards for School Mathematics, and Principles to Actions: Ensuring Mathematical Success for All.
A mathematical trick, as far as I’m concerned, is merely the recognition and application of a pattern. A simple example is the 9× facts.
- 1 × 9 = 9
- 2 × 9 = 18
- 3 × 9 = 27
- 4 × 9 = 36
By looking for patterns, students might notice that 9 times a number is equal to 10 times that number minus the number itself; that is, 9n = 10n – n. For instance, 8 × 9 = 80 – 8. This makes sense algebraically, of course, since 10n – n = (10 – 1)n = 9n, but I wouldn’t expect a second grader to be able to work this out symbolically. I might, however, expect a second grader to correctly reason that this should always work. From that, a student would have a way to remember the 9× facts that is based on experience and understanding, not just remembering a rule that was delivered from on high.
Taking this further, recognizing this pattern helps with multiplication problems beyond the 9× facts. Recognizing that 9n = 10n – n can also help a student with the following exercise:
- 9 × 26 = 260 – 26 = 234
As a more sophisticated example, students might be able to generalize a rule for multiplying two numbers in the teens. I’ve seen tricks like this presented as follows, with no explanation:
- Take two numbers from 11 to 19 whose product you’d like to know.
- For example, 14 × 17.
- Add the larger number to the units digit of the smaller number.
- 17 + 4 = 21
- Concatenate a 0 to the result.
- 21(0) = 210
- Multiply the units digits of the two numbers.
- 7 × 4 = 28
- Add the last two steps.
- 210 + 28 = 238
- And there you have it: 17 × 14 = 238.
Why this trick works is rather beautiful.
17 × 14
(10 + 7) × (10 + 4)
10 × 10 + 7 × 10 + 10 × 4 + 7 × 4
100 + 70 + 40 + 28
10 × (17 + 4) + 28
210 + 28
The steps requiring students to add the larger number to the units digit of the smaller number and then concatenate a 0 to the result are represented in bold above as 10 × (17 + 4). Those steps in the rule are just a shortening of the second, third, and fourth lines in the expansion above.
Presenting this trick with an explanation is better than presenting it without explanation, for sure, but better still would be for students to look for patterns from numerous examples and, if possible, generate their own rules. If you presented students with the following list:
- 15 × 16 =240
- 17 × 14 = 238
- 13 × 18 = 234
- 19 × 12 = 228
would they be able to generate the above rule on their own? Probably not. But that’s okay. The rule is not the point. The point is that there is an inherent structure and pattern within numbers, and students might notice some patterns in the factors and products in the four examples above — for instance, the smaller factor decreases from 15 to 14 to 13 to 12 as you progress from one example to the next; the larger factor increases from 16 to 17 to 18 to 19; and the product decreases by 2, then by 4, then by 6.
Something’s going on there, but what?
Ours is not to tell them why; just give them more examples to try!