## Posts tagged ‘examples’

### Every (Math) Trick in the Book

Linda Gojak has long been a proponent of doing away with tricks in math class (see also *Making Mathematical Connections*, October 2013).

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
- etc.

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, 9*n* = 10*n* – *n*. For instance, 8 × 9 = 80 – 8. This makes sense algebraically, of course, since 10*n* – *n* = (10 – 1)*n* = 9*n*, 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 9*n* = 10*n* – *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

238

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!*

### Making Progress, Arithmetically

Today is 11/12/13, a rather pleasant-sounding date because the numbers form an arithmetic sequence, albeit a trivial one. It’s not the only date in 2013 for which the month, date, and year form an arithmetic sequence. How many others are there?

Several nights ago, my sons asked if they could do bedtime math, but Eli asked if we could do problems other than those on the Bedtime Math website, because “they’re a little too easy.” So instead, I navigated to the MathCounts website and opened the *2013-14 MathCounts School Handbook*. We scrolled to page 9 and attacked the problems in Warm-Up 1.

Things were going well until we reached Problem 8 in the set, which read:

The angles of a triangle form an arithmetic progression, and the smallest angle is 42°. What is the degree measure of the largest angle of the triangle?

Eli asked, “Daddy, what’s an *arithmetic progression*?” pronouncing *arithmetic* as “uh-rith-ma-tick” instead of “air-ith-met-ick.”

I could have just answered Eli’s question by stating the definition:

An

arithmetic progressionis a sequence of numbers for which there is a common difference between terms.

But such a definition isn’t very helpful, since I’m not sure that either Eli or Alex know what *sequence*, *common difference*, or *term* mean. It would have led to even more questions.

Plus, I’ve always believed that kids understand (and retain) more when they discover things on their own. Call it “discovery learning” or “inquiry-based instruction” or any of myriad other names from educational jargon, it just means that giving kids the answer is not the most effective way for them to learn.

So instead, I said, “Let me give you some examples.” And then I wrote:

1, 2, 3

3, 5, 7

Alex said, “Oh, I get it! An arithmetic progression is a nice pattern of numbers.”

So I said, “Well, let me give you some patterns that *aren’t* arithmetic progressions.” And then I wrote:

2, 4, 8

“That’s a nice pattern, isn’t it?” I asked. “But it’s not an arithmetic progression.”

“Oh,” said Alex. He thought for a second, then revised. “You have to add the same amount every time.”

And there you have it. Three examples, and my sons were able to define *arithmetic progression*. It’s not as sophisticated as “a common difference between terms,” but “add the same amount every time” is a sufficient definition for a six-year-old.

So they generated an arithmetic progression with 42 as the smallest term:

42, 45, 48

Eli said, “I don’t fink vat’s enough.” When asked to explain, he said he *thought* that the angles in a triangle add up to 180 degrees.

“Are you sure?” I asked. He wasn’t. Nor was Alex. So I asked if they could convince themselves that the sum of the angles is 180°.

Alex said, “Well, the angles in a square add up to 360°, and you could cut it in half.” So we did:

They then reasoned that each triangle would have a sum of 180°. “But maybe that only works for a square,” I said. “How do you know it’ll work for other shapes?”

Eli suggested that we could cut a rectangle in half, too:

And again they concluded that each triangle would have a sum of 180°.

Understand, this is NOT a proof of the triangle sum formula. When they get to high school and need to demonstrate the rigor that the Common Core State Standards are demanding, well, then we’ll worry about formal proof. But for now, I’m okay with six-year-olds who can demonstrate that kind of reasoning.

They then took another guess, but this time they chose three numbers that added to 180:

42, 59, 79

Realizing that the difference between the first and second terms was 17 and the difference between the second and third terms was 20, they revised:

42, 60, 78

They concluded that the largest angle had a measure of 78°. And all was right with the world.

So why am I telling you all this?

Partially, it’s because I’m a proud father.

But more importantly, it’s because this vignette demonstrates that teaching is an art, and successful teaching doesn’t happen by accident. It’s not easy, as many people believe. What’s easy is the perpetuation of **bad teaching**, a la Charlie Brown’s teacher, or textbooks that simply present information with the belief that students will absorb it by osmosis. **Good teaching**, however, requires content knowledge and pedagogical knowledge, and it demands teachers who can handle unexpected classroom twists and turns and have the ability to adjust on the fly.

A student is convinced that a right triangle isn’t a right triangle because the right angle isn’t in the lower left corner? You better find an effective way to clarify that misconception. (Hint: Don’t use a traditional textbook where every picture of a right triangle shows the right angle in the lower left corner.)

Students think that 16/64 = 1/4 because you can “cancel the 6’s”? Uh-oh. Better find some counterexamples pronto, and help them understand *why* 16/64 can be reduced to 1/4.

Your students don’t know the definition of *arithmetic progression*? Then you better figure out a way to help them define it, and just writing *your* definition on the chalkboard isn’t gonna cut it.

Want to see what good teaching looks like? See Dan Meyer, or Christopher Danielson, or Fawn Nguyen. Or many, many others who don’t blog about it but inspire students every day.

Someday soon, I hope to add my project at Discovery Education to the list of examples of good teaching. Until then, I’ll just keep blathering about my sons.