## Posts tagged ‘algorithm’

### Our Library’s Summer Math Contest

Every summer, our local library runs a contest called *The Great Big Brain Game*. Young patrons who solve all of the weekly puzzles receive a prize. The second puzzle for Summer 2017 looked like a typical math competition problem:

Last weekend, the weather was perfect, so you decided to go to Cherry Hill Park. When you got there, you saw that half of Falls Church was at the park, too! In addition to all the people on the playground, there were a total of 13 kids riding bicycles and tricycles. If the total number of wheels was 30, how many tricycles were there?

First, some comments about the problem.

- I dislike using “you” in math problems. I believe it’s a turn-off to students who can’t see themselves in the situation described. There are enough reasons that kids don’t like math. Why give them another reason to shut down by telling them that they went somewhere they didn’t want to go or that they did something they didn’t want to do?
- Word problems are not real-world just because they use a local context, and this one is no exception. This problem attempts to show an application for a system of linear equations, but true real-world problems don’t have all the information neatly packaged like this.
- Wouldn’t the person posing this problem already have access to the information they seek? That is, if she counted the number of kids riding bikes and the total number of wheels, couldn’t she have just counted the number of bicycles and tricycles instead? It has always struck me as strange when the (implied) narrator of a math problem wants you to figure out something they already know.

All that said, this was meant to be a fun puzzle for a summer contest, and I don’t mean to scold the library. I don’t know that I’d use this puzzle in a classroom — at least, not presented exactly like this — but I love that kids in my town have an opportunity to do some math in June, July, and August.

Now, I’ll offer some comments on the solution. In particular, the solution provided by the library was different than the method used by one of my sons. Here’s what the library did:

Imagine that all 13 kids were on bicycles with 2 wheels. That would be a total of 26 wheels. But since 30 wheels are needed, there are 4 extra wheels. If you add each of those extra wheels to a bicycle, that’ll create 4 tricycles, leaving 9 bicycles. So, there must have been 4 tricycles at Cherry Hill Park.

And here’s what my son did:

If you can’t see what he wrote, he created a system of two equations and then solved it:

2a + 3b = 30

a + b = 13a + 2b = 17

13 – b + 2b = 17

b = 42a + 12 = 30

2a = 18

a = 9

That’s all well and good. In fact, it’s perfect if you want to assess my son’s ability to translate a problem and solve a system of equations. But I have to admit, I was a little disappointed. What bums me out is that he went straight to a symbolic algorithm instead of considering alternatives.

I think I know the reason for this. This past year, my son was in a pull-out math program, in which he studied math with someone other than his regular classroom teacher. In this special class, the teacher focused on preparing him to take Algebra II in sixth grade when he enters middle school. Consequently, students in the pull-out class spent the past year learning basic algebra. My fear is that they focused almost exclusively on symbolic manipulation and, as my former boss liked to say, “Algebra teachers are too symbol-minded.”

A key trait of effective problem solvers is flexibility. That type of flexibility comes from solving many problems and filling your toolbox with a variety of strategies. My worry — and this isn’t just a concern for my son, but for every math student in the country — is that students learn algorithms at the expense of more useful problem-solving heuristics. What happens when my son is presented with a problem that can’t be translated into a system of linear equations? Will he know what to do when he doesn’t know what to do?

The previous pull-out teacher said that when she presented my sons with problems that they didn’t know how to solve, their eyes would light up. They liked the challenge of doing something they hadn’t done before. I’m hopeful that this enthusiasm isn’t lost as they proceed to higher levels of mathematics.

### AWOKK, Day 4: My Puzzles

It’s Day 4 of the MJ4MF **A Week of KenKen** series, and I’m very excited about today’s offering. But first, in case you missed the fun we’ve had previously…

- Day 1: Introduction
- Day 2: The KENtathlon
- Day 3: KenKen Times

The puzzles that appear at KenKen.com and within the KenKen app are automatically generated by something that those folks call the **KENerator**. (Clever, no?) Likewise, the puzzles that appear in the MathDoku Pro app are also generated by a computer algorithm. The benefit is that both of these sources will provide a nearly infinite supply of puzzles. The downside is that computers don’t think as well as humans, so the puzzles range from mundane to amusing, and, in the words of Thomas Snyder, are “too easy and too computer-generated.” Rarely do they fall into the category of truly interesting.

That’s why Thomas Snyder attempted to outdo *The* *New York Times* KenKen puzzle back in 2009, when he presented a new KenKen puzzle every day. His themed puzzles were meant so show “what the puzzle should be” and how to make them interesting. Of the 20+ puzzles he presented, this is my favorite, which he created for March 3 (3/3).

You can see the large numeral “3” formed by the cage along the right edge of the puzzle. Further, every cage includes at least one digit 3 as part of the target number.

I also like his “Perfect Ten” puzzle, where every cage has 10 as the target number.

Like Snyder, I agree that KenKen puzzles are generally more interesting — and, usually, more difficult — when they’re created by a human instead of a computer. The following are several themed puzzles that I’ve created.

This puzzle is relatively easy, but I like that it uses only 8’s and 4’s. In honor of the small town of Eighty-Four, PA — which apocryphally is said to be named after the town’s mile marker on the Baltimore & Ohio Railroad — I call this the “B&O” puzzle.

You may know that 1! = 1, 2! = 2, 3! = 6, …, 6! = 720, and that these numbers are known as *factorials*. Consequently, I call this the “Factorial” puzzle, since the target numbers are the first six factorials.

Of all the puzzles that I’ve created, my all-time favorite is a special 8 × 8 puzzle that I created in honor of the current year. But if you want to see it, you’ll need to check back on Sunday, September 25, for AWOKK, Day 7.

### 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!*