**Posted by Sandy Merz on Monday, 04/07/2014**

I first saw it on Facebook.

A parent's letter to a teacher suggesting that the method used to teach subtraction was unnecessarily complicated and that an alternative approach was readily available.

Next, I saw State Superintendent of Public Instruction John Huppenthal answering questions about education in Arizona. A whiteboard with lots of Xs, Os, and arrows was propped against a chair off to the side. I thought it was a football play. But then a mother held it up explained it was how her child's teacher explained simple computations.

Then I read Alec Torres' piece in National Review showing the Ten Dumbest Common Core Problems that his readers had sent in.

The common thread running through these cases is that parents and children are frustrated and confused by some of the instruction and materials used to teach math in the Common Core Era.

Common Core defenders argue that learning math only begins with memorizing facts, which they make sure to call algorithms. To truly own the material, a student must comprehend the concepts behind the facts, sorry, the algorithms. To be ready for college and careers a student must also be agile in applying the math in realistic contexts.

Most parents agree that a math curriculum should balance facts, concepts, and applications.

As a cautious supporter of the Common Core, I'd say that the validity of the complaints in these stories merits investigation. Good materials and methods should be defended and emulated. Bad materials and methods should be identified and improved.

My guess is that many materials and methods fail because they try to combine facts, concepts, and applications. But facts, concepts, and applications don't blend well.

That's because calculating, conceiving, and applying math represent end members in a tri-polar system. In a polarity you don't look for the compromise between the extremes. Instead, you recognize that you must spend some time at each extreme and learn when to move from one to another. My current Stories from School Arizona post Ed Solutions Are a Breath Away! covers polarity management in some detail. A great resource is Polarity Management: Identifying and Managing Unsolvable Problems by Barry Johnson.

Here's an example of what I mean by moving from fact to concept to application. Back in around seventh grade you probably learned that subtracting a negative number is the same thing as adding a positive number. So, 12 - - 6 becomes 12 + 6 and the answer is 18. The rule may have made sense. But even if it didn't, you still learned the rule and practiced on worksheets. I haven't met many adults who ever thought about why the rule works or how it's ever applied.

When I ask why 12 - - 6 = 12 + 6, they say that's how you get the right answer. When I ask how it works, they say it's the rule. Round and round we go.

To get to the concept they don't need another algorithm to memorize (the Xs, Os, and arrows). They need to remember being five years old.

**Me: **Read this problem out loud: 8 - - 7

**Them:** Eight minus negative seven.

**Me: **When you were five what did you say instead of minus?

**Them:** Take away!

**Me: **So if you **take away negativity**, what happens?

**Them: **You're going to get more!

**Me:** How do you get more of something?

**Them: **You add to it!

**Me:** So do you get why subtracting a negative is the same as adding a positive?

**Them:** Kind of!!

So when does anyone really subtract negative numbers?

Well, did the temperature this winter fall below 0 where you live? Were you happy when the sun came out and took all those negative degrees away?

Paying off a debt is really subtracting negative money. Decreasing withholdings increases my check.

I find in algebra that my students learn the facts by drilling, learn the concepts through reasoned discussions of what the facts imply, and learn practical applications by inferring the concepts present in a quantitative situation.

Take a crack at the following puzzler that I give my students. See how your focus jumps between context, concepts, and calculation as needed. But there is no time when you do a little of each all at once. That's how you manage a tri-polar system.

*"On January 1, Mister started buying a cup of coffee every day for $1.69. He pays with two dollar bills and puts the change in in the change tray in his car. He repeats this until there is enough change to pay for the coffee with all coins. On that day he pays with the coins and puts the change from that in a little pouch. He follows this routine until there is enough change in the pouch to pay for the coffee. He leaves the change from that on the counter. He never misses a day. How much money is in the tray and the pouch on December 31?" *

I can't wait to see your answers.

## 4 Comments

## None?

Sandy,

You got me. I love a good riddle.

I

thinkthe following is right...It would take 6 days for mister to have enough change in the car tray to pay for coffee and he would have 0.17 left to put in the pouch.

Then it would take 10 days for mister to have enough change in the pouch to pay for the coffee. He would have 0.01 to put on the counter.

However, once he has enough change in the tray to pay for the coffee, the remaining change goes into the pouch. Once he has enough change in the pouch to pay for the coffee, the remaining change goes on the counter. Therefore, there is no money in the car tray or pouch on December 31.

Did I get it?

Also, thanks for your great explanation of what it means to subtract a negative. I've given my students some practical explanations but yours is more detailed and will be a perfect review.

:)

Lalla

## Right so far, but you're not quite finished

Hi Lalla,

Thanks for playing. You're right that there are two cycles - the coin tray cycle and the pouch cycle. On the day he pays with money from the pouch and leaves the penny behind he does have no money in the pouch or the tray - but that's not the last day of the year - he's just back where he was on 1/1 and starts over and repeats the whole process from the beginning. That goes on a few times during the year and on 12/31 he's in the middle of a cycle and has some money in the tray and/or pounch. Want to try again?

## Aren't there two other variables?

Is it leap year?

On December 31st, has Mister bough his daily caffeine yet when the money is counted?

## What other variables?

It's not a leap year and he counts the money after he buys the coffee and puts any change wherever it belongs.

I'm not sure what other variables you mean?