*Out in Left Field proudly presents the ninth in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names have been changed to protect privacy.*

**Not Quite Like Old Times**

On the Monday that Tina returned, I was preparing for first period. I was always nervous before the first period bell rang, like an actor hearing the audience as he waits for the curtain to rise and the play to begin. I was particularly nervous today because there was a possibility that Tina might not show.

I heard the usual sounds outside my classroom—kids playing handball against the giant wall of the gym building that stood across from our module. As the school buses arrived, more kids poured into the courtyard. The lonely sound of handball was replaced with what sounded like 10,000 students milling around. I was relieved finally to hear the familiar clacking of high heels on the walkway to the classroom and the sound of the key in the door.

“Welcome back,” I said.

“Thank you,” Tina said, and headed for her desk. She saw the group sympathy card from the class and one from me and put both in her purse.

A math teacher came into the room, hugged Tina and gave her a small gift. The two chatted for a minute and then the teacher left. In the remaining few minutes, Tina tried to catch up on what was going on. “What are we doing in pre-algebra?” She looked tired and tense.

“Just starting Pythagorean Theorem,” I said.

“And algebra?”

“Multiplying binomials using generic rectangles.”

She suddenly brightened and said “Don’t you just love generic rectangles? It’s such a great way to teach how to multiply binomials.” She went on about how much she liked the approach, and how CPM would connect it to factoring later. I was glad to see her perk up and talk excitedly about CPM, even if I didn't care for it.

Generic rectangles are a way to represent the multiplication of two binomials as the area of a rectangle. So x + 5 multiplied by y + 10, can be represented as a rectangle with those binomials as the lengths of the sides:

The students had experience computing areas within rectangles; now they could represent rectangles as above. They compute the areas of the four smaller rectangles, and add up the results to get xy + 5y + 10x + 50. The challenge I faced was not so much in teaching the students how to do it, but how to keep them from calculating (x+5)(y+10) using the procedure known as FOIL, which they had learned in 7th grade pre-algebra.

In fact, later that day in algebra class the issue of FOIL came up in a rather surprising way. I had been in charge of the class. I became concerned when I saw that Tina suddenly looked very tired; she sat down and put her head on the desk. I watched her out of the corner of my eye while I continued working with the students on the generic rectangle problems. One girl, Samantha said “We learned how to do this last year using this method. Why can’t we use it?” She held up her notebook. I began to answer. “Yes, that’s FOIL, but…”

Tina suddenly stood up and shouted: “No! Don’t let them do it!” She then addressed the class: “You are NOT to use the FOIL method yet. I know some of you know it, but you have to understand what it is you’re doing first.”

She was fiercely loyal to the philosophy of the CPM authors who believed that the connection between binomial multiplication and area representation provided the understanding that students need. The authors were also fiercely loyal to the theory that teaching a procedure will frequently skimp on understanding. Well that’s fine, but just use it to introduce the topic, then teach them how it’s done algebraically. Do you really have to spend two weeks working with generic rectangles to instill “connections”, “understanding” and other educational trendy ideas in order to teach them something they’ve already learned in pre-algebra?

In fact, the pre-algebra text had a pretty good explanation for how to multiply binomials prior to bringing in FOIL as a shortcut. They start with the distributive property: a(b + 3) = ab + 3a. Then if "a" is replaced by a binomial such as b + 4, a(b+3) becomes (b + 4)(b + 3). Substituting in the original equation, you get (b + 4)b + 3(b+4), or b

^{2}+ 4b +3b + 12, which is b

^{2}+ 7b + 12.

That’s another thing that made me wonder. How much of this discovery and connection that CPM is bragging about is because of pre-algebra courses that students have had?

All that aside, Tina was happy to see her students again, and they were happy to see her which I was glad to see. She continued on as if nothing had happened. Especially in the pre-algebra class. In that class, I had explained the Pythagorean Theorem, but I was a bit too slow for her, and forgot to give the formula, so she jumped in, just like the old days.

“Let me interrupt and point something out about the Pythagorean Theorem,” she said, “because it wasn’t clear.” These last two words she said looking at me out of the sides of her eyes. I knew she wasn’t pleased. She summarized it as “It’s a

^{2}+ b

^{2}= c

^{2}. Can you say that?” The class repeated it. She then showed them the trick for how to identify the hypotenuse which is the “c” side in the equation: pretend the triangle is a bow, and the arrow goes where the little right angle sign is. The arrow is pointing at the third side: the hypotenuse. Funny how she could be so explicit in the pre-algebra class and yet buy in to CPM’s philosophy in the algebra class, I remember thinking.

But it was good to see her again, and I told her so at the end of the day. “Good to see you too, Huck,” she said. Something didn’t feel right, though. It felt like we both were trying to get back to the way things were before she left—and that we both knew that would never happen.

## 8 comments:

*Two weeks* on generic rectangles? Please tell me I read that wrong or it was a typo. That's 10 hours of class time! Don't they only get 180 hours of class time for math a year? (Give or take.)

Wow.

I've been following Huck's adventures but I think only chimed in once before. I agree completely with Terri's (and Huck's) comments about the length of time spent on this.

We (at WISE Math) are challenged by the ministry folks here that say we're asking for a perfectly good pedagogical method to be stripped out. We counter that nothing could be further from the truth -- we are only asking that pedagogy not get in the way of content. And the generic rectangles approach is a perfectly good example of exactly this.

Huck is exactly right; this lovely visual metaphor serves a nice purpose -- for a few minutes, while the content is being explained. But students must move on and must not be given the impression that this is the end-goal. Dwelling on it does exactly that. And worse, here those sit in our Framework Outcomes as if that is exactly what they were. Well ... let's be honest -- that, effectively, is what they are then.

But it is worse than just time wasted on a lovely metaphor. It is that the metaphor itself is too flawed and limited to cover the subject matter in proper generality.

The subject, after all, is POLYNOMIALS. There are many polynomials, and only a tiny portion of them are linear and have only nonnegative coefficients. Understandably we must start there, and the metaphor works fine in this setting. But it is essential to move off that base as soon as mastery is attained (and REALLY, how long does it take to master multiplication of linear binomials)?

Multiplying linear binomials gives quadratic polynomials. So it is ESSENTIAL that students now understand how to handle quadratics. What about multiplication? Represent (x^2 + 5)(x+10). This is stilll a product of binomials, but it should be clear that the metaphor is very problematic now. In fact, I would avoid it altogether. Even if you insist on this, say moving to 3D visualization with volumes, what about (x^3+5)(x^2+10)?

So, you insist, we'll cross the "higher degree problem" when we come to it. But obviously the moment we're identifying linear polynomials with their graphical representation, we have negative slopes and intercepts, so we're already handling cases that would lead us to products like (x-5)(-x+10). Good luck with the generic rectangle representation for that.

But what is perhaps the most problematic thing here (connected to Huck's complaint that it is reverse development for students already fluent with FOIL) is that rectangles are already used earlier in the curriculum (3-5 years earlier depending on the curriculum) for numeric arithmetic, and the exercises are almost identical. The problem with that is that we see no progression in skill development. In terms of mastery, fluency and broadening of procedural functionality this represents an almost flat line. To those of us who get these kids at a university level and expects them to flawlessly execute moderately interesting algebra the very idea that schools would dicker around with grade-3 level skills in junior high school is frustrating and frightening. It also explains a lot.

Huck responds to R. Craigen:

You're right that negative numbers would be a problem. And when I looked at the problems in the textbook prior to doing the lesson, I saw that a few problems did contain negative values like (x-5)(x+6). Students were focused only on multiplying two values to get the area, however; not what it represented geometrically. The fact that one value was negative and yielded an area of -30, or -5x, didn't seem to bother them. They just wanted to get the multiplication right.

And as far as I was concerned, the less said the better. There were no problems with x^2 as one of the dimensions so that question never came up either. The generic rectangles were used purely as a mnemonic device.

She called it the Pythagorean THEOREM? Conjecture, maybe with enough (two weeks?) of motivating exercises. Otherwise isn't it the Pythagorean AXIOM?

Stupidity has so many ramifications.

I see exactly the same problem in earlier grades with the lattice method of multiplication. Our school uses Everyday Math, and they have spent dozens of class periods over the course of two years wasting time on the lattice method. It might be a nice parlor trick, but it isn't robust enough to bother with.

I think there's a mistake in the diagram. Shouldn't the area of the second subrectangle be 10x, not 10?

Also, Katharine, you spelled "ninth" wrong in your italicized intro.

Also, shouldn't the area of the first subrectangle be xy, not x?

Love Huck's question: "That’s another thing that made me wonder. How much of this discovery and connection that CPM is bragging about is because of pre-algebra courses that students have had?"

There is an oblivion among constructivists that "creative" connections must be based on bsic foundational knowledge for there to be any "connection."

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