- Wow! I didn't realize how uncomfortable I'd be putting my math strategies out there. OK, math folks, have at me. The fact is, if a given strategy is counterproductive in some way for the sake of fleeting success that will not endure, I want to know.

Ulp.

I arrived for a mid school-day session several years ago to be greeted by a 6th grader in tears, with his head on his arms. He had only ever been cheerful or resigned before, never so defeated. He was dyslexic, and had made huge strides in reading comprehension and writing accuracy, as well as math, all school year long. Now he had started doing operations with integers in class, and he couldn't make sense of it. He was super embarrassed to be crying, but also determined to figure it out.

In retrospect, his plight reminds me of my startled realization while starting to learn Spanish, that the accent marks just didn't stick in my brain. I would write a sentence and know there was an accent mark, but I couldn't even decide which word it went over, let alone which letter, and I couldn't hear anything in my pronunciation that gave me a clue. If there had somehow been an equals sign after my "la sopa es para el capitan" sentence, I would surely have gotten it wrong.

I told him it made sense that it was confusing, because those plus signs and minus signs sometimes meant an operation and sometimes changed the value of a digit. We started reading these integer operations problems as sentences, one symbol at a time, and it got him over the hump.

Here's what we said and did (equations were on a white board):

First, I asked him if he knew what each of the pieces was. That is, even if looking at the whole equation was confusing to him, if he looked at the symbols one at a time, could he identify each symbol in order, separately, or was that crazy?

He thought he could do that.

We agreed to read these problems like a sentence, look at one symbol at a time, from left to right, and do what each symbol says to do, on a number line, before looking at the next. Here's what each symbol meant:

- Always start at zero, facing right.
- Digits/numbers tell how far to go in the direction you are facing.
- Plus means face right.
- Minus means face left.
- A negative sign means switch directions

- (On the desk, I often used two pieces of paper to mask all but the part/symbol he needed to look at; on the board I often hid the extraneous info with my hands)

----

Let's start with

3 + 4 = ...What?

*Seven.*

Wait, so that's easy?

*Duh. Yes.*

So are these positive or negative numbers?

*Positive.*

OK, so we just added

+3 and +4

and got +7,

right?

Like this?

But we didn't write those positive signs at first because a number without a sign is positive anyway, right?

*I guess so.*

You mean you're not sure if any of the numbers are negative when I write

2 + 5 = 7

or

3 + 6 = 9 ?

*No, they are all positive.*

Okay, so let's be annoying and make some of those numbers negative in that first one.

It was

3 + 4 = 7

so

-3 + 4 =

3 + -4 =

-3 + -4 =

Oh, and we can subtract too, just to make it worse. Yay!

-3 - 4 =

3 - -4 =

-3 - -4 =

Okay, which one should we start with.

*I don't care.*

Me neither! We have so much in common!

OK, let's take

- -3 + 4

Wait! Where do we start?

*Negative three?*

We could but that could make it confusing later. Let's always start

*before*the first symbol. What's before the first symbol?

*"Before the first symbol?"*

Yeah, the first symbol is that negative sign, right?

*I guess.*

I mean if we read from left to right, the first thing is all the way left, right? I mean left, correct?

*OK.*

So what's on the far left: is it the negative sign or the equals sign? Can you tell or should I show you?

*The negative sign.*

I agree. So if I say "before" it I mean to the left of it, yeah? Because we read left to right. So what's

*before*the negative sign?

*...nothing?*

I agree! So where do we have 'nothing' on our number line?

*Zero?*

Sure! Zero. So let's always start at zero, and if there's nothing to the left I guess we can assume there will be "something" next to the nothing, and something is more than nothing, right? So which way do we go on the number line to mean "more."

*That way?*(pointing)

Yep, to the right. More is to the right; less is to the left on our number line. We count

*up*to the right,

*down*to the left. So far so good?

*I guess.*

So let's always start at zero, facing right, before we read the first symbol.

*OK.*

So what's the first symbol we see?

*Negative sign?*(many would say 'minus')

What does that tell us to do?

*Switch directions.*

Right, so now we are at zero facing left. What's next?

*3*

What does that tell us?

*Move 3.*

Ok so where do we end up.

*3. I mean negative 3.*

Okay now we're at negative 3. What's next?

*Plus.*

What does that tell us?

*Face right.*

Nice! Yes, then what.

*Go 4.*

Okay, so, do you know what -3 + 4 is?

*Positive one.*

Which we usually call...

*One.*

Nice!

-----------

Lather, rinse, repeat...

_______

For that student, on that day, this made it all come together and he went from utter bewilderment to doing a dozen or so correctly in rapid succession, without a number line. It was such a joy to have him leave so confident after arriving so miserable

I also worried that it wouldn't translate easily to multiplication and division (there's a nice discussion of this here), because though it's easy to apply repeated addition to a positive times a negative, multiplying a negative times a positive is more ambiguous, unless you just think of negative groups as sort of negativizing the total. (Ok math people. Be gentle.)

For practical purposes, a student like this one finds (and this one did find) great relief in my encouraging him to just

*ignore*the signs and multiply or divide the numbers, then use his knowledge of the rules for combining signs (different signs = negative; like signs = positive) to put the correct sign on the answer.

However, this clashes with my both/and stance regarding math teaching for understanding. I don't want to deprive a student of deeper understanding by handing them prefab tools for arriving at solutions--which may also deprive them of the core understandings necessary to reason in more complex mathematical situations. Yet, I also don't want a student's math skills or cognitive style to make the insistence on teaching for understanding

*first*a roadblock that they perceive to be standing in the way of their practical need to solve problems, get credit, and feel successful. Creating both/and learning solutions is a delicate, perpetual, highly-individual dance that only seems to work when the student is a key player guiding us to collaboratively build understanding and calculation success, knowing both are vitally important.

_____

Bonus!:

I just remembered, it was with the same student that I first explained pi with string and scissors. And I ran out the door and made a video:

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