This, That, and the Other Thing

Scene from It Figures

From It Figures
Girls' Angle Bulletin
Vol. 2, No. 1.

Good communication. That’s what this blog post is about. That and how math class is a great place to improve communication skills.

It happened a few meets ago at Girls’ Angle. Two members were working together on the following math problem:

How many solutions in positive odd integers are there to the equation

n_1 + n_2 + n_3 + \dots + n_{10} = 14?

The two work together very well and it did not take them too long to figure out that there were 10 solutions in which one variable was equal to 5 while the rest were all equal to 1.

Next, they observed that there couldn’t be more than one 5 nor any solutions with numbers greater than 5. So they set their sights on counting the number of solutions that involved only 1’s and 3’s. They then saw that there had to be exactly two 3’s with all other variables set to 1.

They set upon the task of counting the number of solutions with two 3’s. (If you’re familiar with basic counting principles such as those described in Shravas Rao’s Counting in Volume 5, Number 3 of the Girls’ Angle Bulletin, then you will recognize that the answer is 10 choose 2, but the two members who were working on this problem are both in sixth grade and hadn’t yet heard about the choose function.) They were working together on the same sheet of scratch paper. Somehow, they decided to use X’s to mark which variables would be equal to 3 and dashes to indicate variables equal to 1. They began with:

- – – – – – – – X X,

meaning that the first 8 variables would be set to 1 while the last 2 would be set to 3.

Throughout, the two were constantly talking to each other. They were in their own private world of communication referring to things with the words “this” and “that” and pointing at the scratch paper frequently. As they spoke, both began to see how things would go:

“Yeah…you just go like that…and it’ll go like this…”

“And then it’ll be like this…”

“Right…that’s it…that’s it!”

“Let’s do that!”

If you’re confused, don’t worry, they were too, in a way, though they didn’t realize it yet.

One of them picked up a pencil and, while saying, “So the next one will be,” wrote:

- – – – – – – X X -

At which point, the other said, “Wait…why’d you write that? We already counted that.” The first put on a confused look, “No we didn’t…”

And at this sign of disagreement, they both thought that the other person made a mistake and tried to clear up the other’s confusion. But instead of clarifying things, things grew even more confused. Why? Both were sure that they were correct, so each thought, “she must be wrong!” But, in fact, they were both right!

How can it be that they were both right and yet ended up disagreeing?

It’s because they were seeing two different but valid ways of counting the same thing and had used too many pronouns to realize it!

Member A was thinking:

Member B was thinking:

Count the ways by grouping the solutions according to the position of the rightmost X.  Within each group, fix the position of the rightmost X and slide the leftmost X from the first position to the position just to the left of the rightmost X.

Count the ways by grouping the solutions according to the distance that the two X’s are separated. Within each group fix the distance between the two X’s and slide the two X’s in tandem from one extreme to the other.

So at the juncture where communication broke down, Member A was thinking, “I’ve finished counting

- – – – – – – – X X
– – – – – – – X – X
– – – – – – X – – X
– – – – – X – – – X
– – – – X – – – – X
– – – X – – – – – X
– – X – – – – – – X
– X – – – – – – – X
X – – – – – – – – X

(i.e. where the rightmost X is fixed in the rightmost position) and so the ‘next one to count’ is:

- – – – – – – X X -.”

On the other hand, Member B was thinking, “I’ve finished counting

- – – – – – – – X X
– – – – – – – X X -
– – – – – – X X – -
– – – – – X X – – -
– – – – X X – – – -
– – – X X – – – – -
– – X X – – – – – -
– X X – – – – – – -
X X – – – – – – – -

(i.e. where there is no separation between the two X’s) so the ‘next one to count’ is:

- – – – – – – X – X.”

When Member B saw “- – – – – – – X X -” instead of her expected “- – – – – – – X – X”, she balked.

Because they used so many pronouns, they didn’t realized that their disagreement merely resulted from seeing different ways of counting the same set. And so, despite both ways being perfectly valid, they ended up backtracking and trying to correct the other’s method when, in reality, no corrections were necessary!

How often have you seen people arguing simply as the result of miscommunication like this? In this case, the miscommunication resulted from overuse of pronouns. (Sometimes people who want to be divisive will take advantage of the ambiguity in words to create the appearance of disagreement when there isn’t really a substantive difference of opinion. Studying math is a good way to develop immunity to such tactics, but that’s another story.)

Here, in a math class, then, was a grand opportunity to learn to communicate better. Working in small groups in math class generally leads to many opportunities for developing communication skills. Such opportunities are just as important, and possibly even more important, than the mathematical content. This is also partly why it is good to write out neat solutions to your math problems. Writing forces you to express your thoughts clearly.

One last thing: the two members work together really well and their disagreement was a friendly one…great work, you two! We don’t mind arguments like that at the club at all. They’re a great opportunity to grow. Just please stop using so many pronouns!

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