Math is the art of stating the obvious

Simple questions go a long way towards understanding

Recently I had the following dialogue with a sixth-grader during a tutoring session:

The little girl says:
I have a question I wanted to ask you. We were working on this problem earlier today in class, and I am confused, I don’t know how to do it. It’s 159 – 26.6 because, what do you do? This number 159 has no decimal point. Is it O.K. if you..? Can you put a decimal point, and write an imaginary zero here, like 159.0 even though the zero is not really there?

(I thought it was really interesting she referred to that zero after the decimal point calling it an “imaginary zero.” She even drew it with a series of little dashes, instead of a solid line)

I say:
Yes, that is correct, you can certainly do that. 

She continues:
See, I know how to do 159 – 26 because you just do it, and it is 133 but with 159.0 – 26.6 it’s not going to be the same, or is it?

I say:
No, you are right, it’s not going to be the same. Why not?

She hesitates, and says:
Well, you cannot do 159 – 26 = 133, and then for 159.0 – 26.6 just put the .6 after the result, and say that it is 133.6 because that is not right, or is it?

(She was looking more and more confused as she moved closer to facing her idea that she did not know how to do that operation)

I say:
Exactly, that’s not right. 

She says:
But then, how much is it?

I say:
Well, let’s see, you got 159 – 26 = 133, right? 

She says:
Yes.

I say:
And, what is 159.0 – 26.6?

She says:
I don’t know!

(While saying this she gave me a look like “I just asked you the exact same question!”)

I say:
It’s going to be some kind of number, isn’t it?

She says:
Yeah..

(The look she gave me in that moment I can only translate as “What are you, an idiot? Of course it’s going to be a number. What else could it be?” She is very polite, and she would never say out loud any of the sort but I could just see it in her eyes)

I continue:
How are 26 and 26.6?

She ventures an answer, not sure about what I am asking:
Are they positive numbers?

I say:
Yes, they are positive, definitely, and, which one is bigger, 26 or 26.6?

She says:
26.6 is bigger than 26

(Then the look on her eyes seemed like “I cannot believe you are asking me such simple questions! Is there a point to this? Do you think I don’t know that?”)

I continue:
Now, which one is bigger between 26.6 and 27?

She very patiently responds:
27 is bigger.

I say:
O.K., can you please do me a favor, and do these two subtractions? Just write them all out, please. 

While saying the above I write 159 – 26 and 159 – 27

She quickly writes down both answers: 133, and 132.

I say:
Thank you. Now, since 26.6 is greater than 26, and smaller than 27, would it make sense to say that 159.0 – 26.6 it’s going to be some number in between 132 and 133?

She says:
Yes.

(Then it was apparent she was deeply focused, thinking about that order relationship between the numbers, and the subtraction operation)

I say:
So, it’s going to be 132 point something, right? But point what?

(At that very moment something clicked in her mind)

She quickly scribbles the right numbers on the paper and says:
Wait! Wait! I know how to do this! Yes! It’s going to be 132.4. I got it!

I say:
Excellent! That’s perfect!

There are a few points I want to emphasize in the above dialogue:

1) In my math tutoring, the most relevant factor is not the math itself but how the student is feeling about the math. I did not even ask her how the teacher explained the problem in class. The most relevant fact right then was her confusion, and her confusion was coming from a lack of confidence of some sort. For whatever reason, the teacher’s explanation had not been enough for her to acquire a firm trust in the procedure described by the teacher, or in her own ability to follow all the steps. It was mostly a matter of her stopping herself, holding herself back.

2) It is very likely that, at some point during class, when the teacher first started explaining the problem, she originally had the idea of “just adding the .6” from 26.6, after the 133 from 159 – 26, to get 133.6 (the wrong result). As her initial idea turned out to be wrong, she might have felt discouraged, frustrated, and confused. These feelings may have led her to reject the validity of the procedure proposed by the teacher (“the right way” to do it), and to forget it, blocking it from her memory. It is worth noting that coming up with plenty of wrong ideas is extremely common in math, for everybody, even for professional mathematicians. There is nothing wrong with wrong ideas in math; you just have to keep working until you find an idea that works.

3) When she acknowledged that the possible solution she was thinking about was wrong, and I confirmed that it was wrong, she directly asked me for the solution, and I purposefully avoided giving her the answer. This conscious avoidance on the tutor’s part is absolutely crucial for the student’s self-esteem, and for her future success. This is very important. The best I can hope for when working with my students is to instill in them a strong conviction that they can figure out math problems on their own, by themselves, even when they don’t know what to do.

4) I not only repeated the question back to her but I phrased it in such a way that allowed me, when she said: “I don’t know,” to produce a bafflingly obvious, non-answer answer. That was when I said: “It’s going to be some kind of number, isn’t it?” Getting to such level of obviousness has a few key advantages. To start with, she immediately felt smarter than me. This effect is an invaluable asset in math tutoring, making the student feel smarter than the tutor. Deployed at the right time during the discovery process, this effect can create wonders of understanding.

5) Another advantage of pointing out the obvious is that, when you start seriously asking very simple questions with very obvious answers, all assumptions start flying out the window. It may be very confusing but it is also liberating. It unclogs the mind, and it clears up your vision, allowing you to see what’s there hidden in plain sight, and you were not able to see before because of your assumptions, or because of your mood, or emotional state. Resorting to the obvious allows you to take a long series of tiny, little baby steps towards the solution.

6) Last but not least, another great benefit of stating the obvious is the fact that you move the student from negatively focusing on himself, or herself (“I cannot do this problem, I’m going to fail the class, I am not good at math.”), to focusing on the particulars of the problem at hand, meaning, focusing on the numbers as objects you can manipulate, instead of letting them ruin your life. Focusing on the obvious sets you free from the paralyzing grip of your own emotions, and lets you realize numbers are truly inert in relation to you; you are the one who is alive.