Prague Seminar | 14 March 2018

At the seminar, we discussed conditions for learning mathematics.

Let me tell you what happened in one lesson today and invite you to think about the conditions for learning. What must we do to facilitate learning, deep learning. 

What do you do when you are trying to teach the first day lesson on fractions to a bunch of second graders and a student raises her hand a second time today with the same request "I have a question."

The first question has been handled.

The second question is - "What if there are 5 quarters (fourths) and there are 4 friends sharing them (equally)?

Yeah! 

What if ....?

And that is exactly what happened in class today.

We were figuring out what happens when 2 friends share a piece of square art paper as well as when 4 friends do likewise. They had the square pieces of paper to help them figure out the solutions. They grabbed themselves pencils, rulers and pairs of scissors.

All sorts of things came up.

First they shared some ways of sharing the paper between two friends and I documented the triangle solution. I asked them how they can be sure if the two pieces are the same amount of paper.

Someone said "measure them".

How? I asked.

"Use a ruler."

Before I could respond (I took quite a while to respond, to be honest, about five seconds), someone  said to cut the pieces out. Then put "one on top of the other".

But how can you tell that they are the same amount of paper.

They gave sound methods to check for equality. 

Then they did the same for sharing among four friends. 

Along the way they cautioned about using ruler to draw the lines, also to "fold first before you draw the lines". 

All is well.

Someone used the word "quarter" and I latched in that to show them the symbol for quarter using post-it. 

"I have a question." 

One students asked if we also use "4" when we want a symbol for half.

I showed them the symbol for half.

"I have a question."

Another wanted to know if the symbol is written after the numerator the way fractions are written in words - 1 quarter. 

I showed them where to put the symbols for the words quarter and half.

"I have a question." 

Why did they use 4 and 2 in the symbols?

I asked them if that's a sensible thing to do and got them to explain to each other why they all kinda think that those are sensible decisions.

I even got them to tell me the amounts when I showed them 2 quarters and 3 quarters. A few other things happened.

Then the 5 quarters questions came up.

All I want team to learn today is about equal parts and the naming convention. I was going to wait till grade four to discuss improper fractions and mixed numbers.

"i have a question."

"What if there are 5 quarters and there are 4 friends sharing them (equally)?"

I asked that question and got them to chat while I decide what to do next.

I am not proud of this but I decided to lie.

I pretend I misheard her.

"So our friend is asking what if there are 5 quarters and 5 friends are sharing them equally." Despite the wild waving of her hand, I pretended not to see the girl to posed the question.

The class was able to say that it's easy - 5 quarters, 5 friends ... That's 1 quarter each.

Having settled that - which is division of fractions, by the way - I let the girl protest .... "What if the mystery friend is not there?"

She sure was persistent.

So I packed them off with two options for journal writing.

One, to show in their journal what if 8 friends share a square paper equally. Please use the paper and paste your work in your journal and say how much paper each friend receives.

Two, to figure out the solution to the question our friend just posed. What if four friends share five fourths? Show your thinking and maybe invent a symbol to describe the amount each person gets.