Thursday, March 24, 2016

Talking Changes Everything In The Math Classroom

Talking changes everything. 

That's been my biggest revelation over the past 14 years of teaching.

When I first starting teaching, I didn’t know that math classrooms could (and should) be talking classrooms. I saw them as quiet spaces, where you could often hear a pin drop, and students silently approached the teacher’s desk, but then only when they had a question.

Often we treat math solely as an individual activity.I myself have memories of sitting in a desk, in a quiet room, textbook open, doing problems in a workbook. We didn't talk about our math work, and we didn't collaborate with each other very often.

Assessment happened between us and our teacher, usually at the end of the unit, or on quizzes in the middle of a unit. We made our corrections if need be, but by that time, the class had moved on to the next topic.

When I first started out, I sometimes had to cover for another, very, very experienced math teacher (30+ years). She was very, very good at what she did. Homework was taken up, lessons were given, and students were set back to work again. Kids never talked. I remember how surprised I was by that. They expected the classroom to always be a silent working space. There is a time and place for that, but I think we are better served by talking about their mathematical ideas.

Talking changes everything, and when kids start to talk in the math classroom, they don’t stop. They will talk about the problem they are working on with others. They will come in the next day still talking about the previous day’s work. They will come in eager to talk about prime  numbers, or videos about math they watched on Numberphile. They will wonder aloud about things like infinity, and if pi never ends.

Math classrooms should be talking spaces, because talking spaces are collaborative thinking spaces. Lucy West identifies five types of classroom talk: voicing, repeating, adding on, waiting, reasoning. You can find a number of nice videos here. Discourse, Reasoning, and Thinking (LearnTeachLead).

Of these, reasoning is integral to math class. Is a negative added to a positive always negative? Positive? Why does a triangle have an angle sum of 180 degrees? Which is bigger- 3/4 or 5/8? Why?

I have previously written that assessment shouldn't be an event. Neither should talking. If you haven't already, start with something concrete and manageable, like having kids turn and talk to the person behind them at crucial junctures of the lesson. Get used to having kids explain their own arguments, using white board, or document camera, or good old fashioned chart paper.

Students defending their mathematical arguments:

Drop by the quiet kid's desk, and see if she has anything to say about her math. Kids will surprise you with their thoughts, their strategies. When they get used to it, they will be eager to share with you.

This kid couldn't wait to share his thinking about this pattern:

Talking spaces are spaces for wondering aloud about interesting mathematical ideas. Our classrooms are those wondering spaces, those talking spaces. Let kids start talking in math class, and they won't want to stop. We are used to thinking about "voice" in the writing classroom. How about in the math classroom? Are ready to hear students' own original and unique mathematical voices, as they talk and reason their way through interesting mathematical problems and ideas? 

A student wondered aloud about the value of this part of the $2 coin, so we found out!

Monday, March 7, 2016

Costco-the Proportional Reasoning Store

Check your local curriculum: if it's like ours, sometime around grade 6 students are asked to deal with the interrelated concepts of rates, ratios, and percents. This is part of the growth of thought in the area we call proportional reasoning. In grade 4 (Ontario), students start to think in more relative, and less absolute terms. This often includes starting to see relationships as multiplicative, not additive, as they have been doing since Kindergarten.

I tend to think that unit rates, in particular, used to be more mysterious before Costco became ubiquitous. Their entire business model is built on an economy of scale, or what we recognize as "buying big stuff, so they can sell it to us for cheaper."
The entire store is filled with very interesting possibilities for proportional reasoning problems, particularly due to how they do their price labels.

In this example, you see that the unit sold is actually 2 bottles. Each has 830 mL. Their go-to rate for liquid capacity is price per 100 mL, in this case $0.409.  And of course, how much  you actually pay.

I see lots of opportunities for interesting tasks here. Comparing the price per Litre of various liquids in Costco, for example.

How much is Coca Cola per litre?  How much should it be?

A general question you can ask about Costco prices is: "are they fair?"  You could then compare the price of water, or other liquids at various grocery stores.

As far as Costco goes, fair, and cheap, for me depend on this: can I store the item in my house?  If so, I often calculate the unit rate in my head and decided to make a purchase.

I do think other grocery stores would be cheaper on some items, so it would be interesting to compare, say using the Flipp app or other stores' websites.

This one uses the price per bar as their unit rate, in this case $0.15. The price per kg is also interesting though.

Compare to these granola bars:

Which one is cheaper? Which one would you buy? Why?

There are nearly endless possibilities for interesting math tasks in Costco. We thought it would be interesting to take a bunch of teachers there, and snap away with our phones, interesting materials for math tasks. Would you come?