Partial Metrication/Partial Understanding

Source:; Wikimedia Commons

This post was inspired, as always, by talking to a fellow teacher, in this case about the numeracy that kids need to navigate in the world.

I will begin with a confession: I am relatively innumerate, with regard to Imperial and US Customary units. Not coincidentally, I was born the year Canada went Metric.

If I am cooking a steak, and it's supposed to be 1" thick, I have no solid conceptual anchor for what that looks like, in my mind. I know I am a bit over 6 feet tall, but I only  know I am 183 cm tall because my driver's license says. So, when my sons were born, they were around 9 lb, so I guess that made them about the weight of medium size turkeys, at that time. I never learned mental math tricks for converting between pounds and kilograms. When I travel to the United States, temperatures in Fahrenheit make no sense. Right, except on my house thermostat, on which I know 70 feels cold, if using air conditioning in the summer.

Home Depot is a strange world of things I barely know about. Don't even ask me about that. I know fractions well, but not fractions of inches in the screw section. Is a 2 by 4 really a 2 by 4? I hear not, but how would I know?

The sordid history of metrication in Canada is covered nicely in the Wikipedia article. Basically, it stalled, due to our historical relationship with the UK, and our current relationship with the United States. We are stuck in the middle, seemingly permanently.

Why does this matter?

In Ontario, there is no mention of Imperial units until grade 10, and then only in the Applied course:

 -perform everyday conversions between the imperial system and the metric system (e.g., millilitres to cups, centimetres to inches) and within these systems (e.g., cubic metres to cubic centimetres, square feet to square yards), as necessary to solve problems involving measurement (Sample problem: A vertical post is to be supported by a wooden pole, secured on the ground at an angle of elevation of 60°, and reaching 3 m up the post from its base. If wood is sold by the foot, how many feet of wood are needed to make the pole?)

By this point, it's too late. Numeracy starts at birth. There is also an unbelievably patronizing aspect here: the Applied students are more likely to go into the trades, and therefore are the only ones who "need" to know Imperial units, while the students in the Academic stream march steadily into more abstract territory, with Algebra, and Calculus, as always, the pinnacle of K-12 mathematics education. 

Meanwhile, the goal of the school system is to graduate educated, literate and numerate adults. Being caught between two systems of measurement is not doing our students any favours. 

Representations-Tools for Mental Activity, or End Results of Tasks?

I have worked with a lot of teachers who tell stories of being pushed toward one single correct way of representing their math work by their teachers. If we teach that way, we are probably teaching our students to be too inflexible, too rigid with their mathematical representations. Typically, we focus on the product (written) of the representation, rather than the entire mental process that leads to representation.

Representations are tools for thinking, explaining, justifying. NB: Pape/Tchoshanov, 2001. pic.twitter.com/Nux9jP7Z4b

— Matthew Oldridge (@MatthewOldridge) June 7, 2016

The Pape/Tchoshanov paper is instructive for teachers in getting kids to develop their own mental representations, and to get them out into the physical world.

In their words:

"We use the term representation(s) to refer to both the internal and external manifestations of mathematical concepts."

This is a huge argument for paying attention to your students' thinking, in order to make it visible. We need to help kids visualize representations of interesting mathematical concepts, in order to imagine them into being.

So we have two complementary aspects of mathematical representations:
-the act of representing (verb)
-the representation itself (noun)

This quite neatly corresponds to both NCTM and Ontario curriculum definitions of representation: as both process and product.

The authors propose a "'zone" of interaction between mental and physical interactions.

The nice example used is of "six"-a kid develops an internal mental image of "'six', then connects it to sets of six objects, the numeral 6, the written word "six", the spoken word "six", and so on.

As kids advance in math, we should encourage representational thinking. That is, building and having a repertoire of ways to externalize math concepts. Tables, graphs, tree diagrams, area models, arrays- these are all examples.

The authors put it nicely:

"the development of students' thinking skills requires a multiple representational approach."

Put simply: representations are tools for reasoning.

Two things to think about.

Kids often mathematize math tasks by drawing pictures with little or no mathematical content. Do these count as representations?

"Students often produce representations that lack meaning."

Manipulatives are generally a good thing, as long as they are accurate mathematical representations of the concept, the mathematics you are working on. Daniel Willingham presents this caveat here.

Manipulatives, if not chosen wisely, can get in the way of mathematical representation. Let the tool fit the task (and the math).

The end goal of representation is good communication. Can the student communicate their thinking, in words, or on paper, using mathematical representations?

Even better, can they start to see how different representations are connected? Do they see a table of values, a pattern made with cubes, a graph, and an algebraic expression as fundamentally the same thing?

How do they show the same math in different ways?

Fluency with multiple representations should be a goal of thinking math classrooms; we can help kids to visualize, conceptualize, and make real their understanding of powerful mathematics.