We've all seen math problems like this. Problems where the context or numbers are ridiculous. And what might we say to a kid in this situation? "Don't worry about that, just do the math."
What can that possibly teach children about the role of math in the real world?
In my work I think about approaching mathematics through real and realish contexts. I want to talk a bit about what I mean by realish contexts, why they are so powerful, but also some of their shortcomings.
The Importance of Realish Contexts
When I talk about realish contexts I mean situations that children can imagine and that they can act out physically, with objects, in a drawing, or in their minds. Below are a few examples:
4 children share 6 pizzas. If they share the pizzas equally and each child gets the same amount, how much pizza will each child get?
Parker was making a house in Minecraft. He wanted the floor to be a big rectangle made out of cobblestone blocks. The rectangle is going to be 7 blocks in one direction and 10 blocks in the other direction. How many cobblestone blocks does he need before he starts?
You collected $7 for the fund raiser on Monday. How much do you need to collect on Tuesday to get to a total of $11?
We don't really care about the "real-world" contexts here. There's no real pizza or kids, Parker isn't actually building this house in Minecraft (and if he were, based on what I have seen, he wouldn't really plan it out ahead of time like this), and there is no fundraiser going on. This is why elsewhere (here and here) I have referred to these as stepping-stone problems because we don't really care about the context—it just serves as a stepping-stone for getting at the mathematics content.
These types of problems are still important (like really important) though because children can imagine them therefore they allow children to ground new mathematical knowledge in their existing understandings. We have a lot of research about how using simple contexts like this support children in building understanding of important mathematical ideas (some accessible examples: for whole numbers, fractions, and algebraic thinking).
I would argue that the vast majority of work in the field of mathematics education has focused on the careful use of realish problem contexts to introduce increasingly complex math concepts. As a note, realish contexts can be fairly "abstract" as long as they are accessible to the child's current thinking, like asking how many edges there are in a row of 4, 5, 10, or n hexagons. And therefore they can be used for more advanced content as well.
Four hexagons have 21 edges. We might think of each hexagon as adding 5 plus the 1 initial edge on the left. Or we can think of 2n edges on the top, 2n edges on the bottom, n-1 "middle edges" and 2 edges on either end. Or we might think...
Realish Contexts versus Pseudocontexts
To be clear, there are also really bad uses of the "real-world." These are "pseudocontexts" a term from Jo Boaler that Dan Meyer has written about (here, here, and here). These are situations where the context is so contrived, so meaningless, and so ridiculous that it only emphasizes how disconnected math is from the real world. Use of these contexts actively teaches children to ignore their real-world knowledge. Psuedocontexts often lack the features of realish problems in that they do not provide a good support (or stepping-stone) for students because they are not grounded in children's understandings. Instead, psuedocontexts often come as "application" problems at the end of the chapter.
Pseudocontexts are bad because they teach students (a) to ignore their real-world knowledge, (b) to just apply whatever math techniques they recently learned, and (c) that math does not actually apply to the real world.
The Problem with Realish Contexts
I like to think Courtney, I, and all of Parker's amazing teachers have done something right in supporting his reading and mathematical development. So I was pretty sure that Parker, of all kids, would not fall victim to a pretty famous example of the trouble with realish contexts, the shepherd problem:
There are 25 sheep and 5 dogs in a flock. How old is the shepherd?
Students are really bad at this problem and it turns out that Parker is no exception. He said 30.
This table is from Foegen, A., & Dougherty, B. (2017). Instruction that meets the needs of students with mathematics disabilities and difficulties. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 893–907). Reston, VA: National Council of Teachers of Mathematics.
How does this happen? Parker gets a lot of the great math instruction both at home and at school that I talk about on this blog. He has had a lot of experience with realish contexts, but not really any pseudocontexts. He's also had some past success at questioning how realistic math problems are. So what gives?!
Even when teachers avoid pseudocontexts, the over-use of realish contexts can lead to problems. In a classroom filled with realish contexts kids will still tend to learn: (a) there is always supposed to be an answer to a math question, (b) the information they need is always included in the problem, and (c) they usually have to "do something" to the numbers in the problem to get that answer.
Unless children are more regularly exposed to real problems then this is all math will ever be for them. Instead, they also need chance to engage in asking their own questions about the world, dealing with messy contexts where they need to make assumptions or do research, and confronting problems like this where the typical "word problem game" doesn't work out the way it should.
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