Abandoning the Perfect Match

As you have probably noticed by now, the role of context and applications in mathematics education is a key interest of mine. Over the years I have come to believe more and more firmly that not all mathematical learning has to be in context, and that it is counterproductive to attempt to teach everything in context. Forcing context on mathematics often leads to meaningless and contrived problems that almost certainly do nothing to enrich learning.

On the other hand, I also believe that when clear and relevant connections can be made between mathematics and other content domains or broader everyday-life issues, they should be. Mathematics, after all, was developed to solve real problems, and some connections can be productive. It is helpful, for example, for children to think about putting things together when they do addition and creating equal groups when they divide.

This seems like a very pragmatic, and maybe even obvious, position to take: Use context when it makes sense. Yet many times in the course of my career I’ve been struck by how difficult it is to strike the right balance. Take, for example, the relatively simple mathematical topic of graphing a linear inequality. (Full disclosure: What follows is a retelling of the process I went through to write a lesson on this topic during my curriculum development years.*)

To graph something like x > 3, you draw a number line, draw a circle on the 3 mark, and shade everything that’s greater than 3, like so:
NumberLine1

And it’s pretty simple to come up with a context for an inequality. For example, suppose a library has a rule that a patron can check out no more than 4 books at a time. If b is the number of books, we can represent the situation with the inequality b ≤ 4. A graph of that inequality looks like this:
NumberLine2

But wait… if this is a representation of the context, then we really shouldn’t have any values of b less than 0. You can’t check out -2 books. So maybe this:
NumberLine3

Well, and you can’t really check out a fractional part of a book, so what we really want is this:
NumberLine4

Okay. Now we’ve got a reasonable representation. But creating one brought out some other mathematical ideas, like defining domains and discreteness versus continuity, that I didn’t expect when I set out.

I don’t mean to argue that it was too hard to make a context work in this instance. My colleagues and I were able to finesse these ideas into a lesson that we thought did justice to both what we intended to be the main mathematical idea (graphing inequalities) and these other mathematical ideas. Rather, my point is that the connection between the mathematics and the context wasn’t simple. There wasn’t a one-to-one correspondence.

My point in telling this story is this: Lately I’ve been thinking that perhaps a lot of my troubles — and to the extent that I can generalize, the troubles of the mathematics education community — in connecting mathematics and context is that we’re focused on finding a perfect match. We dismiss potential ways of connecting mathematics to life outside of school because we’re afraid that either the mathematics or the context becomes skewed, oversimplified, or contrived. But maybe it doesn’t need to be. Maybe instead of adjusting the math or the context to achieve a fit, we can acknowledge that the match is imperfect, and take it for what it is.

I started thinking about this last week after reading a book chapter describing a researcher’s experience teaching a class on mathematics for social justice to high school seniors in a Chicago community. Gutstein (2012) describes how he structured the course around “generative themes” that were suggested by the students in the class. I was particularly intrigued by his discussion of a unit on the spread of HIV/AIDS, where students used complex mathematical models to simulate the spreading of the disease and had in-depth discussions about some of the more striking statistics showing the disproportionate effects of the disease on certain populations.

At the time, he was concerned that the mathematics and the context were not well aligned: “[S]tudents did not need to model HIV/AIDS to understand the disease, grasp the disproportionate impact on certain populations, or not blame Black women” (Gutstein, 20102, p. 35). Later, he considered the idea that this may not be a problem, but rather just an example of a different kind of connection: “At times, we use mathematics to explain social things (like the election being stolen), at other times, we use social analysis to explain mathematics (like high AIDS rates). The point here is, I think, that you cannot easily explain one without the other” (p.35).

Sometimes, mathematics directly helps solve a social problem (like explaining the results of the 2008 election). Other times, a context helps establish why mathematics might be useful (like AIDS statistics motivating a study of mathematical models). In either case, though, the match isn’t perfect. Still, the lack of a perfect match does not diminish the connection’s utility.

As for most of the musings I write about in this blog, I am not sure where this realization leaves me or how I would do anything in my curriculum development work differently if I could go back and do it again. I don’t have any easy strategies for opening up my own thinking about connections to context. Instead, I’ll just end with one more example that struck me as I read today.

In an article about ways in which the mathematics education research community can be more attentive to issues of intersectionality — that is, the particular issues and challenges faced by those in two or more overlapping marginalized groups, such as Black women — Bullock (2017) says the following: “The idea of multiple burden speaks to intersectionality’s key concern that racism, sexism, and other forms of oppression, when considered in parallel, appear additive, but those who experience these oppressions in combination endure multiplicative effects” (Bullock, 2017, p. 31).

This article has nothing whatsoever to do with any of the issues I discussed in this post. She wasn’t aiming to make a connection between mathematics and social justice issues (at least not in the same way Gutstein (2012) was). Still, her framing of the issue in terms of a mathematical idea — specifically, the difference between additive and multiplicative growth (5 + 5 versus 5 * 5, for example) — really brought her point home for me. I understand intersectionality better than I did before.

The connection between intersectionality and multiplicative growth isn’t perfect. But it is powerful.

References

Bullock, E. C. (2017). Beyond “ism” Groups and Figure Hiding: Intersectional Analysis and Critical Mathematics Education. In A. Chronacki (Ed.), Proceedings of the 9th International Mathematics Education and Society Conference (MES9, Vol 1) (pp. 29–44). Volos, Greece: MES9.

Gutstein, E. R. (2012). Mathematics as a Weapon in the Struggle. In O. Skovsmose & B. Greer (Eds.), Opening the Cage: Critique and Politics of Mathematics Education (pp. 23–48). Rotterdam, Netherlands: Sense Publishers.

 

*Shout out to my colleagues Andy Isaacs (my most loyal reader and commenter!) and Sarah Burns, who were involved in this discussion.

2 thoughts on “Abandoning the Perfect Match

  1. I’m a big fan of contexts in computing education. I understand when you might not want to use a context (e.g., https://dl.acm.org/citation.cfm?id=1869747). Context makes the most sense in the first stage of Pat Alexander’s Model of Domain Learning (https://en.wikipedia.org/wiki/Patricia_Alexander), makes less sense in stage 1, and makes no sense in stage 3.

    Could you explain the argument for not teaching with context to grade school students in mathematics? If the mathematics can’t be tied to a real-world context, why are we teaching it? If it can be tied to a real-world context but not at the given grade level, why are we teaching it at that grade level?

    What is the relationship between context and “intellectually honesty” as in Bruner’s famous quote, “We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development”?

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    1. Hi Mark,
      Thanks for the comment — you made me think through these issues a little bit more, which is always a good thing.

      When I said that I don’t think all elementary school math needs be taught in context, I am drawing on two kinds of experiences I’ve had in the past: (1) having to write absurd numbers of word problems for particular kinds of mathematical operations (for example, lots of practice problems for adding fractions with unlike denominators), and (2) watching young students engage productively and meaningfully with uncontextualized problems.

      (1) is why I became disillusioned with context as the gold standard. I just don’t buy that, after a certain point, context makes a meaningful difference to kids. The problems become contrived — fast. Moreover, there’s a pattern of research showing that often times kids just pick out numbers from work problems and operate on them without thinking (e.g., http://www.sciencedirect.com.proxy.uchicago.edu/science/article/pii/S0959475297000066). One argument — indeed, the one presented in the article I linked — is that we need to do a better job of presenting the problems and encouraging students to think about them critically. And I agree with that point of view. Generally, I’m in agreement that things SHOULD be introduced in context. I just don’t agree with the insistence that practice problems should be contextualized too. If you want kids to practice adding fractions, just have them add fractions. Don’t propose a context where one measurement is given in half-feet and another in 12ths of feet just to get kids to add fractions. Generally I guess I’m saying that where I think we get into troubles is when we let the math determine the context and insist on generating a long list of contexts. Most of the time, there are meaningful contexts. When they run out, I’m saying my opinion is to let them run out.

      (2) is related to your question about context and intellectual honesty, I think. In my experience, kids engage in an intellectually honest way with all sorts of problems — not just “real-world” problems (which is what I think most people think contextualized problems are or should be). For example, I’ve seen kids engage meaningfully with the question of how much longer a circle’s circumference is than its diameter. They were rolling various cans to try to figure it out, but the problem wasn’t posed to them as being about the distance around and across a soup can. They were just thinking about circles and pi. My point, I think, is that I worry that the constant pursuit of contexts can overlook meaningful explorations like that.

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