Un-structuring mathematics

In two of my courses this semester, we’ve spent some time talking about well-structured domains (WSDs) and ill-structured domains (ISDs) and the ways in which beneficial instruction might look different for each. A well-structured domain is one in which all concepts and procedures can be readily delineated and described. By contrast, “[i]ll-structured domains are characterized by being indeterminate, inexact, noncodifiable, nonalgorithmic, nonroutinizable, imperfectly predictable, nondecomposable into additive elements, and, in various ways, disorderly” (Spiro & DeSchryver, 2009, p. 107). As examples, Spiro and DeSchryver (2009) describe the idea of muscles bending a joint as a complex, but ultimately well-structured domain. There are many facets to this process, but in the end all those facets can be well described and the way in which muscles work is consistent across human beings. By contrast, the concept of justice is an example of an ISD because its application and meaning across instances will vary considerably.

Spiro and DeSchryver’s (2009) principle argument is that highly guided and direct instruction may well be the most effective approach for learning material in WSDs, but such approaches are ineffective in ISDs. Providing definitions and a discrete set of examples of justice, for example, serves to provides students with a narrow understanding of the term and can promote significant misconceptions. Spiro and DeSchryver argue that instruction in ISDs should therefore facilitate “a nonlinear criss-crossing of knowledge terrains to resist the development of oversimplified understandings” (p. 115). Students should be given experiences that help them apply existing pieces of knowledge about justice, for example, in new ways, so that they can think flexibly about the concept.

For the most part, I buy into this idea. I agree that we tend to teach kids to think too rigidly about concepts. Here’s my problem with the discussions surrounding this, though: While history and philosophy are the typical examples given for ISDs, mathematics is almost always used as a go-to example of a WSD. For mathematics, the narrative says, direct instruction is just fine!

This deeply bothers me.

I’m not going to try to make the claim that structure doesn’t play a huge role in mathematics. I completely understand why people think of mathematics as well-structured. Being a mathematician is, in large part, about seeing structure in an ill-structured world. My worry is that references to mathematics as a WSD justify the perpetuation of mathematics instructional practices that are both problematic and deeply entrenched in our instructional system.

Take for example, the standard algorithms for the four basic operations. Claims that direct, highly guided instruction is optimal in mathematics would suggest that explicit teaching of the standard algorithms is fine. Yet we’ve known for years that rote teaching of algorithms, without accompanying opportunities to invent algorithms, is harmful to kids’ number sense and understanding of place value (Kamii & Dominick, 1997). And what about word problems? If direct instruction is suitable for mathematics, it would seem that all those superficial word problems we place at the end of lesson problem sets are just fine. Yet we’ve know for years that students’ school experiences with word problems leads them to dissociate school mathematics with sensemaking — they don’t take context into account when solving word problems (Silver, Shapiro, & Deutsch, 1993).

This has gotten me thinking harder about whether or not I believe mathematics is really well-structured in the way that Spiro and DeSchryver (2009) describe. Does addition, for example, have a straightforward definition? If we’re thinking about it as an operation on abstract numbers, then maybe it is. But if we’re thinking about how it applies to contexts, then I do think it can have many means. Addition, after all, has multiple use cases. It’s useful when you want to find a total (4 blue fish and 3 red fish, how many all together?), make a change that results in more (4 blue fish and 3 more blue fish come, how many blue fish now?), understand a comparison (I have 4 blue fish and 3 more red fish than blue fish — how many red fish?), and so on (Usiskin & Bell, 1983). Given this, is addition any more easily described to students than the ill-structured concept of justice? I am not so sure.

I like Spiro and DeSchryver’s (2009) call to help students “criss-cross” ill-structured domains to avoid oversimplified understanding. I just wish that mathematics were not so often casually discussed as the domain where it does not apply.


Kamii, C., & Dominick, A. (1997). To teach or not to teach algorithms. The Journal of Mathematical Behavior, 16(1), 51-61.

Silver, E. A., Shapiro, L. J., & Deutsch, A. (1993). Sense making and the solution of division problems Involving remainders: An examination of middle school students ’ solution processes and their interpretations of solutions. Journal for Research in Mathematics Education, 24(2), 117–135.

Spiro, R., & DeSchryver, M. (2009). Constructivism: When it’s the wrong idea and when it’s the only idea. In S. Tobias & T. Duffy (Eds.), Constructivist Theory Applied to Instruction: Success or Failure (pp. 106–123). Mahwah, NJ: Lawrence Erlbaum.

Usiskin, Z., & Bell, M. (1983). Applying arithmetic: A handbook of applications of arithmetic. University of Chicago.

One thought on “Un-structuring mathematics

  1. I agree! These two are extremes, and most things are somewhere in between.

    I find that people often see things as one of the two extremes – like things are either so straightforward that direct instruction should be used or so ill-structured that students either get it or they don’t. Yet those things that appear straightforward are often not as straightforward as they think – they are teaching particular skills in particular circumstances rather than the underlying concepts. Likewise, the reason they think things are so ill-structured is that they were able to learn the concept easily and have never had to think about what exactly makes up that concept. That’s what LT’s are for.

    It would be useful to have a lot of examples that are in the middle, in which some aspects are more tractable than justice but yet not as straightforward as addition or long division.


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