Q: What's the purpose of context in math class?

A: To provoke modeling.

Q: What's the problem with the hot air balloon problem?

A: It offers a context without provoking modeling.

"There were 58 geese and 37 ducks in the marsh. How many birds were in the marsh?"

Q: So is the above bird problem pseudocontext?

A: No, because it's provoking modeling.

In my book, a context doesn't need to be realistic if it serves a purpose, and the purpose of context is to give kids a chance to think about something interesting. If a context provokes modeling? Great! What else can you ask of a context?

(You could ask that it be "real world," but that's a desire that I don't share. And neither should you.)

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I was checking out Dan's latest thing, and I came across this comment:

I dunno, this one made me wish the follow-up showed him just throwing the paint away and starting over. There’s just not enough investment in materials & time to make me think past. Plus if 6 tablespoons was enough paint to do the job then 30 is just a waste of paint far in excess of throwing away 6.

Which if the class brings it up organically would be a big win; then you can talk about why to choose one path or the other and it’s some good critical thinking.Pseudocontext, eh? I say, nah, it doesn't really matter if it's realistic. What matters is whether he can do something with his scenario for kids. Context only becomes problematic when it's purposeless.Otherwise it smells like pseudocontext to me.

**Update**: This is what I mean by "purposeless context."
Going after Shell? You come at the king, you best not miss.

ReplyDeleteSeriously, now, I regret my part in promoting the pseudocontext meme somewhat. It gave voice to something a lot of students and teachers FEEL, but it's really hard to qualify reliably or with any validity. (One student's context can be another student's pseudocontext and vice versa.)

Other measures matter more to me now.

"How much does the question puzzle you?"

"How much sense is there for you to make of the context, or is the context fully made sense of?"

"Does it provoke modeling?" is another useful one, though getting consensus on what modeling is isn't easy. Got ten minutes for that post?

I'm pretty hubristastic, but there's zero chance you can bait me into trying to define modeling in a blog post.

DeleteOf course, you're right that "Does is provoke modeling?" is limited in its usefulness by the diverse and fuzzy conceptions of "modeling." Everything that I'm thinking right now is edging into tautology (e.g. "To model is to make sense of context") so I'd better think about this harder.

I think it would be better to say that modeling (i.e., context) is one avenue for provoking intellectual need. But no single one provocation works perfectly for all learners at all times everywhere!

Delete- Elizabeth (@cheesemonkeysf)

FWIW, I think the CCSS did a great job defining modeling:

ReplyDeletehttp://www.corestandards.org/math/content/hsm

The choice to limit "modeling" to empirical scenarios is what intrigues me the most about that definition.

DeleteWhat does modeling look like in whatever the opposite of an empirical scenario is? (Hypothetical?)

DeleteNB. We had to conduct an empirical study here at Stanford to level up. By "empirical" they meant "watching stuff and collecting data." There were professors who considered basically every study empirical, though.

Maybe I'm taking "empirical" too literally. Does a empirical scenario have to be a situation that truly occurs? I assume that the answer is "no," that fictional scenarios can be modeled mathematically.

DeleteMore to the point, what part of the modeling cycle isn't involved in solving your Obscure Geometry Problem? Identifying variables? Creating representations? Validating and improving the representations? (Or can you only validate against real observations or data?)

Right now I'm reading Children's Mathematics (Carpenter, et al) right now, and there's a chapter titled "Problem Solving as Modeling." At an abstract level, this seems basically right to me. I'm having trouble thinking of instances of problem solving that I couldn't finagle into a modeling framework.

I have no problem with the CCSS definition, though. The most important thing about defining modeling is nailing it down so that it can be taught to teachers and stakeholders. The process of creating an empirical model is significantly more concrete than whatever representations you'd have to impose on (say) a number theory scenario.

Empiricism is concerned with phenomena we can observe. The obscure geometry problem is certainly that.

DeleteCertainly you'll find students identifying variables, formulating models, performing operations, and interpreting their results in a lot of problem solving. (Pile patterns, for instance.) Where I see the modeling cycle parting ways with a lot of problem solving is in the validation phase. Give it a look and tell me what you think.

DeleteWhat's the sum of the first 300 whole numbers?After I pick a way to represent the sum of

nintegers, my first move would be to start testing the model against small values ofn. Then, maybe I would use a calculator to sum up the first 300 integers and directly validate my model. If my answer is off, then I would probably go back to the model, tweak it and then try it for some other large sum.If anything is non-empirical in math, I think that this is.* And I have trouble seeing where this sort of problem solving diverges from modeling, as described in CCSS.

*

Though, there are plenty of thinkers who think thatallmath is as empirical as the sciences. Does this matter for our discussion? No, but I got to use my philosophy degree just there so that was pretty cool.