This post is a summary of the conversation following my post why no modelling?, and this response from Dan Meyer. I have edited the comments for brevity; it is not my intention to misrepresent any of the contributors.
Any further comments welcome...
I help students learn math for one reason alone, and it doesn’t have to be your reason also. I want to help students learn to puzzle and un-puzzle themselves... Those puzzles may have sociopolitical importance, but that’s a higher standard than I choose to set for myself.
The definition of modelling doesn’t specify culture, context, or importance. Modelling is mental work, work of a certain character… (this next bit is from ignore the adjectives, focus on the verbs)... Mathematical modelling comprises a huge set of verbs that range from the very informal (noticing, questioning, estimating, comparing, describing the solution space, thinking about useful information, etc.) to the very formal (recalling, calculating, solving, validating, generalizing, etc.). One of the most productive realizations I’ve ever had in this job is that all of those verbs are always available to us, whether we’re in the real world or the math world.
Although the growing emphasis in the education world is to have students engage with, discuss, and provide solutions to some of our world’s most serious problems, at the end of the day, we are working with kids. And sometimes they just want to talk about how many pennies are in the pyramid.
I certainly support Danny Brown’s desire to pose problems of greater importance and hope to incorporate more of this in my own class, but I do not believe that it is a necessary condition for mathematical modelling.
Is a mathematical model a useful, generalizable description of some real world phenomenon, often but not always physical, or is it just a description in algebraic form of a particular instance of that phenomenon ? There must be some real follow up to what I have seen described as “mathematical modelling”.
Dan-Dan-Geoff Thanks for modelling real-life teacher textual collaboration. I’ve been hesitant to say anything lest I sound critical and get shot down.
My best results are from creating “you are there” scenarios, presenting problems, roles, and sometimes name badges that are related to real 21st C jobs they don’t know much about.
The problems don’t have to line up exactly but need to have similar thought processes… My disaffected students grow to understand there is a place for them in STEM.
As you say, I think this comes down to one’s personal beliefs about what maths teaching should/could be – but also what one is capable of under constraints. There is a mismatch between what I would think (maths) education should be, and what I am able to implement.
Perhaps my views on what education is about are not compatible with teaching maths? I make no apology for the fact that I would like whatever lessons a student walks into to effect some transformation on the way they view the world. It feels less than satisfactory *for me* that students come in to my lessons, do some puzzles, and leave, whether this is modelling, doing any kind of maths, or indeed if I was teaching any subject whatsoever.
Perhaps I am being too ambitious in wanting students to come to my maths lessons and gain something of social or political value – this is what you seem to be saying? It might be enough for you for them to just do some maths, but I think what I am really saying is that this just feels a bit… meaningless...
Number and shape and measurement and logic are all part of the world. Change how your kids think about math, and you’re changing how they experience life.
I think what I am saying is that I would like to be able to transform/enrich/enhance the way children experience interactions with other humans, the way they view their place as a member of society. I don’t think learning about numbers, shapes or measuring things is enough (on its own) to do this?
Me, I’m paid to teach Maths. Every time I get distracted into social or political issues, I’m not teaching Maths. And you can’t combine the two. Given that social and political issues don’t have any actual Mathematical component, the student is either focusing on the Maths, or is focusing on the context.
It seems to me you’re taking a tool that can do the thing you want it to do and saying that’s the only thing we should teach students to do with that tool. Moreover that all the other uses of the tool are meaningless or unimportant.
Also I’m not sure your methods will even accomplish your goals. By using heavily contextualized modelling tasks, students risk not learning the transferable skills and practices (e.g. ratio, proportion, modelling, etc.) they’ll need to help them answer other questions in other contexts. Simple as my modelling contexts are, they allow teachers to make transferable skills explicit and transparent.
Yes, I’m inclined to agree with you – both/and not either/or. You are right: as maths teachers, we should teach using a range of approaches in a range of contexts, some of which may have social or political meaning, and others that have other meaning – and some of which have meaning in the maths of and for itself – and of course, meaning is relative, isn’t it…
I think what I am trying to do is imbue mathematics with some meaning according to what I think is valuable, where it may not be desirable or even possible. So either I have to reconcile myself with the fact that maths has some value in and of itself, or maybe teach something else…?
You can’t end every lesson with stunning personal worldview shifts and transformations. But you shouldn’t ignore the effect that the type of work you do in your math class has on students views of themselves and others.
Whether the task suggest it or not, your students are learning about their role in society and in your class every day.
I absolutely agree, this is a fundamental principle of the organisation of my classroom – collaboration and collectivism, a sort of apprenticeship . The *way* we work is socially important.
But given that, I would *also* like the *subject* of our studies to be socially meaningful; this is my problem. I find it hard to reconcile: the way we work is socially important, but the thing we are learning doesn’t have to be?
I think many of us struggle with how the math topics our districts/Common Core require can be made relevant to our students every day. I know I am in a perpetual search to make lessons meaningful. There is no personal criticism for either side of the argument, just gratitude that we can all participate in this exchange of ideas. There is no right or wrong, there is perspective and hope we can feel that satisfaction after facilitating a lesson with impact.