I think it would be useful to include the two paragraphs above the ones you quote:

“Brown: If I am going to spend the time planning and exploring modelling situations in my classroom, I really want to model something *important*, in the sense that it *matters*, perhaps politically, or socially, perhaps having a *local* relevance for the students in the classroom – [link] has been an inspiration (one of the authors, @LaurieRubel also suggested citydigits.org, which looks exciting). There are also some interesting ideas in [link], and [link] looks really interesting. The idea that mathematics can be used to explore social justice, making decisions about important issues, is the kind of thing I am interested in bringing to my students.

In the past, I have created some resources of my own (for KS3/4), such as this on the UK 2010 election, but this quality of modelling experience takes time to plan and explore, and is also difficult to align with the curriculum.”

It’s obviously got to feel pretty raw to see someone post negative comments about your blog which they probably didn’t initially intend to make public. It’s only fair that you respond!

At the same time, it presents Danny and Geoff’s positions more clearly if you include their examples of what they DO consider important and relevant.

They’re asking a lot – Danny later says a problem is meaningful if it “transforms the way they view the world” – but I also honestly think some of your material does achieve that goal, so why explore the aspiration?

Hi Dan

Thanks for this response, a lot to think about there. Also, really interesting comments here above.

Here are my first thoughts about this:

I didn’t really intend to *criticize* personally (although this is in effect what I have done), but rather question the purpose of maths modelling/teaching generally.

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. Taken as a whole, my part of the conversation was me expressing my frustration at the difficulty I have in finding/creating/bringing real meaning – and by this I mean social/political meaning – to my teaching as a result of these constraints. This is the main thrust of what I was saying – that there is a mismatch between what I would think (maths) education should be, and what I am able to implement.

I wonder perhaps if I am teaching the wrong subject? Perhaps my views on what education is about is 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.

Again, this is not a criticism, it is a genuine attempt to start a(nother) conversation about whether we can realistically hope to bring meaningful contexts into maths lessons. I would genuinely love to hear your thoughts on this Dan…

I would like whatever lessons a student walks into to effect some transformation on the way they view the world.

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.

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.

And that’s before you get to the focused students, who want to enter medicine, engineeringetc who will literally hate you for holding back their aspirations to push your own.

There are a number of “things out there” in which costing, estimating and probability have to be dealt with and related to what is “acceptable”. Cost of road improvements against benefits, cost of medical screening against monetary and health benefits, environmental protection and so on. The classic example was use of amniocentesis as a test for Down’ syndrome. Enthusiasm for this waned when it was observed that for every Down’s syndrome fetus identified six healthy fetuses were miscarried. OOPS.

Danny says, “I don’t think learning about numbers, shapes or measuring things is enough (on its own) to do this?”.

It depends on how you learn those things. Look at the video Danny posted from a tutoring session in which a group of students work a problem related to exponents and sequences. The task is not a modeling task, and the content is not a vehicle for transforming a students “role in society”.

Except that when you watch students engaging with a task that they are motivated to understand they are doing all sorts of things that relate to the “way they view their place as a member of society“. I can’t imagine a situation in which a student isn’t both learning something about their place in society and simultaneously asserting some version of their belief about their place in society. It’s happening all the time. It was not the explicit goal of the activity and was not a focus for the activity, but watch the video and think about what students might be thinking about their place in society as they work to understand the abstract, calculation focused, binomial expansion question and you can still see a student gaining confidence, seeking to help others, questioning together and cooperating to accomplish a goal.

As far as modeling goes, one reason many teachers are on the modeling bandwagon is because many of the modeling tasks put students in situations where they have an opinion right away. My favorite is when students declare early on that they are certain of their opinion. They can supply reasoning and support and they are invested. They are not looking at a problem or procedure and complain-asking “when are we ever going to need this?” because there is no need, they believe they already possess all that is required.

In really good modeling tasks the rough/hasty opinion that many students develop early on become visibly lacking to the student at some point in their work on the task. So, now there is a need and, hopefully, the student feels it. I think Dan’s word here is perplexity. And for many of these tasks that perplexity is an end in and of itself. When I fully dig into a task that I am challenged by, I have some ego at stake. The reward for improving my solution can be that I gain confidence in my ability to figure stuff out, to handle challenge, and persevere. Other rewards might be social — that I helped someone else improve their solution, that I saw the task from their point of view.

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.

Dan-Dan-Geoff Thanks for modeling real-life teacher textural collaboration. I’ve been hesitant to say anything lest I sound critical and get shot down. My students are older. Puzzling for the sake of puzzling often works well. 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.

Most repetitious work will likely be replaced with robotics. I relate D=rt problems to queuing systems (egressing buildings & waiting in line). Absolute value is related to metrology (measurement) and tolerance ranges on blueprints. Testing solutions is quality assurance. Curve fitting helps in marketing analysis. The problems don’t have to line up exactly but need to have similar thought processes. Students are surprised there is more “out there” besides police officer, nurse, and mail carrier. My disaffected students grow to understand there is a place for in STEM. Hard work can pay off.

I love Nick, Merryl and Lane’s comments. This is super interesting. Not at my computer right now, but briefly in response to Nick – I absolutely agree, and 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?

@danny I would argue that, when the problems posed in class are worth your attention the lesson has more value to students than just ‘brain teasing’ for the sake of getting a little brain exercise. I think you might have to present a better definition of “socially important” in order for this discussion to evolve, it seems as though you are not satisfied if the lesson changes some conception that the student has of a concept about, say, proportionality or measurement. So, if those are not socially important concepts, are you thinking that your lessons need to change a students conception about race, religion, justice or identity?

To piggy back on @michael’s comment (#6) reasoning, perseverance, problem solving and student’s opinion about their abilities to engage in these activities are socially important. These abilities and students opinions about them do not exist in a vacuum separate from the student’s conception of their self. I’m not saying that every student needs to pursue an advanced math degree and become a research mathematician, but if we could rid students of the ‘I-am-not-a-math-person’ mentality and give them the confidence to reason and engage in mathematical activities a little bit at a time, then the work has importance to the student when they leave our room.

I am rather taken by the passionate expression of doubt about forcing people to study mathematics. I am not in favour of forcing people at all, but it then behoves us to attract people to work on things that they might not otherwise encounter. This is the Vygotskian stance: schools as institutions are responsible for bringing students into contact with ideas, ways of thinking, perceiving etc. that they might not encounter if left to their own devices.

Once forced to do something, I am with you on asking why it should be this or that and not something else. However, it is also the case that applications or uses of mathematics in socially relevant contexts tends to dwell in arithmetic, and not to encounter mathematics.

For example, as a customer, you want to know what something will cost. But as an entrepreneur you need to have a policy so that different customers are treated fairly. Such a policy is essentially algebra. it is a statement of generality.

For example, if you want people to be able to challenge what politicians say and do, you need to instantiate their generalities and ask yourself whether they are reasonable, equitable, etc. This involves specialising or instantiating generalities.

If you want people to be able to think coherently about proposals (such as the EU, but also such as building houses, a high speed rail link, a new football stadium etc.) then they ned to be able to challenge the modelling assumptions. This requires being aware of structural relationships, which is what mathematics can offer (as does physics, but in a mathematical way).

I am particularly concerned that the EU debate is content free; people simply assert things and then claim the other side’s assertions are wrong. This shows tremendous disrespect for voters.

So working on socially relevant issues is valuable. But ‘relevance to me’ means, ‘real to me’, and as the RMP project has shown, well as it has confirmed what has been known for ever such a long time, what can be real to someone has to do with what they can imagine, can grow to imagine, and is not confined to what they already do every day.

So the weakness in your stance is, I think, that while promoting social awareness of inequality etc., that your language can be taken to mean ‘relevance’ is what people already do day by day, and that, I think, will condemn people to not discovering ways of thinking that open up possibilities for them, whether through social critique or through finding fulfilling work.

I think this is a manifestation of the deeper rift between the pure mathematicians and the applied mathematicians. While we say we teach math because it is useful, many see merit in doing math for math’s sake–like art, it is the creative process involved that gives it relevancy.

Math has it tough as a subject because it straddles both worlds, the creative and the applicable. I suppose this is true about every subject to some degree. But with math, the divide is more apparent because the two ends of the spectrum are so far apart: there is little more creative or useful as mathematics.

So, do we teach math like we do art, or do we teach it like we do physics? I struggle with this often. I suppose others do, too, even if its subconsciously. Personally I sit on the ‘pure’ end of the spectrum. I see relevance in the beauty of math without needing any application for it. If I’m not careful, my math class turns into something more like an art class. I have to make a conscious effort to bring in the applications to show math is useful for things.

There really should be two synergistic strands of math class. That’s hard to do with one teacher and one course though. Instead we tend to find one blend and sit there …and its often at our comfort level or personal preference.

This is all good stuff. I spent an adult lifetime doing physics and engineering. With time on my hands, I thought I’d try to figure out why young American students can’t solve problems even though they’re fine with arithmetic. I had the same experience.

It’s taken awhile but I’ve concluded school mathematics comes at problem solving from arithmetic/algebra. The subject of problems for most of us is not mathematics, that’s just a tool to evaluate expressions we formulate in terms of the elements of the problem itself. It is this front-end of problem solving that is creative and difficult. I suggest an approach to solution expression formulation in recent posts to my blog.

From what I can tell school mathematics is pretty much a world unto itself; the rewards for teaching it must come from within the system itself. It is easy to understand frustration felt by those seeking to make a difference for students in the world beyond school. I have two suggestions: (1) Make connections with the problem solving world, and look beyond the mathematicians and cognitive scientists; it’s the subject matter not the methodology. (2) Look for ways problem solving can help you teach. How can you expect students to see any good in what you teach them when you haven’t?. This is not a teach-by-the-numbers thing; it is using quantitative thinking to get at what you want your teaching to be.