A Response to Danny Brown & Geoff Wake: Should Modeling Be Important?

Danny Brown:

Some of the other online modelling resources, such as Dan Meyer’s blog, don’t really fit what I would class as meaningful modelling, and can feel contrived, or of little relevance/import to students’ lives; if I am going to spend the time bringing modelling situations to my classroom, I want to address matters of importance, socially or politically.

Geoff Wake:

Yes, I’m interested see how Dan Meyer promotes a sort of pseudo-modelling that seems to be quite popular among certain teachers. I think one aspect that appeals is that he suggests a narrative that is immediately accessible. On the other hand some of the questions are not particularly meaningfully tackled using mathematics seriously.

I see two tacit questions.

One, should math be important?

And by “important,” I’m using Danny’s definition: relevant to a student’s life, either socially or politically.

See, there isn’t any one agreed-upon definition of “mathematics.” They’re all arbitrary, personal, and cultural. And given finite hours in a school year to spend learning math, they’re all political. They create winners and losers. Class time spent how you’d prefer is time not spent how someone else prefers.

So 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 unpuzzle themselves. Math offers us the opportunity not just to solve puzzles, but to generate them from scratch – just you and your brain and maybe something to write with.

Those puzzles may have sociopolitical importance, but that’s a higher standard than I choose to set for myself. So it’d make more sense for Geoff and Danny to criticize my standard than to assume I’m aiming at theirs and missing. I’m not.

Two, should modeling be important?

I suspect Danny, Geoff, and I would agree more about the point of mathematics than the point of modeling. Their criticisms specifically concern modeling, and the fact that I ask questions like “How many pennies are in the pyramid?” and “How long will it take the water tank to fill?” rather than questions like (I’m guessing here) “Is capital punishment sentencing just or unjust?” or “How should California manage its water supply?”

But there is much more consensus around the definition of “modeling” than “mathematics,” and that definition doesn’t specify culture, context, or importance. Modeling is mental work, work of a certain character, work that I think we’d all agree is uncommon in many classrooms and unfamiliar to many students.

Modeling asks questions about a context. It works to make those questions more precise and tractable. It nourishes those questions with data where none exists. It sets reasonable bounds on an answer before finding a solution. It solves questions mathematically and then tests those answers against the world’s answer.

Basically, “modeling” is a verb and it doesn’t help our understanding of the verb to attach it a priori to adjectives (like “important” and “relevant”) or to nouns (like “capital punishment” and “water supply”). If you want to understand modeling, ignore the adjectives and the nouns. Watch the verbs.

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Chris Evans:

Additionally, we have to remember (as math teachers) that we are not the only teachers and courses these students encounter. I teach mostly 11th and 12th graders, and they frequently tell me about the political conversations they are having in government class or the serious social topic they are writing about in English. I have observed that, although students seem to appreciate these connections to real-world problems, these topics are heavy, and at times students appreciate engaging in “lighter” application problems and activities.

Nick Hershman:

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.

John Mason:

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.

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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

29 Comments

  1. I read Danny Brown’s post, and I do agree that at some point(s) throughout the school year, I would like my students to be exposed to modeling scenarios of the more serious social or political varieties. However, I tend to agree more with Dan that modeling problems can be rich, engaging classroom experiences even when the context is a “lighter” topic.

    Additionally, we have to remember (as math teachers) that we are not the only teachers and courses these students encounter. I teach mostly 11th and 12th graders, and they frequently tell me about the political conversations they are having in government class or the serious social topic they are writing about in English. I have observed that, although students seem to appreciate these connections to real-world problems, these topics are heavy, and at times students appreciate engaging in “lighter” application problems and activities.

    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. Honestly, as an adult, I need a break from serious topics sometimes. Furthermore, although educated adults should be informed enough to discuss some of these serious social and political issues, many of us work in jobs that solve puzzles more similar to the mundane, lighter questions listed above.

    To summarize, 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 modeling.

  2. 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?

  3. 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 modeling”.
    Take for example the very nice basketball one. For this to be a real mathematical model it has to be figured out whether or not the idea (path is a parabola) works in other situations, and if so what physical interpretation can be given to the coefficients. It is also a good idea to at least speculate on the range of situations for which the model “works”, and on reasons why in other situations it probably won’t work.

  4. 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…

  5. There is so much “out there” that is either based on math or can be described mathematically. I am thinking of architecture, bridge design, levers, jacks, self driving cars, internet communication, all of which can be investigated at various levels of simplification. It seems that mechanics has slipped out of USA math education. PUT IT BACK.

  6. 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.

  7. Hi Michael (Dan)

    Yes, sure, everything is part of the world, and changing how someone thinks about anything is changing the way they experience life in some way.

    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?

    Danny

  8. Danny,

    It seems to me you’re taking a tool that can 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.

    It’s as though you liked oil paintings more than any other kind of painting. And all other uses of a paint brush (watercolor, acrylic, etc.) were diminished in your mind as a result, to the point that you thought those other uses were meaningless and unimportant.

    I don’t see any point in trying to convince you you’re wrong. It just seems like a particularly narrow view of what a paintbrush is good for.

  9. Also I’m not sure your methods will even accomplish your goals.

    By using heavily contextualized modeling tasks, students risk not learning the transferable skills and practices (eg. ratio, proportion, modeling, etc.) they’ll need to help them answer other questions in other contexts.

    That is, with City Digits, you may help students understand something about predatory loan practices. But I’d worry that’s all they’ll learn. And they’d need just as much help from you modeling in the next context. Simple as my modeling contexts are, they allow teachers to make transferable skills explicit and transparent.

  10. Chester Draws

    April 1, 2016 - 3:41 pm -

    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.

  11. Chester:

    Every time I get distracted into social or political issues, I’m not teaching Maths.

    Again, this is a statement of values, not of fact. You and Danny both subscribe to a constrained definition of math. (Him: “Social issues only.” You: “Anything but social issues.”) I just don’t see the point of the constraints.

  12. 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.

  13. Hi Dan

    Yes, I’m inclined to agree with you – both/and not either/or – http://www.squeaktime.com/blog/is-a-consistent-pedagogy-possible

    I think that *my* underlying problem is that I am increasingly beginning to think that school maths has little value for many students.

    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 like sociology, or follow a different path – something that appeals is becoming a counsellor where I can talk to students about things that really matter to them – but that is for a different blog.

    Thank you for the conversation, I really appreciate it – you have clarified some thoughts for me..

    Danny

  14. 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.

  15. I love this: “I want to help students learn to puzzle and unpuzzle themselves. Math offers us the opportunity not just to solve puzzles, but to generate them from scratch – just you and your brain and maybe something to write with.”

    I may have to steal that quote and display it my classroom.

    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. Is that a social statement? Not sure…

  16. 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.

  17. Nick:

    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.

    +1

  18. 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?

  19. @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.

  20. Hi Nick, I agree with some of what you say. Of course there is value in reasoning, perseverance, etc., and any lesson in which students emerge more confident/competent in these ‘skills’ than when they arrived is probably a good lesson in that sense.

    But I would argue that these skills are not specific to maths, and could be gained whilst also studying something more directly relevant to the majority of the lives our students will live.

    I don’t think that there is a great deal of value for many people in many of the mathematical concepts we teach at school. I could write a long list of irrelevant-unless-you-want-to-be-a-scientist subjects that every child must study until they leave school. Pythagoras Theorem – why? Simultaneous equations – why? Prime numbers – why? Is it enough to say ‘because they are interesting/beautiful?’ I don’t think so. Is it enough to say because they allow us to puzzle and become resilient? I don’t think so – other more relevant endeavours could fulfil these criteria.

    If everyone really has to study maths to 16/18 (but again, I ask why?), then at least *try* to demonstrate how it might be meaningful when placed in context. Can we not at least *try* to place it in contexts that will at least have some impact on children’s lives – and yes, I am talking exactly about what you mention: social justice, racism, sexism, poverty, inequality, psychology, … – contributing to our children’s awareness of the society they live in.

    So what if children don’t learn Pythagoras’ Theorem if they learn instead how to analyse the data that tells us that people in the UK lost £5bn gambling in 2015, and that they majority of those who gamble, and the majority of betting shops, are in the poorest areas. And so what if they can’t reproduce this knowledge in an exam because they got too interested in the context?

    So what if they don’t learn how to solve simultaneous equations if they learn how to interpret the graph that shows that the UK government is reducing the living standards for the poorest people in the UK and improving the living standards for the richest 10%?

    This is not to say that I don’t think maths-in-itself is not valuable, or interesting *for some*. I just don’t think we should expect all students to be interested in maths-in-itself – we should accept that it is only for some (me and you?), and allow others to opt out if they wish.

    And why is it *not* OK to be *not* a maths person? Why does everyone have to be a maths person? This is very odd – it seems OK that people are allowed to not be sociologist people, or philosopher people, or even be politically aware, but every child should be maths person…?? I say we as maths educators should either accept that maths is not for everyone, or try a bit harder to make it more relevant.

  21. 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.

  22. @Danny:

    While I certainly use social justice aspects in my classroom (we’re doing a concussion study today using angular velocity, and later this year I have planned a density lesson using neighborhood segregation) I feel uncomfortable with the standpoint that math has no inherent social justice in itself.

    It does, and I might recommend the book _Radical Equations_ which is the first place I learned of the standpoint that algebra in itself is social justice. That is, providing stronger mathematical background opens opportunities, and denying that opportunity (deciding that perhaps they don’t need to learn the Pythagorean Theorem, say) passes some judgement about what they are capable of in life.

  23. 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.

  24. Just to clarify, I think of the goal of “art class” (or music and other humanities) is to help students gain an appreciation for the subject and learn to dabble in it a bit. In contrast, the goal of the “physics class” (or industrial and STEM courses) is to problem solve and get things done. There is a lot of modeling going on. In the end of course, most don’t care if you took art, liked it, but still can’t paint. Math does not have that luxury because, unlike the humanities (as great/important as they are), math is *useful*.

  25. 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.