“Real-World” Math Is Everywhere or It’s Nowhere

Amare is looking at these 16 parabolas. Her partner Geoff has chosen one and she has to figure out which one by asking yes-or-no questions.

all 16 of the parabolas

There are lots of details here. She’s trying to focus on the ones that matter. The color of the parabola doesn’t seem relevant. They’re all blue. The window of the graph is the same for all the parabolas.

She focuses on the orientation of the graphs and she asks a question using the most precise words she can given her current understanding. “Is it like a hill?” she asks.

geoff responding "no" to "is it like a hill?"

Geoff answers back “No” and Amare eliminates all the “hill” graphs from consideration. So far so good.

now only 9 parabolas left

Amare is now at a loss. She knows that the graphs are different but she isn’t sure how to articulate those differences. “Is it wide?” she asks.

After a long pause, Geoff answers back “Yes.”

Amare eliminates several graphs, one of which happens to be Geoff’s graph. Their definitions of “wide” were different.

"oh no! you eliminated your partner's parabola!"

Their teacher brings the class together for a discussion of the features the students found useful in their exchanges. The teacher offers them some language mathematicians often use to describe the same graphs. Then they all return to the activity to play another round.


Here is a diagram the GAIMME report uses to describe mathematical modeling (p. 13):

the modeling cycle from GAIMME

I contend that Amare and Geoff participated in every one of those stages.

Here is GAIMME’s definition of mathematical modeling (p. 8):

Mathematical modeling is a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena.

I contend that Amare and Geoff satisfy that definition as well.

Many mathematical modelers would disagree, I suspect, given the reaction to my panel remarks last week.

Polygraph isn’t “real world.” They’re convinced it isn’t. When asked to describe how we know a student is working in the “real world” or not, though, they beg the question with adjectives like “legitimate,” authentic,” or “not mathematical” (essentially “not not ‘real world'”).

They can’t offer a definition of “real world” that categorizes the shapes that are right in front of the student right now as “not real.” They just know “real world” when they see it.

The distinction between the “real” and “not real” world doesn’t exist and insisting on it makes everyone’s job harder.

It makes the teacher’s job harder. She has to maintain two models for how students learn — one for ideas that exist in the “real world” and one for ideas that exist in the “not real world.” But they can unify those models! The tasks that mathematical modelers often enjoy and Polygraph should be taught the same way. That’d be great for teachers!

It makes the mathematical modeler’s job harder. The tasks mathematical modelers enjoy are not categorically different from Polygraph. The early ideas that teachers need to elicit, provoke, and develop in those tasks differ from Polygraph only in their degree of contextual complexity. Instead of telling teachers, “Here is how this task is similar to everything else you’ve done this year,” and benefiting from pedagogical coherence, they tell teachers, “This task is categorically different from everything else you’ve done this year and why aren’t you doing more of them?”

I’m trying to convince mathematical modelers that their process is the same one by which anyone learns anything, that they should spend much less time patrolling borders that don’t exist, and instead apply their processes to every area of the world, every last bit of which is “real.”

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


  1. I generally agree. I don’t think that the distinction between “real” and “not real” exists, and sometimes it feels like a proxy for the distinction between “useful” and “pointless.” There is not consensus about which math falls into each of those categories. Sometimes, I feel like other educators insinuate that math is only useful for finance and applications in science/CS.

  2. “…they should spend much less time patrolling borders that don’t exist…”

    I think this good advice in so many arenas of mathematics (and any discipline of thought)!

  3. Ralph Pantozzi

    March 11, 2019 - 9:36 am -

    I have not been met by the border patrol in my travels, but I have been met by those who would move me to look to do mathematical modeling with contexts that I might have previously shied away from.

    A balanced array of modeling activities is a good one. No need to exclude this task from mathematical modeling, but (as stated) it lacks aspects that other contexts have. One nice aspect of mathematical modeling with certain contexts is the lingering (in a good way) debate about what to do with the answers. Does a result like “this graph is wide” help us make another decision?

    • As far as lingering debate about answers goes, I’m wondering if we’ve found the most precise vocabulary for describing parabolas. Are there more precise words we can use? Parabolas we haven’t seen yet that will make our current language less useful?

      The Polygraph context is lacking something that many other modeling tasks have. I’d call it contextual complexity or something similar. Regardless, I’m unconvinced it lacks so much as it to put it in a different category, much less as to require different pedagogy.

    • Ralph, you bring up a thought-provoking question. I would argue that the ability to participate in the ‘lingering debate of what to do with the answers’ is what is being developed through an exercise like Polygraph. Activities like Polygraph help students to develop the skills necessary to meet many of the NCTM’s standards of mathematical practice. In the process of narrowing down the options based on certain features, students are constructing viable arguments, using appropriate tools strategically, attending to precision, looking for and making use of structure, and looking for and expressing regularity in reasoning.

      I agree with you in that there is no need to exclude this task from mathematical modeling, but further, I believe that tasks like this are essential to the process of mathematical modeling. For if you see modeling as a process, a result like “this graph is wide” leads to comparisons of (and discussions about) the features of a quadratic model, the structure of a quadratic function and what the coefficient in front of the squared term represents (and why), and provides students with the experiences/vocabulary necessary to participate in further applications and modeling contexts.

      Of course, I appreciate a healthy debate and would be more than willing to dive into the question you asked further if you disagree.

  4. I’ve been contemplating this intensely since Dan’s last post. I made a comment there that modeling (as defined by CCMS) is using math to analyze “empirical” situations. Well, empirical is defined as “observation or experience rather than theory or pure logic”. Another distinction might be “applied math” and “pure math” or “application” and “proving”. I’ve wrestled with trying to teach a course that separates these two realms and agree with Dan at least on the fact that the less I try and separate or distinguish between the two the more successful the class goes. After all, what makes math pure or applied? …Circumstances? …Context? Even when a mathematician is proving the most abstract of concepts, they still rely on SOME conceptual understanding that is very real to them and develop ideas in a process that mirrors the “modeling” of an engineer or physicist. Why try to separate the shadows in the cave from the objects that make them when all that really matters is that students comprehend the light.

    To say the process we are calling modeling can’t be used when working in this imaginary universe of mathematical images is odd to me, and I’m curious to hear their justification as to why they want them separated. Why is it a “dangerous idea” for us to lump working in both worlds under the singular title of modeling? Anyone got an answer for this?

    P.S. I highly recommend Mario Livio’s book, “Is God a Mathematician?” for more on this.

    • …And for the record, I DO feel like there is a separation between this real, observable world and the imaginative universe of mathematical objects. Probably because I’ve been conditioned to think that way. Of course, just because someone said it is so, doesn’t make it true. Dan pointing out that to a student the polygraphs are more real than, say, electrical circuits makes me reevaluate my stance. Maybe there’s a fuzzy line here. Maybe its a moving target that depends on circumstance. If either of those are the case, then drawing a line between them is not only difficult, but probably pointless. Its a very philosophical question.

  5. Hmm. I think maybe I’m border patrol here.

    This is not to say that I disagree with your use of this task, and especially not with the idea that it is real. But I do think that it is good for students to distinguish between mathematical concepts and those real world phenomena that we use to model them. No one can see a parabola; a parabola is a one dimensional locus of points in a euclidean plane that are equidistant from a directrix and a focus. It has no width, it extends infinitely, and it has a precision that we can never attain in the physical world. It is an idealization built upon more elemental idealizations (points, lines, planes) and is itself a masterpiece of human imagination.

    You are offering students representations of views of parabolas, physical situations that are suggestive of parabolas, and asking them to describe the physical representations and develop common terminology for classifying the differences between them. Hard core modelers might bemoan the fact that these representations are perilously close to the idealizations, but that doesn’t seem to be a serious flaw when trying to develop students ability to converse with a vocabulary that empowers them to describe idealizations called parabolas.

    On the other hand, presenting these representations and saying these _are_ parabolas, and talking about mathematical idealizations as if they are no more than the physical representations being provided, that strikes me as a dangerous path. Mathematical objects exist only in our imagination; they are not real. the whole goal of mathematical modeling is to match mathematical idealizations with physical phenomena, things that are real, and through that matching process observe more deeply the reality around us. The distinction is crucial because it separates the abstract realm of mathematics from the observable world of reality. The abstractions never perfectly match the reality, not even in the case of this exercise. The power of mathematics derives from appreciating that even the limited connections between the two realms that we can make still give us profound depth of understanding. But, especially as we become adept at modeling, it is profoundly important to retain awareness that the elaborate lines of reasoning that we can apply to arrive at conclusions in the mathematical realm need not correspond to consequences in the real world. So, I believe that at all levels we should be clear about when we are speaking about reality or abstraction.

    So, the distinction between real and abstract (which is not real, in an essential sense) is crucial. “[T]he shapes that are right in front of the student right now” are always real, but they are just representations of mathematical abstractions which are infinitely more adaptable and powerful. And while this distinction may make our jobs as teachers of mathematics harder, it is essential for students to remember that mathematical consequences and real world consequences are not the same thing.

    • …but if these aren’t parabolas (I agree with you 100% on that), then these aren’t “imaginary objects” and we are dealing with the “real world” and so by GAIMME’s definition, this activity is modeling.

      Also, I’m not sure the kids aren’t seeing these as the parabolas they represent. And even if they are, do this activity with mathematicians who understand that these are just representations of parabolas, does that knowledge fundamentally change what they are experiencing? I think that is Dan’s point. It’s not that these ARE parabolas and therefore REAL. It’s that our interaction in problem solving whether it be with mathematical objects or with physical phenomena are essentially the same.

      The paradox for me is that I agree whole heartily with everything you’ve said; and yet, at the same time I agree whole heartily with what Dan said. I realize the issues of being in both camps. :)

    • So, the distinction between real and abstract (which is not real, in an essential sense) is crucial. “[T]he shapes that are right in front of the student right now” are always real, but they are just representations of mathematical abstractions which are infinitely more adaptable and powerful. And while this distinction may make our jobs as teachers of mathematics harder, it is essential for students to remember that mathematical consequences and real world consequences are not the same thing.

      My last two posts have made two claims that don’t depend on each other and your comment makes me realize how much more I care about one than the other.

      (1) There isn’t a “real” and “not real” world.
      (2) All learning is modeling, therefore we should apply modeling pedagogy everywhere.

      I’m much more invested in (2) than (1). Modelers are committed to the real / not real distinction. We all seem intractable there. Their argument against (2) depends on (1), though, which is in my view the real loss here.

    • First, thank you ALL for a great discussion!

      I couldn’t agree more about Dan’s point #2

      As far as #1 goes, I have been conditioned to disagree with it, but I am starting to wonder if it is at all necessary–at least for the general public. I fully realize that mathematical images exist only in the mind and that physical reality we model them with are only shadows, approximating reality and therefore limited to certain assumptions, ect. I’m not sure this is a necessary metaphysical belief to do or apply mathematics. (I’m not saying we should tell students the parabolas they are looking at in a graph ARE parabolas, but if the representation has the same characteristics, I see no point in explicitly bringing it up. Pints, lines, and planes, or tables, chairs, and beer mugs as long as the behave the same who cares!)

      I also realize that, to a kid looking at a bunch of graphs, parabolas may seem more tangible and real than actual physical phenomena. As Dan showed, their thinking is very much following the modeling process. I see this time and time again when students are building knowledge about the abstract world of mathematics, so I see Dan’s point that if we’re doing the same thing, why are we calling it something else? (Here I should probably add that I would hate to see a math classroom avoid seeing how math “gets stuff done”, which I guess would be my personal definition of “real-world” modeling)

      This conversation has raised some questions in my mind:

      Q1-What are the effects of not specifically addressing a distinction between real/mathematical? why should or shouldn’t we draw this line?

      Q2-Supposing there is a good answer to Q1 that we should: Why do we need to distinguish between “modeling” and “problem soling” or “learning” or whatever you want to call it when we are in the mathematical realm? Is there a reason for wanting to distinguish them beyond the “real” and “non-real”? Is it just to ensure teachers are helping students see how math is useful? I certainly see the benefits of “apply modeling pedagogy everywhere”.

      Looking forward to hearing both sides. :)

  6. Annalee Salcedo

    March 11, 2019 - 11:42 am -

    “I’m trying to convince mathematical modelers that their process is the same one by which anyone learns anything, that they should spend much less time patrolling borders that don’t exist, and instead apply their processes to every area of the world, every last bit of which is “real.””

    Featured Comment (#CIME2019 Co-Panelist)

    I am convinced that the process is the same, that all learning, like the process laid out in GAIMME, like the process you illustrated above with the parabola polygraph, is iterative. But to answer the question that MaryAlice asked that wasn’t actually answered during our panel at CIME: I think what is lost when we say “all learning is modeling” is the imperative to bring problems that are “value-laden” to our students. Prior to CIME and this exchange, I would have continued to call the circle-square problem (one of your 3-Act Tasks) mathematical modeling and offer it as part of the evidence that MM has taken hold in my classroom. After CIME and this exchange, I am still committed to doing the circle-square problem but, more importantly, I am inspired to look for and develop problems that are “value-laden”, problems that exist in our unjust world, that can be mathematized, and that compel students to articulate their values and the values they see, brought to light by the MM problem, expressed in the world.

    When facilitating these MM problems with my students, you can absolutely bet that I’ll have them reflect on their process as I often do after other rich problems they work on. And yes, I hope they do see that the process they applied to the MM problem is the same iterative process they applied as they developed, for example, an understanding of the verbal, graphical, and algebraic representations of parabolas.

    So lots to gain to by pointing out “pedagogical coherence” but also lots to gain by clarifying a definition for mathematical modeling.

    • When facilitating these MM problems with my students, you can absolutely bet that I’ll have them reflect on their process as I often do after other rich problems they work on. And yes, I hope they do see that the process they applied to the MM problem is the same iterative process they applied as they developed, for example, an understanding of the verbal, graphical, and algebraic representations of parabolas.

      Awesome. I think it’d be fabulous if you and other modelers, in your outreach with teachers, ran both a modeling task that’s value-laden with lots of contextual complexity alongside a task like Polygraph, so teachers could understand their pedagogical similarities.

  7. I think there is a real distinction between pure and applied mathematics, and it’s one worth preserving. But applied mathematics shouldn’t just be blindly checking whether various functions approximate various data. I think the math edu establishment has put itself in a bit of a bind here, because responsibly applying mathematics to the world involves deeply understanding the context. But deeply understanding a particular context takes us away from what we are currently charged with as math teachers — it takes us into physics, applied history, finance, epidemiology, management science, and so on.

    I understand the difficulty, maybe. We’re trying to find something that is possible for kids to do that has some resemblance to what working people do with math. But in the process it seems we either get a version of modeling that is so BIG it can include all learning and all math. Or else we define modeling so that there is an insurmountable gap between what our students and mathematicians can do. (Applied math is not just picking one of four functions and iteratively fiddling with the parameters.)

    So I think I am OK with the status quo, where mathematical modeling is sort of losing steam as a math edu initiative.

    • But if you take the application out of mathematics, then it is nothing more than a humanities subject. (a cool one, and there’s nothing wrong with the humanities). Without that, does it lose its bite as an essential subject? Should everyone still learn it? My thought is: of course, otherwise you can’t do the applications when they come up in science, et al.

      But lets be honest, the general public has reject mathematics as a humanities subject. And Math edu of the last 50 or so years is largely to blame. Without showing the masses why its worth learning, few are willing to learn it.

  8. I join Rob Root on the border patrol. Or, more specifically, in believing that there is an important difference between the abstract world of mathematics, conjured to existence by thought, and the concrete, physical world we inhabit. As Rob points out, one of the most remarkable features of the world we live in is that these two very different worlds are related and that we can use our thinking in one world (the mathematical one) to understand the other (the physical one). Wigner called this “the unreasonable effectiveness of mathematics” and that’s what’s lost, that is, having students experience and understand that magic, when one attempts to equivocate on the use of the word “real.” Again, as Rob points out, making that connection requires a different set of skills than does working purely in the mathematical world, no matter how much iteration may be a part of both. If we avoid that fact, we do our students a disservice by failing to recognize we’re not giving them the skills to use mathematics in a powerful way to understand the “real” world. As Michael Persham correctly points out, this requires that our teaching goes beyond mathematics and incorporates ideas from physics, chemistry, sociology, etc. I suspect that’s what makes uncomfortable those who want to redefine mathematical modeling in such a way as to be able to remain only in the world of mathematics. It’s not easy to take this leap, but absolutely essential for so many in so many ways. For some more thoughts on this distinction, I offer: http://modelwithmathematics.com/2019/03/what-do-we-mean-by-real-world/

  9. I am reading this discussion for a while now and am unable to sense the underlying problem, maybe because I am not involved in your US system. It looks to me as if this great page is spoiled by some academic discussion.

    The process of learning is always the same: an iterative, cross-linked accumulation. That does even apply to the skill of modeling itself, by the way, and is obvious.

    Another question is whether or not we want to teach to students the application of math to the real world around us, maybe even base our math education on applications. My answer is yes. I am talking about counting, calculations, geometrical objects, physics, statistics, predictions. These are all real-world things that can be treated (modeled, simplified, structured, understood) by math. Can there be any doubt that this belongs to every math education?

  10. Julia Aguirre

    March 12, 2019 - 6:32 am -

    I know it was not your intent to dismiss or disrespect the math modeling work of the math teachers, math educators, mathematicians and industry professionals on your panel and in the audience, including myself. However, the impact of your 5 minute talk did just that, in both tone and content. Furthermore leaving the CIME workshop right after your talk was a missed opportunity for you to learn more about the important connections of mathematical modeling to equity and social justice frameworks, assessment, and to your own work (e.g. three act tasks, use of technology). Taking to twitter and your blog to argue points with limited context for your social media followers is unproductive. I think we have common goals for children and youth to learn rich, rigorous and relevant mathematics. The young people I have worked with want to see and experience the relevance of mathematics to their lives. When given opportunities to mathematize their world through video, images, inquiry or data, they get excited and more willing to engage in analysis of polygons, systems of equations (including graphical representations) or proof. In the interest of moving forward to transforming the experiences of young people learning math, I would ask you to: check your privilege, actively listen to others who may have a different perspective then you, and be open to revising your own assumptions about mathematics teaching and learning. Your platform is powerful. Use it responsibly.

    • Thanks for your feedback here, Julia. I agree that I missed out on the diversity of ideas and perspectives at CIME. My loss.

      I hope you’ll reconsider your perception of social media, however. The conversations and ideas presented at #CIME2019 are too important to limit to a few hundred people who had the resources to attend.

    • Julia Aguirre

      March 13, 2019 - 8:28 am -

      Dear Dan,

      Your response to my comment contains 2 points that need to be addressed.
      1) I agree. The CIME workshop on mathematical modeling has important ideas to share. It would be great to continue a productive conversation at larger conference like NCTM or SIAM or perhaps a podcast, something that offers more capacity to productively engage ideas together rather than the asynchronous format of blogs and twitter. I appreciate those formats as I do your work on them. But, as you know, they are limited in many ways. I will be at NCTM this year in San Diego. I invite you to meet there and we can discuss common ground further.

      2) As for your comments about who attended the CIME conference and their resources, you are right the conference was small, and many participants (many of whom were teachers) were local. However, if you looked on the CIME website as well as the announcements made during the conference, there was funding available to offset costs to attend the workshop including help with childcare expenses. The CIME organizers are sensitive to costs especially since the point of the Critical Issues in Mathematics Education workshops are to bring math teachers, math educators, and mathematicians together. Participant funding is crucial to support inclusive voices.

  11. Daniel J. Teague

    March 12, 2019 - 6:46 am -

    I’m sorry for this overly long response, but I couldn’t write it short (and you could easily say that I couldn’t write it long either).

    In the poem entitled “Poetry”, Marianne Moore describes poets as “literalists of the imagination” and poetry as “imaginary gardens with real toads in them”.

    Mathematicians, too, are literalists of the imagination and like poetry mathematics has its imaginary gardens and real toads, although we more commonly call them mathematical theory (pure mathematics) and mathematical modeling (applied mathematics). This duality we see in mathematics is one of the defining characteristics of many forms of art (form and function).

    The garden and the toads are not the same thing, although there are strong commonalities in how we approach the problems in each area.

    The creative component of pure mathematics is found in conjecture and proof, in the process of mathematical research. This process is played out under the strict restrictions in the idealized world of abstraction and mathematical proof and is precise all the way down to pi (which is never confused with 3.1459). The imaginary garden places no value of utility; truth and beauty, logic and rigor reign here. Every theorem has conditions, and any variance in any one of those conditions means we have nothing, the theorem does not apply. We can create a new theorem with the modified conditions, but it is not the same theorem.

    The creative component of applied mathematics is in mathematical modeling. Mathematical modeling has constraints imposed by the real world; reality is the master. But the flexibility in our use of mathematics gives us freedom that doesn’t exist in the imaginary garden. If the conditions of a theorem are nearly met , then we the results of the theorem may be approximately realized (the entirely of statistics depends of this notion). We constantly use approximations and even incorrect calculations because the goal isn’t getting the correct answer (there is no correct answer), the goal is gaining some imperfect understanding the component of the world being modeled. Instead of the optimal solution, we are after some satisfactory solution that is the easiest to implement. Practices that are essential in modeling are forbidden in theory, and this is a freedom I think students enjoy.

    Certainly, modeling is ubiquitous in life. My perspective is that mathematical modeling is a special kind of modeling and should have a special place in every students mathematical experience. In this respect, if “all learning is modeling” is the same as “all mathematical learning is mathematical modeling”, then the idea to me is dangerous in the sense that students will never engage in the kind of mathematical modeling that I have seen to be so extraordinarily powerful and empowering; its an opportunity lost. Having taught with a mathematical modeling perspective for 30 years, I have had two decades of 10-year reunions and a decade of 20-year reunions.

    Featured Comment

    Without exception, my conversations with former student immediately converges on their experiences with mathematical modeling. We never discuss that day I taught the double-angle formula. The residue of my courses is almost entirely the challenges and excitement that came from mathematical modeling.

    This singularity is why it is clear to me it is different from other mathematical activities the students have experienced. Every year, the conversation is peppered with “it changed my life” or “I finally realized I was good at mathematics”, or “the first math course I ever liked”. Modeling doesn’t focus on being quick, but on being deep and creative and integrating your own ideas with the mathematical principles you’ve chosen. Every student’s real-world is different, which is why it matters so much to them that they can embed their reality in their mathematics. That’s where the wonderful variety of approaches comes from and why mathematical modeling is such a great way to get to know your students’ realities.

    Mathematical modeling calls on the students to use the essential concepts and mathematical tools developed in class in creative and collaborative investigations. A fundamental component of the joy and excitement of mathematical discovery lies in the ability to direct the process of discovery, to have some significant ownership of the problem being investigated and the processes used in making and sharing the discovery. Ownership of the mathematics occurs when students have the flexibility to make decisions about what to solve and how to solve it themselves. This means they are thinking their way through problems rather than just remembering what they were told to do or repeating the teacher’s approach. Ownership is power and power generates agency. Students can be led to create their own problems or, at the very least, help frame the problems they encounter. When given the opportunity, students can engage in thoughtful mathematical activities that highlight the creative aspects of the subject. What I see every year is that students gain confidence in their mathematical identity when the solve problems in their own way. They gain control of the mathematics rather than having the mathematics control them. They use their own minds rather than borrowing the teacher’s or Newton’s mind, so thinking is dominate over remembering.

    • Thanks for your response here, Daniel. Certainly, I’m happy to host comments of any length from y’all folks who have put in so much time and energy into the project of mathematical modeling.

      Without exception, my conversations with former student immediately converges on their experiences with mathematical modeling. We never discuss that day I taught the double-angle formula. The residue of my courses is almost entirely the challenges and excitement that came from mathematical modeling.

      I’m not interested here in preventing you or your students from enjoying your experiences applying the skills of mathematical modeling to situations outside the classroom, situations that are value-laden or that contain lots of contextual complexity.

      I’m interested here in understanding why we don’t teach the double-angle formula in the same way.

    • Daniel Teague

      March 14, 2019 - 2:09 am -


      I appreciate the response and acknowledge and agree with your point. Indeed, the Arise project out of COMAP and the Precalculus and Calculus texts may colleagues and I wrote attempt to do precisely that. But there is a significant difference in teaching from a modeling perspective and students engaging in a modeling experience as described in the GAIMME Report. The difference lies in the locus of control of the mathematics.

      If I create an activity or project that supports students discovery of or exploration of a topic in the curriculum, the activity is necessarily scaffolded or limited in a way that draws from the students the desired result. We can see from the exploration of the pattern of sound waves that the sum of two sine waves can be written as the product of sine functions, which creates the notion of a trig identity. Or by consideration of pollution flowing through the Great Lakes, students recognize that the output of one equation must be the input for the next, and through this recognition, devise a way to create a system of difference equations. As you note, using small modeling activities as a generator of mathematical tools and concepts can be an excellent way to motivate both mathematics and students. Modeling shares this attribute with many other excellent approaches to student engagement. But, in these settings, the student is still following the lead of the teacher; the leading being less visible. The space in which the students can move mathematically is limited because the goal of the lesson is a particular result.

      Mathematical modeling (my definition here) gives the students more room to move and gives them ownership of the problem. Ownership is the power that drives the process. Their ideas and perspectives and values are important and valued. The exploration itself and the effect the process has on the students’ understanding of mathematics and themselves as young mathematicians is sufficient purpose for the activity. The smaller, content-focused modeling activities builds confidence and gives some guidance to students on how to think about and approach the more open-ended projects, but they don’t replace those projects.

    • Mathematical modeling (my definition here) gives the students more room to move and gives them ownership of the problem.

      No doubt.

      I am excited to see “more room,” though, which implies a continuous variable rather than a discrete one. That’s largely my point here, that the binary distinction between real / not real & modeling / not modeling is illusory and creates unfortunate pedagogical binaries.

    • Ralph Pantozzi

      March 16, 2019 - 7:38 pm -

      Quoting Dan M repeatedly here:

      We can and should teach the (fill-in the blank with your desired topic) the same way. Do we want self-identified math modelers or the modeling “establishment” to “apply their skills across all of teaching”? Yes!

      “The tasks that mathematical modelers often enjoy and Polygraph should be taught the same way. That’d be great for teachers!” Yes!

      The thought “The distinction between the “real” and “not real” world doesn’t exist” is debatable. “Insisting on it makes everyone’s job harder” is also debatable. Are modelers “committed to the real / not real distinction”? ¯\_(ツ)_/¯

      Do “we just need them (modelers) to drop this meaningless distinction between the real and non-real world” to pursue this work? I don’t believe so.

      Talking with students about distinctions along the continuum of the “real” and “not real” (feel like I’ve been watching the Matrix movie series here) can advance the work of “eliciting early ideas, provoking students to determine their limits, and helping students develop their ideas further”.

      Across time and context, what appears “real” and “not” to a person can vary, and “understanding” something often makes something “real.” There is a spectrum of the “real” as has been noted. Yes, a set of blocks is “real” (or a picture of those blocks). They’re also both potentially not real at all, just as the Polygraph can be real (or not).

      Graphs or anything on the spectrum of “contextual complexity” are not necessarily real in the same way that a question about a change in the price of a life-saving drug might be, or a decision on which candy bar is a better deal. (Or in recent news, the velocity data of two crashed airplanes.) These are all real in different ways, and all have the potential to be “not real”. GAIMME and other examples push us in a good direction, toward ideas to notice.

      As Dan M has argued in his pseudocontext series, the students know when we’re pretending. Let’s continue to talk to students about what they think is real.

      Will read Dan M’s lit review in detail and be back. Thank you!

  12. I see ” “Real world math” is everywhere or it is nowhere.” as a beautifully succinct expression of the ultimate abstraction of mathematics. Modeling can not cover “everywhere”. It does not converge to abstraction. To appreciate this It is not necessary to do mathematics the way a mathematician does. Maybe it only requires a gut feeling that math has an essence that holds it together for you to find if you need it. The mathematics you use has absolutely no dependence on what it is used for. Its essence comes from mathematics itself.

    My concern is not that abstraction isn’t explicitly taught, it is that it is denied. Maybe it is sufficient to give permission to think about an essence to the parabola beyond orientation, size, or location, to play with in your mind, no need for a grid or a coordinate system.

    Keep up the stimulation.

    • Funny, I was taking it the other direction: even the most abstract mathematical thought gets represented somehow–graphically, symbolically, etc.

      The task of the mathematician is to find a representation that holds true in our reality (has the same preperties), allowing communication and comprehension of the abstract.

      To further the irony, when I first started teaching, I was very heavy on the abstraction and way too light on the modeling. I had to make a conscious effort to leave the world of pure mathematics and enter “contextual complexity”.

  13. People engage in mathematics for different reasons, being propelled by different motivations, and this is what I’ve used to mentally separate pure mathematics from applied mathematics and, as I’ve used the term and heard it used, including modeling as a subset of applied mathematics. There is not necessarily a difference between the mathematics being done, but rather why it is being done, and while that may not be a distinction worth arguing labels over it is certainly one that has pedagogical implications. I’m reminded of Paul Lockhart’s statement that “you have to have something you want to run toward” and this something need not be the same for all people. Conventionally, modeling has been used (I think) to describe situations where we want to run toward a solution to a physical problem and mathematics provides a path. The term pure mathematics has been used (I think) to describe situations where we want to run toward understanding. The journey and destination might even be the same, but the reason we began each journey is different and if for no other reason than describing tasks in a way useful for tapping into student motivation, it might be worth preserving the distinction the term modeling has generally provided.

  14. David Albertsen

    March 13, 2019 - 9:12 am -

    I’m a bit sad to see that some people got upset over Dan’s comments regarding modeling. Teaching math is hard enough without having to choose sides, or tear down well-meaning people. From my perspective, it has been a positive experience to read the arguments on both sides and think more deeply about my everyday practice of teaching math to alternative high school students.

    It seems to me that we need a mix of teaching what my colleague calls “tool skills” and application activities. That doesn’t contradict what I think Dan is saying. Please correct me if I misunderstood, but my interpretation of Dan’s point is that we should place more focus on the value of student misconceptions during the process of teaching. Looking for ways to better understand (model?) why impartial mastery makes sense in the student’s mind at that moment provides the best route for a teacher to provide targeted and useful bits of instruction (e.g. “I think you might have been subtracting instead of adding, multiplying or dividing. Tell me more about why subtracting makes sense in this situation.”).

    Finally, I really want to say how much Dan Meyer and Desmos have improved my teaching over the last few years. I regularly use curated Desmos activities, and make my own custom activities. They provide a visual, dynamic foundation for understanding that helps me guide students through experimenting with their own ideas about the math (what Dan calls modeling/learning?).

  15. Seems like the panelists’ interpretation of modeling is a subset of your interpretation. It looks like you both agree that modeling is a verb (tinkering, playing, collecting data, adjusting initial/boundary conditions, asking “what happens if…”, etc), making “mathematical modeling” the use of math within the action of modeling.

    To me, superimposing a “real-world” condition onto the definition stated above is outside of the boundaries of “modeling” using “math” – hence requiring a new term. In practical terms for teachers/students, the focus should be on the process of modeling with math, regardless of context.

    For example, the “Circle-Square” problem (Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal) and slide 17 of “Transformation Golf” (determining equivalences between series of transformations) would be considered outside the realm of “real-world”, using the panelists’ definition/intuition. However, I’ve had great success (just yesterday) in my classes tackling these problems in a way that was identical to analysing projectile motion and the spread of a disease through a population. If you remove the context and look at the verbs, each problem is quite similar and falls within the definition of modeling with math.

    I feel like there is a common thread here between pseudo-context, “real-world”, and modeling. Something like the discus textbook problem in your recent talk, even though it models a physical situation, wouldn’t be considered modeling because it was already modeled. There’s no act/verb going on there, and is maybe the distinction that could be more useful for teachers to focus on rather than “real-world”.

  16. I appreciate it when Dan pushes back against ideas. Recently it was the idea of “mistakes” with regard to Growth Mind Set. This time it’s modelling.

    I don’t know that I have anything profound to add to the discussion, but after following it her and on Twitter I keep coming back to Math Practice Standard #2 “Reason abstractly and quantitatively” It feels like much of this is tied to the ideas of decontextualizing an contextualizing. It sounds like the other members of the panel require a context in order for it to be modeling. It also sounds like Dan says once you have decontextualized the math then it’s the same pedagogically.

    That’s likely a gross over simplification, but I’m trying to understand what the rest of the panel disliked so strongly. Do they just think that because there’s no physics problem to recontextualize the math to that it can’t be modeling? I’m not sure I buy that. Decontextualizing just sounds like another step in the iterative process that you go through to solve the problem as opposed to something that causes the whole definition to break down. And if it’s just another step in the iterative process then is it that crazy to refer to the whole thing including decontextualization as modelling?

    Like I said, I’m still trying to work through it all…

    • Thanks Scott. That’s my biggest question too. Why do they think this idea of not distinguishing problem solving with a rich context from problem solving without a rich context is dangerous? No one has answered that for me yet. I’m very curious to know their thoughts.

    • Daniel Teague

      March 14, 2019 - 2:02 am -

      Scott and Taslam,

      I encourage you to read again John Pelesko’s contribution to this discussion. John Pelesko and Michelle Cirillo’s blog Model with Mathematics was referenced. Their presentation of mathematical modeling is quite wonderful and may help explain the importance of distinguishing the real-world that we see in mathematical modeling. I have copied several paragraphs from “What do we mean by “real-world”?” that are particularly salient.

      [dm – folks can click the link to read it]

    • Thank you. I appreciate the link. O have read it and it seems to be just repeating things already said. What I might have missed is the argument for why its important to distinguish between the two. Let me see if I’ve synthesized that argument correctly (feel free to correct me, this might be overly simplified):

      1) the argument for not extending the “real-world” to mathematical forms is so that students realize the represantions are not the actual forms and therefore lake the precision of the forms.

      2) the argument for leaving the definition of modeling to “real-world” objects is because students need to build the skill set involved in contextualizing and decontextualize.

      As for the definition of “real-world”, things of substance won’t work because Dan’s graph and Polesko’s triangle representation both have substance. Granted, they aren’t the actual mathematical forms, but Dan is saying the students are modeling the physical graphs that represent the parabolas with verbal mathematical descriptions (which could be refined into symbolic ones). How does this fail to fit your definition?

    • It sounds like the other members of the panel require a context in order for it to be modeling. It also sounds like Dan says once you have decontextualized the math then it’s the same pedagogically.

      Right! Same pedagogy in both cases.

      But I’m also am saying, along with other folks before me, that “contextual” isn’t a binary variable (where “decontextual” and “contextual” are the only settings) but a continuum. Decontextualizing (or “abstracting”) is the act of removing detail. You do that everywhere. When you drive, you don’t pay attention to every detail around you. You abstract away the details that aren’t important for driving. You have decontextualized the world. You do the exact same act when you’re trying to describe parabolas! You decontextualize the set of parabolas to a set of verbal descriptions.

  17. I think I understand Brian’s points clearly enough to see how we differ. A discussion around them could be useful if I can hold up my end.

    I do not think “applied and pure are” two different kinds of math. There is one kind of math, call it mathematics. Applied mathematics applies parts of it to particular kind of problem solving. We agree modeling is at least a part of applied mathematics and therefore it is not a branch of mathematics for me. There are other disciplines with this sort of relationship; eg. physics and applied physics come to mind.

    You say “modeling is used when you want to run toward a solution to a physical problem”. The physical sciences are not built on modeling, they are based on physical understanding. In the absence of that understanding it may be necessary to turn to modeling, but modeling is not a substitute for understanding. Much of mathematics and physics begins as co-understandings.

    I confess I feel strongly that understanding and abstraction are not luxuries at any level of mathematics. American students have long had problems with signed numbers and fractions that have been, and still are, explained by modeling; negative numbers as numerals on thermometers, numbers on a number line the other side of zero from positive numbers, as the downward numbers on a graph, as charged particles. Subtraction is taught as removal. And the students end up mystified by “minus a minus is a plus”.

    Why aren’t we trying to fix the problem students have with signed numbers. It is not difficult to construct them as oriented differences of pairs of natural numbers to abstract out their essence from relations to the “real world”. More or less the same thing can be done for fractions.

    It is not that abstraction is invoked at every use of integers or fractions. It is (1) the utility of mathematics relies on abstraction, otherwise mathematics would depend on the definition of words; (2) we can answer questions like, what is “minus a minus”; (3) when we solve problems, we don’t need to rely on retrieval of modeling relations to plug numbers into, we can sometimes use the deeper abstracted essence of mathematics; “negative integers, and positive ones too, are oriented differences, they are not intrinsic properties of physical projects or directions, nothing is being modeled by anything else.

    Most of us learned school mathematics in terms of relations, by modeling essentially. It can be convenient to believe mathematics is more or less just another natural language where it and the natural world are entangled by intertwining definitions. This is not mathematics, and that is fantastic news for students. Mathematics is powerful, universal, and unique. This does not seem to come across in school mathematics.

    • First, there is some great work by Henri Picciotto and Hung-Hsi Wu on negatives and fractions respectively. Really good stuff! …Especially Wu’s stuff. What he says is very similar to what I think you are saying.
      Henri Picciotto’s webpage: https://www.mathed.page/
      Hung-Hsi Wu’s webpage: https://math.berkeley.edu/~wu/

      And, I agree that the power of mathematics is abstraction. I can’t really add temperatures on a thermometer in a meaningful way (sure I can add their values, but so what?), but I *can* and numbers on a number line. I could even add numbers without a number line, but the representation offered by the number line is there because it is sound. The adding that takes place on the number line behaves the exact same way as the numbers do without it.

      But, I don’t think the problem lies in giving students context to attach meaning to. Otherwise no human ever would learn any math. We wouldn’t have gotten where we are today. But these models need to be sound in their properties. I can’t show students a bunch of marbles in a jar with negative numbers painted on them and expect them to understand what negative numbers are because they don’t share the same properties (in this case the marbles have no ordering). No, I don’t think the problem with students understanding about negative numbers and fractions stems from contextualizing them. It has more to do with misunderstanding there properties.

      For example, Picciotto points out that the “-” symbol actually has three distinct meanings: minus (7-4=3), negative (as in -2), and opposite (as in -x, which could be positive or negative depending on the value of x). we’ve used a representation that is potentially confusing because it is crossing over into other properties: “subtraction” is not “negative numbers” nor is it “opposites”, but they are related. and we chose to represent them using the same symbol because of this.

      Likewise fractions take on several meanings which Wu lists: parts of a whole, the size of a portion, the quotient of the integer division, a ratio, an operator (like “2/3 of…”). This list comes from Susan Lamon’s 1999 work, Teaching Fractions and Ratios for Understanding. Wu goes on to quote Lamon:
      “‘As one moves from whole number into fraction, the variety and complexity of the situation that give meaning to the symbols increases dramatically. Understanding of rational numbers involves the coordination of many different but interconnected ideas and interpretations. There are many different meanings that end up looking alike when they are written in fraction symbol” (pp. 30—
      31 [of Lamon’s work]). All the while, students are told that no one single idea or interpretation
      is sufficiently clear to explain the “meaning” of a fraction. This is akin to telling someone how to get to a small town by car by offering fifty suggestions on what to watch for each time a fork in the road comes up and how to interpret the road signs along the way, when a single clearly drawn road map would have done a much better job. Given these facts, is it any wonder that Lappan-Bouck (1998) and Lamon (1999) would lament that students “do” fractions without any idea of what they are doing?”

      He goes on to say that clarifying what is going on with fractions isn’t obtained by eliminating contextual examples (on the contrary, he pushes for an early introduction to fractions in everyday context as early as 2nd grade), it is gained by having the student “sit down to organize and theorize about” their experience with fractions and arriving at a very clear, precise, sound definition of what a fraction is and what its properties are–which I think is what you are driving at… maybe. :)

      For the source of these quotes, I point you to here:
      I found this a fascinating read!

  18. This topic is so important and the discussions taking place are “pure” gold (pun intended). I recently read Dan’s twitter post and responded to his tweet with a few graphics and an excerpt from a book that addresses mathematical modeling because this topic has been an integral part of writing my books over the last 10 years. At the end of the day, in my personal opinion, I wholeheartedly believe applied and pure mathematics are not mutually exclusive and as a result, both should be honored in our teaching practices. A few exerpts from the book: “you may notice that although the method helps discover relationships and rules, it operates in both a pure sense and an applied sense in that discoveries within number systems may be explored for nothing more than pure discovery with no application. Take, for example, discovering that an ODD number multiplied by an ODD number will always be ODD. Is there any application to context that would prove this discovery useful? Or is it useful because it is, in fact, building a mathematical habit of mind, engaging in a mathematical process of curiosity…” The author continues later with “Applied math employs modeling, in addition to pure math. Consider creating a model to represent a pattern or solve a problem by creating a new system (such as creating the imaginary number i ). That model was invented simply to create a method for solving problems involving negative radicals. That “model,” if you will, contributed to some of the most important constructs in technology. Scientists, for example, use imaginary numbers to understand the flow of fluid around various objects, as well as for uses in electrical circuits and the transmission of radio waves. Does this mean the invention of the imaginary number created the means to understand these scientific phenomena? Or is it possible that these connections and problems could have been solved using other means, or not explored at all? Did it merely allow people to more easily understand the science and math behind these natural phenomena? In many respects, one can easily conclude that applied math and pure math are not mutually exclusive but, rather, co-dependent on one another in many ways. ***Critique: Alexander claims that mathematical modeling is a dynamic process involving great synthesis of ideas used to problem-solve, make predictions, and extrapolate ideas and that true mathematical modeling has nothing to do with drawing pictures, creating a scale model, or using a visual representation to problem-solve “real world problems”. Do you agree? Why or why not? Critique: Isaiah disagrees with Alexander and states that solving problems at any level is modeling because our understanding of mathematics is simply a replica of a perfect system we seek to understand in an imperfect world. Essentially, he believes that every math sentence we write is a model. He contends that a student choosing to create a number line, for example, to represent 3 fish joining 2 other fish, then writing the equation 3 fish + 2 fish = 5 fish to find a sum of 5 fish is mathematical modeling, even just simply 2+3=5. He furthers his point by showing how the student had to solve a real-world problem by making sense of the situation, deciding on a strategy to find the answer, and adjusting the model if necessary. Further, the student can make predictions about his equation used. Can the equation be used to solve other problems involving 3 items joining 2 more? Did the model serve its purpose? Can you make a prediction about 3 dollars added to 2 more using this model? Can this same model work with adding 3 halves to 2 halves? Or 3 negatives to 2 negatives? How can you alter the model to solve other problems? In his opinion, these questions are the precursors to the use of mathematical modeling in higher levels of mathematics, but mathematical modeling nonetheless ….” (McGill 2013-2018) I know this was a long post with seemingly disconnected excerpts from the book, but the point I was hoping anyone reading this would conclude is that there is beauty in both pure and applied math and the process of learning mathematics can be joyous in both areas. Blessings!

  19. The problem here is trying to use the overly philosophical terms “real” and “not real”. Real vs. not real does not matter. What actually matters here is “inherently or intrinsically valued” by humans vs. “valued because of enabling something that we intrinsically value”.

    Knowing parabolas is not (for most people) intrinsically valuable. The food is intrinsically valuable, that one can buy with the wealth plundered due to the ability to strike a castle from 1000 yards out with a cannon by knowing the right angle and speed (to use a crude but accurate example).

    Knowing how to solve chemical equations is not (for most people) intrinsically valuable. The joy one feels is intrinsically valuable, that one can get from making a lithium-ion battery that can power a phone that plays a comedy special on YouTube.

    • What about the joy one feels learning or solving problems about parabolas? I’m reading a book right now, The History of Algebra, for the sole intrinsic value that it brings me joy.

  20. …oh. “most people”…


    But I think that’s why Dan is making his point. Students get excited about solving problems–if well crafted. Historically, so much mathematical work has been done solely for the joy of figuring it out. Let them find intrinsic joy in driving the double angle formula.

    • David Santo Pietro

      March 15, 2019 - 2:26 pm -

      Yes, that’s correct, “most people” do not find math intrinsically valuable. Yes, mathematical work has been done solely for the joy of figuring it out….by those relatively few people who find it intrinsically valuable.

  21. Drop “real”, consider “embodied”.

    Marbleslides is embodied in the way Polygraph is not. Marbleslides evokes physical movements and physical actions: the veeery satisfying, bouncy marbles. Polygraph evokes beautiful visualizations.

    Both are lovely for modeling, and knowing which is embodied has meaningful teaching implications. Yay?

  22. Melissa Allen

    March 31, 2019 - 12:15 pm -

    As a future teacher, it worries me how generations look at “the real world”. We often believe that the “real world” is not math based- that it is business based, conversation based, or humanity based. But it is all those things and math. I think this is why students are drawn away from mathematics and learning from it. Its evolved from this preconceived idea that math, or at least higher level math, is not something used in our every day, real life experiences. While in a normal day, not everyone takes a derivative or graphs a polynomial, it is important to know why we have these representations and outcomes. Math is applicable almost everywhere. From art to business, we use math to make decisions and analyze our world to a much more precise level. I think that when parents and non-educators look at curriculums and ask questions about why students are learning such material, like polygraphs, they don’t understand the bigger picture. Not every student will use this, they’re right on that, but they will see it. If they don’t understand what a graph or a statistical statement is trying to make, they will be confused and walk through life not really knowing what the value of numbers really means.