“The” Answer

This post was inspired by a recent twitter conversation between @woutgeo, @ddmeyer, @cheesemonkeysf (cool handle!), and myself. The 120-character at a time conversation revolved around a comment from @woutgeo:

“…am still mulling whether modeling Q’s can have correct, known answers”

Since this seems to be a common point of confusion, or even contention, I thought I’d talk about this idea a little today. That is, what do people mean when they make statements like “Modeling problems don’t have a single, unique, correct answer.”? If you read the introductory chapter to the new Annual Perspectives in Mathematics Education (APME) volume, Mathematical Modeling and Modeling Mathematics, you’ll find versions of this statement in several places. I’ll highlight two:

There are multiple paths open to the mathematical modeler, and no one, clear, unique approach or answer.

Mathematical modeling authentically connects to the real world, starting with ill-defined, often messy real-world problems, with no unique correct answer.

What do people, in particular, what do those of us who do mathematical modeling professionally, mean by such statements? What are we really trying to say with statements like “no unique correct answer”? There are actually multiple levels to this point, and I think it is worthwhile exploring a few of these levels here today.

At the simplest level, such statements express the point that the answer to a modeling problem is not like the answer to a typical textbook or classroom math problem. When we think of the idea of an “answer” to a math problem, due to many years of repetitive training, what we most often visualize is a number. “The” answer is 586, or 7, or 2+3i, or \sqrt{17}, or something like that. Perhaps, if we’re a bit more deeply immersed in algebra, or trigonometry, or calculus, our default vision of “answer” might be more like x^3 or \sin(3 \theta) or \frac{1}{x} or some such expression. Note that this default vision of an “answer” is some form of mathematical object and tied to that, perhaps so intimately that we don’t see it, is the idea that this answer is easily checked. It’s the result of “doing the math correctly,” and hence, of course, we should only get one such answer. But, when we talk about the answer to a modeling problem, these are not the types of objects we’re talking about. In one very important sense, the answer to a modeling problem is a model. And, here is the first place where this idea of “no unique answer” comes into play. Because models of a given real-world situation can be constructed using wildly different mathematical tools and are based on assumptions made by the modeler, it is often the case, likely even, that two modelers approaching the same problem produce different models, i.e., different “answers” to the same modeling problem. This is the point that the first of the two statements from the APME volume above is really making.

But, there is another level to this idea of “no unique answer” that’s worth exploring. The second statement from the APME volume mentioned above points to this next level. Let’s examine the statement again:

Mathematical modeling authentically connects to the real world, starting with ill-defined, often messy real-world problems, with no unique correct answer.

Here, note that “answer” is not referring to the answer to some modeling problem in the sense discussed above, but is referring to the “answer” to a real-world problem that the modeler is trying to address. Here, the notion of “no unique answer” points to the messiness and inherent uncertainty of the real-world. Because we can never hope to capture all of that messiness or tame all of that uncertainty, our models always remain provisional, approximate, and open to improvement. That is, we obtain “an answer” to our problem, and we evaluate whether or not it is good enough for our purposes, but we never get “the answer.”

Another way to see this is to always keep in mind that mathematical modeling is ultimately, a process. Hopefully, it’s a process that draws us closer and closer to the truth, but like an asymptote, never quite gets there. You can see this point of view and get a sense of the notion that there isn’t one right “answer” or model, but rather a never-ending array of possible “answers” or models, in the CCSS for mathematics:

In situations like these, the models devised depend on a number of factors: How precise an answer do we want or need? What aspects of the situation do we most need to understand, control, or optimize? What resources of time and tools do we have? The range of models that we can create and analyze is also constrained by the limitations of our mathematical, statistical, and technical skills, and our ability to recognize significant variables and relationships among them. 

There is a wonderful one-paragraph story by Jorge Luis Borges that is related to this second point. It’s called “On Exactitude in Science” and in this story, Borges explores the idea of modeling and uses absurdity to remind us that useful models are always incomplete. His story is short enough to reproduce here:

In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattered Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is no other Relic of the Disciplines of Geography.

The point, of course, that Borges is making is that a big part of science, a big part of understanding the real-world, is about making models and that the only perfectly complete model of reality is reality itself, but that such a model is also completely useless! This is true whether we’re talking about physical models like maps or more abstract mathematical models. The magic is that maps and models, approximate, imperfect, and ignoring vast parts of reality are incredibly useful and are our best tools for understanding, predicting, and controlling that reality.




  1. >In one very important sense, the answer to a modeling problem is a model.

    I’m having a hard time seeing this as fact rather than assertion. The model and the prediction it provides are pretty limited without each other. Without the prediction (and its validation in the world) I have no way of knowing if my model is valid. Without the model, I have no way of justifying my prediction beyond luck.

    The “no correct answers, only models” group have a much harder case to make, IMO. Not just because my case essentially includes theirs, but also because the net result of “no correct answers, only models” is to make modeling harder for teachers to implement and less likely for students to experience.

    1. Yes, of course, prediction is important in the cases where that’s why you developed the model. But, the model’s ability to predict (if that’s why you built it) is how you judge your “goodness” of the model, it’s not that the prediction is the “answer” the student has obtained, the model is their “answer,” how you evaluate it is by comparing to reality. The problem with your focus on “answers” and “predictions” rather than on models and process is that the types of questions you’re driven to ask with such a focus are not model-eliciting. Getting the questions right is a key part of the pedagogy of teaching mathematical modeling. If you narrow the focus to be on a one-number prediction, you’re short-circuiting the iterative nature of mathematical modeling and I’d argue that it is that approach that makes it harder for teachers to implement genuinely and less likely for students to experience authentically.

  2. Saying that the answer to the modeling problem is a number seems to me like saying that the answer to a proof is the last line of the proof rather than the proof itself. We would want to understand and see all of the reasoning (i.e., the model), not just the final conclusion.

  3. Yes, the analogy with proof is a good one. It’s not the theorem that is the “answer” in that case, it’s the proof. Of course, one judges the proof against the theorem, but the focus is on the proof. One proof can be better than another, more elegant, or more generalizable. The same is true in modeling. We judge the model against reality, but the focus is on the model. One model can be better than another, predict better, be more generalizable, or explain in addition to just predicting. Good analogy.

  4. Dan – not sure why the comment wouldn’t post. Let me post for you:

    From Dan Meyer:

    Again, the original proposition I’m critiquing is that “modeling questions can’t have correct, known answers.” I’m not arguing that the prediction is more or less important than the model itself. Me, I’m happy to accept that they are both answers. I’m also happy to finally understand that we’re just negotiating the definition of “answer” and I don’t think there’s much value in continuing to elaborate mine.

    But even if I accept the premise that “the model alone is the answer,” for given contexts and questions, I’m still unsatisfied. There are correct, known models.

    For instance, I have a fixed-rate mortgage. If I want to check my bill and make sure I’m not getting ripped off, or if I want to calculate the total cost of ownership of my home, there is a correct, known exponential model to help me answer that question.

    See also Sunday’s Last Week Tonight, in which John Oliver’s team caught his 401k broker using an incorrect model in an Excel spreadsheet. The correct model resulted in a net savings of ten million dollars.

    How am I supposed to square these instances with your insistence that there aren’t correct, known models?

  5. Now, my reply…

    I think you are setting up a bit of a straw man here with your proposition. I don’t know of anyone who says “modeling questions can’t have correct known answers.” I think a much more nuanced reading of these statements is required, which is what I’ve tried to provide in the blog post above. The statements like the ones I mention above are trying to explain what the practice of mathematical modeling is really like, not draw hard and fast boundaries around the nature of modeling questions. I think that is the real point – if we want to really teach kids the art of mathematical modeling, we have to pay very close attention to what those who do mathematical modeling actually do.

    Your mortgage example points to another issue – the word “model” is used in many ways, by many people, with a myriad of meanings. Yes, we might call the way you calculate mortgage payments a “model,” but it’s not one that is obtained via the process of mathematical modeling. That’s a very subtle point. I’d prefer to call examples like the one you mention “applications of mathematics” to draw a distinction. The difference is that in the mortgage case, the entire system is inherently mathematical, how you calculate mortgage payments is simply defined from the start in terms of mathematics. There are no assumptions, choices, or decisions to be made, it’s just follow the definition. I think if you imagine holding up that model to the modeling cycle you’ll see where one need not engage in the process of mathematical modeling to construct or use such a model. Now, there is certainly value in having students work with such models or do such problems, but it doesn’t teach them the process of mathematical modeling.

    This points to one of the key difficulties in teaching modeling in K-12. Mathematical modeling really does require engagement with the real world and real world problems in ways that often push the knowledge needed well beyond the typical mathematics classroom. I think this often leads to a tendency to look for problems that don’t do that, i.e. problems like the mortgage problem that keep one comfortably out of the real world and really, totally in the realm of mathematics. This discomfort is hard for many teachers to deal with, but encouragingly, not as hard for students once you turn them loose.

    Anyway, too bad you are on the west coast, be great to have this conversation over a beer! Certainly lots to discuss!

  6. > I don’t know of anyone who says “modeling questions can’t have correct known answers.”

    I’m responding to Avery’s question above. He’s opened up that possibility.

    > I think if you imagine holding up that model to the modeling cycle you’ll see where one need not engage in the process of mathematical modeling to construct or use such a model.

    I appreciate the application / model distinction. I’ll probably mentally throw a bunch of problems at it over the next couple weeks and see if I can make it hold up in my head.

    This may have no bearing on this conversation, but I think one way in which I depart from a lot of the modeling-friendly organizations like GAIMME, COMAP, and even the NCTM yearbook, is that I see value in students experiencing parts of the modeling cycle.

    Constructing an algorithm for the “square-ness” of a rectangle, for example, allow students to formulate a model, even though there isn’t a lot of noisy data for them to sift through, discard, and retain.

    Similarly, I’m happy for students just to answer the question, “If I’m planning a Thanksgiving dinner for my family, what information is important for me to know to figure out how to set a budget?” even if they don’t formulate a model for those data, calculate an answer, interpret it, or validate it.

    The way a lot of modelers talk about modeling, teachers are either exposing students to the entire cycle of nothing at all. I don’t think that kind of high bar is helping modeling make inroads into classes.

  7. On this we absolutely agree! I wholeheartedly believe that breaking down mathematical modeling into various competencies and having students experience them in parts is the right way to go. That is, yes, we need to expose them to the entire cycle, and teachers need to know what that looks like and what the goal is, but that there is tremendous value in having students work on smaller chunks.

    I explored a little bit of this idea in:
    Michelle and I have spent a lot of time thinking about the thought tools that mathematical modelers wield and how one learns to wield them. Note, this notion of thought tool is exactly the same as thinking of having students experience parts of the process of mathematical modeling.

    I think the GAIMME and COMAP folks, etc., are partially reacting to a fear that people will only do parts and then, only the easy parts. Plus, as far as I can tell from the literature, there is not a great consensus on what the core competencies really are for mathematical modeling. Folks in Germany have done some great work in this area, but still, feels incomplete. But, that is no excuse for not having students experience parts! Although, I too, always feel it necessary to add the caveat that it has to be building to the whole and that both teachers and students need to understand that as well. Also, I guess I’d add that we need to make sure that teachers understand that when they’re doing parts, they’re doing parts, not mathematical modeling. See, I sympathize a bit with the fear that comes through in GAIMME and COMAP etc.!

  8. “I think the GAIMME and COMAP folks, etc., are partially reacting to a fear that people will only do parts and then, only the easy parts. ”

    Yes! And that actually makes sense because doing parts is kind of what has already been happening in the name of mathematical modeling. So they’re trying to push that we really really do modeling now – not the lame stuff that has been tagged as modeling in textbooks. Dan’s MT article actually highlights this well.

    Given that even in the GAIMME report, the one task featured for elementary level modeling is once again the same old bigfoot problem (which is a nice problem!) that has been done and written about from researchers all around the world, I’ll admit that I’m a bit of a skeptic as to whether or not young children can really Model with Mathematics.

    I liken this to proof in mathematics. Past standards documents tend to talk about Reasoning and Proof. In fact, the Sense Making document described reasoning on a continuum with proof being at the higher end of that continuum. With few exceptions, we generally tend to talk about young children explaining, reasoning, or justifying. I think that most people don’t expect young children to engage in proving. I wish that there was some other word or phrase, like reasoning, that we could similarly use to describe “baby modeling” or pre-modeling activities that are perfectly accessible and appropriate for younger children. Instead, the best that we’ve come up with (so far!) is scaffolding it by having students engage in parts of the modeling cycle. That’s not to say students cannot mathematize real-world phenomena – I’m just not sure what genuine Modeling with Mathematics looks like for young kids. And even for older students, it’s definitely challenging to engage in the entire cycle in an authentic way.

  9. I should have added that while I think it’s challenging, I am very happy that MwM is included in the Common Core. Without it, applied math is not present in a meaningful way, and leaving it out fails to represent a fuller picture of what mathematics is really about.

  10. Many a scientific law came from examining the behavior of experimental data and experimental data has measurement error. So why not build your models from data? Start with simple models of stacking objects that produce models of good fit and then consider measurement error. Mathematical modeling is part of the scientific method: data > model > simulation and consider predictions and how they are influenced by error. Data pooling can help average out errors and identify outliers. Revise the model by considering more variables; hence, a multivariable approach. Modeling needs the mindset of how errors influence the model. The Common Core and NextGen Science Standards need to be melded together into a grand interdisciplinary approach. This is all possible to get started with spreadsheets, the gateway to computational science.

  11. I agree, building from data is a key approach, and I certainly agree that mathematical modeling should be viewed as a process in the service of science and the scientific method.

    I would argue though that building from data is only one approach to mathematical modeling. It’s what the Common Core calls “descriptive modeling.” The other approach, what CCSS calls “analytic modeling” builds on the application of physical law to construct models. For example, if I’m trying to model motion, I’m going to start with Newton’s Laws and leverage those to get a deeper understanding than I can get via data alone.

    Love the point about connections between CCSS and NGSS! We should be teaching mathematical modeling in partnership with our science colleagues!

Leave a Reply

Your email address will not be published. Required fields are marked *