Over the last several months, we’ve explored mathematical modeling from a variety of perspectives here in this space. We’ve talked about the CCSSM, the modeling cycle, thought tools used by modelers, and we’ve explored a variety of examples, all with the goal of trying to understand how mathematical modelers think and how we might best train our students in the art of mathematical modeling.

One thing we have not done is follow the path of a typical course in mathematical modeling. Today, I want to talk a bit about what such courses look like, and where we might benefit from typical paths and why we might want to deviate from typical paths. Until recently, at least in the United States, mathematical modeling courses have largely been restricted to higher education. They’ve been taught primarily at the advanced undergraduate level and the graduate level. There are exceptions and there have been efforts to teach such courses at the introductory undergraduate level, but it’s reasonable to claim that the majority of the effort around teaching mathematical modeling has primarily occurred at the upper undergraduate and graduate levels.

This means that the majority of textbooks focusing on mathematical modeling are also aimed at the upper undergraduate and graduate levels. If you search for “mathematical modeling” on Amazon, you’ll, in fact, find several thousand books along these lines. Some are more specialized, some are less specialized, but generically, they are aimed at this advanced audience. If you examine the most popular of these, the ones that focus on “mathematical modeling” rather than “mathematical modeling in X,” you’ll find they all follow a typical pattern and that this pattern is reflected in course syllabi at institutions across the country. Roughly speaking the pattern is one where the texts are organized by mathematical topic first, and application areas within those topics. That is, you’ll see them divided into chapters with titles like “Modeling with difference equations,” “Modeling with ordinary differential equations,” “Modeling with partial differential equations,” and so on. Within each chapter, the authors will introduce an application area, and then show the reader a bunch of models developed in that application area using the mathematics of the chapter heading.

The philosophy behind this approach is one of “modeling can’t be taught, but it can be caught.” That is, authors and instructors largely assume that if they show students enough examples of mathematical models, they’ll eventually catch on and know how to model. I freely admit, that I too was a member of this camp for a long time. After all, it had worked for me! But, in teaching mathematical modeling over the last twenty some-odd years, and especially as I’ve had the opportunity to work more closely with the K-12 community, I found myself growing increasingly frustrated and disillusioned with this approach. In the large majority of cases, it became apparent to me that while a handful of students did “catch it,” the vast majority did not. They could perhaps use models they’d seen before, and engage in somewhat trivial extensions of such models, but when faced with a new situation, they were lost. They hadn’t, by and large, learned how to actually model. They hadn’t become modelers.

I gradually realized that I don’t agree with the “can’t be taught, but can be caught” philosophy and many of the blog posts here have been inspired by my personal change in perspective on this issue. I believe that we can, in fact, deconstruct and understand the process that constitutes mathematical modeling, articulate this process more clearly for our students, design activities to engage students in essential competencies that comprise this process, and end up teaching a lot more students how to actually model. My colleague, Rachel Levy at Harvey Mudd, says that what we’ve been doing for a very long time is teaching “model appreciation” rather than teaching the art of mathematical modeling. I think there is a lot of wisdom in that statement and that we can make this shift, at all levels, from teaching model appreciation to teaching the art of mathematical modeling. As I’ve had the chance to explore the mathematics education literature on this topic, especially the international literature, I’ve learned that there has been progress made in this regard and I’ve worked to incorporate the best of these ideas into my own teaching and try and share some of them here.

But, in the back of my head, there is also this whole “baby with the bathwater” thought that’s been nagging me for a long time. In an earlier blog post we explored the notion of “thought tools.” These are those ways of thinking that are often widely applicable and are characteristically wielded by those who engage at a high level in a particular practice. We talked a bit about some of the thought tools wielded by mathematical modelers. This thought tool perspective pushed me into doing a lot of meta-thinking – trying to observe my own thought processes as a mathematical modeler and compare them with those of students just learning the art.

One thought tool that I find myself wielding frequently and have observed other mathematical modelers utilize as well is “analogical thinking.” That is, when faced with some unfamiliar modeling situation, a modeler will often start by arguing “well, this situation is kind of like situation X, so perhaps we can proceed as follows…” They have, in their head, a repository of modeling approaches for a wide variety of problems. They rely on the miraculous fact that mathematics and mathematical approaches to applied problems are often wonderfully generalizable, and so develop solutions to new problems relying at first on what they’ve seen work before. The uniqueness of each situation requires adaptation and creativity, but they can often grasp a ready starting point from which progress can be made.

That brings us right back to model appreciation, because after all, where did they get this broad experience with various modeling approaches? Well, they’ve looked at and worked through lots of models built by others to tackle lots of different situations. Seems they caught something useful somewhere along the line. The seemingly presents us with a dilemma. The vast majority of students don’t catch modeling by the model appreciation approach, but the experience gained via model appreciation seems essential to be a good modeler. I don’t think this actually a dilemma. I think that we should conclude that we need to engage students in both some model appreciation and some deconstruction and practice of the art and the process of thinking like a mathematical modeler. I believe, and can claim from my own personal experience, that this dual-approach, is a more successful approach.

So, as you think through and work to incorporate mathematical modeling into your classroom, I suggest you keep this dual approach in mind. Spend time giving students the opportunity to practice modeling on new and fresh situations, help them understand the process and the practice, pay attention to developing the thought tools they’ll need and the competencies they must master, but, from time to time, don’t be afraid to engage in a little model appreciation and help your students start to build their own “model repositories” in their heads.

John

This week, I’m taking a break from posting and Michelle is taking the lead with an interesting post about her experience learning about mathematical modeling. As someone who has been embedded in the modeling world for 20+ years, it’s this kind of perspective that I’ve come to value greatly in our work together. Hope you’ll enjoy!

John

In 2009, I attended the 19th ICMI Conference focused on Proof and Proving in Mathematics Education in Taipei, Taiwan. While there I heard a memorable talk given by Anna Marie Conner, a mathematics educator, and Julie M. Kittleson, a science educator, called, Epistemic Understandings in Mathematics and Science: Implications for Learning. In their talk, Conner and Kittleson argued that there are similarities in the processes engaged in to legitimize knowledge claims in mathematics and science, but there are also significant differences that must be recognized and understood because have implications for disciplinary learning. The goal of this work was to promote better understandings among teachers and students of how disciplinary knowledge is established in both mathematics and science. They further argued that examining and understanding these connections will lead to valuable insights for teachers and teacher educators in focusing and sharpening both their own understandings and those of their students. Here are some key points:

  • While mathematical inquiry makes use of inductive searches for patterns, knowledge is established deductively in mathematics.
  • Mathematical proof is the basis of knowledge in mathematics;
  • Establishing truth in science is more nebulous.
  • Scientific knowledge is constructed by the interplay between theory and data, but truth is never conclusively established. (p. 1-107)

Conner & Kittleson provided two classroom examples, one from mathematics and one from science:

The interplay of inductive and deductive reasoning is apparent in both mathematics and science classrooms. To illustrate, consider two classrooms, one geometry and one physics. Both classes are taught by teachers who strive to align their teaching with the standards (NCTM, 2000; NRC, 1996). In the geometry class, students explore characteristics of a figure with dynamic geometry software that models Euclidean geometry, and develop a hypothesis about some aspect of relationships within that figure. In the science classroom, students explore the motion of a pendulum and develop a hypothesis about the relationship between the pendulum’s length, the mass of the bob, and the time it takes for the pendulum to complete one swing. To this point, at least from the perspective of the students, they have engaged in similar, perhaps almost identical, activities. However, it is at this point that the different epistemological understandings of the specific disciplines must engage. For in mathematics, the student must attempt to deductively prove the veracity of the hypothesis. No amount of experimentation will allow him or her, from a mathematical perspective, to establish the truth of the hypothesis, and once it is proved, it is established as true with no need to re-prove and no possibility of contradiction. In science, on the other hand, the student must carefully craft an experiment to determine whether his or her hypothesis is correct. This experiment may disprove the hypothesis, but it will not prove that it is correct. It may confirm the hypothesis, but that confirmation is tentative, and is subject to the possibility of disconfirming evidence. (p. 1-108)

Since I’ve been studying and facilitating professional development on modeling, I’ve had a realization that, while likely obvious to applied mathematicians and scientists, was not as obvious to me, a mathematics educator who is still developing understandings of what it really means to model with mathematics. One of the many realizations that I’ve had is that mathematical modeling, which connects mathematics to the “real world,” behaves more like science than mathematics. How so?

Well, for starters, one engages in mathematical modeling to understand something about reality and/or to help you predict something in the real world. At the same time, the real world is messy, and true modeling tasks are ill-defined.

Because the real world allows for many areas of investigation and requires the modeler to make choices, decisions, and assumptions throughout the process, a mathematical modeling investigation does not culminate in any one “right” model or path to a solution. When people look at the same real-world phenomenon, just as in science, they can have diverse perspectives into the task’s resolution. They will inevitably make different choices and assumptions, and thus, mathematical models are ultimately built based on hypotheses or guesses as to how the real-world system behaves. A mathematical model is therefore judged by the accuracy of its predictions, the power of its explanations, or the simplicity of its implementation. In fact, when evaluating mathematical models, the aphorism of statistician George Box, which essentially says that all models are wrong, but some are useful, can be helpful to keep in mind. That is, we can never “prove” that a mathematical model is “correct.”

As Conner and Kittleson argued, teachers who understand the epistemology of both mathematics and science are in a better position to capitalize on the similarities between math and science and to highlight the subtle and more obvious differences between the two. This has led me to realize that effective teaching of mathematical modeling really does require the development of expertise in both mathematics and science, or at least those aspects of science related to the real-world systems one wants to study in one’s classroom. For this reason, more conversations and collaborations between mathematics and science educators are critical for the development of 21st-century skills.

Michelle

 

Last time, we talked about the notion of “STEM” and in particular, the notion of “STEM” as a meta-discipline. We discussed the idea of organizing STEM activities around the central practices of theory, experiment, and design, or mathematical modeling, the scientific method, and the practice of engineering design. We put forth the notion that a good, integrated STEM activity, would help students to grasp the interrelationship of science, mathematics, and engineering sketched out in this diagram:

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And, I indicated that this time, we’d get concrete, and explore a particular STEM activity. So, today, I want to talk about one such activity and how it might be developed to include all of the elements we discussed last time.

While examining lots of different STEM activities one thing I’ve noticed was the many such activities that are organized around having students build a simple piece of technology. It is remarkable how modern material science and the economies of scale associated with computer manufacture in particular have spawned the modern “Maker Movement” and a inspired a whole new generation of DIY’ers. This has made possible the opportunity for students to design, build, and play with a wider variety of technologies than was possible just twenty years ago. If you haven’t explore this, I encourage you to visit instructables.com and spend a few minutes browsing. It’s a great starting point for inspiration for STEM projects.

For today’s project, I was inspired by the many instructables-based projects around building a speaker. I don’t mean learning how to use an off-the-shelf speaker, or building a case for a speaker, I mean building an audio-speaker from first principles out of wire, paper plates, magnets and such. If you search “speaker” on instructables.com, you’ll find a dozen such projects outlined. Since there was something magical about the idea of plugging my iPhone into something I’d cobbled together out of paper and glue and producing a sound, this sounded like a lot of fun. I was also inspired by the picture I shared last week:

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The idea of a speaker is readily grasped from this conceptual model but we note that grasping the idea relies upon understanding a bit of the physical world. In particular, we need to know that an electric current produces a magnetic field, that this field is time-varying with variations in the current, and that sound is a physical disturbance of air. With those pieces of knowledge, we see that a speaker works by taking an electric current that varies with time as the signal and producing from that a physical motion of a membrane that moves air molecules, creating a sound wave. Even with that background and the conceptual model above, there is still likely to be something very abstract for most students about this level of understanding. That’s where, to me, making this hands-on and having them build and experiment with their own speaker has value.

For my own personal challenge, I decided to attempt to build a speaker using nothing that I didn’t already have laying around in the basement. My parts list came down to:

  • A plastic drink cup
  • Packing tape
  • A hot glue gun
  • A paper clip
  • A small permanent magnet
  • A piece of paper
  • A few feet of magnet wire

It’s likely that you have all but the last item laying around your workshop. If you make it a habit to save the components from old DVD players or CD players, you’ll also have the magnet wire at hand. If not, you can pick up a spool at Radio Shack for a few dollars at most and have enough to make dozens of speakers.

Here’s the finished speaker:

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Note that the design is quite simple.  I cut a hole in the bottom of the cup and put a piece of packing tape over the hole to serve as the vibrating speaker membrane. I then wound the magnet wire around a tube of paper and glued that tube directly to the membrane. Finally, I bent the paper clip into a U-shape, glued the permanent magnet to the bottom of the U and attached the entire U to the paper cup so that the magnet was suspended inside the paper tube. Voila! Instant speaker. To test, I clipped the earbuds off of an old set, plugged the jack into my iPhone, and attached the bare wires directly to the speaker wires coming off of my paper coil of magnet wire.

There is something magical about hearing sound come out of such a simple contraption. But, the sound was quite difficult to actually hear, so I took one final step and rigged up a quick and dirty amplifier:

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For this, I used an LM386 amplifier chip, two capacitors, a 9V battery, and some wire. Running the signal from my iPhone through the amp and then to the speaker I then had a reasonably clear, reasonably loud working speaker. Now, I haven’t given you the most detailed explanation of how to build this speaker or amplifier here, but as I mentioned above, you’ll find dozens of detailed plans on instructables.com. Here, I really want to get back to thinking about this as a typical STEM activity and talk about where you might take this next. Let’s return to our sketch from above:

IMG_0225

Thus far, we’ve been working firmly within the “Design” box in this sketch. We’ve take a little knowledge, i.e. the basic principles of how a speaker works, and designed and built a working prototype. I think it is easy to see how we might structure this type of activity for students and have them arrive at this point. Now, notice to get to this point I’ve used no mathematics at all and very little science. This is where we have to be very deliberate about not having this STEM activity end here and where we have to be very deliberate about helping students see a genuine need for the tools of math and science in this project.

How do we do that? We don’t want to “tack on” some extra math and science. It’s precisely at this point that we want our students to feel that genuine need and to feel compelled and driven to pull out tools like mathematical modeling. Here, and in many cases, I think you can achieve this by asking one simple question – how do we make it better? Now that we’ve got a working prototype of a speaker, how do we improve our design? If you push just a little bit on this point, you’re quickly out of the “Design” box and back to “Questions” that you need mathematical modeling and science to answer. Here’s a few such questions that you might encourage your students to investigate next:

How does the volume of the speaker depend on the number of turns in my wire coil? If I’m going to manufacture these, I’d like to use as little wire as possible, but not too little! Can I build a mathematical model that relates turns to the amplitude of sound?

To test the quality of a speaker, one typically maps out the frequency-response curve of the speaker. That is, as an input we put in pure sinusoidal signals of different frequency, but the same amplitude. We then measure the intensity of the sound coming from the speaker. If we sketched such a curve (experimentally perhaps) for our speaker, what would it look like? Could we build a mathematical model that predicted the shape of this curve and related that shape to the physical properties of our speaker?

The radius of our magnet coil was chosen so that the permanent magnet would fit nicely inside the coil. How would the performance of the speaker vary if we used a different magnet or a different coil? Can we build a mathematical model that related speaker performance to these coil properties?

I imagine that once you start thinking this way, you’ll be generating many such questions and I encourage you to stop at this point in a STEM activity and have your students generate their own such questions. These questions have now led them back into that “Theory – Experiment” box where mathematical modeling is a central tool and opened up multiple opportunities for your students to use their modeling skills in a way that is genuinely motivated by a problem they care about. As they build and analyze their models, the new knowledge they gain should then influence the next iteration of their design. If you can get your students moving through this big loop, Questions – Theory – Experiment – Knowledge – Design, then you’ve got them doing STEM rather than “S-T-E-M.” Good luck! I look forward to hearing about your STEM activities!

John

 

In addition to having the opportunity to work with mathematics teachers implementing mathematical modeling in their classrooms, I also often have the opportunity to work with groups of teachers from across math and science who are trying to implement “STEM” ideas in their teaching or in extra-curricular activities. This has given me the chance to think through the notion of STEM and in particular, to think carefully about the role of mathematics and mathematical modeling in the K-12 teaching of STEM.

One of the most useful articles I’ve found for understanding the “big picture” and history of “STEM” is an article called “Evolution of STEM in the United States” by Professor Emeritus William E. Dugger, Jr., of Virginia Tech. Dugger carefully defines each of the letters in the acronym “STEM.” He offers that:

S – Science, which deals with and seeks the understanding of the natural world, is the underpinning of technology.

T – Technology, on the other hand, is the modification of the natural world to meet human wants and needs.

E – Engineering is the profession in which a knowledge of the mathematical and natural sciences gained by study, experience, and practice is applied with judgement to develop ways to utilize economically the materials and forces of nature for the benefit of mankind.

M – Mathematics is the science of patterns and relationships.

Within these four definitions one already begins to see the interrelated nature of these areas and their dependence upon one another. Readers of this blog will also note his characterization of mathematics as the science of pattern and relationships; again, this way of thinking about mathematics illuminates the deep connections and utility of mathematics for S,T, and E.

Dugger also offers several models of the way STEM is, or can be taught. The first of these is the “silo model,” where we view STEM as “S-T-E-M,” with each discipline being fully distinct and independent and taught with no integration among the four. The second is similar, retaining the silo nature, but emphasizing some disciplines more than others. Dugger labels this one as “S-t-e-M,” with the upper and lower case indicating levels of emphasis. A third model occurs when one of the four disciplines is integrated into the other three. The most common is when the “E” is integrated into the rest. Dugger denotes this as “E -> S,T,M.” Here, the S,T, and, M, remain in silos, but the E is integrated into each. Dugger’s final model is the fully integrated approach, where each of the four are integrated into one another. This, in effect, has us view STEM as a “meta-discipline” and this kind of integration can be simply denoted as “STEM.”

When I work with STEM councils or groups working to create STEM opportunities for students, it’s this last model that I emphasize. For me, this “meta-discipline” or integrated approach is what’s necessary if our students are going to be able to work effectively in the scientific community of today and contribute to the solution of our most pressing challenges. Nature recently released a special issue on interdisciplinarity with the subtitle “Why scientists must work together to save the world.” This is just one more voice in the chorus of voices calling out for bridges between the myriad array of disciplines created as a result of fifty years of hyper-specialization. Helping students experience an interdisciplinary perspective and be prepared to work in interdisciplinary teams seems like a worthwhile goal for those working to create STEM opportunities for students. Note that this doesn’t diminish the importance of the disciplines or obviate the need for deep content knowledge, but rather, creates opportunities to bring that content knowledge to bear on problems that can’t be solved by working solely within disciplinary boundaries.

This, of course, bring up the challenge of actually doing this in practice in K-12. What does a good STEM activity actually look like? The risk is that we create activities where students do a little “S” and a little “T” and then a little “E” and then we throw in a little “M” and we’re back to the “S-T-E-M” model rather than the “STEM” model. To help people think about “STEM” vs. “S-T-E-M” and designing good STEM activities, I encourage them to think in terms of the three central practices of STEM. In shorthand, we can think of these three practices as theory, experiment, and design. You may be more comfortable thinking of these in terms of cycles that you’ll find in the NGSS and CCSSM, namely, the mathematical modeling cycle, the engineering design cycle, and the scientific method. One way to picture the relationship among these cycles or among the practices of theory, experiment, and design is:

IMG_0225

That is, we have questions about the real-world. We use the tools of science to find explanations and build the ability to make predictions. These tools of theory and experiment, or modeling and experiment, generate knowledge about the world. We take this knowledge and use that to design technology that lets us control and modify our world. Doing so often creates new questions that we then again turn to theory and experiment to explore, and so on.

A good STEM activity engages students in this sort of interplay of the central practices. The starting point can be anywhere in the diagram but shouldn’t stay confined to a particular box. In practice, it’s often easiest and most easily engaging to start in the “design” space, that is, to start with what we might think of as an engineering challenge. But, a really good engineering challenge will have students asking questions that they need to turn to theory and experiment to answer. They’ll get a deep sense for how science informs engineering and how engineering challenges lead to interesting scientific questions. And, along the way they’ll engage in mathematical modeling, engineering design, and the practice of science.

Now, this is still all rather abstract! Next time, I’ll make this very concrete and we’ll explore a particular STEM challenge and see exactly how the interplay pictured above might look in an activity appropriate for K-12 students. Till next time!

John

 

Last weekend, I had the chance to spend the day in NYC with my daughter. While most of our day was taken up with a college recruiting event, I couldn’t resist taking her to visit one of my favorite bookstores in the world, the Strand Bookstore at the corner of 12th and Broadway. Housing 18 miles of books, both new and used, the Strand calls out for serendipitous rather than targeted browsing. I’ve never been disappointed and never walked out with anything I would have anticipated in advance.

Last week, I stumbled upon (and bought, couldn’t resist), an out-of-print book called “The Way Things Work.” It’s actually a two-volume set, billed as “An illustrated encyclopedia of technology.” What made it irresistible to me was its structure and the fact that it was about one thousand pages of cool models. Every pair of pages consists of a brief paragraph of text describing a particular piece of technology, and on the facing page are pictures, illustrations, models. I scanned a few pages so I could share them here. Here’s the one on elevators:

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And, here’s one on speakers:

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I wanted to share these pictures, these models, because they reminded me how important such models are, not only in communicating understanding, but in the mathematical modeling process. When we think about the mathematical modeling cycle and the often daunting “Formulate” phase, we often leap directly to thinking about formulating in terms of mathematics. But, that’s not what you’ll find most modelers actually doing in practice. That is, they don’t leap from “Problem” to writing down well-formulated mathematical equations describing a system. Instead, if you watch modelers at work, they’re likely to start by drawing pictures. Lots of pictures. Lots and lots of pictures. The reality is that most of us don’t think in terms of equations. We think in terms of pictures and stories and the process of sketching out a picture of the system we’re trying to understand and telling ourselves the story of the process that’s happening in the system is how we first develop understanding. As we’re sketching and talking through what we believe is happening, we’re simplifying and making approximations. We’re leaving things out of our pictures and putting other things in. We’re developing a conceptual model of what we believe is going on. We’re starting to introduce notation and often, geometry. We’re making guesses and conjectures about what’s driving what we’re observing. It’s usually only then, only after we’ve gained some simplified hold over the system, that we start to translate and turn that understanding into mathematics.

Last week, I was also reminded of the importance of pictures and story in the modeling process by a fellow named Anton (Tony) Weisstein. Tony is a mathematical biologist and teaches at Truman State University in Missouri. Much of his work focuses on developing innovative curriculum materials for the teaching and learning of mathematical biology, and much of this focuses on finding ways to teach the art of mathematical modeling to biology students. Tony was visiting the University of Delaware and I had the chance to attend a workshop he conducted on his approach.

The example Tony used was the modeling of the spread of infectious disease. In mathematical biology, there is an incredibly widely used and wildly successful modeling approach to such problems known as the “S-I-R” model or the “S-I-R” class of models. When introducing his students to these models, Tony starts with a discussion of infectious disease and guides his students to slowly sketch a conceptual model of how they envision such diseases being transmitted. He draws his first sketch like this:

SIR1

The discussion to this point has been about the types of individuals that could be in the population of interest. Tony’s steered the class to identify three key categories within that group. The first, “Suceptible” is represented by the box labeled “S.” This is the group of people who could get the disease. The second, “Infected” is represented by the box labeled “I.” This is the group of people who have the disease. The third, “Removed” is represented by the box labeled “R.” This is the group who has perhaps died, recovered, or otherwise been permanently removed from the population of those who could get or have the disease. Here, Tony stresses something important – he says “let’s start with the simplest possible scenario, we can always complicate things later.” He encourages students to remember or write down all those complications so that they can return and think them through later. Next, Tony focuses the discussion on the process. How can  individuals move from one box to another? When the class says “Someone who is susceptible can get infected,” Tony modifies the picture to include this idea:

SIR2

Further discussion leads to the idea that someone who is in “I” can move to “R” and another modification of the picture:

SIR3

In this way, Tony elicits a complete conceptual model of the situation from the students before ever thinking in mathematical terms. His pictures are, to me, of the same character as the pictures from “The Way Things Work” and very much like the sketches and diagrams you’ll happen upon if you catch a mathematical modeler at work.

I’d encourage you, as you work with your students, to encourage drawing and sketching and storytelling as you and they discuss the phenomenon you’re trying to model. Building these conceptual models and clarifying your thinking by sketching and explaining the process makes your work in the “Formulate” box of the modeling cycle much easier and much more fun.

John

 

 

How many piano tuners are there in the United States? How many golf balls would it take to circle the earth at the equator?  How many ping-pong balls would fit in the Superdome?

We’re often asked if questions like these are the basis for a good modeling exercise or if a student answering questions like these is engaged in mathematical modeling. My general answer? “Maybe” or, “It depends.” Today, I’d like to explore this question a little bit and think through how such questions can be a way to engage students in the art of mathematical modeling and what it takes to investigate one of these questions from a modeling point of view.

Problems like these are often referred to as “Fermi Problems” after the Italian physicist and Nobel Laureate, Enrico Fermi. Fermi was known for his ability to make quick, remarkably good, estimates to answer questions like these. Sometimes we refer to the calculations involved as being “back-of-the-envelope,” indicating a sort of rough order-of-magnitude estimation procedure for obtaining estimates. Sometimes, these types of questions become interview questions at firms like Google, purportedly to test a candidate’s ability to think quickly and reason quantitatively. There is no doubt that this skill of estimation is an important part of what we mean by “quantitative reasoning,” and there is no doubt that quantitative reasoning is an important skill for mathematical modeling. But, is an investigation of a Fermi Problem the same thing as doing mathematical modeling?

To answer this question, let’s take a particular Fermi Problem and think about two different investigations, one which stays pretty clearly embedded in the realm of quantitative reasoning and a second which shows how we might develop such a problem and push our students to genuinely take a modeling perspective on the question. Since ping-pong balls are always fun and generally familiar, lets take up the question – how many ping-pong balls would fit in the Superdome?

Now, one way to think about answering this question would be to simply take the volume of the Superdome and divide by the volume of a ping-pong ball. In pre-Google Fermi days, we’d need to figure out how to estimate both of these quantities, but now we can get good estimates of these two volumes with just a few keystrokes. Since the first numbers that Google gave me were in cubic feet, I’ll work in these units. I found:

Volume of Superdome \approx 130,000,000 ft^3

Volume of a ping-pong ball \approx \frac{4}{1750} ft^3

Dividing, I obtain:

Number of ping-pong balls in Superdome \approx 56,875,000,000

And, I’m done. Generally, with most Fermi problems, one says “wow, that’s a lot of ping-pong balls,” and the discussion ends.

Now, let’s think about what we just did in the context of the modeling cycle.

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We started with “Problem,” i.e., how many ping-pong balls would fit in the Superdome? We can argue that we moved to “Formulate” when we said “divide the two volumes,” but that already feels a little forced.  We could argue that we went through the “Compute” stage and perhaps even the “Interpret” stage when we took our final number and said “almost 57 billion ping-pong balls” as an answer, but that feels even more forced.  And, what about that “Validate” part? What about comparing back to the real-world? What about the whole cyclic nature of mathematical modeling? I’d argue that if this is the type of discussion around a Fermi Problem, then, no you’re not really doing mathematical modeling. That’s not to say that you’re not doing something valuable, but at the same time, you’re not really involving your students in mathematical modeling and you’re not really training them in thinking like a mathematical modeler.

So, the question becomes – what would an investigation of this problem look like if we really wanted to use it as a way of having students experience the art of mathematical modeling in a more genuine manner?

Consider an alternate approach. We begin, as before, by reasoning about volumes. This time, perhaps, we introduce a little notation, and think about the question a little more generally. For example, we might say the number of ping-pong balls, N, that fits in a given space is a function of the volume of the space, V_s, and the volume of a ball, V_b. That is,

N = N(V_s, V_b)

Now, our initial model of this dependence is as before:

N = \frac{V_s}{V_b}

Why is this different than what we did before? Well, in one way, it’s not. We could repeat the steps from before, substituting in the same estimates as above for V_b and V_s and obtain, of course, the same answer. But, expressed this way, we can also more easily think about this functional dependence that so often lies at the heart of mathematical modeling. If I increase the volume of the ball, N decreases in an inversely proportional way. If I increase the volume of the space, N increases in a directly proportional way. This hypothesized functional relationship makes intuitive sense. Written this way, I can also easily think about model validation. While I can’t afford fifty-six billion ping-pong balls nor get away with filling the Superdome with ping-pong balls, I can fill a shoe box. I can actually do that experiment, fully, and compare it to my model’s prediction. This will give me a sense of whether or not my model is realistic and believable. And, if I do that, I’d find that my model overestimates the number of balls I could fit in a shoe box (or any box) by about 25%. This would likely lead me to realize that ping-pong balls don’t pack together without gaps, and I’d be driven around the modeling cycle, and inclined to modify my model, perhaps by including a space filling scale factor:

N = c \frac{V_s}{V_b}

We could now think about and explore how this scale factor, c, depends on the shape of the object filling this space. That is, c takes one value for a sphere, but a very different value for a cube. We could even now think about our modeling efforts from both a descriptive and analytic view point. From a descriptive viewpoint, we could fill different sized boxes with ping-pong balls, gather data, and fit a curve to this data. How does this curve match with our proposed analytic model above? When we think about the analytic model we’ve proposed, we could think about the underlying principles about the real-world we’ve had to posit in order for this model to make sense. For example, we’re relying on a “ping-pong ball exclusion principle” that says no two ping-pong balls can occupy the same space at the same time. We’re assuming that the Superdome is empty and ignoring all those bleachers already filling the space. We’re also assuming the ping-pong balls don’t deform under the weight of those above them. Would this really hold true in the Superdome?

If we returned to the modeling cycle and tested our thinking against the cycle with this approach to the problem, I think we’d be able to convincingly argue that we’re genuinely making use of an iterative modeling approach this time around. Now, that’s kind of interesting – after all, in both cases we’re ostensibly solving the same problem. How can it be that in one case we’re “modeling” and in the other, we’re not? The key is that modeling is not just about your choice of problem. Modeling is both about the choice of problem AND the approach you take to solve that problem. Remember we build mathematical models to explain and predict. In the second approach to our ping-pong ball problem, we’re not seeking a number or an estimate as much as we’re seeking insight and understanding.

If we take the time to do Fermi Problems in the second way described above, and carefully pick the types of Fermi problems we think about, I do think that the investigation of at least some such problems can provide a valuable way to introduce students to the process of mathematical modeling. They have the advantage of being easily understood as problems and of relying on extra-mathematical knowledge that’s likely to be in the toolbox of even young students. I encourage you to attempt some Fermi Problems with your students and try them from a modeling perspective.

John

 

 

I’m fortunate in that my current position at the University of Delaware gives me the flexibility and opportunity to fairly regularly spend time in Delaware schools talking with and working with Delaware teachers. It’s especially fun to speak with teachers who are just starting to incorporate the practice of mathematical modeling into their classrooms. Today, I’d like to talk about a fairly typical dilemma or stumbling block that these teachers face and share some ideas for how to get past common sticking points.

The common dilemma is this – many teachers find a great real-world problem for their students, find a nice real-world data set for them to work with, and get their students genuinely engaged in investigating this problem. The students then grab the first tool in their toolbox, i.e. curve fitting or regression, fit a nice curve to the data, make a nice plot, and then say “Now what?” That is, the teachers we have worked with are very comfortable carrying out descriptive modeling and very comfortable working in the “compute” and perhaps “interpret” or “validate” boxes of the CCSSM modeling cycle, but then find themselves at a loss for what to do next. These teachers generally sense that there’s something missing, but then can’t quite find their way forward with a next step.

I asked Michelle about this common phenomena and asked for her perspective as a former high school teacher in particular. Here’s what she had to say:

This dilemma resonates with me. Because I did not take any coursework in mathematical modeling as a secondary education major (an issue which is a national problem according to this article by Newton et al. (2014), I did not have a good sense of what mathematical modeling was really about. My notions of mathematical modeling were largely formed by my engagement with curriculum materials. This is problematic because, as pointed out by Meyer (2015), many textbook exercises labeled as “modeling” tasks do not authentically engage students in all aspects of the mathematical modeling cycle. It was not until graphing calculators, and subsequently regression problems, became an important part of school mathematics that I really had any opportunities to engage in any form of modeling. At the time when it was introduced into school mathematics, I either had to teach it to myself or attend professional development workshops to learn about calculating these simple descriptive models. However, unlike the teachers described above, I thought I was done once the line of best fit was calculated, and I verified a “good” r-value. It is promising to see that these teachers are aware that there should be more to modeling than this. 

It’s encouraging to know that there has already been a shift in how teachers are thinking about mathematical modeling and that at least in my experience in Delaware, are trying to think beyond curve fitting. But what should a teacher who is faced with this dilemma do? How do you get past that feeling of “Now what?” or take your modeling beyond simple curve fitting? Based on my observations, I want to offer two pieces of advice or two strategies you might follow when faced with this dilemma.

Strategy #1 – Go back to the beginning

It seems to me that a common reason why this dilemma occurs at all is because not enough attention was paid up front to the “Problem” phase of the modeling process. It’s important to remember that what you are doing is trying to answer a question and that question is not a mathematical question! That is, being able to fit a curve to data is not the point of the exercise. Rather, your students should clearly have in mind the real-world problem that you’re trying to solve or the real-world question that you are trying to answer. It’s this question that drives you around the modeling cycle and this question that you should be returning to once you’ve done something like fit a curve to a data set. So, when faced with “Now what?” one answer should always be “What question were we trying to answer to begin with?” Bringing the focus back to your core question is one way to move the conversation forward.

Strategy #2 – Validate and critique the model

What do you do when your students say “we’ve fit this curve to our data, we have a good r-value, and hence we can now predict that in the future there will be X of Y”? That is, what do you do when they’ve done regression and think they now have entirely answered their real-world question? Here’s where you have an opportunity to push the conversation forward by validating and critiquing the model. I’d ask students questions like “how much confidence do you have in your prediction?” or “do you have any reason to believe that the trend you’ve sketched out will continue indefinitely?” Most curves obtained by curve fitting, when pushed beyond the range of data where they were obtained will lead to results that are genuinely open to question. This might be because we have no real reason to expect such trends to continue or it might be because we have no real reason to expect such trends to even exist, as we saw in The Shrinking Mississippi. Opening a conversation like this, allows you to help your students understand the limitations of all models, especially those that are purely descriptive models. This creates an opportunity to return to the start and push your students’ thinking toward analytic models built on a more fundamental understanding of what it is that they are exploring.

It’s important to remember both that mathematical modeling is a cyclic process and that what drives one around the cycle is the attempt to answer a real-world question. When you’ve built a model and are faced with the “Now what?” question, keep the goal in mind, be critical of the models you’ve built, and you’ll find ways to help you and your students be able to say “Oh, that’s what’s next!”

John

 

 

 

Last week, I had the opportunity to spend a few hours working with folks at Delaware’s Appoquinimink school district. The group was a “STEM Council” composed of district administrators, teachers, principles and others committed to ramping up STEM activity and STEM opportunities in the district. It’s always a lot of fun for me to get to work with such a dedicated group of people and to spend time talking about STEM, modeling, and education. A good portion of our conversation revolved around the intersection of the CCSSM and NGSS and today, I ‘d thought I’d share some of that discussion here.

The NSTA published a Venn diagram that makes exploring the overlap in the CCSS and NGSS practices quite easy:

PracticesVennDiagram_Page_1_000

While all of the practices are, of course, crucially important, and while it is interesting to think about all the overlap regions, what’s of most interest to me is the purple region that shows the overlap between the CCSSM and NGSS:

PracticesVennDiagram_Page_1_000

This region is really showing us where the practices of mathematics and the practices of science overlap. It should come as no surprise that this is where the modeling standards live. Readers of this blog are certainly familiar with the CCSSM SMP #4, i.e. “Model with mathematics,” but may perhaps be less familiar with the parallel standard in the NGSS, S2, “Develop and use models.” It’s important to note that S2 is actually broader than SMP 4. When the NGSS says “Develop and use models” they are talking about both mathematical models and other types of models. While one can certainly make a strong argument that mathematical models are the most important types of models that scientists use, it is useful to explore some of these other types of models and to understand a little bit about modeling more generally than as described in the CCSSM.

Why should we bother to think about these “other” types of models? I’ll argue that besides often being useful in and of themselves, having broad knowledge of “models” and “modeling” makes one a better teacher and doer of mathematical modeling. Many of these other models and modeling approaches are more intuitive and more accessible than mathematical modeling and a discussion of these can serve as an entry point into mathematical modeling. At the same time, when we do mathematical modeling, we’re often also, consciously or unconsciously, using other types of models in our process. When I work with groups like Appoquinimink’s STEM Council, we often explore four “other” types of routinely used models so that folks can consciously add these to their toolbox and think about how they use them and how they can be used as part of a mathematical modeling process. Here’s a brief description of the four types of models we talk about:

Scale Models – This is the one with which most people are already familiar and likely the idea that leaps to mind when someone says the word “model.” By scale model, we mean a physical representation of some real world object that is proportionally scaled to some other size. This might be an architect’s scale model of a proposed building, an astronomer’s scale model of the solar system, or a biologist’s scale model of the cell. In each of these cases, the object under study has been represented on a human scale. That is, it’s been built to a size that we can readily deal with visually. It’s been built to a size that we can take in at a glance, see as a whole, and see relevant parts as needed. In building a scale model, one makes decisions similar to those made in any modeling process. We ignore some things and focus on others. We choose what to focus on and what to ignore according as what we’re interested in visualizing or understanding and our models’ utility is dictated by these choices and decisions.

Idealized Models – The notion of an idealized model is certainly one that is used routinely by mathematical modelers, but they also serve as a useful tool for thought experiments in their own right. By an idealized model, we mean one where we conceive of a system as consisting of idealized parts that don’t actually exist in the real world. We might talk about frictionless blocks sliding down frictionless inclined planes or perfectly elastic billiard balls, or perfectly rigid rods, or one-dimensional rods and so on. None of these objects actually exist. There are no perfectly elastic balls or perfectly rigid rods, but if we want to think about the motion of billiard balls or the motion of a pendulum, it is useful to conceive of such idealized objects. We might then draw conclusions just from thinking about systems of such idealized objects (the arc of a perfectly rigid pendulum will lie on a circle) or frequently, the system we conceive of as constructed of such objects becomes the one we mathematize as we build a mathematical model.

Analogical Models – We we think about analogical models, we’re, well, arguing by analogy. We say “A is like B and B behaves as such, so perhaps A behaves in an analogous manner.” If you’re an economist you probably talk about “pumping money into the system” or “turning up the interest rates.” There is no “pump” in the world’s money supply and no knob for adjusting interest rates. What you’re doing is saying the economy is like a machine and money like something that flows through that machine. Pumping and turning knobs then become ways to think about what you are doing to the rather abstract “machine” that is the economy. We’ve talked about “toy models” before and these are too often great examples of analogical models. Our Great Lakes problem had us exploring the flow of a contaminant through three small containers. There, we were arguing that the flow of a contaminant through the Great Lakes behaved analogously.

Phenomenological Models – I always like to draw particular attention to this one because, unfortunately, it is what many people think of as being “mathematical modeling.” And, it’s not identical! Phenomenological modeling is what we do when we fit a curve to data. It’s our way of describing a data set using mathematical objects. We may do this visually or we may use mathematical tools like the method of least squares to do this fitting, but at the end, what we’re doing is describing data. It’s important to realize that we’re modeling data and that this is one step removed from the real world. This is the type of modeling that the CCSSM calls “descriptive modeling,” and so it falls into the class of “mathematical models,” but it’s important to note that it’s but one subclass of mathematical models. An important one, but just a piece of the puzzle.

If you’re a math teacher and you haven’t read the NGSS, I encourage you to do so. At the very least it’s worth reading their description of “Develop and use models.” Working together with our science teacher counterparts is a tremendous way to further the teaching and learning of mathematical modeling and exploring what we have in common (which is a lot!) is a great place to start.

John

 

 

 

In his interesting recent book, Average is Over, economist Tyler Cowan spends considerable time discussing and analyzing the phenomena of “freestyle chess.” While Cowan is interested in the future of the economy, the future of work, and the impact of technology on the workplace, aspects of the freestyle chess movement struck me as a good analog for how mathematical modeling activities should play out in the classroom or in the workplace. So, today, I’d like to explore that notion a bit and introduce the idea of “freestyle math.”

Chess, of course, developed as a competitive game between two players. The relative simplicity of the rules and the absence of chance as an aspect of the game combined with the deep complexity and tremendous variety of possible play has made chess an enduring pastime played by millions for about 1400 years. For most of that time, the standard of play was human vs. human, with exceptionally talented humans becoming “grandmasters,” people who could play at extraordinarily high levels.

In the 1950’s, things began to change. Alan Turing, who we discussed in a previous post on pattern formation, realized that a computer could in principle be programmed to play chess. Turing even developed strategies for such a program and wrote chess-playing code long before the computer hardware side of the world was developed enough to execute such code. However, before long, the hardware side of the world was up to speed and by the 1970’s one could purchase a dedicated chess-playing computer that looked like this:

chesschamp4

I recall an uncle of mine owning one of these early machines and I recall being amazed that the computer could pretty handily beat everyone I knew. But, at that time, at least, really good humans were still far better than the best chess playing machines. This too changed rapidly and in 1997, a computer called Deep Blue, developed by IBM, beat the reigning world chess champion, Gary Kasparov, 3 1/2 to 2 1/2 in a six game match. (Note that 1/2 points are awarded to each side in a draw.)

In the last eighteen years, computers have only gotten better at chess and humans, well, haven’t. Today’s most talented players don’t stand a chance against a good chess program and a good chess program can now be a single app on the phone in your pocket. However, shortly after losing to Deep Blue, Kasparov asked an interesting question – What would happen if instead of human vs. human or man vs. machine the competition were man + machine vs. man + machine. The new competitive sport of “freestyle chess” was born soon thereafter.

In this incarnation, chess is a team sport, where members of the team may be computer programs. In fact, in freestyle chess pretty much anything goes with Kasparov having remarked “Even if they were assisted by the devil, that would probably be covered by the rules.” And, in freestyle chess, something remarkable happened. Human-machine teams could defeat a single top-ranked chess playing computer program. The level of chess played by these man/machine hybrid teams was now higher than that played by either man or machine alone in the past.

Cowan describes freestyle chess matches as a being like a beehive of frenetic activity with human teams scurrying from computer to computer, combining, and analyzing suggested moves from multiple analyses before making a final decision. He also notes, interestingly, that the best freestyle teams are not necessarily made up of top chess players working with top chess computer programs. Rather, a strong freestyle team needs humans with a very different skill set. Yes, they have to understand chess and the basic elements of play, but they also need to understand computer programs, how they work, what their limitations are, what their strengths are, and how to combine the brute-force “number crunching” approach of computers effectively with human intuition.

As I read Cowan’s description of freestyle chess, I was struck by how similar the activity he described is to the activity of an effective student team working on a mathematical modeling investigation. So, I’d like to encourage you to think of mathematical modeling in your classroom as “freestyle math” and I’d like to explore for a moment some of the characteristics of what that might look like. I’m encouraging you to think of “freestyle math” in this “anything goes” sort of mindset. That is, students should be free to tap into the internet, simulation, textbooks, friends, etc., without bound. The focus should be on the goal of understanding the phenomenon they are investigating and in how one gets there, anything should be fair game. So, here are five characteristics of freestyle chess that I believe map in a one-to-one fashion to freestyle mathematical modeling:

The humans know something, but might not be super-experts

In freestyle chess, the human members of the team are not necessarily grandmaster level chess players or even master level chess players. But, at the same time, they know the rules of the game, have an understanding of strategy, and generally play as well as a decent club player (someone who plays regularly). On a modeling group in the high school classroom you want your “players” to have a roughly similar level of expertise in mathematics and have some expertise in the application area they are investigating. You don’t need a team full of math-whizzes, but you do need your students to take ownership of the mathematics, have a comfort with using mathematics, and be able to know what they need to learn.

Collaboration is key

As in freestyle chess, in freestyle math, the ability to collaborate effectively, both with other humans and with computers is a key skill. The ability to work with a diverse team is important and students also need to know how and when to use technology in their investigations. This means our students need to know when to “Google” and when to think. They need to know when to analyze with pencil and paper and when to turn to the computer.

The humans understand the strengths and limitations of computers

An essential skill of freestyle chess players is knowing both the strengths of various computer programs and the limitations of various computer programs. This is at least as important of a skill for mathematical modelers to possess. They should know to turn to the computer to fit a curve to a data set consisting of even a mildly large number of data points. At the same time, they should know what the computer is doing when it is fitting that curve, what assumptions they are making when they take that curve as being predictive, and exactly where the limitations are in what the computer is telling them. I can’t emphasize this one enough – to use a tool effectively you have to know both what it is good for and what it isn’t.

The humans are able to tap into and filter vast quantities of information

In freestyle chess, humans also bring the ability to both gather vast quantities of information and to filter this information. Freestyle math needs the same skill. Even a discerning Google search about a real-world scenario that students might investigate will return hundreds of thousands of links. Students need to be able to quickly and efficiently zoom in on the relevant information and know what information to trust and what to ignore. At the same time, a simple computer simulation can also quickly return reams of data. Students need to know how to organize this information, how to visualize this information, and again pick out what is relevant and what isn’t.

The humans supply creativity and intuition

Humans clearly bring both creativity and intuition to the table and both of these skills are required in abundance in freestyle chess and in freestyle math. In mathematical modeling, decisions, false starts, retrenchments, intuition, and creativity are the norm. Students need to be able to offload to a computer the things it does best and bring to the table the sorts of creative approaches and intuition that they do best.

As you work to engage your students in the practice of mathematical modeling I’d encourage you to think “freestyle math” and look to freestyle chess as a model for what open collaboration between people and between man and machine can produce. I’d also encourage you to think about the skills that such activities demand that go beyond the simple mastery of mathematical procedures, and how freestyle math demands a deep understanding of concepts and the ability to integrate that knowledge in the pursuit of understanding the world.

John

 

 

 

Often in mathematics, and certainly often in mathematics class, we think in terms of direct problems.  That is, we’re given something to “solve,” we’re required to find a path to a “solution,” and that’s where we end up. For example, we might be asked to find the solutions of the following equation

x^2 + 2x + 1 = 0

and we proceed by using the quadratic equation or completing the square or some such method, and arrive at values for x. If we think of this in terms of applications, we often pose direct problems as well. Here, for example, we might say, suppose we drop a ball from a height h, we assume air resistance is negligible, how fast is the ball traveling when the ball hits the ground? Or perhaps, how long will it take the ball to hit the ground?

But, often, in the real world, our problems aren’t formulated like this. Rather, we’re faced with problems that are posed the other way around. That is we’re faced with what are called inverse problems.  In these problems, what we’re given is the solution and what we have to do is figure out where it came from. You’re likely familiar with such problems, but perhaps haven’t thought of them in this way before. One common example is the “sonar problem” or the problem of echolocation. If I’m a bat, what I do is make a noise, and then listen for the reflected sounds that come back to me. What I then try and do is figure out the direction, location, and perhaps shape, of the objects in the environment that caused the particular patterns of reflected sounds that I heard. That is, I’m given the solution, i.e. the reflected sounds I measured, and have to figure out the problem, i.e. what pattern and shape of objects in the environment would cause that set of reflected sounds?

Such problems are an important class of problems that the mathematical modeler must be equipped to deal with. And, such problems, can be a source of challenging and engaging problems for students learning the art of mathematical modeling. Today, I’d like to explore a few simple versions of such problems and how you might use them in your classroom.

One of my favorites is a problem that was posed by Isaac Newton in Universal Arithmetik and was discussed by Groetsch in his text Inverse Problems: Activities for Undergraduates. Newton posed the problem like this:

A Stone falling down into a Well, from the Sound of the Stone striking the bottom, to Determine the Depth of the Well.

The unusual capitalization is Newton’s and apparently the style of the time was to state commands rather than pose questions. Today, we might state the problem as:

Can we determine the depth of a well by dropping a stone into the well and listening for the sound of the stone striking the bottom?

Why is this an inverse problem? Well, what we’re asked to do here is to take information we obtain at the end of a process, i.e. the time interval between us releasing the stone and it us hearing the stone strike the bottom of the well, and to infer how the stone must have traveled and from this deduce the depth of the well.

This particular inverse problem can be approached by solving the direct problem, which requires us to build a mathematical model of the time which elapses between the release of the stone and us hearing the sound of the stone striking the bottom of the well. We can imagine that we release the stone and start a stopwatch at the same instant. When we hear the sound, we stop our stopwatch and call that elapsed time, t_m. Further, we can suppose that t_m can be divided into two parts, the time it takes the stone to fall, t_f, and the time it takes the sound to travel back up the well, t_s. Our measured time is then the sum of these two:

t_m = t_f + t_s

Our job is then to build a model of the fall and the travel of the sound that lets us express t_f and t_s in terms of the depth of the well, h. It’s worth noting that both modeling the fall and understanding the fact that sound travels at a finite speed fall within the scope of the NGSS standards and so, should be within reach of the high school classroom. I’ll skip the details here, but you should be able to arrive at an expression that looks like:

t_m = (2h/g)^{1/2} + h/c

Here, g is the gravitational acceleration and c is the speed of sound in air. A quick plot of t_m as a function of h yields: (plot prepared with Desmos)

g7gmgpouwv (1)

Now, given a measured time, t_m, all one needs to do is draw a horizontal line on this plot at that value of time, find the intersection point with the curve shown, and read off the value of h, the predicted depth of the well. It’s  worth a conversation as to whether or not the h/c term is important here or when it may be important. It’s also worth taking time exploring how an error in the measurement of t_m translates into an error in the prediction of h. The sensitivity to such error that one sees in this problem is a typical feature of inverse problems.

If you do decide to introduce your students to the notion of inverse problems, it is worth spending at least a few minutes sharing with them the many areas of application where such problems arise. I’d suggest focusing on medical imaging as this is likely familiar territory in some sense, but unfamiliar territory in the sense of them understanding that most of modern medical imaging rests on building mathematical models and solving inverse problems.

I’ll leave you with one last inverse problem that is readily understood, but challenging to investigate. Imagine I tell you that I have a container of some sort but that I’m going to keep that container hidden in a box and not let you see it directly. However, what I will do is pour any volume of water you’d like into the container and tell you the height to which the water fills this hidden container for that volume. I’ll do this as many times as you’d like and give you as much data of this form as you want. The question is then this – can you tell me the shape of my container? Suppose I told you the container had rotational symmetry. Would that help?

I hope that this small taste of inverse problems has inspired you to consider this very important class of problems as you work to build mathematical modeling into your classroom practice. As always, we’d love to hear about your successes and challenges with this or other mathematical modeling investigations.

John