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!


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.



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:


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 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, 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:

SDeanCopier15110911060_0001 - Copy

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:


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:


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 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:


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!