Today was the last day of this year’s NSTA STEM Forum and bright and early tomorrow morning I’ll be headed back to Delaware. I want to thank NSTA for a great conference and I want to especially thank all of those who joined me for one of our two NCTM workshops on mathematical modeling. I really enjoyed working with all of you and hope to have that chance again in the near future. I’ll repeat the offer I made at the end of each workshop this week – please feel free to tweet or email anytime with your questions, thoughts, or comments about mathematical modeling! I love thinking about this stuff and love hearing from math and science teachers working to implement mathematical modeling in their classrooms.

At this week’s STEM Forum there were a massive number of fascinating sessions offered. So much so that it felt like for each session I attended, there were a half dozen that I had to miss that I really wish I could have attended! So, today, for those of you who couldn’t join us for our NCTM workshops, I’ll give a brief recap and provide the slides and a few other related materials. Along the way, I want to share and explore some of the excellent thoughts and ideas offered by participants during and after these workshops. I apologize in advance for not having the foresight to write down names! If you recognize yourself below, please drop me a line and I’ll correct this oversight. One more stylistic note before we begin – our two workshops (middle and high school) were essentially the same, with the math explored in the high school session being slightly deeper than in the middle school session. So, below, I’ll just say things like “in our workshop” or “Our workshop began” for simplicity.

For easy reference, here’s a PDF of slides from our workshop: NSTA_STEM_2016_Pelesko_HighSchool

We began the workshop by sharing a short story that appears at the start of the physicist Eugene Wigner’s famous talk “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” I won’t reproduce the story here, but if you follow the link, we shared the first paragraph of Wigner’s talk. We then spoke briefly about this idea, the idea that there is this incredible power, a power that verges on the mysterious or miraculous, afforded to us once we understand how to use mathematics to understand the natural world. We talked about this as the basis for answering the question “How do we put the M in STEM?” and we asked participants to share their experiences and their challenges with incorporating mathematics into STEM activities.

Participants shared some very real and very pressing concerns that we need to overcome if we’re to be successful in putting the M in STEM and if we’re to be successful in implementing STEM overall. These included overcoming the discomfort that many math teachers feel with science and many science teachers feel with math. One participant noted that to overcome this, we need to find time and space for science and math teachers to collaborate and to work together. Absolutely! Another participant noted that if you looked at the NGSS and the CCSSM, they are almost forcing this to happen. These are excellent points and I want to emphasize them here – if we are going to effectively teach students the art of mathematical modeling, it is crucial that we learn to work across math and science. Yes, one can do mathematical modeling using contexts that require little understanding of science, but the full power of mathematical modeling is only really unleashed when we’re involved in a deep scientific investigation of phenomena in the natural world.

Next, we spent some time talking about the overlap in the practice standards between NGSS and CCSSM. In particular, we talked about SMP #4 – Model with mathematics, from CCSSM and Practice Standard #2 – Developing and using models, from NGSS. We spent some time exploring how the NGSS standard encompasses the CCSSM standard, with mathematical models being one type of model talked about in NGSS. The key idea we explored was this relationship, the idea that mathematical models are really scientific models encoded in the language of mathematics.

Then, we spent some time looking at a STEM activity and where mathematical modeling fit into the picture. The activity we discussed was the Great Lakes problem that I’ve talked about here. Participants spent some time working to construct a mathematical model of the Great Lakes system and at the end, shared their results and their thinking.

I want to mention two things shared by participants that I thought were really cool. One participant from Montana, told me about the Berkeley Pit Mine in Montana:


Apparently, this is an abandoned copper mine, huge in scale, that is slowly filling with water. The problem is that the water is incredibly contaminated and that eventually the water level will rise to the level of the local water table. When that happens, backflow will occur, contaminating local and regional water supplies. This is clearly a source of many wonderful STEM and mathematical problems. I’ll think about this one some more and I’m sure will post more about this one. Thanks for sharing!

Another problem shared by a participant, was one she does in her class. In this project, she has students read the book “The Immortal Life of Henrietta Lacks” and then explore changing concentrations of drugs in the body. As her experimental system, she has students start with a beaker filled with water and a certain amount of dissolved salt. They measure the mass of the system, then remove a small quantity of the “drug filled water” and replace it with an equal amount of clean water. Then, they measure the mass again and repeat. Tracking the mass measured each time, they uncover the curve that describes the changing concentration of the “drug” in the system. Mathematically, this is identical to the Great Lakes problem we explored in this session and a great example of the generalizability of mathematics. That is, we often discover that mathematical models we’ve built of one system are able to describe what we see happening in lots of systems. This happens when the underlying processes are the same, as they are here. Again, thanks for sharing!

Thanks again to everyone who participated this week! Please feel free to email or tweet anytime! Looking forward to more great conversations about the art of mathematical modeling.




This week, I’m in Denver Colorado for the 5th annual STEM Forum and Exposition organized by NSTA. Later this week, on behalf of NCTM, I’ll be running two short “NCTM sessions” on mathematical modeling. In these sessions, we’ll explore the question “What should the “M” in STEM look like in a good STEM activity?” Later this week, I’ll post more about these sessions and the STEM forum, but today, I want to talk about my plane ride.

No matter how often I fly, I find that looking out the window of an airplane never gets old. Given a choice, I always choose a window seat, even if it means seat 35F at the very back of the plane as it did yesterday. My choice was rewarded with a cloudless sky for most of the flight and I spent my time alternating between reading and looking out of the window. Now, if you fly from Philadelphia to Denver, or fly any other route that takes you across the mid-west you’ll see large swaths of the country that look like this:


Crop circles! Well, okay, that’s probably not the kind of crop circles you thought of when you read the title to this post. But, they’re still really cool and it’s fascinating to see the patterns laid down over thousand and thousands of acres of the United States. These crop circles are, of course, the result of what’s called “center pivot irrigation.” This is where a pumping system is built at the center of a circle, a long mobile arm of sprinklers is constructed, and this arm pivots around the central pump, irrigating a large circle of crops. If you’ve ever driven through regions where center pivot irrigation is used, you’ve likely seen the sprinkler arms:


Flying over thousands and thousands of these crop circles yesterday, I realized that there are all sorts of interesting mathematical modeling and generally more STEM questions that they present. Here are just a few that occurred to me:

The pivot arm is on wheels all along its length. At various points along this radius, motors of some sort drive the motion. Since this is a radial motion, the motors furthest out must be going faster than the motors closest to the pivot point. How is this motion coordinated? How does this need to increase the motor speed with radial distance impact the largest such circle that’s practical?

Since water is being forced from a central point, the water pressure must drop as we move outward along the radial arm. This means that if all sprinklers were identical along the arm, the crops closest to the pump would get over-watered, and those furthest out under-watered. How does one design a sprinkler system for this arm so that we deliver the same amount of water across the entire circle?

While the arrangement of these crop circles clearly must follow, to a certain degree, local topography, why are they generally arranged in a less-than-optimal packing arrangement? That is, we know that hexagonal packing of circles covers more area than the rectangular packing we observe. So, why do these circles generally follow the arrangement on the left rather than the right in the picture below?


Why are the circles that we see generally all the same size? For the most part there are only two sizes of circles one will see. Why does there appear to be a minimum circle size? We know that if we used circles of various sizes, we could cover more area as in this picture:


Why don’t farmers use circles across a greater range of sizes?

Now, as with any good mathematical modeling problem or any good STEM problem, I imagine that the answers to these questions are complex, and involve multiple factors and multiple constraints. As my flight neared Denver yesterday though, I decided to see if I could at least convince myself that there was a good reason why we don’t see small crop circles by sketching out a really simple mathematical model. This was a “back of the airplane menu” type model, but nonetheless, fun to play with and I thought I’d share it with you here today.

I wanted to see if there was an economic reason why we don’t see circles below a minimum size. That is, are circles below a certain size just not profitable? I assumed that there were three basic costs associated with constructing and running a single center pivot irrigation system:

    \begin{equation*} C_p = \text{Fixed cost of purchasing a pump} \end{equation*}

    \begin{equation*} C_m = \text{Cost of water used, proportional to square of the radius of the circle} \end{equation*}

    \begin{equation*} C_r = \text{Cost of pivot arm, proportional to radius of the circle} \end{equation*}

I also assumed that the revenue one would generate was proportional to the area of the circle:

    \begin{equation*} R = \text{Revenue, proportional to radius squared} \end{equation*}

Putting this all together meant that the profit, P, could be written as:

    \begin{equation*} P = R-C_p-C_m-C_r \end{equation*}

Using my assumptions of proportionality to the radius, r, I could rewrite this as:

    \begin{equation*} P = a_0 r^2 - C_p - a_1 r^2 - a_2 r \end{equation*}

Here, the a_i are positive constants of proportionality. A little rearrangement yields:

    \begin{equation*} P = (a_0-a_1) r^2 - a_2 r - C_p \end{equation*}

Now, a_0-a_1 must be positive, otherwise, the profit would always be negative and there would be no sense in ever having any sort of circle. Knowing that means the general shape of the profit curve as a function of r must look like:


The fact that this curve becomes positive only at some finite positive value of r means that, yes, there is a minimum size below which a crop circle just isn’t profitable. I’m convinced that our farm industry knows what it’s doing when it avoids making teeny-tiny circles. In fact, it seems that we should be driven to make our circles as large as possible (note that real circles cover hundreds of acres) and that there must be another explanation for why we only make them up to a certain size. I suspect that the maximum size of theses circles is dictated either by the demands of designing for the pressure drop across the sprinkler array or the demands of increasing speed of the motors as we move further out along the arm. Or, probably both.

But, it was time to ask my neighbors to get up one last time so I could use the bathroom and then, quickly, my plane ride to Denver was over. I hope you’ve enjoyed my random flight musings and hope perhaps you’ve found some inspiration for some cool STEM problems here in this post. I’ll be sure to post again at least once more this week and share what I learn at this year’s STEM Forum. Till later.