This tweet from a friend of mine is one of my favorites.
8 names are read per min at the CHS graduation, I will be here for another 36min.. slooooow.
— andreabonilla (@andreabonilla) June 9, 2011
My friend has taken a problem from the world that was personal to her, identified the variables that are essential to the problem, selected a model that describes those variables, performed operations on that model, and re-interpreted the result back into the world. And tweeted about it.
That is modeling — the process of turning the world into math and then turning math back into the world. My friend probably wouldn’t wouldn’t label her experience like that but that’s what she’s doing. That’s what people who do math in the world do.
We know how this looks in many textbooks, though.
The amount of time (t) it takes a number of graduates (n) to cross the graduation stage can be modeled by the function t(n) = n/8. How long will it take all 288 graduates to cross the stage?
Here students would simply perform operations on real-world-flavored math while the important and interesting work is in turning the world into that math and turning that math back into the world.
Here is an alternate treatment, one that has students modeling as the practice is described in the Common Core.
Show this video.
Ask: “If I want to set an alarm that’ll let me take a long nap until just before my cousin Adarsh crosses the stage, how should I set the alarm?”
By design, it’s a short video. I’d like it to be boring enough to provoke my friend’s modeling but not terminally boring.
By design, it lacks mathematical structures because we’d like students to participate in the process of developing those structures. They won’t do that unassisted.
Before we get to the algebraic model, we can ask some important and interesting questions.
How long do you think it will take my cousin to graduate? Just estimate.
I asked that on Twitter and received the following estimates:
These guesses interest us in a calculation and also prepare us to evaluate whether or not that calculation is correct.
Sketch the relationship between the number of graduates and time.
Asking students to sketch the relationship, rather than plot it precisely, asks them to think relationally (“how do these two quantities change together?”) rather than instrumentally (“how do I plot these points?”).
Many students will assume the data is linear. But this prompt may invite some students to consider the possibility that the data is non-linear.
Collect data. Model the data. Get an answer.
Ask students to create a table of values. Ask students to plot the data in Desmos. Regress the data. Give them the graduation program. Calculate an answer.
I plotted the first ten names and modeled their times with a linear equation. (“Time v. names read” was my model, though commenter Josh thinks “time v. number of syllables read” would be more accurate.) The calculation for cousin Adarsh’s 157th name is 19 minutes. I would be foolish to rely on that calculation, however.
Ask your students to “Assume your answer is wrong, that something surprising actually happens. Anticipate that something and fix your mathematical answer.”
George Box: “All models are wrong, but some are useful.”
This is where we turn the math back into the world. This is where we make some math teachers uncomfortable, admitting that the world and the math don’t correspond exactly and that the math needs modification.
Watch all of these math teachers make exactly those modifications in the comments of the preview post. They perform mathematical operations and then proceed to describe why the results of those operations are wrong.
- Scott: “Add the bit of time prior to starting and a few seconds for a switch in readers as tends to be customary in larger groups like this … “
- Sadler: “14 minutes and 10 seconds but given that it is better to wake 10 seconds early than miss it, I would submit 14 minutes.”
- Scott #2: “You would probably want to set your timer a little earlier so you are fully awake when your cousin’s name is called.”
- Julie Wright: “As an embittered W, I am aware that there is lots of ponderous gravity for A’s and B’s, then everybody gets bored and speeds things up.”
Validate (or invalidate) the answer.
Commenter Mark Chubb, at the end of his modeling cycle: “Can’t wait to see Act 3.” Act 3 is the reveal in this task framework I call three-act math. It isn’t enough for Mark to simply read the answer in the back of the book or hear it from me. He wants to see it. So:
If you built a linear model from the first ten names, your answer winds up too large. Instead of 19 minutes, my cousin graduates at 17:12, sooner than the math predicted.
In the video, you can hear the validation of Julie Wright’s hypothesis above. The A’s and B’s get a lot of pomp, and then the commencement reader races through the rest.
Many congratulations to Megan Schmidt for her guess and to Scott and Kyle Pearce for their calculation. They all put down for 18 minutes. Special mention also to aga bey for 16.3 minutes. That commenter’s method? “I took the average of all submissions upthread.” Strong!
Again, if mathematical modeling requires the cycle of actions we find here, our textbooks typically only require one of them: performing operations. The purest mathematical action. The one that is often least interesting to students and the least useful in the world of work. So let’s offer students opportunities to experience the complete modeling cycle. Not just because those are the skills that most of the fun jobs require. But because modeling with math is fun for students now.
I ran into this when working up an exponential growth problem for my son’s precalculus class. The CDC had data on the number of Ebola cases which could be modeled with an exponential growth curve at the time. However, the math needed correction because of a sudden increase in cases. The CDC readily admitted they believe the cases were unreported by a factor of 1.5 to 2.5. Thus, a human eye on the data to recognize that and make an adjustment was necessary.
Later, when the curve could be modeled nicely by a logistics curve, the equation was still incorrect in predicting the end of the epidemic. As teachers we would like to be able to button everything up and wrap it in a bow, but the real world seldom works that way.
Matt TownsleyJuly 6, 2015 - 4:42 pm -
If I was a math education professor, I’d share this post with my students (well, after modeling this type of lesson a few times first) and then spend the rest of the semester dreaming up three acts. When we’re not creating these “lessons,” we’d probably be critiquing each others’ acts.
Julie WrightJuly 6, 2015 - 6:43 pm -
I love aga bey’s strategy of averaging everybody else’s answers, and it worked quite well! The calculation median in the box plot looks awfully close, too. (What was it?)
Last year, when my students did estimation180.com, we would enter “too low”/”too high” values that were (roughly) the median “too low”/”too high” choices for the class. (Somebody would suggest a value, people would raise their hands for whether they agreed it was too low/high, and we’d adjust/check a few times till we had a value where about half the people raised their hands.)
This seemed to work amazingly well to give high-quality lower and upper bounds. We could all tell the class median values were consistently better than any individual’s would have been over time. It also meant that everybody had input into the class low/high boundaries.
(The class median probably would have worked well for the actual estimate, too, but they took turns giving that one individually along with their own reasoning.)
XavierJuly 6, 2015 - 10:10 pm -
I worry about the type of models that usually we do. Almost every situation in real life could be modeled by a linear model (specifically, an afine model – the situation could be modelized by a function of the type f(x) = ax+b). Almost every 3-act math is about that.
I’m worry about it. I’m worry about not to find enough situations in which we need quadratics, trigonometric, exponentials,… functions to modelize. Is it because we oversimplify the model or is it because the 99% of situations could be modelized by afine function?
It it’s the second, our students tend to modelize things always with afine functions because “it’s always this way” ;-)
Just a thought,
Clara MaxcyJuly 7, 2015 - 6:59 am -
This is one lesson that is going into my IBL playbook! I am beginning to look for video opportunities for my classes. Every time I watch one of yours, I think of moments I sat and watched, thinking this would make a great! classroom poser – but didn’t take out my camera! (Large thunk of palm to head!)
Dan MeyerJuly 7, 2015 - 8:59 am -
I know that facepalming feeling!
Maybe useful context: for every video or photo I post there are five more on my camera. That’s to say, I capture lots of math, most of it useless for public / classroom consumption. But I find the act of capturing math sharpens my senses to the math around me. Constant capturing prepares me to notice the great math when it crosses my path.
Malcolm RobertsJuly 7, 2015 - 2:11 pm -
Through no fault of their own, here in NSW (Australia) many of the mathematics teachers aren’t aware of the processes of mathematical modelling and hence don’t actively engage at any level in this sort of activity themselves. As a result they don’t notice as readily opportunities to introduce modelling into their teaching and also would feel uncomfortable trying to do so.
We are trying to address this at the university (Newcastle) where I work by introducing into our mathematics teacher training degree a course in modelling (using problems where progress can be made with only school level mathematics). The problem of assessment still remains though. The external examinations sat by our school students don’t involve any modelling activity.
Dan MeyerJuly 7, 2015 - 4:08 pm -
Love to hear more about this. email@example.com if I can be of service.
Not just a NSW issue. I have yet to see a good (valid, reliable, practical) assessment of modeling.
StacieJuly 7, 2015 - 6:27 pm -
“This is where we turn the math back into the world. This is where we make some math teachers uncomfortable, admitting that the world and the math don’t correspond exactly and that the math needs modification.”
I ran into this when working up an exponential growth problem (http://systry.com/precalculus/ebola-2014/) for my son’s precalculus class. The CDC had data on the number of Ebola cases which could be modeled with an exponential growth curve at the time. However, the math needed correction because of a sudden increase in cases. The CDC readily admitted they believe the cases were unreported by a factor of 1.5 to 2.5. Thus, a human eye on the data to recognize that and make an adjustment was necessary.
Later, when the curve could be modeled nicely by a logistics curve, the equation (http://systry.com/precalculus/ebola-2014-revisited/) was still incorrect in predicting the end of the epidemic. As teachers we would like to be able to button everything up and wrap it in a bow, but the real world seldom works that way.
Dan MeyerJuly 8, 2015 - 9:35 am -
Thanks for the example, Stacie. I’ve added it to the main post. I’ve subscribed to your blog also. Looks like you’re posting some great ideas.
SusanSeptember 3, 2015 - 5:40 am -
I am currently doing this lesson with my students. We are extending it to predict their moment of graduation. As such, they need to consider current enrollment, graduate rates, etc. Terrific idea and a great start to the year.