One thing about being new to the Math Twitter Blog-o-Sphere, is that there’s this wealth of old posts that I’ve never read. And last week I fell down the rabbit hole in an extremely fortuitous way. It started with this old post from Fawn Nguyen, which got me really excited to see if I could do Dan Meyer’s Taco Cart and find a way to fit it into the unit on Trigonometry I was just wrapping up.
But as I started to plan it, I somehow found myself reading through Frank Noschese’s concept of “pseudoteaching”. What really got me thinking was his link to a 1-hr long talk by Harvard Physics Professor Eric Mazur about why he had given up lecturing. I was spellbound and it was Friday night, so I plowed right into another one, entitled “Assessment: The Silent Killer of Learning”. The ten minutes from minutes 23 – 33 were about purposeful assessment regarding problem-solving skills and they blew my mind.
Here is Mazur’s idea of an authentic problem from his own real life:
And here’s what a textbook does with this sort of problem:
And so I had my big epiphany going into Taco Cart; they would really make their own model. We watched the video together and I had them each write down a question, then discuss with their group of 4 (which was randomly generated, hat tip to Fawn her recommendation of Instant Classroom). Then I took questions from them and said I’d answer what I could. In the end, the only thing I answered was how far Dan walked across the sand (99.25 meters) and how far he walked along the pavement (171.5 meters). I wouldn’t even answer if it was really a right triangle.
I instructed them to create a model, list their assumptions, and figure out who gets to the taco cart first.
The kids immediately launched into a discussion of how fast they could possibly walk both on the sand and the pavement. One group asked if they could go into the hallway to time themselves walking to come up with their rate. Another group reasoned that you could leisurely walk one kilometer in 20 minutes, so they picked their pavement speed as 3 km/h. Many groups decided that you could walk twice as fast on pavement as on sand.
Out of 7 groups, there were only two groups who had independently chosen the same speeds for Dan & Ben. What this meant is that everyone’s times for Dan & Ben were different. Not only that, in some groups Ben was faster! When they’d all finished the task, they reported back to me their speeds; I gave them the “official” speed and we watched Act 3.
I loved doing it this way because it was valuable (though it took a long time) for the students to decide what the speed was. I also told them that I had written on my paper a formula that I used to help me know if they were right or not and asked them what they thought my formula was. They generated the formulas Tdan = 99.25/Ssand + 171.5/Spavement and Tben = 198.15/Ssand (which some groups diligently notated even though they’d correctly solved the problem without a formula — old habits die hard).
There were only ten minutes left in the class, and I asked them what follow-up questions they could come up with. And like magic they came up with Dan’s two follow-up problems: Where should the taco cart be so that they get there at the same time and What is the actual fastest route. I gave them free choice within their groups as to which problem they wanted to work on the next day.
What they did the next day absolutely blew my mind. I should pause here to say that these are MYP 4 (9th grade) students. They’ve never solved a quadratic equation (though they have worked with quadratic expressions) or graphed a parabola. They’ve certainly never solved a complicated optimization problem. AND YET. They managed to solve these problems.
Most groups immediately turned to trial and error. Some of them went through a lot of trial and errors, even after I said, “is there a better method you could think of?” In the end, almost every group had a correct, unprompted by me, algebraic equation. This poster below kills me because they went through so many trials and tried to find a pattern within their trials. They were ready to give up and I asked them what they were changing every time, since that’s obviously what they were trying to find. That was enough and a light went on. Unfortunately, they only came to these equations (bottom right) with about ten minutes left, so never got to the final answer, but wow:
I don’t know if they could’ve gotten to the answer, but two other groups did, in what I think is a sort of stunning feat of mathematical persistence, symbol manipulation, and creativity when faced with an unknown type of problem (a quadratic equation):
Both groups that solved this used the illegal move of dividing both sides by b, which I explained to them eliminated the trivial solution of b = 0. They both asked, “well, then how do you solve it,” which is really nice — creating mathematical necessity for a method of solving a quadratic.
And magically two groups got the other optimization problem right too! They both managed to generate a formula with T in it and the only nudge I gave them was to say “What if instead of T you had a y?” and they were off and running to graph it. To my surprise as soon as they saw the graph (first they said “We graphed it and nothing came up,” which was the perfect time to introduce the zoom function — they do not normally have access to graphing calculators, but we have one school-wide set that I reserved for the day) they knew exactly what they wanted — where y was at a minimum so they could find the shortest time. I pointed them to the calc button and they ended up with:
I was thrilled and even more importantly, they were thrilled. They would literally CHEER and high five when they discovered they had the right answer. They were all so excited to watch the Act 3 videos. In the end, 4 groups solved the problems, and the other 3 groups would’ve too if they’d had a little bit more time. This was the best lesson I’ve ever given but to be honest, it gave itself. And it demanded so much more of them than any textbook problem, both by making them model, create their own assumptions, and then giving them a problem that was so far beyond the scope of anything they’d ever done.
I could say a million more things about this lesson, but I’ve already blathered on long enough. Just do this lesson, even if it doesn’t seem like it fits with your curriculum! And do the sequels, even if you think the math is a bit over the kids’ heads. You won’t regret it! Thanks to all the people who inspired me to go for it.