In a cooking mag: a 9×2 round pan holds the same as an 8×2 square pan. True? Test with math? Test with water? pic.twitter.com/LDfDheoQyZ
— Marilyn Burns (@mburnsmath) March 30, 2016
On Twitter, I remarked that Marilyn had summarized the entire modeling cycle in a single tweet. But the part of that cycle she summarizes best is the last: validating your answer.
With mathematical modeling, you don’t have to be the answer key. The world is.
If you have total faith in the perfect accuracy of your mathematical models, testing with water may sound unnecessary. For the 99% of your students who wonder if math has any power outside of their textbooks, test with water.
It was one thing to manually figure that out [if some glue could hold up a human], and then another to try the same thing with a bowling ball experiment modeling the same thing. We were able to see if our answer actually held up in that situation, it was a moment that will stay with me forever.
BryanMarch 31, 2016 - 7:42 am -
I really like this tweet, especially for the discussion it could generate. Which one do you think holds the most? Why do you think that? If they don’t hold the same, how close are they? What heights are needed for them to hold the same? So many great questions to explore. Having students wonder on this picture could provide a weeks worth of investigations (in fact I may use this next week).
I agree with your comment about testing with water, but I would say that even that last 1% should as well. It is one thing to be proficient in mathematics and producing an answer, and a totally different thing to actually experience your answer in a real life situation.
I had a teacher ask me a problem about the crazy glue commercial, where the guy glues his hardhat to an iron beam and then hangs from it. He provided us with some information about glue strength and had us determine how much glue would be needed for the guy to do that successfully. It was one thing to manually figure that out, and then another to try the same thing with a bowling ball experiment modeling the same thing. We were able to see if our answer actually held up in that situation, it was a moment that will stay with me forever.
Dan MeyerMarch 31, 2016 - 8:02 am -
“Forever.” Love it. Added to the post.
Joe MakoMarch 31, 2016 - 8:23 am -
Once I saw they were about about the same volume, my next question is which would I choose to bake with? Thinking about what attributes make for a ideal pan can go a few different directions, what is important for me or for others? For example what if we like more chewy crust, or more moist center? What happens if we have a taller pan, will it cook fully in a 5x5x5 pan? How about a zig-zag pan like http://www.bakersedge.com/product_ebp.html or a spiral pan, and with these approaches how would I calculate the outside dimensions to get a similar volume? Which has a better design for human interaction, such as cutting options, ease of making equal sized portions, and the effort to clean? What other questions could I be asking?
Howard PhillipsMarch 31, 2016 - 10:28 am -
Engineering meets Math, head on, this one.
ScottApril 1, 2016 - 6:13 am -
In my experience moments like the one Bryan mentioned are incredibly powerful as key pieces to help students track and refer back to their mathematical learning. References like “Remember when we filled the pyramid with water and poured it into the prism?” provide much better framing for students to be reflective and to recall information than “Remember lesson 8.3 where we learned about the formula for the volume of a pyramid?” even if that lesson had been discovery-based, super engaging, and delightfully well scaffolded.
Something about visceral experiences that offer just a moment of uncertainty – even when you’re confident in your model – are super appealing. Especially in a content-discipline where most students still are offered strictly learning experiences that have some certainty to them and some “best” solution.
MikeApril 11, 2016 - 10:48 am -
I remember physics became really real to me during a freshman lab called the “ballistic pendulum” where the group’s daily grade was summed up in a single shot at a target given a randomly assigned input parameter and a load of initial testing and data gathering.
Largely, it was lost for me again during the all-theory portion of junior level electromagnetic theory and non-homogenous materials carrying electric charge.
Then I swapped major/minor and finished up my bachelor’s really loving graph theory and real analysis.
AveryApril 15, 2016 - 3:27 pm -
I keep thinking about this, and keep coming back to some discomfort with the fact that I don’t *need* “the real world” to verify which baking pan holds more. Theoretical math and some understanding of volume is enough. I don’t *think* that’s the case with the superglue example. I also don’t see how coming up with a mathematical model for “volume of baking pans” will allow me to solve some new (uncontrived) problem in the future (the cycle of modeling), but maybe I’m just not being creative enough. The super glue example seems more complex, isn’t well-defined, and therefore feels more model-ly to me.
Dan MeyerApril 16, 2016 - 6:59 pm -
@Avery, you don’t need the validation step for the pan but I wonder if that statement simply speaks to your familiarity with the model. In which case, perhaps students who are as unfamiliar with rectangular solid models as you are with the super glue model would benefit from that validation.
The modeling cycle, insofar as I understand it, doesn’t seek to transfer a model to a new problem but to refine a model given new data. You think you know what’s going on given the data you have. Then you get a new piece of data and your model either validates or it changes. Nothing to do with new problems as far as I know, though that’s probably nice.
Mai Nu ThaoMay 5, 2016 - 1:41 pm -
Thanks for sharing this. I’m currently at the end of one of my math methods class and we’ve been learning a lot on how to improve our methods of teaching math to make it not only more comprehensible but engaging and interesting. This post was a great example of how to hook students and engage them. Not only will they be testing it out with math but they will also be able to physically pour water into each pan and experience for themselves if they were correct. This problem could have easily just have been another problem on a worksheet but because with simple adaptations and modifications you not only get better engagement but better retention of information. I believe we as teachers have a choice to choose what we allow our students to spend time doing during our classes and by doing these kinds of problems if not all the time then most of the time would be a great way to inspire students to explore math outside of the classroom as well. We have been learning in my class that it is not the worksheets that students will remember from your class but the moments like these when they experience the math that they will remember later on.