Back in April, I had the pleasure of attending a CPAM Leadership Seminar with Dan Meyer on mathematical modeling, where he lead us through Dandy Candies. Dan wrote about this 3-Act recently here. I’ve used several 3-Acts with my students, but this was my first time to participate in one from a “lower-case l” learner’s perspective. I’ve read about “purposeful practice” and “patient problem solving” for several years now, and I know that I have some understanding of what they mean, but seeing them in action from the learner’s perspective is powerful.
A few things struck me during the seminar. We don’t do 3-Acts just for fun. (I knew this, but Dan made it very clear that this isn’t just about engaging students in doing something; it’s about engaging students in doing math. I’m not sure I’ve made that as clear to other teachers with whom I’ve discussed 3-Acts.) As Smith & Stein point out in 5 Practices for Orchestrating Productive Mathematics Discussions, there is pre-work for the teacher: identifying the math content learning goal for the lesson and then selecting a task that is going to provide students the opportunity to engage in that math content. Even with a 3-Act, where we let our students’ curiosity develop the question, we do have an underlying question that will engage students in the math content we want them to know. What they ask might not be worded exactly the same, and it might extend the mathematical thinking in which we want students to engage, but the math is there.
I have said before that I use technology to give every student a voice – from the loudest to the quietest, from the fastest to the slowest. When Dan solicited questions we could explore from the group, I was never going to volunteer mine for the list. (I am not criticizing Dan’s move here … just noting that I find it challenging, both as Learner and learner, to establish trust in a short session with participants that I’m likely not going to see again.) I *might* have participated had I been asked to submit my question somewhere anonymously.
And finally, I really like the opportunity that we had before each question to answer before performing any calculations. I’ve been working on providing this opportunity for my students, but it still isn’t automatic. I have to remind myself to ask students to use their intuition first. As I heard from Magdalene Lampert, “Contemplate then Calculate”.
We were working on Modeling with Geometry (G-MG) when I returned to class after the seminar last year, and so I tried Dandy Candies with my students.
are the heights the same 1
what are the surface areas of the boxes 1
the similarity between how thevolume stays the same and te cross sections change1
Do all the solids have the same volume? 1
are the surface area and volume the same throughout the same changes? 1
do all the boxes have the same volume 1
how many cubes 1
what shapes could be made 1
how does the surface area change 1
same surface area? 1
could the volume make an equal ratio 1
whats the volume of each cube that makes each shape 1
how many different shapes can be made with those boxes 1
Do they all have the shme volume 1
Is the area of any gift formed by the candies the same? 1
What do you *think*? Which package(s) use the least cardboard?
(No one answered more than one.)
What do you *think*? Which package(s) use the least ribbon?
(No one answered more than one.)
What do you *think* are the dimensions for each box?
I enjoyed watching students use appropriate tools strategically while they were working.
And then I sent the polls again.
Which package(s) use the least cardboard?
(Two answered B and D.)
Which package(s) use the least ribbon?
(15 answered B and D; 2 answered B and C.)
Some mistook “better” for “best”, and others are apparently going to cut the candies in halves.
Modeling with Geometry G-MG
Apply geometric concepts in modeling situations
- Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
- Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
- Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Content-wise, students had the opportunity to learn more about modeling with geometry. And they were able to engage more steps of the modeling cycle than just computation.
I am not *yet* writing 3-Acts, but as the journey continues, I am grateful for those who do and share …