We did a really ambitious activity early this year in Geometry. We slowed down an optimization problem for start-of-year Grade 9 students, in order to get every single kid to understand the process. It worked brilliantly, and we are trying now to help them extend the idea to other optimization problems.
I want to document and share this journey, because I think the experiment that we have started is SO challenging and SO worthwhile. We are trying to get the kids to think like mathematicians. By slowing them down.
Here was the first time we formally introduced optimization (after having the kids play around with building their own popcorn containers). If you read it carefully, you might notice that we tried to emphasize a few things: 1. Tactile learning. 2. Justifying their thoughts. 3. Understanding what x and y represent. 4. Understanding how to analyze the domain. 5. Resourceful use of technology. 6. Interpretation of results back in context.
A little while after going through this, we gave the kids an open-notes test, using a different sheet of paper to start. They did well, which showed us that they really understood the different pieces. Then, on the actual closed-notes exam, we worked backwards by giving them a factored cubic equation, and asking them some relevant questions: 1. What is the dimension of the piece of paper that we had started with? 2. What is the domain of this problem, and why? 3. What is the largest box that can be built here, and what are its dimensions? 4. How many different boxes can we build that would have a volume of _____, and what are their dimensions? Again, the kids did brilliantly!
It has been amazing and humbling to see how far along these 9th-graders have come.
The next question that we gave them, which they worked through in small groups, was an optimization problem involving a known perimeter of a rectangle and in trying to maximize its area. They needed to take the problem from start to finish, in writing a system, combining it into a single function, and doing domain and graphical analysis to find the maximum. Then, as usual, interpreting back in context.
Now, the next problem they are tackling has to do with maximizing the area of an isosceles triangle whose perimeter is 30 units. Not easy. Some kids figured out right away that this follows the patterns of other similar problems, where they will try to write an area equation. Some kids started to make tables -- another really great habit of a mathematician! The class, as a whole, needed a nudge to help them figure out how to get the height of the triangles. We stopped discussions today at writing a general height equation as a class, in terms of x, the length of the two congruent sides, and I asked the kids to keep thinking about the rest of the problem.
A worthwhile experiment, indeed! If by the end of the term, they can do even half of these problems completely independently of us, we will be so thrilled. I think the key is to slow them down. By feeding them the understanding in pieces and then giving them another similar problem, we are building the foundation that it takes to transfer the knowledge.