In this video, students are asked to think about the depreciated value of a car that loses 15% of its $5,000 value after a single year, and then find the equation for the value thereafter. 

The teacher does two things in the launch of the problem that should be noted:

1. She situates the car value problem in prior work and points out the salient difference without getting involved in how to do the problem itself, “we’ve done a lot of work with percent increasing, but today, we’re going to do percent decreasing.” By connecting to prior work it avoids students wasting time going in the wrong direction.
2. The teacher also sets the expectation that students are to be able to justify their equation. She will not be telling them if they are correct, instead they’ll have to validate and justify it themselves and show their work. This move shifts the knowledge and authority from the teacher to the students.
   

The students are trying out different ideas and one of them has arrived at the absurdity that the car is worth $.37 and concludes that it can’t be the correct equation. What is interesting at this point is that their response is to laugh. This laughter reflects a good relationship with doing mathematics, there isn’t a fixation on just getting what is right, but also in recognizing what is not working. The students are expecting the answer to make sense, and they’re using their own agency and judgment. 

You’ll notice the student misinterpreted what the 750 represents, thinking that it is the car’s value after one year. The student misread the problem, but instead of correcting her, the teacher asked her a question that required her to do some mathematics. By mentally subtracting 750 from 5,000, the student arrived at the correct interpretation. By giving students the platform to solve the misreading, the student was able to reestablish their agency as a mathematics problem solver.

You’ll see a great question and answer in this segment that reveals critical insight. He asks “how do we get the right answer, but the wrong equation?” His peer responds that the equation only works once. As a group, they are zeroing in on the essential issue that exponential functions resolve.

The student in desk 27 has not said anything and is not doing anything to make contact with the other students. It is hard to know what is going on in his head in terms of if he is working on the problem and what he is thinking about. This is something we commonly see in groups of four where there are places for students to hide and subsequently students can get lost. While groups of four can be useful sometimes, groups of two students eliminate places to hide and require everyone to participate.

The students are doing something that is very common which is using the instructional context. They know they are currently studying a unit on exponentials so they are therefore pretty positive that their equation is going to be an exponential.

Using the context of the instructional program is a scaffold and does not demonstrate an understanding of the mathematics. It’s one of the things that can mislead students and teachers into thinking that they understand more than they actually do. When they get an exponential situation out of context, like on a test at the end of the year, they’ll be baffled because they will not have that scaffold of instructional context available to them. That is one reason we have to cycle back through the things that are out of the particular program context, as you can see, this is a tough problem for a lot of the students.

The teacher included a just in time review of negative exponents. It’s respectful of the problem solving and modeling reasoning that students attempted and trying to make it decrease even though they didn’t do it correctly.

Most people and certainly most Algebra II students can make sense of exponential situations and think about it recursively, where the initial amount decreases by a certain percent on one step, and then the result of that step is fed into the same calculation and then the result of that step into the next calculation. They could approach this problem with recursive thinking, however it will not lead them to the correct equation. Expressing what is going on in this situation as an equation is very tough, it’s analogous to Algebra I or middle school math where you have a linear situation and kids understand it in the form of a table that you add three each time. But understanding it as a multiplicative relationship after n times, it’s something times n is much harder than linear thinking. 

The teacher made a formative assessment judgment that the students were struggling unproductively and she rerouted them onto a path that was more productive. Productive struggle does not mean that you let them struggle until they go off the road and sink in the mud. Productive struggle means letting students struggle so that they can find the ideas they need and find where those ideas are going to fit with their prior knowledge.

The teacher intervenes by building a table using the student's own thinking to get them on a productive path. It is not clear if this will be enough, but she did get to the point where one of the students said it is going to be 15% of this new amount.