Answering these kinds of questions is difficult for most students, even those who quickly get the right answers to problems. The teacher’s questions scaffold Rosalinda’s thinking. When she doesn’t explain why the denominators stay the same, the teacher notes that some students add the denominators, and asks why Rosalinda hasn’t done that. By the end of the conversation, Rosalinda has fully explained what she did, and why she did it. The teacher helps her further ground her thinking in the situation by asking how else 8 crayons might be divided by two. The teacher does note that Rosalinda got four as the answer, but she comments, “I see how you did that,” emphasizing the reasoning through the steps as what is valued rather than just the answer.

Rosalinda has already solved the problem by adding 8 half crayons, so she is choosing an image that has 8 halves and an unknown whole. This represents the solution to the problem—the unknown total would be 4. Her partner correctly chooses the illustration that represents the problem situation, which has a known whole of 8, and an unknown quantity for half. Each confidently and effectively explains their own thinking—a confidence gained through precisely the kind of practice we are observing. Talking with a partner provides the listening ear and response that a teacher cannot possibly provide in sufficient quantity to every student. 

This same approach of using students’ varied strategies to build flexible reasoning can be used in mathematics classrooms at all levels. By starting in the earliest grades, students’ appreciation of mathematics and identity with respect to mathematics competencies can be developed right from the beginning.

Finding the equation for depreciation over time is a much more challenging problem. The move the students make is to apply the equation they used again for each year: 5000 x .15 = 750. Multiply that by .15 to get 112.5. Multiply that by .15 and you get 16.9. After 5 years, the value is 37 cents. They realize something is wrong; they’re just not sure what. They question their original answer for year one. One student suggests maybe the $750 was the total value of the car. 

If students were working alone, they may well have given up at this point. Because they are working in a group, students externalize their thinking and begin to make progress in solving the problem. One student notes that continuing to multiply by .15 will quickly lead to a value of pennies, which cannot be right. He correctly concludes that there is something wrong with using .15. One of the girls realizes that the percent decline has to be applied to a new base each year—the second critical piece of information. But they have not figured out how to combine these two insights into an equation.

She just tells them they are right that their equation is wrong (a nice way to give them positive feedback), and she leaves them to work further on the problem. With this little nudge, students dig a little deeper. One student correctly concludes that the .15 is wrong. Another correctly concludes that the base has to change each year. This is the kind of conversation that deepens learning and builds students’ perseverance. 

Then she gives them a perplexing message: the second answer is right, but the first equation is getting at the important concept (even though it’s wrong). She calls on a student for an explanation regarding the changing base in each year. Notice that the student who explained was not the students in the small group who originally had that idea. But she took the idea offered by her partner and made her own sense of it. With that critical bit of information, students continue to work on the problem. 

Most Algebra II students can make sense of exponential situations by thinking about it recursively. 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. However, this recursive thinking will not lead them to the correct equation. Expressing what is going on in this situation as an equation is very tough. This is true with any significant transition in mathematics that requires greater abstraction—such as learning multiplicative reasoning rather than doing repeated addition. In this case, the challenge of understanding an exponential relationship as an equation is very hard, so it is not surprising to see these students taking this much time and struggling with it. This is not an idea that can be learned in one lesson, it will more realistically require two to three lessons before students understand.

We do not see the end of this class, so we cannot be sure whether the students figure out the exponential function, 5000(.85)^t. What we do know, however, is that students have struggled to make sense of the problem, and the final equation will provide an “ah ha!” moment that is far more likely to stick than if the teacher had provided the equation up front and students had worked individually to solve the problem by applying a prescribed algorithm. Mathematics, particularly in the higher grades, can take an intensity of focus that is hard for many adolescents to muster. Collaboratively solving problems can provide an incentive for students to persevere, and to use each other’s ideas to build their own understanding.

It is not uncommon for students to use lower-level mathematics to try to solve problems that the teacher expects students to solve using newly learned mathematics (again, multiplication problems can be solved using repeated addition as long as the numbers are small enough and the time allotted long enough). In this type of problem in which students are much less constrained—they can propose many situations that can be characterized by the equation. This pair of students reveal both that they have mastered the earlier mathematics, and that they have not yet grasped the concept of a two variable problem.