PROMOTING REASONING TO RICH TASKS
So often an approach to a mathematical problem is formulaic, totally plug and play, and without much attention given to concepts. After all, adding Уг and Уг and getting У does not make any sense if one would only take a few seconds to think about it. The emphasis of thinking, really thinking, about a problem before rushing to get a solution was a major issue for the teachers. For example, another follow-up problem stated that:
Consider the following proportional reasoning problem that the teachers were asked to attempt:
If 1 robot can make 1 car in 10 hours, how long it will take 10 robots to produce 10 cars?
It is far too easy to slip into the mindset that every value in the first scenario has been multiplied by 10. Almost every group had someone do this. However, when that teacher looked at the answer of 10 hours, the realization that the answer made no sense came quickly. One teacher looked at her work, gave a quizzical look, and said, “Now, that just cannot be right. Hmm, why does not that work?” Then, the group thought about the problem and the relationship between the values. The teachers recognized the crucial importance of thinking about the question before crunching numbers.
Mistakes and confusion allowed the teachers to use proportional reasoning and mathematical arguments to do side-by-side comparisons of solutions, or just talk through comparisons of solutions to find where they did not match up. Then, the teachers strategized to determine not only how to proceed but also to determine why one method did not work. For example, “1 robot can make 1 car in 1 hour” does not mean “2 can make 2 cars in 2 hours.” Teachers discussed why a simple “multiply through” technique did not work. Teachers benefitted from these discussions in several distinct ways. First, they began to see that real problems involving proportional reasoning are not simply plug-and-play exercises; they are multi-layered challenges which require analysis, sound reasoning, and understanding of the relationships among quantities.
Second, they recognized the profound importance of conceptual understanding as a baseline for strategizing approaches to problem solving. And, third, they gained an acute appreciation for the frustration of their students who apply incorrect procedures and cannot understand why their answers are incorrect. Additionally, as can be seen in the posters, the teachers gained an appreciation for the validity of multiple approaches to problem solution. Several teachers mirrored that idea in their writings.
After having experienced the Robot problem, the teachers planned a lesson study around this topic and presented the following problem to their students:
Text Box 9.4 A Math Happening 9c: The Robot Problem
If 3 robots make 17 cell phones in 10 minutes, 12 robots can make how many cell phones in 45 minutes?
The various strategies used by students to solve the robot problem can be seen in Figure 9.5. These approaches can be connected to the learning progressions of the CCSSM Standards.
One approach to this task was to set up proportions to solve the problem. This strategy began with the use of ratios in order to describe the relationship between the number of robots and cell phones produced 6.RP.A.1, expressing the given information as 3/17. From this starting place, students showed recognition of the proportional relationships between 3 robots producing 17 cell phones and 12 robots producing x cell phones (3/17 = 12/x) 7.RP.A.2.B and used this to find x = 68 cell phones produced by 12 robots.
Continuing this multistep proportion problem of 10 minutes for 68 cell phones and 45 minutes for x cell phones produced (10/68 = 45/x) 7.RP.A.3 giving an answer of 306 cell phones. Another strategy involving recognition and use of proportions computed how many cell phones the initial 3 robots could produce in 45 minutes based on a ratio of 17 cell phones in 10 minutes (17/10 = x/45) and then scaled up this number
Figure 9.5 Student work showing various strategies for the robot problem. Source: Authors.
(x = 76.5) by 4 (76.5 x 4) because the problem asks about 12 robots instead of the initial 3 6.RP.A.1, 7.RP.A.2.B, 7.RP.A.3. An additive strategy students used to solve this problem was to visually represent a group of 3 robots producing 17 cell phones in 10 minutes.
This “group” was added together four times to find how many cell phones 12 robots (3 + 3 + 3 + 3) would produce in 10 minutes producing 68 cell phones (17 + 17 + 17 + 17). Students continued by creating a table of values 6.RPP.A with increments of 10 minutes showing how many cell phones would be produced by 12 robots at 10, 20, 30, and 40 minutes. The additional 5-minute increment was computed by dividing the amount produced in 10 minutes (68) by 2. The last strategy used by students for this problem was finding the unit ratio of how many cell phones 1 robot would produce in 10 minutes 6.RP.A.3B, 7.RP.A.1. This unit ratio was then used to compute how many units (cell phones) 1 robot would produce in 45 minutes and use this to scale up to the number of units produced by 12 robots.
Our class observations, conversations with participants, review of the team posters, lesson study examinations, and the individual reflections have highlighted several central ideas. Teachers need the opportunity to struggle with problems in order to develop deep understanding of proportional reasoning. While many teachers expressed frustration with the homework problems as well as the in-class problems, they also recognized that their frustration led them to think about proportional reasoning in ways which they had not employed previously.
This led to deeper understanding. Several teachers reported that they now “get” proportional reasoning and are gaining appreciation for the connections between concepts; they attribute this to the experiences of struggling through the investigative problems without the crutch of plug-and-play procedures.
The daily investigations, such as the cathedral problem and the robot problem, led to discussion and exploration of much more than simply trying to find an answer. Teachers questioned each other’s thinking and would not allow unsubstantiated assumptions. The focus was on mathematics of the proportional reasoning, not the answer. Teachers were repeatedly heard asking each other, “please explain that again, I don’t understand where you are going with this” or “why would that be reasonable way to solve this?” Knowing that numerous approaches to problem solution were both possible and valid freed the teachers to concentrate on the soundness of their approaches, resulting in the teachers being able to develop more profound understanding.
Teachers are learning to think more insightfully and to use proportional reasoning in ways which they have not previously employed. This cannot be expected to occur overnight. The same is undoubtedly true for our students. It takes time and practice before the teachers may see connections between the concepts and techniques which they learned at the summer institute. The rich collaboration and communication experience that the institute provided the teachers with not only helped them to think outside their comfort zone but also helped to impact their beliefs and disposition.