Now let's look at a portion of the Staircase problem video segment in greater detail. In this Interactive Activity, you can watch how students create strategies and test them as they work through the problem. We've annotated this with a particular eye toward the student understanding about generalized, closed rules and iterative rules for algebraic patterns.
After you have observed the classroom, reflect on the following questions and select "Show Answer" to see a sample response:
Question: A goal of the Problem-Solving Standard is the presentation of rich mathematical content in the context of problem solving. How does this problem accomplish that?
Although this problem is easy to understand and represent with tiles, it connects to several ideas –– infinite sums, algebraic generalization, and visual representations of the sum of the integers, for instance. It also supports multiple solution methods, a characteristic of good mathematics problems. Both diagrams and tables reveal aspects of the problem, and students can use one representation to check conjectures drawn from inspecting the others.
Question: Problem solving is often thought of sequentially -- reading and understanding the problem, determining the unknown, formulating or choosing a strategy, implementing the strategy, solving the problem, and explaining your work. How do you see this sequence working in an actual classroom?
In this case, the students, like many problem solvers, are doing multiple things at the same time. The teacher's listening, questions, and observations help him determine which aspects of problem solving are being used and what may need special focus. Two key elements that are often worth emphasizing are how and when students apply strategies and formulate new ones on their own, and how and when they generate and record data and draw reasonable conjectures from the data. These two steps are at the heart of problem solving.
Question: Do you see students learning from errors?
Students are checking their own work and others' work. And in their efforts to explain table entries to Mr. Solomon, they do recognize errors and also notice that there is more than one solution method. This classroom environment encourages the students to learn from one another and to see problem solving as a shared experience.