This spring I've been working with a team of NYC teachers in Math for America. As part of a lesson study, we will all teach the same (approximate) activity, video record ourselves, and then discuss how we approached student interactions.

The activity we are basing our lessons on is adapted from Mark Driscoll's 1999 book Fostering Algebraic Thinking, and you can find many variations of it online by searching "skyscraper window problem." One writeup is on the Science Friday website, from a writeup that fellow MfA teachers Bushra Makiya and Jon Koenig did. The basic setup is as follows. The cost for washing windows on a building changes with height, such that the first floor costs $2 per window, second floor $2.50 per window, third floor $3 per window and so on. Imagine a building with 38 windows on each floor. A) how much will it cost to wash all the windows of such a 12 story building? B) how much for a 34 story building? C) how much for an n story building?

I tried this in four of my classes. Two of the four are ICT sections. One is composed of ENL students. The fourth is general population.

There are a number of approaches to each section of the problem. One thing that struck me across all four of my classes was how many students had never heard of ”story" meaning the floors of a building. But such vocabulary issues are easy.

A bigger struggle is convincing students to show their thinking.

Above is an example of what I've learned to call a bald answer. Numerically it is a reasonable response to the prompt, but there is no evidence of the thought that led to it. In circulating through the room as students are working, I am able to probe for deeper understanding. Sometimes it is on a separate paper, which the student mistakenly doesn't believe matters (since they often think math is more about answers than process). But, after the fact, if I receive a paper like the one above, I have nothing.

The most common strategy in my classes was to find the cost per floor, and add them up.

This was also the basis for many missteps. Several students found the cost per window for the 12th story, multiplied by 38. Some students gave that as the answer, others multiplied that number by 12. One student found the cost as far as the 6th floor, got tired of the process, and simply doubled the number (since 12 is in fact the double of 6).

One thing that puzzled me was the lack of drawings. All student attempts that I saw were based on tables or table-like representations. Although some people are able to abstract directly, I feel that a picture is a powerful tool in understanding relationships, even if it is a poorly drawn picture.

If there were a picture, I can imagine the following data going along with a picture of a vertical stack of 12 windows.

The student appears to be adding the costs per window. At the end, that number could be multiplied by 38, for the number of windows on each floor. This particular paper does not get that far. Numbers are scratched out that are incorrect totals. In between steps are missing, so I'm not sure at what stage things went awry. There is at least one error between floors 6 and 7, where the student mistakenly adds $5.50 when it should be an additional $5. Also, the data seems to peter out at the 9th floor.

Had this be executed correctly the student would have calculated $57 for the 12-story stack of windows. An added benefit of this approach would be that the table in this format could be a valuable basis for generalizing to the n-story case.

The basic idea of the situation is simple enough that, I was happy to note, my ENL students understood my broken Spanish well enough to make a credible attempt at solving.

But I was dismayed that none of the students got very far at working the general problem or n stories. A few papers show the beginning of an attempt to grapple with that. One student stopped after the 12-story section saying it's too much work, and there has to be a better way. But he didn't want to make an attempt to find the better way. Almost all students noticed the pattern of additional 50 cents per window per floor. Most students also noticed there was a $19 difference in total cost per floor. But none of them realized that $19 was a second difference, indicating a 2nd degree equation (quadratic). I think this is because, and I tried to guide them with questions about, their approach dealing with each floor as a separate unit. They solved by adding individual floors. None of them saw that the function was talking about an entire building, so the very detailed tables of values that many students worked out had only two values in the "cost per building" column, and that wasn't enough to see a pattern. That could also be why they were not able to see the second difference, even though we have been working on successive differences for the past few weeks in class.

The students also know Gauss's formula for the sum of the first n integers, which can form the basis of another approach to the general problem. But there weren't able to perform (or didn't think to attempt) that algebraic manipulations to get to the point of seeing that situation embedded in the problem.

All in all, I'm happy with the tenacity most of my students showed in grappling with this problem.