Pythagoras Machines

Long a fan of the NHK show Pythagoras Switch, I was intrigued when my brother-in-law suggested we attend the MIT Museum FAT (Friday After Thanksgiving) Chain Reaction event, in which teams of people build elaborate contraptions which each, in a chain, set off the next team's contraption by the agreed interface of pulling a string.

The above is shows the set up of the chain of teams. I was not positioned well to capture video of the actual chain going off, but video of a previous (perhaps 2014 or so) event is linked here. The event is organized by Dr. Arthur Ganson.

We attended, had fun, and afterwards toured the MIT Museum, where Dr. Ganson has a number of his designs on display.

This one assembles and breaks apart a "chair."

This one has a baby moving with seemingly random motion.

This one had a mechanized fly circling a light bulb.

This was a very tall machine which didn't seem to do anything except move in an intriguing way.

This one had a baby's gaze following a moving object.

All very neat. I'm glad to have attended and learned about Dr. Ganson's work. Now I want to try to build my own.

Tangential

People have told me that it's often better to answer student tangential questions than to continue the planned lesson, because their questions are what interest them, so they're more likely to remain engaged if done correctly.

The lesson today was based on something from the CPM Core Connections Algebra 2 text, related to domain and range of functions. The lesson involves exploration of a few functions, realizing how to adjust the graphing window in a handheld calculator, locating points of interest (such as intercepts and local extrema), and noting the relationship between the window settings and domain and range.

The lead-in referred back to a few functions from previous lessons. One involved the negation of a squared expression, and another involved the square root of an expression. Some students noted that the negation of a squared expression would always result in a non-positive value. Other students asked why. Boom! Tangent. Let's go.

One thing I still struggle with is answering questions too readily, rather than guiding students to answering their own questions. Fortunately in this case, a student took over before I could ruin things. He explained to his mates that the square of an expression is always positive. (I did have to butt in and get them to allow a 0, amending the statement to be never negative.) The student then continued that line of logic to say that the negation of non-negative value must be a non-positive. Hooray!

I started moving things back to my plans when another student questioned the square root function. Why can't we ever find the square root of a negative. I think I was warmed up by the first tangent, so I was able to allow students, who felt they understood, to explain. But their explanations fell short (in my estimation) amounting to either hand-waving, or false statements. So I began guiding as follows.

What is the definition of square root?
(silence)
I wrote on the board a=sqrt(b) (except I used the radical symbol, which I don't know how to product here in a blog).
Students struggled with this, and eventually got to saying that it means that a*a=b. So I continued to first line to include the implication.
Then I moved to concrete. Underneath the first line I wrote 2=sqrt(4) --> and asked what that implied.
Students complied by saying 2*2=4. Exactly the setup I wanted.
Underneath I wrote a=sqrt(-1) --> a*a=-1, and asked what value of a could make that a true statement.
Students struggled. Some offered some values for a. We tested, and didn't get -1 after multiplication, or I pointed out that they were giving me two different values for a at the same time. But shortly, I saw the realization and acceptance -- there isn't any (real) number that could make it true.

I think this class really understands these two ideas now, rather than simply memorizing them as facts.

Skyscraper Windows

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.

Building Better Lessons

Last fall (2017) I was invited to participate in a research project that was to involve video-recording a class I teach. (side note: is it still called "videotape" even though there is no tape involved? what's the current word?) I was to choose a Formative Assessment Lesson (FAL) from the TRU library at mathshell. The purpose was to help build a library of lesson videos that teacher training programs could use.

On the day of recording, I was surprised that I wasn't particularly nervous. I had participated in the Measures of Effective Teaching project years ago, and I suppose I was used to the idea of being recorded. So the day of my lesson came and went, and aside for some nice anchor charts that we generated, I didn't think much about it.

Come the new semester, and I joined a Professional Learning Team (PLT) at Math for America titled Building Better Lessons in Mathematics. In this PLT we were going to watch videos of teachers using the TRU FAL, and see what we could learn from them. We had two guiding questions in mind: 1) What are the different ways that students appear to understand and misunderstand specific mathematical ideas? and 2) Imagine we could go back in time to this part of the lesson and put ourselves in the teacher's shoes. Are there questions we might ask or moves we might make to help students think more deeply about these ideas?

This past week, we watched portions of the recording of my own lesson. I remembered some of the dialog, but only vaguely. Fortunately, one of the researchers provided a transcript, which spurred my recollection. Following is the transcript of a portion of my class in which I interact with a grouping of three students. The part of the task they are grappling with card T3 asking them to transform (x,y) to (-x,-y).

S1: He said you have to translate the x-axis to the negative x-axis, right?
S2: Oh, I see, we're reflecting this.
S1: No, that's a translation.
S2: Let me see [inaudible] negative, for this point, this point is negative two ... negative two, three. If you use this method then ...
S1: You have to [inaudible] the y axis.
S2: He [the teacher, me] messed up. He messed up something.
S1: You have to move this figure, to her, you know what I mean?
S2: Yeah, I know what you mean. He mister, I think you made a mistake.
T: Me? Show me. Teach me.
S2: So, for the example her you have x and y, however this one already starts out a negative.
T: Yes, isn't that x value negative two?
S2: Yeah.
T: and isn't the y value three?
S2: Yeah, so wouldn't this be the opposites?
T: What's the negative of negative two?
S1: It's negative four.
S2: Negative? ... Negative two?
S1: Negative negative two, that's going to be negative four.
T: The negative of negative two.  What if I have the negative of negative three? What's that going to be? [writing on paper -(-3)]
S2: Three?
T: Negative negative four? [writing on paper -(-4)]
S2: Four.
T: What's negative of negative six thousand five hundred thirty one? [writing on paper -(-6531)]
S1: Six thousand five hundred thirty one?
S2: Oh. So ...
T: So this [pointing at transformation rule] I don't know what the value is, but whatever it is, I take the negative of it.
S2: So I'm just adding?
T: Same thing with the y, I'm just taking the negative of whatever value it is.
S2: So, flip this all the way over here [pointing from quadrant 2 to quadrant 4] because negative negative makes a positive.
T: Yes. What's really going on, when I'm doing this, I'm moving these points, but I'm moving EVERY point on the plane.
S2: Yeah.
T: I'm just using this [pointing to the polygon in the preimage] ... to keep .. so my eye can focus.

The first thing that jumped to my mind is, I completely missed the error that S1 was making at first. As a result, I was answering a different question. I think now that he was seeing an x-value of -2, and he interpretted x to -x meaning that he takes the x-value (negative 2) and subtracts the absolute value (2) from it, to get negative 4. That is also why he was thinking it was a translation.

Fortunately S1 was in sync with my questioning, which was intent on clarifying the meaning of the notation -x. I think by emphasizing the language "negative of negative" he saw through the orthography of how we write the operation algebraically.

I was happy with my initial reaction to the question "Hey, mister, I think you made a mistake." I try to encourage my students not to be afraid of errors. As Jo Boaler might say, mistakes are when our brain grows. One of the other teachers in my study group also made note of that line, saying he wants to use it in his classroom.

I was not so happy with how quickly to responded during our interaction. I even feel I may have cut off the students. I think I may have already formed my idea of how I was going to explain things, and as a result I wasn't listening fully to our interaction.

Ultimately the students got to describing the move as a flip, which is appropriate for a reflection (the particular transformation was a point-reflection through the origin.

Despite feeling some discomfort in being recorded (who among us is 100% confident of their teaching practice?), I am convinced that the benefit of being able to critique myself is valuable. In the process, if I can provide some materials that help other teacher reflect on their own practice, all the better.