Sunday, April 29, 2018

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.

Wednesday, April 25, 2018

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.

Sunday, May 28, 2017

String Art with Pencils

I have been attending a series of workshops called Mathematical Ideas in Curve Stitching and String Art facilitated by Dr. George Hart of Stonybrook University. I incorporated some of the ideas I learned in my classes on Friday.

At my school, Friday was a sort of "reward" day. Students with some definition of good attendance were taken on a trip to Six Flags amusement park. (The students remaining may have viewed it is a "punishment" day.) I was unsure what numbers I would have left, being as it would be primarily consisting of students who don't attend regularly, on top of which we weren't given a list in advance. So I planned on not continuing the sequence of lessons I had been in, since a large number were likely to be absent. My classes are called Algebra II, and students are primarily 11th and 12th graders, with some 10th graders as well. Students have all passed Algebra I, but in many cases have not yet passed the NY Algebra I Regents exam.

The ideas involved build on some of the ideas discussed by Edith Somervell in her 1906 A Rhythmic Approach to Mathematics. In my 45 minute periods, I narrowed my focus to three short explorations.

I began with a set of Cartesian axes. I provided students with square graph paper and rulers, and instructed them to draw a pair of axes, and the use the ruler in order to make lines as straight as possible. I wanted to axes roughly centered on the paper, but did not instruct so. I knew from experience that most of my students will draw things in the center of a sheet of graph paper, regardless what scale might be appropriate, but I was also counting on some students NOT putting the origin in the center, so we could have a discussion about how that affected our ability to draw the results.

I asked students to name a pair of number that adds to 10, such as 2 and 8. I instructed them to connect 2 on the x-axis to 8 on the y-axis with a straight line. I asked for another pair of numbers, and similarly connect one on the x-axis to the other on the y-axis. I repeated a few times, and the asked students to connect every such pair of numbers they could think of. I waited for students to ask, "How about 0 and 10?" I asked if 0 plus 10 equals 10? Yes. What does it look like when I draw the 0-10 line? Nothing because it's on the axis. So, the full answer is, yes we can use 0-10, but it doesn't add to our figure because it lies on an axis. I similarly waited for students to ask about negative numbers. I asked if there was something we could add to negative 2 that sums to 10. "Yes, positive 12." So we can draw it. In all classes these questions eventually came out. If they had not, I would have asked them, but wanted to wait for students to generate them if possible.

Errors. Many students in the first class connected to two axis values with big looping curves, despite the instruction to use the ruler to make the line as straight as possible, and despite my modeling this on the board up front. A few students, instead of drawing lines, marked the points whose coordinates add up to ten. I made sure to model more resolutely in subsequent periods, and circulated more busily between groups of desks.

Once there were a few students with enough lines drawn to discern a shape, I held one up, and asked the class what they though would form as they continued drawing lines. The actual answer is parabola, but no student came up with this. Some students were convinced it would form a circle. I did not correct them, but instead suggested they continue to think about it as they added lines.

But I did not allow much more time on the Cartesian graph. My main point was to work with circles of numbered points. I distributed copies of 30-dot circles, as shown. Asked what they noticed about the circles. I want to elicit how many dots there are, what the smallest number is, and what the largest number is. Before proceeding, I wanted to make sure they realize that this is one of the cases where we start counting at zero.

I began by discussing "clock arithmetic." I tried using the more mathematical term, modular arithmetic, but each class clearly preferred "clock," so I went back to that. We tried examples until I was confident they knew how to continue around the circle once they reached the highest number.

I asked students to choose a number between 1 and 30. (One student chose 15, to interesting result.) Then I instructed they connect each dot, via straight line, to the dot that was their number higher. If they chose 10, they would connect 0 to 10, 1 to 11, 2 to 12, and so on. I reconfirmed they understood modulo. Where would 19 connect? 29. How about 20? 30. But we don't have a 30, so where do we go instead. Oh.

I instructed them to continue until all numbered dots have been connected. The figure shows what it would look like if their chosen number were 10. Again, I asked how they would describe the shape they had made. But I had a few students with different numbers hold up their drawings. How are they different. A conjecture formed that "the larger the number, the smaller the circle. In some classes students had chose numbers larger than 15, so I was able to challenge the conjecture, and students struggled to explain that behavior. In the class with a student who chose 15 as her number, all lines intersected at a point in the center, and I asked students if that fit with their conjecture.

Errors. Some students had difficulty understanding the mapping idea, and kept connecting the same dot over and over again to different places on the circle. In the first period, I wrote the numbers as ordered pairs, which may have added to that confusion. From then on, I wrote the pairs with an arrow between (e.g. 1-->11, 4-->14), to emphasize the mapping. I had fewer errors with this notation, but still had to correct a few students.

Some students didn't understand that each of the 30 dots had to be the origin of a connecting line, and each had to be the terminal of a connecting line. At the end there should be two lines for each point. These students stopped early, saying they were done. I showed them how to continue, covering all their dots.

Depending on timing, I went two directions from here. If there was time, and for students who finished quickly, I suggested they choose another number, and with a different color pen they repeat the process.

In some classes, time was short so I went on to the next drawing. I distributed circles of 50 dots. This time, rather than addition, we would multiply. Connect 1 to 2, 2 to 4, 3 to 6, and, generally, dot i to dot 2i. Modular arithmetic again is an issue, so I reminded them. What is 2 times 24? 48. What is 2 times 25? 50. But we don't have a 50, so where do we go? 0. I continued for a few more to make sure most of the students understood. I wrote the examples on the board, for later refreshers.

I circulated, helping students. As the figure developed I asked students how they would describe the shape. Many of them said it looks like a heart. I agreed, and told them the math word for it -- cardioid. Most of the students nodded in recognition, saying, "Oh, like cardio?" I selected a few students and gave them a 100 dot circle, and told them to map i --> 3i.

Errors. There were very few errors. Students had gotten used to the rhythm of connecting dot to dot, and calculating. Not a few of my students still lack fluency in calculations such as "times two" but with time it was not beyond them.  I was intrigued that, I view this sort of problem a "productive struggle," and think it is beneficial to let students spend time figuring the answer. But my Assistant Principal was in the room for one of the classes, and he gave the answers to a student who was struggling. I saw that she was just completing the calculation on her notebook, correctly, when he told her the answer. Instead of allowing her to feel success, she took his answer and drew the line. I wish he hadn't done that.

In all classes, by this point, I was short on time. But I had made sure that each student was following instructions well enough that they could continue on their own. I asked if anyone wanted a blank paper to do again. Most students asked for this.

Monday, May 22, 2017

Screens from Unit Equilateral Triangle Pairs


These are the screens that were used to remind students of the altitude of a 30-60-90 triangle with hypotenuse 1. Although in words I had to call it "height" or "altitude," in the drawing I only indicated a question mark.



These are the screens in which I presented the drawing. In both classes I had the same wrong answer that simply doubled the diagonal in the first figure. This shows how I drew auxiliary lines that convinced the students that wasn't correct. In the second screen I had asked the class what they notice about the figures before I posed the problem. The notes are things the students called out.









Saturday, May 20, 2017

Unit Equilateral Triangle Pairs

I have been attending a series of workshops entitled Whole Class Conversations in Mathematics Classrooms, facilitated by Dr. Betina Zolkower of Brooklyn College. We have looked at some open-ended problems (low floor / high ceiling), examined transcripts of some class conversations, and discussed strategies.

Among the problems we looked at, one in particular intrigued me.

The figure shows three "trains" made up of paired equilateral triangles. The first figure can be thought of as a single pair, the second as two pairs, and the third as three pairs. Students can generate a number of questions on their own about these figures, but eventually lead them to the question, "What is the measure of the long diagonal in each figure."

All the ways I know to answer this question depend on knowing that the altitude of an equilateral triangle is half the length of a side times square root of 3.

Spoiler alert:  my discussion of this problem will involve discussing solutions and approaches. If you want to solve on your own, do not read further until you've satisfied yourself. And if that described you, I hope you aren't satisfied until you find a way to solve for the general case of any number of pairs.


 where s is the length of a side. For ease of calculation, we will define our unit of measurement to be the same as the length of a side of a single triangle. Hence, unit equilateral triangles, and s = 1.


I presented the drawing to three of my classes on a day when testing was otherwise going to put things out of synch. All of the classes are called "Algebra II" but they perform at very disparate levels. I was not confident that they would remember special right triangles, so I began all three classes with two reminder problems.

 Given a right triangle with sides measuring 5 units and 12 units, what is the length of the hypotenuse? This was intended to remind students of the Pythagorean theorem.






What is the height of a unit equilateral triangle? This was intended to remind students of side lengths in 30-60-90 triangles. As I spoke, I noticed that I sometimes switched in "altitude" rather than "height." I decided not to worry about it.

*** edit 21 May 2017 ***
I've decided I should NOT say altitude, as altitude relates to specific segments in a triangle. I drew the auxiliary line off to the side, as height, so as not to give extra hints. By using "altitude" I would somewhat undermine that effort.
*** end edit ***

I was surprised that two students used trigonometry to solve this one. sin(60°). I wonder if they would normally do this, or if it was influenced by EngageNY, which creates a strong association between "height" and "sine."

After these warm ups I presented the drawing of the three "trains." In discussing my classes, I mentioned that I labeled the trains, and I don't recall whether I said figure 1, 2, 3 or n=1, 2, 3. Betina pointed out, and I agree, that one should avoid the "n=" notation, as it boxes students in to one way of thinking in a way that "figure" doesn't. I currently think that no labels at all may be best, to leave it as open as possible.

Introducing the situation, I added a dotting line indicating the diagonal in the first "train." I had equilateral triangle graph paper ready, and offered it to students if they wanted. Some students, seemingly as a reflex action, started shouting out numbers. "1." "2." "1.5." My response to each was, "How do you know? Can you justify that answer?"

For the most part, they settled into group work. The room is arranged with desks in fours.

Interesting wrong answers for first "train": sqrt(6)/2. Students are unfamiliar with adding radicals.
Correct answer: sqrt(3)
*** edit 5/21/2017 ***
These students noticed that the altitudes of the two triangles in the pair lined up. So they added. I didn't see any student multiply by 2. Addition is the prefered operation. Some students, using addition, got to (2*sqrt(3))/2 but did not think to simplify the fraction.
This solution works, but does not generalize because the altitudes don't line up in subsequent trains.
*** end edit ***

Once students got the first train, I suggested finding the diagonal for the second, adding the dotted line diagonal to the second figure. Only two groups in one class and one group in another class got this far.

Interesting wrong answer for second train: 2*sqrt(3). Students assumed it's twice as long as the first diagonal. I added the two single-pair diagonals to the second figure, and asked if they think the new diagonal was the same length as those two put together. Students quickly realized that wasn't the answer, and went back to puzzling.

The period ended before anyone got the correct answer, sqrt(7). Correct answer for third train is sqrt(13). Correct answer for the nth train is sqrt(n^2+n+1).
*** edit 5/21/2017 ***
The solution I found was to draw in another "half" triangle at the end of the train. This creates an obvious right triangle with side lengths n+1/2 and sqrt(3)/2. Application of the Pythagorean theorem, with the opportunity for work with squaring a binomial involving a fraction, gives the general solution.
*** end edit ***

All in all, I was very pleased with the level of engagement most students showed. It was something they were curious about, and they felt they should have enough math knowledge to do it. I even noted that one chronic attendance problem student happened to come that day, and even he got into the problem. He never got past working on the altitude of the triangle, but that threshold was low enough that he felt comfortable diving in. Additionally, the ceiling was high enough, none of the students exhausted the potential of the problem. If they had figured the nth case, I could follow with how long is the shorter diagonal? What if I configured the trains in different ways. This is a very rich problem.

Tuesday, May 9, 2017

Don't Use Appropriate Tools Strategically

I have been using technology in my classroom since I began teaching. I don't do everything with technology, but, as appropriate, and when I believe it might enhance student understanding of topics, I use technology.

My most frequent choices are TI-Nspire Teacher Software, which has interactive geometry, and also displays multiple representations of mathematics, and TI SmartView CE, which allows me to demonstrate on a calculator similar to the TI-83+ that students have access to. I have this software installed on my laptop, and I have seldom been assigned a room with SmartView installed on the classroom computer, much less TI Nspire. But that shouldn't be a problem, because I can connect my computer to the SmartBoard projector, and I'm good to go.

Today, ten minutes before my 5th period class, I was informed that it is against school policy to connect non-school equipment to school projectors (and SmartBoards). So, all the interactive things I had intended to do are now forbidden. How about my document camera, I asked. Forbidden.

I have requested that the school install the software I need on computers in the three classrooms I teach in. I have not heard back, but doubt they will do it.

Right now, I'm stewing, trying to figure out how I can plan lessons going forward. I can do screenshots, and embed them in Smart Notebook (which is installed on the classroom computers). But I won't be able to do anything interactive, and I won't be able to respond to anything on the fly. This has me feeling extremely irritated and frustrated.

Sunday, April 23, 2017

Plan Your Lessons After the Previous Lesson

Those who know me are aware that I've been frustrated about a number of things at my current school. I haven't always been able to articulate just what I find so frustrating, other than a general feeling that I am not supported or valued as an educator. This weekend I finally understood one concrete instance of what gives rise to some of the frustration.

I believe that I should have an interactive relationship with my students. I am not simply a lecturer. Rather I interact with the students in my classes, hoping to help them gain understanding of the math ideas I'm teaching.

One tool in the interaction is formative assessment. I am always observing the feedback I get, verbal and non-verbal, and using that to help me make decisions about what comes next -- do I reinforce, move on, back up? If I am not doing this, I may as well be a recording. The ability to assess and modify on the fly is one of the things that makes me the type of teacher I am.

Friday evening, after work, I attended a workshop let by Dr. Betina Zolkower. As a side comment at one point, Betina commented how odd it is to visit classes and see a homework assignment written on the board from before the class starts. Homework, she points out, should be an organic part of the class interaction. Until you know how classroom discussion has gone, how can you know which problems to assign? Unless the homework is just busywork.

Reflecting on this minor but profound insight, I realized one thing that makes life in my school difficult for a teacher. I generally plan lessons before the class, and after the previous class. I don't know exactly how far we will be able to get on any given day (although I can predict with confidence that I cannot complete any entire EngageNY lesson in a single class period). So I can't plan what to do next day until the current day is done.

But my school has a policy -- all copy requests must be submitted before the end of each day. There is no teacher-accessible copy or duplication machine. The only way to get classroom sets of materials or worksheets of demonstrations or instructions is to submit it to the "copy center" (which is actually just a guy who has the job of spending some time each afternoon running copies. If you miss the deadline, sorry, you don't get the copies until the day after.

In reality there's another option for getting copies. That is to print them yourself, on your home machine or at Staples of Kinkos.

The administration pays a lot of lip service to "formative assessment" but in practice, through this subtle technique, they discourage us from incorporating information gleaned from such assessment into each day's lesson. We are guided into mass-planning lessons, blind to interactions with students, because that is the only way we can get copies.