Symbolic Hangings

In the early 2000s I worked for a software company in Stamford CT.  At the time I was in Tech Support, and all of us were in a sort of cubicle farm, each with our own work area, separated by partitions about five or so feet high. People would personalize their work areas, bringing in photos, or plants, or souvenirs of vacations. One coworker, Kim, brought in a few beanie babies.

One day I came in to work, and walking towards my cubicle noticed that someone had taken one of Kim's beanie babies and hanged it by the neck from one of the air conditioning vents. At the time I didn't understand why it upset and angered me so, but I got a chair to stand on, cut it down, put the toy back on Kim's desk, and continued on to my desk. Over the next few days it happened again and again several times. I knew it was only a toy, but the imagery really bothered me. And each time, I "rescued" the beanie baby.

After about a week, another coworker, John, approached me and asked if I'm the one who kept cutting down the hanging victim. Yes, it was me. He became upset with me. How DARE I disturb his hanging scene (that he created with someone else's property)? A few other workers took his side.

If this happened today, I'm pretty sure John would have been fired, or at the very least, disciplined. But at the time, I was made to feel insecure, as though there is something wrong with me for being disturbed by hanging imagery.

Recent events brought this to mind.

Uses for Garden Bamboo

As people who grow bamboo know, it is a prolific plant. Years ago, I planted some Bissett bamboo (Phyllostachys bissetii) and what I believe is yellow-groove bamboo (Phyllostachys aureosulcata). The plants took very well to their locations, and now each spring, roughly on Mothers Day, I'm greeted by many many shoots. This gives me a supply of bamboo "wood" much greater than I can ever use myself, and so, for practical purposes, is infinite.

Phyllostachys bissetii

This year I decided to try using some of the culms as stiffeners in a fencing I was creating for a raised bed vegetable container.

I chose relatively narrow culms, cut them to size, and cleared them of leaf branches. I wove the culms into the plastic fence material.

The fencing I have is too wide to keep chipmunks out, but should be fine to deter woodchucks and deer. I suppose the woodchucks may choose to simply gnaw through the mesh, but at least I might slow them down a bit.

Snarky email that will never be sent

Incoming message:
My [offspring] is interested in taking your summer Calculus I course at [public institution] to transfer the credit to their home [private institution]. The transfer credit evaluation process is apparently quite strict.
Perhaps you could coordinate the syllabus for your course to the one from [private institution] so there won't be such problems in the future.
Regards,
[parent email address indicating a job at the Federal Reserve Bank]

Outgoing imaginary message:
Thank you for your suggestion. I will pass it on to the department chair.
On a side note, I see that you work at the Fed. I have a friend who is very outspoken with views on monetary policy. May I give him your email address so he might share his ideas with you?
Sincerely,

My Twitter Feed

It has seemed to me, more and more of late, that people with nothing to say are saying it on Twitter. There has always been some fluff on Twitter, but it seemed, on balance, less than on another social media platform that I had an account on. And I liked Twitter as a way to take notes, so I was okay with it. But the proportion of fluff has increased to the point that it is less useful, either as a note-taking or note-sharing or news-updating platform.

But perhaps I'm just getting more curmudgeonly as I age. So I will analyze my feed, as of right now.

educator/activist who has become something of a cult-figure, and talks more and more like a cult-leader.
retired news anchor (who remains relevant)
NYTimes opinion on tv shows
math educator/activist who usually has good political commentary or good mathematical observations
538 political commentary
politician
journalist
activist
mathematician/activist again
someone kvelling about her offspring
math educator repost of someone's white-privilege
promoted ad for a bank that I will never use
journalist
math educator prompt about male-privilege
journalist
repost of opportunistic union official
math educator math pun
mathematician
math educator responding to a silly poll
promoted ad for a premium cable channel
educator/activist repost of a snippy quip

Maybe not so bad. Five or so things that I actually thought were interesting enough that I clicked through.

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.