Trying to find math inside everything else

Archive for April, 2015

Whiteboard Desks

After TMC14, I heard a lot of talk about Vertical Non-Permanent Surfaces (though I didn’t go to that session myself). After reading Alex Overwijk’s post about it, I wanted to use the idea in my classroom, but getting vertical boards up seemed challenging considering how long everything takes in school. The researched showed the horizontal non-permanent surfaces were the second best thing, so I decided to take matters into my own hands. I went to the hardware store, bought some whiteboard paint, and got to work.

I put the paper down to prevent drips but, of course, dripped anyway.

Overall the desks have been amazing. The students love getting the markers and working with them, especially because they hate committing things to paper when they might be wrong. (Yes, kids do doodle/play tic tac toe/etc on the desks, too, but I think that’s no different that what they’d do on paper.) Another benefit is how easy it is for me to interact with the students when I go around. Instead of having to write something on the student’s paper or notebook, which always felt intrusive, I can jot something quickly down on the table itself, leaving it to the student to work it into their own thinking. It’s worked great for tutoring (so I don’t have to get up and go to the board). The paint might not last as long as it could when the room is used by other classes who don’t know what’s up (our night school in particular, I’d say), but it’ll last the year, at least, and I’m more than happy to repaint them before next year.

The only picture of student work on desks I had that wasn’t blurry as hell. I need to work on my photography.


A Better Day

Today went so much better than yesterday it’s hard to believe. I think the real problem was that I forgot to give them new seats yesterday, as I had intended, because my bag of numbers to randomly assign them had gone missing. Today, not only did they have new seat, but they also had a real desire to figure out what they hell went wrong yesterday, because I had asked them all to do as much as the assignment as possible for homework and pretty much no one in the class figured it out correctly. But because of that, we got to experience the benefit of using a table and all the kinds of information you could find in one, we got to transform functions and explain what they mean in a situation, and I got to actually work with several groups of students without going nuts. So maybe the assignment wasn’t actually that bad, and yesterday was just off. We’ll see how tomorrow goes.

Post Vacation Blues

The day after Spring Break is rough. Not only am I jet-lagged, but the students haven’t seen each other in over a week and thus are non-stop talking. That probably could have been fine, however, if the lesson I designed was a little stronger. While the Estimation180 intro went fine, the other activity (modeling profit and demand with quadratic and linear functions) was less so. I tried to start with a notice and wonder that mostly flopped. And then, in question 2 (see below) when I asked them to write a function, the one that was easiest to come up with was not in any of the quadratic forms we had already talked about (standard, vertex, factored), so it was less clear what to do with it. I’m going to have to revisit it tomorrow and try again. I feel like today was mostly a wash.


We’d been having a little bit of a suspension problem at my school. It felt like we had a secret police – students would just disappear and no one would know where they went, only to find out upon their return that they had been suspended. This breakdown of communication was bad enough, but then we would find out the reasons some students were suspended. Playing dice together in the stairwell? Yes, sure, gambling is frowned upon, but it’s not a suspension-worthy offense. It’s not even like it was a gambling ring – it was just three friends. Being suspended for being late? How does that even make sense?

As a believer in restorative justice, after I found out about this, I tweeted the following:

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My friend Abbie disagreed, and we wound up having the following discussion, which I think was enlightening on both ends.

Abbie: I disagree. I was bullied severely in school as a child. Mental abuse is scarring and should be treated just as seriously and part of the reason it went on for years was because “they never touched her.”  Bullshit. Mental attacks are violence.

James: I’m sorry you went through that. I’m not saying that mental abuse should not be taken seriously but that suspension is not the answer. In general, Restorative Justice is a more effective approach.

Abbie: I like the idea of such a program but I have serious doubts about the feasibility in many systems. Many of my problems were a failure of administrators to recognize or react at all.

James: But I guess that’s true regardless of which reaction is appropriate.

Abbie: I recognize there are many excellent admin & teachers out there. But my experience is that there are not enough.

James: Now that’s true.

Abbie: And I’m not sure, if I were in my mother’s shoes, that I’d have trusted the school to handle a public discussion. I guess my biggest frustration is that the response is to remove the victim from the situation. Maybe restorative justice can help that? If so it would be a welcome change.

James: That’s true. Even now, Safety Transfers (as per NCLB) remove victims, not perpetrators.

Abbie: It was one of my greatest frustrations with schools.

James: It must also be situational, as well – some have to deal with a reluctance to suspend or take action to support a victim, whereas I am seeing an overzealousness for suspensions in my school. I imagine there is some sweet spot in the middle.

Abbie: Hmm. Yes, that’s tough. It’s not a cure all by any means. My impulse is to always protect the victims first, but mindless suspensions won’t accomplish that. But asking the victim to relive the experience publicly makes me very uncomfortable. Either response extends the experience for them.

James: Well, publicly really depends on the situation. Restorative justice could be as small as one other person (a mediator).

Abbie: Mmmm. All depends on the mediator! If everyone’s trained and on board I could see its positives.

James: True! It’s certainly not an easy system to accomplish. That makes me amend my earlier statement that suspensions can have their place, but definitely not for victimless crimes.

Abbie: Very fair! I can easily agree with that.

Here we have two problems on opposite ends of the scale: a suburban (mostly White) school that does not take enough disciplinary action, and an urban (mostly PoC) school that takes too much. Which of course, is the story of our country.

After that, I ranted about it at the math department meeting and had the other math teachers (one of whom is a dean) bring it up in their grade team meetings, to push back against these policies. I don’t know how effective this was; I have noticed fewer suspensions since then, but correlation is not causation. However, Chancellor Fariña is pushing for a more restorative approach as well, so I have hope for the future. (At a recent meeting, one of the Deans made a comment about how we wouldn’t be happy about the upcoming changes – speak for yourself, dude!)

Hiking and Slope

Last summer, I went on vacation out west to see some National Parks (Yellowstone, Glacier, Craters of the Moon). At Craters of the Moon, all the trails had these lovely signs talking about how steep they were – since one of us hikers had a bad knee, we needed to make sure the trails weren’t too tough. What’s interesting is that they didn’t just talk about the average grade, which many hiking books do (as we learned to our chagrin in Glacier), but also the maximum and minimum. I feel like this is a good opportunity to talk about average rate of change versus instantaneous.

Later on in the trip, we had a discussion about what it means if a trail is twice as steep as another one. If I told you the the next trail is twice as steep as this one, what would you expect? What would it feel like? Then we also talked about whether we were doubling the slope or double the angle. That distinction is tricky because, for angles less than 10°, which are the most common, the difference between doubling the slope or doubling the angle (up to 20°) is less than 1% extra grade.

Anyway, there’s a lot of data here, so I pose to you: what could you do with this?

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Sorting Algorithms and Quadratic Functions

In my intro-level computer science class, we spent the last two weeks before break investigating sorting algorithms and search algorithms. However, because we were kinda burnt out of Java, I decided to do it computer-free. We used small decks of cards instead. To simulate the computer only comparing two values at a time (to limit the kids using their more powerful brains to speed up the process), students were only allowed to move cards that were face-up, and could only have two cards face-up at a time. The first day I had them come up with their own algorithms and count how many steps it took them, steps being flipping a card or moving a card.

In the subsequent days, I wrote up several common sorting algorithms as they would be applied to the cards. For each of them, we kept track of how many steps the process took, which was always the same for some (Selection) but we had to think about best and worst cases with the others.

We then considered how many steps they would take for 4 cards, 5, 6, 7, 8, then n. And so we wound up creating functions to represent the complexity of the algorithm. Many of these wound up being quadratic and linear functions. All of my students had previously taken Algebra and none had problems with the linear, but the quadratic functions sometimes caused problems. But we would work what exactly changes each step, find second differences, etc, to create the functions. And no one thought this was a weird place for quadratic functions to come up – it just seemed like a natural thing that arose when we started investigating the algorithms.

Below I attached the algorithms I wrote up for the sorts. Go get a set of 8 cards and try them out. Can you figure out the function? (Note: for the Worst Case of the Merge Sort and the Quick Sort, it’s a recursive function that doesn’t necessarily have a nice explicit form.)

Walking in San Francisco

I had this same thought the last time I was here, but I feel it could be fruitful. Sometimes there is a route someolace that is “shorter,” laterally, but there are tons of up and down hills between. So is that way really shorter? I feel like you can do something with this: use the Pythagorean theorem to determine how far you are actually walking, determine different walking speeds on different inclines, and then get a topographical map and determine the speed and length of different routes, then check what Google Maps says.

For an example of what I just experienced: because we didn’t really know where we were going, we wound up walking up the giant hill up Powell St and then down the hill on California St, but if we had gone down Sutter first and then up Grant, it would have been much flatter.

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Airport Planning Project

On my flight to San Francisco today, when the pilot mentioned that we have leveled off at our cruising altitude of 32000 feet, we had just passed Scranton, according to the interactive map. This reminded me of a right-triangle trig project I did my first few years, before it was dropped from the Algebra I curriculum.

I first had the idea by doing a Dan Meyer-style textbook problem makeover. When I was looking for trig problems in the textbook I was using, I saw one that was something like this:

A pilot is flying his plane at 5 miles up and starts his descent 300 miles from his destination. What was his angle of descent?

If I were a pilot, what would I need to figure out? Most likely, I would know I need to land at a certain angle – what I would need to determine is when I should start landing my plane. So I turned the problem around. Then I thought, considering what angles we need to climb at, angles we need to land at, and how high up we need to fly, what’s the minimum distance I could get between two airports that have a connecting fight? (Assuming direct paths.) And so and made the following project:

My students had a lot of fun with this project (even if I did get countries named things like Ratchetopia). Things would get tricky sometimes with scale (I think I had them use something like 1 inch = 50 miles), but overall the process went well. However, sometimes it could be paralyzing, having so many choices of where to put the airports.

Sadly, I don’t have any pictures of the projects my students made. (Maybe on my old phone?)

Binge Watching and Block Schedules

About two weeks ago I watched the first three episodes of Unbreakable Kimmy Schmidt with someone, and the very next day I watched the whole rest of the season. Yesterday and today I blew through 14 episodes of House of Cards (though I think I’ll end there). This kind of consumption has always appealed to me; I’ve often told friends that I don’t want to watch some show or read some book until it’s completed, so that I can intake it all as fast as possible. When I read a book, I often stay up late into the night because I can’t put it down. This way, I’m able to just dig down deep and fulling engage myself in the material.

On the other hand, for the past few months I’ve also been listening to Welcome to Night Vale. Because I can only focus on a podcast when I’m walking and not doing something else, and because I used the other podcasts I listen to as spacers between episodes, it’s been a much more drawn out process than the binge-consumption that is typical of me.

The thing is, though the former is more typical and what I say I prefer…the latter way is much better. This was most evident for me from my time in the Harry Potter fandom. Before the 6th or 7th books came out, there are a ton of activity, most of it powered by feverish speculation and thinking about what the future could hold. After the last book came out, though, a lot of it died down. And I realize that think about what the future could hold is a lot more interesting than dissecting what happened in the past (for me).

When I first started teaching, my school, like most schools, had its classes meet every day of the week. The second semester, we tried something where we met four days a week, but one of those days was a double period. Then my second year, we switched to an alternating day block schedule, which persisted for the subsequent 2.5 years.

The introduction of the block schedule felt like such a relief to me. Finally, we could dig deep into the material: I didn’t have to cut a lesson short because of time, there’d be less wasted time on coming into class, clean-up, warm-ups, etc. And I really believed this, until one time I talked with Elizabeth Statmore about how there are time periods during a lesson that are key for learning, but during a double period we don’t actually get double the amount of those time periods. She also talked about how, when they have a class every day, the repeated reference to the content reinforced ideas better than when classes were more frequent.

When my then-new principal wanted to switch back to a daily schedule, I resisted, but mostly because she wanted to switch mid-year instead of waiting for the next year, and as the programmer I would have to figure out how. But by then I was on board. I was okay with letting a lesson end in a cliffhanger, and drawing something out over multiple days. Because I knew it was better this way. And spiraling things together, leaving little bits in each lesson and bringing it all together into a climax – well, that’s the Welcome to Night Vale way.

Those things I’ve binged on, I consumed them and then moved on. It made me happy at the time, but they didn’t really stick with me. But the things that were persistent, that I drew out over time, those were a lot more sticky. And that’s as true for learning as it is for reading and watching TV.

Meet Me Halfway?

Today I met a friend for lunch, and we decided to meet halfway between our apartments. To think about where that would be, I used MeetWays, which was useful because it can find that mid-point based on a variety of modes of transportation: car, bike, public transit, and walking.

The thing is, my friend took the train to lunch and I walked. (I based the location on the transit midpoint, but decided to walk for the exercise.) So what counts at the midpoint then? If one person drives and another doesn’t, do you include the time spent finding parking as part of your calculation? Is it based on time, or distance? If we both travel for 45 minutes, but she’s much farther from home, is that still meeting in the middle? Considering everywhere I could walk within 45 minutes and everywhere she could train in that time, is there really only one middle point, or do those rings intersect twice?

There are the things I wondered as I walked. Are there other good questions to ask? You can probably, at the least, get a decent rates question out of it.