Concepts and Problems

In my classes this year, I’ve been really concertedly trying to emphasize that students need to really understand concepts and explain ideas in written form clearly. Today I’m faced with a conundrum about how students are connecting concepts with the problems we’re doing.

On my Algebra II quiz, I asked:

Explain — using complete sentences and proper mathematical terminology — why \sqrt{-16} doesn’t have a meaning [in real numbers], while \sqrt[3]{-8} does.

I was really, really, really pleased with my class’ answers. In the course of their explanations, almost students mentioned that \sqrt[3]{-8}=-2. Literally on the same page, however, was a set of radicals that I asked students to simplify. One of them was, gasp!, \sqrt[3]{-8}. It was an oversight on my part and I will probably change if I use parts of this quiz next year. Can you see where I’m going with this?

There were a few students would could do the conceptual work — who even showed that \sqrt[3]{-8} was -2 in their written explanation — who didn’t get the exact same question right below it correct.

Color me flabbergasted. (What is that, a pukey yellow?) It’s just so hard to figure out what was going through their heads.

Mathclub Hat Problem

One of the students in Math Club recently put his own twist on the age old hat question: Assume you have n people, each of whom has a red or green hat put on them. They each don’t know what color hat they have on. However they can look around and see everyone else’s hat.

After getting to spend some time in a room looking at everyone else and their hats (they may not communicate in any way), they are each placed in separate cells and asked to say whether they have a red hat on, a green hat on, or “pass.” Everyone wins the game if at least one person says their right hat color, and no person messes up their hat color. Everyone loses the game if everyone passes, or if anyone says the wrong hat color.

The question is: what is the strategy that those wearing the hats should come up with beforehand? And can you come up with a formula giving the probability that n people win with that strategy?

To make the problem clear, let’s examine the three person case. The possible combinations of hats are:

RRR | RRG | RGR | GRR | GGR | GRG | RGG | GGG

The best strategy we could come up with is to say: if you see two opposite colors (a red and a green), say “pass”. If you see two hat of the same color, say you’re wearing the opposite color.

So you’ll lose with RRR and GGG (everyone sees two of the same color, so everyone will say the opposite color).

But you’ll end up winning with RRG, RGR, GRR, GGR, GRG, and RGG. Let’s look at RRG to explain. The person wearing the first red hat sees a red and green hat. So that person says “pass.” The person wearing the second red hat sees a red and a green hat. So that person says “pass.” The third person wearing the green hat sees a red and a red hat, so that person says “green” and is right! So RRG is a winning combination. Similar arguments follow for the other five.

Since there are 8 possible combinations of hats, and 6 of them have a winning strategy, there are 6/8 chances that everyone will come out a winner! (That’s a whopping 75%!)

So we’ve been investigating what the strategy will be for n people wearing red and green hats. So far, we’ve done pretty well. In fact, we’ve even gotten Pascal’s Triangle involved, which is always great.

And there seems to be a consensus among the students (though no proof yet) that if you have any even number of people playing the game, say 8, you can actually get better odds of winning if you ask another person to join in (so you’d have, say, 9 people playing). That seems totally counter-intuitive, that adding an extra person to play the game with you would lead to a better chance of winning. So if they’re right, I’ll chalk this problem up to a win.

PS. We did talk about the Bloxorz problem for two weeks, but students grew bored and tired of it. I still think it’s a great problem. Maybe one year a student will want to do an independent study on it, and ask me to be the adviser to the project.

I survived Parent Night

Even though I was sick — aching and tired — I survived our Parent’s Night last week. I think it was pretty successful, even though I was foiled a few times by parents who tricked me into talking about their children. (I keep a general policy not to talk about individual kids at these events; it’s a time to share what we do in the classroom, introduce myself to parents, and to tell parents what their kids can do to be successful — and how they can help their kids be successful).  I’m still baffled on how they tricked me. I totally blame my weak immune system for my inability to steer conversation away from talking about little Jane or little Jake.

The night had one tragedy — when SmartBoard didn’t work for one of my classes. I knew this would happen; the same thing happened last year. I even told everyone I knew it would happen again. However, luckily, it happened when talking to the parents of my four student multivariable class. The parents all knew each other — these kids had been in the same classes for gosh knows how long — and so we just gathered ’round my laptop and I showed them what sorts of things go on in our class.

Some observations:

(1) Parents tend to start off the night stoic. Their faces won’t let anything through. Cracking jokes or smiling doesn’t phase them. As the night progresses, however, the parents get more laid back, and by our last class, parents have let their guard down. I swear I heard a few of them laugh in my last presentation. I’ve asked other teachers in my school if they have noticed this phenomenon, and it seems pretty universal.

(2) Parents like to introduce themselves (great). Parents like to follow that up by asking “how’s my kid doing?” (not great). First of all, as I said, I don’t like to talk about individual students. Second of all, who is your kid again? Believe me, unless you say “we’re the parents of Joe Schmo,” every time you meet me, I’m not going to know who you are.

(3) I realized I go into these nights actually expecting some gratitude from parents. And when I didn’t get it from more than a handful of parents, I felt a little slighted. Am I a bad teacher for needing those bits of affirmation? I don’t know. But I can’t help how I feel, and that’s what I felt.

(4) One point I made to almost all my parents is the basis for how I approach designing any class: I try to get them to do work which I think is just beyond the level that they think they can do. Of course, I’m not always successful with this, but I do try to push my students just past their perceived limits. Gauging their limits is tough though. I’m doing a really good job with this in Multivariable Calculus this year, but at the moment, I don’t think I’m pushing my Algebra II or Calculus classes enough.

With that, I’m going to eat an apple, and get me to bed, and hope to be ready tomorrow to embark on yet another week.

Parent Night is Almost Upon Us

We have “parent night” on Thursday and there are five things I have to remind myself to do:

  1. Plan to speak for 15 minutes, even though I only have 10 minutes with them. That way you can go “oops, I guess I’m out of time” and send everyone along.
  2. Integrate humor into the presentation.
  3. Talk about the content of the class, the expectations I put on students, the expectations I put on myself, and anticipating any parent questions and addressing them in the presentation (e.g. do I ever allow extra credit? no.)
  4. Do NOT let any parent ask me questions about their individual child. Politely say “I don’t think tonight is the best night to have conversations about individual children. However, I’d be happy to set up a conversation! Here’s my contact info.”
  5. Don’t freak out!

College Recommendations

I’m being asked to write college recommendations. I have a hard time with this, because I view it as such an important responsibility. The colleges my students are applying to are often prettycompetitive and every part of the application is important.

My strategy for dealing with this is to send students desiring a recommendation the following paragraph:

When you have all your colleges picked and the forms gathered, will you give me the forms paperclipped to stamped and addressed envelopes? It would be good to have them at least two weeks in advance of when you want them sent out. That way I can do them all in one fell swoop. Also, when I write recommendations, I usually ask students for two things: (1) things you want me to highlight in your recommendation [math or non-math related], and (2) for you to write a sample recommendation for yourself. Why? Well simply put, it is this: recommendations become strong recommendations if there are lots of specific details/specific instances/stories. And you know things I wouldn’t know — like if you formed a study group or something. Don’t feel like you need to be humble. Just write it honestly and with confidence.

I’ve talked to some teachers who have a form they give to students, questions students need to answer about their experience in the class(es) they had with the teacher.

Do any of you do something that makes writing these recommendations easier? Do you have any suggestions about how to write a strong and honest recommendation?

CD Club Pedagogy?

One weekend ago, I had the fourth cycle of the CD club I organize. It rocked. (You can read about a previous cycle here.) The general idea: a group of 10-15 people meet up at a local watering hole and bring a mix cd they’ve created around a theme. In fact, everyone brings 10-15 copies of their mix, and when we’re all gathered, we exchange them.

The end result: you get 10-15 cds filled with really good music.

The last four themes were:

1. No Theme
2. The Academic Colon: A CD About Some Aspect Of Education
3. Time Travel
4. Stages Of A Relationship

(You can see the tracklistings for each of my four CDs here.)

There is something really awesome about this set up: the work you put into creating and reproducing one thing comes back to you ten fold. And you get to — and want to — engage with everyone else’s work.

Is there any way to harness this model of intellectual exchange in the classroom? To reverse engineer it?

The two key points:

The object needs to be coveted by all participants (e.g. carefully crafted CDs)
The object needs to be easily reproducible (e.g. copy CDs)

Ummm. The best example just popped in my head: VALENTINES DAY CARDS IN ELEMENTARY SCHOOL!

Or a slight variation:

Can we come up with an single large entity that students individually contribute to? So students have ownership in it?

So in the mix CD example: if every person chose a song on a theme — and we made a CD — we’d have a single CD with input from all. Or if we were making a bulletin board, we could have each student bring in one picture to contribute to it.

Before signing off, I thought I’d share one idea that might be useful. Before a big assessment, I could ask students to each make a one-page set of study questions they created, along with their solutions. I could scan them in for students to use to study from. For students, by students. And for the assessment itself, Icould  chose some of the good problems from the study guides to be on it.

Other ideas? Is there a good math project out there that fits this CD club model?