A Portrait of a Math Teacher as an Aging Man

 

The problem above came across my twitter feed this morning courtesy of John Joy (@johnjoy1966) along with the suggestion that this was a problem from a trig unit. John also questioned who this problem would be appropriate for. I told him I would feed it to my wizards in AP Calculus BC. I also had a class coming in right after John posted it so I did not see any of the conversation – this way I could present it to my class with no prejudice about what to say. When they came in the next period I had the problem from the tweet up on the screen with no other support. I simply said that this seemed like an interesting problem and I had not had time to try it myself. I handed out the whiteboards to each desk group – this was their suggestion! – and I got out of the way. I heard them talk about the function being odd so that they knew f (-3) right away. One group found f (6) by imagining it as f (3 + 3). This meant, of course that they also knew f (-6). Progress, right? But nothing about knowing the values of f (1), f (2), or f(3). I asked if they wanted to hear a hint and three students quickly waived off that notion. After another couple of minutes I went to the board and started writing what we seemed to know about the function.   I wrote f (3) = f(1 + 2) and wrote out what the definition of the function suggested. We got an ugly expression for comparing f(1), f(2), and f(3) to each other but it was not promising. I was itching to give them a hint but they were holding me off. One of my students – thinking out loud – wondered if this might be a periodic function based on the values we knew on the board. Another group suggested that it might be a sine function. I hopped at this – another example of how I need to work on developing a poker face of some sort. The group backed up a bit and suggested that they were kind of joking, but I buoyed them up by reminding them of the periodicity suggestion. I finally gave them a vague clue – one too vague to have helped them at all. One of my students during a class warm up a few days before had the back of his book open to a series of formulas and review facts from their study of trig. I reminded the class that I complemented him on that and I pointed out the similarity of the given function to a trig identity involving the tangent function. The kids were a bit flustered claiming that no one remembers these formulas but they sealed the deal right away once they had this fact in hand.

So, what did I learn from them today?

1. I need to work on my poker face.
2. I need to stop giving clues, they are too good to need them.
3. This group of students is super persistent and creative.
4. The small desk groupings AND the randomization every Monday seems to be working.
5. The whiteboards give them space to ‘think out loud’ and effectively share ideas.

 

Man, a terrific day in Calculus thanks to my wizards and my virtual friends who prod my brain with their great problems.

In a couple of weeks I will be presenting a session at the ECET2NJPA conference. I’m pretty thrilled about this opportunity to network with other educators and to sing my love song to the MTBoS community. The presentation I pitched is called Escaping the Tyranny of the Textbook. I presented a version of this previously at a summer conference for the Pennsylvania Council of Teachers of Mathematics. The focus of my presentation is to help build a network of people to support each other in the creation (and curation) of meaningful classroom experiences without having to rely on textbook publishers to always be the resource of activities, worksheets, practice problem sets, etc. Those of us who are engaged in the MTBoS know that there are an ocean of resources available to us – blogs, twitter feeds, the MTBoS Search Engine , various virtual file cabinets, etc. I incorporated a bunch of these links into my presentation to PCTM because I had a pretty targeted audience. I expect a wider range of classroom interests in my upcoming presentation, so I want to broaden the scope of the links and resources I highlight. This is where you, my dear readers, come in. Please share with me here in the comments or over on twitter (remember – there I am @mrdardy) any rich resources for social studies teachers, or English teachers, or World Language teachers, or elementary ed folks, etc etc etc

 

Thanks in advance for any wisdom!

One of my pet peeves is when students use acronyms to mask knowledge that they already have, thinking that this somehow makes their life easier. I cannot count the number of times I have heard Precalculus students chanting about ‘All Students Take Calculus’ as a way to remember what trig functions are positive in different quadrants. This, despite the fact that they have known for years that above the x-axis is positive for y and to the right of the y-axis is positive for x. Clearly (to me,at least) this does not add to their knowledge, it just adds a filter that obscures relationships in their knowledge. So, I tell them a story to express my disdain for these kind of meaningless mnemonic devices.

Years ago I had a hilarious student named Jeanine in one of my classes. One day, when walking to lunch, Jeanine calls out to me and says ‘Hey, Mr. Dardy I have a mnemonic to remember how to spell my name!’ She tells me that she just needs to remember

Just

Eating

Aardvark

Noses

In

New

England

I love this story for a number of reasons. More than 20 years after this happened this still makes me giggle and makes me think of a wonderful former student. It is a true story and my students instantly see how silly it is to have this sort of memory device. I do not know if this has resulted in enough of them abandoning mnemonics but it is a fun story to tell.

This year, I told the story to a class with a student named Genevieve. The class took on the challenge of trying to develop a mnemonic for her name. Here is their result:

Great

Eaters

Never

Eat

Vegetables

In

England

Very

Energetically.

 

This makes me all kinds of happy.

A brief post here as our labor day winds down. As with most boarding schools, we actually have classes on labor day.

Yesterday I wrote about the clever solution that one of my Calculus students presented and, as I guessed, he did make a clear decision about when a power of x can be negative versus when powers of x cannot be. I had him present his solution to the class to start things off today. In my Calculus and my Geometry classes I am using flippity.net to generate random groups at the beginning of each week. I am not brave enough yet to randomize groups every day, I feel like it is important to me to have some comfort (even if it is just a few days at a time) within my small groups. I have also (finally) bought some whiteboards and have one at each table group. It has worked fantastically well in Calculus. The kids have been talking vigorously, they have been enthusiastic about sharing their work out to the whole class. One of my goals coming in to the year was to increase student voice – especially in whole class conversations – and so far I have accomplished that in Calculus. In Geometry I am also shuffling groups at the beginning of each week. They have been better at talking in their groups rather than projecting out. This is not surprising to me. They are younger students and generally not quite as confident as the Calc BC kiddos. However, last Friday I was pretty insistent about having students stand and say what was on their mind and to say what their questions were. I made a big show of sitting down and asking students to stand so that everyone could focus their attention. We had multiple solutions to a problem offered and a great question from one of the students about a particular solution. I think that as long as I can be consistent, and insistent, about stepping aside and having students take the lead in the conversations then I think I can make some progress with these students and help set the table down the line for our department in having a student body that sees participation as a central part of their job.

My other two classes are each super small right now at 5 students each. They will each grow a bit but the whole idea of random grouping does not work with groups this small. We all sit together at a conference table. These are the sections of our senior-level elective called Discrete Math. We are having some good conversations about voting and ballot strategies. I was delighted to have one of my students tell me how excited she is that a number of her classes are all touching on the same ideas. She is in an AP Government class and I love the idea that students see that ideas can work across course departments.