So, we were reviewing this morning in Geometry getting ready to finish up our circle unit. I was reminding the kiddos of the angle and arc relationships we have been discussing. I had been writing things like where x is an angle formed by the intersection of two chords and a and b are the measures of the intercepted arcs. However, today as I drew my diagram for it one of my students suggested I mark is as seen below.
I think I am delighted by this and will write it this way from now on. It feels to me that it is a more natural way to think about this relationship instead of having a coefficient of 2 or a coefficient of 1/2 that might not seem at all intuitive. I then drew the following
I’d love to hear from some other teachers about whether this seems at all like an improvement over the more standard way of writing these equations.
That’s all for now, just needed to get that off of my chest!
So, there have been many scattered thoughts on my mind in the past week but there are also three things that happened that are just completely awesome.
- My pal John Golden (@mathhombre on twitter) steered a number of his teacher training students over to a post on my blog and twelve of them chimed in with comments. Totally cool! One of them decided to follow my blog and I took the time to respond to each of them. LOVE the idea of new teachers in training dipping their toes into this rich world of teachers blogging and sharing. I am also flattered that John thought my virtual home here was worth a visit.
- I woke up Wednesday morning with a message on twitter from a teacher in Louisiana who asked if he could use my Geometry book at his school next year. I am so excited by the idea that this work might be used at another school.
- In my Geometry class this week we are talking about angle and arc relationships. One of my students stayed after class one day this week and she had this to say. “You know, I was thinking, when will this be important? I mean, when will I need to find an arc length like that? Then I realized that the work we are doing to find that length is what is important. Pretty cool.” Wow.
On the heels of learning some right triangle trig I am really trying to develop more proportional logic with my students. Just this week we had a really productive conversation about the following problem.
Being a bit of a bull in a china shop sometimes, I proposed that we should find the height of each triangle, find each chord length and find the height of the trapezoid by finding the difference of the heights. Not elegant, I know. I was trying to make sure that we remembered some right triangle trig. that we remembered our area formula for a trapezoids, and that we try to develop some patience in solving multi-step problems. That was my plan, but as with many school plans, it did not quite unfold that way. One of my students who is a bright and quick problem-solver pointed out that simply finding each triangle area would be enough. I understand that his solution is pretty much the same as mine, but it certainly sounds more efficient. But as soon as I acknowledged that his idea was more efficient than mine another student trumped each of us. She pointed out that I had already asked them to consider the ratio of areas between the two triangles. So, if we know one area, we can automatically know the other area. If we know both areas, we find their difference as suggested by the first student who chimed in. I was so happy that she took my clue from within the problem and that she was clever enough to really save time and energy this way. I made sure to compliment her in class and I bragged about her work to two of my colleagues yesterday. Oddly, this morning when we were reviewing before today’s quiz I reminded her – and the rest of the class – of her clever idea. She had no memory of this conversation. Sigh…
I’m trying to process this and figure out what it might mean for my classroom practice. I understand that I should be more excited by my students’ ideas than they often are. I understand that I will remember context of conversations more easily than they will because I am not dealing with the cognitive load of trying to learn/understand the conversation. I am simply coming at it from such a different place. What I don’t understand is how a student can be so in command of an idea but then not remember the creative process that made her arrive at this clever conclusion. I discussed this in the faculty lounge right after Geometry today and one of her other teachers intimated that this might simply be modesty on her part. I am not sure how much faith I put in that reading.
So, while I am a bit frustrated and confused, I am choosing to focus instead on the positive energy of yesterday’s conversation, on the clever ideas that my students brought to the table, and on the fact that my students did a really nice job on their quiz today.
In my last post I talked about how my students are benefiting from my pals in the MTBoS. Well, here I am to testify again. We are just about to wrap up our study of the Chi-Square distribution in our AP Statistics class. I used to start this unit with a little document I created based on an article in Malcolm Gladwell’s Outliers. In his book he posits the idea that there is a disparity in birth date distribution for players on a junior national hockey team. In the document I linked to I put the roster information into an EXCEL spreadsheet and I just displayed the data and asked my students to notice things. I felt like I needed to stop using this because in each of the past two years I had a number of students who read the book in an English elective and they gave away the surprise before enough conversation happened. So, I put out a twitter call for help and Bob Lochel (@bobloch) chimed in and directed me to a super helpful post over on his blog. I had seen the cool applet for playing Rock, Paper, Scissors over at the New York Times. So, I borrowed heavily from Bob (and made sure to credit him during our class discussions) and off we went to the computer lab. I prepared a handout to help organize my students and I set them loose. I asked (as you can see on the handout doc) my students to play 24 time in four different contexts. Play with random moves generated by a random integer generator or play with your gut instincts and try each against the two modes of the machine on the Times’ website. The NYT claims that the ‘robot’ plays either as a novice with no pre-programmed knowledge of how the game is played or as an expert with data gathered from other players. The novice learns your patterns as it plays you while the expert calls on a large data set of how people behave. 24 repetitions is probably not enough for the novice computer but I had some time constraints that I was trying to work around. After both of my classes played, I created a document with the data on all of the results. The next day I displayed the data and we had a pretty great conversation about the results. An important note – some of my AP Stats kiddos cannot count because the data did not come in in multiples of 24. Sigh
So I tried to start the conversation with a simple question – Should you do better when you think about the game or should you do better by random number generation? This lead to a quick decision that the expected value of a random number generator would be an equal distribution of 8 wins, 8 ties, and 8 losses for each set of 24. Now the table is set for the important principles of the Chi-Square test. Let’s talk about the difference between observed results and expected results. We also had a great conversation about how it appeared that the random number generator actually outperformed many people – especially in the expert mode. We talked about the fact that the expert mode was trying to predict behavior and how the randomness involved here might actually play in our favor.
In the week plus since this experiment I have been able to refer back a number of times to this experiment and it feels like my students have a pretty good handle on their task here. We have our unit test tomorrow so I hope that my optimism will be supported by some data.
In addition to thanking Bob Lochel I also want to thank a new twitter pal, Jennifer Micahelis (@MichaelisMath) who engaged me in a conversation about this experience and prompted me to gather my thoughts and write about it. I definitely will revisit this experiment the next time I teach this unit.