It has been, as usual, a busy year. It seems that this is absolutely the norm in our professions, isn’t it? I am teaching three classes this year – AP Calculus BC, AP Statistics, and Geometry. In the next few days I want to comment on all of these classes as time allows. Unfortunately, this is the week we are working on midterm grade comments for every one of my 68 students AND my mom arrives tomorrow night for a visit as this is grandparents’ weekend here at our school. The two things I want to share today are reflections on the progress of my Geometry class and a class visit I made last week. I’ll tackle last week first.
We have a number of students who finish the Calculus curriculum before they graduate here. Those students – who have has two years of AP Calculus (we teach BC as a second-year course) move on to an Applied Differential Equations class taught by our school’s President. He is a retired Army engineer and he teaches a lovely Problem-Based curriculum using Mathematica. He ran into me in the hall last week and mentioned that his students were presenting the results of their first research problem and he invited me to join them on Friday. I was able to watch as two of my former students presented fantastic work that they had done on their own on a research topic of their choice. One was presenting a supply and demand curve. She explained her choice of parameters and shared the results. Her classmates made note of the similarities to a predator-prey problem that they had worked on together. She eloquently addressed the similarities and differences between the problems, but what most impressed me was her response after one of her first graphs. She remarked that the graph did not fit what she suspected should happen and her slide suddenly displayed the phrase “Question Results” and then she moved on to discuss her modification. I loved the directness and the simplicity of this message. Just because a fancy machine, and Mathematica is a VERY fancy machine, tells you something, it is not necessarily true. I have to incorporate such a simple response into my repertoire when looking at surprising or counterintuitive results. The second presentation was analyzing the forces involved in walking on high heels. It was a fun discussion and I was impressed by the depth that the student found in examining this situation. Unfortunately, our school President is retiring and I suspect that our new administrator might not have an engineering background. It’ll be up to me as department chair to find someone capable of carrying on this kind of high level work with our best math students.
I start each day with my Geometry class. We have had a blast so far playing with GeoGebra in the lab, discussing reflections, rotations, and vector transformations, talking about distance and Pythagoras. I have been pleased with the level of engagement I have seen from the students. They are reading the text (and finding typos!) and they are engaged in class discussions. Now we are getting ready to begin discussing proof. So I want to sloooow their brains down a bit. I borrowed (okay, maybe it is just stealing) an idea from Max Ray in his book Powerful Problem Solving. I asked my Geometry students to write directions for making a peanut butter and jelly sandwich. I wanted to show them just how hard it can be to carefully describe something. I did not have people act out the descriptions, though. I was a little worried about embarrassment. So, I dealt out cards at random to sift the kids and I had plates, paper napkins, knives (plastic ones – just to be safe!), peanut butter ( one table had sunflower butter due to allergies) , and some jelly. I bought both grape and strawberry. I shuffled the instructions and handed them out and we had a terrific conversation about the assumptions made and about being careful with details. We had some laughs talking about how peanut butter might magically appear on knives to be spread and what side of the bread gets put together. Then we came to the description made by a student I’ll simply refer to as E. She typed her description ( which can be found here ) and she did such a lovely job. Her list is detailed and she took great care to break down the actions involved. All of the class agreed that hers should have been the first one read since we decided to go ahead and eat as soon as we dissected hers. She seemed really proud when I asked her for her copy to share out.
i’ve been thinking about this all day. Remember, this is my 8 AM class. I have resisted some activities like this in the past thinking it just was not my bag. However, I am working closely with two terrific colleagues in Geometry this year and they sort of encouraged me to try. I stepped out of my comfort zone and the result – at least this time – was a relaxed and fun class. Kids were engaged, they supported E and gave her some real praise and after class one of my students who has been struggling so far came up to me to tell me how much fun she had. I think I may have earned some important personal capital in working with her and this may be a gateway to building a relationship with a student who does not seem to have much inherent love for my subject. I also think that some valuable points were made about the challenge of explaining what you know to someone who does not know it. I am optimistic that this will help build a bridge toward understanding some important principles of proof in the next week or so. So, thanks to my colleagues for encouraging me to try something that seemed a bit silly to me. Thanks to E for being a model of thoughtfulness and detail. The rest of her work this year has been uniformly outstanding, so I was not exactly surprised by this. Thanks to my students for trying an assignment that probably seemed a bit weird. Finally, thanks to Max Ray for this thoughtful book that got me going on this path.
We are working with isometries in our Geometry class right now. We are looking at vector translations of objects, rotations (primarily around the origin), and reflections over horizontal or vertical lines. I am only teaching one section of Geometry so sometimes I feel like if I don’t get it right, I’ve missed an important opportunity. There have already been a couple of instances where I explain something after school in a way that I wish I had done the first time around. I am trying to keep note and be a smarter Geometry teacher next year.
Today when we were wrestling with rotations about the origin (I am sticking mostly to 90 degree increments) there was a bit of a clamor with the students begging me for rules about how to transform a point in the general (a, b) form. I really want them to try and develop a better intuition so I have resisted the call for these shortcuts. One of my best students, a boy I’ll refer to as M, pointed out that these rotations always maintain distance (again, we are rotating about the origin) so I pointed out to my class that this really limits our choices. We were looking at rotating the point (4, 2) 90 degrees counter-clockwise. Everyone agreed that this would put us in the second quadrant and I used M’s idea to suggest that our destination was now restricted to either the point (-4, 2) or (-2, 4) since the distance was the same and the second quadrant has negative x values and positive y values. A quick sketch convinced us all that the target should be (-2, 4). I felt like we were in sync with each other and that my sketches convinced my students. A group of them came after school to work to get ready for tomorrow’s quiz and they all confessed to not being convinced. I pulled up GeoGebra and tried to show them what would happen. I referenced M’s idea about distance and one of the students reminded me that I had been emphasizing the Pythagorean Theorem more than the distance formula earlier when we talked about distance. This was the breakthrough that I should have had at 8:30 am instead of 2:45 pm. I drew a right triangle on GeoGebra, called up a slider to rotate the triangle and showed where the vertex originally located at (4 , 2) ended up. Then I reminded this gang of help-seekers that points on the x-axis move to the y-axis under a 90 degree rotation. So I convinced them (and this time I am pretty sure that I DID convince them!) that the side of the right triangle corresponding to the distance 4 represented by the x coordinate of our original point now HAD to represent a y quantity after the rotation. We tried a few other points as well and we were humming along. I tested their wits by asking them one more question before they headed off – I asked about a 270 degree clockwise rotation. I was thrilled that one of the guys quickly pounced on this and said it was simply our 90 degree counter clockwise example again. VICTORY!
At least it feels that way now. I’ll see (and report back) after our quiz tomorrow morning.
A beautiful Monday here – the heat finally broke and fall is beginning. I just want to take a few moments to share what’s been happening with two of my courses. I am teaching AP Statistics (my 5th year doing so now – I am finally beginning to feel comfortable), AP Calculus BC, and Geometry, I have already written about my Geometry book that I wrote this summer (you can grab it here if you have not done so already) and I am pretty pleased so far with how the students are responding. We just spent two days in our computer lab so that they could get their hands dirty working with GeoGebra. In my book, Sect 2.4 is the hand-on intro. This section has the fingerprints of @jensilvermath all over it. It was fun to watch the kids poke around and try to discover. It was a refreshing reminder that all of the talk about digital natives needs to be taken in context. There are certainly tech skills that feel more natural to my students than to me. Hell, there are things my 11 yr old son knows better than I do on our laptops. But this kind of exploration does not necessarily come naturally. I also am reminded of the relationship between comfort with material and comfort with exploring. Some of my students accomplished so much more and were so efficient compared to their peers. I saw a direct relationship between kids who feel confident that they understand directions like – create a regular hexagon and then create a circle that contains the vertices of the hexagon. The students who were willing to simply poke around on the pull down menus were quick and happy to execute some simple commands while others just stared aimlessly at their screens. I led class very directly on Friday in the lab and intentionally did not do so today. I have had a couple of students comment that they are enjoying the text and that it feels easy to read. I have also had a few tell me that they do not like my habit of asking questions for which there are multiple correct answers.
I spent about 5 or 6 hours this weekend reading short essays from my AP Stats students and responding to them. I had them read How Not to Talk to Your Kids an article that I first read about 7 years ago. It was my first encounter with the ideas of Carol Dweck regarding mindset and praise. It changed my thinking as a parent and as a teacher. I gave my students two sets of quotes that I found memorable and asked them to pick one from each set to comment on. I also asked them to find a meaningful quote of their own. I was really proud of them. I got some thoughtful responses and quite a bit of personal reflection. When we debriefed in class today most of them said that the article really made them think about their own childhood, what motivated them, and how their parents treated them. Could not ask for much more than that. I also shared with them a brief video from an interview with Richard Feynman. In it Feynman discusses the distinction between knowing the name of something and actually knowing about that something. My students reported, as expected, the depressing news that they feel that their job is often complete when they know the name of something. I was pleased that they expressed, pretty unanimously, that this is not the way it should be – just the way it is.
I gave my AP Calculus BC kiddos some rough problems to deal with on HW last night. On one of the problems they were asked to consider the greatest integer function – to be called the floor function from here on – and they were to graph the relation (floor(x))^2 + (floor(y))^2 = 1. A tough problem for sure but a couple of students nailed it. Most did not, so I fired up Desmos to show them the graph thinking we’d spend some time discussing why the graph was what it was. Desmos gave a nice graph but one of my students insisted it was not quite right.
So, I told them (kind of bragging, honestly) that I had met Eli this past summer and I would tweet out our issue. My last blog post was about sharing my learning with my students and here was another great opportunity. Eli tweeted back with another go at the graph – found here
This happened while we were still in class! I got to show my students this graph and it still was not quite right. I wrote back to Eli again and his response was awesome. He tweeted back that this problem would be a lunch time conversation at Desmos World Headquarters. How fantastic is this? I get to share with my students tomorrow that a problem we shared out to the world was going to result in some brilliant guys reprogramming their fantastic tool so that it would tackle this thorny problem. Don’t know that it gets much better than this. Oh yeah, as an added bonus I was sent another take on the problem by Christopher Danielson. Here is his take
Days as a teacher don’t get much better than this. Oh yeah, I should note that I got so wrapped up in these tweets and talking to my Calc students after school about this problem unfolding that I was late to get my children from their bus. A friend had to call me to get me out of my classroom building to go grab them. For one afternoon math seemed more exciting than seeing my own children.
I am in my 28th year of teaching high school math (with some overlapping years of middle school math thrown in as well) and some would think that I should have it all down pretty well but his point. Luckily for me, this is only partially true. Also, luckily for me I have a great community of support online (I’m looking at YOU #mtbos.)
In the spring of 2010 when I was interviewing with my current school I was told that one of my tasks was to teach AP Statistics. I had never taught stats before at any level. There was a time in my life – in 2001 as a matter of fact – when I stopped talking with a school about a position because they needed a stats teacher. This time I was more confident and more interested in the school so I took on this challenge. I enrolled in a week-long summer institute taught at Fordham in Manhattan by Chris Olsen. I’ve really enjoyed teaching this class but I still feel far less confident in Stats than I do in any of my other classes. I gave our first quiz of the year last Friday over Section 4.1 of the Starnes, Yates, and Moore 4th Edition of The Practice of Statistics. A number of my students were engaged in a pretty heated debate outside the classroom. I was pretty sure that I knew what the answer was but students in favor of two different answers both made compelling arguments. In the past, I might have dug in my heels and stood by my initial guess. Or I might have thrown he question out. Or I might have given everybody credit regardless of their answer. In any of these situations I would have felt pretty unsatisfied and I would not have been any smarter. I was tempted to go to the AP Community page where I have found some pretty helpful folks. However, the feedback cycle there is not particularly rapid and I have to remind myself to go back and check in. Twitter to the rescue! I sent out a plea to Hedge (@approx_normal) and to Bob Lochel (@bobloch) sharing a link to my quiz and begging for help. Hedge replied in a series of about 8 tweets and Bob replied as well. Hedge suggested that I also reach out to Shelli (@druinok) Temple for help as well. In the end, I felt smarter, I realized that my students who had a misconception (a) had a very reasonable misconception and (b) had that because of something I had said earlier. I now know to be more careful with my use of vocab, I know that there are folks who have my back. I was able to show my students the twitter transcripts of these conversations so that they can (a) see that learning keeps on going on even when you are a supposed expert like they see their teachers to be and (b) they (hopefully) see that I am trying my best to be clear and fair in how I evaluate their work. The fact that Bob suggested that each of the two hotly debated answers should be accepted certainly helped.