A Geometry Explanation Idea

Today will likely be a two post day after too long of a layoff from this space. It’s funny, I know that this writing relaxes me and lets my brain breathe in important ways. However, I am too prone to let busy-ness in my life prevent me from taking advantage of that.

In Geometry we are examining relationships between angles and arcs. You know the drill, right? Central angles = intercepted arc measure. Inscribed angles are equal to half of the measure of their intercepted arc. Interior angles (other than those at the center) are equal to half of the sum of their intercepted arcs. Exterior angles equal half of the difference of their intercepted arcs. As I am explaining this yesterday in class I am emphasizing this half relationship over and over but I just kind of gloss over the fact that the central angle does not fit this pattern. I have gone through this explanation in the past but I do not think I was as explicit about the developing pattern with the one half scale factor in the past. So this morning I was thinking about how I can best help my students focus on this detail. I was motivated, in part, by a lovely blog post by Michael Pershan (@mpershan) which, in part, is about managing the processing load of our students. I was struck when reading by the fact that I was asking my students to juggle seemingly different rules yesterday. So I had this idea and it probably is not revolutionary in any way. But I think I want to simply say that central angles are interior angles. They happen to be interior angles that form two congruent arcs. This way I can ask my students to think about the one half scale factor for every angle/arc question that they think about.

Does anyone else out there do this? Do you agree with this idea? Do you think that there is a downside here that I am not seeing yet? I’d love to hear some thoughts in the comments or you can find me over on twitter @mrdardy


Classroom Conversations

I find myself thinking about how to best moderate and encourage classroom conversations. Two blog posts have me looking in the mirror. One of them is one I have written about previously. Ben Blum-Smith wrote at his blog (Research in Practice) about having students summarize each others’ statements. I am still working on making this a teaching move that I regularly go to. It has mostly worked well for me. However, I have noticed that when I do this I almost always have to interject and pass along some value judgment about the response of one student before I can get another to elaborate/explain/restate what was said. Andrew Stadel (over at Divisible by 3) wrote something that really has me thinking. You should read what he wrote by clicking on his name. (The same goes for Ben’s post – you can click on his name to read what he wrote) I’ll try to summarize part of Andrew’s post here, but I encourage you to go read his post. 

Andrew discourages teachers, as I read it, from telling a student that they are correct (or incorrect) and instead urges us to explore reasoning and to try and get the student to tell us why they arrived at the conclusion that they did. I get it – I think I really do. I want to understand their reasoning and, more importantly, I want my students to be able to reason out loud. To think through their own process and have the language at hand to explain that process. What I notice though, is that my students are rarely willing to expound at length on their own thoughts without knowing whether their statement was correct – or at least in the ballpark. Once I give them some confirmation that their original statement had value, then they are willing to expand on what they said. I understand this feeling. I imagine them feeling that they are about to be ambushed if I ask them to say more about what they think when they are unsure of whether they are right or not. I know that I can work on creating more of an environment where this kind of thinking out loud feels safe. No doubt about it. However, I think that there is something kind of primal that I am working against here. It is so natural to be shy about talking out loud when I think I might be wrong. Add in the social pressure of being wrong out loud in front of your peers and that’s a heck of a force to push against. Another item on my list of things to think about and to try to modify in my teaching. I hope that this process never ends.


Two conversations and a Blog Quote

Okay – so I’m thinking out loud here. Hoping some wisdom comes from this exercise and/or from brilliant comments by my dear readers.


Conversation #1

Working with my outgoing Calc BC group and I comment to one of my students that it’s a tough day for him. He’s on our swim team and they had a 6 AM practice Tuesday morning and a meet that afternoon. One of the other students – a member of our field hockey team – says that her team never ran sprints the day before a big meet. Now, it’s important to understand that our field hockey team has won’t he state championship three of the past four years. This student was a member her whole high school career. I take this opportunity and I ask her if she thinks that this strategy (don’t stress out your body the day before an important match) might be carried over to another realm. I am greeted by a quizzical look and I say ‘Maybe you should not cram the night before a big test.’ Another quizzical look. She asks if I am advocating not studying. I say that the daily diligence of regular work and studying is comparable to daily hard practice in field hockey. Then, relax a bit before an important match (or test) and maybe this is a formula for success. I don’t think that many of my students saw that as a winning strategy.


Conversation #2

I just observed a lovely Precalculus class taught by one of my colleagues. The class was working on a variety of word problems – coins, movie tickets, area/perimeter, etc. My colleague is a remarkably calm, zen-like fellow. He sat in a student chair the whole time (sort of invisible!) and asked one student at a time to come to the board. The rest of the class was attentive, offering help to their colleague and generally being cooperative and positive. The teacher kept asking nudging questions of the student at the board. “What do we want to find here?” “How can we relate the number of coins and the value of the coins?” etc. Being an observer in the class (and not stressing out about HOW to do the problems) I saw that my colleague was modeling for his students a lovely strategy for tackling these problems. If each student could play that conversation back in their head as they struggled with any problem, then they would see much more progress. They might still make mistakes, but they’d have a sound strategy for success. What troubled me – and I spoke with the teacher about it the next morning – is the fact that I KNOW that some of them will not ask themselves those questions. They won’t take his advice for attacking these problems to heart. I am not saying that all of our students need to mimc our behavior. What I am saying is that students who struggle, ought to feel that it is a lifeline that is being offered here. When I asked him about this the next morning, I told him that I was impressed by his careful teaching and modeling. His response was something along the lines of ‘I am a big believer in teaching. I just think it works better when learning happens.’ I really don’t think he was being mean or cynical. 



This morning, I awoke to another terrific blog post by @JustinAion over at his blog Relearning to Teach. If you are not familiar with his work, you should change that and visit him. Pay attention to his tweets as well. Life will be better. He closed out his post today with a powerful quote – “Even with everything I’ve seen, done and learned, even with all of the conversations I’ve had with other teachers, I still only feel as though I’m “teaching” when I’m answering student questions or going over examples.

I wish I could scrub that feeling.”


I think that I’ll walk away from my computer now and let these conversations and this quote marinate a bit. I know I have some questions, but I am not sure that I can ask them accurately enough yet.


So, I sort of pride myself on being the type of teacher who creates an environment in his classroom where conversations can flow. I have my kiddos at two large tables where they are elbow to elbow and talk regularly. Sometimes, of course, the conversation strays – bit there are often rich math conversations going on. I posted this quick story over at One Good Thing, but I want to share it here as well.

I presented my BC class (all in their second year of High School Calculus) with the equation of an ellipse centered at the origin and asked the following rather vague question – “Are there any two points on this curve where the lines tangent to the curve are perpendicular?” One girl, Chloe, immediately answered that the tangent ‘on top’ of the graph was horizontal and it would be perpendicular to the tangent on the ‘side of the graph’ which is vertical. I congratulated her and challenged the class to find some other more interesting points. A student asked what the slopes of these more interesting lines might be and then a boy, Sal, chimed in that any number you pick must be the slope of a line tangent to this ellipse. His argument was based on recognizing that between the two tangents that Chloe had mentioned the slopes range from 0 to positive infinity. In other quadrant the slope would range from 0 to negative infinity. If he had mentioned the intermediate value theorem I might have fainted on the spot from joy.

After I posted the story to One Good Thing I read Ben Blum-Smith’s most recent posting (http://researchinpractice.wordpress.com/2013/09/08/kids-summarizing/) and I now realize what an opportunity I missed by simply congratulating Sal instead of getting others to join in and complete the thought process. Read Ben’s post. You’ll be glad you did. I intend to try and incorporate this strategy into my daily practice.