Trying to Understand what my Students Understand

Starting to think about school again and this question has been clanging around in my brain. On my last test for my AP Calculus BC kiddos I included the following question: screen-shot-2016-12-29-at-11-33-34-am

My BC gang absolutely nailed this question. Almost every single one cited concavity for part b noting that a function with positive slope AND positive concavity will increase at an increasing rate while the tangent line increases at a constant rate. So, moving to the right of the point of tangency means that the function has pulled away from the tangent line. They almost uniformly used the language I just used with slight tweaks and maybe a little less detail since they were operating under time constraints. I was proud of them for such detailed answers to an important principle of graph analysis. However, after the happiness faded there was a nagging concern that arose. I worry that they are SO good at citing this language that perhaps they are simply responding to a familiar prompt. I am not here claiming that these talented students do not understand this principle. I am here claiming that I am concerned that I have ‘trained’ them too well in responding to certain prompts, that I have enabled them to simply repeat a claim that I have made convincingly in their presence. I want to do some deep thinking about how I can circle back to this idea and ask this question in a form that is similar enough that it is clear what I am asking, but different enough that my students will have to say something different to betray their understanding. I would love any advice on how to continue to poke at/probe how deeply my students understand this concept. Any clever ideas out there? Drop a line into the comments section or tweet me over @mrdardy


Thinking about Learning (again…)

Been away for a while for a number of reasons. 

I just read an article on the really got me thinking about what learning looks like and, therefore, what teaching means in this context. Read a great quote sometime ago that basically said teaching does not exist unless learning has happened. This is quite a challenge for us, obviously.

I shared the article with our AP Psch teacher and he said it was a valuable read and that he would share it in the future with his students. I think it’s worth a read, but if you don’t want to follow the link the article discusses a famous memory study subject who suffered damage to his hippocampus. This caused amnesia to set in but over the course of his life he was still able to form new memories of a certain sort. Here, I think is the interesting quote

After the motorcycle accident, K.C. lost most of his past memories and could make almost no new memories. But a neuroscientist named Endel Tulving began studying K.C., and he determined that K.C. could remember certain things from his past life just fine. Oddly, though, everything K.C. remembered fell within one restricted category: It was all stuff you could look up in reference books, like the difference between stalactites and stalagmites or between spares and strikes in bowling. Tulving called these bare facts “semantic memories,” memories devoid of all context and emotion.


I immediately thought of my AP Stats students who are always asked to report conclusions in context, but I also thought of my Calculus students. Both of these groups of students have a deep reserve of the qualities that usually mark a student as a good student. However, too often I have conversations where it is clear that much of what they have displayed as learning in many classes might not go much beyond the sort of semantic memories referred to in the pull out quote. Skill such as setting a derivative equal to zero when solving optimization problems, or running a two sample t test rather than a z test are often reduced simply to factual memory with no conceptual anchor. In stats when we ask about rejecting or failing to reject a hypothesis based on a reported, or calculated p value, it feels like a particular student should either ALWAYS get this decision right or ALWAYS get it wrong based on a conceptual idea about what the p value says. However, I have seen too many instances where this decision seems to boil down to not much more than a coin toss as the student tries to remember a rule. If the p value has a meaning related to probability, then the answer should be clear and consistent. It feels to me that the biggest challenge in teaching these days is to figure out how to help my students slow down and think. Really think about the ideas that they are working with. Too often they have been rewarded with good grades without reflecting on what they’ve learned and how it applies to anything. This sounds (and kind of feels) like a criticism of my students and my colleagues. I don’t intend it that way. I intend this as a question for me and my colleagues (both in my building and around the world) and my students to consider. How can we construct our classes in a way that helps to develop understanding for our students in a more meaningful, more permanent way? I certainly don’t pretend to know the answers. I know that the way I run my class works for some. It makes other crazy. Two super quick anecdotes, then I’m off to pick up my little girl.

  • This year when I was reading my teacher/course evaluations that the students fill out I ran across a great written remark. One of the questions asks whether the instructor challenges the student to think critically about the subject matter. This student in question marked that he agreed with the statement and then wrote ‘TOO MUCH THINKING’ I hope that this was meant in a good natured way, but I DO know that I wear some of my students out with my questioning. They often ask me to just tell them HOW to solve the problem.
  • Last year when we were wrapping up Calc BC and working in class on review material for the AP test two students were talking. They did not know I was close enough to hear (or they did not care) and one said to the other ‘last year I knew how to solve these but I had no idea why it worked.’

Here’s to the never-ending struggle to make this all meaningful.