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


My Students are Making Some Smart Guesses

On Friday in Geometry we were continuing our conversation about triangle centers and I asked my students to find the point where medians coincide in a scalene triangle. There is a good amount of algebraic detail in these problems but my students were doing a nice job pushing through this problem. After finding the centroid, I asked them to form a new triangle from the three midpoints we needed when considering medians. We found the perimeter of the original triangle and I asked also for the perimeter of the triangle formed by the midpoints. One of my students theorized that the new triangle would have one-fourth the perimeter of the original triangle. I asked the other students to quiet for a moment to hear this guess. Before asking GeoGebra to check his answer he quickly corrected himself and said he was thinking about area, not perimeter. A beautiful realization on his part that this triangle formed by midpoints would divide the original triangle into four equal areas. Just as we were congratulating him for this guess another students asked about equilateral triangles. He wondered aloud whether the midpoint triangle in an equilateral triangle would form four equilateral triangles. I realized he was asking whether the triangles formed in the scalene we were looking at were also congruent, not just equal in area. A quick question from me confirmed my guess so we drew our attention again to the GeoGebra sketch we had up. He was able to identify where the congruent angles were that allowed us to prove congruence for the triangles.

This conversation was a wonderful way to end our day on Friday. I am delighted that my students are comfortable enough to make these guesses out loud and even more delighted that they are making such good guesses right now. I pointed out how helpful it is to play with GeoGebra to check these guesses and I hope (I hope hope hope!) that some of my students are making a habit of this.

A Delightful Conversation

Last week in my Geometry class we had a fantastic conversation about a homework problem. Here is the problem in question –



I wish that I could take credit for having written this, but I am certain that I ‘borrowed’ it from somewhere. Likely from the fantastic resources shared with me by Carmel Schettino (@SchettinoPBL)

So, this is the kind of problem that I expect only a minority of my students to navigate successfully on their own, but I am convinced that almost all of them will benefit from thinking about a problem like this one, from a little active struggle along the way. I KNEW that this would be asked in class if anyone took the time to do the HW I assigned, so I was pleased that it came up. I started by telling my students that I LOVE this problem and asked them if they could guess why. One student said ‘Because it’s so hard’. I laughed that off and said, yes it is hard but I love it because it ties together a bunch of important ideas. Off we went on solving this. I started by asking a couple of questions that probably seemed a bit irrelevant at first. I asked why they knew that the y-intercept was (0, 3) and that the x-intercept was (4, 0). Before they could answer I made sure to mention that they knew this without looking at the graph. We eventually arrived at the realization that we know whether a point is on the line or not by looking at the equation itself. If a point makes the equation true, then that point is on the line. If not, then not. This is the kind of thing that I think my students know but being reminded regularly sure does help reinforce it. I hope! So, I thought I had set the hook here for the rest of the problem. We talked about what we know about squares and we talked about how to identify points on the square without knowing their real coordinates. We got a little lazy, and I was okay with that,by calling the bottom right corner (x, 0) and the top left corner (0, y). This gave us no choice but to call the top right corner of the box (x, y). At this point I paused and asked them to remind me what needs to be true about points on a line. Then I asked them to remind me of what we know about a square, therefore what we know about x and y for that mystery point (x, y). It wasn’t easy to get everyone to agree with our conclusions, but I think we got there. We agreed that the x and the y had to equal each other. We agreed that the y coordinate had a definition based on x. We agreed that this was an equation we could solve even though it was not a bunch of fun to solve it. After all of this work it felt like the problem should be done, students were pretty sad to realize it wasn’t. We still had a conclusion to make about the triangles created. One of my students was pretty insistent that they needed to be congruent because their angles had to match up. This was not the time to launch into a conversation about similarity and I decided it was not the time to talk about the restrictions of AAA conclusions between triangles. We have talked about equilateral triangles of different sizes and we are (mostly) okay with that, but I felt that that conversation would be a diversion here. Instead, we kept at the calculating and we looked at side lengths. Once we agreed that they were not congruent, I pointed to the slope of the line and talked about the fact that his instinct was foiled by the fact that x and y lengths were not changing at the same rate. The whole conversation took quite some time, might have been 15 minutes by the time the whole thing was done, but I felt that we had done some important heavy lifting.

If you recognize the above problem as your own, feel free to claim it and let me know. Know in advance that I am very grateful for such a rich problem to tie together ideas of distances, slopes, line equations, properties of squares, and triangle congruencies all into one tidy package!


The Decisions We Make – A Postscript

Thank you thank you thank you, John Golden. John commented on my last blog post and gave me some important wisdom regarding my frustration with my own decisions and the decisions that my students had made last week. As expected, the quizzes were subpar. In the class where I had chosen not to explain the permutation notation I made the following grading decision. I graded the last problem as if it were a 10 point problem as advertised. However, when calculating their grade, I counted it as a 5 point problem. So, the students who had learned the notation earned some bonus points while those who had not were not stung quite as severely. Not a perfect solution, but it did open the door to a public conversation about my frustration and about how we might avoid their frustration AND my frustration moving forward. Don’t know yet how that will sink in, but at least it was received as a good will gesture on my part and no one complained out loud that it was unreasonable for me to have expected them to read that definition. We’ll see what happens in the next week or so as we have two more opportunities for showing some learning here.

The Decisions We Make…

I have two sections of Discrete Math this year, one in the morning and one right after lunch. During the fall term, each of these sections had 7 students. We all sat at a single group of desks together and had some great conversations. A number of the students have spoken to me about how much they enjoy this atmosphere. It does not work for everyone of course, some students prefer not to have the expectation of participation, they would prefer to quietly observe and have more time to think before speaking. Our school is on a trimester schedule and this Discrete course is set up as a trimester course where students can move in or out and not have the demands of previous knowledge from this course. So, I have done some thinking about how to make this course modular. One of my sections expanded from 7 students to 16 this term and we are in the process of figuring each other out and how this new group will mesh. One of the students who has been in the class the whole year commented that class seems more quiet this past week. Interesting that more than doubling the size of the class has resulted in a quieter atmosphere…

All of the above is just to sort of set up what our week together was. We started a probability unit this week and so far all of our energy has been spent on counting techniques. When does or matter? When does and matter? What is the difference between them? What is the deal with that ! function anyways? When can we tell whether replacement matters? These are the kinds of conversations we have been having and I have had in-class activities for us to work on together while I have been asking them to do some reading and some HW on their own outside of class. The great Wendy Menard (@wmukluk) shared some fantastic resources that one of her colleagues shared. She was also kind enough to spend some time on the phone last weekend to serve as a sounding board. One of the decisions I made was not to spend much time emphasizing notation together in class. For example, our text explains permutation notation pretty cleanly, points out that our calculator writes 10P4 while you might also see P(10,4). It clearly shows that this calculation is 10!/(10-4)! while also introducing this notation in more general form of P(n,r). In class we had a number of examples of drawing some subset of members from a group, so I thought that the text’s approach and our class approach would support each other. I also figured that any students flummoxed by the text notation would ask me in class what the deal was. So, the first HW question on Wednesday night was this in fact – Which is equivalent to P(10,4), 10!/4! or 10!/6! ? We had a quiz scheduled for Friday and on one of the questions I gave the students the numerical value of P(26,3) and asked for an explanation of how to get that answer. On Thursday I had a couple of review problems thrown together from the textbook author’s supplemental test bank. I planned on starting class by fielding any HW questions then turning them loose to work on the review problems. In my morning class I projected their HW from Wednesday night and had the first HW question on the board. Not one student knew what P(10,4) meant. They asked whether that was a point on the plane. I have to assume that they did not do the reading or the HW on their own. I quickly untangled the notation, pointed out how it matched some other conversations we had and then gave them their review sheets to work on. That was my morning class of 7 students. After lunch I had the book projected with the first HW question. Not one student in that class knew what P(10,4) meant. I decided to remark on the importance of doing the reading and the HW and then just gave them their review sheets and sat down.

One student came by on Friday morning during my free time to ask me a question about the notation and she remarked that it was clear I was disappointed (annoyed?) that no one had done the HW. She wanted to make sure she understood that notation. Each class on Friday began with me answering questions before the quiz and I do not recall anyone in either class directly asking me to revisit the P(n,r) notation at all. I have not graded the quizzes yet but I know that there were a number of students in my second class that either left the question about P(26,3) blank or simply wrote something to stumble into extra credit. A number called me over to ask about it and I said this was something they needed to know. I do not remember my morning class as clearly, they may have been in a similar boat.

So, as I think about this I realize that I made two very different decisions with my two groups of students and I am not happy about either of them. In one class I came to their rescue and explained something that they clearly could have come to terms with – in some way – on their own. In the other class I let my annoyance take over and I did not address the question at hand. I also realize that my students, especially those in my second class had two decisions to make. On Thursday night, after seeing my disappointment/frustration they could have gone back to their reading and either understood it themselves or they could have checked in with me during review on Friday. It is clear that a number of them did not do that. So I am faced with yet another decision when I grade what are likely to be disappointing papers. I feel that I want to get across a pretty clear message about responsibility but I also need to recognize my responsibility here. It is reasonable, I think, to see my role as someone who expands the conversation from the text, not as someone here to simply recite what the text already explains. But I also recognize that I have 9 students who are new to our class and all of them are new to me as a teacher. If they are used to teachers making sure that every question in their text is also addressed in class then my idea about my role might be a bit of a shock and I did not spend much time together on Monday explaining this about myself. However, I also have 14 students who were with me all fall and it is pretty clear that none (or very few) of them did the reading and the HW either.

I am not happy with myself that I let my annoyance get in the way of clear thinking. I am also not happy that I was not more clear with my morning class about my disappointment that none of them had done what I asked. I am not happy that so many students did not do the reading or the HW. I AM happy that I had a student come by and clarify the question for herself while also recognizing that she should have done so on Wednesday night. I feel that including the question when I compute the grade will likely have a pretty significant impact on many grades as it was one of four questions on the assignment. I also feel that it is a reasonable question to ask, but it relies on notation that I did not explicitly present.

I have been reading a number of the DITLife blog posts and there is a constant reminder about the number of decisions that we make on the fly everyday. These are complicated decisions and I know that I hope that I make them clearly. Here is a case where I think I was probably not as clear thinking as I should have been and I will likely need to make a decision about grading that will, luckily, not have to made on the fly. I have a bunch of new students who are only one week into their experience with me. I want it to be a good experience where they grow as scholars. I need to think carefully about how I respond to this disappointment – in my own behavior AND in the decisions they made.