Communication Breakdown (Rethinking Assessment Ideas)

My first post of 2017, good golly where did January go?!?

Our school uses an LMS through FinalSite, a company that manages our school home pages. It is a pretty typical looking LMS. I populate each class with my students so that when they log on to their student portal they see each class they are in (as long as their teacher uses the LMS) and they can see calendars, they can download assignments, they see their HW and upcoming responsibilities, etc. My hope is that students check in pretty regularly (daily is a pipe dream, I fear) at least on Sunday night to scope out their upcoming week. In addition to populating this calendar – usually about a week in advance, but I sometimes lag a touch, I keep a spot on one of my side chalkboards where i highlight upcoming highlights. I lagged on that recently as well. As I hinted earlier, January has been a bit of a blur for reasons I cannot pinpoint. Anyway, last Monday I had a rare Monday test scheduled for my AP Calculus BC gang. Their class met right after lunch and a couple of students came in during lunch and asked if the rumor that they heard was true. The rumor they were referring to was a rumor that they had a test that day. I mentioned that this had been on their calendar for well over a week and confirmed that, yes, this ‘rumor’ was true. Kids got on their phones to notify classmates and they frantically started flipping through their text book.

I kept my calm and I assured them that they would be fine. they had been doing their work (I said optimistically) and that last minute cramming rarely has much positive impact. Most of the class performed reasonably well – one perfect score and one student who only missed one point out of a group of fourteen – but the average was lower than usual and one student in particular was way off of his usual mark. The frustration got me thinking about a number of things and I want to use this space to think out loud about these issues.

First, I worry about communication in this increasingly digital environment. I used to print off weekly calendars and hand them out at the beginning of each week. Some kids would lose them, some would carefully put them in their folders, some would cram them in their backpack never to be seen again. Mostly, kids seemed to know what was coming up or at least kept it a bit of a secret when they were surprised by an assignment or an assessment. Now, I print almost nothing. I post on the LMS. I keep the reminder chalkboard. I send out email reminders through the LMS. I send some occasional emails from my school account with attachments for notes or suggested extra work. I hear repeatedly from students who did not know I had sent an email or that I had posted to the LMS. This makes me wonder how much of the blame lies on me for moving away from printed reminders. I mean, if 14 out of 14 students did not know that there was a test on Monday then part of the blame falls on my shoulders. But, and this is important, something is odd in the student culture around my class if 14 out of 14 students failed to register this fact in a planner or take a look ahead at their upcoming week before reporting to school on Monday. I do want to take a moment here to compliment my students in their reaction to this event. Not a single one complained about unfairness, not a single one said to me that this was my fault, and when they received their grades back not a single student voiced their unhappiness about the situation. It would have been so so easy to point the finger at me and none of them did. This is a credit to their character and willingness to take responsibility. A number of them did ask to take advantage of my policy for reassessing, but no more than usual really.

So, I am questioning my role in communication and the avenues I choose to take advantage of. The other question this raises for me is my attitude about announcements for assessment. I know that many of my colleagues, both in my building and out in the world, have as part of their practice unannounced assessments. I have never done this and it is mostly because I find myself overly sensitive to charges of increasing/causing student stress. I always make sure that there are at least three school days between announcing and administering an assessment. In the case of major unit tests, I want to have at least one weekend between announcing and administering the test. But this incident has me questioning this commitment. Am I seeing a more true reading of student mastery of material if I check in periodically when they do not know that I will do so? Am I bypassing the stress of test and quiz preparation if I just drop a quiz or test in their lap when they show up for class? How do I setup a situation where an unannounced assessment is not such a big deal for the student?

As always, I am seeking wisdom here. If you have made a practice of unannounced assessments, how do you handle that? How do the students respond? If not, is your reasoning similar to mine? How do you communicate calendars to your students? Teachers here use either our school LMS, Google Classroom, Facebook, or old fashioned paper. What are the habits at your school? What really works? Drop me a line here or over one twitter where I remain @mrdardy

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 –

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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.

Assessment Is On My Mind

Our school is making a pretty big change next year. We are moving from a static schedule where we have seven classes that meet in the same order every single day. Our class lengths are either 40, 45, or 50 minutes depending on the length of our assemblies and our ending time each day. Next year we will be on a rotating schedule in 7 day blocks. Each class will meet four times for 50 minutes and once for 90 minutes over that interval. Five classes will meet every day, four of them will be 50 minutes long and one will be 90 minutes long. As a result of this upcoming change we are being asked to do some deep reflection about our practice and our curricular choices. As department chair I have been encouraging my department to think deeply about trimming or eliminating items from our curriculum to make time to think and have more open ended (and open middle!) problems. This schedule change will really push us to do this and I suspect I’ll be thinking out loud on this space as we move through this process. Where my mind is tonight is on the subject of assessment. I was asked to facilitate a group this morning to talk about our assessment habits and goals. Years ago, I moved to a school that had a rotating schedule with a long block. At the time I was not the same teacher I am now (at least I don’t think I was) and I don’t remember being all that thoughtful about the impact on my practice. The fact that I had moved to a new school and just had to adapt probably diminished any sense of dramatic change that I might have felt. Around here right now we are having some pretty valuable conversations and I was part of one this morning.

My takeaways from the meeting are:

  • I want to have more frequent, but shorter, opportunities to check in on my students’ learning. Ideally, some of these would not have any letter or grade attached at all but simply serve as formative checks of their developing understanding of the material at hand.
  • I think that the old model of a 45 – 50 minute timed sit down assessment where everybody takes the same (or VERY similar) tests needs to shrink, at least in my discipline. Our time together will be too precious to take up too much of it silently hunched over a piece of paper with problems, no matter how creative the problems are. We have 154 class days this year that are not devoted to term exams. Meeting 5 out of every 7 days changes that to 110 days. Granted, these 110 days will be a minimum of 50 minutes each (our current longest class) so contact time will feel different. But we are still only going to see our students for 110 days and we need to make those days count.
  • I want to expand my palette. My students’ grades are almost entirely dependent on times tests and quizzes. I know that there are other smart ways to do this, I want to learn more and I want to grow my toolbox.
  • I have been sneaking in group quizzes lately given that my class is set up on pods. I want more of this. I experimented with my Discrete Math elective in the fall. We have a stretch of days leading up to our term exams called test priority days. On these days we are limited by department for who can have assessments so that our students do not have these pile up on them. Each department has two of these days. I chose to have a group test on the first day where everyone chipped in ideas (obviously some students were more vocal than others) and they all got copies of my feedback on their test. Four class days later, after one new topic was introduced, each student had an individual test. My best students performed about the way that they always had, I wasn’t too worried about them. What pleased me was that some of my students who had struggled during the term clearly benefited from the group work not only in improving their grade by the addition of this group test score, but they performed at a much higher level on the individual test. The combination of the work with their peers and the ability to study from that work and from my feedback all seemed to work well together. I want to capitalize on this and try more ideas like this moving forward.

 

So, the reason I am thinking out loud here about this is the reason I always cite. I would love to hear from you, dear readers. Please share successes and failures you have experienced as you expand your assessment toolbox. I want to hear how you wisely spend time with your students on extended classes like the 90 minutes we will have periodically next year. I want to hear what pitfalls to avoid that we might not even be anticipating. In general, I just want to continue learning from all of you!

As always, feel free to comment here or to engage me on twitter where I am @mrdardy

 

Thanks in advance

Hands-On Geometry

I’ve been at this high school math gig for a good long while now but I periodically have to remind myself of a couple of important facts. The most important one is that not everybody’s mind works like mine. Just because I like a certain way of thinking, or dislike a certain way of learning, I should not assume all my students will agree. In fact, I can be pretty certain that all of my students will not agree, there’s too many individuals for that to work.

When I studied Geometry I did not like physical drawings and constructions. In part because I am a bit inept when it comes to controlling something like a compass, but also because getting my hands engaged does not seem to fire too many of my neurons. So, when I wrote my Geometry book a couple of years ago I did not include much in the way of hands-on manipulations. The past couple of years of working through the text with our students has pointed out the weakness of this approach. So, I put my head together with one of my talented colleagues to try and make an activity that would trigger some neurons for those students who come to life when they get their hands busy. I had been using a pretty cool activity I ran across from Jennifer Silverman but I made pretty flimsy paper copies to work with on a pipe building activity where kids had to manipulate bent angle joints with different pipe lengths. It’s a great activity but using simple paper copies dragged the activity down. We invested in some packs of AngLegs this year and my colleague wrote a pretty cool activity modeled off of our pipe building activity. You can find his document here.

I was impressed as each of the seat groups in my class played with the AngLegs making some discoveries about combinations that worked and those that would not. We discussed, without naming it yet, the triangle inequality theorem to explain why some combos did not work. But the real fun, and the clever heart of my colleague’s activity, was when I asked one student from each group to come to the front of the room. When they left their group the remaining group members were given the following task – I slightly modified the original document on the fly – I asked them to make and measure a triangle. Find six measures, the three side lengths and the three angles. They then put the triangle away where it could not be seen. I sent the volunteers back and their teammate gave them three pieces of information. I left it to each group to decide what information to share. Once given three clues the volunteer student needed to manipulate the AngLegs to copy the triangle described. What ensued was a terrific conversation about what information is necessary to guarantee that I have to make the same triangle. We used this as a launching pad to discuss congruence theorems for triangles. I have some great links in the text to some wonderful GeoGebra activities up on the GeoGebraTube site but I know that many of my students do not do these explorations.  I also know that some just need to get their hands dirty, so to speak. Some kids were able to recreate the triangle but admitted that it was a bit of luck. Some stumbled upon the ambiguous case of the Law of Sines without being told that this is what happened. Some realized that they had no choice but to create the correct triangle.

I was really pleased by the level of engagement and I am now thinking about ways to use the AngLeg sets again soon when we start talking about side and angle bisectors. I want to have tables create and draw their own triangles before we stumble into discoveries about concurrence of these bisectors. This will feel, I hope, a little more authentic than me just giving them a prescribed triangle which may feel a bit like I am just luring them into some pre-prepared trap. I think that this activity we ran benefited my students and we have referred to it on a number of occasions already. The grouping of three or four students together at a time helps and allowing them to get their hands busy has helped. Looking forward to loosening up a bit more and letting my students be more tactile in their approach to Geometry. I’ll still show them the GeoGebra and introduce them to Euclid the Game  but I need to remind myself that they are not a bunch of mini Dardys in the room.

Questions about Questioning

I feel I am long overdue to write this blog post. In part, this is due to, you know, life getting in the way. In part it is because I have about three posts swirling around in my head right now. Next week our students are taking term finals so I will have a little more unstructured time and I may finally get around to writing more. That is, if I get around to writing plans for the short stint between thanksgiving and winter holidays.

Today, I am going to try and make sense of a fantastic post by Mark Chubb (@MarkChubb3) that can be found here. In the post (which you DEFINITELY should read) Mark raises important questions about the questions we ask our students AND the purpose, the goal, of those questions. I often tell a story about a student who graduated back in 1993 named Ashley. I had the privilege of teaching Ashley for four years in a row up through Calculus BC. The week before her AP exam I asked her how she was feeling. She told me that she was not worried at all because she knew that if she got stuck on a problem she would hear my voice in her head asking her what that problem reminded her of or what have we done in the past when we have seen this. I was flattered that she had internalized some of the strategies we had worked on together and I felt good that she felt comfort in my leading questions that I had been asking her over the years. She was also a tremendous student who was in a group of talented kids who pushed each other over that four year span. Since then, however, I have begun to question myself about the sort of questions I pose. I still believe that most of my students would be able to effectively work through problems they are presented if they can have an internal monologue that is similar to the conversations we have as a group. What I worry about is whether my guided questions are taking away their agency, their ability to discern what they think is important in a problem. I made it through high school math pretty successfully and I have confidence that I can guide students through this journey. But posts like Mark’s, and conversations I have had through this blog, in conferences, through twitter all push me in the direction of making my voice less central in my class. I have taken great strides in this direction in the past few years, but I still feel that I talk too much, that I initiate conversations and lines of questioning too often. That I impose my sensibilities about what to notice and what to wonder about on my students. The trouble is that many of them are happy to have me, and their other teachers, take on this burden. It is easier, it feels more stable and safe to hear the expert in the room direct the conversation. I know that this is not the best strategy but I too often fall into this trap.

I am going to lift a portion of Mark’s post here to draw attention to the central question about questions that I think he was trying to raise.

 

Funneling vs. Focusing Questions

As part of my own learning, I have really started to notice the types of questions I ask.  There is a really big difference here between funneling and focusingquestions:

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Think about this from the students’ perspective.  What happens when we start to question them?

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After reading this and playing back a number of classes in my head, classes that I was really proud of, classes where I felt that my students had made some major breakthroughs, I realized that I do FAR too much funneling and not enough focusing. An easy excuse is that my students struggle in metacognitive processes, so it is more painful and time-consuming to do this. And, just like my students want the more comforting path of me telling them what is important, I am prone to take the comforting role of the guiding questioner. But my students are not going to get better at monitoring and understanding their own thinking this way. They are not going to take ownership of what they feel is important in a problem this way. They can get to be better mathematicians and be more successful at high school math, but I fear that I am training students to be mini mrdardys instead of being better high school mathematicians as themselves.

 

Our school is moving to a new schedule model next year. We will be on a seven day cycle where in each rotation we meet four times for 50 minutes and one time for 90 minutes. This will force us to re-examine how we run our classes, how we will value and plan for our time together with students. There are many layers of what we will have to examine but for myself, I think that I will be going back to Mark’s post over and over, I’ll be looking at a few classes that I have had videotaped and I will be working out how to hand ownership of ideas to my students. I will be working on how to make sure that the classroom and the class time we spend together is not so dependent on my point of view and my insights into problems. I want my students to leave my class better high school mathematicians, that is absolutely true. But I want them to be better models of themselves as high school mathematicians, not imitations of me. As Ben Folds sang once, I do the best imitation of myself. I don’t need my students to be imitations of me.

 

Linear Functions

Some of my Geometry students are wrestling with being able to accurately write linear functions given information about points and slopes. I am struggling with how to help them overcome this and I have been doing quite a bit of thinking about how we teach this and what kind of sense it might make to my students. I know that I have these fundamental ideas in my head – an equation is a relationship between the variables stated in the equation (it tells me how to turn an x into a y or vice versa) AND the graph of an equation is the set of points that makes the equation true. I know that I say these things and I am fairly certain that previous teachers have said similar things. I know that my students have repeated some of these things and they can (periodically) carry out these operations. Where the mystery lies for me is why this skill can only be inconsistently displayed and I have a couple of thoughts. I am interested in any wisdom you have about this question.

Almost every single one of my students prefers the slope-intercept equation of a line to any other form. Partially because of my history as a Calculus teacher and partially because I favor more direct problem-solving approaches, I am an advocate for the point-slope equation. I consider it a minor success that most of my students answered the first question on their recent quiz in this form. Here is the question: Find an equation of the line that passes through the points (3 , 1) and (5 , 4). Now, I am careful to ask for an equation rather than the equation but I do not know how much of an impression this might leave on any of my students. Almost every Geometry student answered this correctly and most left it in point-slope form. I was pleased. The next question was this one: Find the coordinates of the following points on the line you found in problem #1.

  • The x-intercept.
  • The point with an x-coordinate of 1.
  • The point with a y-coordinate of 1.

 

Here is where things started to fall apart for many of my students. I have been thinking about the mistakes I see in class and on assessments and it occurs to me that there might be a fundamental problem that I do not know how to solve. When a student wants to write the first problem in slope-intercept form I instruct them to first find the slope, then replace the x and y in y = mx + b with coordinates of either given point to find the b value. I tell them that this way is harder, but many want to hold on to that equation form. If they want to approach the first problem with the point-slope form I tell them to first calculate slope then replace x1 and y1 in the equation    y – y1=m(x-x1)  with the coordinates of one of the points they know while leaving the x and the y alone. I am embarrassed that this inconsistency has never jumped out to me before, but why is it that in one equation we leave the x and y while in the other we replace the x and the y with coordinates of a given point?!???!? I have to imagine that some of my students are absolutely baffled by this inconsistency. I wish that they could verbalize that sense of confusion, but I just now figured it out for myself, so why should they be able to lock in on this? So, dear readers I implore you – help me figure out a better way, a more logically consistent way, that I can help direct my students. This is not an intellectual task that is beyond any of them, but I have to guess that a handful of them are so tired of being asked to do this and have sort of given up on the idea that they will ever master this concept. It is way too easy to just write it off and hope it will go away. I do not want this to be their reaction and I want to see them reliably be able to answer these questions.