A Quick Question about Test Questions

One of the classes I teach is a non – AP Calculus class. Many of my students are showing a reasonably firm grasp of the ideas of Calculus that we are studying, but they are plagued by algebraic struggles and fears at times. I was thinking about the following two ways that I could ask them the same question:

I asked this class today how they would feel about these two forms of the question. Most of them quickly agreed that they would feel more comfortable answering question B than question A. I followed up by asking whether success on question A might feel more like success and a handful agreed that this probably was true. I am not sure what to make of this informal poll and I am bouncing this idea around in my head. I’d love to hear what you think and push back with some questions about my motivations/hopes in (possibly) restructuring my assessment in this direction.


I have been urging them to have GeoGebra open nearby when they do their work so that they can look at graphs and ask for derivatives to verify their own work.

Checking in on a new Policy

I recently blogged about a commitment made by our math department. As a reminder to you, dear reader, here is the statement from one of my course syllabi:

Beginning in the 2017 – 2018 academic year, our math department is adopting a policy of expecting test corrections on all in-class tests. The policy is described below.

  • When grading tests initially each question will get one of three point assignments
    • Full credit for reasonable support work and correct answer.
    • Half-credit for minor mistakes as long as some reasoning is shown.
    • Zero credit (in very rare cases) when there is no reasonable support shown or if the question is simply left blank.
  • When grading tests, I will not put comments, I will simply mark one of these three ways.
  • You will be allowed to turn in corrections. Corrections will be on separate paper and will have written explanations of errors made in addition to the correct work and answer. This work is to be in the student’s words but can be the result of consultation/help. These corrections will always be due at the beginning of the second class meeting day after the assessment is returned. You will return your original test along with your correction notes. I will remind you of this every time I return a graded test to you.
  • It is not required that you turn in test corrections.
  • The student can earn up to half of the points they missed on each individual problem.
  • This policy does not apply to quizzes, only to in-class tests.

There are  a couple of items in the learning curve to report on here. I will lead with the positives –

  • We have an after school conference time that many students take advantage of for extra help. My room has been more crowded this year than it has been for years. I take this as a plus sign that students are committed to seeking help and investing in their math studies. Many of them are there talking with me and with their classmates about test corrections.
  • I have had a number of students turn in their corrections the same day that they received their tests. It is a rare thing for students to turn in work days early, it is happening regularly right now.
  • I walked out of my classroom today with a new student who was talking excitedly about how she is really thinking carefully about her work and she is sure that she’ll remember material better because of this.
  • It is faster to return tests since I am simply marking them 100%, 50%, or 0% for each problem.

Now a few negatives, followed by some philosophical pondering about this whole endeavor.

  • I anticipated that there would be very few zero scores on problems. There have been more than I thought but I hope that is a factor of students learning to show some work. This may be a positive as a zero stings a bit and they may be more inclined to be careful in their explications.
  • Some students feel stung by relatively minor mistakes that initially result in a 50% on an individual problem. These minor mistakes turn into 75%. I am trying to point out that relatively major mistakes can also end up at the 75% level but some students feel a bit cheated.
  • Early in the year averages fluctuate quite a bit anyway, but the fluctuations are exaggerated in this system. I see already that overall averages are a bit higher than normal and there is much less variance in scores. However, some students are scared since they have a hard time seeing the long game as clearly.
  • Explaining this to folks outside the department has been a bit of a challenge.


When I was working on my doctorate I had a professor (my thesis advisor) who had a policy that every paper will be rewritten, not just every paper can be rewritten. The way he did this was to return our first drafts with no comments, just hash marks in the margin at certain points of the paper. These hash marks might be there to point out a flaw in our argument or our paper’s structure. They might also indicate a highlight. They might indicate a misspelling or a simple grammar problem. He was willing to discuss these hash marks in his office hours as long as it was clear that we had sound questions about them, in other words we had to prove to him that we had reflected on our writing. I have never thought so much about my own writing as I did in that class. I do not expect my 14 year olds to do this kind of self analysis, but I know that they ARE capable of careful reflection if they are given the time, space, and motivation to do so. What I am seeing when they come to me is that they have looked over their test, they have referenced notes and their HW. They have done careful thinking and they can usually explain their mistake on their own. This is not universal, but it is happening more often than not. Many students who come to my room to work on corrections have almost no question for me. They are berating themselves for ‘stupid mistakes’, they are laughing at silly things they wrote, they are even saying ‘I have no idea what this work means’ I am pretty convinced that this can be a huge growth opportunity for my students. They are being responsible for their own error analysis here and they are writing thoughtful reflections in the form of ‘On problem 4 I made this mistake, I should have done this instead’

We have midterm grade comments looming and as department chair I know I will have some questions coming my way about our reasoning and the long-term impact on grades. I will try to steer the conversations to long-term impact on learning and self-sufficiency. So far the policy has exceeded my hopes in my classes. I will be checking in with my department to get other points of view soon and I plan on sharing some of those conversations as well.


Another New Beginning

Tomorrow morning will be my 31st opening day of school as a math teacher. I am luckily pretty well past the days of nervous anxiety. Luckily I still have the experience of anxious excitement about the task ahead. As I begin a new year, there are three things in particular that I am really excited about. One has to do with school structure, one has to do with classroom policy, and one has to do with my life outside of school. I’ll highlight them in reverse order.

I mentioned over on twitter that I had the great pleasure of taking on a gig as a DJ at a local college radio station this summer. My wife works at a college across the river from where I work and she helped me land this fun gig. I just got the good news that I can keep my slot at least through the fall. I will be on (as DJ Calc – an old in-joke from my past) on Thursdays from 4 – 6 PM ET on wrkc.kings.edu where you can stream and listen online if you are so inclined.

One of the takeaways of the workshop my team did with Henri Picciotto (@hpicciotto) last spring was that we committed to a new policy of test corrections in our department. I teach four different courses this year, three of them are senior and junior heavy while one is freshman and sophomore heavy. They will all receive the statement below with only one tweak. My Geometry kiddos will turn in test corrections on the third class day after receiving their test while the others (Discrete Math, Honors Calculus and AP Calculus BC) will turn theirs in on the second class day. Here is the statement I crafted for a syllabus.

Test Corrections

Beginning in the 2017 – 2018 academic year, the math department is adopting a policy of expecting test corrections on all in-class tests. The policy is described below.

  • When grading tests initially each question will get one of three point assignments
    • Full credit for reasonable support work and correct answer.
    • Half-credit for minor mistakes as long as some reasoning is shown.
    • Zero credit (in very rare cases) when there is no reasonable support shown or if the question is simply left blank.
  • When grading tests, I will not put comments, I will simply mark one of these three ways.
  • You will be allowed to turn in corrections. Corrections will be on separate paper and will have written explanations of errors made in addition to the correct work and answer. This work is to be in the student’s words but can be the result of consultation/help. These corrections will always be due at the beginning of the second class meeting day after the assessment is returned. You will return your original test along with your correction notes. I will remind you of this every time I return a graded test to you.
  • It is not required that you turn in test corrections.
  • The student can earn up to half of the points they missed on each individual problem.
  • This policy does not apply to quizzes, only to in-class tests.

I will definitely be blogging throughout the year about this topic and I’ll be sharing my thoughts and experiences about this change in approach. The baseline message that I hope we will be sending is this : I want you to learn the material at hand and I want you to have an opportunity to show me (and yourself!) that you have learned this material.

The last thing that I am thrilled about is our new schedule. After being here seven years and meeting every class on every school day in the same order for the same amount of time we area adopting a very different new schedule. We are moving to a seven-day rotation schedule. We will meet five classes per day and each class meets five times during a seven day rotation. During that rotation each class meets in each of the time slots AND each class has one 90 minute block and four 50 minute class meetings. I am excited on a number of levels about this initiative.  I have taught at two other schools with rotating schedules and I noticed a couple of clear advantages. You know that sleepy kid in your 8 AM class? That kids is not usually so sleepy for an 11 AM or a 2 PM class. You know that athlete who keeps missing your 2 PM class because of travel obligations tot he team? That athlete is rarely traveling at 8 AM or 11 AM. You know that class that wanders in right after lunch with a mixture of twitchiness because they do not want to sit again or lethargy as they digest their lunch? You do not see them in the same state everyday. I have seen that different students emerge as classroom leaders at different times of day. Most importantly, I have noticed that students (many of them, at least) give a more honest commitment to effort on HW when they are getting ready for four academic classes in a day instead of six of them. This, too, will be a regular topic of conversation in my blog this year.


As always, drop me a line here or on twitter where I am @mrdardy

I’d love to hear from those who have experienced a major change in school schedules I want to have some idea of how to anticipate possible problems this year. I’d also appreciate any comments about our test correction policy. Anecdotes from experience will probably help me and my team as we make this transition.

So, What Kinds of Change?

In my last post I wrote about our department’s terrific two day workshop with Henri Picciotto. One of the major decisions we made based on the time we spent together is that we have decided, as a whole department team, is that we will allow test corrections on all tests in our department. Before I dive into the format of the decision we made, I want to include a couple of important links here with other points of view about assessment policies. The first comes from a new twitter contact Steve Gnagni (@Steve_Gnagni) who shared this interesting document written by Rick Wormeli (@rickwormeli AND @rickwormeli2 for reasons I am not sure I understand!) called Redos and Retakes Done Right and the second is a link Henri shared gathering together some of his ideas about assessments.

So, a little history here about where I am as a teacher and where I, and my team, hope to move. In the past three years I have had a policy in some of my classes. In any class where I have been the only teacher I have allowed test retakes. If you are unhappy with your test score, make an appointment to sit with me and look at what went wrong on your test and sometime within the week that your test was returned, you can take a new version of this test. Originally, I averaged the two test scores but this year I weighted the retest so that the score that stayed int he grade book was two parts retest and one part original test.  I also told students that anyone who scored below a 70% on the assessment were expected to take the retest. I did not do this in classes where I was part of a team teaching the course since not everyone agreed with this policy. The advantages of this policy were that students who were struggling to master material and perform on tests felt that they still had a lifeline. Those students were more likely to follow up with me and try to figure out what went wrong with their original attempt. Students were willing to take the extra time and energy to try and improve and I had reason to believe that material was sticking a bit better for many of my students. The primary disadvantages? This created quite a bit of extra work for me writing and grading reassessments. Some students seemed stuck on a perpetual hamster wheel of assessments and a handful of students were very honest about the fact that they sometimes pushed my assessments down their list of priorities since they knew this lifeline existed. This was a small group of students but enough that I was questioning the wisdom of this policy.

When Henri was with us he spoke passionately about the advantages of students correcting their own work. He talked about a cycle of student reflection and about the burden of careful written feedback on assessments. A sad fact is that most students (we probably know this about ourselves from when we were students) simply look to the grade. While many of us take careful time to highlight problems and write notes or to write congratulatory notes for work done especially well, much of this probably falls into the cracks. I know that I have read research – and I wish I could find it quickly – about the tension between writing comments on papers and writing grades on papers. These two forms of information for our students do not work in support of each other. So, after some conversation with Henri and then a long, productive final faculty meeting in the week after Henri left, we came up with a policy that we feel pretty good about. On unit tests when we grade them the first time we will assign one of three options to each problem. If the problem is done well, clear work and a correct answer (or a minute problem like some minor arithmetic error) that problem will receive full credit. If a problem shows no sign of clear explanation and no clear sign of understanding that problem will receive a zero. The vast world of problems in between these two poles will receive half credit. We will not highlight or circle errors in solutions. We will not write notes about the problem-solving process. We will simply return the paper with an initial grade. We will be able to do so quickly under these circumstances. The students will then have time to take this assessment and rework any problem that received less than full credit. They can earn back half the points that they missed by submitting corrections. The resubmission will have the original paper and two requirements for earning back points. They will need to submit correct solutions AND they will need to submit a written reflection/explanation of what went wrong and how it was corrected. Students can meet with each other, they can ask their teacher for guidance in our extra help sessions, they can look at their notes and their text, in general they can seek any kind of help. Some will inevitably just take the word of someone or something (like Wolfram Alpha) but ALL will be encouraged to take some time to reflect. ALL will be allowed to earn back some part of the points that they missed. ALL will know that test day is not such a high stakes day where it is do or die. There will be some bumps along the way as we train ourselves and our students to take this process seriously. We will have to be very conscious early in the year about establishing standards for what these written explanations need to look like. The student who earned a 60% the first time has a meaningful lifeline. The student who earned an 85% the first time still has motivation to rework and rethink the material. We will need to think about timelines, especially near the end of a grading term, but these are good problems to have and good conversations to make public. Teachers will be talking to each other about this process as we unpack it. Students will be encouraged to talk to each other about math and to seek guidance from each other. This will feel like a serious sea change for our department, I am totally excited about it.

Or, I should say I was totally excited about it. I know that there are different ways to view this process and the meaning of it. I know that we decided that events that we call tests are subject to this correction policy. We decided (for a number of reasons, some more ideologically defensible than others) that short quizzes were not subject to this policy. I know that I will be balancing this with graded take-home problem sets and on these problem sets I always encourage collaboration. So, when Steve Gnagni shared the article above, I found myself doubting some of the decisions we made. I found old reactions about grades being really seriously challenged and I began to doubt whether our decision on process is ideologically pure enough. I also know that this is progress. I will be sharing Rick Wormeli’s article with my team in the fall and we will be checking in with each other on how we feel about the impact of this new process.

I want to thank Henri again and to thank my new twitter pal Steve Gnagni for sharing their ideas. As long as we are all willing to keep questioning ourselves we can continue to help our students grow.

Seeking Better Feedback From Students

At this school and my last one there is a formal process for students to voice their opinions/concerns/suggestions through a course evaluation form. At both schools I have noticed that not enough students take this process as seriously as we want them to. The forms are well-intentioned and detailed in their questions, but they are all Likert-scale questions (with some blank space for expanding answers) and they are hand written forms. I suspect that many students are self-conscious about their handwriting being identified. I think that one of the results of this is that the only handwritten extensions we tend to see are from students who are happy and want to share nice words in depth (we love it when we see these!) or when they are especially unhappy and want to share unkind words in depth (happily, these are much less common.) However, I always walk away feeling that these could be better and more useful. I would love to have a process where these are done online anonymously as I suspect that we will get more willingness to expand on answers. I also want questions that are aimed a little more directly at the concerns of our math classrooms rather than well meaning general feedback forms.

I am convinced that the process is a meaningful one. Letting our students know that their voices and opinions matter is a powerful thing. Letting them know that the adults who have been evaluating them need to hear back on how we are doing levels the playing field (at least a little bit) and this tells our students that they can be part of an honest feedback loop where we can grow based on their experiences/opinions/successes/failures. But I also know that the process can be better than the one we have.

I know I have read a few posts along these lines and I am off to the MTBoS search engine this morning to see what I can find. I would love to hear from anyone reading this about their successes, and failures, in elucidating meaningful feedback from their students.

As always, feel free to drop a comment here or pick up the conversation over on twitter where I remain @mrdardy

Thanks in advance for any wisdom.

Collaborative Classrooms

When I moved up north from sunny FLA in 2007 I made a big decision about the geography of my classroom. In my last school I had seats arranged in groups of 3 or four, I referred to them as pods and my students selected their pod mates. They usually stayed with the same group all year long and built up some real team identities centered around their pod. When I moved to my new school in 2010 I had two large conference style tables brought to my room and had students sitting in big groups of 10 each. Typically, students stayed at one of the two tables all year long, but there was a little bit of movement at times. In 2015 I moved those tables out and got moveable desks and I am back to my sets of 4 seats. However, this year I am using visibly random grouping (using Flippity and happy with it) to shake things up and encourage more of a sense of collaboration across a larger group of their classmates. I begin each week with a new seating arrangement. I have been pretty happy with most of this evolution and I am convinced that these arrangements have increased student communication over the years. However, I saw Susan Cain’s TED Talk not too long ago where she talked about introverts and the impact of all of this collaborative time on their learning and their comfort. Since seeing that talk I have had a bit of an internal struggle about how to try to compensate for the fact that not everyone wants to talk as they learn. Not everyone is comfortable thinking out loud before they have come to some sort of a clean conclusion. Not everyone thinks at the same pace. Luckily for me, Brian Miller (@TheMillerMath) wrote a blog post that has come to the rescue for me with ideas about how to address some of these fears. Brian wrote about a great idea for balancing the principles at hand that seem to be in competition with each other. His post is over at http://www.mrmillermath.com/2017/01/30/alone-time-in-a-collaborative-classroom/ and I strongly encourage you to read it if you have been thinking about some of these conflicts.

Our school is moving to a new schedule next year where we will have five class meetings in each seven school day stretch with one class at 90 minutes and the other four meeting for 50 minutes. I think that these longer spans of time together (currently, the majority of our class times are 45 with a handful of 40 and a handful of 50 based on assembly schedules, etc.) will work beautifully with the ideas that Brian laid out. I am on the fence about the nature of how I want to deal with the earbud/headphone question. I like the idea that Brian has but I am not sure about endorsing any type of paid music services explicitly with my students. I know that almost all of my students come equipped with earbuds and mobile devices everyday (probably a higher percentage than those that come equipped with a writing device to class each day!) so that is not a hurdle at all. I have announced a group quiz for next Wednesday and I know that I want to have at least one day before then where I explicitly have ‘alone time’ for thoughts before reconvening our pod conversations. I am debating whether I need to physically separate seats for this alone time or whether that is too much time and interruption. I’ll be writing about how it goes.

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

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!


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:


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.


Experimenting with Visible Random Groupings

In some ways I think that I am intellectually adventurous, that I am willing to try something new in my classroom. In other ways I struggle with change. I try to make myself feel better about this by reminding myself that we all struggle with this in varying degrees.

This past summer – my third lucky summer at Twitter Math Camp – I finally committed to trying visible random grouping (to be referred to as VRG for the rest of the post) for this academic year. A little background here.

When I moved up north in the fall of 2007 I made a commitment to not have my students in rows and columns. I no longer felt comfortable with most of my students looking at the back of other students’ heads. So, I rearranged the seats at my old school in ‘pods’ of three or four desks. However, I always let students pick where they wanted to sit. As most of us know, even if WE don’t assign seats, the students essentially do this themselves. I comforted myself by thinking about the camaraderie I saw, by listening in on the lively conversations that did not happen when my students sat as if they were in a matrix, and by the fact that I know that I would have preferred life this way as a student. When I moved to my new school I had two long conference style tables so I had two largish groups of students working with each other. Two years ago I ditched the conference table and went back to pods.

Over the past three summers I have heard more and more conversations about the power of rearranging the students, about shaking them out of these simpler comfort zones and encouraging everyone to be comfortable sharing ideas with everyone else in class. Alex Overwijk (@AlexOverwijk) has been an especially articulate proponent. So, this summer I learned about a pretty cool website (flippity.net) where you can build a roster for a class and anytime you want this program will randomize your class. In groups of 3 or 4 or 5, by the number of ‘teams’ that you want, etc. It creates a cool visual that you can project and the kids get rearranged instead of staying in their friendly neighborhood comfort zone. I committed to trying this for a number of reasons, the primary one being my experience the past few years in Geometry. I had been teaching mostly AP and upper level honors classes and these students mostly knew each other for awhile and they were comfortable sharing ideas and debating/challenging each other at times. Not true of my Geometry class. Even last year’s class which was outgoing, chatty, and engaged. They did a great job in their pods discussing ideas but did not do a good job projecting those ideas out to the class. They always wanted to filter ideas through me and, over the course of the year, inevitably fell into some ruts about who took command when I asked them to work together.

Now, enter VRG. The strongest proponents discuss doing this every single day to continually shake things up. I got a little scared of this because I really value the sense of camaraderie that I have seen developing over the years, so I came to what seems like a nice compromise. On the first day of each week, I shuffle the class. I am now ending the fourth week of the school year and I have some observations I want to share. I am particularly motivated to do so by a twitter chat this morning.

There is sound research in the field about VRG and its effects. This research suggests that the positive effects of this practice are most clearly seen when this happens every day. I do not want to discount this and I do not want to feel like a contrarian. What I want for my classroom is for my scholars to not only know everyone else and hear the ideas of their peers, but I want them to be in a zone that feels comfortable and safe. My prejudice is that this zone is more likely to happen if I have some time to get used to my new teammates. What I have seen in four weeks can be summarized as follows (and I will make separate remarks for my AP Calculus BC group and my Geometry group)

  1. In BC Calculus I have also been incorporating whiteboards that the pods write on together. The combination of whiteboarding (and presenting the ideas of the pod) out to the class along with VRG has been pretty spectacular. Again, these are kids that know each other well, but I have been seeing active conversation across table groups to former teammates that is lively. I can step out of the way and let them bounce ideas around as I wrote about yesterday.
  2. In Geometry we have not done as much whiteboarding, I want to improve on this. What I have seen is students talking to people they did not choose. I see them making guesses to/with their neighbor. I have seen students more willing to stand up and talk. I have heard some lively discussion between students and I know it is not just with their good buddies unless they all magically happen to love each other.
  3. I have been able, in both classes, to call on a wider variety of people because even the shy/underconfident/nervous kid has someone in their group whose ideas they can paraphrase. In the past few years I felt that there was more of a posture of looking to one person in each pod to be the spokesperson. I see less of that now.


When I tweeted out my happiness about weekly VRG I was promptly congratulated AND reminded that this would be even better if it was done daily. I may get there, but I kind of feel that this is my 10% moment. That place where I am making a change I know is for the better but I am limiting myself in my own discomfort a bit so that I can still feel sane and effective in other arenas.