Assessment – A Portrait of a Math Teacher as an Aging Man

Thinking About Speed and Time

On first glance, the title of this post has me thinking about my Calculus classes, but that is not the speed and time angle that is on my mind this morning. Yesterday, I finished listening to the newest episode of Malcolm Gladwell’s Revisionist History podcast. The episode (found here) is called Puzzle Rush which is the name of a variant of chess. In the episode Gladwell raises some interesting questions regarding chess, the LSAT, and various places in our society where it seems that speed is valued more than deep thought. He keeps referring to the hare and the tortoise and wonders when the hares got to make the rules. This pod has me thinking about my assessment practice. As I often do, I am going to use this space to think out loud and I am going to hope for the usual outpouring of wisdom here and on twitter to help me work through my questions/concerns.

Earlier this year, some colleagues were having a grumpy conversation about the kids these days. You know, the usual grumpy late winter talk about what is wrong with kids. A totally natural conversation that happens at some point every year. Not a criticism here. However, I did push back a bit and I said that while my current Calc BC kids would be dismayed by my Calc BC tests from 20 years ago, my kids from 20 years ago would also be dismayed by my tests from today. I am pretty convinced that my students today are being asked for deeper analysis of why the math they have learned works the way it does and they are asked to make more predictions and asked to tie together information more deeply. I am also pretty convinced that they are slower in their calculations and in their algebraic manipulations. If my students from today tried to complete in 50 minutes a test I wrote more than a decade ago, many would flounder. If my students from ten years ago tried to complete a test I wrote this year, many would be flustered by the open nature of some of the questions. In general, I think that the thinking I am asking for now is more important. If I still thought that the old ways were more important, I would not have evolved in my assessment practice in the direction I have moved. Where Gladwell has me questioning myself is that there is still a distinct flavor of speed that comes into play. I have a number of students who are still furiously writing when I give them a three minute warning. They are still furiously writing when I give them a one minute warning. Heck, they are still writing as students are passing from class to class in the hallways and I have to bark at them a bit to give up their work. I am somewhat convinced that this might be true no matter how much I shorten the tests. I also admit, not proudly, that I am a little uncomfortable with the idea of a 50 minute class test only taking 20 minutes for some of my best students. I do not believe that speed is the best judge of talent, I know better. But I also suspect that speed is an ingredient in success in many endeavors. What I am wrestling with in the wake of Gladwell’s pod is how do I strike a balance here. I keep flashing back to an essay I read years ago by Dan Kennedy in which he advises ‘Value what you assess and assess what you value.’ I think that there is a very real part of me that values some level of automaticity. Maybe I am being shallow here, but it feels like my best students, the ones who have really mastered ideas, can do so quickly. Maybe I am just fooled into thinking that they are my best because they move quickly? I can keep rambling with this internal monologue, but I won’t bore you this way. I will just jump to some questions that I have for you, dear reader, and I hope to get a nice conversation going in the comments here or over on the twitters where I am still @mrdardy

How do you estimate the time needed for your students to complete a task in class? I have 50 minute classes (mostly) for testing. I generally work on the idea that I should be able to carefully write out my solutions in about 15 minutes. No real science behind this, just accumulated experience.
When writing a test where I am pretty sure that there is one especially challenging (I usually call them interesting!) question, I try to place that one near the front half of the test. Students can, of course, skip around but most just plow through. I want the problem requiring the most thought to be placed where there is still some time for that thought to occur.
When students finish their test, they are dismissed. Is this smart? How do you approach this?
Our schedule, like many of yours I would guess, does not really encourage flexibility with students who might want that simple two to three extra minutes to wrap up work. I have students coming in for their class and I want to respect their time. My students are on their way to their next class and I do not want to interfere with that time. I am uncomfortable, for a number of reasons, with the idea of having them just come back to wrap up later. Any comments/ideas/hacks that have worked within these pretty common scheduling restrictions?

As always, thanks in advance for any wisdom. I am looking forward to a good conversation that will benefit me and my students.

Balancing Group vs Individual Work

For over ten years now, my classroom has been setup for group work and talk. Currently, I have desks in groups of three and I reshuffle the groups after five class meetings using flippity. One of the courses I teach is called Honors Calculus. It is a differential calculus course that is an option instead of AP Calculus AB. What is typically done be the first week of December in the AB course takes us into May. This allows much more time to review algebra and trig ideas and to really dig into the mechanics and principles of Calculus. I don’t skimp on the level of analysis I ask for in this class, we just have more time to settle in. This year, after a conversation in the first trimester, I settled in to a routine where we have group quizzes – I write five versions of each quiz – but we have individual tests. My hope was that this would decrease the level of stress in the classroom, that it would increase the level of communication between the students, and that hearing multiple voices would increase the likelihood of ideas and techniques sticking with my students. What I have witnessed is that this process has decreased the level of stress overall because a handful of students just don’t worry much knowing that they are paired with confident kids who can carry them to the finish line, the level of conversation HAS increased, but only for a subset of the students who end up in the role of explainer, and ideas are NOT sticking. Mistakes made in November are still being made. Skills practiced (or at least skills that have been available for practice) are not embedded. On our most recent individual test about 15% of my kids did not recognize the need to use the product rule when taking the derivative of a product. I have asked a variation of the exact same question for the last three tests and there is no noticeable improvement in answering that question.

There is another feature of our class that is at play here. In the 2017 – 2018 academic year our department adopted a test corrections policy that I wrote about previously. For the 2018 – 2019 academic year the department voted down this policy. I had spent a considerable amount of time and energy promoting this policy and talking about its importance in the learning process. In the wake of this decision I reached an uneasy compromise with the two courses where I am the only instructor. They can review a test when it is returned and they can reassess on up to three questions from that test with the possibility of earning up to half of the credit they missed. There was a lot of debating in my mind and with my students before we arrived at this imperfect solution. This was in place before the conversation with Calc Honors about group quizzes. Looking back, I feel that the combination of group quizzes AND opportunities to reassess provides too much of a sense of safety net and many of my students are pretty clearly not preparing themselves too carefully or they are simply not practicing much. With the level of practice opportunities provided/the number of times to talk together in class/the class conversations led by me with examples and old assessments offered as practice/etc. I simply should not be seeing the test performances I am seeing. I am clearly complicit in all of this due to the decisions I made about assessment and the decision I have made not to collect or check HW practice. In my last post I thought out loud about the idea of frequent, low stakes, skills-based check in assessments. Had a great twitter chat last night with the #eduread crew (prompted in large part by this article ) and I went to sleep convinced that I need to incorporate some of these ideas into this course next year. I also need to remove the added layer of reassessment, it has not worked in conjunction with the group quizzes. I think I probably still need group quizzes separate from the check-in layer of ways for me to see progress AND as ways for kids to feel that they can buffer their grade with legitimate skill progress. I hope that the combination sends a couple of important messages about what I value. I really (REALLY) like the conversations that do happen in the group quizzes. I am more than willing to write multiple versions of quizzes so that conversations can happen out loud without worrying about giving away information. Our discipline, I think, allows this more easily than some others might. I do not want to collect HW daily for all sorts of reasons, but I think that frequent low stakes check ins send a message about the importance of mastery of topics. I think that I need to adjust my problem sets so that they feature more reminders of topics. My kids know how to take derivatives with the product rule. They probably need to be periodically reminded of it in a more tangible way. I also wonder about balance in point values between these three ways of assessing and reporting on my students’ progress. I do not want to retreat into a mode where I am scaring (or bribing) my students, but I do think I need to be more clear and explicit about what I value and balance it accordingly when/where I can.

As always, any words of wisdom here or over on the twitters (where I am @mrdardy) are much appreciated.

Success!

I have had a very active blogging week thinking about (and writing about) my Geometry class. I have three preparations this year, AP Statistics, AP Calculus BC, and Geometry. I’m not proud of it, but I know that my attention to each class varies at different times of the year. Iy’s not a simple matter of 33 1/3 % of my planning energy being spent on one class at any time. Do many of you go through this as well? By the way, how many preps do most folks have?

Anyways, I blogged in December about my discomfort with HW in Geometry and gathered some nice ideas. I blogged about my decisions about changing habits and it has felt like a raging success. Five to seven minutes at the beginning of class of students sharing their work with each other and correcting each other/reinforcing each other/ sharing their miseries, etc. It’s just been a really terrific week with them and I have let them know how much I appreciate their demeanor, their energy, their willingness to share with each other. Today we had a quiz (you can grab it from here) on Sections 6.1 – 6.3 of our text (you can grab that here) exploring centers of triangles. We’ve talked about perpendicular bisectors, altitudes, medians, and angle bisectors this week. We have played with GeoGebra and looked at how, in each case, all three segments have a common point where they coincide. We’ve talked about which ones could coincide outside the circle and those are not popular choices as the best center of the triangle. We had a great lab activity yesterday (you can grab that here) and it developed into an interesting debate where one group of students nominated the intersection of the angle bisectors as the best representation of the center of a triangle while the other three groups all felt that the intersection of the medians was best. As we had a healthy debate I found myself wishing that I had been clever enough to have physical triangles to manipulate. Next year, I want to be prepared with cardboard triangles of various types with these two candidates for center marked out. I dropped the ball on this one anticipating that everyone would feel best about the centroid. What really impressed me was that the group arguing for the angle bisectors had GeoGebra construct a circle that had this incenter as its center and showed that this circle touched all three sides. I was THRILLED that they thought of this argument.

So, this morning I felt confident as my cherubs asked their last few questions before the quiz and the results are in. I have 12 students in this class and 4 of them earned perfect scores with another 4 earning an A on the quiz. Their class average was 93%!!! I’m thrilled by this. I think that this is due to a number of factors.

In general, my students have had more energy this week in January than they did in the few weeks leading up to our winter break.
I believe that the HW strategy has made a positive difference.
I believe that the extensive use of GeoGebra in class is finally spreading to the home. I have overheard a number of students this week make reference to looking at GeoGebra while doing their HW this week. I am a firm believer in the power of these graphing programs and, for my Geometry students at least, I think that this is the best of the bunch.
I worked hard during break planning out this unit for me and for my Geometry team of two terrific colleagues. This thoughtfulness has paid off.

Oh yeah, one final thought. As a long-time Calculus teacher I have a strong preference for lines in the point-slope format. Every one of my students presented at least one of their line answers in this format. Woo-hoo!!!

Reflection inspired by Meaningful Quotes

So, a blog post from Prof Ilana Horn (found on twitter @tchmathculture) came across my reader last week. It was titled ‘First, Do No Harm’ (you should head over here to read it) and this caught my eye for a number of reasons. The first is that one of my proudest moments in the sprawling world of internet interactions came when Tina Cardone (on twitter @crstn85 or over at her blog here) grabbed a quote from me to use in her magnificent Nix The Tricks. The following quote, a comment I made about the use of the dreaded FOIL acronym, is the one she used in an earlier version of her terrific book.

I would say, then, that it is not reasonable to even mention this technique. If it is so limited in its usefulness, why grant it the privilege of a name and some memory space? Cluttering heads with specialized techniques that mask the important general principle at hand does the students no good, in fact it may harm them. Remember the Hippocratic oath – First, do no harm.

I’m excited whenever I see a new post by Prof. Horn, but this one grabbed my eye by its title. Little did I realize that her post would be itching my brain for days at a time when I have little spare space or energy. We’ve been engaged in fall term finals at my school. Otherwise, I would have responded sooner.

Prof. Horn lays out some common practices that do harm – at different levels – to students and to their chances of increasing their competency in the math classroom. I’d like to respond to a couple of them and try to gather the wisdom of the internet (or at least the minuscule portion of the internet that will read this post!)

Timed math tests – Prof Horn links to Prof Jo Boaler here and says that our assessments communicate to students what we value. I could not agree more with this statement about assessments. I speak to colleagues about this all the time. If we say to our students that we value thought and process but then give them multiple-choice tests where points are all or nothing, then the students quickly figure out that we do not mean it when we say we value process. What we do is FAR more important than what we say in this arena. Years ago I read a powerful essay about assessment written by Dan Kennedy (you can find that essay here.) I found many of Mr. Kennedy’s arguments to be powerful ones and I remember that my primary takeaway was that we should assess what we value and we need to value what we assess. I tell my students that I want them to be able to tackle novel problems. That they need to be able to tie together ideas we have worked with and apply them in a new context. I often give problem sets for HW that require them to remember from past lessons and from past courses. I tell them that I don’t necessarily expect everyone to get these problems completely correct, but that I think it is important that they grow as problem solvers. If I never put problems like this on graded assessments, then my students would quickly sniff out the fact that I don’t really value that process very much. However, what also has to go along with that in a graded assessment is a willingness to pay careful attention to their work, a willingness to reward thoughtful work with meaningful partial credit, and some careful feedback either on their written work or in a group setting when papers are returned. (This feedback question is also burning my brain thanks to a recent series of thoughtful posts by Michael Pershan over at his blog on twitter you can find Michael @mpershan – I hope to draft something meaningful soon in response to these thoughts!) The belief that I have that is challenged by Prof Horn here is the idea of speed or efficiency being valued highly. I think that I want to argue that efficient problem solving is a skill I want to value and one that I want to reward. Where this gets tricky is that I know that there are certain problems – meaningful, valuable problems – that just do not lend themselves to quick solutions. How do I balance the desire to see my students think and wrestle with new contexts with the desire to reward efficiency and cleverness? I also teach in a school run by the bell system (I’m certainly not alone there!) and I need to think how to work within that system. I tell myself that I balance the points on my tests so that the diligent student who has gained increasing mastery of facts and skills can still earn a respectable grade even if they fail to connect the dots on the novel problems. This only comforts me to a small degree. I know how much grades serve as motivators (and de-motivators) for my students. I know that a student who feels that s/he has worked hard can walk away from an assessment feeling defeated and incompetent simply due to failing to finish one problem. I know that students can convince themselves that their hard work was for naught and that maybe they just are not cut out for this particular challenge. I’ve been at this a long time now and I still do not have a satisfactory answer and Prof Horn’s post really brought that home to me again. What do you say wise readers? Is it reasonable/valuable/important to reward those clever students who can solve novel problems more quickly than their peers? Should this be a valued skill? If it is, then I believe it should be assessed somehow.
Not giving partial credit – I agree 100% with this point. As a teacher of two AP courses, I feel that part of my task is to help my students be ready for the format and the peculiarities of the AP test in May. Most of my students choose to take these tests and for those who are not yet seniors, they feel that their test scores can help/harm their chances to get into the college of their choice. What this means is that I incorporate multiple-choice questions into their assessments. Now, if I tell them that I value process, how can I feel good about MC questions? Well, I don’t. I have dealt with this two ways and I am not thrilled with either of them. Sometimes I simply value each MC question at such a low point total that mistakes will not have a great impact on their grades. The other way I have dealt with them is to decide what the most reasonable incorrect answer is and give partial credit for this mistake. I am not happy with either path. Any wisdom from others who deal with the (sometimes) reality of MC questions?
In the comments section there are some additions like this one – Grading practices that do not allow reassessment. Again, I am wrestling with this and I have blogged about this. In my two AP classes, where I am the only instructor, I allow retakes on unit tests for anyone unhappy with their grade. I have averaged the two grades. I have read some powerful arguments against this from the SBG crowd, but I cannot find a place where I am happy simply waving off performances. I may get there one day but I am not there yet. I am not at all happy with myself or with my students about the current retake policy I have. I hope that I can construct a more meaningful one by the time our winter term starts in December.

So many thoughts rattling around my brain. Thank you to Prof Horn for agitating me with her blog post. Thank you to her commenters for furthering the conversation. Finally, thank you to anyone who reads this and helps to continue to refine my thoughts and practice.

Help with a Teaching Strategy

I’m still (relatively) new at the AP Stats game. When I was hired here in March of 2010 I was told that I needed to add AP Stats to my toolbox. I’ve loved the course and teaching it has really changed my point of view about teaching/learning. This year, I decided to try something new with this course. My enrollment exploded this year and I wanted to find a way to adjust to this. I had 12 students, then 12, then 19, and now I have 38 students in AP Stats. I wanted to find a way to get/give some feedback in low stress ways (low stress for me AND for them) and I went out to the local office store and bought some marble notebooks for my kids. I have been digging through my old files (assessments I’ve created and assessments from the publisher) and I have been making time for exit slips for my kids. I am trying to do this at least once each chapter – but I have not been as diligent as I should. I pass out the marble composition books that all have a question glued in them. I pick a question from an old assessment and give the students about 10 minutes to work. I explained to them at the beginning of the year that the goal of this exercise was two-fold. First, I could check in and see where their level of understanding is. Second, they would get some feedback from me to guide them to better understanding. No grades – but helpful (I hope) feedback. Seemed like a great plan. I am running out of enthusiasm fro this project as I have noticed that very few of my students seem to be taking this seriously. Students earning an A regularly on graded assignments are turning in work that is sloppy, incomplete, or even just completely blank. What this says to me is that many of my students don’t actually do work (study, read the text, finish their HW) until just before the actual assessment time. I had a direct conversation with one of my students about this. I talked about daily diligence and his response was that he felt that the course calendar – the document with nightly HW assignments and reading assignments – was simply a set of suggestions about what to do and when to do it. He genuinely did not see any problem with the fact that he was not doing his work on a daily basis. He asked why I was concerned with this since he got around to doing his HW before tests. I’m not surprised to hear that some (many?) of my students are only getting around to doing their work right before a test. I live with 80 students in a 4 story boys’ dorm and I see them at work. Many of them are working hard, but not working terribly efficiently. Lurching from assessment stress to assessment stress. I had genuinely thought that my exit slip strategy would help fight this. It’s pretty clear that this is not working this way. I want to have a conversation about this with my class but I don’t want to be a total cranky pants about this. I’ve got some thinking to do. Any clever advice out there?