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!!!
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.
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?