Busy busy start to the year. I want to take a moment to reflect on and remember a couple of important highlights.
We meet all classes for 30 minutes on day one and we have an hour long convocation ceremony. Our Student Body President is always one of the speakers. This year’s President was in my Precalculus Honors course last year. She delivered a touching speech about productive struggle in my course last year – a course she ended up excelling in, by the way. I have heard a number of speeches by adults advising the importance and lasting power of productive struggle. I imagine that this speech by one of their colleagues probably meant more to our students than hearing some grown up tell them about the glories of allowing yourself to struggle through something. The fact that she highlighted the very things I hope my students take away from my class made it pretty special. The fact that the whole upper school community (and my immediate supervisors!) heard it as well made for a pretty special opening day.
Regular Old Day One
My day started with my AP Calc BC team. I had assigned a HW problem I had never done before. They were asked to graph x + |x| = y + |y|
The table group that had this problem struggled a bit and we talked it through as a group. We then called on Desmos to graph it and the result was not what we thought it should be. I did what I do, I tweeted out the problem and within fifteen minutes one of my former advisees tweeted out a fix so that Desmos would agree with us. It is delightful to have that connection with a student who my current students still remember. It was also fun to think that my questions are still interesting enough to warrant his attention.
There will be some rough days this year, there always are. I want to remember days like this so it is easier to get through the tough ones!
On Monday I report for beginning of the year meetings for the 33rd time. As usual, I have thoughts scattered all about and, as usual, I am going to try to use this space to help whip those thoughts into shape.
This morning I read the latest NCTM email and there was an essay included written by President Berry. In his essay, he challenges us to think about our why. Why do I teach math? He suggests that figuring out the why is a HUGE step to making our classrooms more coherent and productive. In the essay he links to a couple of posts and my favorite of them is from David Wees. You can find it here and it is well worth your time.
David’s post made me think about a time when I was struggling a bit with thoughts like these (I have a post about that here ) and I was thinking that the beginning of a new year might be an excellent time to be explicit with my students about the teacher that I try to be and to try and tease out from them the teacher that they feel that they need. I think back to a story about a former student. He was a brilliant student and has gone on to do some serious financial analyst work in his life. He uses math skills and habits of mind regularly in life. When I taught in New Jersey Chris (the former student in question) lived in Manhattan and he and I would periodically meet for lunch. He told me a story one day. He was working in a small office at the time and had been struggling with a challenging case. My memory is that he said he had been working off and on with a certain problem for a few days. He told his boss that he was going to take a long lunch to get away from this problem and clear his head. Chris told me that when he returned he found some post it notes on his file folder with some questions/suggestions from his boss. Chris said ‘Jim, he reminds me of you. He asks questions I would not have though of asking.’ I have considered this to be the best compliment I think I have received as a teacher. This brilliant person – WAY smarter than me – one who I taught for four math classes (he and I started at a very small school) doesn’t remember a certain lesson. He didn’t point to some trip that we went on together (he was an expert Brain Bowl member and math team member, both activities I supervised) No, he remembered that I asked him questions he would not have thought of on his own. I was prompted to think of this yesterday when an old post by Christopher Danielson was referenced on twitter. You can find that post here. Also, well worth your time as is David’s post above.
So, I guess my question here (see what I did there?!?!?) is this – Is it meaningful to my students to have me share some version of the story above so as to clue them in to my priorities? Is it meaningful to share my priorities in a personal way as an avenue to have them think about theirs? After all, the classroom is theirs more than mine. I need to find a way to recognize and respect their needs in a way that supports what I believe (what I think I know) about teaching and learning. I want to be explicit in discussing our goals and it feels that a personal story about what motivates me to do what I do might be a smart way to do this.
Thoughts? As always, please share any wisdom here in the comments or hit me up over on the twitters where I am @mrdardy
I am overdue in writing about a high energy twitter exchange I was engaged in recently. I am going to include a few links here in this post that will help give some background to the conversation.
First, many thanks to those on twitter who are willing to engage and get my brain moving. In this particular story the star twitter pal is Kristie Donavan (@KristieDonavan) who went on quite a twitter tear and wrote a GREAT blog post. First, I will link the article that started the whole discussion.
A colleague shared an article from Edutopia with me. You can find the article here The article is called The Case for not Allowing Test Retakes. Now, the idea of test retakes/corrections is something that has been on my mind for awhile. Two years ago, after a wonderful PD session with Henri Picciotto (@hpicciotto or over at https://www.mathed.page ) our department adopted a policy of test corrections. You can read my original blog post about it here. Well, last year the department voted to move away from that policy based on a number of concerns that they had about how kids dealt with the policy. Many of their points were raised in the Edutopia article linked above. We have some new admins at our school in the last couple of years and there is reason to believe that we will be urged to move back to some form of test corrections or retakes. That is why my colleague sent me the link in the first place. I tweeted out a link to the article asking for insights and boy did I get some. Most vigorously from Kristie. Who sent a tweetstorm and wrote an awesome post. Here is where you can find Kristie’s post, I urge you to read it. So, what I am wrestling with is a real sense of hypocrisy that might be simply the result of a strong but unsound argument presented in the Edutopia article and in other debates/discussions about educational goals, student motivations, balancing workload, etc. When Henri was with us one of the things he said that REALLY resonated with me was this – ‘When you are grading you help one student. When you plan for a class effectively you help all of your students.’ [I admit I might be mixing his words a little, but the message here was clear, spend time and energy planning for your class do not get buried in grading] What he also urged, and I saw it in our policy, was to concentrate on learning not on grades. When we did our test corrections I saw kids dig into their work, they debated with each other why something was wrong and how to fix it. They engaged with their tests when they were returned instead of simply filing them away in their backpack or locker. I truly believe that my students, my youngest ones especially, benefited for the motivation to reflect that the policy provided. In the wake of an overwhelming feeling by my department colleagues that we needed to move on from that policy, I adopted a variation for two of my classes – the two where I was the only teacher. What I did was I wrote a reassessment for every test mirroring skills as closely as I could for each problem. Students were allowed to reassess on up to three of the problems that they originally took and I would average scores from the original and the retake. I wanted to minimize time and effort on their part so that they were not mired in looking backwards while we were still on the move. I also wanted to make it more realistic that we could find time during our day to make this happen. There are all sorts of tweaks I wish I had thought of, but it felt like a good faith way to try and hold on to the benefits of reflection while providing some motivation to do so. However, the time and energy spent on some much rewriting and regrading was exhausting. I found myself getting resentful and not enough of the kids were showing the same kind of benefits I expected. I also am actively struggling with what SBG would look like in my classroom. I admit some ignorance here, but my understanding from some reading and from a workshop I attended about four years ago makes me worry that my assessment strategy would not mesh well. I cannot regularly look at a problem on a test or quiz and put it in a nice box. I tend to write problems that pull different ideas together or put an old skill in a new context. Twitter pal Julie Reuhlbach (@jreulhbach) very kindly shared a folder of assessments that she uses in her SBG approach and I am beginning to dive in and try to figure out how I can make some form of this fit my life. She also hosted on her blog site a nice post about SBG. That post is here
So, here I am with about two weeks left before the beginning of my school year. I am trying to balance what would make sense for me as a teacher in my classroom with what would work for my department and what would work in our school context as we try to figure out the path that our new leadership wants to explore. All of this needs to be framed with our students in mind, they are the point of why we are doing any of what we are doing. I have an additional ingredient in my head that becomes more and more pronounced and that is the fact that my older child is now in our high school. Factors that I had been thinking about in terms of educational philosophy are suddenly feeling more urgent and more personal.
Where do I stand this morning? I worry that many of my students are SO driven by grades and by trying to balance their commitments that they are motivated to reflect and learn more by grades than by almost anything else. They tell me this year after year by saying things like ‘I would do more homework practice if you graded homework regularly’ They say this even after acknowledging that they would learn more and do better if they practiced more regularly. Given this fact (at least I am pretty convinced it is a fact) I want to have a set of classroom practices and policies in place that take advantage of this motivation and reinforces habits in a way that leads to better learning, less stress and, hopefully, better grades so that my students feel a tangible sense of their efforts. I want policies and practices that do not increase stress and put time pressures on me and my students. I want students to feel that there is equity across their classes, not to feel like they lucked into (or were cursed by) certain teachers. I think that some sense of uniformity of expectations is kind of important. I want a coherent set of principles to be visible to my students and their parents, a way to express what I believe is important about our work together.
This month I start my 33rd year of classroom teaching. At one point in my life I thought I would have figured all of this out already. I suppose the job would be less rich and rewarding if that were true.
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.
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.
Been too long since I wrote, all sorts of reasons but none of them meaningful enough really.
I often use this space to air some thoughts and questions and I always value the conversations that ensue either here or over on twitter (where I can be found @mrdardy)
So, here is what I am pondering now and would love to hear some pushback or validation or further questions to help me organize my thoughts. For years – all 32 of them in the classroom – I have told my students that I do not believe in pop quizzes. I said that I do not want quizzes to be seen as punitive, I don’t want them waiting for me to play ‘gotcha’ with them. Similarly, I don’t do surprise HW checks or anything like that. However, I am thinking that I might have been wrong about this. I see (so often!) kids frantically studying (cramming) knowledge into their brains for a short term amount of time with the intent of performing some data dump on their quiz. I have even had students argue that they do not want me to answer any lingering questions from their classmates because they don’t want to forget before the quiz. As if 8 extra minutes will somehow erase meaningful understanding. However, the more I think back on these, the more I realize that the message being sent to me in these conversations is that there is not meaningful long-term knowledge that the students think is their job. Just be able to reply and re-present skills/techniques. I think I do a pretty decent job of asking interesting questions that encourage/allow/demand some real thinking and some really knowledge to be displayed. But if every assessment is announced and planned for and worried about, then I suspect that I am not really getting a meaningful picture of any developing understanding that my students are working on. I wonder if periodic low stakes check ins would be a better use of my time AND a more true picture of what the students are understanding. These check ins would take less time allowing us to have more time to talk/debate/discuss (heck, just BREATHE) in our time together. These would occur more frequently giving me more granular data, more of a sense of continuity in charting their understanding. They would not be a source of stress at home and they might (might!?!) send a different, more meaningful message about what my goals of assessment are. A downside is that these feel like they would be more directed at quick skills check ins rather than meaningful, complex and connected questions. those questions take more time, they might not be at home on a quick exit ticket (or entrance ticket?) type of check in. If I do enough of them – or if I build a system with some drops/mulligans – then any particular ‘bad day’ would not have much of an impact. If I am thoughtful about these and I enact Henri Picciotto’s ideas about lagging HW and think of these as lagging assessments, then the notion of a busy night for school or family activities, would not be a meaningful argument about why a particular quiz might be below par. If I lean in on this idea, I think I would move away from my current practice of quiz / quiz / test rhythm in many of my classes. I would probably feel less stressed about time taken for assessments and would feel that there was reasonable data about student performance and understanding. I have adopted a system of problem sets in two of the three courses I teach, open problems that are sometimes thorny but the students have seven school days to complete them and they are encouraged to collaborate on these assignments. This feature also helps ease the concern about grades to a certain degree.
So, I guess what I am asking dear reader are these questions –
Are unannounced assessments inherently unfair?
Are check ins on developing understanding reasonable data to register and count (in some way) as part of the report on progress that is expected at my school?
Is the habit of cramming an inherent part of the problem that we math teachers see all the time – Fragile knowledge or simple lack of ability to recall and reorganize information that has (allegedly) been learned in previous courses?
Thanks in advance for any wisdom shared here or over on twitter
So, the session I wrote about a few days ago (you can find that post here ) continues to pay dividends. Yesterday my Precalc Honors kiddos had a test. Today we were to begin discussing vectors. I had what felt like a pretty clever idea this morning. I started off by posting this image (stolen from the opening evening problem that Amy and Allyson shared with us )
I got a quick question back asking if the dots were equidistant. I confirmed and then my students began to quietly count. I encouraged them – as I always do – to chat with each other and I was hearing things about medium sized squares, big squares, etc. I suggested that some more formal classification might be helpful. A couple of kids quickly concluded that there are 30 squares to be formed. This is a correct answer under certain restrictions. unfortunately, these restrictions were not placed on the question. A student named Max said 40 out loud, then said 50. This shook up the crowd a bit and people began to dig in. However, they were hesitant to debate Max because he has a reputation (well deserved) for being pretty on point with questions like this one. SPOILER ALERT: I AM ABOUT TO UNVEIL OUR SOLUTION. IF YOU WOULD LIKE TO AVOID THAT AND THINK ABOUT IT FOR AWHILE FIRST, COME BACK LATER.
Still with me? Good, happy to have you. I went to the board and drew a square of side length sqrt(2) and got two great reactions right away. One person called this a diamond but acknowledged it is also a square. Another said we should redefine squares to avoid this. I then stepped out of the way to encourage discussion about sizes of diamonds that could be formed. We had a list on one side of the diagram listing number and size of ‘squares’ and developed a list on the other side of the number of, and size of, the different diamonds. We had some great debates about the parameters here. We decided that the only diamonds had size lengths of sqrt2, sqrt5, sqrt8, and sqrt10. We were unsatisfied with the seeming lack of a clear pattern here. You will see in the picture below how I tried to impose a little bit of order on the counting by making sure that I identified groups of diamonds or squares in numbered sets that were all perfect square integers in their count. What you will also see in the picture (coming soon, I promise!) is that I pivoted the conversation soon to vectors. My Precalc Honors kiddos took a test yesterday and we are prepared to start a new chapter on vectors. I did not particularly advertise that this was the next topic, but it felt like I could pivot in that direction. Many of the kids in this class took Geometry at our upper school with a text I wrote. In that text, I intentionally introduce some vector language early in the year. When I got to school today, I did not intend to pivot from this diagram straight into talking about vectors, but when we were discussing diamonds of length sqrt5 I realized that it was meaningful to distinguish between a horizontal change of 2 with a vertical change of 1 versus a horizontal change of 1 and a vertical change of 2. Time for the photo now and then a little more explanation.
So, in the photo above, a bit of glare there unfortunately, you see a green side of delta x = 1 and delta y = 2. I drew an arrowhead and one student muttered ‘vectors!’ It felt like such a natural trigger and frame to discuss vector notation. Almost instantly kids were discussing magnitude, direction, remembering notation, etc. Man, it was a great way to start the day!
I ended up sharing this problem with a couple of other classes during the day and each time I confessed that my partner and I only found 45 squares and were VERY confident of our answer. Each class figured out where we had gone wrong and they seemed pretty proud that we worked through this all together.
Another opportunity here to thanks Amy and Allyson for the great PD session and I know that I will be pulling some other tricks out of the bag of tools that they provided for us last week.
Back in November we were having conversations at our school about improving our ability to place new students in our curriculum. Every year we have a wave of brand new students who have to move down from our original placement suggestions and it is always a frustrating thing for them and for us. So, I did what I do and I reached out to a number of department chairs at schools like ours seeking advice. One of the people I reached out to was Amy Hand. She is the math chair at Packer Collegiate Institute in Brooklyn, NY. In addition to sharing some wisdom, she also mentioned a workshop she was putting together. Here is the flyer she sent me –
I was immediately intrigued and I went to my admin to pitch the idea. We decided that we could afford to send a handful of folks together there and we ended up inviting our two Geometry teachers and one of our middle school teachers to go with me. I have been on the inquiry bandwagon for awhile but I knew that I could learn some new wrinkles to add to my game. I was excited to bring along three colleagues to listen to someone other than me pitch this idea. I also knew that the power of being in a room together working side by side with colleagues is always a powerful thing. Well, we returned Friday afternoon and I have had a couple of days to let some of the ideas sink in – as well as a couple of days to get caught back up with my life here. I am happy to spend a little time here telling you about what a wonderful experience this was. I feel that there is some positive energy that can help move our department forward in examining how to open up our classroom culture to encourage more open inquiry from and for our students.
Amy and Allyson Rohrbach, a colleague of hers from Packer, put together a really meaningful and packed two days for us. We started with a short introductory session on Wednesday night. This seemed like an unusual idea, but it was a great way to start. We had what seemed like a completely innocent problem on our tables. It was a problem from Brilliant that involved counting squares. I wish I had the image of this problem, perhaps I can find it soon to share. What seemed like a completely innocent problem instead became a lovely conversation about counting procedures with different folks going to poster paper at the front of the room to share their strategies. We got to know each other over snacks and beverages and discussing math ideas, Amy and Allyson framed our work efficiently that night and I think that we all left the room that evening energized for our work the next day. Thursday was the heavy lifting day but even that was paced really well and Any and Allyson kept us shifting gears so we did not feel like we were sitting with one idea or one problem for too long. There was plenty of space to explore and I don’t think that any of us felt rushed. One of the problems we worked on is one that Amy and Sam Shah worked on together at Packer and Sam blogged about that problem here I know that the next time I am teaching Precalculus I will be framing our discussion through this problem. I feel certain that I read Sam’s post when it came out, but working side by side with folks in this environment brought the problem and the pedagogy behind it to life in SUCH a meaningful way.
Again, we had GREAT conversations discussing/debating/explaining ourselves. I certainly have fun listening to my students debate like this but there is a different level of fun when I can get lost in the math myself to this degree. It is also energizing to hear from and share with people that I have never met before. There is such a sense of open curiosity in a carefully designed environment like the one that Amy and Allyson helped create.
I walked away from this event convinced that I will have an ongoing exchange of ideas with Amy and Allyson and I am already discussing a school visit to bring some of my colleagues who did not make it to this event. You can follow Amy and Allyson over on twitter where they are @MathSenseLLC. you can also check out their next workshop which is described here They will be on the west coast for this trip. If this is more in your neighborhood and you want to be recharged in your commitment to inquiry driven education or if you want to be nudged in this direction, I cannot recommend this highly enough.
This year I am teaching Precalculus Honors at my school (in addition to two different levels of Calculus) and I have not taught this course since the 2010 – 2011 school year. Last weekend, as I was planning ahead a bit, I realized that trig angle addition identities were coming our way. I have to admit that I have been entirely unsatisfied with how I dealt with this in the past. Most texts have some sort of distance formula based derivation of the formulas and I have read through them over and over never really satisfied that I could add much to the presentation. I generally presented these as facts and tested out a handful of examples to see that the formulas verified what we already knew to be true from the unit circle. A pretty unsatisfying situation. So, I did what I do. I sent out a call to twitter for help and got the typical handful of helpful responses. One really stood out and I tried it out in three of my classes. Tim Brzezinski (@dynamic_math) sent me a link to one of his lovely GeoGebra explorations. You can find that link here I am including a screenshot below to help you understand what we were able to accomplish due to Tim’s clever design (and his endless willingness to share!!!)
The students are presented with the above image and the very simple facts that this is a rectangle and that the two yellow triangles are similar. The point on the right side of the triangle is movable. A few things right off the bat struck me as wonderful here. We talked about WHY we could know that the yellow triangles were similar. So, we had the opportunity to remember the AA postulate. A student in one of my classes knew that the upper angle is alpha + beta because it is the alternate interior angle of the lower left corner angle. Super sweet! I was going to present a boring conversation about 90 – alpha and 90 – beta on the top. So, I liked that aspect right off. I also LOVED the aspect of how open this construction is AND the fact that it was not at all obvious to my students what we were about to discover. Pretty cool.
I ran this first for my Precalculus Honors kiddos and had each small group discuss where the wages need to go then we put our thoughts together. After a (very) gentle reminder of the structural properties of rectangles, we realized that we had discovered the angle addition formulas for cosine and for sine. An interesting response followed. One of my more curious and driven students asked ‘Don’t we have to prove this?’ I think that this speaks volumes about the natural response to the idea of ‘proof’ in our students. This exercise seemed clear and concise. Couldn’t qualify as a proof, right? Now, I am not fooling myself here. There is still a great deal of simply committing these formulas to memory at the end of the day. But I am convinced (CONVINCED!) that this feels more meaningful now. My kids were able to see and derive for themselves these relationships. They stopped and thought about similarity, about ratio definitions for the cosine and sine functions, and about the structural requirements of calling something a rectangle. I went on to tell them that they do not need to commit to memory double angle formulas because they come straight from here. Most students don’t take my advice on things like this, they feel safer simply consuming memory space with formula after formula, but that is another issue entirely.
After this went SO well with my precalc honors kiddos I unveiled it in my Calculus Honors class. We were just getting to the point where we were dealing with derivatives of trig functions and I knew that the chain rule was about to be laid on top of this. I guessed that this would be a great exercise to jog their dormant trig memories from last year. Again, in each section of Calc Honors, small group conversations led directly to sharing of ideas and a quick dissection of the diagram. I am pretty sure that these conversations woke up some sleeping facts in their brains and I hope it pays off in the form of quicker recall and comfort when we lay the chain rule on top of the standard trig derivatives soon.
Many thanks to Tim and to all the others who shared out ideas when I sent out my call for help. My students don’t really understand how much better their education in my room is due to the network of supportive, smart, creative folks out there. I do make an explicit point of telling them when I am using ideas/activities from others to help make all of this clear. The subtext I hope sinks in is this – If you have an interesting questions, send it out to the world. You’ll get some interesting feedback.
Last week – I know, it’s taken too long to write about this – my Precalculus Honors class started the day with a brief quiz. One of my PCH students named Max finished the quiz early and started sketching on his scrap paper. He showed me a diagram like this:
He described the problem this way – I have a square and a quarter circle coming across it. I also have a circle inscribed in the square. What is the area of these little regions? (I clumsily sketched in those regions on GeoGebra)
Well, it turns out the the topic of the day in AP Calculus BC that day was to be trigonometric substitution for integrals and this problem would be a lovely introduction to the need for this skill. AP BC was meeting for the 90 minute block and I decided that I would introduce Max’s problem, spend about ten minutes dissecting what we could and then hit a bit of a wall where I would introduce this new skill. I was pretty proud of myself and feeling very fortunate that Max thought of this question. Well, as we all know, life doesn’t always work out the way we want it to in school. I presented this problem and told them that it came up in Precalculus Honors. My BC kiddos started dissecting it right away. They concentrated on the lower left corner, they decided we should agree to a side length for the square and off they went. We decided the square should have a side length of 2 so the inscribed circle would have a radius of 1. Avoiding fractions until we HAVE to deal with them is a good plan in general, right? So, the lower region is 1/4 of the difference between the inscribed circle’s area of pi and the square’s area of 4. Good start. Next we convinced ourselves that the two remaining squiggly areas are congruent. It would have been nice if we could drop a line from the point of intersection to divide that region in two but it’s not symmetric. The different radii of the circles intersecting prevents that from being true. So, here is where I figured I would introduce this new technique. I mentioned this idea but the feeling in the room was that we should be able to answer this question using tools that a precalc student should be able to use. I was sitting in the back of the room at this point with my laptop on and a GeoGebra sketch projected on the front wall. Ideas and questions started flowing and students asked for a Desmos sketch like the one below:
Jake proposed this and felt that the added symmetries would be helpful in discussing this problem. I asked if anyone wanted to see a point of intersection identified and we did at first but then erased that point from the conversation. We are about 20 minutes into our 90 minute class now and probably at least 5 minutes behind where I wanted to be but the energy in the room was pretty incredible. Students started going up to different boards and sketching ideas. They asked for paper printouts of the demos sketch and started moving from small table group to table group. People were debating and correcting each other and I just sat there. I was listening, I was tossing out questions, but mostly I was just watching this all unfold. The students were dusting off old trig ideas and old geometry ideas. They were debating the need/desire to have the decimal guess of the point of intersection. One student, Nick, was determined to think about this in terms of proportions and he drew a lovely argument that the area would end up being around 10% of the whole square. His classmates were unconvinced and he argued his point two or three different ways. One student, Colin, broke the region into circular arcs and argued about finding the area of a central angle. He had a great drawing but I did not capture it on my iPad. This conversation kept rambling on over the course of our allotted 90 minutes together. I proposed a couple of times that I could give them a new calculus tool but they kept waiving me off. Noon rolled around and I told them they could go to lunch. Many of them did, kind of exhausted by all of this at that point. One group of three – Nancy, Andy, and Michael – were fired up at this point and were sure that Colin had made some small mistake in his sketch. They produced this –
So, this sketch is pretty impressive in its detail but, more importantly, this sketch happened about 20 minutes after lunch began and after I excused myself to run an errand during lunch. During the 90 minute class, my colleague David from across the hall wandered in a couple of times asking kids to explain what they were doing. He told me that Nancy, Andy, and Michael worked for at least a half an hour of they hour long lunch debating this problem. The other thing that happened while I was gone was that Andy, Kelly, and Michael had modified my Desmos sketch on my laptop pursuing their idea. Their modification is here –
I was feeling pretty great about their perseverance, their engagement, and the amount of geometry and trig that was being remembered in the service of this curious problem proposed by one of my students. I was also more than happy to amend this week’s Calc test by taking off the one problem that relied on the trig substitution technique. I had one more class after lunch (one of my Honors Calculus sections) so I sadly erased some of the work on the board and I described the problem to that group. Some of them had already heard about it during lunch! My BC kiddos were still talking about it even after they left. At the end of the day one of our Differential Equations students wandered into my room. He said ‘I heard there was a good problem today.’ He, Owen, then proceeded to discuss the problem with Andy and Nancy who had come back to the room to discuss this. Owen dove in to the problem debating with Andy and Kelly and he produced these sketches – (the first one got rotated in translation)
I tweeted the problem out, like I do, and a former student jumped in and offered this sketch –
Another colleague, Adam, came by when he overheard this conversation and he attacked the problem using Google sketch up to find the ratio that Nick wanted – it was smaller than his proposed 10% neighborhood.
There is no real ending to this story, the weekend came, life moved on. On Monday my BC class was more focused on asking questions about this week’s test. My Precalc Honors kids were impressed by my enthusiasm in talking about all of this but they did not share Max’s curiosity about the question. I went home feeling pretty great about the sense of play and sense of curiosity of many of my students and my colleagues. While I cannot let everyday roll this way, I need (NEED!) to make sure to create spaces where this kind of magic can happen. I think almost all of the credit for this adventure lies with my students who are interested, motivated, curious, and persistent. I hope that I have helped them along by modeling curiosity and by being willing to let this kind of free range play happen in class.