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.

Thinking Out Loud

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

A Residue of Professional Development

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 simply asked ‘How many squares can be formed?’

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.

The end result after our launch into vector conversations. Note that diamond count is written as 4 + 4 and 1 + 1 for different sizes. Trying to focus on perfect square counts there!

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.

Meaningful Professional Development

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.

Trig Identities

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 front page of Tim’s GeoGebratube link

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.

A Fun Rabbit Hole

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. 


Debating Divisibility

In our Precalc Honors class we are discussing exponential and logarithmic functions now. I want to relate a fun observation/suggestion from a student a few days ago and a debate that fired up in class today.

Our text defines exponential functions as any function of the form y = a*b^x as long as b is positive and not equal to 1. One of my students, a girl named Shailee, suggested that it would feel more logical to simply say that b is greater than 1. This way, functions with a base between one and zero would instead have a negative exponent. This might make it more consistent to think about positive exponents representing growth and negative exponents representing decay. This also feels like a smoother definition for b instead of having two qualifiers, we’d only have one. Kind of a nice suggestion and one that I will be adopting for our class conversations this year.

Today I ran an activity that was suggested by Henri Picciotto when he came to do a workshop with my department in May of 2016. I had a couple of containers of 10 sided dice. They were numbered 0  through 9. I assigned a rule for each of my three groups. One group was to roll all the dice and count the number of evens. They then dispensed any that were not and rolled again. Lather, rinse, and repeat. The idea was that the number of dice remaining should model a half-life for them. The second group was looking for primes. Again, exponential decay with a base this time of 4/10. The last group was asked to look for multiples of 3. Someone asked if 0 counts as a multiple of 3. I reflexively said no but then paused and thought out loud about it. I threw the question out to twitter and we went on our merry way. We gathered data, plotted it on Desmos in a table and asked fro regression equations of the form y = a*b^x. Worked pretty well except one group went from 40 something dice down to something like 8 right away when they were supposed to have a 1/3 chance. We then checked in on twitter where interesting things were being shared. I’ll clip a few tweets below:

A side conversation also occurred when Christopher suggested that the 0 on the die was a 10 not actually a 0. This, of course, would have prevented this whole interesting conversation from happening in the first place. Anyways, this got a heated debate going in class where my students just felt uncomfortable about the idea of 3 being a factor of 0 since this implies, by a simple extension, that EVERY integer is a factor of 0. I guess we all accept without much debate that 1 is a factor of every integer, but this feels off somehow. I went off to lunch to bounce this idea off of some folks and I might have scared a couple of colleagues who are less comfortable with math. A lively debate/discussion at lunch led one colleague to casually say ‘So much just happened there’ When I returned to my classroom and my twitter feed the conversation had moved into a modular arithmetic mode. Here is a taste:

So, let me first say what an honor and a treat it is to share in a conversation like this with my students, my colleagues, and my virtual faculty lounge of folks spread around the globe. It is a mind-blowing thing to think about how much this world of education has changed for me since I took the plunge to going twitter. I am convinced (CONVINCED!!!) that life is better for my students since I did. I also want to say that the idea of modular arithmetic is one that I love to share with my students and I am determined to figure out how to find time to do so with my precalculus students since this debate brought up these ideas. I also have to admit that I am just a tad uncomfortable saying that every integer is a factor of 0. One of the side conversations at lunch went like this : Me – If 0 is a multiple of 3 then that means that 3 is a factor of 0. Rachel (science dept colleague) – If we say 3 is a factor  of 0 wouldn’t we say that 0 is a factor of 0? Me – Uhmmm, this would imply that zero divided by zero is a thing, right? This reminds me of debates I had with a friend from my old college town debating the physical meaning of 0 ^ 0

So, a delightful lunch time conversation, right? Fun to lift the curtain a bit and have my students see a debate unfolding. Fun to get my brain agitated thinking about all of the implications of saying something as simple to my kids as ‘Look for multiples of 3’. Probably a lesson to think a little more carefully about my directions to them!

 

Many many thanks to Henri, Sam, Christopher, David, and Bryan for engaging in this conversation and for giving me the idea of this experiment.

Another Great Student Observation

Yesterday in Calc BC we were discussing L’Hopital’s Rule. I have mixed feelings about this conversation every year because it feels like this powerful idea about comparing rates of change, but I know many (most?) see this as a simple mechanical process. Of course, there are always cases where it gets misapplied just because it is a fun new tool. What is that old saying about a hammer? If all you have is a hammer, then everything looks like a nail, right? Suddenly, every rational function limit should be subjected to L’Hopital. A couple of years ago, I finally settled on a way that feels comfortable to start with the old classic limit of sin x / x as x approaches 0. GeoGebra is a good friend here.   

We all agree pretty quickly that it is clear that there is a limit here, despite the fact that the idea of dividing 0 by 0 causes us discomfort. The breakthrough I had a couple of years ago (and I am guessing many of you have had it as well) is to next look at the graphs separately. 

So, the conversation now centers on the fact that we can only see one graph as our eyes near the origin. An interesting debate arose. Students were convinced that these graphs clearly intersected each other, no question there. What became a bit fo a debate was whether there might be multiple points of intersection. I had the idea of creating a new function. If we look at the graph of y = sin x – x and see that there is simply one root we can put the question to rest. Before I could suggest this I was trying to get someone to voice my idea. Instead, one of my students (Jake S) made a contribution to the conversation. He said he was thinking about the inverse sine graph. He pointed out that if we snip off a piece of the sine graph we were looking at from the minimum on the left of the y-axis to the first Max on the right of the axis we would have a function that was constantly increasing. Since it started off above the line (which is moving at a constant rate) and then ends up below it, it seemed clear to him that there was only going to be one point of intersection. I was kind of floored by the connections he was making on the fly. I do not expect my students to think of inverse trig functions without being directly urged to and even then there is great reluctance. I did not expect an argument that had this kind of subtlety on the fly in the class conversation. I have not thought deeply enough about his argument to determine whether there might be some odd functions that defy his thinking, but this graph sure did not.

I say this on this space periodically and I say this in my life to anyone willing to listen on a regular basis – I am spoiled by the kids in this BC class. I don’t mean to imply that I do not enjoy my other classes. I do. I’m just generally spoiled (fortunate / blessed / lucky / fill in the blank) I get a kick out of all of my classes at different times for different reasons. What I reliably get from teaching the scholars in BC is a daily reminder that I still have an awful lot of math to learn. I know that I know things that they do not yet, but I also know that they are coming to class with questions and insights every day that blow my mind.

New Ways of Thinking About Calculus

I have been teaching a long time now, so when students get me to think about a new way of doing something I am always excited. A super brief post here highlighting two solution techniques suggested to me by my AP Calculus BC students in the past week or so.

We are studying inverse functions and the relationships between their derivatives. We had settled on the fact that the function y = e^x is the function that is its own derivative. We also knew how to differentiate y = ln x based on this fact about the exponential function. I asked about the derivative of the function y = 5^x. I intended to derive a pattern for this derivative using the fact that we had derived to deal with natural log functions. Instead, one of my students suggested that we should think about the e^x function and the chain rule rather than develop a new rule. He pointed out that we know that e^x will eventually be equal to 5 so 5^x is simply a new power of e. He suggested that I write 5 = e^u and then differentiate the function y = e^(ux). I was delighted by this. Rather than add new rules to remember, simply this and rely on derived facts. I will always encourage my student any time that they can whittle down the number of rules to remember. Super excited by this.

On Thursday, I had a quiz in that class and one of the questions involved a rational function. I asked my students to verify that this function had an inverse that was also a function. I expected them to take the derivative and show that it never changed sign. I expected that because that is the way that I have thought about it and because I have taught them to approach this question through this lens. One of my students instead said that if f(a) = f(b) for some values where a and b were unequal, then that function is not one to one. He solved this equation showing that it was only true if a = b and concluded therefore that the function is one to one. Delightful! I tweeted about this and one of the responses congratulated him for relying on a definition instead of a technique. I applaud this as well and I hope to remember this well enough to present it as an alternative approach to answering this question.

Super proud of my students and I love that I get to make a big deal about the fact that I am learning from them as well as them learning from me. The big message, of course, will be that we should all be learning from each other.

How Do We Help our Students Ask Questions?

I have told this story to a number of friends and colleagues over the years. One of my favorite former students – he graduated in 1994 – gave me what I value as the best compliment I think I ever received about my teaching. He was a brilliant kid, school seemed effortless to him. I taught him for four years in a row culminating in AP Calculus BC when he was a junior. He took a math class at UF his senior year. About ten years ago I was living in Jersey and he was living in NYC so I had the chance to see him a few times there. Once we were having lunch and he told me a story. He worked in a small business doing financial analysis and he was frustrated by a problem he had been working on. He told his boss that he was going to take a long lunch to clear his head. When he came back his boss had left some notes for him on the file he had been working on. He told me that his boss reminded him of me. “He asks questions that I would not think of asking myself.” I walked away so happy about this. He did not remember a trig graph or a derivative or the fundamental theorem of calculus (although he probably did remember these things!), instead he remembered that I asked him questions that he did not think of asking himself. I felt SO good after that conversation. I was thinking of it today after school during our afternoon conference time built in to our day. All of our teachers are expected to be in our rooms for about a half hour after the end of the school day and many students make a habit of dropping by to ask questions. I was talking to two of my  Calculus Honors kiddos. This is our non-AP class that takes a deep year-long dive into Differential Calculus. We were looking at some problems on one of the problem sets I wrote and these two young women were saying that they understood the problems when we talked in class but they did not know how to start them on their own. I pointed out that almost all of the ideas in class came from the students, I rarely flat out TELL them how to solve a problem, we work through the question together. What I work really hard on is to ask questions of the students that prompt them to see connections and realize what they know about the problem. I want them to feel the power of being the ones who generate the answer. One of the girls said that she does not know what questions to ask herself when she is home working on these problems. So, the challenge is to figure out how to help her, and others, across that bridge. Is it enough to simply model an inquisitive mindset? Is it enough to be a good role model in persistently asking questions? How can I explicitly help my students develop that instinct and ability to push themselves along a solution path by asking meaningful questions? I would love to hear any wisdom on this front. I am going to share a meaningful quote that I ran across in my days as a doctoral student studying curriculum and instruction:

Genuine enquiry is an important state for students to recognize and internalize as socially valid. Consequently it is an important state for teachers to enact. But it is difficult to enquire genuinely about the answer to problems or tasks which have well-known answers and have been used every year. However, it is possible to be genuinely interested in how students are thinking, in what they are attending to, in what they are stressing (and consequently ignoring). Thus it is almost always possible to ask genuine questions of students, to engage with them, and to display intelligent directed enquiry. For if students are never in the presence of genuine enquiry, but always in the presence of experts who know all the answers, then students are likely to form the impression that there is an enormous amount to know, and that experts already know it all, when what society wants (or claims to want) is that each individual learn to enquire, weigh up, to analyse, to conjecture, and to draw and justify conclusions.

 

John Mason