TMC14 – A Newbie’s Reflections Part 1

I am currently sitting quietly in the Detroit airport and I’ll be here all night. Weather foiled my return home today after 4 pretty terrific days at twittermathcamp2014

I have quite a bit that I want to sort out about this event and I’ll do so over the next few days in a couple of parts.

 

A number of people have already written and many more will about their experiences. There will be posts talking about the transformative effects of this working/learning/playing experience if history is any guide. A number of thoughtful posts have already been made. A couple of them are really touching and reflect insecurity (see here by @MrKent800, see here by @lmhenry9). Glancing at my neglected digg reader I see that there are posts waiting to be read from @pamjwilson over at her blog The Radical Rational, @algebrasfriend over at her blog and others as well. My tweet deck is hopping and I am paying closer attention to much of it because I can place some human names and faces to the tweets and to the quick chatter between folks. I’m going to try and capture my days in Jenks, OK and I hope I am fairly thorough in recognizing and thanking some folks who have been pretty important in this experience. Before I embark, it’s important – at least to me – to note that before arriving in the Tulsa airport Weds shortly before noon that I have physically met only one person who will be at the conference. That person is the charming @JustinAion who wrote a daily blog (honestly, he wrote every school day!!!) over here. I’m pretty outgoing when I am comfortable in a situation. I was worried about whether I’d hit that comfort zone at all in my time in OK on this trip.

In advance of the trip I, and many others, had filled in information in a google doc about our arrival and departure plans and folks started volunteering to take people from the airport to the hotel. Pam (@pamjwilson who blogs here) volunteered to take me to the hotel in her rental car. We had a lovely chat at the airport waiting for our car, ran into another group who we followed to the hotel. This trip was a bit of an adventure as our ‘guides’ made two quick reversals during the trip. We got to the hotel a bit early before rooms were supposed to be available, but we lucked out and got into our rooms pretty quickly. Hungry from our travels Pam and I ate lunch with two of the people from the car we followed. I’m disappointed in myself that I do not remember their names.

A nice nap back at the hotel before heading to a gathering room in the back of the hotel where people were chatting, snacking, playing games, etc. The name tags that we would wear the rest of the week were at the host school so we were introducing ourselves mostly just by first names.  Now, remember that I have never met any of these folks in person – other than Justin who brought piles of games to share so I’m feeling a bit overwhelmed here. I chat for awhile and enjoy myself. I briefly meet Tina (@crstn85 who blogs here) – this was a major goal of mine for this first night. Tina is the reason I came to TMC. She invited me to help her organize and run the Precalculus morning session. She and I had exchanged some comments and emails about her amazing Nix The Tricks project and I was (and am) totally flattered that she asked me to join in. We chatted very briefly but I felt better actually meeting her before our work commenced on Thursday morning. After bouncing around a bit I retire kind of early. I could not help feeling a little bit like an outsider. It’s a feeling that is hard for me to shake during the week. This is mostly on me, by the way. All week long folks were welcoming and friendly, but being brand new to this community in the physical sense and being relatively new to the community in the virtual world as well, it was hard for me not to notice the deep bonds between many of the folks here.

Sleep calls to me now. Although it’ll be sleep on the floor in the Detroit airport, I need to succumb. More thoughts soon on the week’s activities – both in the context of the day’s formal activities as well as on the informal social life of the twittermathcamp experience.

Baseball and my Math Classes

I’ve been sitting on this post for a few days trying to find the time to organize my thoughts. This post is motivated by this recent post by Michael Pershan over at Rational Expressions, by my son’s recent little league season, and by multiple recent conversations with a good friend and colleague who is our drama director. I hope that I can make it all make some sense.

Michael talks about what feedback looks like and sounds like in different contexts. He offers these two examples of how a baseball coach might talk to a player who is struggling with his hitting. 

“Tommy, you haven’t been hitting as well as you could’ve lately, amiright?”

 

“Each time you swung and missed, you raised your head as you swung so you didn’t really have your eye on the ball. On the one you hit hard, you kept your head down and saw the ball.”

I think that any of us (teachers, coaches, parents, people who have lived for a little while…) would agree that the second remark is more helpful. [This reminds me of a former student of mine who was a basketball player. At halftime of a game where his team was being routed his coach was blistering the team questioning their will to win etc. He raised his hand and said ‘Could you tell us what we’re doing wrong and how to correct it?’ Unfortunately, his fired up coach did not take kindly to this interaction.] After I read this – and, again, I’d urge you to read Michael’s whole thoughtful post – I started thinking about the language I use when speaking to my students and when writing to them on their work. I also thought about my son’s recent experiences.

 

I watched my son struggle through his inaugural little league season with good cheer. He just turned 11 last month and had never expressed interest in baseball – much to my chagrin as it is my favorite sport. This spring he announced that he wanted to join the local little league. Luckily, they have a level for kids aged 7 – 10 with a variety of experience levels and they do a nice job balancing the teams. He was on a team with fantastic, supportive coaches. They were consistent, they were enthusiastic, they were patient. When a player made a mistake they pointed it out to him/her right away but they focused on what should have been done rather than simply criticize the mistake. They regularly referred back to practices and reminded the players how they were taught to play. I have seen my son give up on tasks at home and tasks from school in the face of frustration. Despite only getting three hits all season (but two of them resulted in RBIs!) I never saw him frustrated and he never gave up. I think that a big part of the reason why is that he was part of a team and felt that he was responsible to the team and his teammates. The fact that the coaches helped create an atmosphere of positive energy was a huge factor. The fact that the team won all but two of their games made a big difference as well.

 

I have had a number of conversations with my friend Jason who is a drama and English teacher at my school. We have talked about how different perceptions are at school about certain students. I think that all of us who have taught for some time are familiar with the fact that some kids ‘click’ better with certain teachers or subjects. But what we have talked about goes beyond that. I have noticed that there will be students who routinely struggle with follow through and commitment in their academic classes but will be praised for their persistence and determination by coaches and/or drama or music teachers. There are students who won’t do homework, they’ll bomb some tests due to lack of preparation, they might skip a class or an assessment but some of these same kids (generally) do NOT miss practices, they learn their lines or their parts for the symphonies, they stay extra time for practices or they regularly get there early. I know that there are all sorts of factors at play here. But I think that the two biggest ones are these: (1) Few kids CHOOSE to talk Algebra II or Chemistry or US History. They are required to take these (and others, these are just some examples) classes. They have made a conscious choice to play basketball or lacrosse or to play cello in the symphony or to try out for a role in the school play. It’s easier to be committed to something you choose to do. (2) In all of these activities a failure to carry out your job has a direct impact on the chance for others to succeed. If I failed an Algebra II test in school no one else was going to suffer, it was just my failure. If I run the wrong route for a pass play in football the entire team might suffer the consequences if there is an interception. If I flub my lines on stage it has an impact on everyone’s performance. If I don’t learn my part in the string ensemble the performance suffers for the players AND for the audience. 

I think that this second aspect is the most important one. This is where I finally try to get to my big point in thinking about this blog post and sharing it out to the world.

How do we create a team atmosphere in our classrooms? How can we encourage our students to feel responsible to each other and pull all together in the same direction? At my last school I had furniture that was easily manipulable and had my room set up in ‘pods’ of three or four desks put together. My students there took great pride in their pod performing well on assessments. I had a few ‘pod’ quizzes and they worked hard together on those. My current classroom is set up in two long tables that seat ten students each. The students interact,they share ideas and when I give them time to work together they do so well. But this does not create any sense of accountability to each other. Perhaps I’m dreaming and hoping for something that just doesn’t happen inside the classroom walls. I don’t think that I am imagining a grading system where their grades depend on each other in any way (although Dan Kennedy wrote a really important essay where he proposes something slightly along those lines as part of a much more complex argument about assessment) since I know how delicate the relationship between learning and grading can be. I look to the hive mind of the internet for inspiration and suggestions. How have you been able to foster an environment in your classroom where students support each other’s learning and where students feel that they are part of a team where their contributions really matter?

 

 

Moving and my math classes

Haven’t blogged in quite a while now. School year ended and most of my energy has been consumed by writing (we’re doing our own Geom text here at my school!) and by preparing to move. The apartment moving adventure brought me (over and over again) to our local Lowe’s hardware store. The trips to Lowe’s got me thinking about my classroom – especially about the Geometry classes I’ll have in the fall. Let me try and make sense of why.

When he was a young man my dad worked as a carpenter for Macy’s in NY. I inherited NONE of these skills. I am lucky that my wife seems to find me handsome since I am certainly not handy at all. I have had the occasion to go looking for shims, looking for shelf pins, buying paint, etc. What I have noticed is that I have NO ability to understand the basic architecture of this store. I know that there are clues about where to find certain items but I cannot decode them. This got me to thinking about my students as they try to navigate their studies. I made myself go and seek help. I was confident enough to be able to describe what I needed and in some cases brought evidence with me of what I needed to buy. In some cases, as with the purchase of paint, I was asked questions that I was not prepared to answer. Since I knew that I needed to paint to make my children happy about their new rooms, I was determined to make sure that I completed my purchase. When I was asked questions about what kind of finish I needed I was able to simply admit my near total ignorance and simply take the advice of the salesperson. This did not make me feel good about myself, but I knew it was important enough to make myself suffer through this relatively minor indignity. I started thinking about my students and how difficult this situation is for a person’s ego. The difference between my level of knowledge and the level of expertise that the workers at Lowe’s possess created an uncomfortable tension for me. Not because of their behavior, but simply because I felt bad about my lack of ability to process the information expected of me. How different is this from the situation that many of our students find themselves in every day? I would say that there are a few big differences.  I knew I needed to work through this to make my move possible. I am not convinced that my students are always able to impress upon themselves the importance of the struggles they are experiencing. Part of that becomes my responsibility to convince them of this. The more important difference as I see it is the fact that I have no continuing relationship with the workers at Lowe’s. I don’t have to face up over and over to my lack of ability to process what seems natural and obvious to them. My students have to see me every day for 9 months or so. It is easier for them not to admit when they don’t understand. It’s easier for them not to ask for help in navigating a mysterious terrain. 

When i start up again in the fall I need to remind myself of the feelings I had at Lowe’s – especially with my younger students in Geometry. It was important for me to be reminded of how uncomfortable it is to be confused. It was important for me to revisit that feeling. I will be processing that this summer and try to make sure that I can create both a sense of the importance of our academic mission for my students and a sense that it is okay to admit when the fog of confusion settles in. I need to be aware of helping my students understand the structural architecture of their studies so that they can help themselves more than I was able to help myself in Lowe’s.

Bragging About My Students

Two things I want to share tonight. One of them has multiple parts.

 

One of my international students shared a lovely gift with me yesterday. It’s a food treat that her mom sent here for her to share. I have a few food allergies so I was concerned but did not want to tell her because it felt rude. Luckily, there are a number of boys in my from who can translate the ingredient list for me. Pretty cool. Oh yeah, I’m allowed to eat it – no nuts.

 

I blogged a couple of days ago about a problem on my calc BC final. Here is the problem

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:

 

“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”

 

 Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty.

Twenty-four students took this final and a number of them really did a wonderful job in explaining their reasoning. I’m going to present a handful of the best responses here.

  1.  4 x 5 = 12  is number base 18. 4 x 6 = 13 is number base 21. 4 x 7 =   she will say 14 in number base 24.   4 x 12 = 19 number base 39, 4 x 13 = 20? number base 42. If following the pattern, as one of the numbers remains constant 4, another increasing by 1 each time, we get the product increasing by 1 each also. This is possible for number base 18 , increasing 3 each time. This 4 x 13 = 20 in number base 42 accordingly. However, 4 x 13 in number base 42 is one full round with another 10, which in base 42 as there exists another symbol for that suppose A, thus 4 x 13 = 1A and will never equal to 20 this way.
  2. She is using number base to calculate it. Every time 4 times initial number plus 1 and number base that is increased by 3. It will never get to twenty because the number base is always growing as number grows. Number base is growing faster than the number we multiply by. (Note – this one was accompanied by meaningful, but scribbly, calculations)
  3. (This answer starts with all the calculations hinted at in the first answer i presented)  Because the base in continually increasing by 3 and the answer is only increasing by one, the answer will never be able to get out of the ones digit.
  4. The base is increasing and in order to get to 20 the result of the calculation must be EXACTLY twice as big as the base., which is not possible.

 

All of these were accompanied by calculations on the side. We spent about two and a half days talking about number bases and I must admit I was really impressed by the patience that my students had with this problem. Nice way to end the year!

 

Thinking About How to End a Year

So, this morning both of my AP classes took their final exams. I have some questions for the world about this process, but first I want to share something fun from the Calculus BC final. You should know that my students understand that the word fun, when I use it in class, means that a problem is challenging, thought-provoking, unusual, or some other words that they might use but I won’t type. I believe that I mentioned that we ended the year after the AP test with a quick tour of some interesting topics that many high school students don’t get to see. I included a small unit on different number bases. I always start this by writing a series of addition and multiplication facts on the board. However, they don’t know that these facts are base 8 number facts. I usually reveal the secret by showing a picture of Lisa Simpson and tell them that these facts are how Lisa would compute. It’s a fun conversation to have. So, for our final exam I told my students that there would be ten problems. Nine of them would be Calculus problems taken from old tests. they have all of their old tests (all ten of them) so they could be very well prepared for that. I also told them that one problem would come from our last two weeks. After discussing this with my friend Richard – a former math teacher – he sent me the following passage from Alice in Wonderland

Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!    

I played around with this for a while and fell in love with this as a final problem for the year. This is how I presented it to my students:

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:

“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”

            Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty. 

 

I will sink my teeth into grading finals tomorrow since I am on dorm duty tonight. I did browse through four or five of the test as they were turned in and two students really nailed the problem and provided beautiful, detailed answers explaining the pattern. I won’t spoil it here, I’ll let you work through it if you wish to do so.

 

I don’t know how your school works, I do know a bit about the four schools where I have worked. All of them have been independent schools that emphasize the idea that we are college preparatory schools. Each school I have worked at has had a statement in their handbook about the importance and significance of final exams as a college preparatory experience. However, I know that tonight many of the seniors in the dorm will tell me that they do not have any finals left. Today was the first of three and a half days of final exams and many seniors won’t have any more after today. There is a pretty common feeling that final exams will not be pretty and are of questionable usefulness with our seniors who are days away from graduation. This is not just a feeling at my current school. But I really wrestle with this. We say we believe that taking a final exam, preparing and organizing a large body of information for a one-day thorough examination, is a useful skill AND one that is important for college. However, it is those students who are closest to college who are the most likely to have been excused from a final exam. In some classes the final experience is a paper or a presentation that happened last week. But our school, and others where I have worked, carve out quite a bit of time for final exam administration. I wonder whether we could use our time in a more meaningful way. I wonder whether the idea of a final exam makes sense only in certain disciplines or for certain age levels. Is it reasonable for us to ask our freshmen to take exams under the same circumstances that we ask of our juniors and (sometimes) seniors? I don’t see AP scores for my students until July. I like the idea of some capstone where I check in with them in one last, broad examination of ideas. I feel pretty old-school in that regard.

I want to make my assessments meaningful for me and for my students. I am really beginning to doubt whether exam week is such a positive way to do this. I would love to hear from others about how they deal with this question. Are many of you bound to a policy that your school or your department has mandated? I want to be smarter about this and I’d love your help.

Professional Growth in a Connected Age

I’ve been teaching for a long time now. It makes me feel old when I realize I am in my 27th year in the classroom now. I joke with my students that I have been teaching longer than any of them have been alive. When I started teaching the predominant models of professional development were the inservice days at school where the school administrators decided how we needed to grow, the weekend workshops or summer workshops that I would scramble to find funding for, or the one or two day workshops that would cause me to miss school. It’s a different world now. I know I’m preaching to the choir if you are even reading this but this world of twitter, of blogs (both writing them AND reading them), of online simulcast workshops, or improv EdCamps, the list goes on. In this day in teaching I am fully convinced that if you want feedback and you want connections to help you think about your craft and to expand your toolbox – if you really want it – there is an ocean of resources at your fingertips. Literally (since I’m typing this right now!) at your fingertips. Not all of it fits everyone. I know that I am still wrestling with the timing and pace of my twitter feed, but I think I’m getting better at it and I KNOW I’m growing as a result of it. I spent a long time reading blogs, then commenting on blogs before I felt confident enough to launch my own. I have two kids at home so I know how tight time can be, but I also know that the past two Saturdays (that I blogged about separately here and here ) where I spent a combined 16 hours out of the house were worth the time and effort. Luckily Mrs. Dardy is kind and flexible and supportive of this pursuit.

I’ve been thinking quite a bit about this, about how different my life as a teacher is in the past few years. I’ve been in regular communication with one of my former colleagues, Gayle Allen. Gayle (@GAllenTC) hired me seven years ago when my family left Florida and we landed in New jersey for a while. Gayle is a remarkable, energetic thinker and was a great boss. She and I have been engaged in a long conversation about professional growth and one of the results of this conversation is an article that got posted today over at a website called Getting Smart. I know that I am not unique in this journey, but I also know that there are still many of our colleagues who have not taken this plunge. Some because they are not interested in doing so, some because they don’t know where to start. I’m pleased to be able to give shout outs of thanks to Dan Meyer (@ddmeyer) and to Sam Shah (@samjshah) through that article and I’m pleased to be connected again (even if we are nearly 3000 miles from each other) with Gayle.

This summer will see a trip to OK to take part in TwitterMathCamp. This would not have happened if Tina Cardone (@crstn85) had not reached out to me and asked me to join in on the fun. This summer will see me finish an in-house Geometry text for our students. This project would never have happened without the encouragement and advice of Jennifer Silverman (@jensilvermath). This summer will see me work on plans to help a brand new teacher in our high school take the leap from teaching Algebra I in the middle school to teaching Honors Precalculus for the first time. All of these experiences will help me grow as a professional. 27 years at it now and I feel like I still have an awful lot to learn. I hope to be smarter this time next week about this craft than I am right now.

Making Connections

We normally have very few school days between AP tests and graduation. This year we have 8 days left (counting today) so I am trying to have a little exploratory fun with my AP Calculus BC kids. They’ve worked hard on Calculus for two years now and I know that there is plenty more Calculus out there. Instead, though, I’ve chosen to take them on a little tour of some topics that they normally would not see in their high school days. Today’s topic was different number bases (the link is to my classroom document for today) and we had some fun with a series of base 8 addition and multiplication facts. I presented them with a  picture of Lisa Simpson as a clue and one of my students noticed that she only has 8 fingers. I use this as my motivation for this conversation. So, I had this nice little handout prepared that I hoped would guide us through a fun conversation. What I did not anticipate was a terrific question that came up. Let me set it up. We visited a website that converts base ten numbers to base 6 while you type. We had fun playing with it typing in things like the year of my birth and a few other nuggets. I asked them whether a base 6 representation of a number would always be longer than the base 10 representation and we had a nice chat about that. Then a student suggested that I type in a decimal so I typed 9.2 and the website did not like this. It would have been very easy to just shrug it off. My kiddos did not. They pushed me a bit and someone suggested that I write the first few negative exponents of 6. One student wisely suggested that I work on the assumption that the coefficients would all be 1 since 1/5 is SO close to 1/6. Here’s where it got fun. Another student said something along these lines – ‘Isn’t this just going to be an infinite series?’ WOW! I was so so so pleased with this. Connecting to Taylor’s and other infinite series in the face of a (relatively) harmless decimal? So proud I was.

We tried this conversion and another student seemed suspicious of the assumption that every coefficient would be 1 (or 0) so I reverted to binary. Now our task was to convert 9.2 into a binary number. The whole number part of 1001 was easy to sell. Now, we had to convert 1/5 into a series of decreasing powers of 2 (or increasingly negative powers of 2). Well, there was quite a bit of computation involved, some nice guess and check strategy was employed, the TI calculator function of turning decimals into fractions was helpful and in the end we discovered (and were able to verify) that 9.2 in base 10 is 1001.0011001100110011…

I have made a few mentions recently of my battery being recharged. It was awfully nice to see that some of my students still have some juice left in their batteries as well.

Charging My Batteries

Two Saturdays in a row now have been spent with some pretty amazing educators. Yesterday I was one of the attendees at EdCamp NEPA. One of the organizers, Mike Soskil (@msoskil) is someone I ran across on twitter some time ago. We had made plans to meet and lunch together at a tech conference but I had a minor car accident that morning so our lunch never happened. I finally met him yesterday and he was as terrific in person as he is online. He was full of energy and enthusiasm and it spread through the room. The vast majority of the folks there were first time EdCampers. I was a second timer so I felt like an old pro in this group. I’m guessing most of you reading this are familiar with the EdCamp model. You arrive to face a big blank session board and write up anything you want to talk about or listen to and people vote with their feet. Mike asked me to consider running a geogebra session and it did not take much to talk me into doing it. So I ran a session I called Visualizing Mathematics – Exploring with GeoGebra and Desmos. I showed off the work my students did in exploring the average daily temperatures in Gainesville, FL on desmos and we got into a great conversation about the power of this kind of visualization. I shared the story that all of my students missed some of the data in the same way and I got to make my standing joke about how long summer lasts in FL. One of my students asked if we would be more accurate with a city farther from the equator. Another student suggested that being closer to the equator would result in more symmetry. A conversation like this would never have happened without this visual support. I also showed off a geogebra file I created to model Taylor Series expansion as well as one of the files that the great jennifer Silverman (@jensilvermath) shared last week at our workshop. This one was called Quadratic Palooza. I had a small group in my room but they were engaged and asked me some great questions. We had a lively conversation about the power of this kind of visualization and how it enables students to ponder and ask questions that they likely would not have thought of before.

I also sat in on a couple of great sessions. The last one of the day was called Love It / Hate It. The moderator would post a statement about some school related policy/issue and we were to move to parts of the room based on whether we loved it/were on the fence / or hated it. We were to discuss with our group to construct an argument to have with our colleagues in the room.

Some of the sessions had participants taking notes together on google docs and here is the link to the schedule page

 

At a time of year when I am usually dragging (we have 10 class days left) I find my self with my batteries feeling recharged. Instead of feeling a bit sluggish, I am feeling perky and peppy, Love it.

 

A Note of Thanks

So, yesterday I had the privilege of spending about 8 hours working. I know, working on a Saturday does not necessarily sound like a privilege. But it was. I had the pleasure of working side by side with three of my colleagues who work in the same building as I do day to day. Mary and Mary and Kathy all agreed to take a Saturday away from their families and friends and join us in learning to use GeoGebra. I cannot feel them (or you) how much I appreciate their willingness to do so. I also had the pleasure of meeting three people from the Philadelphia area who all agreed to give up a Saturday AND to travel two hours to do so. Ed, and Whitney and Andy all took a chance. They heard about our workshop through a flyer that was distributed on a Philadelphia area teachers email list (thanks Ruth!) and, knowing little to nothing about me, my school, or our presenter, they chose to commit a Saturday in May to come and learn with us. I sure hope to continue hearing from them and to build some real community with them even if they are two hours away. I finally had the pleasure of meeting Justin in person! I’ve been reading his blog (and commenting there) and communicating with him on twitter. It’s been a pleasure feeling like we are building a relationship and it was a total treat to finally meet him in person. He drove from the Pittsburgh area to his mom’s house on Friday. Drove to my school and back to his mom’s house on Saturday and today he is driving back to Pittsburgh. He was full of energy, joy, and ideas yesterday and he helped make the day for me. What a treat! Lastly, I finally got to meet Jen in person. Jen has been completely generous of her time and her knowledge. I”ve been picking her brain via emails and google chats. I’ve been stealing ideas from her and she is the main reason I’m brave enough to tackle a curriculum project this summer. I’ll be writing an iBook for our Geometry course next year and I could not have conceived of doing this without her inspiration. She and her husband Charlie drove down from Connecticut and Jen was our leader in this exploration. We made some fantastic discoveries with each other (and through the help of remote twitter colleagues), we wondered and played. All of us were tired at the end and our brains hurt. That’s a great feeling, isn’t it? Community can mean many things and yesterday I felt that my community exploded beyond the walls of my school.

When My Students are More Clever than I am

This happens often enough to keep me excited in my job. I love the feeling when I learn from my students. Nearly everyday I learn important things about people, about human interactions, about kindness and community. What happened today was a great example of when I learn some math from my kiddos. I blogged yesterday about helping a boy with a L’Hopital’s Rule assignment. What I did not mention was that there was a problem I could not solve. I’m not good enough with typesetting here on wordpress so I’ll do my best here. We were trying to evaluate the right hand limit of (x – 1) * tan (pi*x/2) as x approaches 1. Two clues made me think of L’Hopital. The first is that the student told me they were working on L’Hopital’s Rule. the second was the indeterminate form of 0 *  negative infinity.

We looked at a graph to convince ourselves that there is a limit and then we tried to make this a quotient to fit the rule. My instinct with trig functions is to avoid the cotangent, cosecant, and secant functions simply because I feel more comfortable visualizing the basic cosine, since, and tangent. So I made the product a quotient by dividing the tangent expression by 1/(x-1) creating an indeterminate form of negative infinity divided by positive infinity. A quick ratio of derivatives yielded something even word and the second time around was even scarier. We walked away from the problem – in part because I was trying to do three other things at the same time and in part because he was exhausted from thinning his way through the other examples. I asked one of my best BC students to consider it overnight and had some hope that he would. 

This morning at 8 I posed this question to my quiet BC class of 7 and they pounced on it. One student advised that I rewrite the tangent function as a quotient of sine divided by cosine so that L’Hopital is immediately satisfied. Another advised that I rewrite tangent as a cotangent so that I have a ratio as well. In each case, my students saw a way to rewrite a product in terms of 0 / 0 rather than my way of infinity / infinity. Neither format is lovely but we all seemed to agree that 0 / 0 seems less scary. One round of derivatives on each idea led to the conclusion that the limit was – 2 / pi. Geogebra agreed.

What strikes me is that, despite me showing an undesirable approach AND me asking them to recall something from the past, my morning class was perfectly willing to be flexible and to TRY something. Two sound ideas in about 90 seconds and we were off. I was so delighted to start my way this day.