Category: Uncategorized

Thinking about Learning (again…)

Been away for a while for a number of reasons. 

I just read an article on slate.com the really got me thinking about what learning looks like and, therefore, what teaching means in this context. Read a great quote sometime ago that basically said teaching does not exist unless learning has happened. This is quite a challenge for us, obviously.

I shared the article with our AP Psch teacher and he said it was a valuable read and that he would share it in the future with his students. I think it’s worth a read, but if you don’t want to follow the link the article discusses a famous memory study subject who suffered damage to his hippocampus. This caused amnesia to set in but over the course of his life he was still able to form new memories of a certain sort. Here, I think is the interesting quote

After the motorcycle accident, K.C. lost most of his past memories and could make almost no new memories. But a neuroscientist named Endel Tulving began studying K.C., and he determined that K.C. could remember certain things from his past life just fine. Oddly, though, everything K.C. remembered fell within one restricted category: It was all stuff you could look up in reference books, like the difference between stalactites and stalagmites or between spares and strikes in bowling. Tulving called these bare facts “semantic memories,” memories devoid of all context and emotion.

 

I immediately thought of my AP Stats students who are always asked to report conclusions in context, but I also thought of my Calculus students. Both of these groups of students have a deep reserve of the qualities that usually mark a student as a good student. However, too often I have conversations where it is clear that much of what they have displayed as learning in many classes might not go much beyond the sort of semantic memories referred to in the pull out quote. Skill such as setting a derivative equal to zero when solving optimization problems, or running a two sample t test rather than a z test are often reduced simply to factual memory with no conceptual anchor. In stats when we ask about rejecting or failing to reject a hypothesis based on a reported, or calculated p value, it feels like a particular student should either ALWAYS get this decision right or ALWAYS get it wrong based on a conceptual idea about what the p value says. However, I have seen too many instances where this decision seems to boil down to not much more than a coin toss as the student tries to remember a rule. If the p value has a meaning related to probability, then the answer should be clear and consistent. It feels to me that the biggest challenge in teaching these days is to figure out how to help my students slow down and think. Really think about the ideas that they are working with. Too often they have been rewarded with good grades without reflecting on what they’ve learned and how it applies to anything. This sounds (and kind of feels) like a criticism of my students and my colleagues. I don’t intend it that way. I intend this as a question for me and my colleagues (both in my building and around the world) and my students to consider. How can we construct our classes in a way that helps to develop understanding for our students in a more meaningful, more permanent way? I certainly don’t pretend to know the answers. I know that the way I run my class works for some. It makes other crazy. Two super quick anecdotes, then I’m off to pick up my little girl.

  • This year when I was reading my teacher/course evaluations that the students fill out I ran across a great written remark. One of the questions asks whether the instructor challenges the student to think critically about the subject matter. This student in question marked that he agreed with the statement and then wrote ‘TOO MUCH THINKING’ I hope that this was meant in a good natured way, but I DO know that I wear some of my students out with my questioning. They often ask me to just tell them HOW to solve the problem.
  • Last year when we were wrapping up Calc BC and working in class on review material for the AP test two students were talking. They did not know I was close enough to hear (or they did not care) and one said to the other ‘last year I knew how to solve these but I had no idea why it worked.’

Here’s to the never-ending struggle to make this all meaningful.

Observation of Student Behavior

As part of my ongoing commitment to taking one day per week away from the Calculus curriculum, I spent yesterday playing the game of set and 2048 with my students in Calc BC yesterday. My afternoon class was fully engaged in set offering different answers and we found another site which gives you more than one game of set per day. The I opened up 2048. I became aware of this game int he past week due to constant twitter references. I played it some Wednesday night and shared it yesterday. Well, for about ten minutes or so the entire class was engaged tossing out advice and arguing moves. Then the class started to get more and more quiet. What happened was that my students started pulling out their phones and playing the game for themselves. I like the fact that they were interested enough to make sure that they had the game for themselves. I was disappointed that what felt like a great community conversation devolved into individual focus and lack of communication.

I mentioned this and two students told interesting stories. One girl told me that she and her friends recently received free dessert at a restaurant because the server (or maybe the manager) commented on the fact that none of her friends spent dinner on their phone, they were engaged with each other. A boy told me that his friends have a standing challenge sometimes when they dine. Everyone puts their phone in the middle of the table and if someone breaks down and picks up their phone they get to pick up the tab.

It’s interesting that each of these stories was told in a way that made me feel that the student was aware that their technology sometimes gets in the way of interactions. I wonder how much saying that out loud affects their behavior?

Exciting Opportunity

So, one of the benefits of creating a virtual presence has been that I have all sorts of new friends that I have never met. I look forward to thoughtful exchanges on my blog and on theirs, I chime in every once in a while to the torrent of information that is twitter and I am happy that I’ll be able to meet a bunch of these folks at twittermathcamp 2014 in OK this summer. However, another opportunity to actually meet some of the army of talented math folks on the internet has reared its head. The amazing Jen Silverman (@jensilvermath on twitter and at http://www.jensilvermath.com on the web) will be traveling to my school in Kingston, PA to host a one day Geogebra workshop on Saturday, May 3. Here are some reasons you should think about attending:

  1. Jen does amazing work on GeoGebra, she is sort of a GeoGebra Jedi Master. See this page for evidence.
  2. We are hoping to have a manageable crowd of about 12 – 15 folks here. Enough to share ideas but not enough to get in the way of some direct instruction when you need it.
  3. I’m working on taking care of lunch for everyone – so that is a definite plus.
  4. Oh yeah – it’s free!!!

 

Jen created a lovely flier for this event. If I was smarter about managing my blog I would display it below, but you can click the link to see the document.

I hope that many – if not all – of my colleagues from our middle school and high school can join us and I am reaching out to anyone within a reasonable drive of NE PA to come and join us for a day  of learning and sharing.

 

Different Perspectives

A quick reflection here before I wake up my kiddos.

Yesterday in BC Calculus we had onto of our weekly problem days where we (mostly) put aside our current Calculus work and look at interesting problems that may or may not involve any Calculus at all. Here is problem #1 from yesterday (a problem I borrowed from @bretbenesh.

A mountain climber is about to climb a mountain. She starts at 8 am and reaches the summit at noon. She sleeps at the top of the mountain that night. The next morning, she leaves the summit at 8 am and descends using the same route she did the day before, reaching the bottom at noon. Why do you know that there is a time between 8 am and noon at which she was at exactly the same spot on the mountain on both days? We should not assume anything about her speed on either leg of the trip.

 

One of the things I enjoy most in teaching is seeing/hearing different ways of attacking a problem. When I read this problem I immediately sketched a height v time graph with the base of the mountain and 8 AM as the origin and the top of the mountain, noon as some arbitrary point in the first quadrant. A wiggly sketch connected the points. Day two has a y-intercept of top of mountain, 8 AM and an x-intercept of bottom of mountain, noon. No matter how I connect these points the sketch intersects my other sketch and I see the reason why. I’m surprised by this discovery, but I see it. In each of my 2 BC classes yesterday there was one student who saw through this problem and explained it away so quickly that I was wowed. In each case the student immediately switched to thinking of 2 people rather than one. If one person starts at the top at 8 am and walks down while the other starts at the bottom and walks up then they must pass each other at some point! Simple, clean, elegant. It’s fun to learn from your students, isn’t it?

That Elusive A – ha Moment

On Monday we returned from our two week spring break and we finally took the plunge into Power Series in our BC classes. Oh, by the way, we were looking at snow in our area on the weather forecasts. Great first day back after spring break!

So, on Monday and Tuesday we were dealing with defining Power Series’ and looking at the radius of convergence and the interval of convergence for these series’. My students seemed to be dealing with these problems pretty well. Some number of weeks ago – I cannot even remember right now – I introduced this last full chapter of our text by talking about our ultimate goal of developing Taylor series approximations and I used the function f(x) = sinx as my example weeks ago. I convinced my students that we could create a polynomial the behaves like sinx as long as we were willing to be patient enough. I started off (again, this was weeks ago!) with an approximation of sin(0.1) using geogebra and talking them through the idea that we wanted (more accurately, I wanted) to create a polynomial called P(x) that agreed with f(x) at x = 0, and whose first, second, and third derivatives all agreed with those of f(x) at x = 0. We chose x = 0 for relatively obvious reasons and since they had never seen this argument before they were willing to go along for the ride. So, we finally get to the point now where my students can follow along in the logic rather than simply watch and/or write down notes. They come to class yesterday and I tell them that in our 40 minute class I hope that we can finish 2 problems. This creates some visible unease as the idea of 2 problems each taking 20 minutes generates some snarky remarks about how hard this is going to be. What follows is a summary of the conversation with my second BC class of the day – my much more vocal and active group of the two.

Problem #1 – Estimate, correct to three decimal places, the value of sin(0.1) without using your calculator. I start a conversation about what we might be able to know about this value. We pretty quickly agree that it is positive and small. In my morning class I had a great estimate in degrees of what 0.1 radians might look like and I hope to prod the conversation in that direction. I start by asking what a logical upper bound for the estimate might be and I hope to hear someone say 1/2 since that is the smallest exact sine value they know in the unit circle. Instead, Jon tells me that it has to be less than 0.1 which is true and much more accurate. I ask him why this must be so while a number of his classmates are generating their own guesses. His neighbors are in a debate about why 1/2 is an upper bound for reasons that hover around the unit circle. When I question Jon he tells me that the function has a slop of 1 at the origin and that this slope decreases as x increases, so therefore when x increases by 0.1 y will increase by less than that. Wow. I was SO happy to hear this reasoning and I wanted to make sure that the rest of the class heard it as well. I should have dusted off Ben Blum-Smith’s idea of having another student try to restate but I honestly was not sure how many kids had even heard him. I was standing near him and he was speaking to me while his classmates were involved in conversations with each other. So, I took over and restated his point. I then pushed a bit and asked the class why Jon knew that the slope of f(x) = sinx was 1 when x was zero. Here, my mind was anticipating and hoping that someone would mention the limit of sinx / x as x approaches 0. I might have had to take a break at that point to calm my heart down. Instead I got another terrific answer – we know the derivative of sinx is cosx and we know that cos(0) = 1. I asked a student why we were suddenly talking about derivatives when Jon discussed slope and I was calmly told that the derivative IS the slope and we were ready to march on. The procedure for setting up the system of equations is tedious and time consuming and as I started the problem a number of students were rifling through their notes and found the example we did weeks ago when we generated a third degree polynomial to match up with f(x) = sinx. I was again delighted that they (a) remembered we had done this and (b) could find it so quickly in their notes. So we get the function we want and now substitute x = 0.1 into the polynomial. We have the fraction 599/6000 at this point and Jon is pretty pleased. We see that it is less than 0.1 but just barely. I remind them that the directions asked for an answer in decimals without their calculator so we dust off some long division skills and get to 0.0998. I ask a student to pull out his calculator and give me the four decimal answer that his calculator has for sin(0.1) when he recites the exact same decimals I can see some noticeable smiles on my students’ faces. They are pretty impressed. We are almost there, I can feel it.

Problem #2 asks for a four decimal approximation (I correct myself midstream because of the first problem and what I remember of our morning work) for ln (0.9), again without their calculator. So this problem has a different wrinkle. I have not yet introduced formal notation from their text regarding these series, so they don’t know about the center of convergence yet and we are not assigning the mystery, powerful a t this yet. I’m using the phrase ‘we are concentrating on x = ___’ and we want the blank to be a value close to our target but one where we can easily compute and exact value if we need to. We all agree pretty quickly that x = 1 is where we should concentrate and that ln (0.9) will be negative and small. I’m happy that I have enough discipline now to weave in this kind of ‘what do we know, what do we wonder, what can we guess’ kind of conversation into class regularly now. All this twitter and blog PD is taking hold!

So, we go through the tedious process AGAIN of matching a power series out to the third degree so that P(1) = f(1) [where f(x) is now lnx], P'(1) = f'(1), P”(1) = f”(1) and, finally, P”'(1) = f”'(1). However, we have an interesting decision to make here. For the first problem, with x = 0 as our focus, we all agreed that P(x) = a + bx + cx^2 + dx^3. With x = 1 as our focus now, we were a little anxious about this model. Students quickly offered two solution ideas – replace each x with an x + 1 or replace each x with an x – 1. I have to say I was pretty thrilled with how this conversation was unfolding. Agreement on x – 1 was reached. When I was asked why, I responded with the following two questions – (a) What is the simplest equation of a parabola with its vertex at the origin? (b) What is the simplest equation of a parabola with its vertex at the point (1,0)? Everyone seemed okay at this point. We get our polynomial and evaluate it at x = 0.9 and we arrived at the fraction -79/750. When I did the long division we arrived at -0.1053 and, once again, someone’s calculator matched this exactly. A wave of smiles and nods went around the room. Those elusive moments when you can actually see a group of people lock in on an idea are so exhilarating. It was so much fun to see this group of students attentive and engaged, not intimidated by two problems that each took about twenty minutes each. This class is my last class of the day and I ended the day in a very positive mood as a result of this conversation.

 

PS – Another problem day today. Here is my newest problem set. I borrowed problem #1 from @bretbenesh who was clear in explaining that he borrowed them from all over. Problem #2 is an old favorite and problem #3 is from a recent math league competition. 

Fishing for Ideas

When we return on Monday from our two week spring break, my AP Calculus BC kids will be finishing up their version of their Calc text with the final push of study on infinite series. We’ll be gearing up for our tour of Taylor and Maclaurin techniques. I want to design a final unit to tie together some loose ends from their trig days and formalize their knowledge of vectors. I feel that I can teach the required AP vector techniques in about 2 days but I want to craft something a little larger. I’d like to try and frame this by reviewing important trig highlights first. Our kids do not see DeMoivre’s Theorem in their precalc try unit and I don’t want to send them off to college without that tool. I am dreaming of a way to wrap all of this up in a nice, tidy bow. Trig/complex numbers/vectors as a meaningful and lively final unit. I have about two weeks before I would be starting this off in class and I would appreciate any clever ideas/links/words of encouragement/etc. that I can gather from the collected wisdom of my virtual colleagues.

Why Do I Blog?

Once again, Dan Meyer has me thinking. This time the blame can be passed along to Michael Fenton. Michael raised a question on twitter.  His question to Dan was

Could you have written a list of 5 reasons you blog 5 years ago? And a list of 5 reasons you blog now? Lists match? What’s changed?

As usual, Dan’s page is generating some great responses. You can jump to that page here.

As I write, I am on my first day of spring break. It is a quiet, beautiful morning on my mom’s back porch with everyone else sleeping. Let’s see if I can make sense of this question.

I am a relative newbie to this blogging business. My first post was in the first week of July 2013. This is my 58th post. I started reading blogs with some regularity and subscribing to them a few years ago. The first one I subscribed to for regular delivery was either Dan’s blog or Sam Shah’s. My memory is not sharp enough to recall which was first ( Edit – A little research shows that I subscribed to Sam’s blog in Oct 2010 shortly after relocating to NE PA). I was living in NJ where I moved after leaving FLA. While I was in FLA I was enrolled in a program to earn a doctorate in education. I was taking night and weekend classes while teaching full-time and raising a little boy. I even took a year off from the classroom to be a grad assistant and full-time student. I worked on a super cool research project looking at arts integration in schools and was feeling super energized by the reading I was doing, by my classmates, and by my professors. We headed off to NJ where I was going to carry out my dissertation research (where I was lucky enough to work with this guy as one of my research participants) and I worked with a really inspiring Associate Head/Director of Studies and some other good colleagues there. But, once I was back in the classroom walls, I started realizing how much I missed the level of conversation from my graduate classes. As most of you reading this know, we spend TOO much of our time in our four walls closed off to our colleagues. As many of you have probably experienced, there is not much in the way of structured time during the day where we can have meaningful conversations about what goes on in those walls. As a math teacher, I get too much semi-snarky jokes when I try to talk about my class at the lunch table or in the faculty lounge. There is something about the pace and structure of our days that seems to work against built-in reflection time. So, I found that time and space out on the internet. For a few years I read blogs and occasionally commented on them. I finally took the plunge last July and started my own. In the fall when the MTBoS blogging initiative kicked in for its second year, I was all in. I even took the plunge into twitter based on one of the challenges presented there. That must have been in October. Since then I have sent out almost 1300 tweets. All but about 50 of them have been to people I know only through my math experiences or through their blogs and tweets. The majority of them have been sent to people I’ve never met. Through these experiences, I have had the pleasure of being invited to my first EdCamp – with another on the way. I have been invited by the amazing Tina Cardone to take part in a workshop presentation at this summer’s Twitter Math Camp, I have received tweets as answers to questions from Ketih Devlin and Steven Strogatz, I have shared amazing lessons from people I’ve never met, I have pestered my colleagues with emails and document attachments that I have gathered from the web. I spend the first 45 minutes or so of each morning (after I feed the cats and start some coffee) reading my email alerts from the previous night, scanning what might have happened on twitter after I fell asleep, and checking my wordpress reader. I go to school every morning thinking about something that I might not have thought of on my own. I look forward to this quiet time in the morning before my family wakes up as a time to wake my brain up and recharge it. It’s a way to improve the lives of my students and to help ensure that I don’t feel stale and bored. I am in my 27th year in the classroom and I feel as energized about it as ever and I think that much of the credit belongs to a world of people I’ve never met.

So, I realize that most of what i have just said explains why I read blogs and prowl twitter. Why do I write my blog? A much simpler answer. I think that there are two reasons.

  1. I want to give back – at least a little – to this rich world of ideas.
  2. I want feedback on my ideas as they develop.

Not profound, but it feels good to chime in. Now, I’m off to breakfast.

Circle – Square Problem (Stolen from Dan Meyer’s site)

We are only three (well, really two and a half) days from spring break. Spring itself comes late in NE PA but spring break comes early for us. We are in the thicket of infinite series in Calc BC and I wanted to take the last three days on a tour of an old AP Free Response section. I wanted to send them on their break with the full realization that they know almost everything that they’ll need to know in May. So, yesterday I ran off copies of the two calculator questions from an AP test along with the grading rubric. Figured it would be a good working day while I listened in and roamed a bit. Then at 6 this morning I read Dan Meyer’s newest blog post and immediately decided to scrap my plans. He presented the following problem

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

and, as usual, presented a challenge to his readers.

What can you do with this? How can you improve the task?

I’m not sure how much I improved the task, but I did make some decisions about what to do with it. The first decision I made was to hand my students some blank paper and simply read the question to them. I read it carefully twice. I did not want to visualize it for them, I wanted to see what they would do with this. I was surprised that most of them did their best to jot down the exact words that I said.  (This was in my quiet, small morning group – I am just getting ready to meet with my afternoon group) Almost all of them proceeded to draw a line segment and label a point P pretty near the middle. One student identified the two partitions as x and y. Very algebraic! My more engaged (and much larger) second class responded almost unanimously by drawing a segment. They hardly bothered with the words at all. One student had a segment with a square imposed on one side and a circle imposed on the other. Nice.

Both classes were comfortable (being BC Calculus kids, I expected them to be) with some general statements of the area either interns of a variable or in terms of segment notation. Both classes decided that it would be nice to have a default length for the segment AB and they each suggested 1 as the length. This was interesting, I went immediately to 100 as a nice default value. I was thinking in terms of a percentage of length. My first class let me get away with that and we reached a solution to the quadratic pretty quickly. My second class was insistent that 1 was a better choice. Of course, the critical balance value is found regardless. By the time the afternoon class had met the hive mind was working full bore on this problem. I shared Dan Anderson’s terrific Desmos demonstration and I also showed the the Math Hombre’s GeoGebra sketch of the problem.

As per Dan’s suggestion on twitter, I asked my students to try and optimize the situation. Both classes were very quick to conclude that the largest area is the trivial case of the entire segment being used to make the circle. When we graphed the function that represented the sum of the areas it looked tantalizingly close to the equal area point being the minimum. Sadly, this was not to be. We also discussed another scenario proposed on twitter and that was to have different regular shapes, not just a circle and a square.

I was proud of how open my students were to exploring this open-ended question. I was impressed that they were such careful listeners when I presented the problem to them verbally. Many of them are not note-takers, but they were willing to dive in and play on paper with this problem.

I am left with a some ideas/issues to ponder.

  1. I need to figure out a way to find time at the beginning of next year to give my students the opportunity to familiarize themselves with Desmos and GeoGebra. I wonder how I can structure this appropriately. When they see the dynamic visualizations the conversation opens up.
  2. By my standards, I think I did a pretty good job of getting out of the way today. I still started too many of the conversations, but I let the questions percolate from them. I need to find some sort of mantra or something to remind myself to be more quiet and take more time to let the questions arise.
  3. As with many class conversations, the pace was dictated by a few students who are more eager to share ideas and ask questions. I need to work on respecting this while also creating a buffer for those who take a few more moments to ponder.

So happy I scrapped my plan of canned AP FR questions today. I hope that the students are happy about this as well.

Classroom Conversations

I find myself thinking about how to best moderate and encourage classroom conversations. Two blog posts have me looking in the mirror. One of them is one I have written about previously. Ben Blum-Smith wrote at his blog (Research in Practice) about having students summarize each others’ statements. I am still working on making this a teaching move that I regularly go to. It has mostly worked well for me. However, I have noticed that when I do this I almost always have to interject and pass along some value judgment about the response of one student before I can get another to elaborate/explain/restate what was said. Andrew Stadel (over at Divisible by 3) wrote something that really has me thinking. You should read what he wrote by clicking on his name. (The same goes for Ben’s post – you can click on his name to read what he wrote) I’ll try to summarize part of Andrew’s post here, but I encourage you to go read his post. 

Andrew discourages teachers, as I read it, from telling a student that they are correct (or incorrect) and instead urges us to explore reasoning and to try and get the student to tell us why they arrived at the conclusion that they did. I get it – I think I really do. I want to understand their reasoning and, more importantly, I want my students to be able to reason out loud. To think through their own process and have the language at hand to explain that process. What I notice though, is that my students are rarely willing to expound at length on their own thoughts without knowing whether their statement was correct – or at least in the ballpark. Once I give them some confirmation that their original statement had value, then they are willing to expand on what they said. I understand this feeling. I imagine them feeling that they are about to be ambushed if I ask them to say more about what they think when they are unsure of whether they are right or not. I know that I can work on creating more of an environment where this kind of thinking out loud feels safe. No doubt about it. However, I think that there is something kind of primal that I am working against here. It is so natural to be shy about talking out loud when I think I might be wrong. Add in the social pressure of being wrong out loud in front of your peers and that’s a heck of a force to push against. Another item on my list of things to think about and to try to modify in my teaching. I hope that this process never ends.

 

Challenging Questions

A former colleague of mine, someone who inspires me to think carefully and critically about what I do as a teacher, sent me a great email recently. I am going to excerpt it below. (Note – I added the question numbers so I can more easily refer to them later.)
I’ve been doing a lot of thinking about teachers and data. More
specifically, how teachers are surrounded by all kinds of data every
single day in their schools and classrooms – qualitative and quantitative.
This data can be life-changing for their practice and for student
learning. I’m growing more and more curious, personally, about how to help
teachers tap into that.
 
I’m considering writing something about this. Not sure of the venue or any
of the specifics yet. Right now I’m just exploring these ideas with
teachers like you.
 
I’m particularly interested in data related to teaching skills for
transfer. For example, I recall my students learning lots of great math,
but then being unable to transfer their math learning to a science
classroom where the context is different and where the problems read
differently. It’s a question I’ve always wanted to tackle – how to help
students get better at this type of transfer.
 
That’s where my question for you comes in –
 
1) What’s a skill(s) for transfer that you see come up again and again for
your students, a skill where they seem to learn it well in isolation, but
they then struggle to apply it to a new/different context? The more simple
it is, the more interesting I’d actually find it to be, if you have an
example like that.
 
2) What about grit and/or persistence in problem solving? What challenges do you see there?
 
3) How about engagement? Anything that comes up there for you?
 
I am so curious to dig deeper on these questions in relation to info/data
gathering. Really curious about this. I know it smacks of action research
(not a bad thing) but I’m really eager to find a way to simply it and help
teachers work on something like this.
 
Welcome your thoughts! Let me know if anything I’m asking is unclear.

 Quite a bit to chew on, I have to say. She sent me this email four days ago and I think I am finally able to attack some of these questions coherently. Hold on tight, this might be a long post.

Regarding question 1 – the question about transfer – there is one big area that occurs to me. I think that it is related to the difficulties in bridging the gap between the equation solving skills that students want to rely on and the worlds of graphical representation and written explanations where most teachers seem to want their students to take up residence. A recent example is sticking in my craw right now. I have been giving my Calculus classes weekly in-class problem sets to work on. One of the problems this week was: Given that cos 80 = 0.173648…, find the following values without the use of your calculator (a) cos 100   (b) cos 260   (c ) cos 280   and (d) sin 190. When I chose this problem, I wanted them to recall the symmetries inherent in the unit circle and recognize the relationships between sine and cosine measures of angle that are all 10 degrees off of an axis. While many students (not as many as I’d hoped) seemed to breeze through this, I saw a wide variety of REALLY bad mistakes. Mistakes of the nature of cos 100 = cos 180 – cos 80. Now, it’s important to note that these are my AP Calc BC kids. They are among the best math students we have at our school. In their defense I will note a couple of important points. (i) I was not there the day they were working on it. Some of them probably saw this as ‘busy-work’ to a certain degree. Also, with me being gone they may not have been collaborating as much as usual. If I’m here and see confused looks or hear whispered questions, I’ll dive in an nudge them. (ii) Precalculus was quite a while ago for these kids who went through AP Calculus AB last year. Not in their defense, I will note that they had each other and their texts as EASY references. I’ll also note that the mistakes I saw really made almost no mathematical sense for functions more complex than linear ones. They should know (flat out) that the distributive property probably does not hold over subtraction with a function like the cosine function. However, I have seen over and over again in my 20+ years of teaching that symmetries based on graphical representations of functions do not seem to stick with my students the way that I think that they should. I use Geogebra and Desmos regularly with my students, but their Precalculus teacher did not. I wonder how deeply they tried to internalize this behavior when they were studying their trig. So, this seems to me to be an example of what my friend was asking about. The trig skills I referred to above were certainly ‘mastered’ in their precalculus days, however future applications of these ideas rarely go well in my experience. I fear that the students see trig (and many other aspects of math) as facts to learn, rather than concepts to master.

Another skill that is ‘mastered’ in precalculus is working with the definition of the intermediate value theorem. Another area where graphing sense really comes into play. However, in many of the cases where this theorem is invoked in Calculus, my students seem flummoxed. To a great degree what I see here is that my students thrive when they need to remember a formula or a definition. Where they struggle is applying this idea in the context of a problem such as identifying an interval where a root for a function might lie. In each of these cases I see students who can recall facts when they are taught, who can recall definitions when they are presented, but they struggle with the applications of these ideas. This seems especially true when graphing ideas are involved.

Since the introduction of graphing technology in the classroom (the TI and now web-based graphers) I have had the habit of looking at graphs on my own when working AND when working in front of my students. I know that they usually have computers nearby when working, but they don’t seem to have inherited this habit. I am at a loss as to how to help instill that.

For question 2 I have the following observations. When the work is graded for completion, there is little sense of determination to work through a perplexing problem. Simply writing something down is good enough. When the work is graded for correctness, most of my students will then dig in and really challenge themselves to get the problem right. However, this often is accompanied by a lack of discipline about time. Most of the grit and determination I see is directly related to the impact of that work on the student’s grade. This, of course, is not true of all my students but it is true of the majority of them. There are certainly students who are inspired by challenging problems to explore, but more often I find that my students don’t feel that they have the time for this. The challenge I see here is that the students who make it to these highest classes in the curriculum are often the most ambitious and involved students. I believe that there is an inherent curiosity, but I see it diminished under a heavy workload.

I am sure that I have some more thinking to do along these lines, but here is my first major swing at these interesting questions.