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.


Counting with my Four-Year Old

So, even with all of the school closings we have had, our four-year old girl is about to have her 100th day of PreK this week. Her teacher, the amazing Mrs. K, sent out an email asking the parents to have their lil ones count out a hundred of something before Wednesday’s festivities. Since it is Valentine’s weekend we have plenty of candies around. We decided to have Mo count out 100 Valentine’s M & M’s. She seemed unsure of this large task so we sat with her and encouraged her to think in groups of 10. She confidently counted to 10 and we emptied a small bowl into a larger one. She was confident in her teens and counted 11 – 20 comfortably. Then life got a little interesting. We emptied into the larger bowl and she paused. I’ll write in dialogue form for a little while here –

mrdardy :  What comes after 20?

mo:  30

mrdardy: Well, eventually. What comes between?

mo: blank stare…waiting for an answer

mrdardy:  21?

mo: 21, 22, 23, 24, 25, 26, 27, 28, 29, 30! [A successful decade of counting!!!]

Now, I empty small bowl

mrdardy: What comes after 30?

mo: 40!

mrdardy: Eventually…What comes in between?

mo: blank stare…waiting for an answer

mrdardy: 41?

mo: 41, 42, 43, 44, 45, 46, 47, 48, 49, …50?

empty bowl again

mrdardy:What comes after 50?

mo: 60!

mrdardy: Eventually…What comes right after 50?

mo: 51?

I smile and nod and she’s off to the races again. She gets through the 50s, she gets through the 60s, then something funny happens. As she gets to the end of the 70s she says seventy-eight, seventy-nine, seventy – ten. She says this hesitatingly, almost as if she realizes something is funny but does not know how to fix it. I ask her what comes after seventy, hoping she’ll repeat her earlier mistake, but she’s too wise for that now. She says seventy-one. Sigh… I tell her that seventy and ten is eighty and she repeated the same mistake at the end of the 80s and almost at the end of the 90s. After a pause, I can get her to say one hundred triumphantly.

A number of questions pop in my head here and I am hoping that some who are much more expert in dealing with these questions will visit and share their wisdom (I’m looking your way Prof. Danielson)

My main questions here are

  1. When I repeat this with her soon, how many of these mistakes will she make all over again?
  2. I know that the teens are more in her comfort range, but the odd style of the names of these numbers seems inconsistent with all the other number names. I thought that she’d try to say something like twenty thirteen, etc.
  3. She’s almost the youngest in her class. How much developmental stuff is happening in the 8 – 10 month difference in age in her class?

Curious. I hope to gain some wisdom in the comments.

What’s Wrong With High School? – Does Anyone Really Know?

Just read an article over at Slate today (you can find it by clicking on the word SLATE right here) titled What’s Holding Back American Teenagers. Now, I’m not particularly interested in analyzing the data presented here. Certainly the internet abounds with interesting analysis of the research studies and a long discussion could be had about the validity of the data about PISA results, SAT trends, etc. No, what interests me is the following passage and how it relates to some conversations I have had recently with students. Here is the passage that caught my eye

What’s holding back our teenagers?

One clue comes from a little-known 2003 study based on OECD data that compares the world’s 15-year-olds on two measures of student engagement: participation and “belongingness.” The measure of participation was based on how often students attended school, arrived on time, and showed up for class. The measure of belongingness was based on how much students felt they fit in to the student body, were liked by their schoolmates, and felt that they had friends in school. We might think of the first measure as an index of academic engagement and the second as a measure of social engagement.

On the measure of academic engagement, the U.S. scored only at the international average, and far lower than our chief economic rivals: China, Korea, Japan, and Germany. In these countries, students show up for school and attend their classes more reliably than almost anywhere else in the world. But on the measure of social engagement, the United States topped China, Korea, and Japan.

 We here in NE PA have had a pretty cold winter and have had a number of school delays and a couple of cancelled school days. I live on campus in a dorm of about 80 teenage boys. We recently had a situation where we had a three day weekend, one day of school, and then a school cancellation. The night when the cancellation was announced the uproar in the dorms would have made you think that Oprah Winfrey had just shown up to give everyone a free new car or something like that. The level of excitement and joy expressed – one day after a three day weekend – made me conclude that either (A) The students were overreacting and egging each other on in their joy at missing a day of classes or (B) The students really dislike what they do from day to day in their classes and were overjoyed at the release of one day of desperate drudgery. I am too much of an optimist, and I have too much faith in my school, to want to accept the second option. But the reality of the first option does bother me a bit. I take much of what happens in school personally. When my students do well, it makes me happy. When they struggle, I am frustrated and bothered. When they are happy and engaged in class, I feel good about what I am doing. So, I was motivated to talk with one of the more mature students in my dorm and I raised this question with him. He’s a wrestler, very serious about it and heading to a good university next year to continue wrestling. I asked him how we would feel at lunch one day if he got an announcement from his coach that practice that afternoon was cancelled. He said he’d be pretty bummed since he really looks forward to practice. I think he values the camaraderie of his team, the chance to get better at what he does, and the physical release that his tough practices provide. I shared with him my disappointment at the rowdy joy (shrieking, high fives, etc.) that I had just witnessed at our last cancellation announcement and I told him why I took it kind of personally. That it sounded like our students just hate what they’re being asked to do. He acknowledged that he had not thought about the message that these celebrations sent and said it was mostly about releasing tension. I understand that. I think I do. However, I KNOW that they would not react that way is a game was cancelled or a practice was cancelled (well, at least most kids), so I can’t shake the feeling that there is something more there. I have had similar conversations with kids who are regularly late to class. I know that, in most cases, they are not regularly late to practice, or concerts, or games, or play rehearsals.

So, I don’t know where to go with these thoughts. I want to have a meaningful conversation with my students about the messages they send when they are late, when they are overjoyed about not being together, etc. But I don’t want to lecture them and I don’t want to seem like some completely unrealistic pollyanna. I’ve got some thinking to do and I may use this space for more of it. 

PS – After rereading (before posting) I realize that this may come off as critical of my school or our students. Neither is intended at all. I know the feelings in my NJ school were similar with the weather. In South Florida, we had a year of hurricanes met with joy at school closings. These are pretty common reactions. I am just more sensitive now for two reasons – (a) I live in a dorm and see it up close and (b) I don’t want these reactions to reflect my own children’s lack of joy in their school life. 


That’s right. It’s wintery and snowy here in NE PA and I am thinking about the summer. It’s not because I want this school year to end in a hurry. I am enjoying my classes and students too much for that. No, I’m thinking about the summer because an amazing opportunity has presented itself. Tina Cardone – she of the terrific Drawing on Math blog and the mastermind / editor of Nix The Tricks recently reached out to me and asked me to attend Twitter Math Camp ’14 and to help her in running the Precalculus offering in the mornings that week. I’m flattered beyond my ability to put it in words. Looking at the list of people who’ll be there makes me think I need to go around with a sharpie to get autographs on my lovely MTBoS shirt that Jen Silverman designed. We worked up a description of what we hope to accomplish through a lively google doc discussion. We ended up with the following

Current and future PreCalculus teachers! You are invited to join us for a workshop centered around collaboration. There will be time for looking back (topics from Algebra 2 that need reinforcing), looking forward (what will students need for Calculus) and looking side to side (topics students should study to be proficient in math as well as appreciate the fun and beauty of the subject). Be prepared to rave about your favorite topic and rant about your most dreaded one. After that’s off your chest, we’ll get down to work with enthusiastic and creative colleagues. You’ll win them over with your favorite lesson ideas and you’ll find some exciting ways to present your dreaded topics.

Not sure if your course falls under the title of PreCalculus? Tweet us and ask!

Tina @crstn85 and Jim @mrdardy

The camp meets in Jenks, OK (near Tulsa). I have felt so energized by the connections I’ve been making with many of the people I’ll finally meet there. Can’t wait and I am doing a quiet countdown in the back of my head already.


I found a video recently that was taken of me when I was a senior in college preparing for my internship. Some of my current habits were already in place. I was fidgety and tossed the expo marker around while walking and talking. I still do that – only now with chalk. It’s as if time is moving backwards. The main habit I noticed was that I reflexively asked quite a few questions. Back then they were not very well formed ones, but I already was showing signs of resisting the idea that my job was to tell my students. I imagined myself as some sort of disciple of Socrates as presented by Plato in his dialogues – especially the Meno. I, of course, was not then (nor am I now) as clever as Socrates in my questioning, but I am pretty committed to it. When I was a doc student in the recent past my dissertation topic concerned math teachers’ questioning habits in the classroom. I videotaped one of my lectures then for a doc class. I was in my 19th year of teaching and I was still asking questions of my students all the time. My prof commented by saying, ‘You certainly do ask many questions.’ I even heard myself asking questions such as ‘How do you feel about that?” and ‘What is it about this equation that you really don’t like?” I think that personalizing the math this way is helpful.

So, why am I writing about questioning now? I just read a blog post that I found through twitter. It is over at Selected Reads and you can click through to read it. It is a review of a book called Make Just One Change : Teach Students to Ask Their Own Questions and this book is now on my to read list. The reason this caught my eye is that I recently had the pleasure of spending some time with a former student. I went to The North Carolina School of Science and Mathematics for their Teaching Contemporary Mathematics conference just last week. I was delayed in my return home due to lousy weather in the northeast but a former student named Chris saved me. He was my hero for the night picking me up, taking me out to eat, giving me a place to sleep, and taking me back to the airport. Chris lives in the research triangle area and he dropped everything right away to save me last Saturday night. Chris and I each started at Oak Hall School in Gainesville, FL in the fall of 1987. He was in 6th grade, it was a 6 – 12 school at the time, and I was a 23 year old new teacher who wanted to teach math but was saddled with teaching a computer elective on Apple II computers to middle school students to help round out my part-time job. Chris was one of my students then and I had the pleasure of teaching him four more times before he graduated. In a way, I grew up as a teacher with Chris and a small groups of crazy talented kids. It was a small school and I ended up teaching Algebra II Honors, then Precalculus Honors, then AP Calculus AB and AP Calculus BC to a group of six kids all the way through. Other kids were with us early on, but we kept losing kids to graduation along the way. Chris and two others were juniors in BC with three seniors. 

Over the years I have been fortunate enough to have had a number of students and parents say some really nice things to me about their experiences with me. I have also had some not-so-nice things said, but I want to focus on the good tonight. One of Chris’ classmates was a girl named Ashley. About a week before their AP BC exam I asked Ashley how she was feeling about the upcoming test. She said, ‘I’m not worried at all. I know that when I get confused on a problem I’ll just hear your voice asking me the questions you always ask.’ Now, remember, she had been in my math class for four years so she had heard me ask MANY questions by that point. But, I think back to that conversation often and I am very proud of her for internalizing the habit of asking questions when confused rather than feeling helpless – a reaction we have all seen TOO many times. The other conversation that rings in my ear is one I had with Chris about 5 years ago. I was living in Jersey at the time and Chris was living and working in Manhattan. He was doing some high powered finance work involving derivative calculations (not the algebraic derivatives I taught him!) We would have lunch periodically and one day he was recounting a problem that had been vexing him. He told me that he told his boss he needed to take a long lunch to get away and clear his head. When he returned from lunch his boss had made a major breakthrough. Chris was talking about this and he said  ‘Jim, he reminds me a bit of you. He just asked some questions about the data that I didn’t think of asking.’ I though to myself that if a student as bright as Chris remembered that I asked him questions he did not think of then I must have been doing something right.

I’m proud of these stories and they give me hope that this habit of questioning does seem obvious to some of my students. But I also know that Chris and Ashley are outliers in this respect. I see this every day with students who are really bright, students much smarter than I am. I give them unusual problems and they just give up. They don’t ask themselves the kinds of questions that they need to propel themselves toward some kind of solution. They don’t look things up to remind themselves. These aren’t bad students by any means, but they do not reflexively ask questions. I am hoping that this book might give me some insight.

As I strive to become more invisible, I realize that the best way to do this is to help my students become better questioners. I know what questions to ask and when I am prompting them they almost always make notable progress on even the toughest question. If I can get them to ask the questions themselves or, as Ashley described, hear me asking those questions, then I’ll feel much better about becoming an invisible teacher.


Addendum on a snowy Monday morning – Just saw this article called 5 Powerful Questions Teachers Can Ask Students over at EduTopia thanks to a twitter link. It’s worth a read to seed some important ideas.