Category: Uncategorized

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. 

Summertime

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.

Questions

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.

My School – Follow Up

As I mentioned earlier in a post simply called My School we have a large and active international community at our school. Tonight, our dining hall is hosting a dinner of Chinese food prepared with the Lunar New Year (sometimes referred to as ‘Chinese New Year’) in mind. Today at our school assembly time our chaplain delivered a typically terrific talk, this time with Pete Seeger being the focus of his chat. When he finished – a little earlier than usual – he turned the stage over to one of our international seniors named Oliver. He mentioned that at this holiday time many of our students cannot be with their families to celebrate. He then introduced a video put together by another student who goes by the nickname Bobo. Bobo and Oliver had the idea of having the parents of our Chinese students (Bobo and Oliver are each Chinese but we certainly have many other countries represented here) send their holiday wishes to their children. We saw a five minute video of moms, dads, siblings, etc. all wishing their children a happy and safe new year. It was such a terrific gesture by these two young men and it had a great impact on the room. Many of the adults in the community were brought to tears by how sweet this was. Many of the messages were in Chinese and subtitles were presented for some of them. Our chaplain could not help but remark that the wishes of parents – be happy, do well in school, eat well – all transcend cultures. 

 

What a great moment and a reminder of how terrific this community is.

By the way – remember, we have an opening in our math department. Think about it…

Fantastic Afternoon from BC Calc

So, my afternoon crowd was not to be outdone by my morning crew. I slipped in a subtle reference early in the conversation with them so that they would not be inclined to simply introduce the phase shift idea. I wanted them to have a little practice untangling the mechanics involved in dealing with developing a Taylor series. They were very quick to recognize and agree that the coefficients were based on factorials so jumping from the 5th degree polynomial to the 7th degree was pretty easy for them. When I asked for the cosine they were confident about using even powers instead of odds and came to a conclusion pretty quickly. Where life got interesting was when I showed them Michael’s solution from the morning and discussed why i preferred the symmetry generated by an even powered series instead. I also discussed how Michael’s translation idea might give better results for approximating cos x with negative values of x. That’s when they stepped up and knocked me out. They suggested that we take the 6th degree polynomial approximation we had for cos x and do the following: phase shift by pi radians and reflect over the x axis. I am linking to a GeoGebra file that we created. If you want to dig into that file – here are the explanations of the functions.

a and b are self-explanatory

f is the 7th degree Taylor for sin x

g is the phase shift of this by pi/2 to approximate cos x

h is the 6th degree approximation of cos x

m is the crazy reflection/shift to move the cos x approximation backwards to another portion of the cosine curve.

 

Whew – what a day

Fantastic Morning from BC Calc

We are starting our journey through the study of Taylor Polynomials today. I started with looking at  y = sin x and asked them to find a ‘simpler’ function that behaved like sin x does around the origin. I sort of purposely asked this in a pretty vague way and we had a good chat about what I was asking for. One of my students offered up y = x as an answer. This gave us the opportunity to talk about the limit of sin x / x as x approaches 0. It also gave us the opportunity to talk about L’Hopital’s rule. A pretty good start in my mind. Then life got interesting. One student suggested a cubic function but  I was able to get someone to urge an extension to a quadratic function that might match the sin x graph as the next step. I’m not sure what he (Michael) saw that made him jump to a cubic. He’s a really insightful student. So, I held that off and got to working on a quadratic. We agreed that the quadratic better agree with sin (0) and that the slopes should be the same. Someone suggested that the second derivatives should match as well. This resulted in a quadratic with a leading coefficient of 0. Not so good. It would have been easy for them to give up on this process, but Michael had already suggested the cubic. We had success in finding one and a GeoGebra graph confirmed that this worked over a larger region than the simpler linear function. We jumped into a fourth degree polynomial – again with failure due to a leading coefficient of 0. Here is where things really started getting promising. I asked why this was happening and a different student remembered something about even and odd symmetries. The precise language did not arrive right away, but we were able tp get that together as well. Pretty promising… A fifth degree polynomial was found and it graphed even better than the third degree. The students were getting a little tired of this process so I very quickly convinced them of the behavior of the 7th degree approximation. Michael (he was on fire this morning!) recognized the factorial pattern unfolding so we jumped ahead to the 9th degree polynomial. We were feeling pretty good about ourselves at this point. I asked them what function we might be interested in next and, luckily, I was told that cos x would be our next target. I told them that I would be quiet for the next few minutes while they worked this out for themselves. Normally, I am not at all interested in my students – especially ones at this level – simply mimicking my solution patterns. In this case, I thought that this new process was intimidating enough that they would just try to parrot my work. I was fine with that idea, this unit will take some time. However, my best laid plans were foiled. About a minute after I sat down dramatically Michael asked ‘Why don’t we just replace each x with x + pi/2?’ I was SO HAPPY, but i tried to hide that for a moment. Luckily, he spoke pretty quietly and his classmates were still working. I went back to GeoGebra and wrote a new function in his honor. Taking our last guess of h(x) which was our 9th degree polynomial and writing m(x) = h(x + pi/2) and I displayed this graph on top of the graph of cos x. It was a fantastic match but it did not have the symmetry that we had seen for the sin x approximations. The students who had plowed ahead with the polynomial model gave me their 9th degree solution and we looked at three graphs together. The cos x graph, the shifted sin x Taylor series and the cos x Taylor series. A really terrific conversation ensued. Today is what we call a T day where we have 50 minute classes. This felt like an enormously productive 50 minutes. I hope that the afternoon goes at least half as well.

Late Night Reflections

The NCSSM conference ended around 1 today and my flight back home got cancelled due to snow. Luckily, a former student of mine lives nearby and he graciously has offered his place for me to sleep. We also had a great night out catching up.

 

I’ve been thinking more about @JustinAion’s quote from a couple of blog posts ago. He said that he only feels like he is teaching when he is answering student questions or going over examples. After the sessions here at NCSSM I am even more committed to fighting that urge – the one that Justin put his finger on. I want to be more invisible, give my students more room. Ask more than tell. I want to analyze their thinking more than their answers. I want to think deeply about their questions and misunderstandings. I hope to get better at this and I’ll report back as I try some new ideas out about how to do this. 

 

A quick aside based on a  conversation before one of the sessions. A teacher behind me was talking about his daughter Maya who had recently learned to spell her name reliably. He mentioned that she had also recently learned that her middle name was Sage and she decided that she’d rather be called Sage. She was still spelling her name the same way but she was announcing that she had written Sage now, instead of Maya. I have to imagine that the set of alphabet symbols that we would read as Maya simply read to her as her name. If her name is now Sage, then those symbols mean Sage. I was reminded of my own 4 year old who we call Mo. Over the past five months Mo’s ‘signature’ has evolved from OW to WO to OM to MO. However, whenever she sees something with her name on it she simply says it is hers. She seems to see no difference between them. I wonder how similar this is to some of our students writing down things like (x + y)^2 = x^2 + y^2 and not having any recognition that it does not match anything we’ve written.

Conversation Follow Up

A brief one tonight as I am solo dad while mrsdardy is off at a work event.

 

My AP Calculus BC class has a test tomorrow. Their last HW assignment – some review problems I put together was assigned for Monday night with the idea that it would help guide their studying and that today would be a day in class to discuss any concerns. Well, the class where I had the sprinting is similar to cramming conversation did not seem to take my hint at all. It was clear that plenty of cramming is planned for tonight. even asked the one girl directly about our conversation and she just sort of shrugged and said she’d be ‘sprinting’ tonight. I know that these sorts of habit are hard to change. I mean HARD to change.

 

So, tomorrow morning I teach a couple of classes then head to the airport. I am off to NCSSM for their Teaching Contemporary Mathematics conference. I went there about eight years ago and was wowed by Dan Teague, Floyd Bullard, and the whole atmosphere of the place. This time I am going to make a point of meeting Daren Starnes – the author of the Stats text I use. He’s also the Dept Chair at Lawrenceville and he’s been so helpful in setting up a professional visit for two of my colleagues to go there. They got snowed out Tuesday and have already rescheduled. It’s rewarding to start feeling connections outside of my building and it’ll be nice to  meet him this weekend.

 

Oh yeah, while I’m out on Friday my Calc BC kids will work on Problem Set 3

 

Two conversations and a Blog Quote

Okay – so I’m thinking out loud here. Hoping some wisdom comes from this exercise and/or from brilliant comments by my dear readers.

 

Conversation #1

Working with my outgoing Calc BC group and I comment to one of my students that it’s a tough day for him. He’s on our swim team and they had a 6 AM practice Tuesday morning and a meet that afternoon. One of the other students – a member of our field hockey team – says that her team never ran sprints the day before a big meet. Now, it’s important to understand that our field hockey team has won’t he state championship three of the past four years. This student was a member her whole high school career. I take this opportunity and I ask her if she thinks that this strategy (don’t stress out your body the day before an important match) might be carried over to another realm. I am greeted by a quizzical look and I say ‘Maybe you should not cram the night before a big test.’ Another quizzical look. She asks if I am advocating not studying. I say that the daily diligence of regular work and studying is comparable to daily hard practice in field hockey. Then, relax a bit before an important match (or test) and maybe this is a formula for success. I don’t think that many of my students saw that as a winning strategy.

 

Conversation #2

I just observed a lovely Precalculus class taught by one of my colleagues. The class was working on a variety of word problems – coins, movie tickets, area/perimeter, etc. My colleague is a remarkably calm, zen-like fellow. He sat in a student chair the whole time (sort of invisible!) and asked one student at a time to come to the board. The rest of the class was attentive, offering help to their colleague and generally being cooperative and positive. The teacher kept asking nudging questions of the student at the board. “What do we want to find here?” “How can we relate the number of coins and the value of the coins?” etc. Being an observer in the class (and not stressing out about HOW to do the problems) I saw that my colleague was modeling for his students a lovely strategy for tackling these problems. If each student could play that conversation back in their head as they struggled with any problem, then they would see much more progress. They might still make mistakes, but they’d have a sound strategy for success. What troubled me – and I spoke with the teacher about it the next morning – is the fact that I KNOW that some of them will not ask themselves those questions. They won’t take his advice for attacking these problems to heart. I am not saying that all of our students need to mimc our behavior. What I am saying is that students who struggle, ought to feel that it is a lifeline that is being offered here. When I asked him about this the next morning, I told him that I was impressed by his careful teaching and modeling. His response was something along the lines of ‘I am a big believer in teaching. I just think it works better when learning happens.’ I really don’t think he was being mean or cynical. 

 

Quote

This morning, I awoke to another terrific blog post by @JustinAion over at his blog Relearning to Teach. If you are not familiar with his work, you should change that and visit him. Pay attention to his tweets as well. Life will be better. He closed out his post today with a powerful quote – “Even with everything I’ve seen, done and learned, even with all of the conversations I’ve had with other teachers, I still only feel as though I’m “teaching” when I’m answering student questions or going over examples.

I wish I could scrub that feeling.”

 

I think that I’ll walk away from my computer now and let these conversations and this quote marinate a bit. I know I have some questions, but I am not sure that I can ask them accurately enough yet.