On a Roll

Man, my Geometry students are on a roll right now. Today we went through our same new HW procedure again. I was quiet for the first 5 – 8 minutes of our 40 minute class while my students shared their HW with each other. They were asking each other good questions and catching each other’s mistakes. They are still a little shaky at times on their line equation writing skills and their line intersection skills, but the mistakes they are making are much more of the arithmetic and detail type rather than broad conceptual mistakes about what to do.

Today we were concentrating on medians and they seem convinced that the medians should always intersect inside the circle. Last night’s HW (which you can find here along with all our other HW assignments) was on Section 6.2 and they were finding medians and their point of intersection. I also asked them to find the perimeter of an original triangle and the triangle made by connecting the midpoints of the original. This allowed us to do a little noticing and wondering in class together. Everyone seemed pretty convinced that the perimeter of the interior triangle should always be half the perimeter of the ‘parent’ triangle. We displayed this on GeoGebra and I asked them to pick a certain type of triangle that they wanted to explore. One student suggested that a right triangle would be fun so we moved out vertices to the origin, the x-axis, and the y-axis. The interior triangle was still half the size and now the noticing began. They noticed that the smaller triangle not only was also a right triangle but that its acute angles seemed to be the same as the acute angles of the original. They noticed that the four right triangles formed inside were probably all congruent. They noticed that the centroid of the original triangle was also the centroid of the smaller triangle. Then Tara asked about yesterday’s peek at angle bisectors and whether they would meet where the medians met. I asked if there might be a special triangle they could think of where this is true and Miranda guessed that our favorite right triangle, the 5 – 12 – 13 triangle might be special enough. Sadly, it was not, but I was happy to hear a quick guess at this familiar old friend. Then Julia suggested that an equilateral triangle might fit the bill. I worried about how to manipulate our given GeoGebra sketch to match up and she cleverly told me to start a new screen with a regular polygon. Class concluded by seeing that GeoGebra was confirming that Julia’s guess was correct. What a great 40 minutes! I also made a point of telling them that they were on a roll and I hope it carries over to our GeoGebra lab day tomorrow. This is called Chapter Six GeoGebra activity in the dropbox file I had the link to above.

I should have dwelled a little more on my second stats class yesterday. I was really pleased with the three different formulas that those four groups generated. I was especially intrigued by the group that decided that the minimum number in their sample plus the maximum number in their sample should be a good estimator for the true max in the population. I discussed this idea with my morning stats class and we had a pretty vigorous debate over how appropriate this was. Playing with our TI and drawing random samples of 5 from a group of 342 (kind of like the German Tank problem!) convinced them that this technique actually turns out to be pretty accurate.

It’s easy to actively blog when it’s fun to relate what’s happening in class. I hope I can keep up a reasonable pace, if not daily, for the year.

Beginning New Habits AND a Fun Activity

This morning in Geometry I started by not talking for the first five minutes while my students shared their HW with each other. They talked about their answers, they puzzled over why/where they differed and they talked about using GeoGebra on their own to explore the intersection of perpendicular bisectors of triangles. I was SO delighted I almost wanted to call the rest of the day off.  I did not, though and I’m glad I stuck it out.

We looked together at GeoGebra, reviewed (again) how to find the intersection of lines, we let GeoGebra confirm that we were right. We remembered from yesterday that these lines coincide on the hypotenuse for a right triangle, in the interior for an acute triangle, and outside of an obtuse triangle. After playing with another GeoGebra sketch we all agreed that this behavior made this point of coincidence a pretty poor candidate for the center of a triangle. I pointed out that one of our students had suggested – on his way out of class yesterday – that we should concentrate on vertices rather than midpoints of sides. Again, we let GeoGebra take over and looked at a compromise by constructing a line through a vertex AND through the midpoint of a side. I named this for them as the median. I also displayed that these medians seem to ALWAYS intersect in the interior of a triangle and I named this point for them as the centroid. We all agreed that this name was ‘center-y’ enough. As time ran out, at the suggestion of another student, we asked GeoGebra to construct angle bisectors. It does so, but draws an exterior line as well. They did not complain when I erased them, but I want to examine what is really happening there. It felt a little too much like I was waving away a distraction. We saw that these angle bisectors intersect in the interior as well – setting up a great debate for tomorrow about which center is the center-est. Just thrilled with how they hung together during the intro time and during the quick GeoGebra exploring. Need to commit to both HW review time tomorrow and to revisiting the blur of activity on GeoGebra. I am planning on a lab day for Thursday so that they can manipulate these ideas themselves.

In my AP Stats I tried out the German Tank problem using resources found here at the Stats Monkey site. My two classes dealt with this in pretty different ways. My smaller class (12 students today) worked in 3 groups of four. I made a mistake in responding to one of the first ideas I heard. One group decided to invoke the empirical rule and guessed that the # of tanks was their sample mean plus three standard deviations. I responded positively to them but this simply steered the other two groups into following this lead. In my other class I was smarter and quieter. Here I had four groups of four. One group invoked the empirical rule but they also pooled their three samples together. One group used the inverse normal function on their calculator seeking a point where the area was 0.999. One group added their sample minimum to their sample maximum guessing that they should be (roughly) equidistant from the extremes. The final group doubled their median guessing it should be halfway to the max. I was thrilled with the level of discussion and the variety of responses. A great step forward from yesterday’s disappointment where they largely ignored my Radiolab assignment.

I’ll count this day in the victory column for sure.

First Day Back – A Tale of Triumph and Sadness

My first period class (we call them Bells here, rather than Periods) is my Geometry class. I started by sharing with the the NYTimes story about ‘The Interview’ and was pleased that they quickly attacked this as a system of equation. I had a secret plot for starting with this problem. We are getting ready to explore triangle bisectors of various sorts. I started out with this question for my students: ‘What does it mean to call a point a center for an object?’ Luckily, this prompted a quick recollection of centers of circles along with a nice attempt at remembering a sound definition for a circle. I then asked them to consider what would be the center of a square. One of my students, a freshman named Matthew, quickly proposed that the intersection of the diagonals would be his point of choice. I opened GeoGebra, drew a rather random square and tested Matthew’s idea. We saw that this was in fact equidistant from the vertices. I then asked about distance to the sides. This required a quick conversation to remind them of what we mean when we talk about distance from a point to a line or to a line segment. We quickly came to an agreement that the perpendicular distance was what we wanted. GeoGebra confirmed that this ‘center’ was equidistant from the sides of the square, but I pretended to be troubled that this second equal distance was not equal to the first equal distance. My students quickly overruled me and were comfy with this point as the center. Next, I asked what the center of a triangle might be. I had three students each volunteer and ordered pair as a vertex of the triangle. It turned out that they formed a right triangle. We agreed the idea of perpendicular bisectors (which we had JUST looked at for the square!) was the way to go. Some quick GeoGebra showed that these lines coincided at the midpoint of the hypotenuse. I was pleased that this raised questions. No one jumped to the conclusion that this would always happen and a student named Tara quickly guessed that this was happening due to the original triangle being a right one. I then moved one of the vertices so that the triangle was acute and, happily, we noticed two things together. First, the perpendicular bisectors still coincided as I moved a vertex. Second, they coincided inside the triangle. Matthew then asked to see what happened with an obtuse triangle and we saw the point of coincidence migrate outside the boundaries of the triangle. It was great to notice that they still met at a point, but the idea of a ‘center’ being outside the triangle did not make anyone happy. Matthew observed that this point did not feel very ‘centery’ to him. Awesome stuff. Finally, since we had GeoGebra to confirm our work, it did not seem that intimidating to go ahead and find the coordinates of the point of intersection for these lines. My secret plot of having them think about systems of equations at the beginning of class paid off. Overall, a wonderful way to start the new year. Tomorrow, I’ll try my idea of HW review at the beginning of class and see how that feels.

Unfortunately, the feeling of triumph dissipated quickly. I have two Bells of AP Stats this year and I had asked these students to listen to a Radiolab Shorts episode called Are We Coins? and I gave them a handful of question prompts. I asked everyone to jot down some reaction notes and to bring these notes to class today. In my first class of 11 students I had three who showed clearly that they had listened to the episode. I had zero students with notes. I asked everyone to take out their notes and a number tried to fool me by having a notebook in front of them, but none of these had anything to do with my questions. In my next class of 16 students, three of them had notes and one or two others showed clear evidence of having paid attention to my request. Sigh…

I’ll dwell tonight on the Geometry kids instead and get ready to really dig into this idea of ‘center-ness’ for a triangle tomorrow. A couple already asked, on their way out of class, about using the vertices as anchors instead of midpoints. Should be fun tomorrow. Lots of noticing and wondering and a concession on my part to their need for HW reinforcement. Hoping for another great start to a day.

Thinking About Work and about Homework

I took my mom to the airport this morning which means I have one less reason to start thinking about work again. We start back on Jan 5 – as many of us do, I am sure – and I have a pile of papers to grade and some class planning to do. As I have written before, we are working with a new Geometry book that I wrote last summer. You can find a dropbox link to it here. I have also posted a folder of the HW assignments we’ve written for it here. I’ll be updating the HW folder regularly. I have been compiling a list of edits for the text and I’ll be doing the updating work this spring.

As I think about our next Geometry unit I have really been wrestling with the role of HW and how to incorporate it into my classroom routine. I’ve been mostly teaching AP classes the past few years and I have definitely adopted an attitude that HW is assigned but not graded. I’ll discuss it in class when someone asks a question, but I don’t plan around that activity. These questions from my students pop up on quiz days or on test review days but not too often otherwise. I am under no illusion that the students are doing all of what I ask from them, but their performances indicate that most of them are doing enough to get a grasp of the material. I believe that their understanding and mastery of the material would be deeper with more practice. I also believe that they would practice more diligently if I incorporated HW into my grading system somehow. What I have seen is that any way that I have tried to include this into my grading system encourages copying and shortcuts that are frustrating to me and reward bad behavior. This year, however, I am working with younger students again in the Geometry class. These students pretty clearly crave much more attention to HW than my older kids seem to want/need. Days when I post the last HW on my AppleTV and spend time directly addressing it are clearly appreciated. What I am wrestling with is the feeling that these days also lack some of the creative problem solving feel of days with more open-ended questions driving the conversation. Am I wrestling with the balance between what motivates me and what motivates my students? I know that there is a cliche that most of us become the teachers we had. I certainly think about some of the inspirational teachers I had and try to capture some of that magic. However, I also find myself thinking at least as much about the experiences that were NOT inspirational and try to avoid those scenarios. For me, many of those uninspiring days centered around 20 – 30 minute conversations about last night’s HW. Days where we seemed to be spinning our wheels. I know I was not the only student who had completed the HW from the night before and did not need to have most of those problems reviewed right after completion. I know that I have students who drift away from being engaged when we are going over problems they have already thought about and completed correctly and confidently. I am really struggling with how to best strike a balance between what some of my students crave and what I think is best for them. I think that our time together is best spent on open questions that push our understanding forward, problems that would be a real stretch for them on their own. I want to include those (periodically) on HW assignments and concentrate on THOSE problems when we are together. Given that a number of my students seem to crave the comfort of the routine of doing HW and then going over it together, I need to find comfortable ways to incorporate that without stalling other routines. One way I am thinking of – and I know that this is not original at all – is to have answers posted at the beginning of class and asking students to take the lead on any review questions that pop up. I can have a few kids up at once and cover this material more quickly and in a more student-centered way. This way, we can get to the more challenging questions together more quickly and I can respect the desire to have space for HW review. Since we are writing our own HW assignments, the students have no answer page to look at and this may be an important source of the difference in HW attention desired. My older kids have standard texts with answers in the back of the book for reassurance.

I’ll report back soon after we start again and I’ll write about how this works. In the meantime, as always, I welcome wisdom in the comments section or through my twitter @mrdardy

Hosting a Colleague

One of the professional expectations at our school is that we visit a departmental colleague and an out of department colleague at least once each year. I love visiting other classrooms and my job as chair makes it easy for me to ask in my department. I recently hosted a colleague in AP Calculus BC. She is a history teacher and a class dean and she chose the BC class because it fit her schedule and because I have some empty seats in that room. She joked that she would likely not be able to follow much of our conversation but she was excited to come and watch my rock stars in action. There are two main reasons why I love having someone come to my classroom. First, it reminds me to appreciate just what a high level most of my kids are operating on. In BC it is too easy for me to set REALLY high standards and then be disappointed when a few kids don’t reach them. Having my colleague visit and listening to her talk about the level of conversation – and the speed of the conversations – reminded me of how lucky I am to be working with this group of students. I also always appreciate the questions that my visitors ask about my classroom strategies. The past three teachers who have visited my room are all members of our school’s history department. They all commented on the fact that I use specific students’ names so often in our conversation. I’ll toss out a general question, pause, and then add something like ‘What did you do with this problem Greg?’ or I’ll field a question from a student and turn and say ‘Emma, how did you approach this?’ I am pretty sure that I picked up this technique in my classroom methods courses. I have always taught in independent schools – a story for some other time – and most of my colleagues do not have much background in ed classes as part of their college experience. I wonder if this explains why this practice of mine seems so striking to them. It feels like a natural to me as it allows me to remind my students that I know that they are there and that I want their voices to be part of the classroom.

At a time of year when it is easy to feel tired and run down, this visit really helped give me a shot of energy. How about your schools? Do you have much time to visit classrooms?

Trying to Sort Out Some Ideas

so, I wrote a post on Friday in a time of frustration and took it down later that day. Need to learn to use my drafts folder to sit on these ideas. (Thanks to hermathness and others for this idea)

I have spent much of the past day and a half in bed and will be back there soon, so I am going to divide some of the ides that were on my chest and weighing me down.

i wrote a Geometry text last summer and have been putting together HW assignments as we move along this year. It’s been a tremendous amount of work, but I feel that I am really learning about myself, Bout my students, and about what I value. Instead of just selecting some subset of problems from a text or relying on prefab worksheets, I have really been thinking about what kind of work feels important to me. We have a 14 day period of school between Thanksgiving and Christmas breaks and one of my Geometry teammates offered to take the reins of planning during this stretch. I am grateful for this and it have saved me much energy. Last week my colleague sent out a partial assignment and asked for help in fleshing it out. I scoured through some resources I have and proposed three problems to add on to an assignment where the section of the book concentrated on the Hypotenuse – Leg theorem. One of the problems in particular involved two steel rails that met with no gap and then buckled upward due to heat. I think it’s a pretty cool problem, but I knew that I would have to work it out for most of my kiddos after asking them to ponder it for HW. I am TOTALLY fine with giving my students some problems that I think only a few will complete correctly.  It is becoming obvious that most of my students do not agree that this is a great idea and some of my colleagues disagree as well. I see this as a failu on my part. I have not been clear enough in articulating why I think this kind of struggle is a good idea. I feel that I have a better idea of addressing this with my students than I do with my colleague. She told me she would not give that problem for homework because she did not want to freak her students out. I appreciate her concern about her students, I just happen to believe that freaking them out a bit is pretty important. Especially when it is a low stakes situation like a homework problem.

i ned to frame this conversation carefully and with only four days before winter break, it feels like I ought to sit on this for a little while. As always, I welcome any wisdom you have to share!

 

What Are Grades Supposed to Mean?

I’m troubled today by an email conversation at my school this week. We just came back from a long Thanksgiving break which came after fall term finals (we are on a trimester system.) We have the whole week of Thanksgiving off and some kids did not come back until Tuesday night from their travels. These are some boarding kids and they did not communicate clearly their travel plans with the school. For some combination of reasons they did not return until last night. Two of them are students in my AP Statistics class and we already have our first quiz tomorrow. Now, I understand that these kids goofed up and I know that the administration is dealing with it in some way. Probably either after school detention or Saturday morning detention is in the works for these kids. We received an email instructing us to hold these students accountable for any work missed. I understand that this is a pretty standard response. I’ve been at three other schools and I am certain that they all would issue a similar statement. I am certainly guilty of having enforced such disciplines in the past, but I find myself troubled by this now. Maybe it is my exposure to such a wide world of reflective bloggers and tweeters, maybe it is because I now have children of my own, maybe it is because I live with so many of my students and I feel for them in a way that is noticeably different than my other school experiences. Whatever the reason (or combination of reasons) I find myself troubled by the idea of giving these kids a quiz tomorrow when we have discussed a section for three days and they were only here for one of those days. I know that they could have gotten back sooner, I know that they are seniors and should be responsible. I also know that they are pretty bright kids and might very well perform at a reasonable level tomorrow. I am not feeling that this will happen and it feels to me that a disciplinary mistake will turn into an academic grade punishment. I don’t think that I feel good about this. On the other hand (there always seems to be that ‘other hand’ doesn’t there?) if I give them an extension I know that some of my students – those who came back on time, who were jet lagged or simply tired from travel, those students who would have enjoyed an extra day or two of vacation – would likely feel that students receiving extra time are receiving some extra advantage as a result of their irresponsible actions. I don’t want my students to feel that grades are some sort of competition or some sort of reward system, but I know that many inherently see it this way. For those students, it would seem to be an insult to have an extension offered here. It might even be another sign to them that they should try to ‘game the system’ and take time off when it suits them. I know that I feel that grades should reflect the knowledge that a student is able to display. I don’t think that it should reflect good behavior or bad behavior. I think that there are avenues to recognize behavior issues that are separate from grades, but I also know that most schools feel that pressure on grades is the most effective way to get a students’ attention and the strongest lever we have to modify behavior.

Sigh, I know that there are no easy answers here. I would love to hear any comments here or tweets directed to @mrdardy to share how you deal with these issues. How does your school deal with behavior like this? Are you bound to school policies or do you have the freedom to make individual decisions? Please share your wisdom.

Reflection inspired by Meaningful Quotes

So, a blog post from Prof Ilana Horn (found on twitter @tchmathculture) came across my reader last week. It was titled ‘First, Do No Harm’ (you should head over here to read it) and this caught my eye for a number of reasons. The first is that one of my proudest moments in the sprawling world of internet interactions came when Tina Cardone (on twitter @crstn85  or over at her blog here) grabbed a quote from me to use in her magnificent Nix The Tricks. The following quote, a comment I made about the use of the dreaded FOIL acronym, is the one she used in an earlier version of her terrific book.

I would say, then, that it is not reasonable to even mention this technique. If it is so limited in its usefulness, why grant it the privilege of a name and some memory space? Cluttering heads with specialized techniques that mask the important general principle at hand does the students no good, in fact it may harm them. Remember the Hippocratic oath – First, do no harm.

I’m excited whenever I see a new post by Prof. Horn, but this one grabbed my eye by its title. Little did I realize that her post would be itching my brain for days at a time when I have little spare space or energy. We’ve been engaged in fall term finals at my school. Otherwise, I would have responded sooner.

Prof. Horn lays out some common practices that do harm – at different levels – to students and to their chances of increasing their competency in the math classroom. I’d like to respond to a couple of them and try to gather the wisdom of the internet (or at least the minuscule portion of the internet that will read this post!)

  1. Timed math tests – Prof Horn links to Prof Jo Boaler here and says that our assessments communicate to students what we value. I could not agree more with this statement about assessments. I speak to colleagues about this all the time. If we say to our students that we value thought and process but then give them multiple-choice tests where points are all or nothing, then the students quickly figure out that we do not mean it when we say we value process. What we do is FAR more important than what we say in this arena. Years ago I read a powerful essay about assessment written by Dan Kennedy (you can find that essay here.) I found many of Mr. Kennedy’s arguments to be powerful ones and I remember that my primary takeaway was that we should assess what we value and we need to value what we assess. I tell my students that I want them to be able to tackle novel problems. That they need to be able to tie together ideas we have worked with and apply them in a new context. I often give problem sets for HW that require them to remember from past lessons and from past courses. I tell them that I don’t necessarily expect everyone to get these problems completely correct, but that I think it is important that they grow as problem solvers. If I never put problems like this on graded assessments, then my students would quickly sniff out the fact that I don’t really value that process very much. However, what also has to go along with that in a graded assessment is a willingness to pay careful attention to their work, a willingness to reward thoughtful work with meaningful partial credit, and some careful feedback either on their written work or in a group setting when papers are returned. (This feedback question is also burning my brain thanks to a recent series of thoughtful posts by Michael Pershan over at his blog on twitter you can find Michael @mpershan – I hope to draft something meaningful soon in response to these thoughts!) The belief that I have that is challenged by Prof Horn here is the idea of speed or efficiency being valued highly. I think that I want to argue that efficient problem solving is a skill I want to value and one that I want to reward. Where this gets tricky is that I know that there are certain problems – meaningful, valuable problems – that just do not lend themselves to quick solutions. How do I balance the desire to see my students think and wrestle with new contexts with the desire to reward efficiency and cleverness? I also teach in a school run by the bell system (I’m certainly not alone there!) and I need to think how to work within that system. I tell myself that I balance the points on my tests so that the diligent student who has gained increasing mastery of facts and skills can still earn a respectable grade even if they fail to connect the dots on the novel problems. This only comforts me to a small degree. I know how much grades serve as motivators (and de-motivators) for my students. I know that a student who feels that s/he has worked hard can walk away from an assessment feeling defeated and incompetent simply due to failing to finish one problem. I know that students can convince themselves that their hard work was for naught and that maybe they just are not cut out for this particular challenge. I’ve been at this a long time now and I still do not have a satisfactory answer and Prof Horn’s post really brought that home to me again. What do you say wise readers? Is it reasonable/valuable/important to reward those clever students who can solve novel problems more quickly than their peers? Should this be a valued skill? If it is, then I believe it should be assessed somehow.
  2. Not giving partial credit – I agree 100% with this point. As a teacher of two AP courses, I feel that part of my task is to help my students be ready for the format and the peculiarities of the AP test in May. Most of my students choose to take these tests and for those who are not yet seniors, they feel that their test scores can help/harm their chances to get into the college of their choice. What this means is that I incorporate multiple-choice questions into their assessments. Now, if I tell them that I value process, how can I feel good about MC questions? Well, I don’t. I have dealt with this two ways and I am not thrilled with either of them. Sometimes I simply value each MC question at such a low point total that mistakes will not have a great impact on their grades. The other way I have dealt with them is to decide what the most reasonable incorrect answer is and give partial credit for this mistake. I am not happy with either path. Any wisdom from others who deal with the (sometimes) reality of MC questions?
  3. In the comments section there are some additions like this one – Grading practices that do not allow reassessment. Again, I am wrestling with this and I have blogged about this. In my two AP classes, where I am the only instructor, I allow retakes on unit tests for anyone unhappy with their grade. I have averaged the two grades. I have read some powerful arguments against this from the SBG crowd, but I cannot find a place where I am happy simply waving off performances. I may get there one day but I am not there yet. I am not at all happy with myself or with my students about the current retake policy I have. I hope that I can construct a more meaningful one by the time our winter term starts in December.

So many thoughts rattling around my brain. Thank you to Prof Horn for agitating me with her blog post. Thank you to her commenters for furthering the conversation. Finally, thank you to anyone who reads this and helps to continue to refine my thoughts and practice.

The Mysteries of Students’ Thinking Processes

A busy week of writing letters for advisees, writing a letter of rec for a former colleague, and pulling weekend dorm duty. Back on duty again tonight, so it is three out of four nights now!

Last week was the first time in quite a while that I found myself largely disappointed by my students and I have a couple of questions I want to air out. Trying to understand what students understand through assessment is, of course, one of our big challenges as teachers. People much smarter than I am have been hashing this out for a long, long time. So, I have two stories to share that are each nagging at me.

In AP Stats we are wrestling with probability. Most of my students have had very little, if any, exposure to probability before this class so this tends to be a tough unit. We had a problem on our last quiz that went like this:

Mr. Felps has 28 students in his AP Calculus BC class and 8 of them are left handed. We know that approximately 10% of the population is left handed. Can this situation in Mr. Felps’ class happen by chance?

A number of my students felt that this could not happen by chance. It seemed too unlikely to them. This bothered me a bit since we had looked at some simulations and talked about runs of short duration. We had discussed the law of large numbers and looked at a decent EXCEL simulation. I thought I had covered our bases on this one. But what really flustered me was that the follow up question asked for the probability of 8 out of 28 left handers under this condition. Every one of my students attempted this computation. Almost all got it right. BUT – a number who got it right had just told me that it was impossible for this to happen by chance. Somehow in the span of two minutes they seemed to forget that it was impossible and instead gave me the small percentage chance of it happening. What happens? Why do such good students have these kind of hiccups, especially in assessment situations? Man, it feels as if this is THE golden treasure to find as a teacher. How can we help our students step back and be metacognitive enough to sidestep these mistakes?

The second situation involves my Calc BC crew. We had a test last week and I try not to have too few questions on these tests so that each question does not feel so overwhelmingly significant. i have settled on feeling comfy with 7 questions in a 45 minute or 50 minute class test. Our recent unit on arc lengths and surface areas involve some problems that take a bit of time. To compensate for this while still having 7 questions I threw in what I thought was a gift wrapped set of points. Here is the question I tossed in as a softball for them.

I realize that if I increase my cycling speed by 3 MPH it will take me 40 seconds less time to cover each mile. What is my original speed?

I had students who left this problem completely blank. AP Calculus BC students who were so stymied by this that they did not even write an equation relating the information presented to them. I’ve been wrestling with this for days on a number of levels. It feels like this was an easy gift to them, one that my competent Alg II kids can easily solve. However, this was clearly not the way the problem was received by my students. They felt tricked or ambushed. They feel like it is unfair to lose points on a Calculus test on a problem that does not feel like it has anything to do with Calculus. I sort of sympathize on some level, but I feel that it is absolutely essential for these kids – kids who want to pursue serious, high powered technical degrees and futures – to be able to synthesize and recall old ideas with ease. Man, I am frustrated by this one. I felt I was tossing them a bone and it got stuck in their throats.

I have so much thinking to do (still!) about assessments and understanding what my kids understand.

Wrestling with the Modern World

Sometimes I am convinced that the universe is sending me important messages to sort out. I am not sure if I am always up to the task of making sense of these meanings. In my last post I was wondering aloud about how to incorporate technology into my assessments in a way that made sense. I asked my Calc BC kids to wrestle with a tough problem about circles. The problem made much more sense (to me, at least) when I graphed it using GeoGebra. It allowed me to lock in on a region of reasonable solutions. I asked if anyone out there has logical ways to incorporate this newer technology during assessments. For years my students have come armed with TI calculators. Sometimes they know how to unlock its powers, sometimes they do not. Somehow, the world of GeoGebra and Desmos (and Wolfram Alpha, and and and) seems more dangerous or intimidating to open up to classroom assessments. I worry about how to evaluate my students’ progress when I do not know where/how they found answers. So, that’s one part of what is in my head now. I have struggled with cell phone presence in my school. A little background might help explain where I am. Eight years ago when we moved north I became involved with our local Unitarian Universalist church and I volunteered as a youth group counselor. I attended a number of weekend ‘Cons’ with our youth. One of the persistent messages at these events was that this was an intentional community that was being created for the weekend. The youth were urged to be present to each other and to the event. They were expected to put electronics away for the weekend and they were asked not to engage in public displays of affection. For the most part, they bought into these requests and the energy was palpable. Kids were engaged with each other, they were talking, singing, laughing. It was a fantastic, but exhausting, weekend environment. Just last week I visited a school and sat in on four classes and two assemblies while I was there and did not see one student (or one faculty member, by the way) staring at a screen in their palm or in their lap. Kids were present to each other, to their classes, and to their assembly speakers. I found it refreshing. In my school there is a gathering area right outside my classroom window and I often see two or three kids on one bench all staring at their phones. I know that this is my bias (maybe this bias belongs to others as well!) but I find this dispiriting. In my class, I tend to stand near the door to greet people as they come in and some of them are trudging through the halls staring in their hands and barely aware of those around them. I used to have to spend time getting my classes to quiet down at the beginning of class because they’d be talking to each other as they sat down. Not so much anymore. Again – I know that this is my bias here, but I find this a bit depressing. I try to utilize the language from my UU experiences and since I teach in an independent school I CAN invoke the idea that there is a choice made in being at our school. The reality though is that this choice is often the choice of parents and not my students. At the youth group it was much more a matter of choice by the youth engaged. So, after my school visit I was feeling that my bias was being confirmed and supported by the environment of the school I visited. Then my brains was rocked yesterday by Justin Aion. Justin blogs over at http://relearningtoteach.blogspot.com and his posts (nearly daily ones!) are a treat. I have also had the pleasure and privilege of getting to know him in person here at a workshop we hosted (run by the wonderful Jennifer Silverman) and at twittermathcamp this summer. He is as delightful in person as he is through his blog. Yesterday Justin wrote a pretty moving post (you can find it here) about cell phones and I want to try to address his points as a way to help me clarify my own mixed feelings. His final point is the most important (by the way – read his whole post, don’t just take my highlights!):

If the answers to my tests can be looked up on Google, are they really worth asking in the first place?

I want my students to be creating, to be evaluating, to be synthesizing information.  I want them forming opinions and interpreting answers.  It would be great if they could determine the circumference of a circle from it’s diameter.

It would be better if they could tell me which of the given answers is the most reasonable estimate.

A smart phone can’t make judgement calls.  They can’t interpret answers.

If a smart phone can answer my test questions, I’m asking the wrong questions.

I agree 100% with these sentiments. When I first visited my current school I saw a chapel presentation that completely won me over. It was one of the 4 or 5 major reasons why I am here. Our  Reverend addressed these ideas and won me over. I do not think that this is the real reason why I worry about cell phones or other connectivity issues on assessments or in my class. Justin writes passionately about students doing what he wants (needs?) them to do while still being connected electronically through their phones or their headphones. What troubles me is a persistent belief that I have that we all benefit when everyone is engaged in class. The student who is doing solid math while wearing headphones is depriving their classmates of a strong voice and they are depriving themselves of the opportunity to explain their own thinking or to hear the thoughts of their classmates. I believe SO strongly that learning ought to be social and interactive. Maybe I am just inflating any logical concerns about relating to each other but that is where my heart and my head are right now. I don’t know how to balance what I want, what my students want, what I believe is best for the group as a whole, and the needs of the individuals. I know that there is a sweet spot there and that it almost certainly varies by class – hell, even by time of day.

I have asked my students to have their phones on their desks this year. We know that they are in the classroom and I don’t want surreptitious use in their laps. I ask them to look up stuff, I recognize that some of them use their phone as a rudimentary calculator. I don’t pretend that these don’t exist and I want to encourage honesty and openness about their presence in the classroom. Some students have complied while others have not. I speak patiently (but consistently) with those who keep them in their laps and text friends during class.

I know that I want my students to interact and I believe that they do less of it when they are plugged in to their phone or their headphones. I want students to research and solve challenging problems and I know that they do less of that when they are not connected to the internet through their phones or tablets or laptops. I chaired a committee at our school that helped develop a 1 – 1 program in our middle school. That program should soon bubble up to our high school. I believe in technology. I do, I think it improves learning and depend understanding. I am jealous of my students when I get to display complex ideas with Desmos or GeoGebra because I am old and did not even have rudimentary graphing technology available when I was trying to learn trig and calculus. I cannot tell if my visceral reactions to cell phones is at all logical and I am trying to sort that out. Justin – thanks for making me think and making me uncomfortable. Anyone else out there reading this – please poke at me through comments or through twitter (I am @mrdardy) I want to sort through these conflicts. I want to create an environment that is meaningful for my students AND for me. I sometimes feel like the grumpy old man yelling at kids on the lawn (even though I don’t have my own yard!) even though I don’t want to believe that is me.

sigh… This stuff is hard.