Catching Up with the Past Week

So there are a couple of activities this past week that I want to write about. However, I have been swamped with meetings so I have fallen a bit behind.

In AP Stats we have finished our required curriculum as of 8 days ago. I am a big baseball fan and my favorite team is the New York Mets. They are having a pretty wonderful start to their year (or at least were until the last few days) so last Friday I posed the following question to my kiddos: Given that the most optimistic projection I saw for the Mets’ season had them pegged as an 87 win team, what is the likelihood of their current record (which, if I remember correctly) is 10 – 5? I liked this for a few reasons. First, it concerns baseball and likely would have a positive outlook for my Metropolitans. Second, it was not so focused on the most recent material at hand. My Stats students tend to know recent material well but struggle remembering other procedures that have not been practiced as recently. Third, it generated some nice thinking out loud about what approach to take. Being more of an algebra stream guy myself I immediately placed this in the context of a probability problem and was prepared to go down a Pascal’s triangle/binomial theorem path. Most of my Stats students don’t tend in this direction so their conversation focused instead on comparing proportions – the 87 – 75 projection with the 10 – 5 proportion. They suggested running a two proportion z test and looking at the corresponding p-value. This opened up the avenue for me to sneak in my approach and make a connection pretty visible to them. Turns out that we felt that we had enough evidence to reject the null hypothesis of the Mets being an 87 win team in favor of believing that they will exceed that win total. Their recent 5 – 5 run of games might adjust that but I do not want to know this – so I will not re-run the test right now! After we checked our trusty TI to find the p-value of this test I reminded them of the probability approach and we set up the appropriate term of the binomial expansion. Guess what happened? This calculation matched the p-value of the two proportion z test!!! This is one of those ideas that we discussed but somehow seeing these results side-by-side seemed eye opening to my kiddos. A triumph on a number of levels!

In my morning Geometry class we dipped our toes into an exploration of radians yesterday using the ProRadian Protractor designed by the fantastic Jennifer Silverman (@jensilvermath) and using an activity that she designed. I wrote a follow up HW assignment that my kiddos worked on last night. I also linked to a fabulous web site that allowed my students to explore radian measure and I shared these notes with my colleagues. There is also a lovely GeoGebra applet (also designed by jennifer Silverman) that is linked from the worksheet. I was totally excited to explore this activity with my students and I had a really nice chat with one of my teammates.

I handed out the radian protractors as well as our regular old angle protractors and we had a nice conversation about similarities between the two protractors. We had a lovely discussion about this but, looking back on yesterday , I think I allowed too many clues to seep into the conversation too quickly. Jennifer’s activity is a terrific one and I got in the way by loading too many conversations in at the beginning of the class. By having students come to my screen and try to identify where one radian measure would lie on the circle AND by having the protractors side-by-side I reduced the mystery element that I think should have been part of the classroom activity. I think I took away the opportunity to discover what was happening here. I did have one student give a GREAT explanation of why the quadrilateral radian measure was twice the triangles radian measure. She invoked a proportional idea and referenced our (n – 2)*180 formula. I had a number of students quickly see that the ratios we had been working with before ($\frac{x}{360}=\frac{arc}{2\pi r}=\frac{sector}{\pi r^{2}}$) could be easily extended to add one more simple fraction of $\frac{x}{360}=\frac{\theta }{2\pi }$. That definitely felt like a triumph. So, the lesson I learned here – and I hope I remember it for next year – is to be a little more minimalist in front loading this conversation. I think that we can touch on all of these resources and really let the discovery sink in, but I feel I nudged them a little too much this time around. So the plan for next year is to hand out the radian protractor and work through the worksheet. Then hand out the angle protractor and talk about comparing them. Then, the next day after some time to think, show the web app and have them identify where one radian is. Let this unfold a little more slowly.

Rock, Paper, … Chi-Square!

So I tried to start the conversation with  a simple question – Should you do better when you think about the game or should you do better by random number generation? This lead to a quick decision that the expected value of a random number generator would be an equal distribution of 8 wins, 8 ties, and 8 losses for each set of 24. Now the table is set for the important principles of the Chi-Square test. Let’s talk about the difference between observed results and expected results. We also had a great conversation about how it appeared that the random number generator actually outperformed many people – especially in the expert mode. We talked about the fact that the expert mode was trying to predict behavior and how the randomness involved here might actually play in our favor.

In the week plus since this experiment I have been able to refer back a number of times to this experiment and it feels like my students have a pretty good handle on their task here. We have our unit test tomorrow so I hope that my optimism will be supported by some data.

In addition to thanking Bob Lochel I also want to thank a new twitter pal, Jennifer Micahelis (@MichaelisMath) who engaged me in a conversation about this experience and prompted me to gather my thoughts and write about it. I definitely will revisit this experiment the next time I teach this unit.

It’s Not Just a Dream – The Reality of a Data Project

Right now my AP Calculus BC class is taking their final test of the term. I hope I am as happy grading those as I am thinking about my AP Statistics team right now.

Dreaming of a Good Data Project

Our school hosted a ‘Maintain, Don’t Gain’ campaign through the Thanksgiving and Christmas holidays. Those of us who volunteered to be weighed in before and after were candidates for a raffle if we met the goal of no weight gain. I managed to lose 2.2 pounds and got lucky in the raffle by winning a FitBit Flex. I hooked it up on Jan 28 and I am thinking it will help me in AP Stats next week. My cherubs have a test this Thursday and then five more school days before our two-week break. We hilariously call it spring break even though it feels nothing like spring in these parts. Anyways, I am thinking of downloading all my data into an EXCEL sheet and challenging my AP Statistics scholars to dig into this data. As an added bonus, I know a number of them wear a FitBit as well, so we might be able to get a nice data set out of all of this. What I am wrestling with are the following questions/concerns:

• I do not know how sensitive FitBit is in its calorie counter. I have lost some weight in the past month (yay me!) and I do not know if that would interfere with looking for a connection between steps taken and calories burned.
• I am not sure how consistent FitBit is with correlating steps and distance. Are there any FitBit pros out there who can let me know about their experience with this? You can comment here or tweet me @mrdardy
• I want to ask some structured, guiding questions but I do not want to lock them in to my ideas of what might be interesting. I just do not know how focused they  will be or how sophisticated they are as statisticians at this point.
• Debating whether this is better as an individual project idea or a small group one. I am inclined to think that groups are better here. Any thoughts or advice about this?

So, I am shamelessly asking for help and wisdom here. I thank you in advance for any smart comments/tweets/emails/etc

Working on the Holiday

So, one of the oddities about teaching at an independent school is that days off that are taken fro granted most places are seen as prime days for campus visits here. So, we were in session today. No need to feel sad about this though as we have plenty of vacation time as well. Just another Monday.

But…it wasn’t. It was a terrific teaching day and I want to make sure that I make note of it even if only for my own pleasure.

Bells Three and Six – AP Statistics. My senior heavy Stats class did not come back from winter break with much of a sense of urgency. I did not want to just launch right into a new chapter on the heels of the disappointing chapter test we had on Friday. My Computer Science colleague had recently shared with me information about Sicherman Dice which are two six-sided dice that are not standard dice but their sum has the same probability distribution as the sum of two standard dice. I presented my students with a  challenge. Describe two six-sided dice that replicate the probability distribution of two standard die. No other directions really. I have a rudimentary handout I gave them and you can grab it here. I fielded questions as they chatted about this problem with their neighbors. Can the die have negative numbers? Can the die have fractions? Can the die have zero? I kept replying in the reluctantly affirmative and checked on their progress. Most of them had a pretty logical attack where they would transform one standard die in a certain direction, say subtract two from every face and then transform the other one in the opposite direction. Not very sophisticated, but it was nice algebraic logic. One student was working on fractions trying to balance combinations of 1/3 and 2/3 so that she would always get integer answers. Overall, it was the most focused energy from this group that I have seen in two weeks. I hope that this is an omen for our next unit. They were pretty surprised by the reveal and I am curious, in retrospect, that they never asked if any die could have repeated values as this is necessary for the Sicherman Dice to work their magic.

Bell Seven – AP Calculus BC

We’re just getting ready to start exploring the magic of Taylor Series. We took baby steps today reminding ourselves of the language of arithmetic and geometric sequences and series. I always think that this material is such fun to untangle. Tomorrow we’ll play with GeoGebra and I will try to tease out of them the key ideas about how to make a polygon behave like the sine function. Nothing much else to report here.

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.

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

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.

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.

Catching Up and Looking for Some Ideas

There are a few ideas/questions banging around in my brain. No school tomorrow here so I can relax a little more than usual on a Sunday night. I’ll try to be coherent and I hope to get some feedback here or through twitter (where I can be found as @mrdardy)

On Friday our school had the day off and we have been encouraged to use this as a professional development day by our administration. I chose to travel a few hours to visit a school where an old friend is working. The school does some interesting work in the STEM arena and they balance an IB program as well as AP expectations. I gathered some ideas that I will be bouncing off of my colleagues and administrators, but more importantly I just felt energized. I walked away excited to have made some new contacts, happy about many of the things we do at our school so well, interested in figuring out how to develop cultural pieces to support some ideas that work there, and filled with some ammo to talk about the need for schedule changes at my school. As a young teacher I never visited another school. I have long had the habit of visiting other classes at my school and I never feel like I do that enough. It has only been since I moved north 8 years ago that I started making the effort to visit other schools and I cannot recommend this enough. Where I live I pretty much have to drive two hours or so and I have done that the past two years. Every time I have reached out to another school I have received nothing but positive responses and a generous  expense of energy in making the visit happen. I also want to take this space and time to extend an invitation to anyone who wants to come and see our school in action.

On Thursday my AP Stats classes had a group quiz. I stood at the door with playing cards in my hand. Students took one (blind) from me and were randomly assigned to groups. Each class had four groups and each group had a different quiz. There was one question in common to all quizzes but otherwise they each had five different questions. It was SO much fun to listen in as they wrestled with these questions and as they explained ideas to each other. There were some healthy debates but it never got tense or unpleasant. Our school has a very international flavor and I was especially pleased to hear the voices of my international students in these conversations. So much of the material in this course is based on careful reading and vocabulary and I sometimes worry about whether this gets in the way of these students accurately showing me what they know. Have not graded them yet – that is tomorrow morning’s task – but I fully expect them to shine.