Back in the Groove

Our school has a two-week spring break at a silly, early time in the year. We have been back for a week now and I feel like my students and I are all getting back in the groove again. I know that the dreaded senior slump will continue to pick up momentum but at least I am still seeing some energy and engagement from most of my seniors.

I have a few posts bubbling in my brain and I suspect it’ll be a busy blogging week. Tonight I want to briefly touch on my AP Calculus BC class. We are just settling in to our last major required topic of the year, the Taylor / Maclaurin polynomials. I wrote a little GeoGebra demo (you can find it here) and I started off by showing them (without revealing the mechanics behind the scenes) a polynomial approximation of increasing degree for the trig function y = cos x. We played a little noticing and wondering and saw that at certain stages the polynomial did not change. It did not take long to deduce that this happened at the odd powers of the Taylor polynomial. This led to one student remembering something about the symmetry of cosine, another student mentioning that this was a y-axis symmetry and, finally, a third student mentioning that this is even symmetry. So the lack of development due to the odd powers of the Taylor made a little sense. We then switched to y = sin x (as in the link above) and, unsurprisingly, saw that the even powers seemed to do little or nothing here. We did a little more noticing and wondering watching the Taylor expand on GeoGebra. I should note that all of this was centered at x = 0 (or, in the Taylor notation, we had a = 0) GeoGebra’s sliders allowed us to begin shifting that value and some interesting (and ugly/scary) things started happening to the Taylor equation. My kiddos quickly saw that the equation seemed to be undergoing a simple horizontal transformation – at least in the x terms. The coefficients were changing in some mysterious ways. Finally, we looked at the Taylor series for y = e^x. One of my students asked a great question at this point. He asked – Why are there all those factorials in the bottoms? I skipped this question around the room a bit to see if anyone wanted to make a guess. They quickly observed that exponents in the numerator were clearly attached to the factorials int he denominator but – understandably – they had no solid guesses. Without giving away all the mechanics (we have plenty of time for that) I asked what the derivative of x^7/7! is. I was told it would be 7x^6/7!   Correct for sure, but unsatisfying. I must have made my unsatisfied face because one of my students offered a much cleaner version of that answer as x^6/6!  Again, I did not go into the mechanics at this point, but there did seem to be some sense that this was an interesting thing to note. I was pleased by the power of the graphics of the GeoGebra applet. I know that I could do something similar in Desmos but I don’t know the commands there as well as I do in GeoGebra. I will start class off tomorrow with the power series we derived for e ^ x and I’ll ask for derivatives and integrals of that. Should be fun to see them realize in this format why the derivative of e^x is itself.

Fun to be back and excited to unfold Taylor’s series’ with my students. This was one of the genuinely awe inspiring topics when I studied Calculus. I remember being amazed by this idea and it’s mechanics. I hope I can share that wonder.

Delighted

A quick post here – I want to share something delightful that a few Geometry students did this morning. We had our last test of the winter term today and here is one of the last questions:

Prove that the points A (x, y), B ( x + 1, y + 3), C (x + 4, y + 5), and  D (x + 3, y + 2) are the vertices of the parallelogram ABCD. Prove this is true by one of the following two methods:

  • By showing that one pair of opposite sides are congruent and parallel.
  • By showing that both pairs of opposite sides are parallel to each other.

So, I was hoping that the majority of my students would take the quick and easy option of calculating slopes rather than messing with distances. I also hoped that the coordinates having variables in them would make them slow down, be careful, and remember a touch of algebra. I grade page by page and I have graded three of the papers with this problem on it. One student said ‘We can let x and y be 0 so the coordinates are (0,0), (1,3), (4,5), and (3,2)’ I love this thinking. She avoided the worry of dealing with the variables here. It’s a little slippery to determine just how clearly she was thinking here. She might have just been dodging a bullet. One student said ‘I will first transform this parallelogram by the vector <-x, -y> and then we will have the coordinates A’ (0,0), B’ (1,3), C’ (4,5), and D’ (3,2)’ Now, it is ABUNDANTLY clear that he knew exactly what he was doing. I’m so delighted by this that I felt I should share.

This and my great AP Stats classes today made for a pretty terrific day!

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

When I last dropped by my own blog here to write I was cooking up an idea for my AP Statistics class. I wanted to write a good activity to explore data using my FitBit. I was lucky enough to win a FitBit Flex in a raffle and I’ve been fascinated by it for the past month. I had four volunteers who also shared their FitBit data with me and I put my data (identified) with the data of my four brave volunteers (not identified) and developed what feels like a pretty good activity. Yesterday I displayed my data in an EXCEL sheet before deciding I was better off in a Google sheet. We looked at my data together and with Desmos open we transferred some data two columns at a time and looked at lines of best fit. We tossed around some questions that seemed interesting, we questioned some of the data presented (especially the first data line on the sheet identified as Doherty Data), we made guesses about what relationships were hiding. We discussed the impact of height and weight and just generally had a pretty good time noticing and wondering together. Last night I combined all the data into a Google sheet (which can be found here) and I condensed some of the questions that came up yesterday and wrote them up on a Google form (which can be found here) and today I just set my kiddos loose. We have access to a computer lab so everyone had their own space to work. I dealt cards at random to spread the sheets around, I wandered and answered questions about moving columns on google sheets and how to make Desmos (like this graph) work its regression magic. I had discussions about resting heart rate, about whether calories burned or active calories were more interesting to look at. We remembered the dangers with extrapolation when discussing the y-intercept of these regression equations. We tried to figure out which mystery person might be taller or which one might be heavier. There is real joy in listening in on these conversations, but my biggest highlight today is that I got to show off the spirit of my classroom to a visitor today. He remarked on what a treat it was to witness ‘sense-making’ in action. I want to revisit my questions and make them better next time around, but I am pretty pleased so far. Tomorrow, we’ll start class with about ten to fifteen minutes to wrap up this activity and then we’ll share our discoveries with each other.

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

Fighting for Understanding versus Doing

A pretty interesting conversation unfolded in Geometry this morning. We are getting ready to explore similarity, so I gave the kiddos a quick assignment on solving proportional equations with one variable. This was meant to be pure review. When we started talking about these problems I, of course, heard talk about cross-multiplying, cross products, and even heard one student exclaim something about the old keep-change-flip idea. I decided to stand firm and talk about why we were able to do what we do with these proportional equations. We started simply with the equation like  $latex frac{x}{5}=frac{3}{7}$ One of my students was taking a vocal lead in discussing cross products and I asked her what equation to write next. She told me to write $latex 7x=15$. I agreed that this was correct and most of my students recognized what she was doing. I then asked them to pause and wrote the following equation $latex frac{x}{35}=frac{6}{35}$ I asked everyone what they thought the value of x had to be in this situation. They all seemed to agree pretty quickly that x must be 6. So I got them to agree that an equation with one fraction on each side AND the same denominator demanded that the numerators would be equal. They all seemed to think I was making too big a deal out of this. I then asked them if I could do the following to $latex frac{x}{5}=frac{3}{7}$. I asked if I could multiply the right hand side by$latex frac{5}{5}$ while multiplying the left hand side by$latex frac{7}{7}$. One student protested that I need to do the same thing to each side of the equation. I, of course, agreed with her but I asked her to look more closely at what I was doing. She agreed that I was doing the same thing even though it looked different. Most of my students still seemed to think that I was making too big a deal out of this.  Next came the payoff. I picked the following problem from the homework: $latex frac{x}{4}+5=frac{x}{5}+4$ I pointed out that our cross product idea was not really a comfortable fit here. My KCF student quickly suggested that we clear the fractions out of the problem by multiplying by 20. I agreed that this would certainly work but asked if I could try something different. So I wrote the following equation:$latex left ( frac{5}{5}left ( frac{x}{4}+5 right ) right )=left ( frac{4}{4}left ( frac{x}{5}+4 right ) right )$ I was immediately met with resistance. I begged for patience and made them a promise. I told them that I would carefully explain why I was doing what I was doing and that if they unanimously decided that they did not like this approach, then I would cease and desist. I pointed out that we were, again, doing the same thing to each side even though it looked different. I made an argument that multiplying by smaller numbers decreased my chances of arithmetic mistakes and I pointed out that this technique made the common denominator for the problem obvious. The equation became $latex frac{5x}{20}+5=frac{4x}{20}+4$. I saw some signs of visible relief as they saw that this was now a pretty easy equation to process. Combining like terms gave us $latex frac{x}{20}=-1$ and a conclusion that $latex x=-20$. I then solved the problem the more standard way by multiplying everything by 20 to begin with. I felt like it was a bit of a triumph when they voted that this new technique did not need to be banned from our vocabulary. I know that this is not revolutionary, but I certainly think that I made some strides here. My students are well-trained in mechanics and they know what works. I want to have serious conversations about the ideas behind why these techniques work.

Back at it again tomorrow!

PS – Thanks to David Wees and Zach Coverstone for valuable assistance in learning some LaTex for this post. I hope it looks right when I hit publish

It Takes a Village

I have a math related post brewing in my head, but I want to take a moment to relate a story from this past crazy week. In NE PA we have had cold and snow for days and the last time our high school started at 8 AM was a week ago on Thursday. We had a scheduled day off on Monday , three two-hour delays and a late start due to a faculty meeting. My children go to our lower school about three miles away and they take a bus isn’t he morning. On these delay days I am not entirely confident of timing with my kids on the bus and my students lining up for morning class. So, I tend to take my children to school a bit early and get to class. With other young children on campus we share trips down the avenue. Friday morning as I was waiting in the side stairwell with my two kids, waiting for a neighbor to pick them up, my phone rang. It was another neighbor offering to take my boy in with his daughter since they each have band practice. The idea that multiple people are offering to take my kids to school is pretty heartwarming.

Last night my son’s middle school basketball team was invited to have a scrimmage during half time of the varsity game and then they joined the big boys for a pizza party. A great community night. Tonight in the dining hall my five year old daughter was playing tag with a sophomore in the high school running around the place.

there are certainly some drawbacks to living Ina dorm and being on duty pretty much all the time. The past few days remind me of the benefits of this situation.

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.

1st Bell – Geometry. I was looking forward to returning their excellent tests from last Friday but first I wanted to dip into our new unit. I passed out paper, rulers, and protractors and gave them a simple task. Draw some polygons. I insisted on not defining what a polygon was and I did not reveal why they had protractors. I tasked each of them with drawing six polygons and I saw some pretty great stuff. Complex, crisply drawn concave polygons. Some students stuck to the middle of the road and drew squares, triangles, etc. Then the fun began. I started asking for definitions of polygons and I framed the question this way: Explain to my 11 year old son what a polygon is. Brainstorming began. I heard about the need for line segments as sides, I heard about the limitation that there had to be at least three sides, I heard conversations about polygons that pointed inside versus those that did not. Someone offered up the word concave and I pressed for a definition. Everyone seemed happy about the inside-ness of some points on a concave polygon. I pressed them not to use the word point for where the line segments met and they offered up vertex as a better alternative. We decided that each vertex needed two (and only two) line segments and one boy suggested that polygons that were not concave (we agreed to convex quickly) should have angles larger than 90 degrees. He backed off of that but I will definitely revisit this idea soon. When he tossed out that idea he was greeted with references to equilateral triangles. Well played. Then the highlight of the morning came for me. I drew a figure on the board that was made of line segments, that had two line segments at each vertex but it was clearly not a polygon. The reason why is that my figure was not closed. One of my students used that exact language and I pressed, again with my 11 year old son in mind, what we meant by closed. Miranda said ‘Imagine it has water in the boundaries. If it’s closed, the water can’t get out.’ I thought that this was a lovely image. I then closed my crazy drawing but blocked off access to some regions while doing so. I was quickly told that the water needed to be able to get everywhere. I hope that this image stays with my students the way that it is staying with me. As we wrapped up class in a blur of vocabulary about quadrilaterals one of my students said to her neighbor, ‘What a great way to start the week. We got to sit and draw.’ I’ll count this one as a success.

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.

More Thoughts About ‘Helpfulness’

Tomorrow three of my four classes will be taking unit tests. I have always devoted the class day before a test to review. Over the past 5 – 7 years I have become more and more insistent that a review day should be a day where I am here to answer some questions that students come to class with and to help facilitate some meaningful conversation between my students. What many students seem to believe is that review day before a test is simply a time for me to tell them exactly what will be on the test. I always come to class on these days with some prepared questions in my back pocket and I always dream that those questions will stay there. That is not often the case, and it certainly was not the case today.

My Geometry class, the one I’ve been SO proud of recently, was in pretty good shape. We looked at our last HW together, they had some good questions about that but they could not really generate too many meaningful questions of their own. I displayed the review questions I had prepared and they perked up and were terrific in joining in the conversation. I just came away wishing that the class had been more about them and what was on their mind. In retrospect, perhaps it was exactly about what was on their mind. They are concerned about what I am interested in right now so that they can glean some important clues about preparing for tomorrow’s test. Sigh…

My two AP Statistics classes are in a different place emotionally than my Geometry class is. They are almost all seniors and the energy level that they brought back from winter break is distinctly different than the energy level I see in my Geometry students. I gave them class time yesterday to work on their own or with their neighbor on the review exercises at the end of their most recent chapter and my observation is that there were relatively small pockets of productive conversations. However, there were also quite a few incidents of aimless chatter, obsessive checking of their phones, silly debates, and general non-statistical conversations.

So, I feel that I am asking myself the same question I asked myself on these pages just a couple of days ago. How can I be less helpful in the standard sort of hand-holding way that my students want me to be while actually being helpful to them in modeling smart behavior about how to work, how to be metacognitive, how to be reflective, and how to be more self-aware. Trying to recall who I was when I was in high school is probably not the best exercise in answering these questions. I was a different person then than I am now. I am remembering through a distinctly tinted memory lens and I am not teaching four classes of teenage Mr. Dardys.

Gotta keep thinking and keep pushing.

The Perils of Being ‘Less Helpful’

i am guessing that most people who will read this are familiar with Dan Meyer’s TED Talk. When I first saw this it crystallized some ideas that had been festering for awhile. It also articulated some thoughts about my evolving practice as a classroom teacher. I have shared this video with my department colleagues and have also shared it in class with students. As I have mentioned before, most of the teaching I have been doing at my current school (I arrived here in 2010) has been with AP Students. They are pretty well equipped to deal with the sense that I am being ‘less helpful’ with them. Of course, I want my students to feel supported, but I also want them to be pretty self-sufficient. We have a feature in our schedule where school ends after eight periods ( we call them Bells ) and every teacher is expected to be in their room for Bell 9 which is a time for students to come for extra help. We call it conferencing here. This background sets up my post for the night.

i blogged a number of times last week about my Geometry students and the success they had this week. I also shared the news with my Geometry colleagues. I teach this course with two terrific colleagues, but we do not have very similar principles about a number of items. When I popped in to visit one of these colleagues last Friday she said she was happy to hear that the quiz results were so good and she said that she had four of my students in her room Bell 9 on Thursday and she had to ‘talk them off the ledge.’ I’ve been sitting on this for a few days because I am not at all sure what to feel about this. There are, of course, no rules about who you can visit for conference help. I was working with someone Thursday afternoon and missed an brief, but important, meeting that afternoon, so I know that these students did not go to my colleague because I was not available. I have to think that they went there because they thought it would be more helpful to ask her questions. I honestly believe, I REALLY do, that it is vital to hear other voices explain ideas. This is why my room is set up in the form of two big conference tables. I also wrote last week about the great work they did in talking about HW with each other last week. I fear that my students did not feel that asking me for help would be very helpful. I am not at all sure of how to proceed here. It has been awhile since I taught a big group of students this young and I have spent so much time with AP kids that I fear I am developing a reputation as someone who is less than helpful. It’s kind of ironic that I tout the virtues of this video and share it with students and I am now bothered by the idea that this image might be following me a bit. I know that my Geometry colleagues have a different attitude about the type of help that students need/deserve than I do. I have to figure out how to reconcile my beliefs about how to teach and how to answer questions with my belief that it is okay – even necessary at times- to hear another voice explain things. I just wish my stupid ego did not get in the way.

Any wisdom out there for me?

Success!

I have had a very active blogging week thinking about (and writing about) my Geometry class. I have three preparations this year, AP Statistics, AP Calculus BC, and Geometry. I’m not proud of it, but I know that my attention to each class varies at different times of the year. Iy’s not a simple matter of 33 1/3 % of my planning energy being spent on one class at any time. Do many of you go through this as well? By the way, how many preps do most folks have?

Anyways, I blogged in December about my discomfort with HW in Geometry and gathered some nice ideas. I blogged about my decisions about changing habits and it has felt like a raging success. Five to seven minutes at the beginning of class of students sharing their work with each other and correcting each other/reinforcing each other/ sharing their miseries, etc. It’s just been a really terrific week with them and I have let them know how much I appreciate their demeanor, their energy, their willingness to share with each other. Today we had a quiz (you can grab it from here) on Sections 6.1 – 6.3 of our text (you can grab that here) exploring centers of triangles. We’ve talked about perpendicular bisectors, altitudes, medians, and angle bisectors this week. We have played with GeoGebra and looked at how, in each case, all three segments have a common point where they coincide. We’ve talked about which ones could coincide outside the circle and those are not popular choices as the best center of the triangle. We had a great lab activity yesterday (you can grab that here) and it developed into an interesting debate where one group of students nominated the intersection of the angle bisectors as the best representation of the center of a triangle while the other three groups all felt that the intersection of the medians was best. As we had a healthy debate I found myself wishing that I had been clever enough to have physical triangles to manipulate. Next year, I want to be prepared with cardboard triangles of various types with these two candidates for center marked out. I dropped the ball on this one anticipating that everyone would feel best about the centroid. What really impressed me was that the group arguing for the angle bisectors had GeoGebra construct a circle that had this incenter as its center and showed that this circle touched all three sides. I was THRILLED that they thought of this argument.

So, this morning I felt confident as my cherubs asked their last few questions before the quiz and the results are in. I have 12 students in this class and 4 of them earned perfect scores with another 4 earning an A on the quiz. Their class average was 93%!!! I’m thrilled by this. I think that this is due to a number of factors.

  • In general, my students have had more energy this week in January than they did in the few weeks leading up to our winter break.
  • I believe that the HW strategy has made a positive difference.
  • I believe that the extensive use of GeoGebra in class is finally spreading to the home. I have overheard a number of students this week make reference to looking at GeoGebra while doing their HW this week. I am a firm believer in the power of these graphing programs and, for my Geometry students at least, I think that this is the best of the bunch.
  • I worked hard during break planning out this unit for me and for my Geometry team of two terrific colleagues. This thoughtfulness has paid off.

Oh yeah, one final thought. As a long-time Calculus teacher I have a strong preference for lines in the point-slope format. Every one of my students presented at least one of their line answers in this format.  Woo-hoo!!!