Category: Uncategorized

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.

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.

On Thursday my AP Calc BC class took a test on integration techniques. The last question on the test was this – Divide a pizza of 14 inch radius into three equal portions with two parallel lines. Most of my students wisely chose vertical lines. Two chose lines in the form y = mx + b, a bad choice. I went into this intending to give full credit even without a numerical value for the line equations. Setting up the integral appropriately is where the calculus is in my mind. Here is what I find myself wishing after this test and after looking at their work – I wish that they had access to desmos or geogebra while they were taking this test. I wish that they had something much more powerful than their TI calculators to visualize this, to try out ideas, to narrow down where the solution needs to be. I had to struggle through some ugly algebra and some calculus that should have been cleaner and more obvious. I’m impressed by the patience and perseverance I saw but I am frustrated since I know that better tools can help them work smarter on a problem like this one. How many of you out there have a setup where your students have access to these tools on assessments? Am I overthinking this by worrying about internet access during a test? Should I just trust that reasonably written questions can allow them to show me what they know and allow me to judge my students’ progress? I’m thinking hard about this and I would love some ideas.

Great Follow Up Day

On Friday I wrote about a pretty terrific conversation that came up at the end of our BC class. We had tackled a particularly gruesome integral – (tan x)^5 and I had done so by repeated patience substitution and I chose to let u = sec x and look for combinations of sec x tan x as the du piece. One of my students stayed after and showed me his work where he shoe u = tan x and du = (sec x)^2 He was frustrated and told me that he spent about a half an hour trying to figure out why his answer was ‘wrong’. So, I typed up my solution and had it on one side of a page. (This document and the graphs I created are all linked on my last post from Friday.) I typed up the solution my student arrived at and placed it on the back of the page. I had my students working in teams so they each had my solution and their classmates’ solution in front of them simultaneously. I asked them to examine each of them and explain why they did not agree. I heard some pretty good conversations, most of them simply concentrating on making sure that they even followed each of the solutions. We had talked about it on Friday so it was good to hear them reflecting clearly on that experience. After a couple of minutes, one of my students announced that he proved that both solutions worked. I played dumb and asked what he was talking about. He explained very calmly that since one answer was based on even powers of tangent and one was based on even powers of secant, we could show that they were nearly equal. They seemed to differ only by a constant. I then showed them the Desmos graph I created and the GeoGebra graph I created. Both programs were happy with my solution and with my student’s solution. Neither program was happy with the difference between them. But I showed them that every x input we could guess at in the difference function yielded either an undefined answer or an answer of -0.75

I used this conversation with a number of goals in mind. I want them to get in the habit of talking to each other. I want them to see that there is not just ONE way to do math problems – especially ones as sophisticated as the ones we talk about in BC Calculus. I want them to think about graphs. I want them to utilize resources such as Wolfram Alpha, GeoGebra, and Desmos. I want them to notice and wonder about relationships. They are not yet where I want them to be in these terms, but the more often I remind them and the more often I model this behavior, then the more likely they are to adopt these behaviors.

If I did not believe this, I might not have the energy to keep on keeping on in this job. But I do believe it and I do keep on keeping on.

Thank you world of math resources for my students! Thank you world of recourses for me!!

We Broke Graphing Technology Again – A Success Story

In AP Calculus BC we are doing some pretty unexciting stuff right now – techniques of integration. The problems are (sort of) fun little algebraic puzzles but I find little room for conceptual conversations. Maybe I am just missing something obvious. But today was a bit of a revelation and I wish I knew better how to try and insert equations to tell the story. I’ll just have to use some tortured syntax to get my point across. I put up three pairs of integrals and told them that one in each pair was something they knew how to do before they met me (our school does BC as a second-year calculus course) while the second was one they needed my help with. I had an integration by parts example side by side with a boring old u substitution (the integrands were x cos(x^2)  versus x cos x) and they knew which one they COULD do and we talked through integration by parts. I had a partial fraction problem side by side with a  natural log problem (the integrands were (x – 2)/(x^2 – 4x + 3) versus (x + 1)/(x^2 – 4x + 3)) and again they knew the difference and we talked about partial fractions. I had a trig substitution problem against a boring old square root (this time it was sqrt (9 – x) versus sqrt (9 – x^2)) Then someone asked me a HW problem. They were asked to integrate the fifth power of tangent x. I took off writing and trying to get buy in at each of the many steps. I told them at the end that they knew each of the steps they just did not know which direction to move. I assured them that this was a process they would master with a bit of practice. As I was working, I made the decision to substitute for sec x and set up the answer in terms of that function. A student asked me why he could not use tangent to substitute. I did not have a bunch of time left so I asked him to hold his thought and talk to me at the end. He did. As a result, I made a document we’ll examine as a class on Monday comparing his solution and mine. You can grab that here I went through with math type to show his solution and mine. I’ll leave it to the students to determine why they look different and I hope they come to the conclusion that they are NOT different. To help push the conversation I created a Desmos graph and a GeoGebra graph to show my function (called d(x) in each case) and my students function (called j(x)) in each case, I will erase the f(x) that you can see by following these links because I don’t want to give the game away immediately. What troubled me was that each program dealt with my function and my student’s function just fine. When I combined them the graphing technology broke. I tweeted out to @desmos and received – as usual – a quick and helpful reply. In this case, the reply was simply ‘Thanks for sharing. This will help us make better graphs for the future.’ This is the second time this year that we have found a little glitch and I could not be more pleased with the response I have gotten each time. It is such a great way to emphasize to my students what a connected world we’re living in and how they can reach out and find help. My student said he spent a half an hour trying to figure out why his answer was ‘wrong’ since it disagreed with his text’s answer. I hope after Monday that he will begin to internalize the idea that he can check his answers in pretty powerful ways. Ways that I did not dream of when I was learning this stuff in 1982. What a fun fun experience seeing his work and getting the reply I did from Desmos. Add in the fact that I get a date with my wife at a local farm to table restaurant and the day could not get much better.

Progress Report

It has been, as usual, a busy year. It seems that this is absolutely the norm in our professions, isn’t it? I am teaching three classes this year – AP Calculus BC, AP Statistics, and Geometry. In the next few days I want to comment on all of these classes as time allows. Unfortunately, this is the week we are working on midterm grade comments for every one of my 68 students AND my mom arrives tomorrow night for a visit as this is grandparents’ weekend here at our school. The two things I want to share today are reflections on the progress of my Geometry class and a class visit I made last week. I’ll tackle last week first.

We have a number of students who finish the Calculus curriculum before they graduate here. Those students – who have has two years of AP Calculus (we teach BC as a second-year course) move on to an Applied Differential Equations class taught by our school’s President. He is a retired Army engineer and he teaches a lovely Problem-Based curriculum using Mathematica. He ran into me in the hall last week and mentioned that his students were presenting the results of their first research problem and he invited me to join them on Friday. I was able to watch as two of my former students presented fantastic work that they had done on their own on a research topic of their choice. One was presenting a supply and demand curve. She explained her choice of parameters and shared the results. Her classmates made note of the similarities to a predator-prey problem that they had worked on together. She eloquently addressed the similarities and differences between the problems, but what most impressed me was her response after one of her first graphs. She remarked that the graph did not fit what she suspected should happen and her slide suddenly displayed the phrase “Question Results” and then she moved on to discuss her modification. I loved the directness and the simplicity of this message. Just because a fancy machine, and Mathematica is a VERY fancy machine, tells you something, it is not necessarily true. I have to incorporate such a simple response into my repertoire when looking at surprising or counterintuitive results. The second presentation was analyzing the forces involved in walking on high heels. It was a fun discussion and I was impressed by the depth that the student found in examining this situation. Unfortunately, our school President is retiring and I suspect that our new administrator might not have an engineering background. It’ll be up to me as department chair to find someone capable of carrying on this kind of high level work with our best math students.

I start each day with my Geometry class. We have had a blast so far playing with GeoGebra in the lab, discussing reflections, rotations, and vector transformations, talking about distance and Pythagoras. I have been pleased with the level of engagement I have seen from the students. They are reading the text (and finding typos!) and they are engaged in class discussions. Now we are getting ready to begin discussing proof. So I want to sloooow their brains down a bit. I borrowed (okay, maybe it is just stealing) an idea from Max Ray in his book Powerful Problem Solving. I asked my Geometry students to write directions for making a peanut butter and jelly sandwich. I wanted to show them just how hard it can be to carefully describe something. I did not have people act out the descriptions, though. I was a little worried about embarrassment. So, I dealt out cards at random to sift the kids and I had plates, paper napkins, knives (plastic ones – just to be safe!), peanut butter ( one table had sunflower butter due to allergies) , and some jelly. I bought both grape and strawberry. I shuffled the instructions and handed them out and we had a terrific conversation about the assumptions made and about being careful with details. We had some laughs talking about how peanut butter might magically appear on knives to be spread and what side of the bread gets put together. Then we came to the description made by a student I’ll simply refer to as E. She typed her description ( which can be found here ) and she did such a lovely job. Her list is detailed and she took great care to break down the actions involved. All of the class agreed that hers should have been the first one read since we decided to go ahead and eat as soon as we dissected hers. She seemed really proud when I asked her for her copy to share out.

i’ve been thinking about this all day. Remember, this is my 8 AM class. I have resisted some activities like this in the past thinking it just was not my bag. However, I am working closely with two terrific colleagues in Geometry this year and they sort of encouraged me to try. I stepped out of my comfort zone and the result – at least this time – was a relaxed and fun class. Kids were engaged, they supported E and gave her some real praise and after class one of my students who has been struggling so far came up to me to tell me how much fun she had. I think I may have earned some important personal capital in working with her and this may be a gateway to building a relationship with a student who does not seem to have much inherent love for my subject.  I also think that some valuable points were made about the challenge of explaining what you know to someone who does not know it. I am optimistic that this will help build a bridge toward understanding some important principles of proof in the next week or so. So, thanks to my colleagues for encouraging me to try something that seemed a bit silly to  me. Thanks to E for being a model of thoughtfulness and detail. The rest of her work this year has been uniformly outstanding, so I was not exactly surprised by this. Thanks to my students for trying an assignment that probably seemed a bit weird. Finally, thanks to Max Ray for this thoughtful book that got me going on this path.

A Geometry Revelation to Remember

We are working with isometries in our Geometry class right now. We are looking at vector translations of objects, rotations (primarily around the origin), and reflections over horizontal or vertical lines. I am only teaching one section of Geometry so sometimes I feel like if I don’t get it right, I’ve missed an important opportunity. There have already been a couple of instances where I explain something after school in a way that I wish I had done the first time around. I am trying to keep note and be a smarter Geometry teacher next year.

Today when we were wrestling with rotations about the origin (I am sticking mostly to 90 degree increments) there was a bit of a clamor with the students begging me for rules about how to transform a point in the general (a, b) form. I really want them to try and develop a better intuition so I have resisted the call for these shortcuts. One of my best students, a boy I’ll refer to as M, pointed out that these rotations always maintain distance (again, we are rotating about the origin) so I pointed out to my class that this really limits our choices. We were looking at rotating the point (4, 2) 90 degrees counter-clockwise. Everyone agreed that this would put us in the second quadrant and I used M’s idea to suggest that our destination was now restricted to either the point (-4, 2) or (-2, 4) since the distance was the same and the second quadrant has negative x values and positive y values. A quick sketch convinced us all that the target should be (-2, 4). I felt like we were in sync with each other and that my sketches convinced my students. A group of them came after school to work to get ready for tomorrow’s quiz and they all confessed to not being convinced. I pulled up GeoGebra and tried to show them what would happen. I referenced M’s idea about distance and one of the students reminded me that I had been emphasizing the Pythagorean Theorem more than the distance formula earlier when we talked about distance. This was the breakthrough that I should have had at 8:30 am instead of 2:45 pm. I drew a right triangle on GeoGebra, called up a slider to rotate the triangle and showed where the vertex originally located at (4 , 2) ended up. Then I reminded this gang of help-seekers that points on the x-axis move to the y-axis under a 90 degree rotation. So I convinced them (and this time I am pretty sure that I DID convince them!) that the side of the right triangle corresponding to the distance 4 represented by the x coordinate of our original point now HAD to represent a y quantity after the rotation. We tried a few other points as well and we were humming along. I tested their wits by asking them one more question before they headed off – I asked about a 270 degree clockwise rotation. I was thrilled that one of the guys quickly pounced on this and said it was simply our 90 degree counter clockwise example again. VICTORY!

At least it feels that way now. I’ll see (and report back) after our quiz tomorrow morning.