I’m sitting in a building at Exeter, digesting lunch and waiting for the next session to begin. I’m at what has so far been a really valuable math teacher conference. What impresses me most, besides the amazing and neverending supply of food that they offer, is the population of teachers that come. Many of the teachers I’m talking with have 20+ years of experience in the classroom.

Over the next few days I’m going to use this blog to talk about some of the tidbits of interesting problems I’ve been presented with, to good resources or programs that I was introduced to, to neat ways to present topics in class, to ideas that I’ve been inspired to think about.

I’m going to start with a nice calculus problem — probably good for a AP Calculus BC class, but there is definitely a way I could show this problem to my non-AP class.

Here’s the problem. You’re in a museum and you’re looking at a painting which is hung above eye level. (There is a specific painting which is hung high in the entrance room at the Brooklyn Museum that I think of with this problem.) You are standing some distance away from it. The question is: what is the largest angle () that you can get as you walk forwards and backwards? (See diagram below for setup.)

So to be clear, as you move the eyeball forward and backwards along the dashed blue line, what’s the largest angle you can create? Of course if you walk right up to the painting, or far away, the angle is going to decrease to 0. If you can’t see that, look at the diagrams below.

So of course there has to be some perfect distance that will give you the maximal angle. You see where this is going…

Find that maximum angle! (Use the variables in the diagram below.)

Of course this doesn’t have to be a painting. It could be, as the speaker pointed out, an overhead view of a hockey rink, with the painting being the goal, and the eyeball being the player with a puck. Where does the hockey player have the maximum angle to shoot and make it into the goal?

I want you to have the fun of solving it, but the solution I came up with was:

(I can help you with that if you want. Just throw your questions or cry for help in the comments.)

However, there is something pretty amazing about this problem, something that is powerfully seen with geometry software like geogebra or geometer’s sketchpad. Check out the sheet I made and see what happens as I bring the person close to the picture and look for the optimal angle? When you look at this, try to see if there is a geometry connection to our solution for the largest angle…

Do you see the geometry connection? The optimal angle exists when the circle created by the top of the picture, the bottom of the picture, and the eyeball is tangent to the line of sight. Now my charge to you — which would be my charge to my students — is to (a) explain in words why this is true and (b) use geometry to calculate this optimal angle. You know, this work is an exercise for the reader. I mean, I’m not going to do everything for you. Sheesh.

I wanted to revise my teaching portfolio. I decided to think backwards and start from what the goals of my portfolio were and then see what I could do to achieve them:

• A site for me to “sell” myself, if I ever go on the job market.
• A site for me to archive the evolution of my teaching career. My blog acts as that, in a haphazard way. The portfolio sorts through the detritus and organizes it.
• A site for me to express my personality and teaching style.

What I came up falls short of the third piece, but I think it is successful on the two parts.

If you want to check it out, feel free: Sam Shah’s Teaching Portfolio

And just so you know, my Teaching Philosophy as is now is actually just filler. I mean, I believe all that, but I haven’t written a formal one yet.

If you’re really dying to see what our results are, click here. If you can manage to read the prologue, avoid that mouse button and forge on!

I’m writing this after my second year of teaching. Even though in many ways I’m a neophyte, there is one thing I am sure of. The majority of math teachers out there don’t know how to “do” homework. Myself included. Do any of the following sound familiar?

“I just walk around and look to see who has attempted the homework. I don’t have time to collect and grade each students’ homework.” “I don’t want students to feel penalized if they get home and are completely lost and just couldn’t do the work completely, but I also don’t want them to develop a sense of ‘learned helplessness. I want them to learn to figure things out when they are stuck.” “I want homework to be both a site for practice — so students can naturalize the skills that are introduced in class — and a place for me to know where my kids are at in terms of understanding; it’s a place for students to assess if they know something and it should be a place where I assess the state of the class. Right now it’s not doing either really well.” “I hope that one day homework in my class will partly be about problem solving skills, but at the moment, that’s a pipe dream. It’s just practice of the routine problems we do in class, not really getting my kids to think for themselves. One day I’ll figure out how.”

And of course, the questions:

“How much homework should I assign, if any at all?” “Should I make all my own homework, or just assign problems from the book?” “How much time should I spend at the beginning of class going over homework?” “How much do I really think homework should be worth in terms of the final grade?” “Do I grade homework? If so, on completion or correctness or both?” “How do I grade homework?” “Thinking through the whole homework thing backwards, what really is the point of it? Can I use that answer to come up with the amount and kinds of homework I assign, and how I factor homework?”

These all are things that pop into my head from time to time, and then in the immediacy of creating another lesson plan or writing another email, get pushed to the wayside. I mean, at least no math teacher I have talked to has a system they’re totally happy with in terms of homework. Might as well just do what everyone else does and push on.

And indeed, at least from my 2nd year teaching perspective, this seems to be the general attitude.

So I decided to harness the power of the web, and using Google Docs, my blog, Twitter, and a few emails, asked math teachers to fill out a short survey on how they “do” homework. (My blog plea is here.) The survey questions are way at the bottom of the post, below the fold.

This survey was designed to be open ended, and above all, practical. I wanted to “see” the life of a homework assignment — from its inception to its role in the classroom to its place in a students’ grade. I wanted to let teachers say whatever they wanted to say about homework. The philosophical debates will have to rage elsewhere.

There were a whopping 40 responses. I, in fact, was gunning for 20. I mean, the survey was narrative (so it takes a bit of time to fill out) and restricted to math teachers. So that’s awesome.

Now the question is: what to do with the data collected?

I haven’t read through it yet; I wanted to look at it at the same time y’all did. I’ll be reading it over in the next few days and cobbling together bits and pieces of what other teachers have written about — bits and pieces that will work with my teaching style and in my school — into a cohesive plan for homework next year.

What you’re going to do with it is anyone’s guess. My hope is that you read through the data, pass it along to other math teachers, come back here, and write down your thoughts in the comments below. I don’t expect a conversation will start here, but I’m darn tootin’ hoping one will.

So without further ado, click below for the survey results, or view the PDF below.

(Survey questions are below the fold.)

The questions:

# Pick a class. Think of the class that you feel the most successful with in terms of homework. What math class is it? (e.g. Algebra I, Calculus)
# What is your homework? How do you chose your homework each night (if any)? Is it usually from the textbook or something else? How long do you think most kids in your class spend on this work each night?
# Grading homework. How do you grade this homework? Do you collect it? Glance to see if students attempted it? Do you have students exchange their work? In other words, how do you assess this homework? And how does this factor into their final grade (e.g. homework is 25% of a students’ grade)?
# Homework during classtime. How many times a week does your class meet, and for how long? How much time in each class do you usually spend going over homework? And most importantly, how do you go over homework (if at all)?
# Happy? Overall, are you happy with your homework system? If there are places you aren’t quite happy with your system, what are they?
# Speak now or forever hold your peace! Want to share what the purpose of homework is for you? Have ideas that you don’t use in your class but want to try? Have thoughts about homework that aren’t addressed above? Philosophically disagree with homework? Whatever! Speak!

On the last day of classes, one of the seniors anonymously (but with the approval of the administration) put out a fake newsletter titled The Kaleidoscope. I don’t know why, but I got all swelled up with pride when I was mentioned in it a few times. The newspaper is full of inside jokes — jokes that only people at my school would get, or in the case of where I’m mentioned, jokes that only kids in my classes would get. But since I consider this blog a communication and archiving tool, I’m going to put the exerpts from the newspaper here. (I would also like to say to readers that my school is not religious in any way; it is in a historic church. So “chapel” just refers to meetings that we as a school have in the chapel.)

Last Rights of Last Chapel Gives Seniors the Rare Chance ot Draw Attention to Themselves

During the incredibly nostalgic “last chapel” for the Class of 2009, current study body president David “Nightfire” Palgon took extensive time out of a long block of jam related announcements to outline what this year’s student council had been working on. “After much deliberation, extensive research and statistics gathering, an MSA basketball tournament, four bake sales and a year’s worth of early morning meetings, the student council has outlined in the utmost detail a plan to put new pencil sharpeners in every classroom.” At first the chapel was filled with stunned silence, and then, like the flapping of a thousand pelicans’ wings, applause echoed from newly cleaned, non-religious stained glass windows to rarely used organ, to the strange, hieroglyphic, snowflake patterened lights. In no time at all, students were on their feet, broken and dulled pencils raised above their heads in celebration. “It’s about time!” yelled estatic long time anti-pen advocate and calculus teacher Sameer “Worchestershah” Shah. “I’m going to call my high functioning Aunt Derivative, send her some pie, fibenachos and a DXie chicks album and tell her about the pencil sharpeners! I’ve had the absolute maximum a person can take with pens!”

The article continues, but you get the point. And yes, I am a vehement anti-pen advocate in my classes. I do not, however, have any idea why my middle name is “Worchester” or why there is a picture of a Worchester bottle next to the article with my name under it. But I like it. Another article is about the student-faculty judiciary committee.

In the Court of Lawlessness: New SFJC Disciplinary Strategy Raises Concerns about Questionable Interrogation Tactics

“It was horrible!” chimes a confused [StudentName]. “I was late to school. Not too often–once, twice, eighteen times, and forced to go to SFJC. I had to get to school at 5:00am and when I refused to acquiesce to their demands they forced me to watch videos of old activity periods. I couldn’t take it…”  She buries her head in her hands and bemoans her early sign in.

[...]

“It’s really tough,” mumbles a disgruntled Sameer Shah, who recently misplaced an attendance sheet and was forced to follow a sophomore around for a whole day. “The new SFJC punishments, which apply to both students and faculty,” Shah continues, “effectively involve a role reversal.” If a faculty misbehaves he or she must do homework, papers, study for tests, worry about finals, do clean-up duty, and has sign out privileges revokes. But if a student breaks a rule he or she must grade papers, attend TALL Tuesday afterschool meetings, get fired, or in the most sever instances, serve as cafeteria monitor for middle school lunch.

I have to say that this newspaper is hilarious, especially if you got o my school… It’s like the author(s) wrote down every funny thing about our school — from our strange Pelican mascot to the fact that it has taken over a year to work through the red tape to get new new pencil sharpeners in each classroom — and wordsmithed it to priceless gems. I literally was laughing out loud at every sentence. They tap into that very thing of what it means to really go to and live our lives at our school. That’s a hard feat to do. It’s also why you probably read everything above and were like “ummm, Sam, these are NOT funny.” To that I bite my thumb at thee.

Nearing the end of the school year, but when things are still in full swing, I ask students to write a letter to themselves… to themselves at the beginning of the year. The letter should outline things that they learned about the class, about me, about whatever in hindsight that would be helpful for them to know to be successful in calculus class. [1]

I promised them — crossed my heart, hope to die, and all that — that I wouldn’t look at them until after final grades were in, so they should be SUPER honest.

What I get from these letters is not only insights into my students, but more than anything else, deep insight into me as a teacher. It also reveals a lot to me about my about students. Without further ado, here are the relevant excerpts [2] of each of these letters which I’ll be handing out on the first day of class next year. (And before you ask, some of the things in these letters are inside jokes between the class and me.)

[1] At the beginning of this year, I handed out excerpts from the previous class’s letters.

[2] I pretty much included everything relevant… the good, the bad, and the ugly… the stuff I cut out either identified the student or were fluffy sentences that had already been captured.

UPDATE: Survey results here!

We talk a lot about lesson plans on our blogs, but there is the huge issue of homework that we don’t discuss frequently. Partly because the discussions easily lose concreteness and get philosophical (why do we assign homework? how do we make sure the homework as meaning?).

However I’m struggling with how I’m doing homework in the classroom everyday. I can get dragged into a too-long discussion about homework problems in class, and I still don’t know how to grade homework in a way that I am not killing myself by collecting it everyday.

The few teachers I’ve talked to about this have shown me that this problem — of finding a good homework system — is pretty universal. A recent post at Kiss My Asymptotes reminded me of this too.

It’s like no one I know has a good solution on how to “do homework well” in the math classroom. That’s crazy to me. I mean, how can that be? And I just know some people must have good ideas/methods. Which is why I’m just casting a wide net looking for what everyday teachers do in their everyday classrooms. If I can collect a bunch of different ideas from different teachers, I can put that out there to show the different ways we do homework. Maybe we can pick and choose bits from other teachers that inspire us!

Because of this, I thought:

We should all band together and make a list of how we deal with homework in the classroom.

So I’d love you forever if you could spend a few minutes filling out this short survey, and I’ll post everyone’s responses soon.

Without further ado: THE SHORT SURVEY!

Inspired by Kate Nowak’s new portfolio, I am going to be making a new teaching portfolio this summer. I wanted to solicit advice.

Three questions:

1. What would you want to see in a portfolio if you were checking out a math candidate to teach in your school?
2. What are any math teachers portfolios that you have seen that you’ve enjoyed looking at?
3. What place do you think would be a good place to host it?

I’d love for you to put any advice in the comments.

More about my thoughts below, if you care to see them.

Here is where I am now. I made a portfolio last summer, but I’m not totally happy with it yet.

The fundamental thing about the porfolio is that I want it to be less formal, chalk full o’ information, and exhibit my personality. FUN AND LIGHT AND VISUAL AND COLORFUL!

Things I’m thinking of including:

• my life trajectory (how I got into teaching)
• resume
• projects I’ve done with students
• letters of recommendation that I used to get my current job
• my blog (which is the best professional development that I’ve done) and links to some favorite posts
• my use of technology
• classroom projects
• smartboard lessons I’ve used in the classroom
• my evaluations from administrators who have observed my classroom
• professional development / curriculum development
• transcripts (undergrad, grad school)
• non-teaching contributions to my school
• my Massachusetts teaching license

Any other ideas? I’d love any ideas. I’m mulling:

• photos of me teaching
• video of me teaching
• student feedback
• a “me in numbers” page, and a “me in words” page
• a short piece about my love for math
• my twitter username (and some of my favorite tweets) – since I mainly twitter with math teachers
• a “the formal, short, and dirty” page of what a traditional portfolio might have

How I precisely present all of this information is going to be the fun challenge. I’m thinking: venn diagrams, colorful charts, links, etc. It’s fun to dream of it, but I know it’ll be trying to learn how to actually create this half-formed vision.

An interesting thread/question on AskMetafilter:

Hi there,

I have a student who for whatever reason has very recently gotten a serious case of silliness/defiance. What concerns me is that when he is kept in for detention he finds the whole thing to be a big joke and giggles constantly. Even within detention he is defiant by getting up and walking around, etc. Clearly, he would be a candidate for a suspension or office referral, but for various reasons I’d like to avoid that route for the time being if possible. Obviously I’ve spoken with his parents.

I would like to reel him in mainly because normally he works pretty well, but also because he is capable of distracting the whole class. I worry that he’s actually trying to see if he can get sent to the office for some reason. At the same time, it could be a physiological thing (13–14 years old) or a genuine desire to get under my skin. The last option seems very bizarre as up until this point in the year I’ve had a great rapport with him in class.

To be specific, I don’t think that he gets the teacher – student distinction very well (almost all students when spoken to one on one have enough respect to discuss the issue if they are spoken to politely and openly – with him he just laughs right in my face).

So, any suggestions on other things I can do to get him to be respectful again? His parents have agreed to have him do extra homework, but I’m keen to know if there are other ideas. I really don’t think detention is the right fit for him and besides which, I have to supervise it and if he’s giggling the whole time it’ll really undermine the authority. The other thing I’ve done is isolate him from the class and have him do his work individually for the time being.

Thanks for any suggestions!

The responses (35 as of now) are varied and interesting. Some are, imho, awful, and some are heading in the right direction. Of course, without more context, it’s all shooting in the dark. But it was such a well-responded to thread with a lot of different perspectives… Blog posts — except for the most popular ones — don’t generate this kind of “hive response.” I wonder if there is a site out there that teachers use for this sort of discussion/advice? Where anyone can throw out a question or offer advice to other questions? Or are a thousand disparate and not-super-popular blogs enough?

The math finals are given next Monday. And I’m really curious about how my Algebra II students will do, especially as the year comes to a close.

But I will say this: I predict that the average score on the final exam will be 85%. Why? Because according to my new grading program this year (EasyGradePro), the average EVERY QUARTER was  85%. Crazy. And at least for the first three quarters, there were exactly 4 (of 15) students in the A range. (I’m not done calculating the fourth quarter grades yet.) I bet it’s mostly coincidence, but it also suggests that I’m keeping the course pretty consistent in terms of difficulty level.

Back to final exams.

My biggest concern this year was that we didn’t have a midterm (because of the school tragedy). And I didn’t do much time throughout the year reviewing topics from the first semester  [1]. To battle this, I assigned each student a semester 1 topic, and told them to make a study guide of that topic for their classmates.

The other teacher of the course and I thought this was a good idea for a number of reasons.

1. We wanted to have one more assignment which focused on student communication. (That’s something we’ve been emphasizing this year, but we need to ramp it up next year.)
2. Our students are 10th and 11th graders, and we think that now they are ready to take ownership of their own studying. We didn’t want to provide them with a study guide or packet of problems. We wanted them to figure things out. (I would, to put things in context, probably not do this in 9th grade.)
3. We thought that students would desire to do a more thorough job on their study guides if they knew they were for their classmates too.

I taught students how to use Equation Editor in MS Word and I asked them to use either www.graphsketch.com or their virtual TIs for graphing. And then they went at it.

Now I have a confession. I had talked with the other Algebra II teacher about doing this, but we both sat on our hands until the last possible moment. And since I didn’t want laziness to get in the way of our students’ success, I banged out the instructions and rubric in an hour. But I’m so glad I did. The project needs a lot of work for next year, but I think we’ve got ourselves a winner.

My instructions/rubric: Final Exam Review Project (PDF)

I put all the study guides on a website for students to access. Since the guides have the students’ full names, I’ve password protected the site. But you can see what it looks like here (Image of Algebra Two Website). Students can download and print out the individual guides from that site.

The good (what worked well):

• Students seemed to get into this project. One said, after working in class on the study guide, “wow, this is actually pretty fun.” I think part of the fun was using Equation Editor (they liked that), and part was creating something collectively.
• Most students did actually work really hard.
• Most students were actually really good at explaining their concepts clearly. The ones which were more didactic (e.g. “Now let’s make things a little bit harder…”) and sounded like someone speaking/teaching were the best!
• Students seem to be actually using them. During the review days, many were using them. There are 15 students in my class. As of now, the site with the study guides has been accessed 261 times. The site has only been up for four or five days. So students are coming to the site and looking at a guide or two, and then coming back and looking at more guides later.

The problems (what to fix for next year):

• I didn’t get to have a good discussion on what makes an effective study guide and what makes  a poor study guide. I should have talked about tone, layout, clarity, etc. Also, I could have students make these guides earlier in the fourth quarter, and have them exchange them with a partner for critical suggestions for improvement.
• I should have shown an excellent study guide, an okay study guide, and a bad study guide. Luckily for me, if I do this next year, I can use examples from this year!
• Many guides were turned in with mistakes. Almost all had mistakes, in fact. I read through each one of them carefully, and noted all the mistakes, and returned them. I hadn’t anticipated this many errors, so I gave students 1 day to fix the errors and turn them in again, to raise their grade by up to 5 points or if they didn’t make any changes, to lower their grade by up to 5 points. Most errors were fixed. But next year, I must insist upon a comprehensive draft.
• I insisted in students typing everything — because some have terrible handwriting and I also wanted them to get familiar with Equation Editor. (I was horrified by the fact that my seniors in Multivariable Calculus didn’t know how to use Equation Editor; I want to make sure all my students know they can write math on a computer!) However sometimes that requirement got in the way of clarity. There were some guides that had parts that would have been much better if there were some things that were handdrawn in. So, for example, if there was a tricky step in a series of algebraic manipulations, putting a handdrawn arrow to that tricky part and saying “CAUTION! Be sure to flip the sign of the inequality when * or / by a negative number!” would be more effective than typing it out after all the equations are worked out.
• I need to come up with a better way to talk about the number of practice problems required in each study guide, and talk about how these questions should be representative of the types of problems that we did in homework or got on assessments.

But yeah, although I haven’t yet had a chance to talk with my students about if they are finding these guides useful, I have to say that it so far appears like they are somewhat successful. At the very least, I know each of my kids have mastered at least one topic from first semester and are able to articulate that topic pretty darn well. And I can say that at least in terms of students using other students’ work, this is much more successful than the video project I did last year, that I was too busy to repeat this year. (See the videos here.) [2]

[1] Next year, I have to remember to build review into the course more formally. I planned on doing it this year, and then it got lost by the wayside. But I’ll tell students that a previous topic not from the current unit will be tested each assessment (and I’ll tell them the topic). That way they’ll be forced to periodically review topics so it won’t all be a shock at the end of the year.

[2] Argh! I can’t believe I didn’t show my students some of the good videos from last year! ARGH!

Last week my multivariable calculus class turned in their final projects, and made presentations. Of all my classes this year, by any metric, this course was my most successful. I loved seeing their final projects, and the amount of work and dedication they devoted to them. The best part: they were super proud of their projects too.

I made a post about coming up a list of fourth quarter final projects a while back. My big fear with these projects was senioritis. The projects are designed to be largely self-directed, and if a student got lazy, well, …, that would spell disaster. Luckily, none of the students in the class fell prey to that dreaded disease. My kids are great kids, so that helped. But also I let them pretty much have free reign on their projects and kept emphasizing they should pick something they WANTED to have FUN with. Lastly I met with them weekly to help them out and keep tabs on their progress — prodding them a bit here and there.

Without further ado, the four projects:

1. One student actually created a harmonograph (a device which draws damped Lissajous curves).

Yes, that is his video of his harmonograph.

2. One student researched Maxwell’s Equations and read A Student’s Guide to Maxwell’s Equations (Fleisch). He produced a written paper explaining the integral and differential forms of Maxwell’s equations.

3. One student created a giant wooden and wire sculpture (titled “The Visualizer”) which illustrated a lot of what we’ve learned about curves in 3-space. Namely, he illustrated arc length, vector equations for curves, curvature, and the tangent, normal, and binormal vector with his sculpture. He also wrote an associated paper which is to be used with the sculpture to examine these ideas in more detail.

4. One student took foam board and cut a whole bunch of figures (from the simple square to weird and complex shapes). He then calculated the center of gravity of these figures theoretically, and tested to see if the figures would balance at that point. Then he extended this by making figures with non-homogeneous density, calculated the center of gravity of these figures theoretically, and tested to see if the figures would balance at that point.

I mean, seriously, look at that. Amazing. These kids got into it, because it was their own thing. Because they weren’t really worried about their final grades. (I let them grade themselves.) I am going to miss these miscreants a lot next year.