Mathematical Communication

Make it Better: Memory Modeling

“A monk weighing 170 lbs begins a fast to protest a war. His weight after t days is given by W = 170e^(-0.008t). When the war ends 20 days later, how much does the monk weigh? At what rate is the monk losing weight after 20 days (before any food is consumed)?” <– That’s an actual problem from our Calculus book, which I find very amusing. Though it doesn’t really fit Dan Meyer’s definition of psuedocontext, I just get a kick out my mental picture of a monk sitting in a dark room taking a break from protesting the war to scribble away on a notepad trying to make predictions with an exponential model… There are so many word problems that force “real-life” situations into the convenient framework of whatever math topic is being presented in that section. I guess these are supposed to demonstrate to students how useful and relevant math is, but I think we all know that students just find them to be tricky and unyielding disguises to math that they generally know how to do.

There was one word problem that fit an exponential decay model to someone forgetting information, so I decided that instead of just doing the word problem, we would test the model by recreating the experiment. The day after we had a midterm exam, instead of handing back their corrected test, I put them in groups and gave them the following list of 50 three-letter syllables that I generated with a random number generator:


I gave them 15 minutes to memorize as many as they could and then tested them by having them write down all that they remembered. Then, I handed out the midterms and we started going over them. About 5 minutes later, I had them write down as many of the syllables as they could again. Then, we went over a few problems on the midterm… then another memory test…. then more midterm… then another memory test. They had absolutely no idea why we were doing this, so each time they groaned and complained. And they groaned even more when I opened class the next day with another trial. And then again two days after that… And then a last time a week and a half later. All without studying the list after the original 15 minutes.

Finally, I revealed the purpose of the whole experiment. We collected data and used GeoGebra to fit various models to their data. There were four different mathematical models to choose from that I found from various psychological studies (which I had loaded into a GeoGebra file with sliders so that they could move the various models around to fit their data). Each student picked the one that they thought fit their data best (a function to calculate how many words they would remember over time), took the derivative of that to calculate their “forgetting function” (a function that tells them how fast they are forgetting words at any given time), and then used both to calculate how many words they will remember in a few weeks and how fast they will be forgetting them at the point.

We graphed all of their functions on the same axes (y-axis = number of words remembered, x-axis = time in hours) to analyze which model was best and analyze how their memories compared to their classmates. The results are below. The different colors correspond to the model that each student chose.

        CLASS 1 –

        CLASS 2 –

Now, the clean final result of that graph hides how messy the model fitting part was. Though some students’ data fit well, some didn’t, at all, which was actually really nice. They really struggled trying to fit the model and hopefully realized that a lot of these models that we are dealing with in cooked textbook problems aren’t as powerful as they purport to be. If I could do it again, I would have them use more mathematically sound ways of fitting the models than just eyeballing it (I hadn’t really considered this and realize now that, though it would be an investment in time, it would make the whole thing much better).

But besides doing some authentic math that was individually tailored to each student, my favorite part of the experiment was the followup meta-cognitive discussion. We ended up having a really great conversation on how best to memorize these random things, which then led to a great discussion about how to learn and study best (especially how you should go about studying math). We talked about how some people put the words in context by using a story, some people made patterns by grouping similar items together, and the ones that didn’t do very well talked about how they just tried to memorize these random unconnected things by rote memorization. Many also noticed that throughout the closely connected trials on the first day, their number memorized actually went up, so we talked about how assessment can actually help you learn something too (in addition, of course, to regular practice).

Make it Better.

I have one simple question this time: the thing that I really didn’t like about this experiment was that it was entirely teacher centered. They were in the dark about what was going on (for experimental purposes) until the day that we collected data, fit models and did some quick calculations. How can I make this more student-centered and add elements of inquiry? I have a few ideas, but I wanted to see what other people thought.


  1. Word document with list of random words
  2. Excel spreadsheet for collecting data
  3. GeoGebra file with various forgetting models, ready to drop data in
from @bowmanimal 

Email breakdown 2010-2011

Last year I had archived 2,270 emails to and from (and about) students. (I make a separate folder for each student and file everything related to that student.) The breakdown from the data last year was here.

This year, the total comes to 3,364 emails. The reason for the increase is probably due to the reassessments in calculus… students had to send me an email justifying their request for a reassessment.

The student with whom I had the least communication to/from/about had 16 emails. The student with the most had 192 emails. Each student on average has 71 emails.

Edublog Nominations

Drumrollllll please…

Here are my 2010 Edublog Award Nominations…  (my 2009 nominations are here)

Best individual blog: Shawn Cornally’s Think Thank Thunk

Shawn’s blog arrived on the scene in Februrary 2010, and has quickly won over the math teacher blog community. He’s a staple read. Why? Clearly he’s a compelling writer, with posts chock full of quotations inspiring enough to lift and emblazon on classroom banners. But it’s more than that — than the inspiration dripping from every post for the readers to hold out our cupped hands and collect. It’s that Shawn arrived on the scene with a simple philosophy about the classroom, and built a whole curricular and assessment-based regime around it. The philosophy is this: LET KIDS LEARN! GIVE THEM FREEDOM AND THEY WILL LEARN TO THRIVE. This philosophy lends itself perfectly to Standards Based Grading, and his work, coupled with Dan Meyer, Matt Townsley, and others, have led to a wholesale shift in the math edublog community. At the very least, spearheaded by Shawn, Standards Based Grading led to serious, questioning, and respectful conversations around the nature of assessments. And for some of us, it has led to a wholesale reorientation of how we view the classroom and how we think about learning. Did I mention that Shawn only started blogging in February 2010? His blog is a Tour de Force.

Best individual tweeter: David Cox / @dcox21

Reason: He’s always there, and his snarkiness is always on, ready to go! If you need any convincing, check out any of these “Favorite Tweet” posts. He also tweets about teaching.

Best new blog: Shawn Cornally’s Think Thank Thunk

Reasons: listed above

Best resource sharing blog: Mimi’s I Hope This Old Train Breaks Down

Reasons:  I don’t think Mimi gets the recognition she deserves. She has created countless amazing resources for her classroom, and shares them. These worksheets and activities are inspiring enough that… well, let me put it this way… she teaches Geometry, and I dread the day that I have to teach Geometry, but knowing I have Mimi’s stuff at hand makes me think teaching it could almost be fun.

Best teacher blog: Kate Nowak’s f(t)

Reasons: Kate is our Fairy Blogmother. Her blog doesn’t have a singlular focus, but that’s part of its charm. Kate blogs about whatever, and not knowing what’s coming next is part of the appeal. One post might be on special right triangles in dollar bills, and the next her trying to work through the muck and mire of Standards Based Grading. Kate also spends time drawing new people into the blogging community, and she even wrote a post on how to start a math teacher blog that I dare say spawned quite a few new faces in our expanding circle. Her blog is a perennial staple, read and admired by all.

Best use of a PLN: Riley Lark and his Conference on Soft Skills

Reason: Riley Lark created what I think is the first Virtual Conference for the online math teacher community. In general most of the posts that we write, and read, deal with curriculum — and how we get our kids to know it, and how we know they know it. Riley saw a gap in the conversations. We don’t talk about those other things we do as teachers to be effective: those things we do to connect with students so they know we’re there for them, and we want them to be there for us. The relationship building that goes on in the classroom… we all do it tacitly. He asked us to make explicit what we do implicitly. This conference started with 5 “speakers” and blossomed into 17 “speakers,” each writing with our own voice. Riley called upon our PLN and our PLN responded.

Lifetime achievement: Dan Meyer’s dy/dan

Reason: He’s part of the first generation of math teacher bloggers. His cause celebres have changed, but his overarching goal is the same: to get others to think about ways to engage students in the classroom. From focusing on design, to focusing on assessing, to focusing on his What Can You Do With This (WCYDWT) series, to focusing on pseudocontext, Dan has asked provocative questions since 2006. He asks teachers to re-look at what we’re teaching, and asks us to pique the curiosity of our students. Specifically, his mantra (though he has never put it this way) is don’t put the cart before the horse. Don’t teach the concept and the method of solution, without motivating the need for the concept. Grab your students’s curiosity by showing them they need a concept, so they want to learn the concept. And let me tell you, it is easier said than done… but Dan makes it look effortless. He has inspired the second wave of math teacher bloggers, which have since inspired the third wave. Yes, in online terms, 2006 – present is a lifetime.

Reading Math

My Algebra II kids don’t like to read the textbook. Heck, neither do my calculus students. This isn’t surprising. It’s extra work and it’s hard. My class also makes it hard for them, because I do not use the textbook as a skeletal structure for the course. I teach mainly out of my own materials, and use the book more as a supplement.

But that doesn’t mean that I don’t want them reading math. Kids are never taught to “read” a math textbook. If they ever do approach a math textbook, they approach it like a history book. The read it linearly. They also read it passively. Their eyes glaze over. They read words, but they don’t try to connect the words to the equations or pictures. They don’t read with a pencil in their hands. They hope for some Divine Knowledge to descend upon them simply by having the book open and their eyes on it.

That doesn’t work. We all know this. Reading math is an active thing.

And so recently I’ve started talking with my class about it. To start this process/discussion, one that I hope continues, I gave my students a worksheet to fill out (see above). I love the honesty with which they responded.

For question A, some representative responses:

“I read what was assigned to me but did not read anything extra.”
“I find that textbook reading is pretty boring, so I don’t do it unless I have to.”
“I did not because I had assumed I wouldn’t learn things I needed. All I would do was look at examples.”
“No, I find it difficult to understand math when reading it in paragraphs; it makes more sense to me with a teacher.”
“I did not generally read my math textbooks. I did, however, always look over the example problems.”

Some responses for Question B:

1. The writing can be confusing, wordy, and not thorough
2. The book is BORING
3.  Small print
4. Too many words for math
5. Outdated examples

Some responses for Question C

1. Everything is all in one place
2. Have a glossary
3. Can read at own pace; refer pack to the text when I get stuck
4. Sidenotes! Diagrams! Pictures
5. Real life examples
6. Definitions clear
7. Key terms are highlighted
8. Wide range of example problems with step by step instructions
9. Colors!

I hope to do more as we go along. I might have them learn on their own, using the textbook (and the online video help) a whole section or two. There’s no reason they can’t learn to use the book to be independent learners. I will give them class time and photocopies of the section they need to learn, and they will have to figure things out by the end of the class for a 3 question quiz.

I also hope that by the end of the year, we can use their critique of math textbooks for them to write their own textbook. Okay, okay, not quite. That’s way too ambitious for me. Two years ago I had my Algebra II kids write really comprehensive Study Guides for the final exam. This year I might ask my kids to pick some of the hardest material and create their own “textbook” for it. They’ll get to write it in pairs, and then they can share their finished product with the rest of the class. That will probably happen in the 3rd for 4th quarter.

Anyway, I thought I’d share. Since I like to emphasize the importance of mathematical communication to my kids (though I don’t do it nearly enough), I thought I’d talk about this one additional component in addition to getting students to talk and write math… READING MATH!

WELCOME TO THE INTERNETS, a series of tubes

Kate “I’m not snarky” Nowak and Sean “the squirm” Sweeney are working on putting together a…

“Welcome to the internet, math teachers”

page on the internet. As much as I like them, I feel like this project needs a new working title. Because that’s just sad.  In the meanwhile, you should fill out their survey here.

Full confession: I haven’t filled it out yet. But I’m going to, today. So there. THERE. I PUT YOU IN YOUR PLACE FOR CALLING ME A HYPOCRITE.

PCMI as a learning community

Note: This post was started in Late July, and abandoned until now in Late August

I promised three posts post-PCMI on what I’ve learned — on math talk, on lesson study, and on PCMI as a learning community. Now that it is late August, I don’t know if I can do all three, at least not immediately. So I am going to focus on the learning community formed at PCMI, via the way the program was structured.

I first found out about PCMI on twitter. I am pretty sure it went something like this:

samjshah: anyone have ideas of good summer programs for math teachers?
tweep1: PCMI
tweep2: PCMI
tweep3: PCMI & exeter
samjshah: did exeter, should check out PCMI i guess

I hadn’t heard of it. And no one really could explain to me what it was, and why it was so powerful for them. Many people on in the edublogosphere talked about going there a few summers. To me, that’s advocacy enough, because if you’re a teacher, and you attend a 3 week professional development program in Utah, and then want to do that again, well, that’s saying something to me loud and clear.

Without getting too meta or analytical or anything, I think PCMI showed us how a serious, committed learning community could look like. It mirrored what we would have wanted our school experiences to have been, and had qualities that we wished our schools had. That’s what was so powerful. We participated in this professional development around creating an engaging and effective math community, while actually participating in an engaging and effective math community.

This is how a typical day would look.

I would wake up around 6:30 and take a bath and read. Okay, that sounds like a strange way to start the day, but this was my own private heaven, because I love reading in the bathtub and I don’t get to read much during the school year. Then I would mosey on over to breakfast, where we have a daily buffet — a veritable cornucopia of muffins and eggs and cereal and soy milk and coffee and bagels and other stuffs. Over breakfast, I would generally sit with other math teachers.

You see, at this point, I’ll interrupt and say PCMI actually has a bunch of parallel programs running at the same time. Researchers have a program. Undergrads have a program. Grad students have a program. University teachers have a program. So we high school teachers are just one of a few different strands of PCMI.

Over breakfast, we’d discuss our schools, math problems, what we did the previous night, books we’re reading, TV shows we love, weekend plans, Justin Bieber, whatever! The whole three weeks definitely had a summer camp atmosphere because almost no one knew each other and so there was always lots to discover about everyone. There were about 60 teachers in the program.

Then we’d rush over to our morning problem solving session, which lasted a hair under 2 hours. There would be quick announcements, and then Morning Shorts. These are 5 minute presentations given by participants. Here’s mine. Then onto problems. We’d sit in groups of 5 or 6 (one of those people was a table leader, but they weren’t privy to any of the problems or solutions we’d be working on beforehand). We’d be given a packet of problems and then just set off to go. Nothing else. No formal lesson. Just compelling problems. Sometimes our group would work independently, sometimes we’d talk, sometimes we’d get off track (but super rarely). Connections were made, informally, independently, at our own pace.

The problems were made by Darryl Yong and Bowen Kerins, and are online here (look for the Hand-Outs section). The philosophy of the class is pretty well summed up by their “rules”:

Bowen and Daryl made each subsequent problem set after watching us, and seeing where we were at. We were going, each, at our own pace. To make sure we could keep up with the course, and not fall too far behind, there was a core idea that was put at the start of each problem set — called IMPORTANT STUFF. We had to get through that (we always did), and then we could just work on whatever. But the next day’s material would only really require us to know the IMPORTANT STUFF.

The problem sets themselves were the most well-crafted set of problems I’ve ever been given, in terms of scaffolding. I don’t think you can see really how these problems are so amazingly scaffolded until you actually work through them. Because you will start seeing cycling back to old material, little hints about connections you’ll be making (no connection was ever explicitly given to us in the problem set — we had to do all that heavy lifting on our own), and a general ramping up to some really frustratingly engaging problems.

They were also really funny. With each of our names included in at least one of the problem sets — which actually gave us a nice feeling to see in print. And lots of jokes and puns.


Ha, this was the title of the 8th problem set. Notice the use of our names in the problem set! (And each table had numbers, hence the Table 8 in the title.)

I think I got a stomach cramp laughing at the marginalia. Um, it’s okay if your answer has some p in it? Get it? Also, notice Caro(l)’s name in the problem set? (I cut off the l, but it was there, I’m sure.)

And of course the stupid  math humor I’ve come to love so much.

I would just like to point out that all these examples came from the same problem set! So imagine this, every day! Fun! Times!

Okay, so we’d work, and then in the last 15 minutes, or maybe in the middle, Bowen would talk about what people were seeing, he’d maybe throw up a Sketchpad applet he made, or a photograph of some of our work and have us explain it to the rest of the group. And then we’d move on, the next day.

There was one other thing that made this setup so well, minus the self-directed pacing, the well-crafted problem sets, the ability to collaborate. It was that we would only be with our group for 3 days. Yup, that’s it, 3 days. Then we’d be assigned a new group. The end result? After 3 weeks (15 days), we’d have been in groups with almost everyone attending.

Frequent group switching was one of the ways that I think our community was built so quickly, and so powerfully, in 3 weeks.

After some coffee and cookies (that I refrained from eating, thank you very much), we’d then go into our “Reflection on Practice: Connections to Research” groups. For these, we were put in groups of 5 or 6 again, but then we’d have  a few groups in a room instead of all of them. So instead of having all 60 of us in one room, we’d only have 15 or 20 so. We had two leaders, who led us through various exercises, reflections, video-watching and transcript-reading sessions, etc. I would write more about this part of the program, but we learned a lot about a lot of things, so it would be like trying to capture a hundred butterflies fluttering about in a room. Possible, but time-consuming.

For me, the most major theme we hit upon was “math talk” — the purposes of it, how to encourage it, how to evaluate it, and the rewards of it. That’s too big to tackle for me in this post. I’ll leave it at that, until some future time.

We switched up groups and leaders in “Reflection on Practice” session every 5 days.

Then lunch. Mostly we were allowed to sit where we wanted, while we enjoyed more (really good, really filling) meals. At least for me, most conversations were based around teaching, since we had just been given lots to think about in the morning. A few times, we were assigned lunch tables. At first the thought irked me. I’m an adult! But the point was to mix all the different programs together, so we could talk to undergrads and university teachers and everyone inbetween. It was actually fun. For the most part the conversations were enjoyable and engaging, and the few times they weren’t, they were benign and innocuous.

Finally after lunch, we’d embark on the third major part of the day: our individual project groups. There were six different groups (and within some of those groups, sub-groups!):

Reasoning from Data and Chance
Exploring Discrete Math
Investigating Geometry
Implementing Lesson Study
Visualizing Functions
Learning the Math of Image Processing

My group had 9 people in it (2 of them group leaders), and focused on Japanese Lesson Study. It was here we got to create. All groups created some final product. And dang if it isn’t fun to create with other people.

My group went through a sped up cycle of Japanese Lesson Study. We put all our work on a wiki (slightly messy, since it was used a lot), focused on achieving these three goals:

  1. Students will develop a conceptual understanding of converse, inverse, and contrapositive statements, and will be able to use multiple models (specifically Euler diagrams and sentences) to reason about these statements.
  2. Given an assumed true conditional statement, students can distinguish and clearly explain the truth values of the inverse, converse and contrapositive statements — using counterexamples to show the falsity of statements.
  3. Students will develop an appreciation for the precision of language, and usefulness of if-then statements.

We actually got to teach this lesson twice (once to other teachers, and once to real students), and revise it once. One of the most memorable and exhilarating moments at PCMI was watching the lesson that we argued and slaved and nit-picked over come to life when taught to students. The kids were engaged, and I could see them slowly come to understanding on their own. We had come up a list of possible confusions and a list of strategies to employ if they happened. Watching that unfold was… well, you could see the impressive class that results from using collaboration, backwards design, and a focus on student understanding. And seeing kids smile, and work through frustration (productive frustration!), and get that deep a ha moment, that was powerful.

It wasn’t a perfect lesson, but it was perfect enough. A thousand times better than what I can produce in my own classroom. It was an example of the type of lessons and the type of teaching I want to do, where there is less lecturing, serious math discourse, and the  teacher is merely a guide while the students are in the drivers’ seats.

I gave a little spiel about the use of Wikis in lesson study at the end of PCMI. Here is a clip:

Much like our teaching in the classroom, the nature lesson study is organic and evolving. Fundamental to the lesson study process are the dual ideas of: collaboration and continual improvement. The wikispace provides a well-suited home for this sort of work. When we meet, we never quite know what ideas will jump out, what we are going to pursue, what ideas will become central to the lesson, and what ideas will be jettisoned. But it’s important in this type of collaboration – where the creativity of multiple minds comes into play – to have a way to organize these ideas.

As I said, each working group created things which we shared with each other. And seriously, it was all amazing stuff. And then, at 3 something, we were done. That was the end of the required part of the day.

In the evenings, there were a number of fascinating lectures, informal discussion groups, formal classes — all optional, most interesting. Evenings were the social times, where we would BBQ and go into town and watch Veronica Mars and karaoke and eat at a restaurant and play RockBand and go on walks/hikes and …

And the days would repeat.

I know we had a self-selecting group, but it was a learning community at it’s finest. The program was heavily structured — as you can see, it’s a full day. And you couldn’t skip classes or arrive late. You didn’t get to choose groups (except foryour working group, kinda). But within those strict parameters, you had an informal, playful, intense, passionate atmosphere. To me the most defining features of PCMI was the group work — which had groups switch constantly (some every 3 days, some every 5 days, but our working groups stayed constant). We also had lots of different activities (we weren’t in a lecture hall all day) which broke up the time. We were given breaks and informal times to talk about what we had learned. We weren’t really lectured to at any point; it was about thinking and sharing, reflecting, and collaborating. We learned by doing, not by being talked to by Almighty Gods of Teaching. Most importantly, the program was designed for us to be engaged. Clearly. The designers made a conscious effort to be rigorous and interesting. It was differentiated, good for people at all different stages in their careers. And it worked.

If it tells you anything, I spent 3 of my 10 week summer vacation at PCMI. I am going to forego applying to the other program that I have been coveting, in hopes that I can return to Utah next summer.

A Benefit and Peril of a Laptop School

Tomorrow my Algebra II students are going to take their final exam. The year has come to a close. I’ve taken stock of the year in a number of ways. Here’s another summary of my year (a la my SmartBoard notes data).

In order to keep myself organized throughout the year, I make a separate email folder for each student. All emails to, from, or about that student (e.g. with parents, with deans, with their adviser) get archived in this folder. I compiled the data and saw that I’ve done a lot of individualized emailing. To be precise, 2,270 individual emails. I teach only 43 students, so this comes to about 53 emails per student. Most aren’t extensive emails — sometimes it’s just them asking me a quick question, or trying to schedule a time to meet, or me asking them if anything was up. The picture of how these 2,270 emails were distributed is below.

The number of students is on the left, the email range is on the bottom.

I don’t really have many conclusions.  I just wanted to see what the data would look like. The students on the right — who I emailed a lot lot lot with — were the ones I was trying to reach the most. So it’s nice that there was a correspondence between this data and that.

I do spend a lot of my time on email. As for whether it is worth it, well, that’s still up in the air. I think it is.