# “Explore Mathematics”

I teach an Advanced Precalculus class, and I love my kids. This is my second time teaching the course, and I get a rush seeing the kids dive into whatever we do with full intensity. Because the curriculum we teach is so chalk full of things, we don’t really get days where I can go on tangents and have students explore things that I think would be of interest to them.

Earlier this year, I was struck by this post by Fawn Nguyen. It’s rare that I read something and it just keeps rattling around in my brain, and won’t let me forget it. (Thanks Fawn, for being an annoying bee attacking my brain!) If you’re too lazy to click the link, the TL;DR version:

Fawn has her kids go to Math Munch and explore and play with mathematics it based on what interests them. She has her kids keep track of what they do with this sheet:

What I loved about this? It gave kids the freedom to explore mathematics that interested them. The assignment was fairly low-pressure.

I wanted to do something similar. I knew I wanted it to be low-pressure to do, fairly easy to grade, and really focus on what the kids want to do. Thus, Explore Mathematics! was born.

[.docx]

Students are asked to engage with mathematical things that they are interested in during the third quarter. There are two deadlines, so they are working on them continuously and not rushing at the end to finish them. (Also to make marking them easier for me.) There is a low-pressure grading structure, which reinforces the notion that this is more about just engaging and less about “doing the right thing.” In total, I’m making it worth about half a normal test.

I don’t know exactly how this is going to turn out. But I’ve already had a student present a piece of mathematical artwork he’s made, and I’ve had a couple fun conversation with kids about things they’re thinking of doing/looking at. I hope this fosters a lot of fun mathematical conversations between me and the kids about the things they’re finding (and of course, among the kids themselves).

The biggest concern is making this assignment not seem like or become busywork for the kids. I don’t want it to seem like added work just for the sake of extra work! That’s the fine line I am trying to navigate — sort of “forcing” kids to carve out some time here and there in their busy schedules to get exposed to the cool things out there. I have to figure out how I can create this feeling in the kids. Maybe that means I will give up some classtime for them to work on this every-so-often, to show them I value this sort of exploration. Wish me luck on this.

# Switching Up Groups

Today I switched up the groups I had in my Precalculus class and my two Calculus classes. All three classes have kids sitting in groups of 3 (and occasionally 4). They were with their original group members for the first half of the quarter. Now they are getting new group members.

I know this is a little hokey (and I tell them that), but I really want groups to get started on the right foot. I want kids to be thinking about how they interact in groups. I want kids to work at being a good group member.

So I did the following things. After they sat in their new groups, I gave them 20 seconds of silence where they were going to decide the very first thing they were going to say to their new groupmates. Then they went around and said whatever it is they wanted to say.

Then I had them write their names on a notecard. On the front, they were asked to write something they thought was a strength of theirs when working in a group. Some examples:

Good at finding new ways to explain things to people, or simpler ways

I’m pretty articulate and willing to help if someone needs it.

I’m a good listener.

I listen to your answer and compare. I do not think mine is always correct.

Not leaving people being / staying together as a group.

Then on the back, I had students write something they weren’t good at in a group setting, but wanted to work on. Some examples:

Sometimes I don’t ask questions when I might need to.

If I’m stuck I don’t ask for help enough.

If you’re moving too quickly through a problem for me, I typically won’t say anything.

I usually speed ahead.

Then I had kids go around and share what great quality they are going to be able to share with the group, and something they are going to pledge to work on with the group. What was nice is that in those few minutes when they were talking, I saw other people acknowledge and listen. One group noticed that together their strengths and weaknesses worked well together!

I know it’s hokey. I told them I knew it was hokey. But kids shared good things with each other. And maybe it’ll help a few kids be more aware of the others in their group too.

# Senior Letter 2013

Each year I write a letter to my seniors and give it to them on the last day of classes. I have done it since my first year of teaching. And I keep doing it, year after year, but this was the first year I questioned why. It’s been a long year for me, and for reasons not worth going into, I just didn’t feel I had it in me. I was exhausted, drained, and I didn’t think my kids would get much out of it. So last week, I sighed and declared to myself I wasn’t going to write one.

But even though I was firm in my declaration, I felt unsettled.

And I realized: I don’t write the letter for them. I write the letter in honor of my own high school English teacher who handed me a letter at graduation, a letter I still have today. It was thinking of him in my first year that inspired me to write that initial letter. And each year since, now understand it is my way to pay homage to him.

So at the last minute, I decided to write one. It says the same thing I always say: learn stuff, because the world is ah-maaaahzing, and if you can see the world through curious eyes it becomes so potent and energizing. Because of the lateness of the decision to write it (which also involves printing it out for each student with their name on it, stapling and signing them, adding something they filled out on the first day of my class, and throwing in two personalized business cards with different things on them… all in an envelope), I had to crib some of my favorite language from last year’s letter. And although I tend to have the same general message each year, this year I realized I was hinting at something new: there is something really sacred about knowledge, and throwing away opportunities to learn stuff is a choice you make… so go to college with your eyes on the prize.

I wasn’t going to post my senior letter this year, because I just felt it was rushed, I don’t think I got my points across well, and about 1/3 of it was cribbed from last year’s letter. I don’t know how writers do it! But today, a senior who I have great respect for, who I have come to know fairly well throughout his years at school, told me with such earnestness that he really appreciated the letter, and that he felt like there was something about articulating the passion of learning that his teachers felt that really was powerful for him.

I feel awesome that one kid sincerely took something away from me writing it. And that is enough for me.

# A High School Math-Science Journal

In my first year of teaching, fresh from my haze from history grad school, I remember approaching the history and English department chairs about creating a high school level journal for those subjects. I mean, our school has a literary magazine, and also even a publication for works in foreign languages (seriously!). But nothing for amazing critical analyses and interpretations in English and history. I figured having something like this might encourage students to revise already excellent work for publication, and also make the audience of their paper be an audience of more than one. I even contacted the literary magazine student editors to see if they would feel like the journal would encroach on their domain (they said no). For reasons that are still quite beyond my understanding (because I still think it’s an amazing idea), both department heads rebuffed my idea. (Also, if they said yes, they would have gotten an enthusiastic first year teacher who would have taken on all this work!)

And so, I let this idea pass. One of many that I have, think are awesome, and then languish and die, either due to my own laziness or due to external circumstances beyond my control.

Until last year. When I was thinking: I’m a math teacher. Why not start a math and science journal? It’s so obvious that I don’t know why the idea didn’t hit me over the head years ago. So I found a science teacher compatriot who I knew would be interested, and we came up with an initial plan. And at the end of last year, we presented it to some students who we thought might have been interested (as this was something that is something that has to be for them, by them… if they don’t want it, there’s not point in doing it… it’s not about us…). They were, and we were officially off to the races.

We shared with the students the following document we made, with a brief outline of one vision for the journal. But with the understanding that this was their thing so their ideas reign supreme. This was, in some sense, a mock-up that the science teacher and I made to show them one possibility. The one thing that the science teacher and I were really aiming for in our mock-up was that the journal shouldn’t just be for superstar students. We wanted to come up with an journal that has a low barrier of entry for students submitting to the journal, and that if a student has interest or a passion for math or science, that’s really all they need to get started. To do this, original and deep research wasn’t really the primary focus of the journal. So here’s our brief proposal:

The additional benefit of having this journal is hopefully it will cause curricular changes. Teachers will hopefully feel moved to create assignments that go “outside of the box” — and that could result in things being submitted. Students who express an interest in some math-y or science-y idea (like why is 0/0 undefined… something that came up in calculus this week) could have a teacher say “hey, that’s great… why don’t you look it up and do a 3 minute presentation on what you find tomorrow?” … and if they do a good job, encourage them to write it up for the journal. Or a teacher might assign a group project on nuclear disasters, and encourages the students who do extraordinary work to submit their project to the journal. (Which can be showcased by teachers the following year!) Or a student who notices a neat pattern, or comes up with an innovative explanation for something, or who wants to try to create their own sudoku puzzle, or decides to research fractions that satisfy $\frac{1}{a}+\frac{1}{b}=\frac{2}{a+b}$. Or whatever. Knowing there is a publication you can direct the student to, as a way to say “hey, you’re doing something awesome… seriously… so awesome I think you kind of have to share it with others!” is going to be so cool for teachers. (As a random aside, I was thinking I could enlist the help of the art and photography teachers, because of the overlap between math and art… They might make an assignment based around something mathematical/geometrical, which students can submit…)

I honestly have no idea how this is going to turn out. What’s going to happen. How the word is going to get out. If anything will be submitted. If kids get excited about it. Lots of questions. But I have a deep feeling that the answers will come and good things are going to happen with this.

I’m soliciting in the comments any thoughts you might have about this. If your school does a math journal, a science journal, or a math-science journal, what does it look like? What works and what doesn’t? Do you have a website/sample we could look at? If you don’t have one, and you are inspired and think of awesome things kids could put in there (e.g. kids submitting their own puzzles! kids writing book reviews of popular math/science books, or biographies of mathematicians/scientists! getting kids to create photographs or computer images of science or data visualization or just making geometrical graphing designs! trust me — brainstorming this is super fun!) I’d love to share any and all ideas with the kids involved with this project at my school.

# My 2012-2013 School Planner

I know it is Thursday, but I have never really been good with working on a schedule. (I am not the teacher that has a unit outline and homework to give my kids at the start of each week, because how the heck am I going to know where we are in five days let alone one?) So forgive me this fault of posting my Made4Math Monday on Thursday.

Anyway, I’ve posted about this each year. I figure I’ll do it again. My school has a rotating schedule, where we meet kids four times a week (50 minutes each). We’ve had this schedule for the five years I’ve been teaching at my school, and I still don’t know it by heart. I have gotten the class period start-end times down, but which class I’m meeting when is always a mystery. Also, with my brain, scheduling meetings with kids is something that has to happen in my planner so I don’t double book.

To help me out, I designed and published my own 88-page weekly planner. A copy of my weekly schedule for 2012-2013 is below (missing, of course, the 10th grade team meeting, the two “duties” I will be assigned, math club, and the weekly meetings I have to schedule with the other Calculus and the other Pre-Calculus teacher… so don’t be so jealous of all the free time… it’s really not there.)

The cover for the planner is:

Simplicity.

What’s nice is after my first year, there were a few other people who wanted to order planners too, so I have put up a blank planner (meaning: without my classes inputted in them) for them to buy. I don’t make any money off it or anything.

## How I Made & Ordered Them

In order to make them, I had to use Adobe InDesign, which I had never used (the school laptops which we are issued come with it already installed). So I spent hours, years ago, making the original grid, picking the fonts, and working on the design. [1] But those hours were worth it, because even though bits and pieces change each year, I have been super happy with the look of it. (I created the file to be A4 paper size, because I wanted it to be slightly bigger than regular paper so I could slip a sheet of regular paper in there without it sticking out.) Then you convert your file to PDF. Just remember: if you’re going to design your own, make sure you have a blank sheet before you start your calendar pages. This way the entire week will be on facing pages.

So after I made the planner, I used lulu.com to order it. You just upload your PDF and specify what you want. I get saddle stitch (fancy way of saying heavy-duty staples). Designing your cover on lulu is really annoying — their “cover wizard” is difficult as all heck to use. But even if you order a black and white planner (as I do), a color cover comes with it. And if you are a photo person, you can have a photo background! Each year, the price of printing between $7-$9. With shipping it comes to around $15 plus or minus a few bucks. And viola! Your own fancy planner! ## How I Use It I basically just use it to schedule my time. A page from my planner from two years ago: Also, because of my horrible brain, I require kids to email me to set up a meeting. I’ll have none of the coming up to me after class asking to set up a meeting (usually I have to run to another class, so that doesn’t work for me), or accosting me in the halls. I simply don’t know when I’m free. So they email me with all their free periods for three days and I find the first common free that works for both of us. (Usually it is the next day… rarely I have to go a couple days in the future, if our schedules are so opposite.) And that’s it! My planner, from creation to use. [1] You simply need to upload a PDF file to lulu.com to get it published, so if you are good with Word, you could even create something nice in there! # Business Cards, Stickers, and Cards: Oh my! There has recently been this awesome arts & crafts trend that has been happening with math teachers on blogs. Lots of talk about hobby lobby and the container store and puff paints and spray paint chalkboards. I used to be crafty, but no more. I’m too manly to be crafty. [1] But I do feel like I have a fairly good sense of design. So I figured I’d join in with my first installment of Made 4 Math Monday. I should probably preface this by saying this is not math specific. I went to one of those personalization stationary websites, and ordered a lot of stuff. My favorite of the things I ordered were specially designed business cards. I ordered some fun ones for me to use when I’m out on the town, but I also designed some for me to carry around the school, in the classroom or outside the classroom. It’s definitely a novelty thing that will wear off, but I’m amused by them. Can you imagine being a kid and having a teacher sort of flick you a card as you’re leaving? Or just hand you one in the hallway? Or if you were being funny bad (but not seriously bad), have someone hand you one of these? The first card is more like a motto I want to instill in my kids, especially because I’m doing standards based grading. In order to get free shipping, I had to order more. So I designed stickers (I’m getting 24 of each sticker). I should say I had limited time to design them so they aren’t exactly the best things I could have come up with. But they’re good enough. Now that I’m looking at these, I’m wondering if I couldn’t make better stickers which weren’t all about achievement. But maybe a sticker for someone who just bombed something… something to make them smile, and feel encouraged. Or for someone who clearly put a ton of effort into something even if they didn’t do superduper. Anyway, thoughts for the future. I still had a little more money to spend for my free shipping, so I also ordered 10 cards (the outside, then the inside). The card is not my favorite, because I could have done a much better job and come up with a much better concept if I had more time. But here’s the thing. I always keep blank fancy cards at my desk to write thank you notes to people who have helped me out in small and large ways. Unfortunately, I forget I have them, so it hasn’t become normalized behavior. But since I will have fancy personalized cards, I think I will be more excited to use them and so I have high hopes this could become a thing that I do, instead of just intend on doing. I hate having to wait to see how all this merchandise looks (it’s going to take up to two weeks to get to me!), but I’m uber excited about it. As for costs, I had a groupon that cost$20 for $80 worth of merchandise. But for some reason, I had$10 off my groupon. I also had to order an extra $25 for free shipping. So for all of this (I ordered eleven different business cards, even though I only showed a few here), I only ended up paying$35.

And with that, my first Made 4 Math Monday is out.

[1] False.

# Algebra Bootcamp in Calculus

So it was the Old Math Dog who pointed out that I never wrote a post explaining how I deal with the issue of kids not knowing basic algebra in calculus. I started this practice two years ago (when I also started standards based grading) and I have seen a remarkable difference in how my classes go from my life pre-bootcamps to my life post-bootcamps…

An issue in any calculus course — and I don’t care if you’re talking about non-AP Calculus or AP Calculus — is the student’s algebra skills. They might see $\frac{1}{4}x+\pi x -4=0$ and have no idea how to solve that. Or they might not know how to find $\tan(\pi/6)$. Or they might cancel out the -1s in $\frac{x^2-1}{x-1}$ to get $\frac{x^2}{x}$. It depends on where they are coming from, but I can pretty much guarantee you that every calculus teacher says the same thing to their classes on the first day:

Calculus is easy. Algebra is hard.

In my first three years of teaching calculus, I started with how all the books started, and all my calculus teacher friends started: a precalculus review. Then we went into limits.

The problem with that is that we might review some basic trigonometry, and then we wouldn’t see it again for months. And by then, they had forgotten it. And who could blame them. The precalculus review unit at the beginning of the course wasn’t working.

As I transitioned into Standards Based Grading, I looked at everything I taught really closely, and I honed in on the particular skills/concepts I was going to be testing. And since I’d taught calculus for a number of years prior, I knew exactly where the algebra sticking points were. Thus was born The Algebra Bootcamp.

Before our first unit on limits, I carefully analyzed what things I needed students to know to understand limits to the depth I required. I then looked at all the skills and thought of all the algebraic things, and all the old concepts, they would need in order to understand limits. And from that, I crafted an algebra bootcamp, and I made SBG skills out of just those limited skills.

For example, here was our first bootcamp (which, admittedly, was longer than most of the others, because we were settling in and I was gauging where the kids were at):

and I did the same for other units… just the targeted prior knowledge that they tended to not know or struggle with…

Notice how they tend to be very concrete and specific? Like “rationalize the numerator” (because I knew we were going to be doing that when using the formal definition of the derivative) or “expand $(x+h)^n$ using the binomial theorem. Very specific things that they should know that they are going to be using in the following unit. It’s kind of funny because it is a hodgepodge of little (and often unconnected) things, and they have no idea why we’re doing a lot of what we’re doing (why are we rationalizing the numerator? why are we doing the binomial theorem?) and I don’t tell them. I say “it’s our bootcamp… once training is over you’ll see why these tools are useful.”

It is called “bootcamp” because I am not reteaching it from scratch. I’m reviewing it, and I go through things quickly. I only do a few of them in the first quarter and maybe the start of the second quarter. By that point, we’ve done what we needed to do, and they die off.

The reason that this has been so effective for me is because students aren’t having to relearn old topics/algebraic skills while concurrently learning the ideas of calculus. We review these very specific things beforehand so that when we approach the calculus topics, the focus is not on the algebraic manipulation or remembering how to find the trig values of special angles or what a piecewise function is… but  on the larger picture…. the calculus.

Remember: calculus is easy, it’s the algebra which is hard.

So we took care of the algebra beforehand, so we can see how easy calculus is.

My kids in the past two years have made so many fewer mistakes, and we’ve been able to really delve into the concepts more, because I’m no longer fielding questions like “could you review how to do X?” Doing this has also forced me to think about what the purpose of calculus class is. The more I teach it, the more I take the algebraic stuff out and the more I put the conceptual stuff in. For example, I don’t use $\cot(x)$, $\sec(x)$, and $\csc(x)$ in my course anymore  [1], because I wasn’t trying to test them on their knowledge of trigonometry. Doing these bootcamps coupled with standards based grading has forced me to keep my eye on what I really care about. Students deeply understanding the fundamental concepts of calculus. And I think you can do that without knowing how to integrate $\sec(x)\tan(x)$ just fine. [2]

[1] With the exception of $\sec^2(x)$ for the derivative of $\tan(x)$.

[2] I teach a non-AP calculus, so I have this luxury. But it’s nice. Each year I strip more and more stuff off the course and add in more and more depth. And I am glad that I understand depth to mean something other than “more complicated algebra in the same old calculus problems.”