Mathematical Communication

Yet again, following f(t)’s lead

[UPDATE: Posts on starting blogs… Kate’sMiss Cal.Q.L8’sRiley Lark’s (scroll down)… all definitely better than my two cents…]

I can’t help it. I really dig Kate Nowak. (In the platonic way, obvi.) And everything she wants to do, I want to do too.

(On that note, I have a competition idea that I’m contemplating rolling out for the summer…)

(Kate, don’t say you want to eat a vat of fresh tomatoes, please. I HATE tomatoes.)

She gave her few cents about starting a math teacher blog — and things to do and things not to do. I thought I’d piggy back on that and give some totally unsolicited advice of my own.

Actually, I think it would be good if all of us who blog do this. [1]

1. Don’t worry about your blog name. I know a few people who want to start blogs but agonize over getting “the best, most punny, insightful name that captures the essence of what you want to do.” That’s cool. I think I spent an eternity trying to find the best Google Voice phone number, so I get it. I remember I spent at least 2 hours trying to come up with a killer name. But I’ve known people who’ve agonized over it to the point where they never got started. So agonize, but give yourself a short deadline. “By the end of today, I will have started my blog. No. Matter. What.” A rose by any other name…

My story: I gave up on my search for ‘the best name’ and just went with a cool fact I learned (that you can have a function which is continuous everywhere but differentiable nowhere). Totally makes no connection to what I write. So what.

2. Choose WordPress. I know everyone says it doesn’t matter whether to choose or I’ve used both and I have a strong preference for The themes are sleeker (in my opinion), there are more control options, and most importantly, you can easily type equations (y=ax^2+bx+c). I am a huge, huge fan.

My story: I’ve written in livejournal in college, started two blogs (one of them a group blog) using Blogger in grad school, and now I’m a committed devotee of WordPress. However, maybe Blogger has more options since I abandoned it? I won’t knock it, but I know WordPress is awesome.

3. Write like nobody’s watching. Okay, this piece of advice might either sound obvious or counter-intuitive. But it’s the one I most believe in. And I assume you want to blog because you want to engage with others, right? You’ve been out there reading stuff, and you’re like, “ME WANT TOO!” At least that’s what happened to me. But guess what? If you blog for yourself, you’re going to want to write stuff — and it won’t be a chore.

And write about anything and everything related to teaching that you want. Don’t feel restricted to post only about this or that. Make your blog less about being your blog and more about whatever you want to say, and let it grow organically into whatever it turns into. Try not to make it into something — let it grow into being something.

In other words: blog for yourself.

My story: I kept my blog private for four (or more?) months. I was writing for me. I eventually got fed up with just leaving comments responding to others, and never really getting to say anything of my own. So I made it public. But I just kept on keeping on. Writing about whatever I felt like. If I cared about getting a readership, I wouldn’t have posted about multivariable calculus. (Something I post about a lot, actually.) My blog was and always will be (until I grow tired of it) an archive of my teaching.

4. Corollary: Keep at it. Guess what? Don’t be concerned about blog stats and visitors. Remember, you’re writing for you. People will see it. I promise. Okay, yes, you’re going to start looking at the stats. You won’t be able to help it. And you’ll feel good when the numbers are up and the numbers are down. That’s cool. I mean, who doesn’t want to be popular? But I guess the message is: be popular on your own terms. Or another way to put it: be yourself.

My story:  My first post was in August 2007. I’ve written 422 posts since then — not including this one. The average number of visits for my first couple months was 15 and 14 visits/day respectively. It took me until July 2008 (11 months) before I broke the 1,500 visits in a month. It took Kate Nowak 10 months. What you’ll notice is that if you keep chugging away at it, your numbers will go up. Just by the sheer fact that you have been writing more. So more people will stumble upon it. And more google searches will end up on it.

5. Watch what you write. Okay, so I said write for yourself, and I went on and on about it. But you are making this public. So the best rule of thumb: don’t write anything you wouldn’t want your kids, your administration, or a potential employer to see.

My story: I stick to this. I don’t write when I’m angry. I sometimes write when I’m disappointed – but mainly with myself.

6. Perfection isn’t attainable. I know some bloggers talk about having a ton of posts in draft form. Writing and revising. And revising. Heck, if you have an idea, just take the 30 minutes to pound it out and press PUBLISH. Agonizing sucks, and isn’t worth it for something you do for fun. As a lark.

My story: I write, I publish. Sometimes two or three things in a day! Sometimes only once in two weeks. I don’t let a schedule dictate anything. But sometimes, when it’s been a week or two, and I haven’t written anything and I have some spare time, I try something. I sit down in front of my laptop and for 10 minutes, I think if I have anything to write about. Usually I come up with something. The only time I let time linger between when I write something and when I post it is if I hesitate pushing that publish button — which is a sign to me that there is something in it that I’m not comfortable with. Usually something that relates to #5.

Obviously this is what works for me, and it may not be your style. I just really want to say: DON’T STRESS ABOUT IT. Just have fun with it, and don’t worry about it too much, and have fun with it. Oh wait, I said that twice. Well, I meant it.

[1] I sometimes think that some ed grad student should stumble upon our little community and write about its evolution from 2006-present. I got this thought probably because I was trained as a historian for a few years, before I became a teacher, and this is exactly the type of grad student seminar research paper that a sociologist or information scientist who joined our seminars talked about writing. Then we’d talk about Foucault and I’d want to bash my head against the heavy wooden seminar table while I attempted to figure out what “the form making a sign and the form being signalized are resemblances, but they do not overlap” meant. I digress. These posts would be good research fodder for the grad student. [Update: this post and comments at dy/dan would also be good fodder.]

Senior Letter 2010

On Wednesday, I had my last day of classes with my seniors. I write letters to my senior classes each year and hand ’em out on this day. I don’t know if they throw ’em away, or take what I say to heart, or something in between. But I don’t write it for them.

I print it out on school letterhead, and include their “Who I am” sheet they filled out on the first day of classes with an index card with goals they had for the upcoming year.

I hope they stick the letter and “Who I am” sheet in their yearbooks and forget about ’em until years later. I did that with a letter my high school English teacher gave to me, and I treasure that.

A binomial expansion throwdown. You in?

Oh k8, my k8, has thrown down the gauntlet. Or in more modern day kid-speak, she asked you to “BRING IT ON!” (That’s Kate Nowak, for y’all.)

A while ago, she scoured every nook and corner online for videos teaching the binomial expansion, or for some ideas which make the teaching of it… well… not excruciatingly boring. Actual videos that didn’t make her want to stab her eyes out, they didn’t quite exist.

So she’s asking you to: make one. Anything that’s better than what’s out there.

You have weeks to do it (deadline: May 27th). She’s offering some sort of t-shirt prize. I’ll sweeten the pot. If we get 7 or more video submissions, I’ll buy the winner a copy of that Lemov book that the New York Times article featured a few months ago (as long as you don’t live somewhere with crazy shipping costs). And if you own that (or don’t want it), I’ll buy you some insanely cool math book. Yes, this is my own money. No, I don’t know why I’m doing this, since I’m pretty poor.

So when Kate says “BRING IT ON!” I hope you enter so you can say “IT’S ALREADY BEEN BROUGHTEN!”

Also, if you have a math or math teacher blog and want to spread this around, that would be super duper awesome.

Evolution of my narrative comments

In my school, we write narrative comments for all our students twice a year. In order to prepare for my first quarter comments, I looked back at my comments of years past. Although I don’t think my comments are exactly where they should be, I was pretty proud of the long way I’ve come when writing them. [note: information has been changed in all of these.]

They’re not amazing yet, and I know what I need to work on, but I’m happy to see how far I’ve come.

1st year teaching
Stu is a pleasure to have in class. This quarter we have had 4 major assessments: three quizzes and one test. Stu’s grades on these were 13.5/15, 18/25, 59/100, and 43/50. Stu’s homework grade is 95%. Clearly – from her homework grade – Stu spends quality time on her work, which is really important for understanding. On the chapter 1 test, Stu scored a 59% which I know must have upset her. Instead of being frustrated and angry, Stu made an appointment to see me and talk through it. Her improvement was clearly evidenced on the next quiz where she garnered an 86% (43/50). She should be proud of this accomplishment. I continue to encourage Stu to ask questions in class when she’s confused and also to continue to make appointments to individually go over some of the material she finds challenging. Let’s hope this upswing continues into next quarter.

2st year teaching

Stu is a joy to have in class. The earnestness with which he engages with the material in class, working through problems or asking questions, is a boon to any teacher.  I asked students to write a reflection at the end of the quarter, and his was incredibly thoughtful. He wrote “The awareness of my understanding helps me to ask informed questions in class and is crucial to my classroom involvement.” The entire class benefits from these questions.

We have had four major assessments this quarter: a quiz on functions (47/52: A-), a quiz on exponents, logarithms, and trigonometry (35/42: B), a quiz on limits (42/51: B-), and a quiz on limits and continuity (27/33: B-). He has also completed all his homework assignments assiduously. On the first quiz on limits, Stu seemed to have some difficulty understanding the difference between “zeros” and “asymptotes” when doing sign analyses of rational functions. On the second quiz on limits, Stu’s difficulties revolved around proving a function was continuous everywhere (using the fact that it was a composition of two continuous everywhere functions). I encourage Stu to review these quizzes. If he has any questions about how to do these problems, he should meet with me!

His final quarter grade is an 86% (B+).

3st year teaching

Delightfully funny and always striving to do better, Stu is a student who focuses intensely in class to ensure that he understands the material daily. From what I’ve seen so far, it appears that Stu has a strong command of mathematical ideas and abstraction, and he picks up on ideas fairly quickly. When given problems that check student understanding in class, Stu endeavors to get an answer — and is always willing to help those around him also. The questions he raises are good, and I encourage Stu to keep up the volunteerism. The questions he asks benefits the class as a whole.

In a reflection I had students write near the end of the quarter, Stu noted that his way of approaching homework wasn’t working. He said “Initially, I didn’t realize I had to show all my steps and write neatly, but I now know what I need to do, which shows improvement on my part. I’m working hard to be more thorough.” Not only is this important in Algebra 2, but learning to correct mistakes in any class is important because it encourages students to be active learners, not passive learners. Stu certainly is an active learner.

We have had three major assessments and two pop quizzes this quarter. On the major assessments—on sets, inequalities, and absolute value equations; on absolute value inequalities, factoring, and exponents; and on polynomials, domain, and rational expressions—Stu earned 54/60 (A-), 59/70 (B), and 48/50 (A) respectively. On the two pop quizzes, Stu earned 11/12 and 6/7.

Clearly his performance this year has been consistently strong, and I encourage Stu to continue working at this level at the very least. However, I always try to push my students to achieve more than what they think they are capable of achieving, and I know that Stu can do even better.  I am happy to meet with Stu to talk with him about how to achieve this.

In Algebra 2, homework is divided into two parts: daily check for completion, and our binder check for correctness, neatness, and organization. Stu has done all the daily homework, and earned a 25/40 (D) and 45/45 (A+) for the two binder checks.  The binder check is done to encourage organized, active learners, who are expected to correct mistakes. Stu clearly has learned how to do so as the quarter progressed, and I encourage him to continue checking his work for the rest of the year.

Stu gave himself 4/5 for his classroom engagement grade.

Writing in Algebra 2

One of my pushes this year is to get my Algebra II students to write math better. Last year I put “explain this” problems on a few exams and wasn’t so impressed with their responses. This year I am teaching my kids to write responses.

On their first assessment, I put a question similar to one we talked about in class:

Explain to someone who doesn’t know a lot about math why you can never find an x which would make |25x+5.1|-5=-6.

The responses were disappointing across the board. There were bits and pieces of gems, but nothing complete. Not a single student was able to construct a well-written response. Things I received included:

  • The other side of the equation is negative, leaving no possible solution to the problem.
  • You can never find x because the answer is negative and an absolute value problem with a negative after the equal sign is not possible.

So what I did was type up the following document and passed it out a few days after the assessment:

We talked about the vagueness of the responses, the use of pronouns like “it” and making references to “the other side of the equation,” and most crucial, the lack of reference in almost every solution to the original equation. How can you answer a question about an equation without even talking about the equation?

My favorite moment of the discussion this generated was when one student raised her hand and critiqued her own solution, and then said: “I wrote this and don’t even know what I meant.”

On the next assessment, without telling them I was going to do this, I threw the exact same question down. It was on. I saw my kids reread their responses after they wrote them, and really pay attention to their writing. Let me tell you: it all paid off. On this second round, most students got full marks. (On the first assessment, almost no one got full marks, or close to it, for that matter.)

Here are some random smatterings of their thoughtful answers:

  • You could never find an x to make the absolute value equation above true because you would have to subtract -5 from -6, which still gives you a negative number. |-25x+5.1|=-1. An absolute value equation cannot equal a negative number because absolute value is the distance from zero and is always positive [my correction: or zero].
  • In this absolute value equation there is no solution because any number in the absolute value has to be 0 or a positive number. And if you subtract 5 from 0 or a positive number, there is no possible way that can equal -6. So there is no solution to this equation.
  • An absolute value of anything can never be equal to a negative number, since it expresses a distance. When this equation is simplified, it becomes |-25x+5.1|=-1. If the ‘-1’ were replaced with a positive number, you could find the answer [for] x. But since it is a negative, you already know that is impossible.

I am continuing to ask them to express themselves through writing. On that same assessment where I asked them to repeat the absolute value problem, I also asked the following two questions, to which I got some really nice writups.

The following two questions build upon each other. The solution to part (a) will very much help you explain part (b).

(a) Explain why a^2a^4=a^6 without using your exponent rules. Explain it to someone so they can understand it simply!

(b) Explain why a^ma^n=a^{m+n} is true. You can assume m and n  are positive integers. Explain it to someone so they can understand it simply!

I still have to do more work with this, but I just wanted to say: it is worth it to talk with your kids about writing. One 15/20 minute conversation has already yielded great dividends for me.

Why Twitter?

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On November 18th, I decided to give Twitter a try. I wrote:

So I’ve decided there is possibly a vibrant teaching community that I’m not familiar with, because I had decided to ignore Twitter while getting the year in order. So here I am, going to take the plunge. […] I found a whole bunch of blogs by math teachers that I follow regularly. Let’s see if I can find the same on Twitter. 

It is now May 10th. I have made 741 tweets. I follow 71 people. And I check twitter multiple times a day. 

On November 18th, I didn’t “get” it. No one could explain to me why twitter was worth trying. But people on the blogs I read were talking about it. Before writing it off as inane… I mean, why do I care what a math teacher in Northern California had for lunch?… I gave it a shot. My goal for this post is to share with you how I use twitter, and why I continue to use twitter. 

One: I joined twitter to be involved with the math teacher blogger community. Turns out, most of the people writing the blogs I follow regularly have twitter accounts. I didn’t know that so many people were on twitter before joining. So these people, who I sporadically communicated with by commenting on a post here or there, have become people I communicate more regularly with. I solicit ideas from them and I share my ideas with them. The dialogue, short and sweet, is continuous. Like a bird chirping in the electronic zeitgeist.

Two: I get to solicit advice and share frustrations. And I get to give advice. 

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Three: I don’t know much about the people I follow, but I do know we share a set of values about teaching math. We love what we do. Why else would we want to talk with others who are the same. Not that I don’t have great colleagues in my school, but I am the only teacher for three of my four classes. I like to have someone to hash out ideas with. These people on Twitter are those people.

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Four: Links, links, links! I post links relevant to the post I’m writing on my blog. But I tweet lots of random math links that don’t seem to fit in what I’m doing now. Cool things that I think other math teachers might find useful. And others do the same. When I first started twittering, this was hands down my favorite benefit. Plus I get links about non-math related things too. Like when someone linked to the entire 5 seasons of Angel which were on sale for $57 at Amazon for one day.

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Five: I actually like hearing about the ordinary, math and non-math related things that my twitter friends post. Ummm. Okay, I know that these people aren’t my friends. And that I’m not ever going to meet them in real life, for the most part. But I’ve actually come to care when someone’s kid is angry at them or when someone’s husband was in the hospital. It brings the people behind the blog posts to life. Picture 7Picture 8

Six: I didn’t used to do this, but I have started doing this: when I write a blogpost, I tweet about it for other people to learn about it.

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Seven: I have discovered new math teacher blogs out there by looking at the followers of some of the people I follow.

Eight: This doesn’t apply only to Twitter, but also the blogs I read. I’ve noticed that having other people care about what they do makes me care about what I do. I want to do well that much more because of them. I honestly can’t say that I would have the drive for continual improvement and spend the time thinking through things as much if it weren’t for this little community.

And that’s my story with Twitter. I can see how someone wouldn’t find it useful. But to the nay-sayers out there, I will say this: I went in thinking I probably wouldn’t find Twitter useful/interesting/fun. It was only after I was following math teachers and joining in the conversations did I actually say “hey, this is actually pretty rad.”

To visit my Twitter Page, click here.

Just some good books about Math, for those who like Math

The math department, every year, gives awards to four students (some with some monetary compensation for college, some not). I was put in charge of thinking of some books to give with these awards. I sent my initial thoughts to my department head:

For the Math/Science award, I suggest:

*D’Arcy Thompson’s On Growth and Form is full of beautiful prose, and relates the sciences to mathematics. The actual science is wrong, but it is considered a classic piece of literature.
*Anthony Zee’s Fearful Symmetry about the important — crucial — role that mathematical symmetry plays in modern physics. A super-well written book for the layman.

For all other awards, I put out there:

*Silvanus P. Thompson’s Calculus Made Easy has a deceptive title. And it was written in 1910. But almost all accounts agree it is one of the best textbooks around. Even for those who might have thought they understood the conceptual undergirdings of calculus, this book will illuminate them further, without any obtuseness.
*Douglas Hofstadter’s Godel, Escher, Bach is standard reading for all math lovers everywhere.
*Calvin C. Clawson’s Mathematical Mysteries is one of the best and most accessible popular math books I’ve read.
*G.H. Hardy’s A Mathematician’s Apology is quite good at explaining what a mathematician actually does philosophically when he works, written by one of the most important mathematicians of modern times.

My final recommendation differed slightly:

Award 1: Timothy Gowers’ The Princeton Companion to Mathematics

Award 2: Douglas Hofstadter’s Godel, Escher, Bach; Thomas Kuhn’s The Structure of Scientific Revolutions; Bruce Hunt’s The Maxwellians; Silvanus P. Thompson’s Calculus Made Easy

Award 3 & 4: G.H. Hardy’s A Mathematician’s Apology

I really enjoyed thinking through which books might be appropriate. Also I didn’t want to give something I hadn’t read. But this process reminded me of all those books about math out there that I haven’t read (yet), but have really want to. Like Polya’s How to Solve It and David Foster Wallace’s Everything and More.

I posted this book award stuff on twitter, and got some great reactions. (Read from the bottom upwards.)



And then I thought: hey, you all must have a favorite math or math-y book that you’d want to have your favorite students read. I’d love to know your favorites! (Plus this list could help inspire me to do some quality reading this summer!)