Monday, July 5, 2010

An article, a blog, a performance, a realization...

My friends and colleagues at PCMI are going beserk for blogs. The amazing Sam J Shah did a perfect presentation about blogging and tweeting with other math teachers and I think it must have indirectly sent a bunch of those amazing people here. Saturday morning I found a comment on a recent post from the talented, kind and newly blogging Clint Chan. He recommended some articles and I have been happily reading math ed articles in bed since.
The first I read was Lessons learned from detracked mathematics departments by I.S. Horn.

In the first paragraph of this article, I read the following:
"Mathematics is an academic domain often perceived as beyond the reach of educational reforms...This is due to the conventional wisdom that mathematics is unique among the disciplines in its lack of adaptability to more open-ended styles of teaching and learning. How can we teach mathematics for understanding, for example, if the subject is made up of discrete facts that need to be memorized?"

This first paragraph stopped me reading: this introduction is just not my perception, nor the framework that I have been trained within. Specifically, the idea that mathematics is a "subject made up of discrete facts that need to be memorized" felt *almost* unfamiliar. So while the question of how to teach mathematics for understanding is a huge question in my life, my paradigm is utterly tied to the assumption that mathematics makes sense and so largely doesn't need to be memorized. I need to express my gratitude for the lifetimes of amazing and daily work of many (100s? 1000s?) researchers and educators who have helped shift the conversation about mathematics education within my life and across the globe. I want to thank them right here and now - with sincerity, emotion and endless repetition - for shifting the conversation before I arrived and for the momentum they created that has kept me moving forward. Among the many questions I get bogged down by, the question of how to teach a subject that is defined by memorization is just not among them.

Today, I finally got back to the article and found it worthwhile and thought-provoking. Horn analyzes some of Jo Boaler's research, trying to figure out what makes de-tracking work. The two most compelling points to me were: 1) teach curricula based on big mathematical ideas, connections and meaning, rather than a sequential progression through procedural skills, and 2) distinguish between teaching kids how to do math and how to do school. According to the article, these are two key ingredients to higher student performance, deeper understanding, and more enjoyment of mathematics. Duh. But awesome.

Here are some of my favorite nuggets:
1) A characterization of "group-worthy problems"
(a) illustrate important mathematical concepts,
(b) include multiple tasks that draw effectively on the collective resources of a student group,
(c) allow for multiple representations,
(d) have several possible solution paths.

2) A new definition of math: "A tool for sensemaking: Students need opportunities to understand mathematics through activities that allow them to make sense of things in the world."

3) A useful distinction: "Teachers avoided commonly used terms like canceling out to describe the result of adding opposite integers such as –3 + 3. Instead, they preferred the phrase making zeroes, as it more accurately described the mathematics underlying the process."

4) A HW accountability structure:
"At the front of each classroom was a homework chart laid out much like a teacher’s roll book, with students’ names in a column along the side and the number of each homework assignment across the top. Although actual grades were not posted, completion of homework was represented by a dot."

5) A nice detail:
"All...math teachers had a large sign with the word YET placed prominently in their classrooms. In this way, when a student claimed to not know something, the teachers could quickly point to the giant YET to emphasize the proper way to complete such a statement."

6) An important acknowledgement:
"Figuring out how to operationalize slogans like teaching for understanding is a challenge when teachers have not had opportunities to develop understanding themselves; are pressed toward the competing goal of curriculum coverage; work in isolation from their colleagues; and work in systems that value summative over formative assessments."

7) A refined idea:
Noticing whether or not students respond "sensibly" - I haven't integrated this one totally but it feels juicy. Teach for sense-making and hopefully kids will respond with some sense?

8) A great suggestion:
Looking at "fast" student's weaknesses. Are they just doing what they have to do get through the work, or are they making connections, trying to understand the purpose of the activity.

Finally, I wanted to mention Blackboard Jungle, this movie I watched the last half hour of a few nights ago. Apparently the first movie which employed the delights of rock n' roll in it's soundtrack, this 1955 education flick tells the story of Mr. Dadier, an English teacher at a tough boys school. Mr. Dadier rails passionately against his colleagues' complacency and strives to get the young men in his class excited about stories. His kids harass him, harass his wife, even threaten him with physical violence. Of course even in the worst moments (as with so many ed movies) only one kid spoke at a time (so they can really deliver those lines, I know), but there was something more tender and honest about the lonely struggle of this teacher who was trying to shift the paradigm he'd entered into.

Imagine what it would have been like to start teaching in 1955 rather than 2005! Wow. Despite the 55 years difference, I recognized the frustration, the fear, the despair, the passion. I, like Mr. Dadier, do feel like I'm trying to change the conversation, still pushing against the grain, still trying to do what feels impossible, and sometimes I even feel very alone. In the hardest moments, I scold my colleagues, I get discouraged and I feel sorry for myself.

Reading math ed research, being at PCMI, being welcomed by math teaching tweeters I've never met, watching Dan Meyer's Ted Talk, writing this blog, all reminds me 1) that I'm not alone, 2) that I'm not the first, 3) that this is somehow how it's supposed to be (at least for the last 55 years) and 4) that the reason I get to struggle with this stuff is thanks to all those who came before and laid the groundwork: it's a privilege to be able to fight and think and despair about all the stuff I do. I've just got to remember to enjoy it.

May your Mondays be refreshing and delightful. Happy Independence Day!


  1. Wonder how big I should make my YET poster.

  2. The need for mathematics seems so obvious in today's world of cell phones, i-pods , and internet access yet most of my students think that these things fell from the sky. Making the connection between the big ideas in math and the multiple ways to represent them may have potential. I am anxious to try this out.
    R. Stock

  3. Would that it were true, mell. More accurately, they show the need for engineering (software and electrical, mostly). As a mathematician banished to the world of engineers, I'd be hard-pressed to find many around me who know much beyond calculus and some basic statistics, and even those at more of a functional than a conceptual level.

    I hate to speak the uncomfortable truth, but deep understanding and "big ideas" in mathematics are practically useless in today's economy beyond the extent to which they smooth the acquisition of practical skills. Not that I think they shouldn't be taught, but you can't make that case on the basis of cell phones and internet access.

  4. Even though I have limited experience ( one year) as a teacher; thirty years in engineering,  some of your comments were very thought provoking.   Lack of adaptability to changes in teaching styles.  The teaching of math is very adaptable to the use of computer technology as a learning tool.  Discrete facts that needed to be memorized. Not mathematics. Its all about analytical thinking and problem solving.  It makes sense. Great comment.  Does adding the negative cancel out or make zeros. Its all about problem solving. Seems semantic.  I really like the peer pressure idea of the HW chart and will definitely use it in my class. YET. Another very profound way of showing students that learning is a process, which involves a little frustration and a little struggle. That ‘s where the development takes place. And fast pace curriculum coverage where students of different abilities move at an accelerated pace, versus a slower pace with real understanding