Making math concrete

Steven Strogatz, an applied math prof at Cornell, has started a series on the NYT site to re-teach basic math from an adult perspective, with the goal to show “why it’s so enthralling.”

I love the second installment, which explores the way that using rocks in place of digits (i.e., six rocks rather than the number “6”) can provide unexpected insight on seemingly tough problems:

For example, instead of adding just two odd numbers together, suppose we add all the consecutive odd numbers, starting from 1:

1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

The sums above, remarkably, always turn out to be perfect squares. (We saw 4 and 9 in the square patterns discussed earlier, and 16 = 4 × 4, and 25 = 5 × 5.) A quick check shows that this rule keeps working for larger and larger odd numbers; it apparently holds all the way out to infinity. But what possible connection could there be between odd numbers, with their ungainly appendages, and the classically symmetrical numbers that form squares? By arranging our rocks in the right way, we can make this surprising link seem obvious — the hallmark of an elegant proof.

The key is to recognize that odd numbers can make L-shapes, with their protuberances cast off into the corner. And when you stack successive L-shapes together, you get a square!

What makes a great teacher?

If you’re interested in teaching, this article by Amanda Ripley in The Atlantic is a must-read. For years, people have speculated about what makes a great teacher.

But now there is data. It has been gathered painstakingly by Teach For America for over a decade, and it covers hundreds of thousands of kids. TFA linked the test scores of students to their teachers, so that TFA can spot patterns in the data: Which teachers are causing big boosts in the kids’ scores–for instance, advancing them by three grade levels in one year? And what traits do those star teachers have in common?

Read the article for the full scoop. But here’s one bit that fascinated me. The data suggested a “profile” of a great teacher, and so naturally, in hiring new teachers, TFA wants to match that profile. So they’ve begun to build an “outcomes-based hiring” system, and in doing so, there were some surprises:

Once a model for outcomes-based hiring was built, it started churning out some humbling results. “I came into this with a bunch of theories,” says Monique Ayotte-Hoeltzel, who was then head of admissions. “I was proven wrong at least as many times as I was validated.”

Based on her own experience teaching in the Mississippi Delta, Ayotte-Hoeltzel was convinced, for example, that teachers with earlier experience working in poor neighborhoods were more effective. Wrong. An analysis of the data found no correlation.

For years, Teach for America also selected for something called “constant learning.” As Farr and others had noticed, great teachers tended to reflect on their performance and adapt accordingly. So people who tend to be self-aware might be a good bet. “It’s a perfectly reasonable hypothesis,” Ayotte-Hoeltzel says.

But in 2003, the admissions staff looked at the data and discovered that reflectiveness did not seem to matter either. Or more accurately, trying to predict reflectiveness in the hiring process did not work.

What did predict success, interestingly, was a history of perseverance—not just an attitude, but a track record. In the interview process, Teach for America now asks applicants to talk about overcoming challenges in their lives—and ranks their perseverance based on their answers.

Steven Farr, the guy who is leading this research at TFA, has a new book out — Teaching as Leadership — that suggests ways teachers can improve their effectiveness. I just ordered it from Amazon — can’t wait to check it out.

Google and memorization

Here’s Google CEO Schmidt, in response to a question about whether Google is “dumbing down” kids:

Kids use [Google] all the time because it’s a new way of learning. When I was growing up, in Virginia, they made me memorize the names of all the capitals of every county in the state. Completely useless information. So kids today are going from knowing everything to being able to search very quickly. The kids need to learn how to search because they’re going to have to search everywhere. They’re going to have search everywhere on devices that they carry with them.

Schmidt is right about the cult of memorization in schools. Most teachers do a great job of making lessons come alive, but when it comes time to measure what the students have learned, out come the standardized tests (thanks to state and federal requirements). And what’s easy to test in a multiple-choice format? Memorized information. So a student’s understanding of the Confederacy’s war strategy is funneled through questions like, “In what year did the battle of Gettysburg take place?”

There are (at least) two problems with that. First, if you’re the teacher, and you’re running low on class time, what are you going to teach — what’s on the test (i.e., factoids) or the “big-picture” stuff? Obviously you’ll teach to the test, because you’ve had it drilled into your head that the test scores are THE representation of your students’ learning in your class. If their test scores aren’t solid, you’re a bad teacher. So, yes, given a tradeoff, you and I both would teach the Gettysburg date.

Second, facts fade. Very very quickly. I propose a test to lots of school administrators who are in love with recall-type tests as an index of progress. Two weeks after your kids make solid scores, give them a surprise re-test. Same questions, same answers. And get ready to weep. All of us know and experience that our memories fade quickly — just see the forgetting curve literature — and yet we’ve designed assessments that seem to presume memories are permanent, like files stashed on a hard drive.

Google is the perfect real-world memory aid for students. It makes it easy to retrieve the factoids that will inevitably be lost from memory. It makes it so easy, in fact, that it’s foolish to obsess about teaching the factoids. If a student knows that there was a battle, during the Civil War, that represented a turning point, and she can articulate why, and she can discuss the factors that led up to the pivotal battle itself, isn’t that a picture of success? Would anyone think she’s less smart, or less aware of history, if she Googled the dates and the place names?

The Partition of Africa

Here’s a story we received from Peri Chinoda, an AP & Honors World History Teacher at Hume Fogg Magnet High School in Nashville:

Background Information: In 1885 the Chancellor of Germany, Otto Bismarck, convened a meeting attended by 13 European colonial powers including the USA and the Ottoman Empire. The Africans whose land was to be divided among the Europeans were not invited. The Europeans agreed on a number of things to ease colonization and prevent fighting among them.

Activity:Two or three students were asked to volunteer to bake cakes in the shape of the African continent. On the day of the lesson, I convened the conference with all other students representing European colonial powers and those who baked the cakes representing the Africans.

The “Europeans” gathered around arranged conference tables. The Africans were either sent outside the room or made to sit at the corner of the room.

The Europeans cut the cakes and divided the pieces among one another. The “Africans” who baked the cakes were not allowed to eat the cakes. They just watched other students eat the products/fruits of their labor.

After this activity: (a) The Europeans/students were to write down how they felt about eating the cakes while the people who baked the cakes were watching. The Africans/students were also to write how they felt when other students ate their cakes and they did not. (b) They were to read about the colonization of Africa, and write an essay reflecting on European Colonization of Africa.

Students talked about this experience for a long time. The lesson stuck on their minds for a long time. Even parents called the school expressing their appreciation for the experience their children went through to understand the process, effects and the moral issues involved in colonization.

Tammy Is a Quitter

Here’s a story from Dave Rendall, who has a blog called the Freak Factor. (I love his post that argues that if you’re getting rejected, you’re doing something right.)

I hadn’t seen Tammy in almost a year, when she approached me in the hallway. I was there to teach an evening class for non-traditional students. She told me that her cohort was about to complete their last class and invited me to join the celebration. When I arrived, she was anxious to share some news.

Tammy had taken my course in Organizational Behavior the previous fall and one of the topics is change management. The classic model for this concept is Kurt Lewin’s force-field analysis, which helps managers to envision the driving forces pushing change and the restraining forces acting against change. By understanding and manipulating these forces, effective changes can be achieved. However, this is a relatively abstract concept and can become very complex in the context of an organization.

In order to make this concept stick, I use a few of the SUCCES principles from MTS. I start by keeping it simple. Instead of applying this model to an organization, I start by asking students to choose a meaningful change that they’ve been wanting to make in their own life, but haven’t started yet. Selecting a change that matters to them also creates an emotional link to the activity.

They write this change in the middle of a piece of paper and then I ask them why they want to make this change. These reasons are the driving forces and are listed on the left side of the paper on arrows pointing to the right. We then consider the barriers to making the change. These restraining forces are listed on the right side of the paper on arrows pointing left.

To make the activity more concrete, I try to physically illustrate the action of the two forces. I stand in front of the class with a chair and ask for a volunteer. The chair signifies the change, I am the driving forces and the volunteer is the restraining forces. I push the chair and the other student pushes back. The chair doesn’t move. It is “frozen.”

This illustrates the importance of “unfreezing,” which is the first part of Lewin’s change model. “Changing” occurs when driving forces are strengthened or added and when restraining forces are weakened or removed. Before explaining this, I ask the students how I can get the chair to move. The suggestions usually include the four options listed above. Without even reading the text, students can figure out how the process works in the physical world. In fact, during one class a student’s 10 year-old son was in the room. When I asked how to move the chair, the room was silent. The first person to respond was the young boy. He said, “Add more force!” He was exactly right and I was very pleased. I had made Lewin concrete and simple enough for a child to understand.

The activity is also credible because it offers a testable credential. Students are asked to assign numerical values to the strength of their driving and restraining forces. The cumulative scores for each set of forces shows why they haven’t made the change yet (not enough driving force and/or too much restraining force). We then work to increase driving forces and decrease restraining forces. The students always come up with creative ideas that they can apply to their own life. We discuss these as a class so students can see for themselves how it can work in their situation and those of their classmates.

I also share stories of how I’ve used this model to create change in my life. Each time I teach this concept, I choose a change that I want to make and work through the exercise along with the students. This creates a growing list of stories of success and failure, which brings us back to Tammy.

She wanted to start her own business. During the course of the exercise, she explained her driving and restraining forces. I asked if it was possible that her current job might also be a restraining force. Since she liked her job and was paid well, she did not have a lot of natural motivation to go out on her own. Even though she had a good job, it might actually be a barrier to achieving her change. I don’t recall her response at the time and I didn’t think much about it or hear anything from her until ten months later.

When I went to her classroom, she explained that she quit her job shortly after class and started her own business as a Spanish language interpreter. The business was even more successful than she anticipated and she was very happy. She credited the Lewin exercise for giving her the necessary insight and motivation to make a major change in her life. Needless to say, this is a story that I now share with classes to demonstrate the potential power of applying Lewin’s force-field analysis.