Back to School

It is about time for most active teachers to return to their classroom. While I am retired, the beginning of the school year still holds a special place in my routine. My schedule will change soon as tutoring requests pick up, workshops and math teacher meetings increase in regularity, and I start to get more questions about this and that.

Still, the first day is the most important day of the year, and I miss it. I always worked hard to have an interesting lesson that would surprise my students, excite them about the possibilities for the coming year, and give me a chance to meet them. I did not go over rules nor did I have non math getting to know you activities. We did math. I wanted to set the stage for them. It worked.

I made sure the first lesson and every one thereafter was based on what I consider to be my most important in sight about how mathematics is learned and what I think teachers should be doing in their classroom.

We must cover content. We must teach students how to logically approach problems that they have not seen before. If we can also get students interested in math, then we are successful.

It took about ten years for me to figure this out, but the these tasks end up being connected and the outcome often leads to excitement and interest. The idea is to have students solve problems that lead to an understanding of the concepts and content we want them to master. Here is how I was able to manage this.

I gave students a problem to work on. It was one problem and everyone was expected to work on it. Often it was related to the work they did yesterday and flowed into the concepts I wanted them to learn today. It was one problem, one they could all get started on but perhaps could not solve, hence the phrase, work on instead of solve.

As they worked on it, I walked around and listened and watched. I did not help anyone but I paid careful attention to their progress. I tried to give them a problem they has not seen before. They understood that their task was to figure out what to do.

I encouraged them to work alone at first and then discuss their thoughts with neighbors. Sometimes they did arrive at a solution, sometimes they did discover an important idea, and those were exciting moments for all of us. I knew exactly which students had figured it out because I was watching and listening while they did.

Sometimes they did not see the breakthrough idea and could not solve the problem. When I observed that they were about to go off task, I might give a hint, or I might have a student who was on they way suggest a strategy, or I might even demonstrate a procedure for solving the problem. However it worked, I had their attention because they had worked on the problem and wanted to know what they might have done to solve it. contrast this with the traditional method of showing students how to do a problem, when they haven't worked on it. They probably don't even recognize the key idea because they never had a vested interest in solving the problem in the first place. When I start to explain, they want me to explain because they have exhausted their resources.

I believe one of the worst habits teachers have acquired is the belief that they have to explain something before a students can try to solve it. Another bad habit is to give students a worksheet with several problems on it. This encourages students to write down answers rather than ponder the problem at hand. They lose focus and really don't want to discuss the problem as much as they want to finish the worksheet.

It takes a lot of work to get students used to this style. they would complain that I wasn't explaining things. They were frustrated. They were confused about what they were expected to do. But I am stubborn and persistent and eventually there would come a day when I would tell them it was time to discuss the problem, and some of them would say, "No, wait. We almost have it." Then I knew that I had converted them from passive note takers into active problem solvers. I would just stand there and smile and tell them I guessed they could have a few more minutes.

One good problem, let them work on it, discuss it, consider alternate solutions and common errors, then move on to the next problem.

If this is an advanced class, there might only be two or three good problems in a given class period. In a more ordinary class, there might be fifteen in a class period, each a challenge for the students, each within reach.

Not only did it work, but teaching this way is great fun, because there is another unexpected part. Sometimes students think of an approach that I had not considered and I learn to become a better problem solver from them.

Have a wonderful journey with your students.

The Talent Equation

In the last few months, I've found myself getting into arguments with people about talent; in these arguments, I hear myself saying things like "I don't believe in talent."  Less tendentiously, I should say that I'm skeptical about talent, but even that isn't quite right.  So here's what I've come to believe.

First, a basic question: what is talent?  When someone gets an A on a test with little or no effort, we say either "the test was easy" or "she's really talented."  When someone puts years of effort into lab work -- or proving a theorem -- and comes out the other side with an amazingly mind-bending result, we say "Sure, she worked hard--but she was also brilliant to have seen ... "  In both cases, talent is what bridges the gap between the effort somone has put in and the results that come out.  We can express that thought in something like an equation:

Talent + Hard Work = Great Achievements.

As with any equation, we're tempted to apply algebra--in this case, subtracting the hard work from both sides.  

Talent = Great Achievements - Hard Work.

Note what this equation doesn't say: that great things can be achieved without hard work, if only you have talent.  Rather, the equation reminds us that talent is explaining something:  why Jane can work in the lab for ten years and come up with only incremental results, while Jeanine works in a similar lab for ten years and revolutionizes her area of research.  In the classroom, when we say "Jimmy is more talented than Jeff" what we usually mean is something like "Jeff studies for a test and Jimmy doesn't, but they get the same grade."

Notice that the word "talent" (or phrase "more talented than") doesn't necessarily refer to any particular characteristic or set of characteristics, even in a single field such as mathematics.  Some kids have good number sense; some are good at seeing patterns; some are good at recalling any idea or problem they've seen once; some can gnaw on a problem for hours or days without getting overwhelmed; some have great metacognition about what they need to improve and how to improve it.  All of these are talents, and for any combination, I can think of a student I've taught.  I think teachers in particular toss this idea (as well as "smart" or "bright") around without even thinking carefully about which talents or strengths they are trying to refer to.  In my view, simply calling someone talented (or "smart") is intellectually lazy.

It's also useless, to the student--because it doesn't reinforce the actual behaviors that need reinforcing, or point to other areas in which he or she could improve--and to other students--who are led to believe that student X has some "special thing" that nobody else has--and to the teacher, whose job is to strengthen the student, not just praise him or her for what he or she already does well (right?).

Talking about "talent" or "smarts" is actually counterproductive because it reinforces the false notion that great achievements are primarily the result of talent, when in fact virtually every great achievement is the result of years, sometimes decades, of sustained hard work.  Genius/smarts/talent are only invoked to explain why Einstein (or Marie Curie, or Andrew Wiles, or...) achieved the great results they did when other people who worked just as hard got nowhere.  But because we can't point to one or even a few specific things that talent consists in, talk about "talent" doesn't help us identify people who will be successful ahead of time.  And in fact I think that we as math teachers are pretty bad, demonstrably bad, at identifying those people ahead of time, especially when those people are girls, or slightly-rambunctious boys, or nonwhite and nonasian.  (See Lee Stiff's article here.)

So when I say that I'm skeptical about--or I don't believe in--talent, what I'm saying is that talent-talk misses the point rather badly.  The real point is hard work, which for some people will result in great achievements; because hard work doesn't do that for everyone, we explain the disparity as a combination of luck, timing, and talent.  Any task worthy of the name requires concentration, effort, and persistence. Talent isn't the point; hard work is.  Going back to the second equation for a minute, we can rephrase the algebra as an analect: talent is the capacity to achieve great things through hard work.

Now that's something I'd hope we'd try to teach everyone.

Hard work

I have no doubt that productive hard work reaps benefits. Malcolm Gladwell, in his book Outliers, claims ten thousand hours of work are necessary in order to become an expert. His examples range from The Beatles to Bill Gates and even talks about child prodigies like Mozart. I am convinced.

Many of those who put in extra time did so because they enjoyed the work. A friend of mine who is an excellent piano player has told me that when he was a child, he couldn't wait until he got home so he could play the piano for several hours. Over the years, I have noticed that many outstanding math students get very excited when telling me about one of their mathematical discoveries. They didn't do the work that led them to these discoveries because they thought it would make them better at doing the work as much as they did that work because they enjoyed doing the work. I see two consequences to this observation.

The first one is that teachers need to put more emphasis on interesting, challenging, intriguing problems; if they do, the mandated test scores will take care of themselves. Drills and practice just are not going to motivate more practice. They will drive children away from our subject and convince students that they are not good at math, and that they don't even want to be good at it because it is boring. (I just googled "boring math class" and got 46,000 hits.)

The second consequence is related to recent work by Keith Devlin about the potential impact of elctronic games on middle school education. I heard him speak at NCTM and read his book, Mathematics Education for a New Era: Video Games as a Medium for Learning. Among other things, he points out that people willingly play video games for hours on end, and that when they do, they get immediate feedback. They also expect to fail at what they are doing for quite awhile.

I can recall several open ended questions and puzzles I posed to my students that kept them thinking for years. Several students inquired when they were seniors if I would explain to the what was wrong with the "proof" I showed them three years before that all triangles are isosceles. Likewise there are several puzzle-like questions I have asked that students sometimes solved many months after the question was first posed. This sort of persistence only happens when students are surprised and puzzled by something they care about.

A teacher's main job is to pull students in to the subject being taught, to ask them interesting questions, and to let them learn. That means the teacher's main job is to provide students with interesting tasks that students can relate to. Some of these tasks should be problems that relate to their world, and not just school. Some of these tasks should be puzzles and open-ended questions that might take weeks to solve. Perhaps just posing a situation with a photograph or a short video and asking them to ask questions about it will engage them. I am particularly taken with the idea that we need to let students ask some of the questions as part of our regular routine, instead of only requiring them to answer the questions we ask.

Two weeks ago I had the pleasure of being at a talk by Dan Meyer. He raised some of these same questions about getting students involved. In particular, he pointed out that we often break a problem down into so many little parts (scaffolding I think it is called) that the student isn't sure what the real point of the question is and never has to figure out how to solve it. Part of being interested is to figure out what needs to be done. We are so afraid that students will fail that we drag them through every step of the problem in hopes of keeping them from being frustrated. Being frustrated is part of the fun and a valuable part of the experience.

If students work hard because they want a good grade, or because they want to please a parent or teacher, or because they have bought into the belief that hard work will get them a good job, they will stop working as soon as they have made progress toward their goal. If students work hard because they find the work challenging and stimulating, they will keep working no matter what. I think the latter motive for doing work is worth cultivating.