MathSage - About Mr. K - Contact - BlogRoll -
Skip to content

Questions: The pipe dream

There's this trope, mostly appearing in martial arts fiction, of a learning process where tasks that appear menial and pointless are actual lessons that impose the skills or knowledge needed without the student being aware that they're learning anything.

My favorite of these is in The Matador series by Steve Perry1. In it, neophytes try to follow an Arthur Murray style sequence of footsteps. Each successive step is not only more difficult than the previous, but it build and expands on the skills needed in all of the previous steps.

In my dreams, this is how I could teach math: A sequence of problems, carefully constructed, such that each problem provides a small measure of growth from the previous problem - so close that no instruction or direction is needed, but enough of an advancement that the student needs to make a cognitive leap to get there.

I realize this is a pipe dream, for several reasons. Students learn at different rates and with different styles, some need a lot more reinforcement of concepts than others do - there is no one ideal path.

Furthermore, a lot of what we teach in math is not the enlightenment of the concepts, but conveniences of notation. There is no real intuitive way to grasp those - we as teachers need to impart those conveniences onto their existing understanding.

Nevertheless, I believe there is great power in providing just the right problem to a student. Rather than having to convince and cajole a student into knowledge, a well designed question can shake up their assumptions, and provide the spark of inquisitiveness that so many teachers wish their students had.

Key to developing these questions is an understanding of how students develop theories and understandings, and how they can develop incomplete or flawed theories that nevertheless function adequately for a significant subset of the problems they face2.

I'm hoping to keep this as an ongoing subthread in this blog, and I hope that it may actually someday develop into a fleshed out philosophy that is concrete enough to share and defend.

1 I think I just outed how horrible my taste in recreational literature is. I have no apologies - crappy SciFI is a staple of the geek world. (back)

2 cf Polya. (back)

2010 01 27 Comments (1)

Questions 1

My students sit at tables instead of individual desks. To reduce the incidence of cheating, I create two versions for each test. I've learned in the process that similar looking problems actually can vary quite a bit in their effectiveness.

Here's my first example:

-4 - 9 =

-9 - 4 =

Similar problems, same answer, yet about 20-30% of my students will answer one correctly, and the other incorrectly.

I'll leave the which is which and why as exercises to the reader.

2010 01 26 Comments (6)

I’m Helping!

Had a discussion with another math teacher here the other day. We were both in favor of increased observations in our classrooms, both by administrators, and by other teachers. We both think that having more feedback would allow us to focus our efforts at growth.

Where we differed was in how we'd like to have observers behave. I was in favor of straight observation - primarily empirical evidence, along with possible suggestions of roots or paths to follow.

The other teacher was in favor of spontaneous coteaching. I've certainly been in situations (especially early on) where having another teacher jump in provided strong learning opportunities, and It's not that I don't think I can't learn anything from anyone else.

It's just that my teaching style has drifted towards being less helpful. As long as a kid is working something through, I don't want to give them the answer. I want them to figure it out.

I've spent the past 5 years learning to be really patient - I can sit for minutes with an attentive observing look on my face, without giving away anything about whether what they're doing is right or wrong. I've also gotten pretty good at jumping in with a question whenever progress seems to slow down, anything from "What do you need to do next?" to "Why did you do these last three steps?" All of which serve to focus, without actually giving anything away.

I've had an administrator come into a room and try to help out the kids in this situation. It seemed to me that it was almost more a display of the administrator being able to solve a problem than it was of them being able to teach it. Sure, the answer got on the paper faster, but they'd done just about everything except tear the pencil out of the students hands and write the answer down themselves.

I had to let them be proud of their assistance, but then I had to work up a new problem for the kids to do, and let them go through exactly the same struggles they would have before.

2010 01 25 Comments (0)

Side effects

I teach two remedial prealgebra classes.

One of the biggest hangups for these kids is that they never learned their basic multiplication tables. It's hard to develop any sort of number sense if you can't even make a quick guess at how many Xs are here:

X X X X X X X
X X X X X X X
X X X X X X X
X X X X X X X

So, to help them out, I let them use a multiplication table on their quizzes. But, to help make it a learning experience, they have to make their own.

And not just once, but every time. It's become a ritual - before I pass out the test, I have them clear their desks, give them the blanks, and give them a bit of time1 to fill them out and then give them the tests.

I'll eventually have to teach them to be able to do this on a piece of blank paper, so that they can repeat the process with the scratch paper they get for their standardized tests in a couple of months.

There is a surprising fringe benefit to to all of this, though. These kids are the low kids. A large part of their lack of academic success is due to their inability to focus or settle down. It's always been a problem that the minutes before a test are a struggle in setting a quiet tone. This alleviates all of that - I don't have to ask them to be quiet, they do it automatically as they fill out their tables, and then by the time I'm ready to pass out the test they're already focused and ready to go.

1 The amount of time varies across the year, and is based on a bit of teacher subterfuge. I tell them I'm going to start the test when 80% of them are done with the table. I actually start passing it out when 80% are about 80% done. This adds just a little bit of pressure, but has resulted in their times to fill this out dropping from about 5 minutes at the beginning of the year to about 2 minutes by the end of the first semester. (back)

2010 01 22 Comments (5)

Patience

There's a different lesson here than I expected.

This Tuesday, I had one of my former students return to my classroom. He had gone off to a charter school, and lasted less than 3 months. The post was going to be a rant about charter schools being able to pick and choose their students, compared to the public schools.

This kid had been a pain in the butt. He was the only outwardly defiant student I had at the beginning of the year. It didn't surprise me at all that he'd been booted.

So, this story could have had an ugly ending. If I'd acted on my instincts, we would have been right back where we left off.

But we're not. I've got enough good teacher habits now, and enough experience dealing with incoming OTs, that I almost automatically played the second chance card with him. And it paid off. He's not the most diligent student ever, but he's going through the motions. He's cooperative, and quiet. It's only been 4 days, so it could still all change, but for me this was a big lesson on sticking to the game plan, and not going with your gut.

2010 01 22 Comments (1)

KenKen

I like puzzles. I used to do sudoku puzzles. I tried bringing them into the classroom, even though it wasn't math, because I figured it would be better than word searches or mazes for those times when you need to give some of the kids a filler activity. It didn't work all to well - my low performing kids aren't interested in that level of pattern analysis.

I've since started doing KenKen, and my sudoku habit has dropped by the wayside. I don't use it for the kids. If I had to have an education related excuse, it's that it keeps my mind sharp with all of the skills I need to be able to develop a complex algebra problem with a appropriate level solution for one particular kid's skill level.

But mostly it's just a fun way to spend a couple of minutes each day.

The two sites I use are The NY Times site and the Official KenKen site. The rules share some minor similarities with those of sudoku, but involve doing actual arithmetic, rather than just pattern matching.

2010 01 12 Comments (0)

The Eyeballing Game

I'm not teaching geometry this year, but one of the biggest hurdles the kids had to overcome was that they're expected to have some facility with recognizing midpoints, angles and so on, but that they're not allowed to assume that what they see actually is that way unless it's labelled as such.

Matthias Wendel has made an eyeballing game where he gives you geometric constructions that are a bit off, and you try to fix them. I do a little bit better than average, but two things become clear very quickly:

1) You can improve your guessing ability through practice.

2) No matter how good you get, you still won't be able to tell perfectly every time.

This solves a duel purpose - it explains, viscerally, that what you think you see is not always actually what you think it is, and secondarily it shows why, even if you don't get it perfect, you need to be able to show congruency or angle measure in order to make a mathematical argument.

An added bonus in the process is that you get a good understanding of some basic geometry principals. My guess is that's it's probably worth about a half hour of computer lab time, with perhaps some reflection thrown in to round out the period.

2010 01 10 Comments (1)

The 20% problem

I want to respond to a comment left on one of my other posts:

I want the kids to learn and retain it for more than the test. I am a first year teacher, and my biggest struggle is getting the kids to do homework or even study for a test. Without a major makeup work date at the end of the grading period they would all fail.

I am in a classroom where the kids managed to get rid of 2 teachers last year. It has been a struggle and I am a teacher of old school. Do your homework, study for tests and above all use pencil. Take responsibility for your own actions. I some how managed to be asked to teach 4 different preps (every prep at my school but geometry). What was I thinking?

First, I want to acknowledge the frustration. Four preps is a lot of work. Kids who have learned that school is a "get rid of the teacher" game is work. A lot of experienced teachers would be frustrated by that.

But.

But.

The first thing that stood out to me was "I am a teacher of old school."

My observation is that just about every person who wants to get into teaching math probably succeeded at the old school. It's only natural that you want to do what worked for you when you went to school.

But if you start asking people, you'll start hearing the same thing over and over again:

"I was good at math, until I got to algebra."

I've heard that a lot. I've heard it from corporate VPs. I've heard it from principals. Counselors, history & english teachers. All sorts of people, most of them successful, and many of them never understood algebra. My completely pulled out of my ass guess is that only about 20% of us are suited to learning algebra the traditional way.

You can insist on doing things the old way, but you don't also get to insist on being successful for more than 20% of your students1. If you think that's okay, I hope you grade that way too.

Math is hard. Teaching math is even harder. We're in a world where only 20% ever got it, and the other 80% think nothing of pointing that out to your students. It's impossible to get kids interested in a traditional curriculum when your principal is willing to say to them that they never understood math either.

So, how do we fix this2?

First, to paraphrase Ms Vandergriff (my first department chair): You are responsible for what happens in your room. It's not the kids, it's you. That taking responsibility thing mentioned up there? Successful teachers do that. They realize that kids learn, or don't learn, because of what they do in a classroom3.

Second, suspect everything that you do. If it doesn't work, throw it out. I love the idea of homework. But, in practice, the only kids who did it were the ones who didn't need to because they understood it already. So, until I can figure out a way to make it effective in my teaching, I don't assign it. I used to insist on pencils - now I just insist that it's legible. I've seen a very effective teacher insist that all math be done in pen - that way the students see a record of their mistakes and the corrections, and learn it's a process rather than just getting the answer.

If explaining stuff to them doesn't work, stop explaining, and start finding better questions to ask. If they're learning from the make up work you assign, stop doing what you were doing, and teach the way you do with the make up work. If they're not learning from the make up work, then don't give them credit for it.

Finally, change one thing at a time. Try new things, but if you change too much too fast, you won't be able to tell which effects are the results of what causes. Worse, your kids won't have any idea of what to expect, and will act out even worse than they do already.

This isn't a recipe for success. It's a lot of hard work, and you'll have a lot of failures along the way. Eventually, you'll reach 30% of the students, then 40%. And at some point, you'll look back, and realize that doing it old school isn't nearly as important as figuring out what works.

1 That 20% drops to 0% if you're teaching a remedial class. (back)

2 Short of changing the way schools work in this country, which I am not going to even try to do. (back)

3 Not entirely true. There are a whole class of kids who would do just fine if you throw them into a classroom with a textbook. These students are called honors students. These students are probably also the 20%. Perversely enough, a lot of the teachers who teach them get a lot of credit for not doing anything very special. (back)

2010 01 08 Comments (3)

Where in the hell have I been?

So, I suck at New Year's resolutions.

We'll see how this one (to resume regular blogging) sticks.

I'm at a new school. I mentioned that. I'm loving my new school. The teachers are collegial, the kids are kids, and my administration takes the job of creating a good learning environment for kids seriously. It's been easy for me to just go into the classroom, and teach, and enjoy the heck out of the job.

I'm also not doing a lot of stuff that is, to me, new. And it's hard to write about the same old same old.

But I got called out by Kate. And I still see stuff on a regular basis (like that price of liquids graph that I totally would have beat Dan out on if I'd only been posting) that is totally shareable.

What really got me back, though, was a poignant comment on one of my own threads. There's a need there - people want to learn. They just need good teaching. There's a bit of hubris in assuming that I can fill that need, but at this point I'm willing to consider that a professional danger.

2010 01 07 Comments (1)

Little People

The last time I effectively taught adding & subtracting with negative numbers I was on to a good idea, but there were a couple of faults - only the kids actually walking the number line were engaged, and they lost their positional cues as soon as they went back to their desk.

So this time, every kid gets a go:

Get a bucket of Lakeshore kid counters, and print out a bunch of number lines1. Have a talk about the difference between positive/negative and adding/subtracting. (i.e one thing is a property of a number, the other is something you do between two numbers). Once they kind of get it (it doesn't need to be perfect, they just kind of need to know that there is a difference), go through a couple of example expressions just having them identify whether the expression involves addition or subtraction, and whether each term is positive or negative. Once they get pretty good, have them them fill in the hint blanks with walking forward/backward2 for positive/negative, and facing right/left for adding/subtracting.

Now comes the easy part. They find the first number on the number line, decide which way their kid should face, and whether he should go forwards or backwards. By carefully choosing mirroring problems, they can see both the difference and the similarity between swapping both operation and sign. Eventually sneak in problems that drift off of the number line, and let them start proposing rules for how to solve those problems.

1 Numberline pdf (back)

2 A couple of kids in my third period were fixated on forward=right, backward=left, regardless of which way their kid was facing. With them I had better luck using walk/moonwalk. Feel free to demonstrate as your ability allows. (back)

2009 09 17 Comments (5)