What Happens in a Math Class?

I think there are two dramatically different perceptions about what is supposed to happen when students are doing mathematics, be it in class or when they are doing their homework. I think this difference colors the way mathematics learning and teaching is perceived and causes considerable confusion when students, teachers and parents discuss progress. One point of view is that the goal is to get the correct answer to the problems posed, basically that the problem is a “test” to see if the solver can get the correct answer. The other belief is that the problems have been posed so that the solver will learn something by working on the problem, and that working on the problem will make the solver better at doing math.

I was observing a first grade mathematics lesson last week and students were learning about how to use “near doubles.”  Essentially this was a formal lesson that encouraged students to do problems like 13+14 by thinking that twice 13 is 26 so 13 +14 =27, or one more than 26. Or they could reason that twice 14 was 28 and so the sum would be one less than 28. As I observed, I thought about how wonderful it was that these students were learning about the structure of addition, as well as a handy method for finding sums that many adults use regularly, rather than just memorizing a rule for adding two digit numbers. This has to empower them as they learn more and more mathematics.

I walked around and observed that virtually all students got the correct answers to the addition problem. I also observed that many were doing it backwards, they were writing down the sum and then the doubles fact that would help them get that sum.  As in the example above, the student solves 13+14 = 27 first, and then the student solves it backwards, using 13+13 = 26 (a well-known doubles fact) and then 26+1 = 27.  Some had the correct sum but a doubles fact that did not make sense to me, or was not a doubles fact but some other fact of addition. (For example, they might have 10+10=20 as their doubles fact. While this can be used to determine the sum and uses doubles, it shows me that the student has not grasped the concept of near doubles)

My take away from this observation is that often students are not focused on anything other than getting the right answer, one way or another, while the point of the lesson was to make the computation easier as well as more meaningful by learning about the structure of addition instead of just memorizing a rule. In fact, using doubles makes a boring addition problem into an interesting challenge.  It then struck me that there is a dichotomy in the world of mathematics education that has serious consequences throughout the learning experience of an individual. I would like to think that the learner was always focused on what the point of the lesson was, on what the creator of the lesson wanted the user of the lesson to learn and understand at the end of the lesson. I suspect that the learner is often not even considering that but is focused on answer getting as opposed to finding a really great way to get to that answer. I think many parents also have that same belief, even without realizing it. If their child gets right answers on tests, they are happy. And what could be wrong with that?

For one thing, the student who only has one way to approach many problems may find it boring, and may not realize that there are techniques needed later that benefit greatly from thinking about a concept in many different ways. An example in later mathematics that comes to mind is the many different ways to create a graph of a line, or solve a system of equations. The student who has mastered only one way is severely handicapped if the situation does not fit comfortably into the solution method they have memorized.  For another, the student who learns one way to do it may find math to be dull and uninteresting, while the student who explores many different ways of looking at a problem will, I believe, find mathematics as an outlet for creativity and inventive thought. I believe that students who enjoy what they are learning, understand that they have a certain amount of control over how they proceed, and know that there is a utility to what they learn, will work much harder and learn more than those who just do it on order to get the right answer and therefore  a good grade.

This explains something that has troubled me for some time. I frequently hear from parents that their child is not challenged by the curriculum that is being offered in their school. When I have looked at the curriculum I find that it is often very rich and full of many challenges. Perhaps the reason for this disconnect is that when the student looks at  the material, the only thing the student is thinking about is how can I get this done and get a right answer, while I am looking at how many different things a student can learn from the variety of approaches taken.

The problem is, that if I am right about this, it does not give me an immediate plan of action to correct it and we educators believe that it is our job to correct things that are not working. Perhaps that is food for another blog, but at the least I am interested in you comments about my thoughts.

Thoughts on Formative Assessment

It has been many months since I have written something here. I am no longer in the classroom observing and noticing and so am not sure I have things to contribute I haven't already written.

Last week I observed some classes and I noticed something, something I think is worth sharing.
This was a second grade class in a very good school, with a good teacher. The students were capable and attentive. They were learning addition strategies. The context was fascinating to me.

Each student had a page with photographs of fish. They were excellent photographs and the page was welcoming to a reader. Next to each fish was an identifying letter, from A to K, the length of the fish in inches and the weight of the fish in pounds.

The teacher posed the following question:"If fish E ate fish B, how much would fish E weigh?"  Students thought for a moment and then the class decided that the appropriate strategy was to add the two weights. The teacher was diligent about units, so when a student said add 4, she admonished the student to say 4 what, until the student said 4 pounds. I really liked this aspect of the activity. It emphasized the importance of units as well as reminding students they were talking about weight.

They then did another problem just like it. This was followed by the following question, " If fish F ate a fish and then weighed 64 lbs. what fish did fish F eat?" I was pleased with the direction of the discussion, and especially impressed when many of the students quickly answered the question correctly.

The students then were asked to do several more similar problems, on their own and in groups.When they were done, they were asked to create their own question. I then walked around the room and noticed that the students were able to answer the questions, show their work, and create a new question.  I decided to interject something a little different, so I asked one table  the following question: " If fish K ate fish B, how long would fish K be?"  I expected laughter. They did not laugh. Some of them added the weights. I asked about units and what question I had asked, and they agreed that the answer would be the sum of the lengths of the fish. I went to another table and had the same experience. Teacher then got the attention of the class, and I asked the question to everyone. No one got the correct answer to my question. I asked them if they had ever eaten Twizzlers? Many had. After some discussion we agreed that a Twizzler was about a foot long. I asked them if they then got a foot taller. A few said yes, most said no, but no one wanted to change their answer to the fish story.

I will let you draw your conclusions from this experience, but I reiterate, these were able students who were being well taught. I conclude that this would happen in most first grade classrooms. I am not sure why but I find it a bit frightening and am not sure what we can do about it.I m curious to read comments from others. Please help me to understand.

Testing, Testing

As my ITRW friends know, I'm in my second "semester" of a graduate program, which means that I'm buying textbooks, frantically getting readings done ... and taking tests.  Last week was a doozy:  I had two weeks, and a maximum of two weeks, to pass the statewide online test certifying me as a "teacher-evaluator".  What did I learn from this process?

1. For starters, as high-stakes as this test is -- every principal, assistant principal, suburban department chair, etc. has to pass it OR LOSE THEIR JOB -- we had two tries.  That is, you could take any portion of the test, fail it, and after 24 hours take it again. The cynical among us would say "Of course you get two tries, otherwise there wouldn't be anyone left to turn on the lights in schools." But the reality is this: when it's really important to get person X up to a particular level of competence in skill Y, we almost always give them multiple tries. How many residents are dismissed for missing the vein the first time? And how many times do we give students the opportunity to retake a test at no penalty?
2. Second, we were encouraged to do the training together; in fact, virtually everyone from the network chiefs on down told us to take the tests together. Most of the hardest test consisted of watching videos and arriving at performance ratings (along the eight components of Danielson's Dimensions 2 & 3) for the class, and watching with other people--and talking with them about what we saw--was incredibly valuable. I wound up doing the videos on my own (have you tried to schedule a six-hour testing window during the day, with other people, when you have kids?) but the times I did sample videos with other people I learned much more than studying on my own. How often do we provide structures and encouragement for kids to work together on their tasks--and assessments?
3. Third, we didn't have to study: once you opened the training modules, you could go straight to the assessments. You could also dip into any of the lessons in any order. So when you felt ready, you could go ahead and try the test. (Remember, you get two tries.) How often do we give students this option?

But the most important thing I learned from the experience wasn't about designing assessments, or even structuring instruction. My big takeaway was this: I was terrified. Even with everyone I know assuring me that I'd pass, even knowing that my score beyond pass/nopass was irrelevant, even knowing that my practice tests were all well above the mark, I found the experience of waiting to take the test and then actually getting going practically unbearable. 

Now, we don't tell high school students that they'll lose their jobs if they don't pass Friday's math test. But we do tell them that they need straight A's to get into "top colleges" (and if we don't tell them this, they certainly hear it from everyone else around them), and so an A (or B+) is, for our top students, effectively the passing grade--a much higher threshold than I had to cross. And as adolescents, they're not as good at dealing with stress as most adults (or at least as the "optimal" adult). What I'm getting at is this: I left my office convinced that most of our students--and most of our strong students--find testing incredibly stressful. And while I may have "known" that before, I'm not sure I felt it. I'm not sure what to do about this, but I know that I'll think about testing very differently this year.

So maybe that's the pedagogical payoff for attending graduate school.

The Way, Way Back

As we wind up this year, I've been thinking about last summer's awesome movie The Way, Way Back, and not just because it's about a fourteen-year-old on summer vacation.  The movie starts with a long scene introducing Duncan (the aforementioned fourteen-year-old) as a kid who's just kind of, well, a shlump.  His mom's boyfriend thinks so, and while we realize that this adult's criticism is way too harsh, you have to see his point: Duncan is sullen, introverted, and, apparently, completely nonspecial.

Over the movie--and I won't spoil the whole plot for you--Duncan finds a second home at a local water park, where he takes on a new identity as "Pop 'n Lock", the park's indispensable factotum and MVP.  We see him going back and forth between "Duncan," still-awkward fourteen-year-old, and "Pop 'n Lock", master of every detail and every need in the park.  Although Duncan gains a little confidence at home, his mom (and his mom's boyfriend) don't get to see any of his alter ego until ... well, I won't give it away.  It's good.

The point is that I think we have a lot of students like Duncan, students who are mostly unremarkable, even substantially less than remarkable, at school, but who totally shine in some part of their life outside of school.  This is the kid who is super-responsible and on-the-ball at her job at Starbucks; or who's a realistically-aspiring professional dancer; or who keeps track of two younger siblings and three cousins, does the grocery shopping, and puts food on the table every night while mom is at work. These kids are stars. But what we teachers see is the late homework assignment, the "C" quiz, the half-checked-in, half-checked-out stare at school. We don't see the stars.

More often than not, these kids are not middle class, and not white. One of the privileges many (certainly not all) white, middle-class kids experience is not having these kinds of burdens, or if they have them, they often know that they're as much assets as deficits, and that the student's job is to alert me early about the constraints and plan with me to work around them. They send us emails like "Hey, Mr. K, I wanted to give you the heads up that we have rehearsals every afternoon and evening for the next two weeks, so I might wind up one or two assignments behind. Can you tell me what I should focus on the most if I have to do triage?"

I think other students don't know that we care, and often don't think that we necessarily should care.  Many of my minority students have an overt "no excuses" attitude, which is refreshing except for when there really are extenuating circumstances. Yes, if I'd known your sister was in the hospital all last week, I would have been happy to let you take Friday's quiz on Monday.  But on the positive side, I don't think they know that we want to see them at their best--even if their best isn't what we see at school.

I don't have an easy solution for this problem. I think it helps to tell kids what we look for and want to know about, and if we coach them on the kinds of things that might qualify as reasonable exceptions.  (A long time ago I stopped writing "late assignments will not be accepted" on my assignment sheets, and started writing instead things like "under ordinary circumstances, I won't accept this assignment late," with details that described some truly extraordinary circumstances.)  And I think it helps to ask kids about what's going on in their lives outside of school. But the most helpful thing may be to remind ourselves, when we see a kid who seems like a Duncan, that more often than we might think, there's probably a setting in which that kid is really a Pop'n'Lock.

Cutoffs, and the last week of classes

So now that CPS is in all-out last-week-of-classes mode, I have a few gripes. They all relate to cutoffs.

Remember cutoffs?  When you got a hole in the knees of your jeans, you'd just cut them off into shorts, and they'd look like this:

After a while, though, the ends would start to fray, and they'd look more like this:

So you'd trim up the shorts, and the cycle would begin again.  Eventually they would look like this:

OK, maybe not exactly, but you get my point.

The problem with the end of the year is that we have this urge to stop teaching just one or two days before the end--either because we're tired, or because we feel like the kids are tired, or because grades are already either submitted or essentially impossible to move much, so why bother giving another test, and why bother teaching when you're not going to give a test?  There are a hundred rationalizations, all of them bad.  Because then when there are three or four days left, you say "Oh, well, in a couple of days I wouldn't be teaching anyway, so ... " and the end of the year frays up a little more.  And so it goes, until suddenly you're spending an entire WEEK (or TWO weeks!) of instruction in games, "free time," movies, semi-reasonable documentaries ... anything but bona fide teaching.

What's actually wrong with this?  A few things:

• Not teaching because there's no test (or tests are over) just teaches the kids that the test was the only reason you were teaching in the first place.
• We all wish we had more classroom time.  Well, we do: this week, a solid 250 minutes per class.
• Just because the curriculum for the test is over and done with doesn't mean the subject is over and done with.  What kind of teacher doesn't have a favorite poem, theorem, artist, historical event, ... that just doesn't fit into the regular curriculum?  (In fact, for me, a life-changing moment was when, the last week of school, my Brit Lit teacher read us The Love Song of J. Alfred Prufrock aloud.) There's always more to teach--it's not like we're in danger of running out of poems, problems, or paintings.
• Finally, how is it fair that we make kids go to school and then waste their time?  My kids can play video games, watch movies, read books, doodle, and mess around with their friends just fine on their own--they don't need state-compelled school attendance to do any of these things. If the law is that kids are compelled to be in school this week, then fairness and respect for them as human beings demand that we make it worth their while.

It's not so hard.  This week, my students are finishing up a discussion about "shape space", where we measure the distance between different shapes to talk about fractal limits; learning about Markov chains and why regular Markov chains converge; and giving short presentations about explorations they did with fractional linear transformations, queuing theory, and hyperbolic geometry. They're advanced.  In other classes, I've used the time to read a cool book (like Arcadia--thanks, Mary, for the idea!), or do interesting problems we never saw before, or think about mathematical puzzles.  Or you could just get a head start on one or two interesting ideas that the kids will see the next year.

I know I'm spitting in the wind -- Emma told me yesterday that she'd "figured it out: when teachers don't want to teach, they just go on Netflix, put in a documentary, and voila--instant lesson!" That experience makes me sad.  But if we all just held the line ... or the hem ... 

Gender Games

Two issues of gender and games that have come up in the last weeks ...

1.  My oldest daughter reports some frustration that in one of her classes, the teacher divides them into teams every day to play games (no, this is not PE, so be kind) and that, every day, one of the iterations is boys versus girls.  This way of forming teams seems like no big deal to most people, and to me that's a pretty big deal. It bothers my daughter because most of her friends are boys, but even more so because it reinforces our society's idea that there's a huge inherent difference between boys and girls.  We wouldn't separate kids by short and tall, or straight hair versus curly hair -- so why does boys versus girls seem to "make sense" to so many people?

A radio show I listened to a long time ago made the point that even in addressing classes as "boys and girls" reinforces that preoccupation with gender roles.  Don't believe it?  Imagine replacing "boys and girls" with "white children and black children".  If describing the kids by gender doesn't make a difference--as many people seem to think it doesn't -- then why does it feel so funny to use race?  Only because when we separate boys and girls in this way, we are reminding ourselves that gender differences are crucial in ways that we're less comfortable reminding ourselves that race differences are crucial.  (Although if we were honest and not all "it's the end of racism", we'd have to admit that race plays a huge role in kids' experiences in school.  But that's a different blog post.)

The point is this:  dividing kids up by gender except for activities that require gender separation (bathroom use, for example) is just lazy; it makes some kids uncomfortable, and should make all of us wonder why this difference is the one we keep reinforcing in our schools.  Don't do it!

2.  It's the end of the school year and my kids are handing in portfolios; the assignment includes a short essay on "How I got here", namely, into a hyper-advanced college-level post-calculus class while still in high school.  This year's crop follows a trend I first noticed four years ago.  Virtually every boy says something like "The first time I really loved doing math was when I was competing against [my best friend/my classmates/my archenemy/another school] in [some math competition]."  Whether it's a simple race to finish multiplication tables or a full-blown Mathcounts meet, this experience is clearly one that has turned the boys I teach on to mathematics.  On the other hand, almost none of the girls even mentions competitions; when they do, it's almost always negative.  (One girl--now a math major--once wrote "I decided I hated math when my teacher had us play this stupid game called 'Around the World'.")  And my own daughter loves math but hates competition.  So there's some interesting data.

My friend Cathy O'Neill, aka mathbabe, wrote on this issue a couple of years back, "math contests kinda suck."  As a contest author, I have to agree with the majority of her argument: our society's reliance on math contests for identifying and encouraging mathematical talent discourages a lot of girls, who either don't like contests, or don't like doing math the way that you have to in order to be successful at most math contests.  That's why we need more math circles and other places where girls can get their hands dirty doing awesome mathematics--Chicago's new math research symposium is one such.  There's a place for math contests in the pantheon of cool, challenging math activities--but the pantheon has got to get a lot bigger than that.

Guest Blog: She likes math, but she hates math homework

Emma, 6th grade, likes math (does math circle voluntarily on Saturdays, for example), but hates math homework.  When I asked her why, she sent me the following well-thought-out response--unedited by me (okay, I deleted two commas).  Math teachers, take note!
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As a 6th grader, here are the reasons I hate math homework, and solutions to the problems:

1) Quantity- Homework is supposed to be for us to practice what we've learned. So if you know the material, then it's a waste of time to do 20 similar problems when all you need to prove that you know the material is 4 or 5. And if you don't know, then you'll practice incorrectly, and it really won't help you to learn how to graph inequalities if you're repeating incorrect steps for any number of problems greater than that needed to prove that you know the material.

2) Quality- Having similar problems doesn't help anyone. Sure, if you want to memorize steps, it may be helpful, but in order to understand the math behind the equation and prove why you need to take the steps, you should have a variety of problems. Because in order to understand how to use a method, you should take the time to practice with different types of problems, or else you run across a slightly different problem and you don't know how to do it. 

3) Comprehension- Having problems to solve like "x+2=5. x=?" doesn't help you learn math. That's arithmetic. Arithmetic is something that you can do with a calculator. Math is knowing why x=3. And just showing your work doesn't help, because again, you're just showing your calculations, which again, could be done with a calculator. What helps is asking, "Why does x=3?" Because then you have to look up from your calculator and think about what makes the problem work.

4) Grading (as a class)- When homework is being graded as a class, then you should again focus on the logic hidden behind the problem. Ask a kid to come up to the board and prove why their answer is correct (having fewer problems would also prove to be helpful here). When you go over them by just stating the answer, it doesn't help the kids who got the answers wrong. Unless they just made an arithmetic error, they still don't get why it works, still don't understand the material, and now only know that they were wrong, which doesn't help. And as teachers, your job is not to make kids pass or fail, but to get them to learn something.