I Don’t Think the Bus Driver Hates White People

I have an 8:00 meeting this morning, so I’m going to take the early (6:36) bus in. I almost never go in this early, so to be safe, I am already downstairs putting on my shoes at 6:29. Sometimes I joke that I live so close to the bus stop that if I hear the bus from the second floor window of my house, I still catch it as it goes by. That’s an exaggeration, but not by much. Now I hear a sound outside that could be the bus, but I know it’s not, it’s never this early. Still, I peek out the door.

It’s the bus, already on the corner. Crap.

I dash out. The bus stop is diagonally across from my house, and fortunately, the bus is waiting at a red light. This gives me a chance: I run down to the corner and across the street with the light. Now the bus and I are facing each other, I wave, cross the other way when the light turns as the bus waits an extra couple seconds, and get on. I am feeling good about the world and my place in it.

Still, 6:29, that’s super-early. Unless maybe this is the previous bus, running super-late? I think I should say something to the driver. “Excuse me, which bus on the schedule is this?” It’s the morning, everything is rushed, who knows if I actually said “Excuse me.”

The bus driver, not one I’m familiar with, is not pleased. He barks at me: “It’s whatever bus it is when I get here!” I look behind him, see the bus is mostly empty. I know from experience that if it were the previous bus, running late, it would be overcrowded, so this must really be the 6:36. I give the driver my ticket and try to explain: you’re running way too early, people who count on this bus will miss it, then the next buses are overcrowded and run late, etc. I ask him to wait a few minutes to get back on schedule. He remains very hot, says (yells) that I better stop talking to him, threatens to call the police if I don’t get away from him. The good thing about him yelling at me is that the bus is still not moving. A couple people, breathless from running down the street, get on the bus while we’re going back and forth. I count this as a small victory.

A large man walks up the aisle from the back of the bus. I am hoping for a little support from a fellow commuter. “You better fucking stop talking to him so he can go!” he yells at me. “You’re holding the bus up! He’s going to call the cops!” This guy looks like he’d like nothing better than to slug me. “He’s calling the cops!” The bus driver is not calling the cops. My fellow passenger, though he is louder than the driver, does not actually take a swing at me. I sit down in one of the many empty seats.

The bus doesn’t move.

I take my book out of my bag and try to disappear into it. It is eerily silent. In moments of suspense, time feels suspended. The bus still doesn’t move.

Finally, after what must have been only a short wait but didn’t feel that way, the bus huffs and trundles forward. I check the time. It is exactly 6:36. We have been waiting for just a few minutes.

I mull things over as the bus heads into the city. I feel good that I managed to stay calm through the whole episode. Maybe this will be a good story to tell the children — the importance of keeping your cool. It occurs to me that the bus driver ended up doing exactly what I asked him to — waiting to leave until he was scheduled to, to the minute. Or was that just a coincidence? Something makes me decide to talk to him again, though I can’t tell exactly how or why I make that decision.

The bus pulls in to Port Authority. I am going to wait for everyone else to get off and then try to approach the driver. As the people ahead of me are getting off, I hear the usual end-of-trip courtesies: passengers say thank you, driver says you’re welcome or have a good day. That’s encouraging: this isn’t one of those guys who’s so nasty that people give up on saying thanks at the end of the ride.

Once is everyone else is off, I get off too. The driver is standing to the side. “Thanks for waiting a few minutes at the bus stop,” I say to him. “I wasn’t trying to pick a fight, I just…”

The driver is looking me in the eye. “Yeah, I know,” he says. “I get what you were saying.” I talk to him about the schedule, he says he understands, sometimes they get off schedule and it’s hard to keep track. We are talking to each other like people now. He says he’s sorry, and he seems totally sincere. I tell him my name, because that seems like a talking-like-people thing to do, and he tells me his. I was hoping to make peace, but this is more than I expected; in under a minute, our conversation has turned comfortably fraternal. I am feeling good about the world again as I head inside the bus terminal.

Now another man slides over to me. He is fiftyish, tall, thin, with slightly graying hair, wearing jeans and a blazer. “You know,” he says,”I saw the bus driver yelling at you, I got a video on my phone. I’m going to call the bus company.”

I don’t actually want him to call the bus company. I tell him that I just had a good talk with the driver, and that he seemed very direct and sincere. My companion isn’t impressed. “He’s just afraid for his job,” he says. He tells me that the driver has done this (I think he means run ahead of schedule) once before, that people were running for the bus and he didn’t care. He looks at me a little conspiratorially. “I think this guy just hates white people.”

Well. The driver, like the majority of bus drivers I encounter, is black. I am white. The guy I’m talking to now is white. The passenger who was yelling at me earlier is white. Most people on the bus are white, with a few exceptions. What is the logic here, how do you decide that a black man driving a bus full of mostly white people hates them because they’re white? Does the white man who came up to yell at me before hate white people too? Why do we look differently at angry black people than at angry white people?

The man I’m talking to is going his way and I’m going mine, and there isn’t time to ask these questions. But I as I walk up 8th Avenue on a sunny end-of-summer morning, I realize that I’ve lost the tidy story I was going to tell my kids about the virtues of keeping calm. And perhaps a bit of my confidence in those virtues as well.

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First Day of Spring

Don’t look for the start of spring in the calendar. You can’t predict ahead of time when it’ll come, yet you always know it on the day, when something makes you rush outside without bundling up first, and once you’re out, you start inventing reasons to stay out longer, not reasons to come back in.

This year the start of spring came on a Sunday, about a week after the official date. The snow had mostly melted a couple weeks earlier, and the temperature was rising steadily through the early afternoon. My sons and I decided we needed to give the local ball field a try. It can’t be that muddy, can it? We took our baseball things — bat, gloves, and balls — and headed over.

It wasn’t muddy at all, and more than warm enough to play. We tossed a ball around to warm up, then took our regular spots — me pitching, one boy batting, the other catching. Despite the winter layoff, they remembered what to do, where to stand, how to swing. Within a few minutes they were as comfortable with their new metal bat as they had been with a wiffle bat by the end of last summer. They swung and missed often, but got their share of hits too, a few balls going past the infield, boys running happily around the bases when they made contact. It must have looked fun, because pretty soon other kids started wandering over to play. A friend from the school bus, two chatty third graders we didn’t know but who soon seemed like old friends, one quiet boy who knew to take over catching when my sons wanted to go out in the field, still another who was too shy to ask but looked like he really wanted to play.

A batting order was formed. After a couple cycles through it, my younger son announced he wanted to pitch. I was skeptical — but OK, sure, give it a shot. We agreed on how far he should stand from the plate, and he started firing balls in. I stood over to the side and watched. He was no worse at getting the ball over the plate than I had been, and the batting rotation went on — swings, misses, balls hit up the middle, balls hit straight up in the air. Everyone knew to run when they hit the ball. Boys who weren’t up to bat stood out in the field, chased balls, tried to throw to each other, to tag the runner — without much grace, for now. Out here, in the developmental leagues of Montclair, NJ, it’s very much a hitter’s world.

No longer needed in the game, I could look around, taking in the whole park — the field, the pond to my right, the playground to my left. We have been regulars since we moved to town, almost seven years ago now, when the first boy was just a year old and the second was just a plan. How many times have we celebrated the first day of spring here? Around this time two years ago, my older son was in kindergarten, and we came out on a sunny day to find that all his friends from school were at the park too. The kids ran off to play, the parents stood and chatted, everyone happy to see each other, and suddenly my embarrassing suburban fantasy, of a real community and life lived at a sustainable pace, crystallized into something real and true. And then last year, we came here on a similar day, expecting more of the same, only the kids got bored at the playground after half an hour, and went off to join another set of kids playing baseball on the field. They’d never played before, didn’t know what to do, couldn’t hit the ball, and left in frustration, which led directly to a summer’s baseball education through backyard wiffle ball games. And now here we were a year later, starting up a game of our own, pulling in other kids. Oh what Proust could have done with a playground and a ball field.

In the actual day, time passed in a more humdrum manner. First one kid got called home by his parents, then another, and finally it was just the three of us on the field again. I went back to pitching. More time went by, and we all started to drag a bit. I would miss the plate more often than not, and when I did get the ball over, the boys didn’t put level swings on it, and hardly connected any more. Time to go.

We walked over toward the car. I carried the bat, the boys carried gloves and balls. We bumped fists, patted shoulders and backs, complimented each other, and meant it: Good game. And as we walked, I went back even further in time, to pre-parent days, evening endings to long afternoon frisbee games when I was still a student, walking off the field with friends, sweaty and exhausted, arms and legs sore, barely able to see in the approaching darkness. Good game. My sons and I walked off the ball field together, side by side, and it felt like the first day, not just of spring, but of something else too.

Should You Opt Out of PARCC?

Today’s post is a discussion of education reform, Common Core, standardized testing, and PARCC with my friend Kristin Wald, who has been extremely kind to this blog. Kristin taught high school English in the NYC public schools for many years. Today her kids and mine go to school together in Montclair. She has her own blog that gets orders of magnitude more readers than I do.

We’re cross-posting this on Kristin’s blog and also on Mathbabe (thank you, Cathy O’Neil!)

ES: PARCC testing is beginning in New Jersey this month. There’s been lots of anxiety and confusion in Montclair and elsewhere as parents debate whether to have their kids take the test or opt out. How do you think about it, both as a teacher and as a parent?

KW: My simple answer is that my kids will sit for PARCC. However, and this is where is gets grainy, that doesn’t mean I consider myself a cheerleader for the exam or for the Common Core curriculum in general.

In fact, my initial reaction, a few years ago, was to distance my children from both the Common Core and PARCC. So much so that I wrote to my child’s principal and teacher requesting that no practice tests be administered to him. At that point I had only peripherally heard about the issues and was extending my distaste for No Child Left Behind and, later, Race to the Top. However, despite reading about and discussing the myriad issues, I still believe in change from within and trying the system out to see kinks and wrinkles up-close rather than condemning it full force.

Standards

ES: Why did you dislike NCLB and Race to the Top? What was your experience with them as a teacher?

KW: Back when I taught in NYC, there was wiggle room if students and schools didn’t meet standards. Part of my survival as a teacher was to shut my door and do what I wanted. By the time I left the classroom in 2007 we were being asked to post the standards codes for the New York State Regents Exams around our rooms, similar to posting Common Core standards all around. That made no sense to me. Who was this supposed to be for? Not the students – if they’re gazing around the room they’re not looking at CC RL.9-10 next to an essay hanging on a bulletin board. I also found NCLB naïve in its “every child can learn it all” attitude. I mean, yes, sure, any child can learn. But kids aren’t starting out at the same place or with the same support. And anyone who has experience with children who have not had the proper support up through 11th grade knows they’re not going to do well, or even half-way to well, just because they have a kickass teacher that year.

Regarding my initial aversion to Common Core, especially as a high school English Language Arts teacher, the minimal appearance of fiction and poetry was disheartening. We’d already seen the slant in the NYS Regents Exam since the late 90’s.

However, a couple of years ago, a friend asked me to explain the reason The Bluest Eye, with its abuse and rape scenes, was included in Common Core selections, so I took a closer look. Basically, a right-wing blogger had excerpted lines and scenes from the novel to paint it as “smut” and child pornography, thus condemning the entire Common Core curriculum. My response to my friend ended up as “In Defense of The Bluest Eye.”

That’s when I started looking more closely at the Common Core curriculum. Learning about some of the challenges facing public schools around the country, I had to admit that having a required curriculum didn’t seem like a terrible idea. In fact, in a few cases, the Common Core felt less confining than what they’d had before. And you know, even in NYC, there were English departments that rarely taught women or minority writers. Without a strong leader in a department, there’s such a thing as too much autonomy. Just like a unit in a class, a school and a department should have a focus, a balance.

But your expertise is Mathematics, Eugene. What are your thoughts on the Common Core from that perspective?

ES: They’re a mix. There are aspects of the reforms that I agree with, aspects that I strongly disagree with, and then a bunch of stuff in between.

The main thing I agree with is that learning math should be centered on learning concepts rather than procedures. You should still learn procedures, but with a conceptual underpinning, so you understand what you’re doing. That’s not a new idea: it’s been in the air, and frustrating some parents, for 50 years or more. In the 1960’s, they called it New Math.

Back then, the reforms didn’t go so well because the concepts they were trying to teach were too abstract – too much set theory, in a nutshell, at least in the younger grades. So then there was a retrenchment, back to learning procedures. But these things seem to go in cycles, and now we’re trying to teach concepts better again. This time more flexibly, less abstractly, with more examples. At least that’s the hope, and I share that hope.

I also agree with your point about needing some common standards defining what gets taught at each grade level. You don’t want to be super-prescriptive, but you need to ensure some kind of consistency between schools. Otherwise, what happens when a kid switches schools? Math, especially, is such a cumulative subject that you really need to have some big picture consistency in how you teach it.

Assessment

ES: What I disagree with is the increased emphasis on standardized testing, especially the raised stakes of those tests. I want to see better, more consistent standards and curriculum, but I think that can and should happen without putting this very heavy and punitive assessment mechanism on top of it.

KW: Yes, claiming to want to assess ability (which is a good thing), but then connecting the results to a teacher’s effectiveness in that moment is insincere evaluation. And using a standardized test not created by the teacher with material not covered in class as a hard percentage of a teacher’s evaluation makes little sense. I understand that much of the exam is testing critical thinking, ability to reason and use logic, and so on. It’s not about specific content, and that’s fine. (I really do think that’s fine!) Linking teacher evaluations to it is not.

Students cannot be taught to think critically in six months. As you mentioned about the spiraling back to concepts, those skills need to be revisited again and again in different contexts. And I agree, tests needn’t be the main driver for raising standards and developing curriculum. But they can give a good read on overall strengths and weaknesses. And if PARCC is supposed to be about assessing student strengths and weaknesses, it should be informing adjustments in curriculum.

On a smaller scale, strong teachers and staffs are supposed to work as a team to influence the entire school and district with adjusted curriculum as well. With a wide reach like the Common Core, a worrying issue is that different parts of the USA will have varying needs to meet. Making adjustments for all based on such a wide collection of assessments is counterintuitive. Local districts (and the principals and teachers in them) need to have leeway with applying them to best suit their own students.

Even so, I do like some things about data driven curricula. Teachers and school administrators are some of the most empathetic and caring people there are, but they are still human, and biases exist. Teachers, guidance counselors, administrators can’t help but be affected by personal sympathies and peeves. Having a consistent assessment of skills can be very helpful for those students who sometimes fall through the cracks. Basically, standards: yes. Linking scores to teacher evaluation: no.

ES: Yes, I just don’t get the conventional wisdom that we can only tell that the reforms are working, at both the individual and group level, through standardized test results. It gives us some information, but it’s still just a proxy. A highly imperfect proxy at that, and we need to have lots of others.

I also really like your point that, as you’re rolling out national standards, you need some local assessment to help you see how those national standards are meeting local needs. It’s a safeguard against getting too cookie-cutter.

I think it’s incredibly important that, as you and I talk, we can separate changes we like from changes we don’t. One reason there’s so much noise and confusion now is that everything – standards, curriculum, testing – gets lumped together under “Common Core.” It becomes this giant kitchen sink that’s very hard to talk about in a rational way. Testing especially should be separated out because it’s fundamentally an issue of process, whereas standards and curriculum are really about content.

You take a guy like Cuomo in New York. He’s trying to increase the reliance on standardized tests in teacher evaluations, so that value added models based on test scores count for half of a teacher’s total evaluation. And he says stuff like this: “Everyone will tell you, nationwide, the key to education reform is a teacher evaluation system.” That’s from his State of the State address in January. He doesn’t care about making the content better at all. “Everyone” will tell you! I know for a fact that the people spending all their time figuring out at what grade level kids should start to learn about fractions aren’t going tell you that!

I couldn’t disagree with that guy more, but I’m not going to argue with him based on whether or not I like the problems my kids are getting in math class. I’m going to point out examples, which he should be well aware of by now, of how badly the models work. That’s a totally different discussion, about what we can model accurately and fairly and what we can’t.

So let’s have that discussion. Starting point: if you want to use test scores to evaluate teachers, you need a model because – I think everyone agrees on this – how kids do on a test depends on much more than how good their teacher was. There’s the talent of the kid, what preparation they got outside their teacher’s classroom, whether they got a good night’s sleep the night before, and a good breakfast, and lots of other things. As well as natural randomness: maybe the reading comprehension section was about DNA, and the kid just read a book about DNA last month. So you need a model to break out the impact of the teacher. And the models we have today, even the most state-of-the-art ones, can give you useful aggregate information, but they just don’t work at that level of detail. I’m saying this as a math person, and the American Statistical Association agrees. I’ve written about this here and here and here and here.

Having student test results impact teacher evaluations is my biggest objection to PARCC, by far.

KW: Yep. Can I just cut and paste what you’ve said? However, for me, another distasteful aspect is how technology is tangled up in the PARCC exam.

Technology

ES: Let me tell you the saddest thing I’ve heard all week. There’s a guy named Dan Meyer, who writes very interesting things about math education, both in his blog and on Twitter. He put out a tweet about a bunch of kids coming into a classroom and collectively groaning when they saw laptops on every desk. And the reason was that they just instinctively assumed they were either about to take a test or do test prep.

That feels like such a collective failure to me. Look, I work in technology, and I’m still optimistic that it’s going to have a positive impact on math education. You can use computers to do experiments, visualize relationships, reinforce concepts by having kids code them up, you name it. The new standards emphasize data analysis and statistics much more than any earlier standards did, and I think that’s a great thing. But using computers primarily as a testing tool is an enormous missed opportunity. It’s like, here’s the most amazing tool human beings have ever invented, and we’re going to use it primarily as a paperweight. And we’re going to waste class time teaching kids exactly how to use it as a paperweight. That’s just so dispiriting.

KW: That’s something that hardly occurred to me. My main objection to hosting the PARCC exam on computers – and giving preparation homework and assignments that MUST be done on a computer – is the unfairness inherent in accessibility. It’s one more way to widen the achievement gap that we are supposed to be minimizing. I wrote about it from one perspective here.

I’m sure there are some students who test better on a computer, but the playing field has to be evenly designed and aggressively offered. Otherwise, a major part of what the PARCC is testing is how accurately and quickly children use a keyboard. And in the aggregate, the group that will have scores negatively impacted will be children with less access to the technology used on the PARCC. That’s not an assessment we need to test to know. When I took the practice tests, I found some questions quite clear, but others were difficult not for content but in maneuvering to create a fraction or other concept. Part of that can be solved through practice and comfort with the technology, but then we return to what we’re actually testing.

ES: Those are both great points. The last thing you want to do is force kids to write math on a computer, because it’s really hard! Math has lots of specialized notation that’s much easier to write with pencil and paper, and learning how to write math and use that notation is a big part of learning the subject. It’s not easy, and you don’t want to put artificial obstacles in kids’ way. I want kids thinking about fractions and exponents and what they mean, and how to write them in a mathematical expression, but not worrying about how to put a numerator above a denominator or do a superscript or make a font smaller on a computer. Plus, why in the world would you limit what kids can express on a test to what they can input on a keyboard? A test is a proxy already, and this limits what it can capture even more.

I believe in using technology in education, but we’ve got the order totally backwards. Don’t introduce the computer as a device to administer tests, introduce it as a tool to help in the classroom. Use it for demos and experiments and illustrating concepts.

As far as access and fairness go, I think that’s another argument for using the computer as a teaching tool rather than a testing tool. If a school is using computers in class, then at least everyone has access in the classroom setting, which is a start. Now you might branch out from there to assignments that require a computer. But if that’s done right, and those assignments grow in an organic way out of what’s happening in the classroom, and they have clear learning value, then the school and the community are also morally obligated to make sure that everyone has access. If you don’t have a computer at home, and you need to do computer-based homework, then we have to get you computer access, after school hours, or at the library, or what have you. And that might actually level the playing field a bit. Whereas now, many computer exercises feel like they’re primarily there to get kids used to the testing medium. There isn’t the same moral imperative to give everybody access to that.

I really want to hear more about your experience with the PARCC practice tests, though. I’ve seen many social media threads about unclear questions, both in a testing context and more generally with the Common Core. It sounds like you didn’t think it was so bad?

KW: Well, “not so bad” in that I am a 45 year old who was really trying to take the practice exam honestly, but didn’t feel stressed about the results. However, I found the questions with fractions confusing in execution on the computer (I almost gave up), and some of the questions really had to be read more than once. Now, granted, I haven’t been exposed to the language and technique of the exam. That matters a lot. In the SAT, for example, if you don’t know the testing language and format it will adversely affect your performance. This is similar to any format of an exam or task, even putting together an IKEA nightstand.

There are mainly two approaches to preparation, and out of fear of failing, some school districts are doing hardcore test preparation – much like SAT preparation classes – to the detriment of content and skill-based learning. Others are not altering their classroom approaches radically; in fact, some teachers and parents have told me they hardly notice a difference. My unscientific observations point to a separation between the two that is lined in Socio-Economic Status. If districts feel like they are on the edge or have a lot to lose (autonomy, funding, jobs), if makes sense that they would be reactionary in dealing with the PARCC exam. Ironically, schools that treat the PARCC like a high-stakes test are the ones losing the most.

Opting Out

KW: Despite my misgivings, I’m not in favor of “opting out” of the test. I understand the frustration that has prompted the push some districts are experiencing, but there have been some compromises in New Jersey. I was glad to see that the NJ Assembly voted to put off using the PARCC results for student placement and teacher evaluations for three years. And I was relieved, though not thrilled, that the percentage of PARCC results to be used in teacher evaluations was lowered to 10% (and now put off). I still think it should not be a part of teacher evaluations, but 10% is an improvement.

Rather than refusing the exam, I’d prefer to see the PARCC in action and compare honest data to school and teacher-generated assessments in order to improve the assessment overall. I believe an objective state or national model is worth having; relying only on teacher-based assessment has consistency and subjective problems in many areas. And that goes double for areas with deeply disadvantaged students.

ES: Yes, NJ seems to be stepping back from the brink as far as model-driven teacher evaluation goes. I think I feel the same way you do, but if I lived in NY, where Cuomo is trying to bump up the weight of value added models in evaluations to 50%, I might very well be opting out.

Let me illustrate the contrast – NY vs. NJ, more test prep vs. less — with an example. My family is good friends with a family that lived in NYC for many years, and just moved to Montclair a couple months ago. Their older kid is in third grade, which is the grade level where all this testing starts. In their NYC gifted and talented public school, the test was this big, stressful thing, and it was giving the kid all kinds of test anxiety. So the mom was planning to opt out. But when they got to Montclair, the kid’s teacher was much more low key, and telling the kids not to worry. And once it became lower stakes, the kid wanted to take the test! The mom was still ambivalent, but she decided that here was an opportunity for her kid to get used to tests without anxiety, and that was the most important factor for her.

I’m trying to make two points here. One: whether or not you opt out depends on lots of factors, and people’s situations and priorities can be very different. We need to respect that, regardless of which way people end up going. Two: shame on us, as grown ups, for polluting our kids’ education with our anxieties! We need to stop that, and that extends both to the education policies we put in place and how we collectively debate those policies. I guess what I’m saying is: less noise, folks, please.

KW: Does this very long blog post count as noise, Eugene? I wonder how this will be assessed? There are so many other issues – private profits from public education, teacher autonomy in high performing schools, a lack of educational supplies and family support, and so on. But we have to start somewhere with civil and productive discourse, right? So, thank you for having the conversation.

ES: Kristin, I won’t try to predict anyone else’s assessment, but I will keep mine low stakes and say this has been a pleasure!

Fall Math Notes I: The Area Table

This fall, I’ve been teaching a math workshop for grade school kids (grades 3-6) in my town. Once a week for an hour and a half, covering the typical topics for kids this age: multiplication, division, place value, fractions. We’ve wrapped up now (last week was the last class), so I wanted to note down a few impressions before I forget it all.

One thing that worked pretty well was identifying multiplication and area. I started one of the first classes with the following exercise: take a grid (say 10 by 10). Pick a box in the grid, count the number of boxes that are above and/or to the left of that box, and write that number in your box. (Alternately, draw a rectangle extending from the top left of the grid to the box you picked, and count the number of boxes in that rectangle.) For example, if you pick the box in the 4th row (counting from the top) and 5th column (counting from the left) of your grid, then the boxes above and to the left are marked in green in the picture below, and there are 20 of them:

original-937559-1

You don’t need to do anything more here than count boxes, but of course the point is that 20 boxes = 4 rows × 5 columns and also that 20 is the area of the green rectangle (each 1 by 1 box has an area of 1 square unit, so 20 boxes is 20 square units of area).

I had the kids repeat this for every box in the grid. The number you write down in each box is the product of the corresponding row and column, and you end up with the good old multiplication table. I liked how this worked out for a few reasons:

  1. It was a way for the kids to figure out the multiplication table themselves, and get it right. No memorization, and little required in the way of prerequisites or arithmetic skills. (When you’re working with kids with different backgrounds, that’s a very important advantage.) The kids were pretty enthusiastic about doing it.
  2. It was a big enough table that counting boxes over and over got tedious, so the kids started to look for shortcuts. They noticed right away that the numbers in each row increase by the index of that row (e.g., 5th row = counting by 5’s). That helped them fill the table out pretty quickly. It also gave us an excuse to talk about why that worked (each time you take a step to the right in the 5th row, you add 5 boxes).
  3. When we were done, we didn’t just have a multiplication table, we had good geometric intuition to go with it. Meaning, we knew how to think of the table as a family of overlapping rectangles, and every number in the table as an area! (I almost wanted to call the thing the area table instead of the multiplication table, but decided I shouldn’t saddle the kids with made up terminology.) Then we could find more patterns in the table, and try to explain those patterns geometrically. For example, a 5 × 5 square has one more box than a 6 × 4 rectangle (25 = 24 + 1), a 7 × 7 square has one more box than an 8 × 6 rectangle (49 = 48 + 1), etc., and we could explain that by moving boxes around. For another example, if you just stick to the nested squares, the number of boxes you need to add to each square to get the next square grows in a linear way (4 = 1 + 3, 9 = 4 + 5, 16 = 9 + 7, 25 = 16 + 9, 36 = 25 + 11, and so on), and you can see and count the extra boxes explicitly. There are hints here of algebra (x2 – 1 = (x + 1) (x – 1)) and even calculus (derivative of x2 is 2x), and you can get at them just by counting and moving boxes.

More to come (Division! Fractions! Place value!). You can’t wait, can you? Meanwhile, happy Thanksgiving.

Suspense as Suspension of Time

I figured the ballgame would be over by the time I needed to head out. I was going over to Tierney’s Tavern in Montclair, NJ (I hope your town has a place like this) to see my kids’ awesome music teacher Myrna play a solo set on guitar. The Giants-Nationals playoff game I had been keeping tabs on was winding down (1-0 Nats after 8 innings), apparently in plenty of time for me to get to the show. But then the visiting Giants (my favorite National League team, going back to my Bay Area days) tied it up on a two-out double in the top of the 9th. The game was well into the 10th inning, with no sign of ending, when I drove off to the tavern.

At Tierney’s, I found a cozy stool with a good view of the stage, the bar, and — importantly — the TV over the bar. Myrna was playing some seriously hard-edged rock and roll, and the Giants looked like they might take it in the 12th when they got a man to third with one out. But the rally died (popout, groundout). Yusmeiro Petit came on to pitch for SF, Myrna wrapped up, and Thee Volatiles took the stage at Tierney’s. Petit, a second line starter who didn’t quite make it into the shorter playoff rotation, seemed shaky at the beginning, then settled in. He looked like he could pitch for a while. Thee Volatiles were tight from the first chord, filling the room with the kind of garage rock I used to hear all over Boston, and could never resist, in the 80’s. (These guys are from New Jersey, but once you know a certain kind of sound, you recognize it anywhere.) Petit pumped in strikes, the Nats’ pitchers kept pace, and Thee Volatiles banged out chords: a symphony of forward momentum.

Songs and innings raced by. Thee Volatiles finished up. Most of the crowd had come to see them, and now began to drift out of the room. I wondered how many people were left at the ballpark. The night’s last act, Karyn Kuhl (say her name out loud to realize how great it is) took the stage. She had an electric guitar and a small rhythm section: bass, drums. Within five minutes, I was transfixed: I had been expecting a local stalwart, and here instead was the second coming of PJ Harvey, just arrived in Jersey to play a private gig for 30 people. The music left the garage, headed out into vast spaces. The room appeared emptier but wasn’t, really; the sound pulsed, flowed, filled every space it could find, mocked everyone who had left early. Things didn’t feel so linear anymore: was it the 15th inning now? The 16th?

In an instant, the game comes back in sharp focus. Brandon Belt, up for the Giants, starts his swing — smooth, controlled, deliberate — and the other players on the field fade away. Somehow the ball is on a tee, and then it’s headed for the upper deck in right field, a line that turns into a parabola, gravity’s rainbow at the ballpark. 2-1 Giants. Suddenly we are back on the clock, starting to count down: midnight approaching, three outs left for the Nats. The visitors celebrate, wearily but defiantly, in their dugout. This game has made everyone old.

We head to the bottom of the inning, the Nats’ last ups. Turns out it is the 18th: we have played nine innings and then nine more, an impromptu doubleheader. Petit, who has carried a starter’s workload (six innings) after all in this unscheduled nightcap, is out of the game, rookie Hunter Strickland in to pitch for the Giants. I saw Strickland give up two towering homers to the Nats the day before, so I am more than a little terrified.

Strickland gets one out, then faces Nat leadoff hitter Denard Span, a .300 hitter up for the eighth time. Two quick strikes, then three balls. Now one foul ball after another: straight down, off to the side. The count is already full, but the at bat has infinite capacity, getting fuller still with every pitch. Karyn is singing about ghosts leaving. The music swirls, builds. Song and at bat go on, pitcher and batter take their time, another ball is fouled off, jagged notes twist in the air. The suspense, on stage and on TV, is killing me. At that moment, hearing the music, seeing the game, I realize I’m so tense not just because I don’t know what will happen, but also because I don’t know when. There’s so much suspense because time is suspended. Suspense, suspended, pend, meaning hang. Time hangs, and everything is uncertain.

Eventually the clock starts back up again, as it must. Span hits an ordinary grounder to first base, the second out of the inning. Karyn pivots into the Ramones’ “I Just Wanna Have Something to Do,” dedicates it to Thee Volatiles because it is straight ahead punk rock, forward momentum again. At some point midnight comes. One more song, one more out (a fly ball to right field), and it’s over, in Montclair, NJ and in Washington, DC. Sometimes you don’t know if you can explain what you’ve just seen and heard, but you know you have to try.