Moment By Moment

I got busted for this in my writing workshop last Wednesday. I commonly get busted for this. Thankfully, it is not (yet) a cookie-worthy offence[1]. Briefly stated, “moment-by-moment” is when a writer describes every action in minute detail.

“The sound of hoofs stopped. As Frodo watched he saw something dark pass across the lighter space between two trees, and then halt. It looked like the black shade of a horse led by a smaller black shadow. The black shadow stood close to the point where they had left the path, and it swayed from side to side. Frodo thought he heard the sound of snuffling. The shadow bent to the ground, and then began to crawl towards him.”

JRR Tolkien. The Fellowship of the Ring.

Sometimes, like in the passage above, this can be a good thing. A fight scene written in moment-by-moment is riveting. Moment-by-moment drags out the action which is why it’s great in an exciting scene since it can build tension. But in normal, run-of-the-mill, non-action scenes dragging the pace is not good. It’s a big mistake. Readers get bored. And bored readers become non-readers, at least of your book or story.

Something new was mentioned last Wednesday about moment-by-moment that led me to a slightly new perspective. A participant said that the real problem with bad MbM is that it contains a series of actions that do not support the continuation of the story. In other words, the things the characters are doing don’t advance the plot.

This got me thinking about teaching procedures in math class. Yes, this is how my mind works. I started to make a connection between writing a scene out in minute detail—first this happens, then this happens, then this other thing—and teaching a multi-step mathematical procedure—first you do this, then this, then you do this other thing.

Aha! I said. To myself. On the train. Very embarrassing really[2].

Is it any wonder students don’t remember procedures all that well when they are taught them by rote? It’s moment-by-moment teaching. The action (i.e. the steps of the procedure) that the characters (i.e. the numbers or maybe variables) are doing is not advancing the plot (i.e. the interesting problem my teacher should have assigned me instead of this crappy worksheet).

The obvious next question (at least to me) was “how can we harness the powers of MbM for good and not evil?” Well, first off, 1) we’ve got to make students care. Then, 2) we have to make this procedure a must-have for solving the problem. When we have arrived at that point, students will be on the collective edge of their seat waiting to see what happens next—well, theoretically at any rate. Given some of the reactions to the water tank problem, it seems that this idea isn’t nearly as farfetched as it sounds.

I admit to being a little uncertain about how to effectively implement steps 1) and 2) on a consistent basis, but once I’ve got those down, this moment-by-moment teaching is going to be a breeze.

[1] In my writing group certain writing no-no’s mean you’re bringing cookies the following week.

[2] Not really. But it’s funnier this way.


Welcome to Mathland; Leave Your Common Sense at the Door

 “To do well in math class, children know that they have to suspend reality and accept the ridiculous problems they are given. They know that if they think about the problems and use what they understand from life, then they will fail. Over time, schoolchildren realize that when you enter Mathland you leave your common sense at the door.”

Jo Boaler. What’s Math Got To Do With It? Penguin Books, 2008.

Dan is at it again. Making me think hard, when all I want to do is collapse on the couch and watch Law & Order: SVU. His post about pseudocontexts got me excited to participate in his problem-hunt assignment. I emailed my (somewhat randomly selected) problem from a well-known publisher, who will remain nameless, to Mr. Meyer and was a little surprised to hear back from him that my problem was not, in fact, a pseudocontext.

O-kay. Either I completely misread Dan’s post (and Jo Boaler’s book), or the way Dan and I are thinking about this is different. So, what exactly is a pseudocontext?

The way I see things, a pseudocontext is a way of framing a problem that doesn’t make any sense, either with respect to what we know about the world, or with respect to what we know about this particular problem.

Pseudocontext can be a context that is completely arbitrary to the problem. In other words, it doesn’t really matter if you’re dealing with volleyballs, or roller skates, or Fraggles. As Dan puts it, “If you can replace the real-world units with “mystery number” or nonsense units and not lose any kind of interest or engagement you’re dealing with pseudocontext.”

In addition, I think that pseudocontext can also be found in the “make-believe contexts — contexts that students were meant to believe but for which they should not use any of their real-world knowledge” (Boaler) that are far too commonly found in math textbooks. For instance, a problem where you have a plethora of detailed information, but you don’t have the one obvious fact (that you would have if you were operating within the bounds of normalcy) which the problem is asking you to find.  As an example:


Okay, in what reality am I where I know 1) the cost of the meal, 2) the percentage I paid in sales tax, and 3) the percentage I paid in tip, but I do not know the total amount I spent each night? This to me is a pseudocontext, because it places me to a world that doesn’t match up with my real-life experiences.

Now I will admit that the mathematical calculations I’m going to need to perform do make sense in the given context. Perhaps some of you would argue that this means that this is a non-example of pseudocontext. I say, the context itself is unrealistic. I’m having a hard time differentiating between this example and the classic age problem—where I am four years more than half my sister’s age (or whatever)—in the sense that both problems withhold information that you would have access to if you were in that situation.

Sometimes a problem can become a pseudocontext because it is poorly written. I had a great example of this come up in my Math 2 class last week. Students were looking at a Markov chain involving taxi cabs, all of which started out each morning in the garage at Northside. After looking at some interesting patterns showing how the taxis moved around the city, students were asked where all of the cabs would wind up eventually. Logically, many groups said that they would wind up at Northside, because that’s where the garage is. Well, yeah[1]. Of course, that wasn’t what the writers of the problem intended when they wrote it, so I had to do some backpedaling and rephrase on the fly. In this case the context of the problem was ignored in the wording of this particular question, thereby rendering the context irrelevant to finding the answer. If you can—or have to—throw out the context in order to solve a problem, you have a pseudocontext rearing its misshapen head.

What’s the big deal? Why is everyone so down on pseudocontexts? Well, other than the fact that they are simply lame, one of my all-time favorite math people, Avery, points out that “pseudocontext can also change what is being taught/assessed. This is bad news if you think you’re teaching/assessing one thing and it turns out you’re teaching/assessing something completely different.” There was a problem about chickens in my former school’s textbook that always left my city-kid students scratching their heads. Not about the math involved, but because the context of the problem was so unfamiliar to them. I’m not saying that we need to remove any unfamiliar situations from our activities, but when we use these types of problems to assess student understanding we are setting students up to fail for the wrong reasons. It’s brutal to realize that a student bombed a quiz/test not because he/she didn’t know the mathematical concepts, but instead because that student had a misconception about the context of the problem.

Put this way, the use of any context at all sounds a little intimidating. Maybe we shouldn’t use contexts at all. But contexts, used appropriately, can be really awesome. See Amanda’s super-cool lesson on finding the height of the water tower next to her school. Personally, I think Amanda’s lesson satisfies Jo Boaler’s criteria for good use of context, that is, “A realistic use of context […] where students are given real situations that need mathematical analysis, for which they do need to consider (rather than ignore) the variables.”

In a comment on Dan’s post, Ben Blum-Smith summed things up nicely. “The real test of whether a math problem is “relevant” is not “do you use this in ‘real life’,” whatever that means, but “do you want to solve it?” It’s not that you want to solve it because it’s relevant; wanting to solve it is what it means to be relevant.” A good context may or may not encourage students to work on a problem. A pseudocontext will disengage many (if not all) students, because it forces them to disassociate themselves from the world for the duration of the problem. It makes what they already know worthless. And feeling like what you bring to the table has no value is not my idea of a positive learning environment.  

[1] This is one of those moments when I realize my students are much, much smarter than I am. Having been well trained, I went into automatic math-question-translation mode and overlooked the obvious.

Blog Re-organization

This post is not about math, teaching, or anything remotely akin to either of those things. It is simply information about how I am going to post my fiction writing pieces. If you don’t care about this, delete this from your reader now.

Still reading?

I have recently restructured my little blog. It seems that most readers are interested in the teaching and math-y things I’m writing (which appears to be most of what I’m writing). An unforseen side effect of this has been that it makes me hesitant to post my fiction and creative non-fiction pieces because people might feel that that’s not what they came here for. Boo-effing-hoo. But I have arrived at a solution. From here on out every short story or other writings that I post will get it’s own page! So, if you’re interested in reading those (Hi Dad!) you can find them on the page tabs. If you’re not interested, don’t tell me about it, just ignore.

Things I (try to) Never Say

This problem is easy.

Whenever I hear someone saying this, it kind of makes me cringe. I have had instances where I have been working–quite happily–on a problem for half an hour, maybe even longer, in a professional development situation and someone comes up to me and says “oh, that one was really simple.” How does that make me feel? Well, I’m an adult. I don’t judge my own self-worth on other peoples’ opinions. But students, especially when it comes to math, aren’t usually so grounded. I try not to say this myself, and I try to get students to not say it either. Or at least I make them explain why they found the problem easy.

You just have to memorize it.

Translation: I can’t think of a way to make this idea meaningful to you, but learn it anyways.

Because it will be on the test.

‘Cause that’s the mother of all motivational devices, isn’t it?

Remember what we talked about yesterday.

So not helpful. Because if they did actually remember this, they probably are already thinking about it. And if they don’t remember, me telling them to remember is not going to magically make the neural connection in their brain start firing. This is probably the one that I slip up on the most–just did it, well, yesterday.

That’s just the way you do it.

Translation: I’m too busy right now to explain the reasoning behind this idea to you.

Alternate definition: If I take the time during class to go over this in detail, we won’t have enough time to do the lesson I have planned for today.

Sorry about that quiz grade. I guess you should study harder next time.

With my new SBG system, I’m not even allowed to say this. And I love that.

I’ve already had one student pass a skill re-check (that’s what I decided to call them) and I have another three students who have made appointments to come see me next week.

I’m sure I’ve left off several things that I just can’t think of right now. How about the rest of you? What is on your list of things to never say?

First Quiz Re-do

I had my first student re-takes this week.

First off, confession: I feel like I’m doing this SBG thing a little half-assed. I’m not making up new assessments, I didn’t give a spiel to all of my students about it in the first week of school (or the second, or the third…), and I haven’t really been as explicit as I would like to be regarding how and when retakes will be available. There are various reasons *cough*excuses*cough* that I have for all of this, which mainly center around the fact that I’m trying to navigate my way through a new school while I’m doing this.

But anyways, back to the newsroom. We had a quiz about right triangle trig in my Math 3 class on Monday. Three questions, asking to find side lengths and angle measure using the appropriate trig ratio. Most students totally aced it. I had 2 students who missed the problem about finding angle measure–who both just flipped the ratio upside down, 1 student who missed two problems, and 1 who missed all three. I was pretty ecstatic about this, since they took a “practice quiz” on Friday and, ah, didn’t do so well. On Tuesday, when I passed back the quiz, I told my class that anyone who got a 3 or less on either of the two skills needed to come by in tutorial to 1) have a conversation with me and then 2) to show me that they knew how to do the skill now.

So that very day Joe comes to my office. He sits down and tells me that he knows how to find the angle, but that he’d just made an error in writing down the ratio. I know this to be the case, so I pull out a piece of scratch paper and make up a problem on the fly and hand it to him. He spends a few minutes working, gives it back to me and I look it over. Wrong answer. I explain to Joe what he did wrong and show him how to correctly solve for the angle measure.

Joe: “Can I take it again?”

Me: “Not today. You can come back tomorrow though.”

Joe: “Why not?”

Me: “Because I want to make sure you actually know it, not just that you have it for this one second.”

Flash forward to Wednesday afternoon: The Return of Joe. New problem. Joe works. He gets it right…and then says to me:

“Y’know, it was interesting. Yesterday, I thought I knew it. But I didn’t.”

And then the Hallelujah chorus started playing in my head.

Whose Fault Is It?

An interesting opinion piece on the NYTimes website about why America is number 11 (as listed in a recent Newsweek ranking). Conclusion: our students aren’t motivated.

The article quotes economist Robert Samuleson, who says (among other things), “Motivation is weak because more students (of all races and economic classes, let it be added) don’t like school, don’t work hard and don’t do well. In a 2008 survey of public high school teachers, 21 percent judged student absenteeism a serious problem; 29 percent cited ‘student apathy.’ ”

As a former public school teacher I acknowledge that this was a true statement for my own personal experience in the classroom (when your parents are paying thousands of dollars to send you to private school, you’re not allowed to be unmotivated 😉 ). And not seeing the actual survey data, I can’t comment on it’s validity. But there’s a part of me that retorts: “Okay. So what?”

If it’s true that the thing bringing us down is unmotivated students, then the next step would be asking: what are we going to do about it? And on a practical level, what can I, as a teacher, do to motivate my students?

Way back when, I used to teach in Seattle, and I worked with a really cool group of teachers–both in my school and in a couple of other schools–who had a video club. One of the things this group did when they first started (alas, before I had joined) was to create a list of reasons why 9th graders struggle. This list was later referred to as “The Wall” because that’s how much space it covered in giant Post-Its. But they didn’t stop there, as overwhelming as this wall of struggles probably was. They then went through each item on the list and circled every item where they could do something to make an impact.

I think it’s really easy to say “students aren’t motivated” and throw your hands in the air and give up. And it’s really hard to ask yourself those tough questions and look at what steps you could take to change the things you can, and attempt to put aside the things you can’t.


The devil is in the details, or so they say. But what details are they talking about? Some details are vital to the stories we tell. What would James Bond be like if he drank club soda? How would we recognize Pigpen without the cloud of dust hovering above his head? Hamlet wouldn’t be terribly interesting if he was known for his decisiveness.

On the other hand, some details are simply filler. I think “meaningless” is coming on a bit strong, but that was actually the first word that came to mind. Does it really matter that the curtains are red instead of green? Maybe it does. But if it matters, you’d better make me care passionately about those damn curtains. If I don’t care, I’ll probably forget, because some details are forgettable.

All these thoughts about details in storytelling somehow relate to math of course. Doesn’t everything?

So, forget about the curtains—they really don’t matter—and think about how this relates to teaching: If I don’t care, I’ll probably forget, because some details are forgettable.

My job as a teacher is to make my students care about math. Oh, and to teach them content too, you say? Once more, with feeling: If I don’t care, I’ll probably forget, because some details are forgettable.

What exactly do we teach students? Procedures, formulae, facts, an amorphous blob called “problem solving skills”. Those long dry lists of state standards written in the least kid-friendly language possible. That is what we teach, right?

So…what do students actually learn? Ahh. Well, I taught them the quadratic formula, so they must have learned it. Right? Umm, not exactly. Did they need to know the quadratic formula? Were they wanting it, pleading for it, musthaveitrightnow? Did they care? Okay, some of them might care because you told them to care, or because they know it will be on the quiz next week and they care about doing well and earning their points. But did any of them feel fire in the belly, where they needed a tool to solve a problem they really-truly wanted to solve but couldn’t figure it out without this magical thing? If not, are they really going to remember this formula in a month? For the final? In ten years?

Do I want them to? Do I care?

That last question has been something that I’ve been thinking about for a long time. What do I want my students to remember forever? I admit it—I’ll go on record as saying that I don’t particularly care if my students know all of their trig identities in a decade. I don’t remember them either. I can look them up if I need them. The entire online math community can shun me now if you so desire. But this raises a big hulking question. If the trig identities aren’t what I’m asking kids to learn, what the heck am I getting paid for?

I was introduced to the language “habits of mind” when I was a proto-teacher, not yet pulling a paycheck. I think by now, most teachers have at least heard about this idea. If not, see the expansive list at Without Geometry (though, seriously, if you’re reading this post you’ve already seen it—all 11,000 of you). But these are the things I want my students to NEVER EVER NEVER forget. I want them to look for and see patterns, and to explore new ideas, and ask insightful questions about things that confuse them. That’s what I’m teaching students for reals. That’s what I’m hoping they are learning. If they aren’t learning those strategies and skills, then I don’t deserve to have my job. Period.