Wednesday, November 27, 2013

Out of Class Interventions - Never Look Back

Andrew wants to spark a conversation about intervention strategies that work, and I've got something small to share.

In my teaching life so far, "intervention" has always meant "a time to meet with a kid outside of class." For me, that always seemed to be basically a waste of time. What can I do for a kid in forty minutes that I couldn't do in two months?

I'd use SBG. I'd say, look, you've got seven standards that you haven't mastered. Let's do two a week for the next month. Let's meet on Monday during lunch, and I'll tutor you in those skills. Let's reassess on Thursday. And every once in a while a kid would pull it together, but most of the time he would stop coming, or he wouldn't be able to study on his own, or he would still be getting lost on the new material as he's reviewing the old stuff...

Last year I basically begged people on twitter to show me a better way, and Frank Noschese sent me a document that made a small, but important difference in the way my interventions went. The most important part of that doc was the second line of this table:

After reading this, I immediately stopped going over old material with kids, and instead spent our time prepping them for the upcoming week's lessons. 

The theory is simple. In a weekly session, it's usually unrealistic to help a kid learn large swaths of material that they're struggling with. But it is totally realistic to help a kid understand tomorrow's class. That just requires a little bit of foresight and the careful selection of examples. And if the kid gets Tuesday's class, then they've got a decent shot at Wednesday. And we can build an area of strength for this kid, and that will be our start.

I don't want to paint too rosy a picture here. By the time you've got a regular intervention with a kid, it's often going to be rough going. Still, looking ahead worked much better for me than looking back.

Tuesday, November 26, 2013








Tuesday, November 19, 2013

Is Your Own Math Work Shareable?

(Lots to disagree with here, kids. Get excited!)

Students don't like to write about their reasoning. They don't present their work in a way that allows anyone else to comprehend their path to a solution. But we want kids to write about their reasoning. Conflict! Drama!

Why do kids hate writing about math? Here are some possibilities:
  • Kids care more about answers than about the underlying reasoning
  • Kids are lazy
  • Kids don't know how to write clearly about math
Without a doubt, these factors all play their part. But I think that there's something else going on.

Here are a few pages from one of the notebooks that I do math in:

There's a ton of scribbling, some diagrams, some arrows, all of which make a ton of sense to me right now but would take a significant amount of work for anyone to decipher.

I tend not to do math in a way that would make sense to anyone else. The question is, should I? What would I gain by presenting my notes in a cleaner, more legible way?

I can see two reasons for writing this up in a cleaner way. 
  1. For sharing my thoughts with others.
  2. For clarifying my own thoughts.
Let's immediately dismiss the first possibility as unrealistic. These aren't new discoveries, and they aren't even new ways to think about old problems. This is me trying to understand some ridiculously well-trodden math. Why would I share this?

The second possibility is a more serious one. And, look, I'm numero uno in line for the "Writing Helps Me Think More Clearly About Things" rally. (I'm getting clearer on these ideas as I'm writing it right now which is fairly meta.) But writing a clear statement about math is only occasionally what I do when I want to get clearer on a mathematical idea. When I want to test my learning, sometimes I rederive my result. Sometimes I try to tackle a new, but related, problem. Sometimes I take a walk and think about it.

What's behind my own tendency to skip the writing-up process when I'm doing my own math? As before, it might be laziness, it might be a lack of skill. Upon reflection, I think that it mostly has to do with my preference for problem solving. I typically figure that if I don't understand something, then eventually it'll get in the way of my ability to solve problems. Sure, I could prevent that by trying to sort everything out now, but that wouldn't be nearly as much fun as trying to solve another problem. "I'll understand everything with the depth that it currently needs," seems to be the principle by which I usually operate in my own math work.

So, here are my (somewhat loaded) questions:

  • What does your own math work look like? How often do you create something that could be shared with others as part of your learning process?
  • Should I be changing the way that I do math in my notebooks?
  • Should we be asking students to practice math in a way that differs from our own?
  • Does writing about reasoning typically solve a problem for the teacher or the student? 
  • What exactly is the value of writing one's reasoning in math, as opposed to articulating it in speech or in thought?
Lots, lots to disagree with here. Please do! Let's get closer to the truth together.

Sunday, November 17, 2013

Research Review: "Interest -- the Curious Emotion"

Inspired by Dan, I've been struggling to come up with a framework for what people find interesting. In making a list of all the things that I found interesting over the course of a day, I noticed that I tended to ask questions that were relatively easy to answer. This made me think that the ease of answering a question is a crucial factor in how interesting it is.

This morning, I'm reading some actual psychological research on this whole issue. This literature review is called "Interest -- the Curious Emotion," , and it's by Paul J. Silvia. Here are some highlights, lightly edited for blogability:
"In my research, I have suggested that interest comes from two appraisals. The first appraisal is an evaluation of an event’s novelty–complexity, which refers to evaluating an event as new, unexpected, complex, hard to process, surprising, mysterious, or obscure...The second, less obvious appraisal is an evaluation of an event’s comprehensibility."
In other words, whether we find something interesting is jointly determined by its newness and its comprehensibility. What's "comprehensibility"? It's a sense of whether a person has the "skills, knowledge, and resources to deal with an event."

Here's how he thinks his theory plays out in an art museum:
"Consider, for example, a group of college students meandering through the campus art museum. Some of the students find the modern-art gallery interesting: The works strike them as new, different, and unusual, and—thanks to a few classes in art history—they feel able to get what the artists are trying to express. But most of the students, such as the students forced to attend as part of a class assignment, do not find the modern-art gallery interesting. The works strike them as unusual but also meaningless and incomprehensible: They do not know enough about this art to find it interesting."
Silvia's "comprehensibility" is a lot like what I meant by "ease." If you push me, I would have to admit that the questions that I asked myself weren't all easy, but I had high confidence that I would be able to answer them. I'm happy to drop my "ease" for Silvia's "comprehensibility."

Has he done research in math? Yes he has:
"[Subjects] spent more time viewing complex polygons than they did viewing simple polygons."
Anyway, the paper's really worth checking out, and it's chock-full of citations to a really cool body of research. Here's one line worth remembering:
"New and comprehensible works are interesting; new and incomprehensible things are confusing."

Wednesday, November 13, 2013

Things That I Found Interesting Today, Ctd.

Following up on yesterday's post, here are some questions that I found myself wondering on the way to work:

On leaving my building: What sort of symmetry is there on the building's door?

On the subway: For what percent of a year have my wife and I been married?
In the elevator: The weight capacity is listed as 2500 lbs or 13 people. How much are they assuming that each person weighs?

In all these cases I was intrigued to figure out the answer, and I felt satisfaction when I did.

(In the elevator on my way home I also saw this, booo NYC.)


What's remarkable about these questions is how easy they all are for me. They are not very challenging. And I really like math! I do it in my free time. I teach it. That makes me for sure in the top 1% of mathy people on the entire planet. And I know way tougher math than lines of symmetry, percentage conversions and division.

You might have thought that lots of questions occur to me over the course of the day of varying difficulties. But that's not what's happened for me over the past two days. There seems to be an intimate connection between the questions that strike me as interesting and the questions that I'm capable of answering successfully.

I'm essentially asking questions that I've seen asked before.


To temper this a little bit, some tough questions did occur to me later in my day. I'm doing some work with a 7th Grader as part of an independent study, and she's really into infinite series. While preparing for our time together, I asked myself whether I could measure the speed at which a sequence of decreasing fractions approaches zero. That was tough for me, and I didn't know how to take it. I tried to directly compare consecutive elements in the sequence, but that didn't really seem so informative.

I gave up really easily. And then I caught myself, and then I tried again. And then I gave up again when it wasn't really yielding anything.

Why did I find this question interesting? I think, for a moment, I thought that it would be easy to answer, and that I would have a bit more valuable knowledge under my belt. When it turned out that I didn't have easily accessible knowledge, I more or less gave up, and then it took all of what I know about learning to get back to work. And then I still gave up.

I have two take-away thoughts: (1) I need to work harder and (2) Questions rarely stay interesting if the road forward is unclear.


Danielson says that questions are evidence of learning. Maybe part of the reason why is that we usually ask questions when we're fairly secure that we could figure out the answer.

Or not? Again, thanks for kicking around these ideas with me while I try to make sense of it all.

Tuesday, November 12, 2013

Things That I Found Interesting Today

[This is a very, very tentative post. You should consider this a formal invitation to rip it apart in the comments, but, yeah, I want to put a little bit of distance between Future Michael and this thing.]

Here is a partial list of questions that I found myself thinking about today:

  • Had I gotten any emails or tweets after I turned my computer off?
  • What's the best way to understand conjugacy classes?
  • What sorts of things do we find interesting? What sorts of things do people get curious about?
  • What would my students do if I gave them a period of free-choice math?
  • What do the people look like in the subway car running parallel to mine?
  • What's the song that's coming out of that classroom?
  • When it snows, why does it harder to see the tops of tall buildings than the middles?
  • What was my wife's day like?

Dan Meyer has been thinking about what makes pure math tasks interesting, likable or enjoyable. I think that this is going to push him to a general theory of engagement, and he's asking folks to describe what makes their most likable pure math tasks so interesting and enjoyable.

This is worthwhile, but I think that reflecting on what interests our students will only take us so far. The problem is that we have so little access to what our students find interesting. It's hard for us to get into their heads.

On the other hand, it's really easy for us to get into our heads. Here's what I suggest: carry around a pencil and paper with you for the next few days, and every time you find yourself curious about something, mark it down. Then, after a few days, try to understand what sorts of things you find interesting. These can be math things, or they can be non-math things.

Based on the sorts of things I found myself curious about today, I'll toss out a couple early conjectures:
  • I almost always find myself curious about questions that I'm actually able to answer. I almost never find myself really curious about a matter that there is a low chance of me figuring out. 
  • I find myself most interested in questions whose answers are rare or uncommon. I suspect that this is the reason why I don't find easy questions interesting; it's because I perceive their answers to be common, cheap and readily available to others.
  • You can usually predict how interesting I'll find a question by asking two further questions: (a) How difficult will it be for me to figure this out? (b) How valuable is the answer of this question to me? (This value often comes in the form of other people being impressed with me.)
I'm not especially confident in my tentative ideas, but we'll see if they hold up as I pay closer to attention to the things that I find interesting.

Monday, November 11, 2013

Exponents Without Repeated Multiplication

The Problem With Exponents Education 

Kids have trouble learning exponents. In 5th and 6th Grade they regularly multiply the base and the power instead of performing exponentiation. In 9th Grade they think that negative exponents must yield negative results, and in 11th Grade they struggle to figure out what in the world a fractional power might mean. And, if you push them hard enough, they'll often reveal a tendency to still multiply the base and the power.

What is the source of all these troubles? One answer is that exponents are introduced to kids solely as repeated multiplication

You might not find this troubling. You might say, "Hey, that's a fine mathematical definition. After all, isn't multiplication just repeated addition?"

In math we deal in abstractions, and abstractions are best understood from multiple perspectives. Sometimes it's helpful to think of multiplication as repeated addition. But is 3.4 times 5.7 really best understood as repeated addition of 5.7 some 3.4 times? Instead, you might think that there are times when area is a good model for multiplication. Other times call for scaling as our model. Arrays are often helpful. The number line is often useful.

The point is that, in multiplication, we have multiple models that help us glimpse aspects of what is essentially an abstraction. For thinking about the Distributive Property, you want the array model. But arrays are less helpful for decimal multiplication, and there you really want to be talking about area with your students. For multiplication of negative numbers, you want transformations of the number line. Different models for different moments.

We have one way of thinking about exponents, and it's not terribly effective all on its own. We need more models for exponentiation, and we need to think about replacing "repeated multiplication" as our students' first exponents model.

Start With Squaring and Cubing

In the early grades, students should be able to talk (talk!) about squaring and cubing a length, and that's it. You don't worry about the notation. You don't talk about anything besides lengths. You don't talk about powers. You don't do anything besides squaring and cubing lengths, and getting kids used to that language.

You might kick that unit off with a visual pattern such as this one:

You make sure that kids can draw the next step. You ask them what the 7th picture looks like, and you ask them how many bricks are in that picture. You use these problems to give them practice with multiplication and you use the language of "squared" persistently.

Then you toss this in front of them:

4th Graders will find these problems much more challenging. They'll have trouble drawing 3D models, and you should teach them to sketch drawings of cubes, an important Geometric skill that's often never taught to kids explicitly. And, though you start by asking them how many cubes are in the fourth picture, pretty soon you're asking them "What is 4 cubed?"

And, if your kids are anything like mine, they will absolutely not figure this out by multiplying 4 by 4 by 4. Instead they'll start trying to calculate 16 times 4.

This difference is subtle enough that you might dismiss it, but you shouldn't. 16 times 4 is an entirely different conceptual model for cubing than 4 times 4 times 4 is. You and your class should make the relationship between "Something cubed" and "That same thing squared" explicit by the end of your study. 

This, and nothing else, should be kids' first exposure to exponents.

The Next Models

Of course the Geometric model of exponentiation will hardly suffice for kids in the long run. But it's the foundation, it's where we start. And what we're going to do for kids is extend the operation of squaring and cubing to other powers,to fourth and fifth and seventh powers. To do this, we'll need another model for exponentiation.

For extension to other powers we'll rely heavily on Geometric series, our second model of exponentiation. If we want to, we can even be fancy and explicitly connect the old model with our new one:

And, again, the spoken language that we use is pretty much the point here. We're going to start talking about the picture after the cube as the "Fourth Power of Three." More generally, we're going to talk about powers as entries in these patterns.

What's going on here is that we're moving toward a third model, which is a recursive definition of exponents, with squaring and cubing as our anchors. We'll define "3 to the fourth power" for students as multiplying "3 cubed" by 3. After all, isn't that how we moved from 3 squared to 3 cubed back in our earlier work?

Still not convinced that the recursive model is a worthwhile investment? Think ahead to how hard it is to convince kids that negative exponentiation doesn't have to produce negative results, and think about how helpful these patterns will be for that.

Of course, now that you have extended exponents beyond squaring and cubing, you might find it interesting to revisit that earlier model with your kids. Can we make sense of raising a length to the first power? Can we make sense of raising a length to the fourth power?

We have three models -- Geometric, Geometric Series, and Recursive Definition -- and we haven't said a word about repeated multiplication. So where is repeated multiplication in all this? It's festering in the classroom. Some kids have figured out that you can skip steps in the recursive definition using repeated multiplication. Kids have shared this as one of a few computation strategies for doing exponents calculations. (Others are skip counting to follow the Geometric Series or using exponent properties to take shortcuts, e.g. 3^5 is 9 x 27.)

And, then? Then you make it explicit. You talk about the repeated multiplication definition, because that's a really important model for exponentiation also. In particular, it's one way to see that negative exponents often create fractional values. (Though, as evidenced by student errors, it's not a particularly effective tool on its own.) Seeing exponentiation as repeated multiplication makes simplifying expressions easier. (But, again, this is an area that is currently riddled with student errors.) So we should teach repeated multiplication to kids explicitly, and this should happen some time in (how about) 6th Grade.

But the foundations for exponentiation need to be laid several years earlier in a child's education if we want to really help our kids avoid all those pesky, Algebra-killing errors in high school.

Friday, November 8, 2013

Yarmulke Tales!

If you've ever seen me in person, you know that I wear a yarmulke on my head. Here's how that's impacted my life, lately:

  • One of my 4th Graders pulled on my arm during dismissal. I looked down. "Mr. Pershan! Mr. Pershan!" She pushes a tiny girl in front of me, and there's pen doodles all over her hands and face. "This is Sarah. She's not Orthodox, but she's Jewish. And she's really weird."
  • During lunch duty, some kid who I don't know wished me "Shabbat Shalom!" on his way out of lunch.
  • A bum threw a penny at me on the subway.
  • A 2nd Grader who I don't know (standing in line, waiting to get into the music room) asked me why I was wearing a yarmulke on my head. I shrugged. He said, "It's because you're Jewish!" Bingo!
  • A 3rd Grader who I teach ran up to me before class. "I KNOW WHAT YOU BELIEVE" he said as he pointed and grinned at me. I asked him what he meant. "I KNOW WHAT THE THING ON YOUR HEAD IS!!!!!"
OK, and this last story is from the beginning of the school year. I had to miss a day for a Jewish holiday (Sukkot) and I told my 4th Graders that they could expect a sub the next day. This was on the second week of school, so I told them that I was a religious Jew and that the thing on my head, etc. Then I told them if they ever wanted to ask me questions about it that they should totally feel free to. 

Yeah, of course they had some questions. "Do you always wear it?" "Do you wear it when you shower?" "What happens if you take it off?" "Is it something that gets passed down from generation to generation in your family and it's actually really old?" (No.)

I had been teaching at a Jewish school for the past three years, so none of this ever happened. It's weird, but pretty adorable.

Tuesday, November 5, 2013

The Double Ferris Wheel

It's really hard to find models and contexts for Unit Circle Trigonometry. Like, really tough. The one go-to that everybody uses is the Ferris Wheel, which is great, but it's practically all that we have.

"Oh, no!" you'll say. "What about all that astronomical stuff? What about the percentage of the moon that's visible on a given night?"

Well, two things. First, why would you want to model the percentage of the moon that's visible with a sinusoidal function? If I really want to know what the moon's going to look like on January 17, 2015, then I'm just going to subtract a bunch of 29.5 day intervals from 1/17/2015 until I land back on my data.[1]

The second problem is this:

The moon's visibility isn't sinusoidal. Then again, of course it isn't. If it were really sinusoidal, then its orbit would be circular.[2] 

What is a Trigonometry teacher to do? Practically nothing interesting in the world shows truly circular motion. (Oh, pendulums?) And even the things that do show circular motion are rarely worth modeling.

We're stuck with Ferris Wheels. So, find cooler Ferris Wheels.

We watched the video, and I asked them to graph height vs. time on a Post-it.

I tossed these under the doc camera, and we narrowed down our options. I chose two at a time and asked the kids to compare them, which usually resulted in us throwing out one of the graphs. When we got stuck, I suggested that we separate the two wheels and figure out an equation that determines the motion of each. After some thinking, a kid suggested adding the equations together. I asked her to show us what she meant, and she produced this:

This lesson was fun, tough, and a genuine context for some Daily Desmos-style sinusoidal modeling.


This year I find myself just going nuts with Ferris Wheels and rides. We've already studied square-shaped tracks and rides, inspired by this lovely visualization:

What's next? There are a lot of rides out there, but many of them seem to be versions of this Double Ferris Wheel. Maybe the next step is to get weirder. Like, what sort of ride would have this height graph?

I'm running out of rides. Ideas? 


[1] In other words, all you need to model is the periodicity of the moon's cycle. There's nothing that pushes people or students to pay attention to its sinusoidal nature.
[2] In class we model the visibility of the moon, and I tried to escape these problems by asking "Is the curve of the moon's visibility sinusoidal or not? Is it like our Ferris Wheel's motion, or is it a different pattern?"

Saturday, November 2, 2013

The Six Acts of a Mathematical Story

Act One: Introducing The Central Conflict

"What's the most ridiculous looking book I can walk out of here with?" I asked. The librarians were incredibly helpful. And I lugged this enormous dictionary back to my classroom and hid it in a filing cabinet.

I told the 4th Graders that I had a surprise for them. In the time it took for me to pull this monstrosity out of my filing cabinet and drag it to the middle of the room, the kids already started asking how many pages long it was. I didn't say a word. I just went to the board and started recording guesses.

Act Two: Confronting The Central Conflict

The first round of guesses were all in the 10,000s, so I took a folder and stuck it after the 100th page. I told them where I'd placed the folder, and we took another round of guesses. The kids seemed split on whether this data made their initial guesses too high or too low. (You can see the lines connecting the initial guess and the second guess.)

That was interesting, so I moved the folder to the 500th page and took a final round of guesses. You can see those in the third column.

Act Three: Resolution of the Central Conflict

[Insert picture of last page here. I'll get it when I get back to school.]

The dictionary ends on page 3,210, and we marched that page around the classroom. (Wikipedia says that it has 3,350, though I don't exactly know how they're getting that number.)

Act Four: Those Clips That Sometimes Show Up After The Credits?

"What would be some interesting math questions that we could ask about this book?"

"How thick is each page?"
"How many words are defined in the dictionary?"
"How many words are in the dictionary, including everything?"
"How many letters are there in the dictionary?"
"How many pounds of ink went into that dictionary?"
"How many atoms are in the dictionary?"
"How many of those Arabian Nights books on the desk would fit into the Big Book?"
"How many words are on a typical page of the Big Book?"
"How many words are there that are not defined in the dictionary?"
"How many pixels is this on a computer screen?"
"How much room is there on each page?"
"How long would it take to read the entire dictionary out loud?" [My question.]
"How many words long is a typical definition?"

The bell rang, and the kids went home, or to some other class, or whatever it is 4th Graders do when they're not in my classroom.

Act Five: Falling Action? Rising Action???

Everybody got a dictionary page. I photo-copied them and handed them out.

How many words are defined in the entire dictionary? What information are we going to need to answer that question? If we could figure out how many words are defined on a typical page, then we could do some crazy multiplication to settle the bigger question.

There's our line-plot. Most kids figured that, based on the data, either 84, 85 or 86 was the typical number of words defined on a page. 

(One girl -- the duck, actually -- insisted that because she had counted 143 bold words on her page, that this was the typical number. I think that she just didn't like the idea that her counting was somehow in vain. Either that or she was burned out from Halloween.)

Act Six: The Sixth Act

There are around 276,606 words defined in the dictionary, though this is tough multiplication for my kids to do.

I'm having trouble finding out exactly how many entries there are in the 1934 Second International Edition, but Wikipedia reports that the Third International Edition "contained more than 450,000 entries, including over 100,000 new entries and as many new senses for entries carried over from previous editions." So that should mean no more than 350,000 for the Second Edition, right? Not bad, kids.

With the rest of class we tried to knock off a few more of their questions. Did you know that linguists estimate that there are roughly 1,000,000 words in the English language? We figured out (roughly) how many words are not defined in this dictionary.

We also had a fun conversation about how to figure out how thick each page of the dictionary is.

[Six Acts]

There you have it. This was a ton of fun, and I certainly recommend checking your local library for a book as silly as this one. I'd love to help those without access to a physical copy to play with this. I uploaded some of my pictures to 101qs, but please let me know if you think of certain pictures that would make this problem more useful to you.