How Teachers Use Howard Gardner

The second chapter of Jack Schneider's From The Ivory Tower To The Schoolhouse focuses on Howard Gardner's multiple intelligences theory and how it made its way from Harvard to Pinterest. How did the theory catch on, and why does multiple intelligences theory continue to resonate with teachers when research ideas so rarely do? 

In answering this question, Schneider looks most closely at the rhetorical usefulness of multiple intelligences to teachers:

"The theory seemingly possessed scientific credibility and provided teachers with straightforward concepts and clean language to use in resisting what they viewed as a misguided and top-down policy move." (p.62, see also pp. 57, 64) 

For a moment, though, let's take a different perspective than Schneider's. In addition to whatever ideological value multiple intelligences has for teachers, teachers clearly find it useful in our work with students. Look at Pinterest or Teachers Pay Teachers and you'll see hundreds of pictures, posters and documents attesting to its usefulness. Useful how? It seems to me that teachers primarily find MI helpful in establishing culture and signaling values to students.

Consider the poster at the top of this post, clearly designed and directed to children. Also consider that there is an entire genre of activities and lessons invoking Gardner and intended for the first days of school. Teachers use multiple intelligences as an introduction to a course or as a quick icebreaker.

These activities are put at the front of the course for a reason, and it's the same reason that the posters end up on the walls. Teachers tend to use multiple intelligences not for planning or for assessment but rather to communicate their core classroom values with students: a promise to see each student as an individual, a sincere desire to do what's best for everyone, a commitment to focusing on learning. 

A major theme of Schneider's book is that in order to enter the classroom, a piece of research needs to be transportable -- memorable, catchy, easy to remember, easy to share. His point is that transportability is necessary for catching the attention and minds of teachers. But once we note that teachers use multiple intelligences to signal values to students, we might see transportability differently. Part of the success of Gardner's theory is that its succinct catchiness made it more useful for teachers in signalling their values to students. In other words, we can turn Schneider's idea on its head. Teachers aren't just subject to the memorable idea of multiple intelligences. We are also savvy recognizers of the usefulness of a catchy idea for our work with students.

Schneider doesn't spend much time detailing the usefulness of multiple intelligences to teachers. This is surely because he sees other, more powerful actors at play: policy makers, independent school leaders, PD consultants and authors. Teachers are subject to all these forces, and there isn't much room to discuss the classroom in this truly exciting narrative.

But there might be another factor keeping Schneider's discussion away from the classroom. In his recap of this chapter, Raymond Johnson brings up learning styles:

If there was one thing I wish Schneider would have expanded upon, it would have been the commingling of multiple intelligences and learning styles. It's addressed in a couple of paragraphs but only briefly.

If you're a teacher, you might find Raymond's request puzzling: "What's the difference between multiple intelligences and learning styles?" But the ideas are substantively different. Gardner's theory is just an attempt to describe the nature of intelligence, it's not a theory about how people learn best. (See Gardner's piece, "Multiple Intelligences Is Not Learning Styles.")

Throughout this chapter, Schneider is careful to talk about multiple intelligences, not learning styles. After all, he wants to understand how Gardner's theory caught on so widely. Learning styles has its own history, and Schneider isn't writing it.

But to a large extent it's through learning styles that Gardner's theory has become pervasive in the teaching community. If we look only at multiple intelligences, and not at learning styles, we'll fail to see many of the ways in which Gardner's theory became useful to teachers. In doing so we run the risk of overemphasizing the ways in which teachers are subject to the forces of education, rather than as agents of history ourselves.

Raymond Johnson and I are reading Jack Schneider's new book, "From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education." This post was in response to Raymond's recap of Chapter Two. Previously: Chapter One.

The Hexagon of Proof

Following up on the work of Serra and De Villiers, and in the spirit of recent discussions about the success Bloom's Taxonomy has had in penetrating classrooms, I present the Hexagon of Proof.

There are six components to the Hexagon of Proof. Learning is a messy affair that doesn't follow any sort of strict hierarchy, so a math classroom should involve all six of these aspects of proof. Still, if teachers find that their students are having trouble proving things in some area in math, students may benefit from time spent disagreeing over or debating some related mathematical propositions.

The idea is that the reasons that are needed for proof can be developed through a variety of contexts that kids are more familiar with, such as arguing with each other over something controversial. Once a reason have been exposed, though, the reason can have a life of its own in a proof.

Here's how this might look in class:

Disagreeing - Hook the kids into a disagreement. "Draw two triangles. Measure their angles. Do you think that all triangles have that many degrees?"

Debating - In the face of disagreement, ask kids to defend their views. "Some folks here aren't sure that all triangles will have 180 degrees. Why do you think they do?"

Convincing - Give kids a chance to win over their peers. "And what do you respond to your skeptics?"

Explaining - In the face of agreement, explain why something is true. "So, why do all triangles have 180 degrees?"

Teaching - Explain something whose explanation you don't yourself require. This might involve pretending, as in "We know that all triangles have 180 degrees. How could we teach this to a 4th grader who doesn't know this?"

Proving - Use the traditional language and structure of mathematics to prove that something is true, whether or not its controversial or needs explanation. "Prove that all triangles have 180 degrees."

Bloom's Taxonomy and "The Ease of Appearing to Adopt it Without Changing Practice"

Other aspects of [Bloom's Taxonomy's] pragmatic appeal, and particularly the ease of appearing to adopt it without changing practice, were less intentional.

The taxonomy gained traction in K-12 classrooms because teachers were repeatedly exposed to it, because they believed it made sense with regard to how it addressed issues of student ability, and because they saw it as relatively straightforward to implement. 

Jack Schneider is worried about educational research. He knows why most educational research has no impact on teachers and teaching -- most researchers aren't writing for impacting teaching anyway -- but he wants to understand why, sometimes, research does infiltrate the classroom. In the opening chapter of From The Ivory Tower to the Schoolhouse he looks at Bloom's Taxonomy as an instance when, against the odds, a piece of university research did become widely known among teachers.

In Schneider's analysis, part of what made Bloom's Taxonomy so attractive to teachers was that they saw it as just reaffirming what they were already doing. "Teachers could continue what they had always done, making minor adjustments to the taxonomy, their practice, or both," he writes. This was a crucial aspect of its popularity.

So Bloom's Taxonomy was a hit, a blockbuster. But at what cost? In the spirit of "Bloom's Taxonomy" or "Gardner's Multiple Intelligences" (the subject of his next chapter), let's enshrine this as "Schneider's Dilemma."

Schneider's Dilemma: A piece of research will only became widely adopted by teachers if it is seen as reaffirming their existing practice. But if a piece of research already fits in with teachers and educators' existing beliefs, then how can it improve practice?

Schneider -- clear-eyed and erudite -- sees this possibility, and doesn't see it as majorly problematic. Of course, Bloom's Taxonomy meant many different things to many different educators, and he even anticipates critics who would suggest that "the taxonomy has reinforced the status quo in schools." Despite this, he thinks that Bloom's Taxonomy is a true success story:

Yet whatever the challenges and unintended consequences, the taxonomy has stimulated thought and discussion among teachers over the course of many decades -- a rarity for a piece of educational research. It has given teachers a common vocabulary for talking about educational objectives, served as a framework for considering the arc of student development, changed the way that many teachers think about lesson design, and oriented many teachers away from rote memorization and toward the development of thought.

Maybe. I wish that Schneider had worked harder to make the case of the taxonomy's real impact on the profession. What's the path from a universally-compatible taxonomy to making meaningful change? How did the taxonomy effect teachers' practice if it was adopted precisely because it was seen as fitting with practically all existing teaching philosophies?

I can speculate here. Maybe Schneider thinks that the taxonomy was sort of a subterfuge (a virus?), secretly and subtly altering teacher's worldviews over the years. Or maybe, once it was entrenched in teacher's vocabularies, its meaning could be refined over time, so that the particular advantage of the taxonomy is that it gives teachers a manageable, bite-sized concept to focus on.

Or maybe "blockbusters" don't really lead to much change to the instructional core of teaching. Maybe progress comes through other avenues, and widely popular research will inevitably face some version of Schneider's Dilemma, defanging it on its path to glory.

Raymond Johnson and I are reading Jack Schneider's new book, "From the Ivory Tower to the Schoolhouse: How Scholarship Becomes Common Knowledge in Education." We'll each be writing posts on each chapter. This was Chapter One.

Grading Systems vs. Feedback Systems vs. Incentive Systems

I know that I'm teetering very close to over-saturating the internet with talk of grading and feedback, and I promise that I'll take a step back, but first I want to share a challenge I received on my last post.

Honestly, it's been hard for me to discern what you're trying to write about and what it is you'd like to discuss - SBG as an educational movement? The tension between SBG and grades? Calling on SBG advocates to defend their system so you can understand it/tear it down?

That's mathymcmatherson pushing me to be clearer about my argument. In response, here's a brief summary of where I stand, without much in the way of justification.

• Your feedback system should not be your grading system. SBG is certainly a grading system, but many teachers think of SBG as also providing important feedback to their students. This is not a good idea. 
• Your grading system should not be used to motivate kids. Grades -- and really all extrinsic rewards like badges, points, declarations of incompetence or mastery -- should not be used to motivate your students to learn math, because you want kids to have their own motivations to do good things, and cheap rewards are known to backfire. Yet many teachers use SBG to motivate learning.
• We know what good feedback looks like, and it's very good for learning. Good feedback helps learning by giving kids a chance to reflect and reengage with problem-solving, conversation, and all that good stuff. The Shell Center recommends questions as a particularly effective form of feedback. This is miles away from getting a "novice" on "Solving Linear Equations" or what have you.
• None of this is a reason not to use SBG. My real problem is with the way people think about feedback and incentives. I think that the way people talk about SBG reveals some problematic views on feedback and motivation, but none of this is an argument against using SBG. Use SBG if it helps, but your book-keeping isn't good feedback and you should expect any changes in motivation to come with some troubling side-effects.

I have other things that I believe, but these are the crucial bits.

Why I Find Standards-Based Grading Fascinating

People ask me all the time, "Hey Michael, why do you spend so much time harping on Standards-Based Grading?"*

* No one asks me anything of the sort, we are all insignificant specks of sand in this grand sandbox that is the internet and even shouting doesn't make us heard as much as fulfill our need to feel as if we're heard and therefore of some consequence to the universe.

The short explanation for why I find SBG interesting is that I have strange and somewhat confusing interests.

The longer version goes like this: I was trained to teach on blogs. I cobbled together enough experience working with kids in camps and summer programs to convince someone to hire me as a math teacher, but I surely had no clue how to teach so I responded by doing the same thing you do when you can't remember how to get to the airport: I googled it.  Quickly I landed on Dan and Sam and Kate's blogs and I clicked and clicked and read as much as I could.

Standard-Based Grading was abuzz, and I brought it into my classroom from Day One. That means that I have a different relationship to it then a lot of other people who transferred onto the SBG-bandwagon. I tend to associate SBG with some of the decisions that I made in my first few years teaching, and I tend to be very critical of my first few years teaching. ("You're in your 4th year, michael, it's still your first few years teaching." OK.)

So much for psychology. But lately I've been reading a bunch about the history of education reform, and I'm fascinated by SBG as an instance of classroom innovation. Why do teachers like SBG? Why do districts and administrators like it? What problems does SBG solve? We are smarter, but not much smarter than teachers in the past. So why is this innovation emerging now?

I have no firm answers. But things I've learned? I have things that I've learned.

First, I know that teachers justify SBG in a variety of ways, some of which are actually incompatible with each other. I did some research and collected a bunch of justifications that bloggers and tweeters use for SBG, all collected in this doc. Here's an especially interesting contrast that I found:

• SBG Provides Incentives for Students To Learn More, Through Remediation and Reassessment
• SBG Helps Kids Focus On Learning Instead of on Points

If SBG is helping kids to ignore points, then how is it also using points to motivate learning? Either points are being ignored or points are being leaned on heavily.

The answer, of course, is that different teachers have different reasons for using SBG, to the extent that some people have flatly incompatible motivations for SBG. I recognize this pattern from some of my historical reading. It makes sense, right? An innovation needs a broad base of support to be popular, and if it can draw the alliances of educators with widely varying needs and philosophies then it's more likely to stick. (Of course, it also becomes open to the critique that it stands for very little, but it seems to me that these are competing tendencies in many reform movements.)

By the way I'm obviously not a historian, just some fool who's read a few books. I'm sharing what I'm thinking, not what I know.

Here's the other thing about SBG: it seems to be a classroom innovation very much of our times. We live in the age of Standards-Based Reform efforts, a time when high-stakes testing has made a close correlation to standards a vital concern to all teachers of public schools. I wonder: can this partially explain the broad appeal of SBG? Can it really be a coincidence that SBG emerges in the years of No Child Left Behind?

The story, perhaps, goes like this: in a time when standards-based reforms became a living reality for teachers, a grading system that could more closely align classroom assessments to state assessments starts seeming natural. Because there's widespread dissatisfaction about grading, teachers then use this opportunity to graft on their particular concerns to the innovation in grading. Some teachers wish that there was no grading. Others want grades to be more carefully aligned with incentives. Still others just want to be able to ease communication with parents.

SBG then starts defining itself against "traditional" grading, making it hard to criticize SBG (since everyone has some problem with traditional grading). Since SBG is such a disparate camp, it becomes hard to keep track of what SBG stands for and this aids in its rise, but ends up meaning very different things to very different people, following the same path that "projects" and "problems" and "occupations" and so many other instructional innovations have tread.

So much for amateur history. The last reason why I find SBG so interesting is because I find conflicts between what I'm learning about effective assessment and some especially popular interpretations of SBG. Finding the right tone and angle to criticize a popular ideology is an interesting rhetorical exercise, and one that I think is important because, in my (current) opinion, some of our best have gotten this one wrong in an important way.

Dan is our very best.

Take some of the justifications for SBG that I collected in my doc, put them under the microscope and they end up squirming a bit.

• SBG Helps Kids Focus On Learning Instead of on Points: How? By assigning them points for everything that they know? 
• SBG Provides Incentives for Students To Learn More, Through Remediation and Reassessment: One thing that we know is that extrinsic incentives for good behavior often backfire in remarkable ways. (Click that link to read about how financial incentives for parents picking their kids up from day care on time lead to parents showing up later.) Grades should not be used as an incentive. 
• SBG Assesses A Student On What They Currently Know, As Opposed To What They Knew In The Past: This seems flatly impossible. Grades are static, knowledge is dynamic. In any event, are we really claiming to be able to represent what a student knows in a series of skills and numbers? And what do we care about accurately measuring a student's learning, anyway? Will this help the kid's learning? How?
• SBG Provides Superior Feedback to Students: Superior feedback comes without a grade, so Standards-Based Grading would seem to be a poor candidate for providing better feedback. Maybe SBG gives better feedback than "traditional" grading, but since we know that kids ignore all feedback that comes in anything as easily fixated-on as a number I can't see much advantage to providing many numbers instead of one number.
• SBG is an Appropriate Response to Standards-Based Instruction: We also know that a major problem in math instruction is that it comes across as piecemeal and disconnected. Good math instruction is deeply connected, and I can't see how either Standards-Based Instruction or an approach to assessment that atomizes mathematical knowledge can create the deep web of learning whose creation we're trying to foster.

The sheer variety of things that SBG can stand for makes any sort of criticism very very difficult. SBG can become deeply entrenched. But perhaps precisely because it can be safely defended, SBG becomes an excellent forum for hashing out some crucial disagreements concerning assessment, in general. Criticizing any particular plank of SBG is non-threatening, since defenders have many different planks upon which to shift. It's in this context that we can maybe make some progress, getting people with very different uses for SBG to agree that, yes, students learn more from feedback that is completely dissociated from grades. After all, criticism of using SBG for student feedback doesn't mean the end of SBG and another radical change in grading practice. Maybe this is how we make progress through educational reform, even if not through the educational reforms themselves.

It seems to me that SBG is an innovation with a broad base of support, a loose and shifting definition, and a strong connection to tendencies in the wider educational reform movement. There are a network of justifications for it, and this network provides it with a strong defense as well as diffused results when implemented widely. At the same time, I'm betting that the popularity of SBG gives us a safe context in which to debate the essentials of assessment, and perhaps to change a few hearts and minds on some important matters.

Update: Confused by this piece? You weren't the only one. Here's a clearer restatement of the crucial bits.

How I'm Giving Back Today's Quiz

The kids took the quiz on Friday, and they're getting it back today. Here's how they're getting it back.

Sticky Notes - I don't do this for every question on the quiz. I'm going to pair the kids up and give them time to find any disagreements between their work. Often I bank on those disagreements coming up on their own. I'll write a question on a sticky note if I think kids should spend more time thinking about some problem, and I'm worried that they'll just fly through the question on their own.

In this case, I was worried that they didn't spend time thinking about a careful proof or explanation, so I wanted to slow students down and encourage them to work more on a proof.

 

Their Answers, Collected And Mirrored Back - For this volume question, I couldn't think of a great question to ask on a note, and I'm worried that they will only consider the answers their partners have and end up missing out some of the subtleties of their answers. Since their work was all over the place, I decided to pool all their answers and mirror them back. 

I could have printed and handed this out, but instead I'm going to project it and ask kids to decide which of these answers were correct while they're reviewing.

Nothing - If there's a baseline of good thinking about a problem in the quiz responses, and there's just a few mistakes then I'll either write a sticky note or do nothing. I'll do nothing for a few reasons: (1) I often don't have enough time to write feedback on every question for every kid and (2) if there's a strong baseline of correct answers in class, then I can be fairly confident that important disagreements will come out when the kids are reviewing questions. Sometimes I'll scribble down a note to myself to make sure that a student has had a certain conversation with either me or her partner.

Whole-Class Questions - My kids have such a tough time using similar triangles in problems. Only one kid in my class was able to pull this question off with any confidence.

Instead of going around to groups and pushing them to catalog the sort of relationships between triangles that they might be seeing here, I'm going to do that with the entire class. Then I'm going to encourage them to separate the potentially similar triangles from the main diagram, and then I'll give them time to try this problem on their own. 

So I'm using a variety of feedback and all of it is directed towards giving my kids more time and support to problem solve. This is fundamentally different from the sort of feedback that most folks give, which is directed towards reflection in preparation for future problem solving.

(Related post: Questions Are The Best Feedback)

Creating Trigonometry

Once we started studying the steepness of ramps, I asked kids to compare the steepness of different ramps. Following the excellent CME Geometry some of the ramps that I gave my students had only the angle of inclination given, others had the heights and widths given, and others gave two other sides of the ramp.

The kids came up with three major strategies for comparing the steepness of ramps:

1. Draw all the ramps and find their angles, and then just find their angles.
2. Find the ratio between the height and the width of the ramp.
3. Compare the ramp length to the width.

I asked students whether the first two rules agreed with each other, and they discovered that they did. Huzzah! Two confirmed ways to compare the steepness of ramps.

But we had a major annoyance: how do you find the height/width ratio if all you know is the angle? How do you know the angle if all you have is the height/width ratio?

So we started keeping a conversion chart on the side of the wall. We've been adding a few triangles a day for the last couple of days, but it's time to move this process along.

It's all a little bit half-baked at this point, but here's what I'm thinking:

• In class today, we break into groups and start to make more of the conversion table. 
• We start keeping personal tables instead of the class one, because the list is getting too long. Each group decides how they'd like to organize their list. Do you order them by angles? By ratios? Both?
• Each group creates a few problems that can be solved with their tables.
• Then we swap tables and problems with each other. We try to solve your problems with your tables.
• We compare conversion tables, and then talk about what we'd like to improve to make our stuff more usable.
• Then I hand out a trigonometric values table and ask them to figure out what they're looking at.

If I had to guess, my problem will be that this lags a little bit, so I'm interested in ways of adding some structure to help kids feel like they're moving along at a decent pace. I'm very interested in your help sharpening this plan.

Breaking Handicap Laws

That's an awful wheelchair ramp. That shouldn't be legal!

On your own, write down rules for wheelchair ramp builders to make sure that the ramps are safe. 

If kids are stuck, show them a steep ramp, and ask them to explain what's wrong with it. Push them to generalize: Can you draw a few ramps that aren't allowed? A few ramps that are?

Then I put them into pairs.

Make the most dangerous ramp that you can while still following your partner's rule. Be cruel here: try to break the spirit of the law while still meeting its requirements.

After a few minutes, we pool all of the kids' rules together.

My favorite requirement was that one kid said that the ramp had to "lead somewhere," so that it didn't drive you off a cliff or anything. That's awesome.

Then we looked at the actual law.

With your partner, decide which is tougher: our class' law, or the American with Disabilities Act requirements.

What really stood out to me is how much larger our angles are than the actual requirement. It leads me to think that our understanding of steepness on the page doesn't really match up with our physical experience with steepness. This was a nice intellectual way to realize this, but we should probably find a more visceral way of experiencing it.

What I've Learned About Asking Kids To Pose Their Own Problems

Source: Illustrative Mathematics

Last Year Pershan: "What does the next step look like? What would the 10th step look like?"

This Year Pershan: "What are some interesting questions we could ask about this?"

And here's what the kids came up with:

"What comes before [the first picture]?"

"How many cube blocks would you need to build this [third picture]?"

"What the next one going to look like?"

"I think it's going to be the power of 3."

"What would the _____ step look like?"

"How many squares could go in the 4th step?" (To clarify, he was asking how many squares could fit on the grid of the 4st step in the pattern. So that includes all squares, 1x1, or 2x2, etc.)

---

Analysis:

At the beginning of the school year, I said that I was going to help my students learn to pose their own mathematical problems. As you might predict, this failed repeatedly for me until it didn't, and here's what I've learned since then.

There's an enormous difference between "Any questions?" or "What do you wonder?" and "What's an interesting question we could ask?" There's value in all of these prompts, but when I started this year I wasn't attuned to their differences. "What do you wonder?" makes a play at natural curiosity, and comes off as not so different from "Are you wondering anything?" It's easy for a student to answer this sort of question in the negative, but it would be sort of awesome if they were in the habit of wondering about things all the time. "Any questions?" comes off as concern for the kids, not different from "Can I help?" 

"What's an interesting question we could ask?" is different. It's less about natural curiosity and more about imagination and the way a student sees the mathematical world. Coming up with an interesting mathematical question is often difficult for kids, it can involve a great deal of creativity, and it gives me a window into their mathematical world-views.

I see this as a distinction between "question asking" and "problem posing."

As X gets more familiar, asking questions about X gets easier. This is a bit of a "duh," but at the beginning of the year I was trying to prompt kids to ask spin-off questions at the beginning of studying some object, scenario, or problem. This failed, and I think part of the reason why is that they weren't familiar enough with the sorts of questions we could ask (and answer) about these objects. As the year has gone on, and kids have seen what sorts of questions we ask in class, asking questions has gotten easier for them.

For a couple of kids, this can be a great way to get them to create their own extension problems. For a few -- maybe two or three? -- of my students, I always have to worry about them finishing some task quickly. Worry is the wrong word, maybe, but as I'm planning I'm always thinking about whether I have enough interesting stuff for them to think about. Once we got in the habit of posing our own problems, though, it's been nice to share the responsibility of coming up with an interesting question with the kid. We look for an interesting follow-up problem together, and it feels like a truer collaboration with a student than most anything else that I do.

This has worked better with 4th Grade than with high school, though it's also worked in Precalculus a bunch. I just feel like I should be upfront about this.

A Question That Builds Understanding Of "Power"

Is this a power of something?

How could you change the picture so that it's a power?

---

When I posed this question to students, the first answer that I got was "Yes! It's a power of three!" quickly followed by "No it isn't!" 

The next thing the kids came up with was "Yeah! It's the first power of 54!" This was expected. As part of our work on powers in the past week kids had worked with power patterns, and realized that any number can come at the start of one of these patterns.

Then I asked kids to pair up and to find a way to change this picture so that it was a power of something. Here were some of the answers that the kids gave when we came back:

...a power of 3.

("That could be the 1,274 power of one," B said.)

...a power of 2.

It took us a surprisingly long time to get to this:

Here's what I like about this task:

• It works out the language of "power" without worrying about which power it is.
• It's not calculation heavy, and I think that heavy calculations can get in the way of learning the language.
• It's a nice open question that allowed for a lot of different approaches.

We did this as part of a Quick Images activity, which I learned about from the TERC curriculum. (Video with 1st Graders here.) That's a fun game on its own: flash a dot-image for a second, then ask kids to write down what they saw and how many dots there were in total.

Exponentiation Is Like Place Value, Unlike Addition

You get two digits: "2" and "3," and you get to use them only once. Using the language that appears in school math and these two digits, how many different numbers can you denote?

Here are 16 ways that I came up with to do something with those two digits.

It seems sort of silly to group all of these together. I mean, "32" is an entirely different sort of answer than "2+3."

Let's distinguish between two categories. On the one hand you have the operations, the things that perform some sort of process (or mapping) on two different numbers that yields some third number.

I think that adding, subtracting, multiplying and dividing are all clearly examples of performing some operation on 2 and 3. So all of these go into my "operation" pile.

It really seems to me that "23" and "32" don't belong in this pile. I did some other trick with my starting digits to get 23 and 32. I didn't really perform some process on the digits 2 and 3 to produce a third number. Instead, I used those digits to express a new number. It was just smushing the digits together -- I wasn't combining them with some operation like addition or multiplication.*

* Math Major sez: Oh really? You just performed the mapping (a,b) --> (10a + b). Yeah right, Math Major, yeah right. 

This second trick of mine is more about using those starting digits two express a new number. I'm not even treating my starting digits as numbers in their own right, not really. What I'm really doing is treating them like letters of the numerical Alphabet, and lining them up to "spell" some number.

It's obvious to me that "23" and "32" were produced by "spelling" a new number with my starting digits. I think that I also did this with the decimal point.

I've got two categories now: the results of performing some process on numbers, and the results of expressing some number with digits. 

This leaves 6 of my 16 results uncategorized.

Next up, I want to tackle the fractions. Some subtlety is needed here, because you can see the fraction bar as a division symbol and it wouldn't be a mathematical sin. It's a bit of a fielder's choice, in that sense. Still, I think of fractions as numbers in their own right, not some sort of quotient waiting to be revealed. I'm going to sort them into "numbers that I expressed" instead of "numbers that I got through performing an operation."

That leaves radicals and exponents. And it's the exponents that I'm really interested in.

If you think that "2 to the third power" means something like "multiply 2 by itself 3 times," then I think that this number pretty obviously goes in the "result of an operation" pile.

The alternative, though, is to think of "2 to the third power" as quite a great deal like "23." It's not so much the result of operating on the numbers 2 and 3, as much as it is using those digits to express some number.

So, which pile does it go in?

Christopher Danielson has pointed out that there is no really good way to talk about the "result" of exponentiation. For addition you get the sum, for division you get the quotient, but how do you talk about what you get out of exponentiation? 

I take Danielson's observation as partial evidence that our number language has stacked the deck against thinking of exponentiation as a operation. Instead, the way we talk about exponentiation suggests that it's best seen on the model of "23" or "32," as using digits to express some number.

If this is true, then it partially diagnoses the major problems that students have in learning about exponents. Their teachers have been teaching exponentiation as a process, when it's really a feature of the expressive power of mathematical language.

The Separation Of Learning And Grades Is Antithetical To SBG

Ilana Horn writes that amazing math programs make a clear distinction between doing school and doing math. By making explicit what aspects of "doing school" are necessary for students, these schools are able to separate their impact on doing and learning math.

In line with their goal of increased participation, the teachers were explicit that learning to be a student was an important part of their curriculum, and they came up with structures to support that learning. At the front of each classroom was a homework chart laid out much like a teacher’s roll book, with students’ names in a column along the side and the number of each homework assignment across the top. Although actual grades were not posted, completion of homework was represented by a dot...At the same time that they emphasized traditional student skills like doing homework, they did not confuse failure in class with students’ intelligence or ability.

In this way, teachers were able to help students do the things that are necessary for school, but not let that get entangled with their feelings about math or doing math.

How does SBG compare to Horn's model? SBG is premised on the idea that grades should represent how much a student knows. I'll link to Dan here because this was such an influential post for so many people:

The students record the new concepts and their scores on their Concept Checklist. (Template here.) They have now started a self-monitoring process which will continue throughout the year. This is one of the most beautiful parts about this system, that every student knows exactly what she does and doesn't know.

This runs completely against the separation of learning and grades that Horn advocates.*

*This is what I wrote in the original draft. Ilana Horn was kind enough to clarify that I've muddled her position. Check out her comment below, or her clarification on twitter.

I stopped using SBG because I found that it was suffusing every aspect of assessment and feedback with grades and their negative effects. We should be trying to push grades into a corner, out of sight, but SBG wants us to instead dangle them in front of our kids on a weekly basis.

(Score-keeping: This is a change from my previous position that "SBG is worth it." I no longer think that it is, both because the tendency to too-neatly divide math into standards is an unhelpful way to think about learning and because grades and learning need to be separated. I think that I'd rather have grades be determined by occasional tests and homework than by learning, but here I'm less sure of myself. What I currently do for quizzes is supported by my school's no-grades policy, and I'm sure that's influencing my direction here.)

Questions Are The Best Feedback

The above paragraph appears in every single Shell Center lesson. It's how they recommend giving feedback to students.

"Help students to make further progress by summarizing their difficulties as a series of questions."

They don't suggest that you correct their work for them.
They don't suggest that you give students an answer key so that they can reflectively analyze their earlier work.

What they suggest, instead, is to treat a piece of written student work the same way that you'd treat any student idea. Ask a question, give them some time to stew on it. Give them a chance to improve, not in a week when they sign up for a reassessment, but right then, on the same problem that they had trouble with. Don't take away the chance for them to make progress by giving them the answer key.

The changes that I've made to the way I give feedback on quizzes are an explicit attempt to mimic the Shell Center recommendations. I collect the quizzes, I make a copy of them, I use the copy for my own record-keeping needs, and I give them back their original with questions the next day. I just hope that I'm making Malcolm Swan proud.

They Need To Talk About Powers Before They See Exponents

A power of 5

Is multiplication really repeated addition?

In a recent post, Christopher Danielson argues that this is the wrong question to ask.

It is a strong and presumptuous claim to say what an idea is. In recent years, I have come to an understanding of why repeated addition is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it is.

Similarly, I argue that while there's nothing wrong with thinking of exponents as repeated multiplication, it's not the best way to start thinking about them. Instead, we need a different conception of exponents to build on.

What is that conception? Writing about multiplication, Danielson says "Multiplicative structure is captured better by this idea: A times B means A groups of B."

What's the analogy for exponents? "A to the power of B means, ummm, B nestings of A? B recursions of A? B doublings/triplings of A?"

With multiplication, we can rely on the informal language of groupings to ground multiplication. But every day experience provides less clay to be shaped into the precise language of exponents. Presumably, this is what makes exponents so hard for kids and teachers of kids.

All of this leads me to two claims about teaching exponents meaningfully:

1. "A to the power of B means B _______s of A." Effective teaching of exponents will ensure that kids have something to fill in that blank. 
2. The best noun to fill that blank is simply "powers." As in "A to the power of B means B powers of A." 

(I'd say that a "power of A" is an element in the Geometric sequence of A.)

What follows from this is that we have to teach kids what powers are before we can teach them about exponentiation. 

There are no special obstacles standing in the way of this task, though. "Power" is essentially a new piece of mathematical language, and that means our job is going to be giving our kids lots and lots of opportunities to talk about powers before showing them exponents.

How do you create situations where kids end up talking about powers? That's the most important teaching problem here, and it's the one that deserves careful and creative thought. I'm sure you have ideas -- I'll share mine soon.

(In case you're keeping score: I'm now recanting this post and its enthusiasm about grounding all of this in Geometry.)