Tag Archives: curriculum

The goods [#NCTMDenver]

Good turn out for my session Saturday morning (EIGHT O’CLOCK!).

Thanks to Ashli Black (@Mythagon) for the shot of title screen.

I’ll get some more details up here sometime soon. In the meantime, here’s the handout (.pdf). And here’s the slide deck (.zip, and which—to be honest—was just a photo album on the iPad; the simplicity of this was liberating).

Here are Alison Krasnow’s notes from the session.

road.to.calculusOne last thing…this is the absolute best form of session feedback, as far as I am concerned—getting to read someone else’s notes on the session speaks volumes about what participants experienced (in contrast sometimes to what I think we did).

The slides:

UPDATE: This talk has been adapted to a paper submitted to Mathematics Teaching in the Middle School. I’ll keep you posted on its progress.

Beyond the Textbook wrap up

What does this have to do with mathematics?

I had a question at the beginning of the day on Thursday, which I shared through Twitter.

The question got louder in my head as the day progressed. From my perspective, a tremendous amount of time was being invested in designing the platform for a mathematics textbook-of-the-future while not very much evidence was being presented that any of our work reflected knowledge of mathematics for teaching.

My worry continued to deepen that we were designing a better platform for delivering Khan Academy content.

Considering that my critique of Khan Academy has nothing whatsoever to do with the platform, and everything to do with the pedagogical content knowledge of the instructional designer, this was fast becoming a problem.

So I sought out some sympathetic ears in a lull in activity. I hit Frank Noschese and Chris Harbeck with a vulgar version of this question: What in the world does this have to do with mathematics?

Angela Maiers took me up on this question by arguing that, essentially, Mathematics has nothing to do with this, and that’s the way it should be.

In the end, it turns out that the two of us had very similar concerns. An example helped to bridge the gap. That example follows.

At heart, multiplication is about same-sized groups. Whether you write five groups of three as 5×3, 3×5, 5(3) or some other way, multiplication structure is about some number of same-sized groups.

water.bottle.array

We can use multiplication to count the water bottles in this photograph because they are arranged in an array—rows and columns.

But many children do not count things this way.

We can know this by observing children as they count. It is quite common for children to count an array by circling around the outside, or even in a seemingly haphazard order. Even very skilled counters may not notice the unique structure of an array.

A common counting sequence for a child who does not use the rows-and-columns structure of an array

A common counting sequence for a child who does not use the rows-and-columns structure of an array

If they do not notice this structure, they cannot use it.

If they cannot use the multiplication structure of an array, they miss out on an opportunity to use arrays to develop the commutative property of multiplication. One view of the array below is as five groups of three. The other is as three groups of five. The array makes those groups for you, and it suggests that a groups of b will always be the same as b groups of a.

commutative.property

The array support the general argument that ab=ba for all whole numbers.

If they cannot use the multiplication structure of an array, they miss out on an opportunity to use arrays to develop the associative property of multiplication. One view of the collection of shoes below is as four groups of three. A different view is as four groups of six.

shoe.rack

How many shoes? (Credit to my student, Marissa Brown, for the photo. She submitted it for a class assignment.)

If you see four rows of three, then we can express the total number of shoes as (4•3)•2. If you see four rows of six, we can express the total number of shoes as 4•(3•2). Of course these are equal—each of them correctly counts the number of shoes on the shoe rack.

Therefore, (4•3)•2=4•(3•2).

And again, the deep connection between (1) multiplication, and (2) the structure of rows and columns suggests a more general argument.

There was nothing special about 4 rows, nor about 3 pairs, nor about the fact that these were pairs. Anytime we have A groups of B groups of C, we can compute either (A•B)•C or A•(B•C).

That is the associative property of multiplication.

So What?

But what can we use this property for? What good is it?

For one thing, it’s good for mental math.

Quick: what is 6×60?

If you are like most of us, you unconsciously multiplied 6•6, then by 10. You used the fact that 6•(6•10)=(6•6)•10. You used the associative property of multiplication.

And Javier, in an IMAP video, uses it to figure 5•12. Go there and watch for it.

Did you catch his implicit use of the associative property?

He knows that:

blog.beyond.textbooks.3

Or dig this. What is 35×16?

Use the associative property twice:

35x(2×8)=(35×2)x8=70×8=(10×7)x8=10x(7×8)=10×56=560.

This is about number sense; it’s about the numerical relationships that form the heart of mathematics.

But it’s also about the inner working of paper and pencil computation. Let’s say you want to multiply 35×16 by the standard American algorithm. Then you would, at some point, say to yourself “3 times 6 is 18”. But that 3 doesn’t mean 3. It means 3 tens. The fact that you can treat it as a 3 is due to the associative property of multiplication.

Division, by contrast, is not associative. (a÷b)÷c is not the same as a÷(b÷c). This explains why we do not operate digit by digit in the standard long division algorithm.

There is much, much more.

Contrast with what Sal Khan has to say about the associative property of multiplication.

Khan knows this property. But he does not know (1) that an array is an important representation that can help to establish this property, (2) that children need to be taught to see the multiplication structure of an array, (3) that—at 1:55 in the video—he is using the associative property to do the computation 12•30.

Et cetera, and on and on.

This video demonstrates my concern perfectly. Too much attention to delivery method (exercises! badges! energy points! sympathetic narrator!) and not enough attention to mathematics, not enough attention to how people learn mathematics.

Bringing it home

And—to be frank—if Discovery Education doesn’t have someone paying extremely careful attention to all of this throughout their beyond-the-textbook writing process, they’re not going to produce something that will have an impact on mathematics teaching and learning in this country.

But if they do? Perhaps the sky is the limit.

I have been through a brainstorming/prototyping process before that was very much like Thursday’s session. That other one didn’t have the same attention to the possibilities of electronic student materials that this one did. If Discovery can get both parts of this right, they could create some exciting stuff.

I believe they want to do that. I really hope they can.

Course design question

Brian Frank observes:

Lots of physics majors get by with strong algebra skills.

Same story in Calculus, of course. And you can substitute “calculator” for “algebra” to describe a whole mess of other math courses.

Shouldn’t we be ashamed of ourselves when we see this happening? Shouldn’t we be designing courses (and course sequences) in which this is not possible? Isn’t it time we (as a field) stopped living by last century’s textbooks, allowing students to skate by on last century’s skills?

Should we be designing courses that require critical thinking; courses that require students to really, deeply learn the material?

I know Brian’s working on that problem. Shouldn’t more of us be reading his work and taking his example?

This is what I’ve been saying…

I ran across this from Koeno Gravemeijer, quoted by Jeffrey Choppin in the Mathematics Education Research Journal:

Older design principles take as their point of departure the sophisticated knowledge and strategies of experts to construe learning hierarchies… The result is a series of learning objectives that can make sense from the perspective of the expert, but not necessarily from the perspective of the learner.

This is where I have struggled with the Common Core State Standards. That business about complex fractions last spring? This is what I was talking about. My recent fussiness with Hung-Hsi Wu? Ditto. Even that business about logs and quotients; it was about the difference between the views of mathematics content seen by experts and by novices.

The handcuffs of NCLB

While taking a break from the work I was supposed to be doing recently, I found myself typing these words to a colleague:

It occurs to me that if the testing environment were not so toxic right now, we’d be ready for a real revolution in providing content. The technology is available to do a lot of open-source mathematics teaching; the ideas and community exist. But Larry and Stu aren’t really in a position where they feel they have the right/obligation/opportunity to move away from the published curriculum very far. If they toe the line and make modest progress on test scores (or none at all), they are covered. If they bust a move and see anything besides phenomenal gains in their first try, they have reason to be very, very worried. And furthermore they can’t count on collegial and administrative support along the way.

Those are the handcuffs that frustrate me. Without the constraints of NCLB, we wouldn’t need the big publishers much longer.

Within the constraints of NCLB, we get Khan Academy (which gets called ‘revolutionary’ but is not) and teachers are free to reform grading (SBG anyone?). But we don’t get meaningful curriculum change; least of all open-source curriculum.

Larry and Stu (annotated edition)

Aaron B. says:

You’ve got me interested in the opinion that you aren’t explicitly stating.

And Dan Meyer says he was:

Mostly confused. Those two [Larry and Stu] are strange. (via Twitter)

So forthwith I offer a deconstructed, annotated version of the conversation. All links from the original post have been deactivated in order to avoid a second round of massive pingbacks. My comments include a few new links; see the original post for the original references.

Larry and Stu have been teaching high school math for 10 years. They take a load off in the teachers’ lounge at lunchtime.

I am always thinking about what my own students (whether the context is middle-school, college or professional development) take away from the experiences we have in the classroom. How does what I say match what they hear? And vice versa.

In a particularly vivid example, I recently worked with a group of teachers on a Connected Mathematics implementation. Several of these teachers had been to Michigan State and had met the authors of the curriculum. One of these teachers claimed definitively that Betty Phillips had told her that the units in the curriculum could be taught in absolutely any order. Problem is, Betty is our loudest voice on the importance of the order of mathematical ideas and the sequence of units in the curriculum. Surely what Betty said is not what this teacher heard.

So I’m always curious about the relationship between what is said and what is heard. I wonder about the takeaway message, in common business parlance.

Larry and Stu are two typical high school math teachers with enough experience to have settled in to a standard American mode of mathematics teaching, but also not so deep into their careers that they aren’t interested in what else might be out there. I imagine them being good at what they do, caring about students and proud of their work.

Larry: Y’know Stu, I’ve been doing a lot of reading on the Internet. There’s some great stuff out there. And I’m convinced that Math Class Needs a Makeover.

Surely someone has emailed Larry and Stu a link to Dan Meyer’s TED talk. I didn’t see it until February of this year but most of the people I talked to afterwards had already. In May, a friend posted it to Facebook.

Stu: What do you mean, Larry?

L: We gotta make math real-world.

S: Huh?

This is not new language. Many people have burned many calories thinking about what it means to make math “real-world”. The Dutch Realistic Mathematics Education program has done the best job of defining the “real” part of “real-world”. Something is “real” if it can be brought to life in the imagination of the student. If students can operate on it in their minds, it matters little whether the thing actually exists.

Dan recently was on the cover of an Education Week supplement with the phrase, “Real-world learning” in large type next to his photograph. He expressed discomfort with this phrase (calling it “a phrase I try to avoid”). Meanwhile, Karim over at Mathalicious unambiguously defends the real world as “an underused launching point in most math classrooms”.

I understand Dan’s hesitance to adopt the “real-world” label as it has been used and abused to justify all sorts of instructional decisions. It is perhaps only a bit more meaningful and a bit less bullying than “good for kids,” as in “I just want to do what’s good for kids.”

L: OK, see there are these Russian Dolls. And, oh yeah, these escalators. And there’s a whole mess of these graphing stories. Kids watch these videos and they make a graph showing the relationship between time and height, or time and distance, or heck-there’s even one on time and time!

I have indicated on this blog and in the comments on Dan’s that these are all brilliant contexts and great contributions to the field. After I met Dan back in February, my mind was buzzing for weeks. I spent a lot of time sorting out what exactly I could learn from Dan’s work. Ask my colleagues; I wouldn’t shut up about it.

S: Sounds interesting, Larry. Tell me more.

L: Right, so you show this video and kids ask the question. You don’t have to give them some phoney-baloney task, the video leads them right to it. (By the way, do you know what exogenous means?)

I am all about purging the mathematics curriculum at all levels of phoney-baloney tasks. But here is an interesting tension I have come across in following Dan’s work.

Dan is working very hard to help math teachers to understand what it means to pose a problem in a compelling way to students who are not predisposed to seeing the world mathematically. I’m 100% on board with that agenda, and I have learned a lot from it.

I am more critical than he is of the accepted, enacted curriculum in the U.S. I love his visual reformulation of the house painting problem as a counting beans problem, for instance. But I’m not willing to give the math curriculum a pass on the value it assigns to the work equation. There’s the tension for me.

My office mate and I have debated many times the value of this very equation. I argue that the work equation unnaturally emphasizes time per job, rather than the more natural job per unit of time. So if I’m painting a house, I am more likely to be operating on the “fraction of a house I can paint in an hour” than “the fractions of an hour it takes to paint a house”. And if I think about houses per hour, I can’t use the work equation.

I am more interested in eliminating the work equation from the standard curriculum than I am in retooling the problem we use to get at the work equation. Lounge chairs, deck of the Titanic, etc.

And the “exogenous” thing is a good-natured jab at my man, the erudite Karim who wrote the lovely sentence on Dan’s blog, “I think the prohibition on exogenous questions may be a bit too strict.”

S: No idea. But, OK, kids ask the question. Cool. Then what?

L: Well, then you give ‘em whatever information they need to solve the problem. And you let them work, or maybe some days you lecture. But the point is, they’re motivated to do the work because they asked the question and they identify with the context.

Here’s another tension for me, and it has been a common topic of discussion over on Dan’s blog. Let’s say we craft this perfect mathematical problem, kids ask the question and really, badly want to know the answer. Then what?

This question is outside the domain of Dan’s WCYDWT work. He’s working on setting up the task. That’s good. And yet it is a very real problem of practice to identify how students are to learn some mathematics from the task. It requires an entirely different set of skills to move students forward mathematically than it does to pose the problem in the first place.

I think Larry and Stu could very well show the escalators video, get everyone to the point of wondering how long it takes Dan to go up the down escalator, and then not really know where to go from there. If they are going to implement these ideas in ways that support student learning, they’re going to have to improve other aspects of their practice as well.

S: Good. Go on….

L: OK. Then after they answer the question, you don’t just look in the back of the book for the answer, you show the answer in the video. They see the results of their calculations. If they’re right, they get validated. If they’re wrong they know they’re wrong. They get that math solves real problems.

This is a lovely feature of WCYDWT, I think. If the problem has been posed with multimedia, the solution to the problem can be found there too.

By the way, I am frustrated that I cannot get my escalators answer to agree with reality within 3 seconds. Even making the most generous assumptions about things we can’t quite see in the video I’m off by at least 3 seconds. I want to be right!

S: Nice. Do we do videos every day?

L: No. That would get old. Once every couple of weeks or so.

S: What do we do in between?

L: Well, what we’ve always done.

S: Oh, so not all of Math Class Needs a Makeover? Just 10% of it?

OK, so here’s where I tip my hand and get Dan’s goat (see comment on original post). Of course I know that he doesn’t think only 10% of Math Class Needs a Makeover. Of course I know that his laser focus on one aspect of teaching doesn’t imply endorsement of everything else.

But the use of “Math Class” is ambiguous and I have played with that ambiguity in my mind as I’ve thought about Dan’s work. Does “Math Class” refer to a single class session? Or does it refer to the mathematics curriculum in a more general sense? Larry and Stu are sorting through this ambiguity in their conversation.

L: Maybe. Or maybe we flip the classroom; I’ve read about that, too.

There are others on the Internet working on changes in the ways we teach math. Perhaps you have run across Khan Academy?

Most of the discussions of “flipping the classroom” that I have come across have been founded on a set of faulty assumptions about the kind of makeover math class needs. Dan’s scope is limited to one class period, one problem at a time. I agree with his assumptions about how people learn-by being interested in the topic at hand, by being puzzled by something and wanting to figure it out, by connecting new knowledge to prior knowledge, etc.

But KA and much of the larger “flip the classroom” rhetoric is founded on assumptions that students learn best when they are told clearly and repeatedly and by practicing what they have been told. So Khan tells clearly in his videos and he can tell repeatedly because students can watch the video as many times as they like. No need for the teacher here. Then the teacher becomes the tutor who helps students with their practice in real time. If you buy that this is what learning mathematics is about, then flipping the classroom is a substantive improvement. But if you question the curriculum, or if you question the effectiveness of telling, then KA and flipping don’t have much to offer you, I’m afraid.

For me, there’s not much interesting in these ideas. I’m not opposed to them, I just disagree with the assumptions they represent and I don’t see that they offer much for me to learn. So I’m not gonna put in the kind of work I did for Dan.

fin

The main thing to understand from all of this is that I respect Dan’s ideas and work enough to spend time critically questioning them. Not dissenting so much as asking hard questions.

An imaginary conversation

Larry and Stu have been teaching high school math for 10 years. They take a load off in the teachers’ lounge at lunchtime.

Larry: Y’know Stu, I’ve been doing a lot of reading on the Internet. There’s some great stuff out there. And I’m convinced that Math Class Needs a Makeover

Stu: What do you mean, Larry?

L: We gotta make math real-world.

S: Huh?

L: OK, see there are these Russian Dolls. And, oh yeah, these escalators. And there’s a whole mess of these graphing stories. Kids watch these videos and they make a graph showing the relationship between time and height, or time and distance, or heck-there’s even one on time and time!

S: Sounds interesting, Larry. Tell me more.

L: Right, so you show this video and kids ask the question. You don’t have to give them some phoney-baloney task, the video leads them right to it. (By the way, do you know what exogenous means?)

S: No idea. But, OK, kids ask the question. Cool. Then what?

L: Well, then you give ‘em whatever information they need to solve the problem. And you let them work, or maybe some days you lecture. But the point is, they’re motivated to do the work because they asked the question and they identify with the context.

S: Good. Go on….

L: OK. Then after they answer the question, you don’t just look in the back of the book for the answer, you show the answer in the video. They see the results of their calculations. If they’re right, they get validated. If they’re wrong they know they’re wrong. They get that math solves real problems.

S: Nice. Do we do videos every day?

L: No. That would get old. Once every couple of weeks or so.

S: What do we do in between?

L: Well, what we’ve always done.

S: Oh, so not all of Math Class Needs a Makeover? Just 10% of it?

L: Maybe. Or maybe we flip the classroom; I’ve read about that, too.

fin