# Category Archives: Curriculum

## 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.

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

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.

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.

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:

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.

## Beyond the textbook

In an odd and interesting turn of events, I find myself sitting at MSP waiting for my flight to Washington, DC to attend Discovery Education’s second Beyond the Textbook conference.

The invitation arrived by email out of the blue about a month ago. I recognized the conference from Audrey Watters’ participation last year, but was very unclear on how they found me.

The reply (edited mostly for length) came from Steve Dembo:

I believe real innovation comes from cross-pollination.  I’m a big believer in the Medici effect, which is why I want as diverse a group as possible…It’s not an easy task by a long shot and I spent days researching people outside the folks I know personally or by reputation already.  I don’t know the exact path that led me to you, but I did spend a lot of time finding one person I respected, someone like a Dan Meyer for example, and then searching through the people he followed on Twitter and that he referenced on his blog.  And from those people, I’d do the exact same thing, looking both for ‘influencers’ as well as people who I thought had interesting ideas that might make a meaningful contribution to the conversation.

So where do you fit in?  You used to be a middle school teacher and we’re going to try to key in on 7th grade.  You’re a professor and have a background in curriculum design already.  And most importantly, people that I trust and respect, trust and respect you.  And I refused to hold your Spartan connection against you (I’m a Hawkeye).

That’s impressive. I have certainly met a lot of people who claim to use social media to do something besides communicate directly with their audience. But this guy is clearly the real deal.

As I think about the upcoming two days, I find myself thinking about the idea of a textbook.

I suppose the category of publishing into which Connected Math fits is Textbook, but I don’t really think of it that way.

It has been years now since I abandoned textbooks in my math courses for future elementary teachers.

And I am designing a new Educational Technology course at my college this semester, which I will teach in the fall. I began the formal planning for that course by digging into a couple of the publisher texts for these courses.

I lasted about 15 minutes before abandoning that part of the work.

Textbooks conjure up—for me—pre-digested second hand material. Textbooks assert, summarize and question.

Then I read Mindstorms and found a text that I could use to introduce major ideas in the course.

So for me, it is important for us to remember that going “Beyond Textbooks” does not have to mean abandoning books, nor texts.

For me, going beyond the textbook means identifying the resources necessary for introducing the material students need to grapple with, and to support them in their efforts.

My students would be happy to tell you that I have not found this perfect set in any of my courses, and that some of my courses are in better shape than others. But none of my students (I hope) would claim that they are bored by the text materials on offer, or that the courses based on these materials consist of regurgitated second-hand knowledge.

So as we think about moving beyond the textbook, let’s celebrate the possibilities for student engagement and the liberation of students’ minds. But let’s not forget that texts and books are resources for building knowledge and questions.

In late developing news, Frank Noschese showed up with Heads or Tails Double Stuf Oreos as a gift. Good man.

## Syllabus

Here’s a draft syllabus for March’s online course.

This is the first draft and will not be the official one. Participants will get the official one shortly before the course begins.

In the meantime, have a look and let me know what you’re wondering about.

It’s going to be a good old time.

## Syllabus

The Mathematics in School Curriculum: Functions

Pilot spring 2012

### Instructor

Christopher Danielson
blog: http://christopherdanielson.wordpress.com

### Course Goals

The goals of this course are to broaden participants’ knowledge of (a) curricular approaches to function relevant to the middle school, and (b) the ideas behind the formal mathematical idea of function.

An important assumption behind the content of this course is that item (b) above encompasses both formal/logical components and psychological components pertaining to how both sophisticated and naïve learners think about these ideas.

### Course Format

This course takes place entirely online with no requirements for synchronous participation. Regular, daily participation will be essential but time of day for this participation is at participants’ discretion and convenience. See Principles of the course below.

This course will run using the learning management system Canvas from Instructure.

### Course materials

All materials will be provided as downloads or links through Canvas.

### Principles of the course

• Full participation will mean agreeing to spend about an hour a day for the duration of the course. The “hour” is an average and is at your convenience. Course activities will include working through mathematics tasks, reading articles, seeking resources and participating in asynchronous online discussions.
• But full participation is not just about seat time. It is about committing to learning, and to supporting the learning of your classmates.
• We are here to learn; this will require critical examination of what we think we already know. We cannot be possessive of old ideas—we need to be ready to expand them, to let go of them when necessary, and to welcome new ones.
• We should seek to appear curious, not smart.
• We all bring expertise; we should seek to share ours, and to take advantage of that of others.
• This is not a pedagogy course. We will examine mathematics and curriculum quite closely, but implications for teaching are not the direct product of our activity. Conceptual insight is. Instructional implications will follow. These may require long term fermentation before ripening.
• We should base our arguments and claims on evidence.
• We should ask honest questions, and lots of them.
• Discussions are not ever closed. Continue to contribute to old discussions as we move forward; it would be lovely to have each discussion be a record of our developing thinking.

Approximately one hour per day for the duration of the course is expected. The “hour” is an average and is at a participant’s convenience.

This course is ungraded and not for college or graduate credit.

All participants adhering to the principles of the coruse above and completing all assignments will be issued a certificate for clock hours towards relicensure. Participants requiring additional documentation of their participation should email the instructor with necessary details.

### Summary of activities

Introductory activities: Reading principles of the course, introducing ourselves and exploring the online platform.

Discussion: What is a function? Participants will discuss their own understanding of functions, the ways that they and their students think about functions, and the relevance of these ideas to middle school curriculum.

Tasks: Participants will work a number of paper-and-pencil mathematics tasks involving function ideas. These tasks either come directly from elementary and middle school curricula, or are adapted from them. Sources include Everyday Mathematics, Connected Mathematics and Mathalicious.

Reading: Vinner, S. (1992). The function concept as a prototype for problems in mathematics learning. In E. Dubinsky & G. Harel (Eds.) The concept of function: Aspects of epistemology and pedagogy. Mathematical Association of America.

Discussion: Participants will work to integrate the ideas from the initial discussion with those in the tasks and the reading by considering the question, What images do you carry around pertaining to function? together with the implications of these images.

Task: Participants consider functions graphed in polar coordinates. They begin with a game from Connected Mathematics to develop polar coordinates, and move to simple (i.e. constant and linear) functions.

Create: Participants create a product for public sharing. This may take any number of forms, including (but not limited to):

• a blog post reflecting on experiences as a learner and/or implications for instruction,
• a lesson plan (for any audience),
• an interpretive dance,
• a work of visual art,
• etc.

The exact form of the product is not important. The important thing is that it adhere to the spirit of the assignment, which encompasses these two criteria: (1) it should be made public (i.e. shared beyond the course participants), and (2) it should incorporate one or more ideas of the course pertaining to function.

To complete the course, the product—or a link to, or a photograph or other description of the product—must be submitted through Canvas.

## CMP3 samples are online

If you’ve been following along for a while, you may know that I worked for two years with the Connected Mathematics authors in a revision of the materials responding to the Common Core State Standards.

My work for the project finished this summer. In the meantime the author team has been writing, and Pearson has been continuing development of the commercial product. I understand it may a while yet before we can hold student books in our hands, but electronic samples are now available on the Pearson website.

It was super interesting for me to look carefully through the offerings, as I was present for much of the development of the project. But it’s really hard for me to look at it with fresh eyes.

I’d be curious how y’all think Pearson has managed the tech-development side of things. Be sure to explore the “Online simulation” and report back.

## Systems of linear equations

The way I see it, there are two types of systems of linear equations problems:

1. Those in which each equation represents a function relationship between two dynamic variables. We will call these racing problems, and
2. Those in which each equation represents partial information about two or more static relationships and we seek to infer information from the system. We will call these unknown value problems.

Racing problems are pretty easy to cook up and to make plausible. One person gets a head start, but runs more slowly. There is a greater start up cost for service A, but the unit rate is less than for service B. Et cetera. We make some simplifying assumptions (e.g. that rates are maintained throughout the race), but with a bit of finesse it’s not too challenging to make these seem reasonable, and thus to avoid the dreaded pseudocontext.

But unknown value problems are a different beast. Here are two classics of the form:

Boat in the river. Kerry’s motorboat takes 3 hours to make a downstream trip with a 3-mph current. The return trip against the same current takes 5 hours. Find the speed of the boat in still water. From Bittinger, et al., College Algebra: Graphs and Models.

Juice blends. The Juice Company offers three kinds of smoothies: Midnight Mango, Tropical Torrent, and Pineapple Power. Each smoothie contains the amounts of juices shown in the table. On a particular day, the Juice Company used 820 ounces of mango juice, 690 ounces of pineapple juice, and 450 ounces of orange juice. How many kinds of smoothies of each kind were sold that day? [Table omitted-you get the point] From Stewart, et al., College Algebra, Concepts and Contexts.

Variations involve canoes and known distances instead of known current speeds, or (worst of all) known differences in canoeing speeds, but unknown canoeing speeds. They also include varieties of gasoline sold, together with known totals, but unknown breakdowns. These are all completely phony, and what are we doing measuring boating speeds in miles per hour, anyway?

It took many trips through the land of College Algebra before I could put my finger on the difference between racing problems and unknown value problems. Part of the difficulty is that unknown value problems often masquerade as racing problems, as in the Boat in the River problem.

[Note: This confusion between dynamic racing problems and static unknown value problems may well be what I find so compelling about Dan Meyer's escalator problem, and what several of my colleagues find so baffling and uninteresting about it. Also, these things may be due to other factors.]

When I wrote my Oreo manifesto, I was on the verge of a breakthrough on these matters. And now I offer the results of this breakthrough to you. The key question for me was this: What are some scenarios in which we really do have information about sums of parts, without knowing the values of the parts?

I have two such scenarios, each of which breeds many real-world problems.

### Scenario 1: Nutrition labels

These are the Oreo problems. If we accept—as Chris Lusto has demonstrated decisively—that Double Stuf Oreos are in fact doubly stuffed, then we can use nutrition labels to answer questions such as, Are there more calories in the stuf of a regular Oreo, or in a wafer? What about fat? Nutrition labels give us information about the calorie (or fat) content of the whole cookie; we need to infer the calorie (or fat) content of the constituent parts.

Having mastered that technique, we can move on to Ritz Crackerfuls. The Big Stuff Crackerful has “75% more stuff” in the middle. Again, data from the nutrition label allows us to use a system of equations to infer the caloric content of the crackers and of the cheesy stuff.

Then it’s on to milk. One percent milk has 100 calories per cup. Two percent milk has 120 calories per cup. (These are approximations, of course). So how many calories should be in a cup of skim milk? How many calories should be in a cup of pure milk fat (the answer is surprisingly large)? And what percent is whole milk, anyway, given that it has 150 calories per cup (this one is surprisingly small if you don’t know the answer already)?

### Scenario 2: Prices of mixtures

E85 is 85% ethanol, 15% gasoline and is cheaper than regular gasoline in the Midwest. Regular gasoline, though, has ethanol in it too-typically 10%. We should be able to use a system of equations to compute the underlying prices of pure ethanol and of pure gasoline (again, I get that there are simplifications involved here), and then to predict the price of gasoline with 20% ethanol, which will be required in Minnesota sometime in 2013.

A simpler version of this comes from my trip east this past summer. In rural North Carolina, I found a gas station that proudly announced that one of its two pumps dispensed “Ethanol Free” gasoline, while the other warned that its gasoline contained 10% ethanol. The former was more expensive (and does not exist in Minnesota, which is what made the sign remarkable to me).

At my local butcher shop, 90% lean ground beef costs \$3.89 per pound, while 85% lean ground beef costs \$3.69 per pound. What does this say about the underlying per-pound price of beef fat? How about of pure lean beef?

It occurred to me for the first time last night that I could apply the pricing techniques to Oreos. That is, I began to wonder whether the Triple-Double Oreo is fairly priced. We should be able to infer the price of a serving of stuf, and the price of a wafer, then calculate the expected cost of a bag of Triple Double Oreos. My experience is that all bags of Oreos are priced the same, regardless of contents. So is it fair? I don’t know. But I’m gonna find out.

### Conclusion

So there you have it. Two scenarios, each with multiple examples, in which to situate your unknown value systems of equations problems. You no longer have an excuse for assigning the Boat in the River problem.

I’m watching you.

I’ll know if you do.

### Postscript

A further distinction between racing problems and unknown value problems is that racing problems are usually best modeled with slope-intercept form while unknown value problems are usually best modeled with standard form.

It can be hard to see the Oreo problem as a function relationship (the number of calories in a wafer depends on the number of calories in a unit of stuf? Not really.)

Similarly, it seems weird to describe the running of a race in standard form. $y=5x+20$ can describe someone who got a 20 meter head start, and who runs 5 meters per second. But to rewrite this as $-5x+y=20$ obscures these facts. Why should the sum of the distance and the opposite of 5 times the elapsed time be constant at 20?