Parent letters

Here is an email that a friend of mine—father of a first grader in the Minneapolis Public Schools, and math-teacher-on-parental-leave—received from DreamBox, an “adaptive learning platform” for K—8 math.

Congratulations! [Child] successfully completed a group of DreamBox Learning lessons.

Are you familiar with the concept of compensation? This is a strategy that can be used to make addition problems “friendlier”. To use it, just subtract a “bit” from one number and add that same “bit” to the other to create two new numbers that are easier to add mentally.

[Child] learned this strategy by completing a series of lessons using the special DreamBox tool, Compensation BucketsTM. For example, when shown the problem 29 + 64, [Child] turned it into 30 + 63. Towards the end of this unit, [Child] was adding 3-digit numbers with sums up to 200!

On the Run: Friendly Numbers

Take turns supplying two numbers to add. The other player has to make the two numbers into a “friendlier” equivalent expression. For example if you say 38 + 27, [Child] might say 40 + 25, or 35 + 30.

Remember, your encouragement to play two or more times per week for at least 15 minutes each time will help [Child], because knowledge is built most effectively when concepts are presented regularly. You can always check [Child]’s latest academic progress on parent dashboard.

Best regards,
The Teachers at DreamBox Learning

I’ll let my friend say what he found so great here. I agree with every word.

I just wanted to share this Dreambox support email I got today as [Child] was working. This is perhaps a more impressive feature than the software itself.  Not only does it provide a very simple explanation of what skill she is learning, it provides the jargon needed to have a conversation with her: “Bits”. As a middle school math teacher that was always a roadblock for a lot of parents: the language parents didn’t use themselves when they were in math.

Compare against the report I receive on a weekly basis from XtraMath.

Screen Shot 2016-01-25 at 10.12.23 AM

You should know that DreamBox is a for-profit corporation that charges for its services, while XtraMath is a non-profit that provides its services for free while soliciting donations to continue the work.

But here’s the thing: XtraMath is not a free version of DreamBox.

Math fact practice with an emphasis on speed is not a version of conceptual development. It’s a totally different mission based on completely different understandings about what it means to learn mathematics.

DreamBox is trying to help my friend support his daughter’s mathematical development. There is an actionable message: Play with this idea that relates to something she’s been working on.

XtraMath is giving parents “the information they need to know how well their children know their math facts, and the progress they are making toward mastery.” But what are parents to do with that information? How do they support their children in moving forward?

Ultimately, this is a question about data in ed-tech. I’m not here to single out XtraMath; it’s just the case that’s in front of me each and every week. There are lots of ed-tech products out there trying to do what XtraMath is doing.

When kids do things on computers, it is easy to collect data about what they do. It’s easy to turn that into a report or a dashboard.

It is much more difficult to determine what it all means. But data without meaning isn’t useful.

For instance, what does Tabitha’s report above mean? What am I to understand about Tabitha’s mathematical knowledge based on this report? This is the major representation her school offers of her progress in third-grade mathematics. What am I to make of it? More to the point, what is a less mathematically knowledgeable parent to make of it?

But what is a school to do?

I say provide more meaning, less data.

Don’t include free as a primary criterion for adopting materials.

Create and invest in things and experiences that support all caregivers in supporting their children’s mathematical development.

Lay off the speed-based fact practice and find ways get kids talking, thinking, and doing.

And what is an ed-tech company to do?

Invest its limited resources in crafting materials consistent with research on children’s learning and on parent communication.

My favorite quadrilateral

I am a known fan of triangles and hexagons. I have also been having quite a bit of fun with laser cutters at a local maker space.

A while back I wondered what it would be like to decompose a triangle and play with its parts. So I cut up a triangle and got busy.

2016-01-15 21.09.28 copy

My play started simply.

Things quickly got more complicated, with symmetry, patterns, and tilings.

I saw happening in myself what I kept telling people I saw in children at Math On-A-Stick last summer. The longer I persisted, the richer the ideas I had. These are in a bowl on the dining room table (along with my favorite pentagon and some materials for Tabitha’s decimal study), available for play whenever we like.

2016-01-24 11.41.22

You may not have a laser cutter, but you certainly have access to a compass, cardstock, and scissors. I recommend getting down to business so you can play with my favorite quadrilateral.

Project Pentagon

Pentagons are taking over my life.

You may have heard the announcement this summer that mathematicians found a new tiling pentagon. Previously, there were 14 known classes of convex pentagons that tile the plane. Now there are 15. Maybe that’s all there is; maybe there is another class, or even infinitely many classes, remaining. No one knows.

My Normandale colleague Kevin Lee brought some samples of this new pentagon to Math On-A-Stick this summer, mere days after the announcement. This led to discussing the nature of sameness of the pentagons with my father, which led to further reading, and so on…

I am now drawing an example of each of pentagon type using Geometer’s Sketchpad and Adobe Illustrator, cutting them out of wood on a laser cutter, and then figuring out how they go together. No phase of this project is simple.

I consider a pentagon “solved” if I have at least once figured out how it tiles.

I have successfully drawn and cut pentagons 1 through 11. I have solved all of these but number 9.

The project is making me think a lot about learning.

For example, tonight I was working on pentagon number 8. I solved it.

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These sets of four can continue to go together in a way I see and can describe.

But that’s not the only way to view the solution. Maybe someone else solves it using sets of three.

2015-12-16 21.20.15

This is the exact same arrangement—the same solution—organized differently. The threes are meaningful here, whereas the fours were meaningful in the first solution. Which is better? Which is right?

Another solution uses sixes.

2015-12-16 21.21.30

With that set of six pentagons, you can tessellate by translation only. The three pentagons at lower right are the beginning of the next set of six. Each of these has the same orientation as its corresponding pentagon above it. Does that make it a better solution?

I’m thinking a lot these days about the kinds of questions I’ve posed here. I’m trying to sort out my answers to a larger question:

What is (or should be) the relationship between informal outside-of-school math, and school math?

I have given a couple versions of a talk that asks four basic questions about people’s mathematical activity that occurs outside of school:

  • Is this math?
  • Is it school math?
  • Do we value it?
  • Why or why not?

I invite you to join me on this journey.

I’ll keep you posted on the pentagon project.

A new Calculus Activity Builder activity

Let me bring you up to date, in case you have not been following along.

I am on leave from my community college teaching this year, and am working at Desmos remotely from St Paul.

A large chunk of my time involves working on the pedagogy side of Activity Builder, which we released this summer.

Activity Builder lets you build a classroom activity using one of three basic screen types: graph, question, and text with image.

From time to time, I’ll take the opportunity to turn something I’ve done in the classroom before Activity Builder and make an online version. I did that yesterday. (Here is a link if you want to play along as a student—I recommend doing that!)

It’s a simple little calculus activity on the surface. You see a function that is graphed on the coordinate plane, except that parts of the graph are obscured by large black circles.

ivt.1a

There are four such graphs, and I ask the same three questions of each one.

  1. Behind which circle(s) must there be roots for this function?
  2. Behind which circles might there be roots?
  3. Behind which circles is it impossible for there to be roots?

After each round of questions, you have the opportunity to move the circles aside to see for yourself whether there are roots.

This is a little routine I developed as a Calculus teacher to spur conversation, and it contrasts with a standard textbook approach, which asserts the importance of three conditions for knowing there are roots:

  • continuity on the interval in question, and
  • a sign change between the interval’s endpoints

In that spirit, you are told in this activity that the first three functions are continuous. You are not told that the last one is.

In a classroom setting, I’ll discuss these examples once students have worked through them. In that discussion, I want to get students to verbalize the following things:

  1. There are sometimes roots where you don’t expect them (Screen 8).
  2. There are sometimes not roots where it looks like there really ought to be.
  3. If the function starts negative and becomes positive, it has a root.
  4. And vice versa. (Screen 4)
  5. AS LONG AS THAT FUNCTION IS CONTINUOUS!!!!! (Screen 16 for crying out loud)

Only after that am I ready to state the Intermediate Value Theorem.

This activity illustrates a curricular principle I sketched out recently, which is that lessons build on students’ experience, and help them to structure that experience mathematically.

This activity creates an experience for students, and then it’s my job to help students structure that in a formal way—through statement of and exploration of the Intermediate Value Theorem.

I’m not a big fan of providing structure for things students haven’t experienced. Typically they see no need for it, and struggle to incorporate these structures into their view of the world. Also, students end up lacking meaningful mental images for representing and triggering the formal structures.

This is theme that plays out in all of my work, by the way. Math On-A-Stick, Oreos, Talking Math with Your Kids….all are predicated on Experience first, structure later.

 

The tale of Tabitha and the two division problems

Consider these two division problems:

Problem A: 22 cookies. Each kid gets 10 cookies. How many kids can get a full share? How many are left over?

Problem B: 22 cookies. There are 10 kids. How many cookies does each kid get? How many are left over?

These are not copied verbatim from Tabitha’s third-grade homework this week, but the numbers and context are the same. (Forgive me; I didn’t think about the potential for large-group discussion until the homework went back to school.)

The point is this: One of these problems was very easy for Tabitha, and the other was very challenging.

Do you know which is which?

We talked about this on Twitter today. (Click through for some really outstanding discussion….seriously.)

I have written about the two major types of division problems before, and they are relevant here.

Problem A was a snap for Tabitha. She skip counts well, and she is a whiz with place value. How many 10s in 268? Why 26 of course! This is the sort of thing I’m talking about.

So Problem A above is a piece of cake for her. This problem—for Tabitha—is very clearly asking How many tens are in 22? For her, this isn’t really even a question worth asking. Each kid gets one ten. There are two tens. QED.

Problem B doesn’t submit to this strategy in an obvious way. It requires her to keep track of 22 things as they get shared among 10 kids. One for you, one for you, one for you, etc. That’s taxing work, and so it’s a much harder problem for her.

When we discussed this problem together the other night, I made the argument that you use up 10 cookies each time you give everybody one cookie. I wanted to help her see how her strategy from Problem A would be useful in Problem B, while respecting that—for her—the sameness of these two problems is not at all obvious.

What’s the moral of the story? Let me know your thoughts in the comments.

A quick plug for Estimation 180

Estimation is more than rounding.

Most of the time we don’t teach this, but it is.

Tabitha (8 years old) had a homework assignment the other night that asked her to imagine she had $100 to spend in a catalog, and to make a list of things she would like to buy from that catalog. She found the latest American Girl catalog and got to work.

There was a table to fill out with three columns.

  1. Description of item
  2. Actual cost of item
  3. Estimate

A couple minutes later she asks, What’s the estimate if it costs five dollars? Should I write $5.01?

She has discerned that estimate means write down a number that is not the exact value.

But that’s not what estimation is about at all. Estimation is about finding a number that makes sense, and not worrying about whether it’s the exact value or not.

The image below seems to be going nuts on the Internet today (despite my exhortations to the contrary! Oh, Internet! When will you learn to listen to me?)

estimate

“Is this reasonable?” is a great estimation question. Rounding is one way to answer the question. But if a kid can quickly find a number that makes sense and it happens to be a precise number, then we probably haven’t asked a good estimation question. Rather than mark it wrong because the kid didn’t round, we should ask this kid a more challenging question next time.

What does a good estimation question look like? What would be more challenging?

I’m glad you asked.

Estimation 180. Thinking of a number that makes sense is much more interesting when you have to bring your knowledge of the world to bear.

height

Is 75 inches a reasonable answer for the difference between the father’s height and the son’s? Is 75 centimeters reasonable?

Oh! And here’s the first in a nice sequence about numbers of pages.

Kindergarten questions

I am spending a bit of time a couple days a week in kindergarten this year. It was part of the now-changed sabbatical plan, but important to me to follow through on.

Today was my first day. It was awesome.

The young ones are working on patterning. AB patterns, AAB patterns, ABB patterns and ABC patterns. I’ll leave the curricular questions for when I know more. Today I’ll take these activities at face value, which is to say: this is the mathematics these children were working on today.

The children were instructed to use square tiles to make an ABC pattern. If you haven’t spent time studying curricular approaches to patterning in early elementary, this means that they were to use the tiles to make something such as this:

Screen Shot 2015-09-17 at 7.12.35 PM

Color is the only variable attribute of the tiles the children were using.

I had several interesting conversations about this task with children today. The one I want to report is the following.

I made this pattern:

Screen Shot 2015-09-17 at 7.17.32 PM

I asked the girl sitting next to me whether I had made an ABC pattern.

Girl: Yes.

Me: How do you know?

Girl: [blank look; long pause]

Me: Show me how you know it’s an ABC pattern.

She carefully points to each tile, saying one letter per tile in the following way.

Screen Shot 2015-09-17 at 7.20.53 PM

She pauses.

Girl: And you need another C.

Learning is messy—beautifully messy.

I left today with two big questions on my mind; each relating to this exchange, and to others not documented here.

  1. What would these children have done if asked—prior to instruction—to make a pattern with their tiles?
  2. How does this kind of patterning work interact with learning to count?

I hope my readers will see that these are not questions I expect to be answered in the comments. I hope you will see that these are big and important questions worthy of wondering about for days, weeks, and beyond. I hope you’ll join me in wondering about these questions, and the consequences of potential answers to them.

I argued a while back that learning is having new questions to ask. I hope you’ll join me on my learning journey.