## The future of math

Nothing important here. But in case you’re into this sort of thing, we had some fun on Twitter this morning imagining the future of math. Click on through if you like.

## 5 reasons not to share that Common Core worksheet on Facebook

You are browsing Facebook on a Sunday evening. Someone has shared a baffling piece of math homework that was sent home with their child. Accompanying commentary bemoans the current state and future trajectory of mathematics teaching and learning, and lays blame for this at the feet of the Common Core State Standards.

You are baffled by the worksheet too. You are about to click the Share button.

But here are five reasons that’s a bad idea.

### 1. Credentials are not a trump card.

Almost invariably, the parent who shares the worksheet cites a degree as a credential for the critique. “I have a Bachelor of Science in electronics engineering,” wrote the parent in the recent “Letter to Jack”.

The math you need to know to be an electronics engineer is different from the math you need to know to be a math teacher. I am quite certain that electronics engineers use math that I have not studied, and similarly I use mathematical ideas that they have not studied.

When my teacher friends watch the following video of my son Griffin, they tend to see that a number line would be the right thing to draw to capture his thinking, and they tend to know what is coming when he goes to solve the problem on paper. They tend to describe my son’s work as demonstrating competence or proficiency with subtraction, but suffering mechanical errors when solving with paper and pencil.

Non-teachers who view this video are less likely to see a connection to the number line, and they tend to consider Griffin’s knowledge of subtraction to be weak.

The difference is that teachers have a different kind of mathematical knowledge from electrical engineers. Not necessarily more or less knowledge—different knowledge. This is because different mathematical knowledge is required to do their job.

Deborah Ball refers to this different knowledge as mathematical knowledge for teaching. It is what mathematics teachers know who are more successful in their work. This knowledge includes common errors with standard algorithms, as well as their sources [start at about 2:30 in this video]. It includes common correct, alternative ways of performing and showing computations.

Moral of the story: You may not want to look to an electrical engineer as your primary resource for the current state of math teaching.

### 2. It is probably misinterpreted.

Homework time can be stressful. This is not new to Common Core.

Parents are trying simultaneously to be helpers and enforcers. When a child does not understand what appears to be something simple, tempers can flare

We parents are not at our most rational at these times, and this may prevent us from fully understanding the goal of the task.

When Frustrated Parent wrote to Jack, he committed two important errors of misinterpretation: (1) He assumed that Jack’s method was being taught as a preferred algorithm for subtraction, and (2) He assumed that something unfamiliar to him must be complicated.

These are totally understandable. I do not hold Frustrated Parent in contempt for his frustration.

I am simply asking the rest of us to resist sharing without asking critical questions.

The strategy shown on this particular worksheet is counting back

## The hierarchy of hexagons, continued

In defining Bobs, Stacys, and the like, did you run into situations where your definition admitted shapes the students didn’t actually want. For example, once you defined a Stacey as a hexagon with three congruent acute angles, did you draw some other Stacys and have students blurt out “wait, that’s not a Stacy — that’s not what I meant!” If not, how did you privilege the *definition* over some sense of “I know a Stacy when I see one.” Is it because their definitions were based on some property they liked about one example, rather than trying to say what was the defining quality of some *group* of hexagons?

Yes, yes, yes.

This instructional sequence is all about moving through the van Hiele levels.

An important learning goal for these lessons is for students to separate what it looks like from its mathematical properties. That means we need to talk about this very issue.

Early in the process, a central challenge is identifying precisely the property the student had in mind. This requires a use of language that can be unfamiliar and strange. Does “has a right angle” include rectangles, which have several right angles? Or did you mean “has exactly one right angle”? That sort of work comes first.

Then we look for other shapes that have this property. If those early-discovered shapes violate the spirit of the original intent, the student may object and will be invited to revise. If she wants to add another property, then I will usually suggest that this is a second class of shapes, and that the one she was really after is the combination of these two classes.

So your shape is special because it has all sides the same length AND it has at least one right angle. Let’s do this…let’s call hexagons with all sides the same length equilateral and let’s give a name to hexagons with at least one right angle…Who has a name for us?

A “sally”? Good. From now on, a sally is a hexagon with at least one right angle.

Now this shape is both equilateral and a sally. Let’s give this special category of shape a new name.

Et cetera.

That’s OK on day 1. Our goal on day 1 of the lesson sequence is to have a set of between five and eight named classes of shapes that have interesting interrelationships.

I need to be the judge of when we have enough, and whether they are of sufficiently interesting variety (I have screwed this up before). I reserve the right to add in some properties that I know will be interesting. For example, I make sure concave and equilateral make it into the mix somehow—either by student introduction or by my own.

After day 1, we need to move away from what things look like. We will be operating only on the properties as we have defined them. If we accidentally left ambiguities, we can plug those holes as clarifications. But these definitions cannot otherwise change. It is important to notice that shapes with very different appearances can share important properties.

It is important to notice that mathematical properties of shapes behave differently from the look of a shape.

For example, this semester we defined a class of hexagons this way.

windmill is a hexagon with three acute angles and three angles greater than 180°.

The iconic windmill is this one.

Much later, as we were in the process of trying to decide whether a windmill can be a utah (a utah is defined as a hexagon with two sets of three parallel sides), we happened upon this most un-windmillish object.

In trying to push the limits of windmill-ness, we started to consider shapes that had the right properties but that looked nothing like the original shape.