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.

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.

 

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The Twin Cities Shapes Tour

I recently put out a call for K—2 classrooms in which I could talk shapes with students. As a result each of the next several Mondays (Presidents’ Day excluded), I will be in a different early elementary classroom somewhere in the Minneapolis/St Paul metro area.

Last week I was at two schools: Dowling in Minneapolis and Echo Park in Burnville. I talked with one kindergarten class, three first grade classes and four second grade classes. I have learned a lot.

In particular…

Young children find composing and decomposing shapes to be much more compelling than adults tend to. They nearly all saw the bottom-right figure here as being a square and four circles. Adults can see that, of course, but we are more likely to think “not a polygon”.

On that note, I am now quite certain that we spend way too much time having young children sort polygons from non-polygons. That bottom-right shape has many more interesting properties than that of not being a polygon.

For example, a class of second graders on Friday were variously split on the number of “corners” that shape has. Is it 0, 4 or 8? Second graders can understand each other’s arguments for and against these possibilities.

These arguments can lead to the reason that mathematicians use vertex instead of corner. “What exactly is a vertex?” is a much richer and meatier mathematical question than “How many vertices does this shape have?” But if that latter question only comes up with respect to convex polygons, then it is unproblematic and not interesting for very long.

So imagine for just a moment that the lower-right figure has 8 vertices (and it wouldn’t be too difficult, I now believe, to get a classroom full of second graders to agree to this perspective, whether it agrees with the textbook definition of vertex or not).

Now kids can work on stating exactly what makes a vertex.

And what makes a vertex is going to be awfully close to what makes a point of non-differentiability (large point at apex of figure below).

I’m telling you: in twenty minutes with second graders, we can get very close to investigating things that are challenging for calculus students to describe. My point is that second graders are ready to do some real mathematics, and that sorting polygons from non-polygons is not the road to it.

Other things I found interesting:

• When kids give us something close to the answer we expect, it is easy to fool ourselves into thinking they understand. Example: on the page below, one boy said about the lower left shape that “if you tip your head, it’s a square.” A couple minutes later, it occurred to me that there might be more to the story. I asked whether the shape is a square when your head isn’t tipped, or whether it only becomes a square when you tip your head. He confirmed that it’s the latter.

• Another second grade class was unanimous that the one in the lower right doesn’t belong because it’s not a square. When I asked “is the lower left now a square, or does it only become a square when you tip it?” the class was evenly split. This was surprising to both me and the classroom teacher.

• Diamondness is entirely dependent on orientation in the mind of a K—2 student.

• The 1:1 correspondence of sides of sides to vertices in polygons is not at all obvious to young children. I sort of knew this but saw it come up again and again in our work.

• A first grader said that the spirals below didn’t belong with all the other shapes we had seen that day because “you can’t color them in”.

Even the unshaded ones that had come before could have been colored in, you see. These spirals you cannot color in even if you try. What a brilliant and intuitive way into talking about closed figures—those that can be colored in.

 

The hexagons are here! [#nctmnola]

Forgive the delay. Here are pdf files of the hexagons we built for use in my hierarchy of hexagons lessons. You should be able to open and edit them in Adobe Illustrator. Consider them CC-BY-SA.

Set 1 (pdf)

Set 2 (pdf)

Shout out to former students Jen Carlson, Nadaa Hassan and Brenna Magnuson for collaborating on these.

Hexagons by Christopher Danielson is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Update: Below is the current complete set, with added hexagons from former students Ruth Pieper, Brandon Schwab and Mona Yusuf.

The latest “Common Core” worksheet

You have seen this on Facebook.

My annotated version (Click to enlarge)

Ugh what a mess.

Please share the annotated version widely.

I’ll say what I have to say (comments closed) and move on. If you wish to discuss further, hit me up on Twitter or pingback to the blog. Want to talk in private? Click the About/Contact link up top.

Also, Justin Aion—middle school teacher extraordinaire—wrote up his views on the matter. You can read them over in his house.

Here goes…

The intended answer

Dear Jack,

You only subtracted 306 from 427, not 316. You need to subtract another 10 to get the correct answer of 111.

Sincerely,

Helpful student

The purpose of this task

I cannot say whether this was the right task for this child at this time because I do not know the child, the teacher or the classroom.

I can say the following:

• Analyzing errors is a useful way to encourage metacognition, which means thinking about your thinking. This is an important part of training our minds.
• The number line here is a representation of a certain kind of thinking—counting back. The number line is not the algorithm. The number line records Jack’s thinking. He counted back from 427 by hundreds. Then he counted back by ones. He skipped the tens. We can see this error because he recorded his thinking with a number line.
• Coincidentally, the calculation in question requires no regrouping (borrowing) in the standard algorithm, so the problem appears deceptively simple in its simplified version.
  
• This task is intended to help students connect the steps of the standard (simplified) algorithm with reasoning that is based on the values of the numbers involved. Why count back by three big jumps? Because you are subtracting 300-something. Why count back by six small jumps? Because you are subtracting something-something-6. Wait! What happened to the 1 in the tens place? Oops. Jack forgot it. That’s his mistake.

So what?

The Common Core State Standards do require students to use number lines more than is common practice in many present elementary curricula. When well executed, these number lines provide support for kids to express their mental math strategies.

No one is advocating that children need to draw a number line to compute multi-digit subtraction problems that they can quickly execute in other ways.

The Common Core State Standards dictate teaching the standard algorithms for all four arithmetic operations.

But the “Frustrated Parent” who signed that letter, and the many people with whom that letter resonated, seem not to understand that they themselves think the way Jack is trying to in this task.

Here is the test of that.

A task

What is 1001 minus 2?

You had better not be getting out paper and pencil for this. As an adult “with extensive study in differential equations,” you had better be able to do it as quickly as my 9-year old.

He knows with certainty that 1001 minus 2 is 999. But he does not know how to get the algorithm to make that happen.

If I have to choose one of those two—(1) Know the correct answer with certainty based on the values of the numbers involved, and (2) Get the correct answer using a particular algorithm, but needing paper and pencil to solve this and similar problems—I choose (1) every time.

But we don’t have to choose. We need to work on both.

That’s not Common Core.

That’s common sense.

[Comments closed]

Brief thoughts on being ready for calculus

A smart friend (whose permission I have not asked) read an article of mine that will be published in Mathematics Teaching in the Middle School sometime soon. The article is based on my NCTM talk last spring, titled “They’ll Need It for Calculus”.

This friend asked by email:

For clarification: are you arguing that the sorts of problems that you point to will help students better understand calculus, or that these sorts of problems will help students do better in their calculus classes?

I was pretty sure that you were making the first argument, but not the second.

My reply, which I stand by, is this:

That these two things are different from each other is a pretty damning critique of the whole affair, is it not?

You know what will help them do well in their calculus classes? Memorizing about 20 of these: