Tag Archives: diagrams

Diagrams, week 5

Working through a series of rational number workshops for 3—5 grade teachers. My colleague, the incomparable Anne Bartel, drew these diagrams to help illustrate an important difference in two ways children might solve a problem in which 4 things are shared among six people.

To wit: Many kids will share out halves first, and then deal with what’s left over (top), generating a sum for an answer: \frac{1}{2}+\frac{1}{6} Others will partition each of the wholes and distribute one of these pieces to each sharer (bottom), yielding \frac{4}{6} . Or sometimes \frac{4}{24}. We should be on the lookout for these different solutions and use them to generate mathematical conversation (esp. the importance of saying what we have \frac{4}{6} or \frac{4}{24} of.

Diagrams, week 1

It is officially a thing now to post a photograph from our classrooms every day of the school year. I can’t keep up with that.

But I have written about the importance of differentiating between diagrams and decorations. And I have a swanky new iPad (Thanks TED-Ed!) So I’ll aim to get a diagram from my teaching up here each week. Some will be mine. Some will be my students’. Some may represent thinking transparently. Others may be more challenging to interpret.

All will be examples of representing mathematical thinking with pictures.

Today we worked on the Oreo problem in College Algebra. Specifically, given information freely available on the Nutrition Facts labels on regular and Double Stuf Oreos, and given the assumption that Double Stuf Oreos are in fact doubly stuffed, we asked:

Where are there more calories in an Oreo? In the wafers or in the stuf? And how many calories are in each?

Along the way, I drew this diagram to represent a student’s words. The A, B, C and D were his letters, as were what they referred to. I just drew the picture for all to see.

It’s not just me (Smart Boards)

I’ve been catching up on some podcast listening. Here’s Audrey Watters talking to Steve Hargadon on their weekly podcast back on January 29:

…I think there are lots of systems in play that don’t actually want…I mean the folks who sell you interactive whiteboards don’t really want the content to be accessible elsewhere because then why the hell would you buy an interactive whiteboard?

And here was me recently on the relationship between software and interactive whiteboard hardware:

…here’s something strange about Smart Boards. [The ability to capture student work is] not a feature of the board; it’s a feature of the software. But Smart fancies itself a hardware producer, so it hasn’t designed the software to do much of anything without the board.

If we were designing the ideal piece of software to do what you’re suggesting, I’m not at all convinced that it would require a Smart Board to run on, and I doubt that it would look very much like Smart Notebook at all.

I taught myself to use a Smart Board. I have presented professional development sessions around Smart Boards. I have used Smart Boards in my classes-including College Algebra and math content courses for future elementary teachers.

I’ll say it directly. An interactive whiteboard is a crummy tool, massively overpriced. The software has been designed to sell the hardware, rather than as an excellent interface that stands on its own.

In short, Smart Boards suck.

There is a small amount of meaningful additional functionality that an interactive whiteboard brings to the domain of classroom presentation media. In math, this has mostly to do with manipulating visual images-moving this rectangle onto that one to compare their areas and the like.

Here’s a crummy video of the one lesson I really do want a Smart Board for. It’s from Connected Mathematics: Bits and Pieces II.

[WTF] Understanding student thinking

I debated whether to begin the year on a positive note. I had fun with Oreos (more coming in that department, by the way). But now it’s time to get serious.

I posted this last May:

Alexs Pate, the author of Amistad, visited my college last year to speak about rap, writing and a whole mess of stuff that was on his mind. I ran across a one-sentence note I made during his talk:

Writers need an empathetic imagination with their characters.

Rephrase this as:

Teachers need an empathetic imagination with their students.

Of course I would add that textbook authors need this also.

So I’m using Stewart’s Calculus, Early Transcendentals (7th Edition; don’t get me started on editions). It’s my first time teaching Calculus 2 so I’m reading the text extra closely-in an attempt to model critical reading skills for my students and to imagine what will be easy, helpful, challenging and unhelpful as my students engage with the text.

On the surface, I imagine that about 90% of Calc texts are more or less identical. The curriculum has certainly converged. (heh) Limits, then derivatives, then integrals.

So it’s really only at the level of minute detail that most texts differ. Do these details matter? I’m gonna say yes. We are trying to get students to read the text. In order to convince them that this is a reasonable thing to do, we have to convince them that the text’s author gets them as the audience.

So here’s a thought experiment.

Consider the following problem (an example in the text): A tank of water in the shape of an inverted cone is filled with water to a depth of 8 m. The tank has a base of radius 4m and a height of 10 m. How much work is required to empty the water by pumping it to the top of the tank?

Do you have a picture of the cone in your mind? Have you oriented it in space so that you can begin to label variables? If so, then you’re ready…

  1. Where did you put the origin?
  2. What did you label the axis along which you’ll integrate?
  3. Which direction is positive on this axis?

Stewart gets it wrong on all three questions:

  1. The origin is at the top of the cone.
  2. The vertical axis is labeled x.
  3. The positive direction on the x-axis is down.

Yes, I understand that we’ll get the same answer either way. But if we are trying to teach students, we need to choose examples from which they can learn. This involves understanding something about how they think and then using this knowledge to make good instructional choices.

Nitpicking, you say? But did I mention that this is the SEVENTH EDITION?!? And have you seen the guy’s house?

"I had no idea you could make any money writing books. That was not a motivation at all. It was a surprise, but it enabled me to build this house. And I’ve got to continue to work to pay for the house. The house’s cost ($24 million) is double the original estimates."

I don’t begrudge the man a nice home. But that cone is sloppy writing. When the most commonly adopted Calculus text does such a poor job of meeting students where they are, I get angry.

Maybe we’ll get a better one in the eighth edition.

Now back to work.

Diagrams and decorations

I recall a poster in math classrooms of my youth that implored me to “draw a picture” as part of the problem-solving process. A useful strategy, to be sure. But it turns out that it’s a learned skill.

Pictures that are useful for demonstrating or examining the mathematical structure of a problem are special. We don’t all make them intuitively. I have asked the future elementary teachers in my courses to draw a picture that might be helpful in solving the following word problem:

Tabitha has five baskets of apples. Each basket has eight apples. How many apples does Tabitha have altogether?

I frequently get back something of this form:

I refer to this as a decoration, and I contrast it with diagram. A diagram demonstrates mathematical structure; it represents mathematical ideas differently from a symbolic form. A decoration makes the symbols look prettier or more contextual, but does not on its own represent the underlying mathematical relationships.

We can decorate our diagrams. The inclusion of an apple in the picture does not preclude it being a diagram. But it doesn’t necessarily make the picture useful for solving problems either.

With that in mind, which of the following are diagrams and which are purely decorations?

Exhibit A

Exhibit B

Exhibit C

P.S. Extra credit to anyone who can find or take a photograph of the “draw a picture” poster. I seem to recall it being one in a set of five or so problem-solving posters.

A visual launch-probably interesting only to CMP teachers

Connected Mathematics-a 6th through 8th grade mathematics curriculum-is designed with a teaching model in mind: Launch-Explore-Summary. In my professional development work, I help teachers to understand the teaching model in general, and I demonstrate effective techniques for particular problems.

For a few summers now, I have been demonstrating a Launch for a problem in the 8th grade unit Frogs, Fleas and Painted Cubes. I came up with the idea after leaving the middle school classroom, so I have not used it with students. But every summer, teachers ask where they can find written instructions for it and I have had to say that they do not exist.

My summer teaching partner has reported to me that it has been effective in her classroom, so I am now motivated to write it up and share it widely. In what follows, I will assume that the reader has access to the Student and Teacher Editions of the unit. The problem in question is Problem 2.1 of Frogs, Fleas and Painted Cubes.

The original launch

In the original problem, students consider the effects of a land swap in which a square piece of land is swapped for a rectangular one with the same perimeter. So a 5 unit by 5 unit square, might be swapped for a 3 unit by 7 unit one. Students record their information in a table and look for a pattern in the relationship between the areas of the two pieces of land.

In designing my alternate launch, I wanted to achieve two ends: (1) increasing access for visual learners, and (2) increasing the generality of the results.

(1) was important to me because I have been working in my own teaching on visual representation in mathematics, and because the problem asks us to represent something geometric in a table. It seemed like a natural place to increase the visual components of the problem.

(2) is important as a teaching principle. The original problem asks students to consider a variety of squares, but to always change each dimension of the square by 2 units. Thus a 5 by 5 square becomes 3 by 7, and a 6 by 6 square becomes 4 by 8. But there is nothing special about 2 units. A more general pattern emerges if we consider different numbers of units.

The new launch

For this problem, students will work in groups of 3 or 4. Each group receives a bunch of colored (say pink) inch grid paper and a bunch of white inch grid paper. After setting up the context as in the Student and Teacher Editions, have each student draw a square (with whole number side lengths) on a sheet of the white grid paper. Students should coordinate to make sure that each student has a square of a different size from their groupmates.

Each group is assigned a number from 1 to 6, repeating numbers if there are more than 6 groups.

Using the pink grid paper and the group’s assigned number, each student draws a new rectangle. Say my group’s assigned number is 3. Then I will transform my original square (say it was a 5 by 5 square) into a new rectangle by increasing one dimension by and decreasing the other by 3. My new rectangle would be 2 by 8. This preserves the perimeter. What does it do to the area?

To investigate this question, each student tries to cover his/her original white square with his/her pink rectangle. Students will need to cut these pink rectangles apart in order to cover as much as possible of the white square.

Then each group puts their covered squares onto a large poster paper and these are displayed around the room, in order of the assigned numbers (see below).

Now the class is ready to do the mathematical work of investigating the relationship between the assigned number, the size of the original square, and the area of the pink rectangles. Some possible observations that students might make and questions they might pose include:

  • It does not matter what size square we start with; the difference between the area of the original square and the area of the pink rectangle is the same within each group.
  • The group’s assigned number matters-as the assigned number increases, so does the difference between the areas of the original square and the pink rectangle.
  • The area of the pink rectangle is always less than the area of the original square. We can see this because we never quite cover the white with the pink.
  • Several of these images show a white square peeking out from behind the pink. Is it always possible to rearrange the pink so that a white square peeks out from behind?
  • When will it be possible to arrange the pink so that it is in the shape of a square (with whole-number side lengths)?
  • etc.

After some work observing patterns and asking questions, now students should be ready to make the table in the text. However, students should alter the table to match their group’s assigned number, and they should conjecture how the tables of groups with other assigned numbers should look.