Discovery by Dragging

A few days ago I led a webinar on the Common Core and Sketchpad for Sketchpad beginners, and I showed four Sketchpad activities aligned with both the Content Standards and the Standards for Mathematical Practice. I mixed it up a bit by showing two activities in which students manipulate prepared sketches, and two activities in which students start with an empty Sketchpad window and create their own construction to manipulate and to make conjectures.

The first activity was Quadrilateral Pretenders. Here’s one of its pages—a prepared sketch that students can use to develop and refine their own definition of a parallelogram:

How can you tell the difference between the quadrilaterals that are always parallelograms and those that are only pretending?

In the rest of this post I’d like to reflect on some things that I didn’t discuss explicitly during the webinar. (You may have some comments of your own; please post them! Webinar attendees will note two changes in the sketch: I changed the question on the screen to “How can you…” and I used “Pretend” as the name of the button that makes the quadrilaterals all pretend to be parallelograms again.)

Cognitive Demand and Student Ownership
The questions we ask students, and the information we provide, are very important. The question “How can you tell the difference…” is not a yes/no question, and it’s not even asking students to categorize the figures they see. As a “how can you” question, it’s asking them to describe their thinking, and it’s implicitly asking them to do something. We may have to encourage them a bit, and we might even have to ask a student questions like “What do you think you can do to these figures?” or even “Can you read me the directions out loud?” to get them started. (Or we can offer a hint, as I did with the button above—to be used only if the student really needs it.)

Standards for Mathematical Practice
By giving students ownership of the task and the responsibility of figuring out how to proceed, we’re using this activity to address at least six of the CCSS Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.

These standards remind us of the importance of reasoning and sense-making, of all the factors that go into our students’ ability to give meaning to the mathematics they experience.

Discovery by Dragging
Because students create meaning best through actual experiences, dragging is a powerful way to explore mathematical constructions.
For instance, from the starting position of quadrilateral WXYV, drag the vertices in turn: first W, then X, then Y, and finally V. What do you notice? What do you wonder? Similarly, try each of the four vertices of quadrilateral CDEF. Why do you think some of these vertices behave differently from others? How might you learn more about the mathematics behind these behaviors? These questions emerge out of curiosity about why mathematical objects behave the way they do, and they can lead to investigations that may be even more valuable than the ones envisioned by the original writer of this activity.

Reflecting on the Activity, and on the Webinar
With the pressures we experience as teachers today, we sometimes get carried away by our lesson plans, our objectives, and the timeline we’re expected to follow. I know I did in this webinar: I wanted to be sure I covered four activities in different styles and on various topics. And we have to remind ourselves of the importance of slowing down, of giving students time to develop perseverance, of prodding them to formulate a definition with greater precision, of encouraging them to discover for themselves the structure that relates the different categories of quadrilaterals. In the end it’s the Standards for Mathematical Practice that we need to emphasize; it’s the quest for reasoning and sense-making that we have to instill and nurture through our teaching.

I’m not entirely certain how I’d change my webinar presentation based on these reflections; it’s a tough format, with over 100 attendees and a remote rather than direct connection. But I’d have done better if I’d presented only three activities, and spent more time exploring how best to use the parallelogram activity to address various Standards for Mathematical Practice. This blog post is my effort to fill in that missing piece.

Thanks to Rick Gaston for suggesting the title theme of this post, to Daniel Scher for encouraging me to write it, and to the Math Forum (from whom I shamelessly stole my “What do you notice? What do you wonder?” questions).

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