Thursday, April 14 • 8:00am-9:15am • 3008 (Moscone West)
Visualizing Mathematics Concepts: A Key to Making Connections
Description: Making connections, both within a concept and between concepts, is an important part of developing understanding. Discovering concepts through visual representations can provide a powerful entry point into making these connections. We’ll explore a variety of tasks that can engage high school students to “see” the mathematics.
Here’s an overview of the workshop. I’ll make reference to some slides and activities that are in the links above.
As participants were coming in, I wanted them to work on this Tug-of-War problem from Marilyn Burns. We could have just presented a system of equations in 3 variables, but that would greatly limit the number of students who can access this problem. One could take a symbolic approach to solving this problem, but the context, especially visually, allows one to focus on the algebraic thinking, and actually develop notions of substitution.
Rather than speak to why I think visualizing is so powerful for making sense of mathematical concepts, I wanted the participants to experience it as learners. For more of the why, check out this great new web resource from Jo Boaler.
Visualizing supports making connections – and that process helps to guide the sense-making and reasoning. It is critically important that students own this. Connections won’t be made by telling them or showing them. Consider these types of connections:
- Concepts to context (note: I deliberately didn’t say “real world”)
- Within a concept (e.g. the various representations of a function)
- Between concepts (e.g. number to algebra, geometric patterns)
Pattern exploration, such as the binomial squaring on the left, can be an effective vehicle through which learners can discover mathematical relationships. But we need to go further. Once learners ‘see the what’, we need to ask WHY does this make sense. And this is where visualization offers a powerful way to do so, such as with the area model shown on the right. This is the final version of an area model that students could develop through an inquiry process.
The Find That Square activity (see blm in handout, or use geoboards) is not simply a geometric activity, but it begins that way. Learners quickly find the perfect square areas, but after a few minutes someone will exclaim, “I found 8!” (or some other non-perfect square). This triggers a response in the classroom, “What? How?” Let them struggle – they’ll eventually realize the orientation can be rotated, and then embark on finding all of the other areas they can. In debriefing, we discuss not just which areas were found, but also how they know. For example,
Here are 2 different ways to visualize and area of 5 square units (9 – 4 or 4 + 1). Some use Pythagoras if they know it, but I like to use this activity to launch into a discovery of the Pythagorean Theorem.
The above TI-Nspire activity allows one to vary the dimensions of the green right triangle. In so doing, they notice the square on the hypotenuse always consists of 4 triangles which are congruent to the green triangle, as well as a square in the middle (with area 0, 1, 4, 9, etc). A wonder: how does the area of the blue square relate to the dimensions of the green triangle? Play with this – then generalize the area of the square on the hypotenuse.
Speaking of squares, several years ago I had a visualization idea for simplifying radicals. Rather than expound upon it here, check out my blog post, and scroll down to Exploring Radicals. After the workshop, several of the participants told me this was the highlight for them.
After a quick number trick, which we made sense of using algebra tiles, I used the above slide (which had animations) to introduce the idea that with visual patterns (shout out to Fawn Nguyen), learners can see and represent the patterns in different ways. Out of which came our final exploration, Border Tiles:
What I love about visual patterns is that the visual patterning makes sense of the terms of the symbolic representations. There are many different ways to “see” the number of border tiles and to “see” the number of center tiles, e.g. I see n^2 + n, or n(n+1). Be sure also to check out this from Marilyn Burns.
We didn’t get time to visualize dividing fractions, but it begins by considering the meanings of division. A quotative meaning of division can be very helpful in making sense of this operation. For example, consider how many fourths go into 2/3:
Finally, I’d like to thank Jill Gough for coming to my workshop and creating this awesome sketchnote:
Please feel free to share your own ideas or questions in the comments. I’m always looking for new ways to represent typically abstract concepts more visually.