I was really pleased to read Jo Boaler’s latest paper, “Seeing As Understanding: The Importance of Visual Mathematics for Our Brain and Learning.” You can download a copy at the youcubed website.
In her paper, Jo and her colleagues make the case for helping students “be strong with visuals, numbers, symbols and words.” The paper ends with a call to action “to expand the ways we think about mathematics, and to teach it as the visual and multidimensional subject that it is.” I think that the paper is an important read with a compelling message. And a bonus is that it includes specific suggestions for putting research ideas into practice.
The border problem is included in the paper (see page 9) as an example of how to teach students “visual algebra through pattern study and generalization.”
The border problem is one of my all-time favorites, an investigation that I’ve introduced to students many times in classrooms in grades five through 8, typically over a span of several days. If you’ve never tried it, I think that you and your students are in for a mathematical treat.
For a quick visual overview of the lesson, watch this video of me teaching the border problem to Cathy Humphrey’s middle school students. It’s an oldie (as you can see from my hair color). (The video clip is included in the three-DVD set, Mathematics for Middle School, and is included with permission from ETA hand2mind, Inc.)
Also, to provide you support for thinking through the investigation and planning instruction, I’ve made a five-day Border Problem lesson plan available. This is an excerpt from A Collection of Math Lessons, a Math Solutions resource that I wrote with Cathy Humphreys. The resource is also an oldie—first published in 1990—and the chapter presents a detailed vignette about the border problem that includes specific teaching suggestions along with samples of student work.
I hope this serves you and your students. I’m interested in hearing how the investigation goes in your classroom.