MTBoS Mission 1: How rich is rich?

I easily careened through Sam’s detour barricades, first by believing that my rich problem isn’t rich enough, and then by lacking a sense of personal ownership of my classroom, which was easier to do since I’m not yet a classroom teacher. (I’m certified and working as an edtech in a middle school, sometimes along with a dedicated seventh-grade resource room teacher.) I like this prompt because it has helped me realize that richness is in the mind of the beholder.
Our kids are wounded; they rarely or only barely get to gleam. Maybe a few school years spent mostly addressing insufficient skills development via flat, skills-based worksheeting and workscreening exercises have persistently dulled their appetite for mathematical curiosity and wonder, two important dispositions that are hard to develop in the absence of math success (see: vicious cycle). Recently, our number rockets have been struggling to understand area and perimeter. They have made progress, but dealing with the area of “chipped” rectangles is still a challenge for them. This activity aims to 1) promote analytical problem-solving skills and 2) focus thinking on the concept of area. It includes a kinesthetic activity because that’s important, helpful, and always inviting. Students will be given a chipped rectangle area task and a set of 10 rectangles cut out of grid-paper. They will begin by finding the component paper rectangles that fill the task shape’s area (for two of the tasks, there are two sets of component paper rectangles), and continue as guided.
I hear the questions because I have also asked some of them: Isn’t this too guided? Wouldn’t it be better, more open-ended, and potentially more empowering, to show less and allow students to discover processes and relationships by themselves? Yes and yes. But. Kids will not benefit from invitations that they are too scared to accept, or that they are too nervous to like. An image comes to mind: once in a Dallas, TX neighborhood, I saw a young child riding her small bicycle. It had only one training wheel, the other presumably having been removed after some time spent riding with both. Brilliant, I thought; a brilliant mix of guided and self-controlled exploration.
So, my intent is to guide our students to experiences of success with mathematical reasoning and to nudge them into developing the habits (mathematical practices) they will need if they will ever want to risk looking for mathematical structure, and maybe there find beauty.

[Materials here: ten sets of constituent paper rectangles for the three rectangle problems.]

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