Mathematical Tools

It's so interesting to me when micro-themes start to emerge from my Twitter feed. Most recently, I've noticed many of my tweeps grappling with the use of mathematical tools, both the ones we would consider more traditional like hands-on manipulatives, and the high tech devices, apps, and programs that are more recently showing up in math classrooms. This morning, Malke Rosenfeld posted this question:
This seems like a simple question, but it's an incredibly important one to consider. I think about the word "tool" and how it's really the name of a category of objects we use in certain ways to achieve certain outcomes and not a precise description of any one object. When it comes to math, what makes an object (or representation, or strategy) fit into the category of "math tool"? My simplest answer is that if we are able to use something to support our mathematical thinking in a way that wasn't possible without it or in a way that makes a strategy more efficient or a concept more clear, it's a tool.

It's not just the intent of an object that makes it a tool, it's its actual use. I've been in plenty of classrooms where fraction tiles, connecting cubes, pattern blocks sat on shelves not being used at all. On the flip side of that, I have been in classrooms where the use of a high tech "tool" has been applied with intent to further student thinking, but in a way that actually undermines it or unnecessarily complicates it (some interesting commentary from Tracy Zager on that here).

My assistant superintendent always says, "Whatever you do, know why you're doing it" and this idea of intent is the foundation for the use of math tools, both low and high tech. I would add to that saying "know why you're doing it, and evaluate whether your intended outcome is the actual outcome." This guides my use of math tools with students. I recently had the chance to teach a week of Summer Math Camp (you can read more about that in Jamie Garner's blog here) where we zeroed in on addressing unfinished learning in the area of fractions for rising 4th and 5th graders. Fraction tiles, both student made and pre-created and labeled, were an important tool during that week. Our intent was to build students' conceptual understanding of fractions by creating hands-on experiences with building and reasoning about fractional relationships. The outcome blew us away. Our students showed immense growth by the end of the week, and I attribute much of that growth to our use of fractions tiles as a tool. It transformed our instruction in a way that would have been impossible without it. On the other hand, I shared with Tracy Zager that I recently watched a 4th grade class grapple with completing a multi-digit addition problem on devices that required them to use a mouse to hand-draw numbers while the rest of their class watched (the teacher was displaying responses on goformative). It was painful, to say the least, with the use of technology complicating what should have been a straightforward task for 4th graders.

So this brings me to some of my newest learning related to tools, the SAMR Model. The SAMR model was developed to help teachers gauge how they are implementing technological tools into their instruction. It stands for Substitution, Augmentation, Modification, and Redefinition. There's a great blog and intro video here to grow your learning on SAMR. The model distinguishes between tools that enhance our instruction vs. those that transform it. I wonder about how we might use the ideas behind the SAMR model to help us make decisions about the lower tech tools we employ in our math instruction and when we might switch them out for a high tech tool, or evaluate whether a tool has actually been used in a way that enhances student understanding.

I look forward to continuing this conversation! What are your thoughts?


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