In my last post, I shared notes from the California Math Council meeting last weekend. I focused on a couple of talks about the use of technology (Asilomar Notes: Tech). Today I write about a different sort of tool, the story table. Shira Helft and Taryn Pritchard’s Asilomar workshop introduced us to this powerful representation of algebraic expressions, which has applications to the study of functions throughout high school. I would not be surprised if it could also be used in middle school, and in the exploration of equation solving strategies.
Here is an example:
Each column corresponds to a step in getting from x to 3|x – 4| – 6. Note that we’ve already improved on the usual approach to order of operations, which is usually discussed as a way to interpret the final expression. This table gives us insight as we go in the other direction, starting simple, and moving towards increasing complexity. As the words “story table” indicate, we are telling the story behind the expression. If that is all we got from this representation, it would be a lot. But that’s only scratching the surface: story tables help us deconstruct almost any function and get a deeper understanding of it. In this example, as we move from one column to the next, we can see the impact of each operation on moving and stretching the graph, and we can see that the symmetry of the graph starts in the third column (why?)
Shira and Taryn gave examples involving linear functions, quadratics, third degree polynomials, trig functions, exponentials, logarithms, and they challenged us to think of more. And they did it effectively: instead of doing a lot of talking, they gave us plenty of time to do our own explorations.
Note that the story table can be used in many ways, depending on which part of the table is revealed, and which part is left for the student to fill out. There is no way to spell all this out in this blog post. If Shira and Taryn ever write this up, which they should do as soon as possible, I will link to it.
For now, I will just say that I learned more about teaching algebra in this one session than I had in any conference presentation in ages — perhaps ever. This is because story tables are not just a good tool to teach a particular topic: they are a good tool to teach many, many topics. As Shira and Taryn put it, “when you can, use a knife”. There are lots of specialized kitchen tools: peelers, corers, slicers, pizza cutters, zesters, melon scoops, and so on. But a knife is a flexible tool, which can be used to do many things, in many situations, often replacing specialized, one-use tool. They suggest that as teachers, we should be judicious in selecting tools, and prioritize the ones with the widest range of applications, the knives.
I will use the rest of this post to discuss tool selection. If you’re familiar with my work as a teacher and curriculum developer, you know that I practice and promote a tool-rich pedagogy. In fact, that was part of my talk earlier in the day at the same conference! (More on the talk: Reaching the Full Range.)
In all my writings about tools, I may not have discussed a good strategy for tool selection, so here we go. Let’s start from Shira and Taryn’s advice. They gave an example of a knife : the rectangle model of multiplication. (I call it the rectangle model, not the area model, because for young children, it is an array of objects rather than a continuous area, but that’s another conversation.) That is indeed a good example, as it spans arithmetic, algebra, and calculus. Here are some other examples of multi-use tools:
- Electronic graphing, which these days is best exemplified by Desmos, and can be used from pre-algebra to calculus.
- Algebra manipulatives, especially the Lab Gear, which covers too many topics to list here, but check out my books.
- The geoboard, suitable to teach about slope, area, the Pythagorean theorem, and more.
- Function diagrams are an unfamiliar representation, and thus they are resisted by some teachers. Still, they are a powerful tool in understanding domain, range, composition, iteration, the chain rule, and more.
By all means prioritize such tools! And definitely add story tables to the list, as they are a truly brilliant and multifaceted tool.
However I don’t agree that we should limit ourselves to such a list. As one masters knife-level tools, there is nothing wrong with also adding specialized tools to one’s toolkit as one gets further along one’s career path. For example, I’ve used pattern blocks to introduce angles, geometric puzzles to illustrate scaling, the circular geoboard for inscribed angles, and the ten-centimeter circle for basic trig (for those last two, see my Geometry Labs.)
There are also electronic tools which on the one hand are fairly specialized, but on the other hand are so powerful they are definitely worth learning about. They are tools not just for the student, but also for teachers and curriculum developers. Here are three of those:
- Snap! for programming and more. I’ve used Snap!’s predecessors (Scratch, and all the way back to Logo) to introduce basic programming ideas, some fun turtle geometry concepts, and some deep math and computer science: fractals, recursion. I’ve also designed tools and games using accessible computational environments (See for example my games, coded in Snap! by Parisa Safa: Signed Number Arithmetic, and Complex Number Arithmetic. Slow to load, but worth the wait.)
- GeoGebra for geometry. Actually, GeoGebra also has graphing, spreadsheet, and a computer algebra system, all in one application. But it is mostly a phenomenal all-purpose tool for geometry, and a great environment to create worthwhile applets that zero in on specific concepts.
- Fathom for statistics and probability.
Actually, listing all these tools gets me back to Shira’s and Taryn’s advice: since you can’t learn them and use them all next week, prioritize! The only reason I have such a long list of tools in my repertoire is that I’m old, and I’ve had plenty of time to learn them, and to develop activities for them. But hey, if you’re planning on being a math teacher for a while, keep an open mind about learning new tools. More tools means a more varied classroom, more visual bridges to concepts, more student initiative and responsibility, multiple representations of the most important ideas, a better way to preview and review material, … I would recommend you start with story tables, but I don’t have materials for you. Let’s hope Shira and Taryn provide those soon!