At NCTM in April, Dan Meyer was posing some tough questions about math teaching brought up for him by a really cool interactive article by Bret Victor. Something about the article reminded me of a co-teaching experience I’d had in a 5th grade classroom recently, and reflecting on that experience helped me think about how I’d answer Dan’s question, which was something like, “what is the role of math teachers when technology can do what it does in Bret’s interactive article?” It might help to realize Bret Victor is the man behind the Kill Math project.

The Story
So I was teaching 5th grade kids about area and perimeter using this scenario: you have 36 meters of fencing and want to build a rectangular frog pen using all of it. What are some different pens you could make? If each frog needs 1 square meter of space to flourish, how many frogs can your pen designs hold? Which design holds the most?

One traditional model of teaching suggests that what’s hard for students when solving word problems is getting rid of the fluff and decoding the underlying abstract mathematics hidden in the context, and that if the teacher can restate the problem in mathematical language, it will support the students to solve successfully. Here’s what I observed when we used that model:

The next period we tried an alternate model, in which the context was used to elicit the students’ concrete ideas, and the concrete ideas were valued. The teacher helped the students organize their ideas and look for patterns. In short, the teacher avoided abstraction that the students didn’t suggest, while supporting organization, pattern recognition, and referring back to the concrete.

Once we established that when frog farmers say “pen” they mean fenced-in-space-for-keeping-animals-safe, not ink-based-tool-for-writing, there was enough going on in the context that the students had some ideas about how to draw different pens, check if they fit the farmer’s specifications, and how to try to fit the frogs into the pens.

My Reflections
As the Common Core points out with the Mathematical Practice “reason abstractly and quantitatively” one piece that’s really at the heart of mathematics is moving among and making links between different representations of quantity (or shape) and relationships among quantities (or shapes), including abstract representations of the quantities and relationships.

Bret Victor’s work is technology that allows more people to make more of those connections, and make them more strongly better, assuming they can make sense of the technological tools. I think our job as teachers, then, is mostly the job of making sense of the tools: why abstract the problem this way? How does the abstraction map to reality? How does it break? And hopefully to prepare some percentage of our students to be the ones to design and improve these tools and their next generation. It’s really exciting to me to look at his tools; I see them as giving more of my students access to having and sharing really powerful ideas, and I see math teachers as having a role in helping students to become people who can use these tools to solve problems and communicate about them confidently.

I am thinking of a metaphor based on how Blogger or WordPress have changed education in writing: every generation since we were writing by carving wood and stone has faced the challenge of how do you make information legible, useful, engaging, and reach lots of people? The more technology we have, it seems like the more people can try to tackle that challenge, and the less time we have to spend identifying who will be our stonemasons or scribes or printmakers or computer coders and training them in the mechanics. The more time we can spend on the creative, interesting, individual tasks of making each piece of content as legible, useful, and engaging as possible. Again, that’s really exciting to me as a teacher — I get to spend time with students thinking about their ideas, their specific piece of writing (or math) and how best to tackle the messy problems of trying to fit what we generally know to specific peoples’ needs. How fun — it certainly requires both general knowledge of what tends to work (e.g. representing change over time on a Cartesian plane with time as the x-axis and other things on the y-axis) and the habits of mind to apply knowledge and push the envelope (e.g. understanding that it makes sense to ask how we can best show the relationship between time and our unknown variable visually).

A big challenge is to think of what this means for classroom teachers right now, as these tools are being invented. Here are some things that feel really true to me:

• Put the strategic right in the center of “use appropriate tools strategically” and recognize that what we call “algebra” in school is a tool. When is it strategic to use? Why has it had the impact that it has on the world? What’s worth knowing about it?
• Stop telling kids what’s good for them and show them. Trust that quantitative and spatial abstraction is interesting and useful and spend a lot of time generating contexts that show and motivate learning the powerful tools. No kid would ever persevere at piano if they’d never heard music; few would practice scales if they’d never heard for themselves the tricky fast scales hidden inside tough music they’re trying to master.
• Figuring out how to assess students’ ability to move among, generate, and compare different representations and abstractions in the service of solving problems. What does it mean to get good at that? What are the 10-20 big ideas across math at all levels that define just what it is to abstract or quantify a situation (eg the real number line and coordinates which map physical space to quantity, which are at the heart of understanding Bret Victor’s car driving tool)?
• Being clear and honest that fluency and drill and practice and lecture belong in math class but that in the absence of fitting into big ideas about quantity, space, relationship, and representation they won’t serve our students. The reason they won’t is that the better technology gets the less demand there will be for people who are good at number crunching and symbol manipulation and the more demand there will be for people with heuristics and strategies and big ideas for solving particular problems.
• Keep a balance between investigation into pure and applied mathematics. The world needs dreamers and doers in all domains, and they have to start coming from every classroom, not just the demographically college-bound.

My ideal classroom has students working to solve particular problems that I set up for them and using those problems to identify tools they don’t have. If I think they will be able to use them and remember them after hearing them once or twice, I tell them. If I don’t, I set up experience for my students to learn how to re-invent (or invent) them. And then we ask what new questions we generated or if it’s time for me as the expert to define another challenge.

That means needing a deep well-articulated bank of challenges, a sense of their scope and sequence and different paths through them, clear ideas about what mastery means that are aligned with college and business and citizenship demands, and support for effective intervention that supports not just specific tools like using calculators, solving 2-step equations, graphing, and making a guess and check table, but also habits of mind like abstraction and persevering and evaluating for reasonableness.

That’s a big challenge for the designers of curriculum and support material, and one that I think has only sort of been taken up in any really useful way… if it had been, fewer teachers would spend time re-inventing that wheel! I’m excited about the power of online collaboration to help share the materials teachers have invented and reinvented, and really excited about the power of the internet to help us tag, categorize, comment on, critique, and improve new and existing resources.

Hi! Who out there in math-teacher-twitter-blog-land has played DragonBox yet? It’s an iPhone/iPad app that teaches the rules of algebraic manipulation through an intriguing, almost context-less, rule-based environment. You can download it for $2.99 or read about it on GeekDad over at Wired here: http://www.wired.com/geekdad/2012/06/dragonbox/

I’m asking because I’m really intrigued!

First of all, I want to play in the environment (I want to invent a subtraction operation, introduce the distributive property, play with addition of fractions, etc.). What breaks? What becomes less clear/mathematically sound? What is improved?

Second of all, I am really surprised by the total lack of context, especially that there’s no support to think of why we’re isolating the box and why we have to do the same think to each side sometimes and each group other times. Every context I try to associate in my head is confusing/incomplete when we get to higher levels. But clearly there are reasons we add to both sides and divide/multiply every group by the same thing, all about preserving equality. What does equality mean in this game?

I’d love it if you would play the game (with your kids, I hope) and tell me what you think! Is the lack of context a plus? What happens as kids play? What would a “sandbox” area look like?

I’ve been reading some of the SBG (standards based grading) post-mortems folks are posting at the end of the year (like this one and this one). One theme (and it’s come up in my graduate classes to) is that the kids who take advantage of opportunities to reflect and revise are the kids who are already doing okay. Those are the motivated kids, the ones who “get school” and know how to earn good grades, through some alchemy of learning and caring and doing their work and taking notes and studying and getting extra help.

Getting the lowest-performing students in for help and re-assessment/revision is a lot harder. It made me think of “Multiplication is for White People”: Raising Standards for Other Peoples’ Children which argues that kids tune out of school to protect themselves from constant messages of being not good/smart enough. And that anything labeled as remedial is another blow to those kids, not to mention they don’t believe they can get better at anything school-related. Plus, as Lisa Henry points out, a lot of kids in struggling schools have work or family responsibilities during out-of-school time.

So here’s my crazy idea. What if we hired or recruited the lowest performing kids to tutor the middle and higher kids? I know, it probably wouldn’t work because kids know who has status and they’d balk at being tutored by a low-status kid. But maybe they could tutor younger kids or something… Anyway it gives us an opportunity to celebrate the kids least celebrated, to work with them closely on learning habits, and they can tutor by asking teacher-questions, like “how do you know?” or “what does this remind you of?” or “what is your best estimate for the answer and why?” and if they get stumped they can go to Khan Academy or something and show a video (which is what typical peer-tutoring often looks like: watch me while I do this slowly and pause me to repeat when you get stuck). The kids they’re tutoring are the ones who “get school” and they can refer back to their notes or ask to pause the video or do all those other good-student habits, and the low-performing tutors help with persistence and eliciting their tutee’s thinking and asking good questions.

Anyone want to try it?