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?

Several teachers I’ve worked with want to set a culture in their classroom that increases not just the quantity but also the quality of mathematical talk. It made me wonder… how do I recognize good mathematical talk and how would I help students see they’re doing it right?

Good mathematical arguers:

• use definitions to justify their use of mathematical objects & processes (e.g. when deciding if something is a rectangle or not, they cite the definition)
• clarify the assumptions they’re making or the criteria they’re using (e.g. I decided to do this because…)
• are accountable not just to why they think they are right, but also why other solutions are wrong, or why their answer may not be unique
• consider multiple cases
• use simpler examples to justify themselves
• think about whether their answer is reasonable and fits the original context of the problem
• is accountable to everything they noticed in the problem
• uses units and names of quantities, not just values (numbers) to talk about their thinking (e.g. I divided the total number of apples by the number of bags, not I divided 12 by 4 and got 3).

What else?

(Thanks to Ken Templeton, @ktempleton04 for the find!)

I just watched Eleanor Duckworth’s TED talk about “When Teachers Listen and Learners Explain,” especially starting at minute 10:00.

http://www.youtube.com/watch?v=1sfgenKusQk

Please watch it and tell me what you think about this question: what would have been different if they’d used actual chocolate milk?

(and yes, answers like, “it would have been stickier” or “they would have gotten stomach aches” are important answers!)

One thing we’re working on in the classrooms I coach in is how lessons might be formatted differently when you’re helping kids learn a new skill vs. having them practice something they know. Lots of classrooms have the warm-up, work on your own/in a group, go over, exit ticket model with very few variations, whether the “work in a group” phase is completely new learning or a review of a concept you’ve mastered. That can make it hard for kids to:

• realize what you expect. They get in the, “the teacher asked me to work on this so he must think I know what to do. I don’t. Augh!” mindset which leads to blind guessing, giving up, and defeat.
• use appropriate tools. They try to apply algorithms when you want them to guess, draw, make lists, or reason. And they guess, draw, make lists, or reason when you want them to practice algorithms.
• store new learning in the appropriate category. If you think kids are going to discover a new tool from your activity, and kids think they’re rehearsing something they’re supposed to know how to do, the experience will be filed in the “hard stuff I can’t remember” file instead of the “cool things I discovered and want to use later” file.

So… here’s a possibly illustrative pair of lessons.

Lesson 1: Rehearsing old stuff

The background: Students had learned the Pythagorean theorem in middle school, and most had mastered it, even finding missing legs given the hypotenuse and other leg.

The lesson:

• 2-minute drill: students have two minutes to solve a problem or list 3 things they notice about it on an index card. The problem is finding the missing side of a right triangle. Students turn in their index card and the teacher does a formative assessment while the students do the following:
• Go over the 2-minute drill problem:
  • Students hand off the whiteboard pen to one another as they take turns putting “noticings” on the board.
  • When all noticings are recorded, a student who has an idea to begin the problem solving writes their idea on the board.
  • If other students disagree or can add to the problem solving, they raise their hands, get the pen, and add to what’s on the board.
  • When the problem is solved to everyone’s satisfaction, and the teacher is done assessing students’ index cards, he shares their 2-minute drill stats and asks students about any aspects of the solution that he finds unclear (or thinks other students may find unclear).
• Give students a worksheet of “finding missing sides in right triangles” to work on in their small groups. During that time, meet with each group to make sure they are having success. Pay special attention to finding missing legs problems.
• Go over a finding missing legs problem as a class if multiple groups are struggling on it. Ask students, “what’s different about this problem?”
• When students finish, give them a word problem in which the Pythagorean theorem would be helpful but no diagram of a right triangle is provided. Solve as a class, in a teacher-led discussion.

Lesson 2: Learning new stuff

The background: After the prior lesson was implemented, the teacher and I discussed it and he mentioned that the students struggled a lot with the word problem with no diagram given. Given a right triangle, their Pythagorean skills were triggered, but looking for right triangles (say in squares, kites, isosceles triangles) was a real struggle.

I wondered what strategies students can use to help them with that specific skill, and he decided drawing a careful diagram would help.

I noted that this was a chance to try a lesson format for getting good at a strategy and applying that strategy to a particular content area (word problems about right triangles). The format I suggested was a Smartboard version of a “gallery walk”

The Lesson:

• 2-minute drill on a Pythagorean word problem.
• Students go over the 2-minute drill question, generating together a picture and the work with the Pythagorean theorem.
• Students are given another Pythagorean word problem, but this time are asked to work in their small groups to “come up with a picture that could help you solve the problem”
• As students work, teacher circulates and lets groups know, “I would like you to put your picture on a new slide on the Smartboard.”
• Once each different picture is on the board, the teacher asks students to compare and contrast the pictures. Student ideas include:
  • That one has a lot of labels
  • Those three are the same, just some have more labels
  • That one’s the best because it has a, b, and c labeled already.
• Finally, the teacher says, pick one picture that makes sense to you and use it to solve the problem.
• As groups get answers, they compare their solution paths with other groups, comparing both pictures and work.
• Each group eventually comes to the correct answer through comparison with other groups and prompting from the teacher to check, “is that answer reasonable? Does it make sense in the story?”

The Comparison:

The main difference between the lessons that I noticed were:

• In the recall-based lesson, skills weren’t broken down for the students. They were expected to know and apply a whole bundle of skills, a reasonable expectation given their middle-school curriculum, but which got hard when the students didn’t have the skills (e.g. when students couldn’t draw a picture for the word problem).
• In the lesson for learning, students were asked to explicitly compare multiple approaches to the same task.
  • A consensus emerged, that labels are good in mathematical diagrams, that will probably have more oomph than hearing the same reminder from the teacher.
  • Generating and comparing multiple approaches sets the tone “I don’t expect you all to do or think about this in the same way… but I do expect you to compare and make use of different ideas.” Students are in a learning, not just doing, mindset.
• In both lessons, students participated (and led the 2-minute drill process), but only in the lesson for learning was work from all students solicited and made public for comparison; and only in the lesson for learning were the relative merits of multiple types of drawings discussed.
• In the lesson for learning, much less teacher input was needed. The students drew all the pictures, decided which to use, and decided whether or not their answer made sense. When students struggled in the recall-based lesson, the teacher led students through the problem. He realized that students needed a different experience than being led through the problem to master creating diagrams for right-triangle word problems.
• Overall, the students were more on-task and engaged, and used more resources available to them, in the hard parts of the lesson-for-learning. That makes sense because when students are in the “I should know this but I don’t” mindset, it’s easy to disengage when they don’t know. That students stayed engaged and tried different ideas in the lesson-for-learning shows the teacher was effective in setting a tone of learning, not practicing, for this portion of the lesson.

And also…

In an ideal world, there would have been more time and attention and the end of the lesson for learning in reflecting on the learning. A discussion on what makes a math diagram useful, what people noticed that led to certain effective diagrams, what students want to remember for the next time they have a word problem with no picture, etc. would have helped cement the “this is a learning moment” idea and also the “file this under cool stuff to use again” practice.

A lot of talk on the math-ed-web-o-sphere has focused on effective techniques for engaging students in solving problems (e.g. Dan Meyer’s 3 Acts) and especially on perfecting the hook: the juicy image, movie, story, question, etc. that gets students wondering and conjecturing.

In the part of my job in which I am a math coach, we’ve been having a lot of conversations about structuring the curriculum to make big ideas and connections more prominent. One hypothesis is that students in “math class mode” focus on solving this problem and that other problem and never “how do these problems fit together and help me learn about interesting stuff?”

If that’s the case, a good hook for a unit can help students learn in service of a question that’s important to them, which is a powerful and sticky and organized kind of learning. But how do you plan for that? What does it look like?

Another way to ask that is, “if a good question is Act 1, and solving it is Acts 1 – 3, what Acts might be the rest of the unit? How do they relate to Acts 1 – 3?”

Leaving aside completely for now the challenge of picking good questions, I tried to make a lesson planning framework that would help me and the teachers I work with use the good questions we come up with to make coherent unit plans that hang together around an interesting (dare we say “essential”) question.

Here’s a blank-ish version and a sample version having to do with statistics. The sample is pretty long because I used the same question to do three parts of an entire month+ long unit on statistics (displaying data, measures of center, and measures of spread).

At ATMOPAV session 6, we compared what we noticed, what we wondered, and how we worked on some of the math questions we had about this story:

Feeding Birds: You have 7 cups of birdseed. You use 2/3 of a cup of seeds each week.

Lots of people agreed to try the problem with their colleagues and/or students and share noticings, wonderings, and strategies. Check the comments to see what the results of our experiments are!

1) Students ask mathematical questions (that in turn drive their learning)
2) Students and teachers seem happy to be in class and proud of their contributions
3) Students and teachers relate what they learn to what they already know (in math and in their lived experience) and to what they want to know more about/do better (in math and in their lived experience)
4) Students and teachers collaborate to learn and make sure everyone is learning
5) Students and teachers communicate their ideas fluently out loud and in writing
6) Students solve novel problems with a variety of strategies, tools, and representations
7) Students and teachers ask not just “am I right?” but “how do I know?”
8) Students and teachers ask, “what other math/patterns/generalizations can I discover from this?”
9) Students and teachers ask, “what are we learning about? what’s the big idea? what do we need to practice to get better?”

What’s missing? What’s superfluous/not fundamental? How do you know that your classroom looks like you want it to? How does this relate to what you assess?