Hi Mathtwitterblogosphere,

I’ve missed you. It’s been almost a year since I last blogged here, but there’s a pretty good reason for that — I was putting together a manuscript that has been accepted for publication. The book is the collected wisdom of the Math Forum on facilitating activities to help students unlock their mathematical problem-solving potential. It’s called Building Understanding Through Problem-Solving and the Mathematical Practices and will be coming out from Heinemann in the fall. Putting together the book used up all my words (and time, and wore out the ‘e’ key on my computer) but now that it’s done I’ve got lots to say!

There are lots of ways to say what the book is about, but one of them is that the book is about things that teachers can do to support students developing both the disposition and skills to look at a math problem (any math problem, not just the awesome ones) and think, “I have things I can try!”

The topics range from building classroom cultures of listening and valuing ideas, to supporting students to communicate their thinking for different audiences, to activities that help students break down that wall of resistance to anything that looks like a math problem, to support with key problem-solving strategies like guess and check, change the representation, or make a mathematical model.

One huge reason for writing the book is to try to step into the gap (still wide, but narrowing) that is left when we focus all our attention on concepts or on skills. As math teachers we are getting more information on how students best learn skills (like math fact fluency, how to divide fractions, or how write valid equivalent expressions), and increasing attention on how to ground those skills in concepts (like understanding that division means “how many of these are in those?” or knowing that two expressions are equivalent “when the two expressions name the same number regardless of which value is substituted into them“). However, there’s more to doing math than knowing and calculating — there’s the doing part. Stuff like looking for patterns, generating and testing hypotheses, generalizing results, etc. The stuff that’s in the Standards for Mathematical Practice (1 page), not just the stuff that’s in the content standards (lots of pages).

The other stuff — practices, doing math, problem-solving strategies, methods, whatever you want to call it — is the fun part. It’s the “glue,” or the “verbs,” or the “story” while the skills and concepts are the words or objects that we do stuff with. How we get students to be doers, not just consumers, of math is a fun and interesting question. Fun idea: concepts and skills can be delivered through telling. Doing math just plain old can’t — it has to be learned by doing (and my hypothesis is that through doing math the concepts and skills  can also come along for the ride a lot of the time, and will usually be better retained and connected).

The book we’re publishing can be thought of as an activity guide (with student work and stories) for getting students doing math, and the beginning of a set of ideas for how we can think about what it might look like for students to get better at doing math — how we can track students’ progress and help them become better and better young mathematicians.

So, please accept my apology for not blogging, and I hope it turns out that the book is useful and interesting.

Sincerely,

Max

I’ve been taking Keith Devlin’s Massive Online Open Course (MOOC) “Introduction to Mathematical Thinking,” and it’s been really fun to see how my brain’s mathematical thinking skills line up with the course assignments. It’s especially fun to see the times when ideas are new to my brain and how that’s different from when they’re familiar.

For example, Assignment 1 was about reasoning from definitions, proof by counter-example, proof by induction — those are tricks of the trade I’m familiar with. I can read math problems and see how those tools are applicable, I know what to focus on when we’re proving properties from definitions, etc. I wasn’t flummoxed by stuff like what does it mean to divide 0 by 2, because I could work comfortably with the idea of 0 as being just another number that’s divisible by 2 — I was working on a higher level of abstraction.

But then again, on Assignment 2, I was flummoxed! I didn’t know what to pay attention to, and different stuff popped out at me. The assignment was about connecting natural language to formal logic (like if Statement A is “it will rain tomorrow” and Statement B is “it will be dry tomorrow” then is the plain language statement “It will either rain tomorrow or be dry all day” equivalent to the formal logic statement A v B (A “or” B)?) I got quite tangled up in the question of can it rain and be dry on the same day, and trying to nitpick the words in the sentences and find some hidden trick or meaning — because I didn’t have any idea what the basis of comparison was, or what it would mean to make a logic statement equivalent to a plain English statement. On the homework, I found myself confused and looking for “key words” to translate little parts of sentences into logic. In short, I felt like a student who struggles with word problems because they don’t know what it means to mathematize something, and so they are focused on different details than an expert would, and use different rules and tricks to do the mathematization. For an expert, it feels obvious, how to apply this heuristic, but for the student seeing it modeled, it’s not clear what details are salient to the expert.

I had an ah-ha! moment with Assignment 2, but more accurately, I had two or three ah-ha! moments. Two or three times this week I’ve struggled and asked my peers for help and had someone show me a truth table. Ah-ha! I said — to find out if two statements (whether in logic, plain English, or both) are equivalent I need to find out if they have the same truth table. And I could answer one question. But then when I became stumped again, I didn’t think of truth tables, because the question felt different!

I hope that now that I’ve reflected on truth tables and what it felt like to need one, and the (seemingly) different contexts that they helped me in, I will now have truth tables as a mathematician tool. Just like I have proof by induction, and showing sets are equivalent by proving an arbitrary element from one has to be in the other and vice versa, as mathematician tools that leap out at me.

Reflecting on my experience as a learner, I noticed that:

a) I needed to struggle with the problems repeatedly, without help, before I really cemented my learning.

b) I needed models of explicit tools from people who were smarter than me (about this kinda thing), but I couldn’t make sense of the tools and when to use them the first two times.

c) I needed to compare my ideas with other non-experts, to articulate them to myself and others.

d) I needed to persevere through enmeshment in all the wrong details, and be able to come up for air and entertain ideas about what other people saw, not just the alluring details. And be exposed to people just above my level, and their ideas, not just expert ideas.

I wonder how many of those features my classes and work with teachers have. Are they all necessary? Are they sufficient?

So I just finished two workshops (an hour-and-a-half apart, if you drive fast), and both of them were versions of workshops I’d done successfully several times before, on topics I’m really familiar with. As I was leaving the office, I said to my boss, “y’know, I’m feeling much more prepared than I was last year. I hope I haven’t jinxed myself by being so prepared.”

I think I did jinx myself. I feel like I learned something about good preparation vs. what y’all might call pseudo-preparation (preparation that feels good but doesn’t lead to learning).

Before I write about what happened, though, I have to say that learning something about good teaching through the experience of knowing that you just gave a bunch of students an experience that could have been way better… well, it feels pretty crummy. Even being able to blog about it and maybe contribute to the collective wisdom of math teachers, it still feels pretty crummy to think of the students who are going to have to lead peer-mentoring sessions next week and not only do I suspect that they aren’t as prepared as I could have helped them to be, I don’t even know how prepared or unprepared they are. Blech.

This is not to say, by the way, that they didn’t learn anything. I think they had some good experiences and I know I said a lot of stuff that if they remember it will be really helpful. It’s just that… well, let me tell you what happened.

I had done this workshop for the peer leaders before and I had done stuff with them I liked. So I prepared by planning out the list of things I was going to do. I made copies and found materials and planned out what I wanted to say and how I wanted to make connections among the activities. I knew the content I was going to cover, what I wanted to write on the board, etc. I was really well prepared, way better than last year.

Last year, I had prepared on the fly, as I was driving from Philadelphia to Dover. Prepping while driving meant no writing, no looking for problems, no making handouts. So what I did instead was visualized the workshop. I had imaginary conversations (out loud! yay for talking to yourself while driving!) with the students. I thought through the logistics of each transition over and over, planning what I would say and how the students might respond.

So this year, when I was running out of time to do all the activities and say what I wanted to say, the narrative in my head was all about what I wanted to do and say. I wasn’t as focused on the students and what I wanted to hear from them — I didn’t have a plan in my head for how to listen to my students. And so I was wrapping up the workshop and realizing, I haven’t listened TO them. I’ve listened FOR what I wanted to hear to be able to say my next thing, but I haven’t been having dialogue.

I’d characterize what I spent the morning doing as pseudo-prep. Pseudo-prep for me is planning what experience I’m going to have, what has to happen in the lesson, what I want to say and cover. That only partly works because it’s not preparing to work with my students (as they say, I was preparing to teach content, not teach students). For me, an alternative way to prep seems to be to take long drives before teaching… meaning, to think through dialogues with students, to imagine what I might hear from students and different alternative paths the lesson might take based on those different dialogues. Somehow, I need to find more ways to prepare myself to track and attend to what I want the students to learn, experience, and talk about, and fewer ways to track what I want to say and do.

Summary:

Pseudo-prep, for Max, means (and this will probably be different for you since we all have different processes that help us get ready to do stuff):

• Focusing on coverage
• Preparing a sequence of activities I want to be sure to do
• Planning out what I want to be sure to say and write down
• Focusing on my actions, not the students’ experiences
• Planning only one possible sequence of events
• Not asking myself, “what do I want to learn about my students’ views of this?” and instead asking, “what do I want to tell my students about my views?”

Alternatively, ways I can prep that actually help me do good workshops and lessons:

• Focusing on what I want to learn about my students.
• Focusing on how I will track any shifts in their views.
• Planning different activities that I might use.
• Thinking about what I might hear from students that I could use to diagnose what they currently think & feel.
• Thinking about how the activities I have planned move students along a journey towards more nuanced ways of thinking about mentoring vs. tutoring.
• Having imaginary conversations with students where I think about every crazy thing they could say — so I can feel calm when listening to them say those things!
• Focusing on the logistics of flexibility — how can I support myself and my students to be comfortable if I decide to do something I didn’t make a handout for. Can I project it? Have them take notes? Send them a summary by email later? What will work best?

I’m thinking that these two kinds of preps can actually take the same amount of time, and the latter works better for me and my students. What does pseudo-prep look like for you? What have you learned about how you prepare best from reflecting on those crummy feelings after a lesson that you know could have been better?

It was an enormous honor and privilege to be welcomed to Twitter Math Camp the way that I was. As an enthusiastic but inconsistent Tweeter, and a math educator who’s been out of the classroom longer than I was in it, I was nervous that I wouldn’t be welcomed.

In fact, as everyone else covering this monumental event has mentioned, the spirit was one of openness, welcoming, and generosity. Even though I was a surprise to many other folks there (whether we’d never met in the twitterblogosphere or they thought from my tweets I’d be older and wiser IRL), as soon as I got there I was welcomed warmly by Lisa and Shelli (our fearless leaders who got STAFF t-shirts!), and fell into easy conversation with my car team, Glen (whose wife grew up in the same teensy-tinsy town my mom did) and the always outgoing Marsha.

There were lots of things that made this one of the best PD experiences ever:

1. We spent so much of the day just doing math together and whatever came up, came up. In the Math 1 group, our fearless leader, Sara, whose experience with rich tasks, is, well, rich, led us to do some math, talk about it, and when we strayed too far from actual math, led us back into doing the math again.
2. We moved back and forth super easily, as Elizabeth said, between talking about what worked and why it worked. There were people there who were fountains of knowledge of how to make specific things happen with kids in class, and people there who were fountains of the deep, thoughtful reasoning behind specific classroom occurrences. For example, in Math 1 we talked a lot about units and slope (all that back and forth over slope between our very own Karim Ani and Sal Kahn is like old news to us Math 1 folks). The conversation moved effortlessly between sharing classroom approaches to getting kids focused on units and interpreting problems, comparing and critiquing what worked, and unpeeling the math to understand why it worked — why are units useful for introducing slope? What do mathematicians do to understand the given information in a problem? How does focusing on units relate to algebraic reasoning? And what specific graphic organizers, questions, and activities get kids doing those things?
3. Everyone took the time to get to know each other as individuals. There was no one-size-fits-all. It was a lot of, “you would like this because,” and “I would do it this way but I see how that works for you…” And as part of getting to know each other as individuals, we cut loose together. A LOT. That was fun.
4. Everyone treated each other with respect, and you could just tell that down to their core, each person respected themselves and their students too. We were all passionate about kids learning, more than anything. That made it easy to put ego aside and listen to each other and share our own ideas. I know when we go home, we’ll all be treating our students and colleagues that way too.

Oh, and the other thing that made this best ever was that we were in St. Louis. Do you know what else is in St. Louis? The City Museum. The City Museum is about the least museum-like place you have ever been in your life. When you think about a museum you think about walking around, looking at things, reading, not really touching anything. Even at a science museum, maybe you will touch a few things, turn some gears, crawl through one giant-sized version of something. The City Museum is the opposite of that. For example:

• At the City Museum, there are holes in the floor. On purpose. For you to drop into and crawl around the basement on your belly. Why not?
• At the City Museum, when you are done crawling around the basement on your belly, you might squirm your way up through a tube to discover you are emerging out of the mouth of a giant stuffed elephant.
• At the City Museum, there are several two-story tall slides.
• At the City Museum, there is an old fighter jet on top of a tower of industrial scrap metal that you can climb up to, walk around in and climb over.
• At the City Museum, there is a ball pit and a ferris wheel on the roof and slides that I swear are steeper than a 45º angle of elevation… 60º maybe even…
• At the City Museum, there are rope swings, and scaffolding to climb, and dark narrow passages to squeeze through, and holes to climb into and out of, and slides to go down, and an orderly chaos of people of all ages (yes, grandmas included) doing all of these things.
• If you feel you must learn something, you can always admire the collection of doorknobs arranged by the type of symmetry they display.

If you have never been to the City Museum and you can get there, you must. If you have never been to Twitter Math Camp and you are a tweeting math teacher, you must. I think that about covers it.

At Twitter Math Camp, one of the highlights for me was the “My Favorites” presentations. They started off with a bang… or maybe a pufffft…..whap! That’s the sound of Hedge @approx_normal shooting me with the marshmallow gun she taught us all how to make. And I’m still pondering what Glenn (@gwaddellnvhs) showed us on the last day about geometric interpretations of imaginary roots of quadratics.

Somewhere in the middle I got to talk about one of my favorite things: pausing the student/teacher interaction (by doing it online) and then practicing diagnosing students and asking really good questions. I was asked to blog about it, so here goes… Read More→

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!)