Lots and lots of things matter a lot in classrooms. Curriculum matters, standards matter, assessments matter, the physical set-up of the school and classroom matter, the school climate and culture matter, the various cultures the students come from matter… But I’m coming more and more to agree with Steve Leinwand that good instruction a) really really matters, and b) is something that teachers can and should have control over.

Perhaps surprisingly, that’s a controversial statement. First of all, there are folks who want to make political statements about teaching and argue that the only way to improve learning in schools is to end poverty or extend the school day or to write harder assessments or to fire more teachers or to pay teachers based on test scores or… And then another kind of controversy is among teachers themselves, as they talk about what their jobs are. Should teachers be writing standards? Writing curriculum? Writing lessons? Writing assessments? And if so, what kind and how many and when?

I’ve been working in two pretty different school districts. Both have high-needs populations and would like to see improvement in their test scores. Both have a lot of initiatives going on, and a lot invested in those initiatives. PLCs, data-driven instruction, computerized formative assessment benchmarks, literacy across the curriculum, students in career-based academies, lesson-planning and curriculum mapping packages, and more.

Each district gives teachers an average of 70 minutes of teacher planning time per day (not counting lunch, because people should get to eat lunch without multi-tasking). There’s an expectation that some of that time will be in structured collaboration, and some of that time will be individual work-time. What the teachers I talk to want to be doing, first and foremost, and don’t ever have time to do, is:

• The basics (before the school year, perhaps): Looking closely at the existing curriculum (i.e. with the the book open in front of them, reading carefully and talking over the details) and figuring out:
  • What is being taught/emphasized
  • How kids are expected to figure it out based on the experiences
  • Why it works that way
  • How to tell if it’s going well
• Implementation details: Working with the curriculum they’ve got to figure out:
  • How to engage their kids in the experiences they’re supposed to be learning from
  • What scaffolding their kids will need to get from the initial experience to the expected outcome
  • Anticipate multiple methods students might come up with for making sense of the experience in the curriculum
  • Anticipate and structure activities that incorporate multiple approaches and multiple representations that kids can/will use to make sense of their experience
  • What will be hardest for the kids and what kids might do/say to show they’re struggling
  • How to meet kids’ struggle so it’s productive
  • Planning concise interventions for when struggle gets unproductive
  • Avoiding sloppy language that gets in the way of precise understanding
  • Good questions to engage students, focus them, redirect them, etc.
  • Key vocabulary that will help students organize their thinking and communicate better with others
• Classroom management and routines: Looking at the curriculum experience and content to decide on & find or invent:
  • Figuring out how to make complex experiences (Barbie Bungee! Centers! Algebra Tiles!) go smoothly
  • The right blend of whole-group, small-group, and individual work for their kids on this experience
  • Classroom routines for engaging kids in the work & the discourse to most efficiently experience and harvest the fruits of the curriculum tasks
• Looking at student work: Differentiating instruction, adjusting whole-group instruction, attending to the social-emotional needs, etc:
  • Reflecting back on student talk, written work, etc. to figure out if the kids learned from the experiences
  • Making decisions about revisiting topics, moving ahead, doing small-group interventions, asking kids to stay after school, etc. based on kids’ work
  • Noticing individual students and how they’re feeling, engaging, thinking, and learning
  • Noticing the group and group norms, behaviors, etc
  • Planning social/emotional interventions as well as mathematical interventions
  • Making notes about what to do differently next year

And sometimes, because teachers are creative and wise and fun:

• Identifying gaps in student understanding, numeracy, etc. and figuring out if and whether to address them, and if so, finding additional resources to do that
• Bringing in an extra activity that they found online, heard about at a conference, dreamed up, etc. because it seems like it would swap in or supplement nicely something in the curriculum
• Making time for students to pursue projects or topics that they are interested in — sparking mathematical passions

Here’s what the teachers do instead during the structured collaboration time, which eats up most of their “planning” time (leaving about 0 minutes for making copies, grading, using the restroom, jotting notes on what just happened in class that you want to remember for next period, tomorrow, or next year, posting the homework online, responding to parent calls and emails, attending IEP meetings, making handouts/visuals, organizing manipulatives, setting up the classroom for today’s experience, and all the other tasks teachers need to do as individuals):

• Teachers writing or aligning curriculum maps, scope and sequence, etc. Unless teachers are getting extra money and extra release time and extra professional development and working with experts because they like this sort of thing, it’s the wrong level of granularity. It’s like asking physicians to be medical researchers, too. People who write curriculum are smart enough that teachers shouldn’t have to rewrite it to make it usable (one hopes, and if not, buy a different curriculum) and it’s not that hard to make sure the stuff on the end-of-year test is covered before that test by re-ordering chapters with a sparing hand. Teachers should be focused on implementing curriculum to meet the needs of their specific group of students, which is NOT trivial!
• Teachers writing tests, called formative assessment tests, but which aren’t used to figure out why kids are struggling and what to do about it. If all the time is spent writing traditional tests and quizzes that won’t be recorded in the gradebook, and not talking about why students are making mistakes and what can be done to help them the next day then writing paper-and-pencil formative quizzes is not so useful.
• Teachers meeting to learn about new school climate, routines, rules, initiatives, etc. This stuff is important, but good instruction is probably the main contribution to school climate. If classrooms aren’t places where kids are invited to work hard and supported to succeed, then they won’t want to be there, which impacts climate. Of course, like eggs and chickens, it’s hard to get kids to work hard and succeed if their overall experience of school is chaotic, unsafe, etc. I’d caution administrators to make sure that teachers have LOTS of time to plan & reflect on instruction, and be frugal in the amount of time spent engaging teachers in face-to-face meetings about school climate.
• Teachers going over statistical data about which broad topics students don’t know. Data-driven instruction is only as good as the data. I think most teachers already know what their students don’t know. I think it’s harder for teachers to know what their students do know, and why their students can’t do the things they can’t do. Do students have “buggy algorithms” that they think work? Are they giving up before they begin? Do students know they don’t know but are trying something anyway? An exercise I love to do is ask, “What way of thinking about the math makes this student’s work coherent to them?” I can’t easily do that if all I know is that they chose C instead of B on this problem, and I definitely can’t do that if all I know is that 25% of them missed 80% of the “measurement items”
• Teachers filling out paperwork, like lesson planning templates. Teachers are often good at following instructions and crossing t’s and dotting i’s because they’re told to. It’s how you survive a job in a bureaucracy! Lesson-planning forms are only useful if they’re part of a focus on good instruction, and every nuance of the form is pointed towards encouraging dialogue about good instruction, and grounded in teachers’ concepts of good instruction. I’d lean towards introducing forms the way I’d introduce any procedure: start with the concept of the good instruction, solicit teachers’ current lesson-planning methods, talk about what works well and what we could improve, and generalize to a form that works for our community and our definition of good instruction.
• Teachers meeting in cross-discipline groups to discuss student behavior problems or school-wide, non-instructional or supplemental initiatives. Again, fine stuff, if there’s time for it. But right now, there’s not enough great math instruction happening in so many high-needs schools that I’d prioritize instructional planning and reflection over all this stuff. Squeeze in the conversations about tricky kids, how the 4th grade is doing overall, or cool cross-curricular projects after planning and reflecting on good instruction. Otherwise, planning and reflecting on good instruction (which is kinda the whole point of being a teacher) is what gets pushed to the side.
• Teachers doing work for extra initiatives such as finding readings for the reading across the curriculum initiative. Again, cool stuff, but not as important as good instruction.

(Almost) none of the above is a total waste of time, but none of it is the stuff that research on teacher collaboration actually suggests is useful. Considering how little time teachers get to plan, let alone collaborate, it’s frustrating to me how much of that time is wasted time. It’s also frustrating to me how much time teachers spend writing lessons or searching the web for lessons. I’m certainly very guilty of that — I tend to start with a vague sense of the topic, Google for it, and then try to do what I find on Google. My main focus is on what to do, not how to do it.

What I should be doing is starting with the curriculum, figuring out what kids are supposed to be doing & learning from that doing, and then applying my particular expertise to how to get my kids from A to B. It’s really concrete — should they be reading the questions and writing answers, or is this piece best done orally? Should students work in small groups on the main task and then share out, or should I lead the group through a demo and then have them try on their own? If small groups are right for this task, what will it take to keep them working productively, and how long can I expect each group to stay on task? Will it work if the groups are random? Will I need to make sure Keion doesn’t work with Marvin, or can they get it together today? If a group finishes early what should they do? If a group gets stuck, should I tell them what to do, ask a question, have another group share out, or something else? How many different “kinds of stuck” can I anticipate and how many different responses can I prepare? How long can I wait for a group to catch up before moving on? Will students’ struggles with integers get in the way? Will I let them use calculators? If students use Guess and Check to solve all the problems on this page, what are three things I could say/do to help them look for and find a more efficient approach? The list goes on and on… And I haven’t even scratched the surface of student-work questions like, based on what I asked them in their exit ticket yesterday, who needs more time grappling with yesterday’s concept? How can I help Deshaun get more fluent with division facts so he can make sense of one-step equations? What does this class believe counts as a right answer to an inequality — do they get that it’s a range of numbers that make the inequality true? If not, is there something in my language that could help them access that, or is there some challenge I could engage them in that would bring that point home? How does my curriculum make that idea “pop” and why did they miss it the first time through? Why does Emily consistently evaluate 2x when x = 5 as 25, but can solve 2x = 10?

If teachers don’t have enough time to ask, answer, reflect on, and revise their thoughts about the questions above, then we shouldn’t be filling their time with things other people get paid to do, like writing curriculum, writing fancy-schmancy benchmark tests, looking at data that’s not useful on the individual student level, or discussing which minutes of the day the bathrooms will be open to students.

Lots of times when we ask questions in math class, they fall into 2 categories:

1. Procedural/Right Answer questions, e.g. “What do you call the longest side of a right triangle?” or “What did you get for number three?” or “What is the mode of this data set?”
2. “Higher-Order Questions” aka hard questions, e.g. “Why do you think someone might have come up with that [wrong] answer?” Or “Which of these is correct? Defend your choice.”

In my experience, even though we want all kids to be able to answer both types of questions, they’re both tricky. For the first type, kids either know what I’m looking for or they don’t, and so I either get a few loud kids participating or awkward silence, and often devolving into off-task behavior.

For the second type, look out! Talk about awkward silence and devolving into off-task behavior. Kids look at me like I’m crazy when I ask them to synthesize, justify, explain, etc. And they wait. They can’t out-wait me (I am the king of outlasting the awkward silence) but they sure do try.

So I’ve been trying to come up with questions that are good, math-y questions that don’t fit in either of those categories. I want questions that every kid can answer, by virtue of being a human (and therefore reasonably observant, semi-rational, interested in other humans, and decently resourceful). I want questions that kids see some need to answer, or are interested by. And I want questions that get kids doing some intellectual work that will help them do more work. And that doesn’t shut them down. Oh, and that helps me figure out what’s going on with them. And that aren’t questions I already know the answer to. Here are some:

• What do you notice about ______?
• What are you wondering?
• What’s going on in this ______?
• What’s making this hard?
• On a scale of 1-10, how easy is this for you? How come?
• What’s one thing you remember about ______?
• Here are three different ______. Which do you like best? What’s one thing you liked about it?
• Tell me one thing you thought about problem three.
• What’s the first thing that pops into your mind when you see this?
• What’s the fourth thing that pops into your mind when you see this?
• What do you think a mathematician might notice about this?
• If you saw this image/story/statement on a math quiz, what question(s) might go with it?
• If your math fairy godmother appeared right now and offered to give you one helpful hint, what would you ask her for?
• How confident are you in the work you’ve done so far?
• The answer to the problem you’re about to work on is ______. How could someone have figured that out?
• Have you ever had an experience like the one in the story?
• What do you think the person in the story might be feeling?
• Why do you think I showed you this?
• What’s one thing you like about what she just said?
• What’s one thing you’re wondering about what he just said?
• What’s your best guess for the answer to this problem?
• What is an answer that is definitely wrong for this problem?
• Make a prediction. What do you think will happen…
• Without writing anything down or calculating or thinking too hard, could ______ be the answer?
• What’s your gut feeling?
• Do you have a reason or a gut feeling (or both)?

And from the comments/Twitterers:

Dan Meyer:

• “What do you think an incorrect answer would look like?”
• “What more information do you need here?”

This Google Doc from Justin Aion of questions he uses to help his students become better readers in math class.

Max Hoegh:

• “How would you explain this to a ___________?”
• “How would you explain this with a drawing?”

Ed note: In part because some of us teach 10-year olds, but also because I think that explaining math is a constant process of revising and adjusting based on audience feedback, I left the audience of “How would you explain this to…” blank. I like the idea of playing around with different audiences for different explanations. Like, “How would you explain this in a Tweet?” or “Send a friend who missed class today a text message about what they missed.” or “How would you explain this to a friend? How would it be different to explain it to an enemy?” I even know of a teacher who pasted her class picture from 3rd grade on a chair and will drag that chair to the front of the room when she wants kids to explain something clearly and step-by-step.

In general, I’m trying to push myself to ask more questions in which I’m not trying to get the kids to say the thing I need them to say. Instead, I’m trying to find questions that get kids to put into words the things they need to say — to let me know what’s on their mind, what their current working model is, where they’re stuck and what they’re ready for. I can make predictions but I never know exactly where a kid will turn out to be, and so I try to maximize what I can learn about them, while using questions that let them know I value them and really want to hear their ideas (not them stating my ideas for me!).

In the Powerful Problem Solving book that’s coming out this month (!) we included the famous “Make a Peanut Butter and Jelly Sandwich” activity, in which students write instructions for making a sandwich and then their teacher or partner acts the instructions out very literally — e.g. “put the peanut butter on the bread” is interpreted as “place the (unopened) jar of peanut butter on top of the pile of bread.” Or “spread the peanut butter on the bread” doesn’t imply “with a knife” so you’re scooping out globs and globs of peanut butter and smearing it all over all sides of the bread with your fingers.

I got to visit a classroom where the activity was being implemented. The activity is in a chapter on good math communication and focuses on the important of revision. Watching the activity in action, I was struck by the subtle differences between focusing on precision and focusing on revision.

If you focus on precision, this can become a kind of “gotcha” activity. An activity in which the teacher sets up the kids by saying, “hey, this is really simple, everyone knows how to make a PB&J, so of course you can explain it…” knowing that they won’t be able to explain it to the alien the teacher is going to pretend to be, without warning them. The message the kids might take away is “writing in math class means painstakingly explaining your work to someone pretending to be an idiot” which is clearly not fun. There’s a reason the word “pain” is the first syllable in “painstaking!”

Because in fact, giving instructions that teach someone how to do something is NOT easy. The tricky part of giving instructions is figuring out what the other person does and doesn’t know, and tailoring (aka revising) your instruction to meet their needs.

When I watched the “Make a Peanut Butter and Jelly Sandwich” activity, the teacher had a GREAT launch — she showed a picture of an alien and explained that on Bob the Alien’s planet, they’d picked up radio transmissions of “Peanut Butter Jelly Time.” Bob wants to know what this amazing experience of  making a peanut butter and jelly sandwich might be, since the aliens so enjoyed hearing the song!

So right away the students were in a mindset of needing to figure out how Bob the Alien thought. After the teacher acted out some instructions as Bob, the kids started to say, “Wait, Bob has no common sense!” and “Bob is taking these directions SO literally!” and then “Wait, can I change my instructions?” or “I need to revise this part.”

The activity was structured to have lots of revision moments built in — once after seeing “Bob” in action on some sample directions and then again after having a peer pretend to be Bob. The students revised other people’s instructions, not their own, to help make the revision not personal, and more about thinking about what they’ve learned about Bob.

The teacher’s language can help reinforce that we’re revising based on new data, not just recognizing that we should have done better the first time. The teacher can ask, “What are some different ways you think Bob might interpret that? How would you change your instructions if Bob did this instead of that?” The teacher can also ask, “What did you think about Bob before he read the first directions? How did your thinking change after you saw how he interpreted them?”

Writing instructions on how to make a Peanut Butter and Jelly Sandwich for an alien can be a great experience that helps students understand what revision is, why we revise, why feedback from others is an important part of revision, and why explanations might need to take different perspectives into account. The key is to make sure that the expectation is that we will get new information about how the alien thinks and revise based on that. This is not a “haha, you thought you knew how to write instructions!” It’s a “wow, that alien sure does interpret things weirdly, I guess I’ll have to try again now that I know that!”

Finally, it’s nice to have the students be the ones to articulate what they learned from the experience. After the activity, it’s neat to wonder, “What does this have to do with math?” or “What math experiences does this remind you of?” One thing I thought I might do is make an audience-o-meter with “Bob” at one end and “Myself” at the other and try to think about different audiences we might write mathematical explanations for, and the levels of detail, amounts of revising, etc. that we might need to include depending on where we are on the audience-0-meter.

PS — with the rising prevalence of peanut allergies you might want to make cream cheese and jelly sandwiches, or use American cheese slices, or do instructions for brushing your teeth…

PPS — Another cool part of the activity is when the students get to be Bob. They’re playing with the mathematical skill of coming up with a counter-example, of finding other ways to interpret a mathematical definition or instruction… which is so important! It’s sort of like they’re practicing the skills in this amazing play-let based on key moments in modern Geometry.

If you’ve spent any time around the Math Forum folks, you’ve heard of “I Notice, I Wonder” — two little phrases that we use to start students talking mathematically. We’ve seen the questions “What do you notice?” and “What do you wonder?” used to launch lessons, get kids doing careful reading during problem solving, help kids give each other constructive feedback, support students to look for patterns after doing an activity, and encourage reflection and extension after a project or activity. They’re powerful questions because everyone has something they can notice (note that it’s not “what do you know” or “what do you think” — it’s a much more fundamental level than that!). And wondering is plain old fun!

A buddy from Twitter Math Camp asked about the value of noticing and wondering for high school. She knew it could be powerful but wondered if colleagues and students would feel it was beneath them. Here are some things I’ve noticed, and some thoughts about them:

• High school students, especially juniors and up, are the shyest noticers and wonderers.
• By high school, kids (especially kids who are good at school) are very attuned to what the teacher wants them to notice, so they often say, “I don’t notice anything,” or “What do you want us to notice?”
• Noticing and wondering often starts with “what can I get away with” type noticings like, “I notice the graph is blue,” or “I notice your drawing isn’t very good.”
• A good prompt goes a long way with high school students in particular — they have a harder time suspending their disbelief than, say, third graders.
• It’s harder for high school students to make noticing and wondering a habit — they tend to be more likely to compartmentalize and think of it as an activity someone has to direct them to do rather than a skill.
• High school students, like all people, feel valued when their ideas are heard, recorded, and made use of — so they can get a lot of value from noticing and wondering.

Based on my noticing here are some tips for noticing and wondering with high-school students:

• Go multi-media. Start with pictures or videos. Some good places to find pictures and videos are:
  • Some of my favorite pictures for doing math with on the Internet: http://mathforum.org/blogs/max/pictures-for-the-lindy-scholars/
  • http://mathforum.org/blogs/pows/ (search around for the pictures and videos)
  • http://mathforum.org/pow/support/videoscenarios.html (though honestly other than Charlie’s Gumballs and Val’s Values, these are more for younger students. However, you might challenge high school students to make better videos).
  • Any of Dan Meyer’s 3 Act Math Tasks: http://threeacts.mrmeyer.com
  • Any of Andrew Stadel’s Estimation 180 images: http://www.estimation180.com
  • Any of Fawn Nguyen’s visual patterns: http://visualpatterns.org
• Make it clear that everyone has something to say and everyone’s things are valued, by:
  • Not commenting at all on students’ noticing and wondering, just listening with a welcoming expression.
  • Writing EVERY noticing or wondering down, whether it’s “relevant” or “right” or not.
  • Asking, at the end, “are there any noticings or wonderings that you’re wondering about?” and then encouraging the authors to clarify as needed.
  • After solving a problem or doing an activity that you launched with noticing and wondering, ask, “How did we use our noticings and wonderings?” and go back through them to value the contribution of each.
  • When something comes up that a struggling student had noticed, foreground that moment to help give that kid more status. For example sometimes a student notices something “obvious” but then later on that obvious thing turns out to be a key to the solution — value that contribution!
• Be explicit about the skill you’re teaching. Here are some ways to do that:
  • Ask students to notice and wonder with different lenses on. Choose a picture and ask “What would a scientist notice? What would an artist notice? What would an athlete notice?” Then ask “What would a mathematician notice?”
  • After noticing and wondering, once everyone’s voice has been heard, ask, “Which of these did you use math to think of?” and “Which of these could we use math to explore more?”
  • After everyone’s voice has been heard, talk about how as a group they’re getting better at noticing and wondering.
  • Look at noticings and wonderings from another class (people share lists on blogs and Twitter a lot and you can compare your list to theirs).
  • Notice and wonder about an example or image from the text, and then see if you noticed everything that the text pointed out about the image/example.
• Use student wonderings to drive lessons to make the class feel more student centered:
  • Encourage silly, creative, and fun wondering by valuing even off-the-wall wonderings (like when someone wonders “Does Sally have a tapeworm?” when you do a problem about Sally eating a whole pizza, encourage more thinking and discussion about tapeworms and the math behind them).
  • Choose a student wondering to explore, rather than the question you’d originally intended.
  • If student wonderings don’t make sense to explore that day, come back to them later, support the students to answer them on their own, and/or choose a different scenario where you and the students DO wonder the same things.
• Help them remember to use noticing and wondering:
  • When they’re stuck.
  • When they’ve got a possible answer.
  • When someone else is explaining.
  • When they’re reading a textbook.
  • When they’re reading a math problem.
  • When they’re looking at a math image like a table or graph.
  • All the time!

And as for how to help colleagues experience and appreciate noticing and wondering:

• Use your own students as guinea pigs and videotape or record the session. When students notice cool things or wonder something awesome, share that (innocently)!
• Math teachers love noticing and wondering about math-y images like this: http://mathforum.org/blogs/pows/free-scenario-filling-glasses-wcydwt/ so get them doing it as a fun exercise, and then thinking about how it can help students.
• Send your colleagues to http://101qs.com to get them wondering about math images and videos.
• Share Annie Fetter’s Ignite talk about noticing and wondering: http://www.youtube.com/watch?v=WFvYZDR4OeY
• Share some of these blog posts about noticing and wondering or with examples of noticing and wondering:
  • http://blog.mrwaddell.net/archives/808
  • http://kalamitykat.com/2013/02/19/intro-to-projectile-motion/
  • http://resolvingdissonance.wordpress.com/2013/02/15/noticing-and-wondering/
  • http://oldmathdognewtricks.blogspot.com/2013/02/noticing-and-wondering.html
  • http://justyourstandarddeviation.blogspot.com/2013/02/notice-and-wonder.html
  • http://blog.constructingmath.net/2013/02/analyzing-student-questions/
  • http://mathreuls.pbworks.com/w/page/63615099/Business

So, Math Twitter Blog o Sphere — if you’ve noticed and wondered with high school students, what have you noticed and wondered about them? What’s unique about the high school experience, and what helps high school students and their teachers value noticing and wondering?

Perhaps because I’ve watched every step of this saga of Michael’s Journey Into the Complex Plane with hawk-like attention, I’m totally down with what he’s trying to do in his blog post about how he might introduce students to complex numbers. He’s looking for a genuinely perplexing, easy to formulate question that students have the werewithal to begin to answer.

I think he’s nailed easy to formulate, and his 17 pages of work show that students do actually have the arithmetic know-how to answer this. I think students will be perplexed by this, but I do wonder about where are the parts where students need specific mathematical habits of mind & skills to be enabled to persevere.

Some things that need to be in place for this to work:
Students need to be able to hold on to the ambiguity of multiplication as an operation on the plane and the (shorthand) idea of multiplication of a point by a point. Or we need to have a language that unambiguates that. E.g. 3 * -2 is “where does 3 go under the transformation that takes 1 to -2?” [and then commutativity is NOT obvious].

If we keep using the shorthand of multiplication of a point, by a point, students need to be comfortable with having multiple physical representations of the same operation, or we need to train them in one that we want them to use. Again, I’m not sure which is better, but I’m leaning towards really hammering and making both sensible and automatic the idea of a twisting, scaling slide rule kinda thing (i.e. multiplication of real numbers is rotating and dilating the real number line, and adding real numbers is translating the number line left or right).

Also, depending on your definitions of dilation of the number line based on points, you don’t need the rotating idea until you introduce complex numbers, because the signs of your points will take care of that (dilating by a negative ratio includes a rotation in GeoGebra or Sketchpad, but you can define your dilation based on length, not position, and then you do need a rotation. I made a collection to help play with that idea here: http://www.geogebratube.org/collection/show/id/5056. It’s not fun and visual, but it is mathematically intriguing to see how the points are defined).

Here’s where the habits of mind really come in. If we ask students to extend their understanding of 1D operations on the real number line to 2D representations, they need to be able to:

• Understand that generalizing means making a coherent system that doesn’t “break math”
• Decide on the rules that we want to define not breaking math to be
• Generate conjectures about what a generalization might look like
• Test those conjectures
• Persevere through multiple conjectures and tests
• Accept a definition of multiplication that is not their initial intuition and may even trouble their 1D understanding of multiplication
• Persist through defining a generalized multiplication to mastering said multiplication, both geometrically and algebraically.

Most students have never been asked to conjecture possible definitions for an operation, and have never been exposed to the idea that mathematicians posit the existence of objects and operations and then test to see if they break or not. Which is too bad because that’s a lot of what mathematicians do, and something students are capable of, but getting students to the point where they’re willing to define mathematical operations or objects for themselves and then persevere through playing with possibly broken objects long enough to find one that works, is hard.

[It's sort of like giving a kid a huge pile of boxes to open on her birthday, with the caveat that most of the toys she'll find are missing pieces and will never work, but once she's done opening & testing them all she'll have found some AMAZING working toys and learned a lot about how toys work. This is why math class is not a birthday party, it's HARD fun, much more like learning to ride a bike (ouch!) then opening birthday presents.]

Students also need a robust enough understanding of operations that they get what it means to not break math. They need to expect commutativity and associativity and the distributive property (which most kids don’t understand, let alone value!). They need to compute fluently with positive and negative numbers, including distributing. A robust understanding of the geometry of transformations would be nice too.

All images from http://www.ics.uci.edu/~eppstein/junkyard/spiraltile/

Finally, the transformation of the plane that relates closely to complex multiplication is the beautiful Spiral Similarity, which results in lovely spiral tessellations. Could a launch perhaps be based on telling some technology to make spiral tessellations for you, and then making the connection among algebraic and geometric definitions of transformations, and finally generating a robust set of algebraic rules for exploring and defining spiral dilations and 2D translations and then connecting that to the transformation composition that takes 1 to -1. See more about Spiral Similarity here: http://www.ics.uci.edu/~eppstein/junkyard/spiraltile/

PS — on contexts, perplexity, motivation, etc. see Riley Lark: http://larkolicio.us/blog/?p=787

Update — I’ve been meaning to share this article for a while; it’s not completely relevant here but I like it: Research Mathematicians as Learners And What Mathematics Education Can Learn from Them — it’s about the doing of mathematics as mathematicians see it and the opportunities for students to do that kind of thinking. The second is a book about elementary math concepts and how they relate to higher math, as told through the eyes of what used to befuddle research mathematicians when they were elementary students: Shadows of Truth: Metamathematics of Elementary Mathematics.

We were supposed to listen to students talk about painful/unpleasant experiences in math and make a concept map showing the connections and themes in what we heard. My highlight — the way creating a concept map both helps my peers/instructors learn about my thinking and was a learning experience for me. The act of displaying my thoughts helped me see new connections.

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.