If you hang out with educators on a regular basis, then surely you have heard someone say a variation on, “My kids come to me not knowing stuff, and that makes it hard for me to teach them enough new stuff.” If you hang out with helpers-of-educators regularly, then you hear these helpful helpers saying things like, “Tough, you have to teach the kids you have.” And “Don’t remediate, accelerate.” and “You can’t go back and teach everything; if they’re behind it’s their responsibility to get extra help,” and “You have to differentiate and make your lesson work for everyone,” and a whole other mess of conflicting things. Because (duh) this is a hard problem. Personally, when I have a hard problem I like to look at a specific case, and guess what! One happened to me the other day. So let’s play “What would you do?”

I was guest-teaching a class that had been studying exponential growth (doubling and tripling, mostly), writing recursive and explicit rules, making tables, and graphing. Today, we were introducing the concept of compound interest as an exponential scenario, and the teacher’s goal was to have the kids recognize just that — compound interest can be modeled with exponential functions (and is cool and important, because free money!).

I was expecting that the hard bits would be the vocabulary of compound interest, the concept of investing, and the concept of the bank giving you more money each month because you have more money each month. And all of those things were hard for some subset of kids, but what was really hard for the majority of kids was doing mathematical operations with the quantity 8%.

Here’s the lesson plan:

• Pair kids and ask them to discuss, “You won the lottery and there are two prize options: would you rather have $10,000 now or $20,000 in ten years? And what would you do with the money?”
• Individuals answer the above question by making a graph of the money they’d have over the next 10 years based on their choice.
• Share a few different graphs on the document cam and discuss the story they tell.
• Pretend to get a call from a banker offering to hold the $10,000 in a CD earning 8% interest, compounded annually.
• Ask kids “What would you do now? Take $10,000 now, $20,000 in ten years, or invest the $10,000 in the 8% interest rate account for 10 years?”
• Define the question: will earning 8% interest on $10,000 over 10 years earn you more money than just waiting and letting the lottery folks give you $20,000 in ten years.
• Turn students loose to use a table (strongly encouraged), graph, or rule to help them answer that question, using their knowledge of percents.
• Come together as a class to compare & discuss tables, and convert tables to graphs and tables and graphs to rules to help us understand what’s going on in compound interest situations.
• Practice on another compound interest situation.

So it turned out that the students right away thought of investing. On their warm-up graphs, some saw investment as a linear scenario (I’ll get $1000 a year from investing in the stock market) and others saw their growth rate increasing as their money grew (exponential graphs).There was engaged discussion about spending vs. saving vs. investing, and more than one student knew what compounded annually meant when the banker called.

Here was the part I wasn’t quite prepared for. When 8% was mentioned, I right away heard fear about the use of percents, and students asking tentatively, “doesn’t that mean we have to divide something by 8 or zero-point-eight?” Maybe half the students knew that 8% was the decimal 0.08 (most went right away to 0.8, which sounds a lot like point-zero-eight but is a lot different!), and only one could state clearly that 8 percent was 8/100 which was 0.08, since 0.8 was 8/10 or 80/100. And worse, even once we’d established that 8% could be typed on a calculator as 0.08, more students wanted to divide than multiply (I guess because they wanted a smaller number)?

So the conversation about making the table wasn’t around what I was hoping — identifying those students who’d found 8% of the principal but forgot to add it to the principal and so who got 10,000, 800, 64, for the balance, and identifying those students who’d found that she made $800 in interest the first year and so reasoned she’d get $10,000; $10,800; $11,600; $12,400; etc. Then we could have talked about linear vs. exponential growth and what that has to do with compounding, and we could have talked about 8% vs. 108%, and gotten into some key ideas that would let us make sense of an exponential rule.

Instead we got bogged down in 8% vs. 80%, and number that didn’t make any sense, like $10,000 / 8 or .8 or .08. Even though we talked through those issues, I lost a lot of kids in talking through those issues, and we didn’t have very good conversations about the other, more connected-to-the-concept-of-exponential-growth conversations.

So, what would you have done, with 20/20 hindsight? Here are my questions:

1. Can a class of students, the majority of whom don’t understand how to calculate with percents, learn compound interest? Should they be asked to? If so, how? If not, what should we do instead?
2. Would there have been a way to differentiate this lesson so that more students got through the right amount of it at their own pace? What could I have expected everyone to have gotten? What could I have done to support the most struggling? The most advanced?
3. Could technology have helped students focus on the concepts, rather than calculations? How? What technology?
4. Is there a way to present this concept more conceptually, so we avoided getting bogged down in the calculations? Were the students conceptually ready but lacked fluency, or were there underlying conceptual issues I needed to address?
5. How long does it take a good understanding of percents to sink in? Did I need to stop and teach a conceptual foundation for percents and then come back to compound interest? What would that do to my pacing guide (already pretty far behind with snow and 4 rounds of state testing instead of the 1-2 we used to do)?
6. And finally, does thinking about this particular story give us any more insight into the general question of teaching kids who come in with gaps in their understanding, fluency, or both?

Talking about race, class, gender, sexual orientation – stuff like that – in relation to math class feels like tricky territory. Race, class, gender, sexual orientation, language, physical ability, religion — they shouldn’t matter for student achievement. Does focusing on social identities in thinking about math class help or hurt students who are outside of the mainstream?

I think it’s really important to be aware of and responsive to students’ outside-of-math-class cultures and experiences. Here’s why. I want all my students to work as mathematicians in my class. I believe that all students have a playground in their brain for shapes and quantities, that all students can wonder, observe, conjecture, explore, refine, reason, explain… And I also believe that all my students are pretty novice about these things. I have to be able to listen to them and hear them as mathematical thinkers, even when their expressions are very, very novice — and therefore different from what I’m used to hearing.

Students’ outside-of-math-class experiences change the “accent” in which they speak math. Not necessarily their actual accent (though some of our students have accents or speak math in a different language than we do!) but more the way they think and express their thinking.

We don’t realize we all have accents. We think of people with accents that are strongly different than our own as being “weird” “different” “cool” “special” — and rarely think of ourselves as being people with accents (unless we have an accent that’s different from the people around us so it gets pointed out to us a lot).

I have a hard time recognizing my students doing math and being mathematicians when I am distracted by their “accent,” and I’m more distracted by it the more different it is from how I conjecture about, reason about, and explain quantities and shapes. In fact, I don’t always realize I have an accent — I believe I’m fluent in math and speak the universal language of math. Which makes what my students are saying and thinking sound “weird” “special” and even wrong.

Here are some concrete examples that I’ve read about or experienced:

• Dealing with uncertainty and the issue of right/wrong — different cultures, and in the US, different economic classes, have different ways of thinking about uncertainty, especially when it comes to uncertainty in school. In some cultures (for example, the latte-sipping liberal elite!) school is a place to learn to argue, to poke holes in accepted ideas, to get troubled and cause trouble. In other cultures school is a place to learn the right way to behave, talk, etc. School is your ticket into proper society… so encountering uncertainty, challenging authority, poking holes, etc. is bad news. This doesn’t mean that students whose cultures relate differently to authority and knowing should be told what to do and made to just memorize procedures… but it does mean that they’ll engage differently with the challenge of checking on their own whether a solution is correct, that they will need explicit coaching & parameters around times that they are debating the relative merits of different arguments when there is a teacher in the room who could be the voice of correctness but is choosing not to be.
• Written vs. oral communication — like everything, there is as much variation among individuals of the same “culture” as there is between cultures on this… but… that said: the role of writing vs. talking is different in different cultures. In some cultures the word on the page has a different kind of authority than the spoken word. In some cultures reading and writing is more of a communal, oral exercise where people talk about what they’re reading or writing as they read and write it. In some cultures girls are brought up to be more reserved, to not blurt — and so they may find it easier to use writing to marshall their thoughts before speaking. In some cultures, you talk things out before you put anything down on paper. So a “think-pair-share” task where students write silently and then share orally may go awesome in one classroom and bomb in another, because the students in the other classroom would have to “talk-think-write” to be successful on the same task.
• Consensus building & arguments — this one I’m more out on a limb about. I don’t know the research and have only read anecdotes but… it seems to me like one thing that happens in math class is there’s a tension between knowing things because you’ve deduced them through rigorous logic, knowing things because you have faith in the person telling them to you, and knowing things because they seem right inductively. Back in the Dolciani days, in theory everything was deduced through rigorous logic, and in practice a lot was taken on faith from the teacher. Student-led investigations these days in theory begin with intuition and inductive reasoning (wow, in all these examples, making a triangle just knowing the side lengths guaranteed that my triangle was exactly the same as yours) and end with rigorous deduction (that will always be true because I can prove there will always be a rigid transformation that maps the three sides onto their corresponding sides and since rigid transformations preserve shape and angle measure, all the corresponding parts will be the same). In practice, I think a fair number of investigations build up students intuition and inductive reasoning, but then the conclusion that this always works is taken on faith again. And it seems to me that there’s a cultural piece at work here. What is the way to win an argument in students’ out-of-math culture? Convincing others? How? By logic? By coercion? By force of personality? By showing lots of examples? By offering counter-arguments? Which is more important — being right or everyone agreeing? If students’ out-of-school experience is that agreeing is important, and that it’s not polite to bring a counter-argument if everyone else agrees, then where does that leave them in math class? How do we honor their experience of argument and also help them participate in doing math, as mathematicians?

It’s so, so important to me to meet mathematicians and math teachers from many cultures, and hear them talk about their experiences with school mathematics, their children’s experiences in school mathematics, and their students’ experiences with school mathematics, to help me hear my own accent.

As a white, American, mostly East Coast, mostly male, mostly middle-class person, I am so used to argumentation and picking things apart and valuing logic and trying to see “why does that work?” and “will that always work?”, and doing that kind of argument everywhere from the dinner table to friends’ houses that when I don’t see those same “accents” in my students I think they aren’t mathematical. I don’t see my students’ use of different kinds of mathematical knowing as they use their spatial intuition (not my strength) and can just “see” why something works, let alone when they decide to agree on a conclusion that doesn’t follow deductively because it preserves the feelings of everyone in the group (even if some of them “know” it doesn’t work). I don’t hear the mathematics in my students’ accents so I don’t know how to support them to be accountable to the math even as they’re being accountable to their group, or accountable to their lived experience of the world.

So… if you have a different perspective about your students’ math, about your own mathematical habits of mind, if you come from a culture where school math is a little different, where arguments happen differently, where authority and rightness are interpreted differently… I want to talk to you at TMC14, and I want to talk to you explicitly about your background and experience and how it’s made a difference in your math classroom!

One of our PoWs this week involved finding three consecutive numbers that added to 63. The four main approaches were:

1. Divide 63 by 3 and use that as the middle number
2. Guess and check, generally starting in the teens or low 20s
3. Write an equation: x + (x + 1) + (x + 2) = 63*
4. Think of 63 and 60 + 3 and find three consecutive digits that add to 3 (o, 1, 2) and then divide the 60 into three 20s, yielding 20+0, 20+1, 20+2. (This was the rarest strategy).

I’m always trying to define for myself what the heck Mathematical Practice #7, Look for and Make Use of Structure, means in actual kids’ thinking. I think strategy #4 is a particularly good example of a middle- or elementary-school version of making use of structure, taking advantage of place value.

I wonder what would have happened if the three consecutive numbers had added to 60 or 61.

I wonder if the fact that the structure they found doesn’t make every type of problem easier invalidates it as a strategy somehow?

Hmm….

Update: an upper-elementary student used a strategy that combined algebraic representation and looking for structure:

Y + (Y+1) + (Y+2) = 63.
Y + (Y+1-1) + (Y+2-2) = 63 – 1 – 2
Y + Y + Y = 60
3Y = 60
Y = 60 ÷ 3 = 20

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