Dan who is former high school math teacher updates on innovations, ideas, and news in math teaching. Dan Meyer has a solid blog that appeals to both my inner nerd as well as the greedy little kid who wants something he can take and apply.
This was new. I was on a raised platform with seven middle school students to my left and six to my right and several hundred math teachers surrounding us on all sides.
This wasn’t a dream. The MidSchoolMath conference organizers had proposed the idea months ago. “Why don’t you do some actual teaching instead of just talking about teaching?” basically. They’d find the kids. I was game.
But what kind of math should we do together? I needed math with two properties:
The math should involve the real world in some way, by request of the organizers.
The math should ask students to think at different levels of formality, in concrete and abstract ways. Because these students would be working in front of hundreds of math teachers, I wanted to increase the likelihood they’d all find a comfortable access point somewhere in the math.
I asked the students to tell each other, and then me, some quantities in the video that were changing and some that were unchanging. I asked them to describe in words Adam’s height above the ground over time. Then I asked them to trace that relationship with their finger in the air. Only then did I ask them to graph it.
I asked the students to “take a couple of minutes and create a first draft.” The rest of this post is about that teaching move.
I want to report that asking students for a “first draft” had a number of really positive effects on me, and I think on us.
First, for me, I became less evaluative. I wasn’t looking for a correct graph. That isn’t the point of a rough draft. I was trying to interpret the sense students were making of the situation at an early stage.
Second, I wasn’t worried about finding a really precise graph so we (meaning the class, the audience, and I) could feel successful. I wanted to find a really interesting graph so we could enjoy a conversation about mathematics. I could feel a lot of my usual preoccupations melt away.
After a few minutes, I asked a pair of students if I could share their graph with everybody. I’m hesitant to speculate about students I don’t know, but my guess is that they were more willing to share their work because we had explicitly labeled it “a first draft.”
I asked other students to tell that pair “three aspects of their graph that you appreciate” and later to offer them “three questions or three pieces of advice for their next draft.”
I like how they show he took longer to go up than come down.
I like how they show he reached the bottom of the slide a little before the video ended.
I think they should show that he sped up on the slide.
Etc.
If you’ve ever participated in a writing workshop, you know that workshopping one author’s rough draft benefits everyone’s rough draft. We offered advice to two students, but every student had the opportunity to make use of that advice as well.
And then I gave everybody time for a second and final draft. Our pair of students produced this:
Notice here that correctness is a continuous variable, not a discrete one. It wasn’t as though some students had correct graphs and others had incorrect ones. (A discrete variable.) Rather, our goal was to become more correct, which is to say more observant and more precise through our drafting. (A continuous variable.)
And then the question hit me pretty hard:
Why should I limit “rough-draft talk” (as Amanda Jansen calls it – paywalled article; free video) to experiences where students are learning in front of hundreds of math teachers?
My students were likely anxious doing math in front of that audience. Naming their work a first draft, and then a second draft, seemed to ease that anxiety. But students feel anxious in math class all the time! That’s reason enough to find ways to explicitly name student work a rough draft.
That question now cascades onto my curriculum and my instruction.
How should I transform my instruction to see the benefits of “rough-draft talk”?
If I ask for a first draft but don’t make time for a second draft, students will know I really wanted a final draft.
If I ask for a first draft, I need to make sure I’m looking for work that is interesting, that will advance all of our work, rather than work that is formally correct.
How should I transform my curriculum to see the benefits of “rough-draft talk”?
“Create a first draft!” isn’t some kind of spell I can cast over just any kind of mathematical work and see student anxiety diminish and find students workshopping their thinking in productive ways.
Summative exams? Exercises? Problems with a single, correct numerical answer? I don’t think so.
What kind of mathematical work lends itself to creating and sharing rough drafts? My reflex answer is, “Well, it’s gotta be rich, low-floor-high-ceiling tasks,” the sprawling kind of experience you have time for only once every few weeks. However I suspect it’s possible to convert much more concise classroom experiences into opportunities for rough-draft talk.
To fully wrestle my question to the ground, how would you convert each of these questions to an opportunity for rough-draft talk, to a situation where you could plausibly say, “take a couple of minutes for a first draft,” then center a conversation on one of those drafts, then use that conversation to advance all of our drafts.
Larry Berger, CEO of Amplify, offers a fantastic distillation of the promises of digital personalized learning and how they are undone by the reality of learning:
We also don’t have the assessments to place kids with any precision on the map. The existing measures are not high enough resolution to detect the thing that a kid should learn tomorrow. Our current precision would be like Google Maps trying to steer you home tonight using a GPS system that knows only that your location correlates highly with either Maryland or Virginia.
If you’re anywhere adjacent to digital personalized learning – working at an edtech company, teaching in a personalized learning school, in a romantic relationship with anyone in those two categories – you should read this piece.
Berger closes with an excellent question to guide the next generation of personalized learning:
What did your best teachers and coaches do for you—without the benefit of maps, algorithms, or data—to personalize your learning?
My best teachers knew what I knew. They understood what I understood about whatever I was learning in a way that algorithms in 2018 cannot touch. And they used their knowledge not to suggest the next “learning object” in a sequence but to challenge me in whatever I was learning then.
“Okay you think you know this pretty well. Let me ask you this.”
I’m absolute junk in the kitchen but I’m trying to improve. I marvel at the folks who go off recipe, creating delicious dishes by sight and feel. That’s not me right now. But I’m also not content simply to chop vegetables for somebody else.
I love the processes in the middle – like seasoning and sautéing. I can use that process in lots of different recipes, extending it in lots of different ways. It’s the right level of technical challenge for me right now.
In the same way, I’m enamored lately of instructional routines. These routines are sized somewhere between the routine administrative work of taking attendance and the non-routine instructional work of facilitating an investigation or novel problem. Just like seasoning and sautéing, they’re broadly useful techniques, so every minute I spend learning them is a minute very well spent.
For example, Estimation 180 is an instructional routine that helps students develop their number sense in the world. Contemplate then Calculate helps students understand the structure of a pattern before calculating its quantities. Which One Doesn’t Belong helps students understand how to name and argue about the names of mathematical objects.
(Aside: it’s been one of greatest professional pleasures of my life to watch so many of these routines begin and develop online, in our weirdo tweeting and blogging communities, before leaping to more mainstream practice.)
I first encountered the routine “Two Truths and a Lie” in college when new, nervous freshmen would share two truths about themselves and one lie, and other freshmen would try to guess the lie.
We ask the student to write three statements about their object – two that are true, and one that is a lie. They describe why it’s a lie.
Here are three interesting statements from David Petro’s circle graph design. Which is the lie?
The shaded part is the same area as the non shaded part.
If these were pizzas, there is a way for three people to get the same amount when divided.
If you double the image you could make a total of 5 shaded circles.
And three from Sharee Herbert’s interesting parabola. Which is the lie?
The axis of symmetry is y=-2.
The y-intercept is negative.
The roots are real.
Then we put that thinking in a box, tie a bow around it, and slide it into your class gallery.
The teacher encourages the students to use the rest of their time to check out their classmates’ parabolas and circle graphs, separate lies from truth, and see if everybody agrees.
Our experience with Challenge Creator is that the class gets noisy, that students react to one another’s challenges verbally, starting and settling mathematical arguments at will. It’s beautiful.
So feel free to create a class and use these with your own students:
BTW. Unfortunately, Challenge Creator doesn’t have enough polish for us to release it publicly yet. But I’d be happy to make a few more TTL activities if y’all wanted to propose some in the comments.
Taylor registered her Twitter account this month. She’s brand new. She’s posted this one tweet alone. In this tweet, she’s basically tapping the Math Teacher Twitter microphone asking, “Is this thing on?” and so far the answer is “Nope.” She’s lonely. That’s bad for her and bad for us.
It’s bad for her because we could be great for her. For the right teacher, Twitter is the best ambient, low-intensity professional development and community you’ll find. Maybe Twitter isn’t as good for development or community as a high-intensity, three-year program located at your school site. But if you want to get your brain spinning on an interesting problem of practice in the amount of time it takes you to tap an app, Twitter is the only game in town. And Taylor is missing out on it.
It’s bad for us because she could be great for us. Our online communities on Twitter are as susceptible to groupthink as any other. No one knows how many interesting ways Taylor could challenge and provoke us, how many interesting ideas she has for teaching place value. We would have lost some of your favorite math teachers on Twitter if they hadn’t pushed through lengthy stretches of loneliness. Presumably, others didn’t persevere.
Interesting looking at my early #MTBoS tweets. Most of the time I got no response at all. I wonder why I kept at it.
So we’d love to see fewer lonely math teachers on Twitter, for our sake and for theirs.
Last year, Matt Stoodle invited people to volunteer every day of the month to check the #mtbos hashtag (one route into this community) and make sure people weren’t lonely there. Great idea. I’m signed up for the 13th day of every month, but ideally, we could distribute the work across more people and across time. Ideally, we could easily distinguish the lonely math teachers from the ones who already experience community and development on Twitter, and welcome them.
So here is a website I spent a little time designing that can help you identify and welcome lonely math teachers on Twitter: lonelymathteachers.com.
It does three things:
It searches several math teaching hashtags for tweets that a) haven’t yet received any replies, b) aren’t replies themselves, and c) aren’t retweets. Those people are lonely! Reply to them!
It puts an icon next to teachers who have fewer than 100 tweets or who registered their account in the last month. These people are especially lonely.
It creates a weekly tally of the five “best” welcomers on Math Teacher Twitter, where “best” is defined kind of murkily.
That’s it! As with everything else I’m up to in my life, I have no idea if this idea will work. But I love this place and the idea was actually going to bore a hole right out of my dang head if I didn’t do something with it.
It’s the muscle that connects my capacity for noticing the world to my capacity for creating mathematical experiences for children. (I should also take some time in 2018 to learn how muscles work.)
Right there you have an image created by Brittany Wright, a chef, and shared with the 200,000 people who follow her on Instagram. Loads of people before Ilona had noticed it, but she connected that noticing to her capacity for creating mathematical experiences for children. She surveyed her Twitter followers, asking them to name their favorite banana, receiving over one thousand responses. Then on her blog she posed all kinds of avenues for her students’ investigation – distributions, probability, survey design, factor analysis, etc.
But Ilona ran a marathon and I want to run some wind sprints. I need quick exercises for strengthening that muscle. So here are my exercises for 2018:
I’m going to pause when I notice mathematical structures in the world. Like flying out of the United terminal in San Antonio at last year’s NCTM where I (and I’m sure a bunch of other math teachers) noticed this “Suitcase Circle.”
Then I’ll capture my question in a picture or a video. Kind of like the one above, except pictures like that one exist in abundance online.
Civilians capture scenes in order to preserve as much information as possible. That’s natural. But I’ll excerpt the scene, removing some information in order to provoke curiosity. Perhaps this photo, which makes me wonder, “How many suitcases are there?”
In order to gauge the curiosity potential of the image, I’ll share the media I captured with my community. Maybe with my question attached, like Ilona did. Maybe without a question so I can see the interesting questions other people wonder. You may find my photos on Twitter. You may find them at my pet website, 101questions.
I want to get to a place where that muscle is so strong that I’m hyper-observant of math in the world around me, and turning those observations into curious mathematical experiences for children is like a reflexive twitch.
(Plus, that muscle will be more fun to strengthen in 2018 than literally any other muscle in my body.)
BTW. Check out the 3-Act Task I created for the Suitcase Circle. It includes the following reveal, which I’m pretty proud of.
Suitcases - Act 3 - Vimeo
BTW. The suitcase circle later turned into Complete the Arch, a Desmos activity, which has some really nice math going on.
Top Chef. Project Runway. The Voice. Live competition shows have introduced audiences to the worlds of cooking, fashion, and singing — and opened a window into the intricate craftsmanship that these industries demand. Now it’s time for one of America’s most under-recognized professions to get the same treatment. Hi, teachers!!
Two teams of math teachers will teach a lesson to a live audience and receive judgment from a panel of “teacher celebrities.”
Teaching is and should always be a collaborative endeavor. Competition is what causes rifts among staff, encourages teaching in silos, and prevents us from growing together.
Good teaching requires complicated decision-making based on a teacher’s long-range knowledge of a student and of mathematics. We should reach for any opportunity to make those decisions transparent to the public, who will always benefit from more education about good education. But a live event with an audience you don’t know and can’t interact with individually will necessarily flatten “teaching” down to its most presentational aspects, down to teachers dressing up in costumes, down to Robin Williams standing on desks in Dead Poets Society.
I asked teachers what kind of TV show would do justice to the complexity of teaching, if The Voice and Top Chef were the wrong models. Jamie Garner and James Cleveland both suggested The Real World, which seems dead on to me.
Maybe more like The Real World - stick a bunch of teachers in the same house, and working at the same school.
Rather than a game show, I envision that it would be more like The Real World: Math Class. This would allow for a development over time of understanding of the work of a classroom, not in one hour segments focused on competition. 1/2
The Real World a) isn’t a competition, b) allows for characters to develop over time, and crucially, c) isn’t a live event. It is edited. You don’t watch the cast members do anything mundane. In the case of teaching, we’d love for the public to understand that good teachers assess what students know and adjust their instruction in response. But no one wants to watch a class work quietly on a five-minute exit ticket in real time. So the show would edit quickly past students completing the assessment and straight to the teacher trying to make sense of a student’s thinking, involving the audience in that process.
The challenge I’d like to see the folks at Chalkbeat take up is how to make those invisible aspects of teaching – the work that takes place after the bell – visible to the public. The work of presenting is already teaching’s most visible aspect.
2018 Jan 1. Chalkbeat’s Editor-in-Chief, Elizabeth Green, clarifies her rationale for launching the competition and responds to some concerns raised here and on Twitter. She describes lesson study as the touchstone for her Teach Off and how she’s had to alter that format to fit SXSW.
It’s a really interesting article, full of references to the education scholars who have inspired her work for a decade. But I still tend to think she and the members of her design team have underestimated the magnitude of those compromises and how they’ll distort the approximation of good instruction her audience will encounter.
Depictions of mathematics in TV and film generally lack nuance. When Hollywood doesn’t hate math, it reveres it, genuflecting before the eccentric, generally white male weirdos taking up space in its highest echelon – your Will Huntings, your John Nashes, etc. – with little in between.
But Arthur nails the nuance in “Sue Ellen Adds It Up,” and reports three important truths about math in ten minutes.
We are all math people. (And art people!)
Sue Ellen is convinced she isn’t a math person while her friend Prunella is convinced there’s no such thing as “math people.” You may have this poster on your wall already, but it’s nice to see it on children’s television. Meanwhile, Prunella is convinced that, while she and her friend are both “math people,” only Sue Ellen is an “art person.” Kudos to the show for challenging that idea also.
Informal mathematical skills complement and support formal mathematical skills.
Sue Ellen says that she and her family get along fine without math everywhere “except in math class.” They rely on estimating, eyeballing, and guessing-and-checking when they’re cooking, driving, shopping, and hanging pictures. Prunella tells Sue Ellen, accurately, that when Sue Ellen estimates, eyeballs, and guess-and-checks, she is doing math. Sue Ellen is unconvinced, possibly because the only math we see her do in math class involves formal calculation. (Math teachers: emphasize informal mathematical thinking!)
We need to create a need for formal mathematical skills.
Sue Ellen resents her math class. She has to learn formal mathematics (like calculation) while she and her family get along great with informal mathematics (like estimation). Then she encounters a scenario that reveals the limits of her informal skills and creates the need for the formal ones.
She’s made a painting for one area of a wall and then she’s assigned a smaller area than she anticipated. She encounters the need for computation, measurement, and calculation, as she attempts to crop her painting for the given area while preserving its most important elements.
Nice! Our work as teachers and curriculum designers is to bottle those scenarios and offer them to students in ways that support their development of formal mathematical ideas and skills.
This is my best attempt to tie together and illustrate terms like “intellectual need” and expressions like “if math is aspirin, how do we create the headache.” If you’re looking for an elaboration on those ideas, or for illustrations you haven’t seen on this blog, check out the video.
The Directory of Mathematical Headaches
This approach to instruction seriously taxes me. That’s because answering the question, “Why did mathematicians invent this skill or idea?” requires a depth of content knowledge that, on my best days, I only have in algebra and geometry. So I’ve been very grateful these last few years to work with so many groups of teachers whose content knowledge supplements and exceeds my own, particularly at primary and tertiary levels. Together we created the Directory of Mathematical Headaches, a collaborative document that adapts the ideas in this talk from primary grades up through calculus.
It isn’t close to complete, so feel free to add your own contributions in the comments here, by email, or in the contact form.
Lisa Bejarano’s post Two Kinds of Simplicity offers a useful idea about teaching complex fractions, but much more interesting to me are the three kinds of knowledge she puts to work in her class.
Knowledge About Teaching
Lisa has read widely from sources online and offline and has a great memory. So when she asks herself, “How am I going to teach [x]?” she can quickly summon up all kinds of helpful posts, essays, books – even the mental recording of previous classes she’s taught on [x].
Knowledge About Students
I stopped to think about how this would work with my class.
Lisa has taught long enough and knows her students well enough that she can test each of those resources out in her head, all during the lunch break before class. You can see her swiping right and left on each of them – “Yeah, maybe this idea. Definitely not that one.” – as she sees her students in her imagination. I’m sure Lisa is open to the possibility that her flesh-and-blood students will differ in surprising and awesome ways from her mental model of those students. I wouldn’t bet against her intuition, though.
Knowledge About Math
She ultimates decides to start her precalculus students with the elementary school analog of their lesson, turning an abstract fraction division problem into a more concrete one.
Then, as her students acquaint themselves again (or in some cases for the first time) with helpful models for that division, she builds back up to the abstract version of her task.
Lisa is only able to move up and down the ladder of abstraction like this because she knows a lot of math – specifically where it builds from and towards. If she doesn’t know that math, her options for helping her students basically shrink down to “let’s solve a few together.”
Finally
I don’t know if it’s possible to practice what Lisa is doing here. It’s knowledge, the tightly connected kind you get when you spend thousands of hours in math classes, reflect on those observations, write about them, talk with other people about them, and then use them to inform what you do in another math class.
It’s possible, even easy, to spend the same number of hours without acquiring that tightly connected knowledge.
It’s something special to see it all put to use.
BTW. My guess is a lot of those knowledge connections were tightened because Lisa is a dynamite blogger. On that theme, let me recommend The Positive Effects of Blogging on Teachers, an article which does a great job describing ten reasons why teachers should think about blogging.
It’s wonderful to be here. I spend most of my days with people who don’t fully get me. Wife, friends, dog – none of them gets me like you get me.
None of them understands the feeling of mathematical epiphany that motivates my professional life, the sudden transition from not knowing to knowing.
One of my earliest mathematical epiphanies was the realization that if you let the number of sides on a regular polygon increase without bound, you get a circle.
And that all the relationships you find in a regular polygon have analogous relationships in a circle. For me, that realization was literally a religious experience. I finished that limit on the back of a church bulletin while a churchlady glared at me.
So on the one hand it’s great to be in this room – I am among my people – but on the other hand it’s really uncomfortable to be here because you all make me really aware of my privilege, and aware of how many people are not in this room.
The economic 1% gets a lot of grief lately and whether we know it or not, whether we like it or not, we are all also in the 1% – the mathematical 1%.
In 2014, 2.8 million degrees were awarded in US universities – bachelors, masters, and doctorates – and 1.1% of them were in mathematics. If you change the denominator to reflect not advanced degree holders but anyone with a high school diploma our elitism becomes even more apparent.
I was on Instagram last night checking out the #MAAthfest hashtag along with The Rich Kids of Instagram. While there are fewer yachts, bottles, and shrink wrapped stacks of bills on the left, and maybe more plaid and elbow patches, there is still the same exuberant sense of having arrived. We have made it.
And just as the economic 1% creates systems that preserve its status – policies like the mortgage interest deduction for homeowners, discriminatory lending policies, and lower taxes on capital gains than income – through our action or inaction we create systems that preserve our status as the knowers and doers of mathematics.
When someone says, “I’m not a math person,” what do you say back? Barring certain disabilities or exceptionalities, everyone starts life a math person. Infants can recognize changing quantity. Brazilian street vendors develop sophisticated arithmetic algorithms before they set foot in school.
It is our action and inaction that teach people they are not mathematical. So please consider taking two actions to extend your privilege to the other 99% of humanity.
First, change the definition of mathematics that people experience.
[Here we explored together Circle-Square, a task that involves questioning, estimation, intentionally declaring wrong answers, recalling what you know about circles and squares, computing an answer, and verifying it. You can watch it.]
Now I don’t want to suggest to you that this is the experience that will change a person’s definition of mathematics and extend our privilege to the 99%. I just want to suggest to you that you just had a very different mathematical experience than the people who encountered that problem in its original form:
Mark an arbitrary point P on a line segment AB. Let AP form the perimeter of a square and BP form the circumference of a circle. Find P such that the area of the square and circle are maximized.
That experience offers people only a certain kind of mathematical work. You recall what you know about perimeter, circumference, and area, compute it, and verify it in the back of the book.
Those verbs are our mortgage interest rate deduction, our discriminatory lending policy, and our tax advantages. Through our action and inaction, society has come to understand that math is a merry-go-round revolving endlessly through those three verbs – remember a procedure, compute it, verify it.
You might think, “Well that’s what math is,” but the definition of math isn’t a physical constant in the universe. It’s defined by people, just as people define the ways that wealth and power accrue in the world. That definition is then underlined, reflected, and enforced in public policy, curriculum, and syllabi.
So, second, let’s change the definition of mathematics in public policy, curriculum, and syllabi.
To begin with, let’s eliminate policies that require intermediate algebra for college study.
The facts as I understand them are that:
College completion is increasingly essential to even partial economic participation.
College study is generally predicated on a student’s ability to pass a mathematics entrance exam. In the California State University system, that exam is heavily weighted towards intermediate algebra, problems like these, the majority of which depend on the recollection of an obscure and abstract procedure:
Students fail these exams in staggering numbers (68% nationally) placing into “developmental math” courses, courses which cost time and money and don’t offer credit towards graduation.
Those courses are disproportionally composed of African American and Latinx students.
Only 32% of students in developmental math ever take a math course required for graduation.
It’s hard to imagine a machine more perfectly configured for the preservation of mathematical privilege.
Those statistics would bother me less if either a) I believed in the value of intermediate algebra, b) better alternatives weren’t available. Neither is true. That intermediate algebra has little value to the majority of college educated professionals hardly requires a defense. As Uri Treisman said, “The most common use of algebra in the adult world is helping their kids with algebra.”
I am sympathetic to the argument, however, that we shouldn’t choose college requirements solely because they’re useful professionally. College should offer students a broad survey of every discipline – a general education, as it’s called. That survey should generate intellectual interest where perhaps there was none; it should awaken students to intellectual possibilities they hadn’t considered; it should increase the likelihood they’ll speak favorably about the discipline after college.
Those goals are served poorly by intermediate algebra. And better alternatives to intermediate algebra exist to serve the CSU’s desire to “assess mathematical skills needed in CSU General Education (GE) programs in quantitative reasoning.”
Specifically, statistics.
When 907 CUNY students were assigned either to remedial algebra, remedial algebra and supplementary workshops, or college-level statistics and workshops, that latter group a) passed their course in greater numbers (earning credit!) and also b) accumulated more credits in later courses.
So we should be excited to see the California State University drop its intermediate algebra requirement for graduation. We should be excited to see a proposal from NCTM that reserves intermediate algebra concepts for elective courses in high school. But we should regard both proposals as tenuous, and understand that as people of privilege, our support should be vocal and persistent.
We can choose action or inaction here. Through your action, the definition of math may change so that it’s accessible to and enjoyed by many more people, so that many more people understand themselves to be “math people.” I want to be clear that own privilege will diminish as a result, that we will become less special, but that humanity as a whole will flourish. Through your inaction, or through your tentative, private support for initiatives like these, the existing definition will endure, along with the existing distributions of privilege. Choose action.
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