What Math is Happening?

I stumbled upon some blogs doing 180 Day of Math.

I’m not that ambitious.

But I think it’s fun to document some of the math that’s happening.

Week 1 – AMDM and Algebra 1

How many different colors do we need to color a map? Other get-to-know-you-awesomeness. And building tall towers. Follow the criteria for success!

Week 2 – Algebra 1

Preassessments, set-up materials, more teambuilding, analyze a pattern and start writing about math!

Week 2 – AMDM

Find the pattern for Triangle Numbers (presented visually). Then practice writing up a problem/solution. And presenting.

P.S. I’m really glad Dan Goldner and his Pre-Calc class exist because I’m modeling my Advanced Mathematical Decision Making class off of it. I think the seniors will like the independence.


Authentic Audience


Next year, my math class will hopefully involve each week ~2.5 days of problem-solving, followed by ~1.5 days of writing/videoing/making some product about a chosen problem.

Over the past months, I’ve gone back and forth with being concerned about motivating WHY we have to make products. I mean, I plan to talk to students about why explaining your work is good for you and good for others. But those reasons aren’t always the most compelling.

Then, I read Drive which (if I remember correctly) posits that if you have met people’s need for competence and autonomy, purpose is less important, though awesome (and I plan to have tons of choice and mastery all over my class). And I took Developmental Designs, which identifies student needs as competence, autonomy, relationship, and fun. So I worried a little less about motivating the why of products.

But then I was talking with (one of my) my awesome student teacher from last year, who was thinking about an authentic audience (which would so much more clearly motivate why). And I got thinking again.

For awhile, I’ve wondered why so many high schools have Literary Magazines but no math equivalent. And then, post talking with said former-student-teacher, I thought — why not make it happen?

So, to start (baby steps) to address the authentic audience/why we make products issue, I am committing to the following for this coming year.

1. As often as possible, involve choice in what product students make to explain their math work.

2. As often as possible (which will be less so than #1), define or have students define an audience for their product (or maybe this can be all the time and just often be “other students who need help” or “myself for reference” and then work on getting other audience in the future … hmmm).

3. Have a classroom website (thanks, Mom and English Teaching Vegan for helping me get mine started last night!) where students somewhat frequently post products for parents/anyone to see. (I think this is a cool first step and am interested in developing it more in the future, maybe with purposeful interactions with  … someone(s).)

4. Twice during this year, convene a group of interested BPS math teachers and their students and a bunch of submissions to produce a “Math Mag” either in print or (more ideally) online.

I think these steps are doable (though suggestions for how to structure the students-posting-on-class-website would be welcome) and a step towards making our math output authentic. What do you think?

Same Problem, Multiple Days (Part 2)

I’ve been mapping out Unit 1 more specifically, and I feel I may have partially resolved my own query about spending more than one day on the same problem (in a way that incorporates some of the commenters from my first post!).

Many of the weeks are set up the following way now (after Kogut and I got some time to talk during our awesome retreat).

Monday: Community building, week set-up, and launching of several problems
Tuesday: Students continue working on the several problems.
Wednesday: Students wrap-up their work on several problems, choose one to make into a product (paper, video, pamphlet, flipbook, etc).
Thursday: Peer-feedback and revision on the product.
Friday: Year-long project, Unit project, summarizing, big picture, community building again etc.

So on Monday-midday Wednesday, students will have some number of problems (3-6 or so) that they can spend their time on. Then Wednesday and Thursday, students will choose a problem to spend more time on, making their work more formal and complete.

How does this help resolve my earlier question? First, this system means that a student controls how much time s/he spends on each problem earlier in the week — if someone gets into a problem, they can stick with it or if they want to do a bunch of problems in slightly less depth, they can do that too. Further, if a student does multiple problems on Monday-Wednesday, they will have to spend more time with one of them, but will get to choose which one it is and will get to work on it in a creative way. Hopefully all this will mostly lead to meaningful engagement with problems, whether in passing or in depth and will remove the angst of my trying to get the whole class to dive deeper on the same problem in the same way at the same time.


What’s the Purpose?

Awhile back, Dan Meyers posted about how a teacher’s stance about why we study math will strongly affect the classroom s/he creates. In particular, he talks about the different classrooms that will come from a stance of students will be interested if they see that this math is useful for a job versus this math is fascinating and cool and makes me have questions. (Read his post; he says it better.)


I’ve been thinking about this a lot. I’ve spent time during my professional life feeling compelled by each stance. Now, I believe both have their place. But in particular, I’m excited about talking to students about this!

If I were to break up the math that happens in my version of Algebra 1, I would put it in 4 categories.

1) Math is fascinating/fun (ex. Eric the Sheep).

2) Math is useful, in two fairly distinct categories.

  • It can model what we do in real life in a way that’s illuminating even if you wouldn’t actually “do this” (ex. modeling real life linear functions in a variety of ways)
  • This is how people actually use math in their jobs (ex. exhibition)

3) The Geometry teachers will have my head if I don’t teach you this (ex. solving one-variable equations with lots of steps).

I plan to discuss these categories with students and then share with them what math we’re doing and why. I think owning the purpose will aid student buy-in a lot. I can say things like, “We’re going to model guitar lessons using math. You probably wouldn’t actually do this in real life, but let’s see what you notice because it might make some things clear.”  Or, “This is how social scientists actually use the math we’ve been learning.” Or, “This has absolutely no real life applications except problem solving is good for your brains and this is cool.” Or, “Yup, we’re going to learn a content skill right now and practice it. It’s going to get algorithmic for a minute, but here’s why this thing is so important to have down, and I promise I’ll only do this to you occasionally.”


Same Problem, Multiple Days

Some of the best days in my classroom this year were the first days of students grappling with a good (low threshold, high ceiling, high cognitive demand, content-illuminating) problem. Students were engaged, discussing, modeling, struggling, defending their thinking, etc.

Many of these problems require multiple days. By the end of the first day, students may have “gotten” parts of the problem, may have even generalized, but they weren’t (according to me) done. Many students though, despite high interest/engagement on the first day, demonstrated much less interest in returning to the same prompt.

The great thing about these good problems is how much you can do with them! So how do I teach/motivate students to sustain their interest over multiple days with the same prompt? Will discussing the value of sticking with the same problem (the why we do this) and a class norm of “you’re not done until you’ve used at least two methods and two (three?) representations” do the trick?


Making Choice Real

As I’ve read some fun books about making the tasks we do in school more authentic/genuine/engaging, one common theme is student choice. Image

But …

How do we make choice consistently meaningful? There are a few projects I do already where choice is (fairly) meaningful. When we study data for our Exhibition, students get to choose one of the variables they are investigating (and their choices range from % of population in prison, to MCAS scores, to physicians/1000 people). That choice seems meaningful. Often I allow students to choose a method for solving; that also seems meaningful.

But when I imagine students working on the everyday tasks next year, I struggle to imagine how choice could work. Here are my concerns.

First, I’m banking heavily on students completing tasks in different ways and teaching each other those ways in order to meet a myriad of Algebra standards. How will students compare methods if everyone is doing different problems? Perhaps it’d be fine if a critical mass of students were doing each problem and then together had to come up with a variety of methods.

Second, how in the world do we make the choice meaningful? Say I do put out 4 different word problems — what’s the process for having students read all the problems and having them choose the one that would be most interesting/accessible for them?

Literally as I’m writing this, I’m thinking to myself, well, it seems I do have a plan …

1) Offer choice in topic when it makes sense (i.e. Exhibition).

2) Choice in method is a fundamental value and so will (almost?) always be assumed.

3) Choice in task is perhaps most meaningful for me when there are few enough choices that student-to-student teaching can happen and when the choices are quickly processed by students (so more visual or topical than verbal).

“Students don’t know things!”

In conversations with anyone from members of the math department to random strangers in coffee shops (all of whom have opinions about teaching) to family members, people often bemoan to me the idea that, “students these days don’t know things.” The complainant will often follow this statement with a list of content that students don’t know – how to simplify radicals, where India is, how to identify a complete sentence, etc.

And you know what, it’s sometimes true. Some students don’t know how to simplify radicals* (unless they learned in middle school). But I am only worried by what students “don’t know” when I am trying to teach them to solve a specific type of problem in a historically-identified efficient way. When I give students a rich, low-threshold, high-ceiling task (which by design means there are multiple ways to enter the problem and that students can take their thinking far), I get to see what students know. They can enter the problem and move through the more challenging questions at their own pace/using their own method. They can compare their method with others, argue about and defend an answer, and find another way to answer the same question.

Basically, the problem isn’t about students not knowing things, it’s about me not asking the right questions often enough. And accessing prior knowledge doesn’t (have to) mean trying to remember the CMP book that teaches the topic you’re about to address or writing a contrived word problem about food to relate something to students’ lives. Accessing prior knowledge can mean offering up a really good question** and letting students pursue it any way that makes sense to them (and they will!). When I allow students to do that, I get to experience all the things students know. Then, I can work on connecting all those ways and building up students’ problem-solving toolboxes.


p.s. I painted a slightly idealized picture of low-threshold, high-ceiling tasks. But it’s mostly accurate.

* And who the hell cares?
**Eventually, hopefully, I’ll figure out how to tap into students’ questions.