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.



And now for Geometry…

Much my teaching career has been focused on algebra.  You know, the subject that most adults feel they should tell me that they hated and never use… sigh.  Well, while teaching algebra, I have always taught other things as well, this year, my primary class (in time) will be…geometry!  (But, with my residents and my work with Langer, who knows, algebra may still take most of my time…)

So now I need to think deeply about geometry.  And read many thing about this visual land of mathematics.  Hooray geometry!


But what about the year-long projects?  I truly have NO idea what a year-long project would look like in a geometry classroom.  Sure, in an algebra classroom, systems of equations are PERFECT: these problems use ALL of the objectives from the year and have multiple entry points.  But what does that look like in geometry?  

…And what about the units of study?  I have outlined some sort of thinking about that but there seem to be MANY (seemingly unrelated) topics mushed together, which didn’t particularly help my little ones last year.  Are there lovely curriculum maps that exist (and make sense)?

And focusing on the mission-driven work of community leadership and the multi-dimensions of social justice? Last year we looked at tiny houses (environment) and shipping food donations (resources for all) but all in all, the work wasn’t driven in the mission, for the most part.  Ideas of how to do this more?


Oh, I have much to read, write, and plan about… huzzah geometry!



Math for Social Justice

I wish someone had said this to me 6 years ago. Maybe it’s obvious, maybe controversial (I’m actually not sure), but I figured it’s worth mentioning.

Math for social justice definitely is using mathematics to analyze social systems and real-world situations through a critical lens (i.e. racial patterns of home ownership or average pay rates).

Math for social justice also, definitely, is posing interesting problems (related or unrelated to real life) to students and encouraging/facilitating/coaching them to use their personal understanding and knowledge to puzzle through it. Or, posing rich situations and having students write/answer their own questions.


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.”


Howard Gardner and I Should Be Friends

I realize that everyone and their mother has to read Gardner for education grad school. Or at least learns about his multiple intelligence theory. But I am currently reading a book about something different, his book “The Unschooled Mind” (1991). Why am I telling you this? Because Gardner is fairly radical in ways beyond the multiple intelligences and I’m enjoying reading him, despite the fact that it’s going at a snail’s pace. (If he also was problematic, feel free to let me know!)

His basic idea in this text (at least so far) is that when we are small, we are able to learn lots of things (language! walking! sneezing!). We also build ourselves a schema for understanding the world that mostly works. Then we go to school where people try to teach us things and for most … none of it sticks. Or very little. And so we often, out of the classroom, revert to our pre-school intuitions, most of which are incomplete, inaccurate, or unsophisticated. (His example: What forces are acting on a ball that someone has tossed in the air? Most people imagine a force besides gravity/friction.)


A few people will, Gardner argues, reach “disciplinary understanding” where we can apply our deep conceptual and skill knowledge to novel situations. But that’s just because of their own interest/motivation. School doesn’t really foster this “disciplinary understanding.”

Perhaps, knowing me as you do, you can see why reading about Gardner’s premise had me jumping up and down. Plus, he’s snarky and dry.

  • Explaining how entrenched our young-selves’ understandings are, “in nearly every student there is a five-year-old “unschooled” mind struggling to get out.” (p. 5)
  • Defining school learning, “students simply respond, in the desired symbol system, by spewing back the particular facts, concepts, or problem sets they have been taught.” (p. 9)

I am totally looking forward to reading more about how to bridge intuitive learning and school learning (or even better, turn most of school learning into intuitive learning where students revise their thinking and lose the naive parts of understanding). This past week, Kogut and I spent the week holed up in a cottage, working on curriculum. Here’s hoping that my work during that time (Unit 1: Linear and Non-Linear Patterns) encourages students to discover and add to their intuitive schema rather than regurgitate meaningless symbols.


Genuine Systems of Equations (Help!?)

Kogut and I are on retreat in lovely Maine, listening to pandora extensively, working on re-writing our curriculum to be awesome in all the ways we’ve talked about so far.

In some ways, it’s going really well. Kogut (see below) is taking over Unit 0 and planning how we will get to know the students and start teaching them all those vitally important skills of working together, persevering through frustration, discussing math, using various tools, listening to each other, taking breaks when needed, etc.

I’m working on Unit 1 (linear, then non-linear patterns), using a combination of “math is cool!” awesome visual patterns and “math is useful” real-life patterns. Mapping our foci over the course of the weeks (anything always goes, but sometimes we’ll require some tabling, and then some graphing). And thinking about a myriad of ridiculous/fun products students can make to communicate their understanding (flipbooks, videos, write-ups, self-help pamphlets, etc.).

But. We are both into the idea of having a year-long project (problems that students can return to over the course of the year, that use various tools from throughout the year and allow students to showcase sophistication in the math as they learn more). Systems of equations seems like a pretty ideal topic to both of us — you can use tables! equations! diagrams! graphs! technology! to solve systems. But. Finding non-contrived systems problems. I can’t stomach the problems that pretend we wouldn’t already know the answer, or that are just totally irrelevant.

So I googled “genuine systems of equations” and got this blog post by Dan Meyers. Which mostly served to make me feel more hopeless because … yeah, I agree. And I have such a hard time finding genuine/authentic systems of equations problems (yes, I know about cellphones).

Here are three sample problems that might be acceptable. Do you have suggestions? Or more problems? Keep in mind that we will not require that every problem be solved in a specific way just a variety (most problems: solve 3 ways; some: 2 or 4 ways). We will also use some of Dan’s 3-act system-y problems, but one of our goals is to relate as many problems as possible to Community Leadership …

Problem 1: (This is in the realm of math is cool, not actually useful. It is borderline to me.)

Burning Candles.  People are lighting candles at a vigil to honor those who died in the Bangladesh Garment District Factories Collapse.  There are two different types of candles: green and red to represent their flag.  The red candles are 6 inches tall and burn ½ an inch per hour.  The green candles are 7 inches tall and burn ¾ of an inch per hour.

(a)  A photographer wants to take a dramatic picture when all of the candles are the same height; when should she take the picture?

(b) The local fire department wants assurance there will be organizers with all of the candles throughout the night until all of the candles have been extinguished (or have burned completely).  If the vigil starts at 8 PM, when should the organization expect to have volunteers shifts end?

Problem 2: (I think this is ok … though maybe stupid because the answer to whether Consalvo ever raised as much as Conley will be answerable by September 24th.)

Fundraising.  Dan Conley entered the Boston mayoral race on April 3rd with $866,000.  This amount was about twice as much as his closest rival, Rob Consalvo.  Use the internet to determine a reasonable amount each candidate fundraised per week.

a) Use what you find to determine whether Consalvo had as much campaign money as Conley at any point in time between April 3rd and the Democratic Primary on September 24th.

b) What do you think about corporations or unions donating large amounts of money to campaigns?

Problem 3: (This is a somewhat simplified but fairly real scenario.)

Minimum Wage.  According to the US Census, the average cost of living for a one-parent, two-child family (FAMILY A) in Boston is $5,260 per month.  Assuming a family could get a childcare subsidy or arrange for free childcare, the average goes down to $3,769.  The cost of living for a two-parent, three-child family (FAMILY B) in Boston is $7,058.  If we assume one of the parents provides childcare, the average goes down to $5,006.  Assume the parent in Family A works a minimum wage job, $8.00/hour.  Assume one of the parents in Family B works a slightly above minimum wage job, taking home $9.50 per hour.

a)  After how many hours will each family “break even” (i.e. earn enough money to cover their expenses)?

b)  After how many hours will the families have the same amount of savings/debt?

c)  Read this:  What should the government do to help this situation?



What needs to exist at the beginning?

There is a joke that floats around that aludes to the idea of making a beautifully organized classroom will make the school year perfect.  As if the children will walk in and think to themselves “ooooh, look at the boarders on that bulletin board!  And the scissors are labeled!  I will do my better than my best to corporate with everything this lovely lady has to say!”  I mean, it’s pretty amusing.  But true to some extent.

I teach ninth, tenth, and eleventh graders at a 9-12 high school.  The tenth and eleventh graders are, more often than not, students who I have already had the pleasure of teaching for at least one year; they know what they are getting into when their schedule declares I am their math teacher.  But the ninth graders, oh the ninth graders, have no idea.  I tell my former students to warn my new students about their upcoming adventures.  Most respond with something along the lines of “I won’t even know how to begin, oh maybe I’ll tell them the story about when you…”  

In the first days of school, we are checking each other out.  What will happen if I do this?!  Students test out behaviors like trying not to read, helping their peers with communicating their thinking, or following me around for more hints.  I test out jumping around, not giving reasonable answers to many questions, and taking students’ words at face value.  Both parties are trying to glean as much information about the other as possible without being painfully obvious.  (I mean, I am probably able to write a best-selling book of student questions (and how I answered them) from days 1 and 2.)

Surely the students are also looking at the environment they will be working in for the next 10 months.  I can’t vouch for students looking carefully at the coordination of the manicured classroom but I know they look to see what sorts of supplies are available and how used/cared for they appear, what sort of wander space is at their disposal, and how clean common spaces are.  

I’ve recently continued to think deeply about the ACTIONS of the first days of school (and not just how I should set up the bookcases).  I am trying to record what I notice (and note) in a variety of mathematical, teambuilding, and building routines activities.  I am thinking deeply about their purpose for me and the students.  Will students be able to make a better estimate of the adventures if we do A versus B?  Will the scholarly routines we want to happen consistently throughout the year (without too much prompting) really be driven home with this modeling or demonstration or discussion?

Unit 0 is just the beginning.  And simultaneously it is a big deal (to set up the culture of our classroom, the school, and mathematics) and not a big deal (so students don’t freak out by day 3).  Oh, Unit 0.  So, I am going to continue to think deeply about the routines, systems, activities, objectives, standards, goals… And attempt to implement.  More adventures to come.