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!




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?



Year-long Projects vKogs

Perhaps I am a bit comfortable thinking about a system with year-long projects in my classroom next year…mostly because of what much of my “instruction” looks like in my classroom now.

For the classes that I am the singleton teacher, much of the class time is driven on the idea of assessment as learning.  (I mean, I also love assessment for learning, but much of my time is not spent there.)  I give my little ones a page with the outline of the unit/week/chunk of time with the content and scholarly behavior objectives.  With the assistance of adults/peer tutors, the students monitor their progress and think deeply about their progress towards mastery of my (the units?) predetermined objectives and of their personal goals.  


Then for the day to day situation, we follow a predictable schedule.  The Do Now has a review of skills (for instance, creating tables from an equations).  The students work on the DN until it is accurate and then retrieve their unit folder.  They work on the next step of their unit work/project.  As the students complete a chunk of work, they receive feedback (and not advise).  The students make their desired revisions and then submit for assessment as they work on the next piece of the unit.  Students make seek or offer help from/for their peers throughout the class period.  The students in the minutes of class, write about their progress from the day and orally describe what their next steps will be.

There is some sort of end date that everyone is working towards – but the end of the project looks different for various members of the class based on their needs and strengths.  Then we have portfolio days where the students write about the unit skills and their progress in many skills.

Recently, Langer’s residents separately came to visit one of my classes that uses this process.  Both residents commented that the class seems to run itself and the students seem to know what to do.  For an outside observer, this probably seems true.  In order for this work to be successful, students and teachers need to be able to go with the flow and be able to set and meet personal deadlines based on a larger deadline and knowledge of oneself.  I often wish the the students could stay on task for longer stretches of time, or recognize that that stamina is probably a good thing to develop.  We have many conversations about what members of our community need and what we are able to provide to each other.  We talk about how to provide feedback, use feedback, and reflect on trends in feedback.  


As I think about year-long projects, I think that this system that I have already used for a while will work for much of the time.  I imagine that students will be able to have year-long folders (that are ongoing) and unit folders that are more immediate based on the current work.  Because of the importance and focus of systems of equations in the Algebra 1 curriculum, I feel that a series of leveled systems of equations problems would work well for our thoughts of year-long projects.  As students work on developing new skills, they return to problems and enhance their problem solving processes and solutions.  Students will be able to employ multiple representations and problem-solving strategies to attack problems and make decisions. 

Of course, it would be lovely to have a diverse array of types of problems that students need to be able to figure out.  Over the summer, among other things, I’m going to peruse and carefully consider the course that would use this textbook as the primary source of mathematical education.  Perhaps I’ll be inspired to rethink everything…

What Constitutes a Reward?

I’ve been reading (slowly, ever since I realized I should take notes on my for-work reading, a task much harder on the T than just reading …), Drive: The Surprising Truth About What Motivates Us by Daniel Pink, recommended to me by several people. I think all the teachers should read it (in our copious spare time). So far, it’s focused mostly on intrinsic versus extrinsic motivators, and how external rewards often decrease internal motivation.

That’s really important for teachers, no?

So here’s my current wondering. I’m doing some sort of class meeting/circle next year. It’s happening. My current plan is to do this (at least) on Mondays (main purpose: build community, transition back to school check-ins, launch the main math task of the week) and Fridays (main purpose: close the week, celebrate community leadership, celebrate completed work).


So I got the idea of celebrating work from ROLE Reversal, the same book that recommended Daniel Pink (along with my coworker). But, is celebrating completed work/final products, etc a reward? That would then decrease the intrinsic motivation of students to attempt more problems/revise more products? If so, I don’t want to celebrate completed work! But if not, I think it’d be a way to build community and get excited about math.

Hmm.  (Maybe I should keep reading?)

Year-Long Project Ideas (vLanger, v1)

(Yes, that means this is my idea, Kogut probably has other ideas, and that this is IN PROGRESS. In case you couldn’t use context clues to figure out what vLanger, v1 meant …)

Side note: I am traveling to Philadelphia this break to visit some friends from college.  Yesterday, I went to the library to get another book for the trip (though I have to do lots of other actual-work things on the train, of course).  In the hopes of finding a new inspiring book, I went to the education section and found, “Teach Like Your Hair is on Fire.”  While I am only a bit into the book (and definitely don’t agree with all his curricular methods or his tone all the time), the beginning of the book has the same premise as the Alfie Kohn and Mark Barnes books I mentioned earlier: treat students as people.

It is so refreshing to read these books.  I kind of want to sneaky-require all new teachers I know to read one of these books, so they don’t get lost in the world of classroom management through control while forgetting students are people.  I have many stories about this, but one in particular I remember from the first day of my teacher training program.  Anyway.

End side note.  Superficial connection: at least two of these authors ALSO talk about meaningful, year-long projects.  Which I’m very exited about instituting next year.  I have two ideas (that aren’t mutually exclusive) so far.  I would love to hear other ideas or comments on these ideas.

Idea #1 – Solve {insert some somehow non-arbitrary number here} Choice Tasks by the end of the year

For a while now, I have been collecting cool tasks that aren’t tightly connected to my curriculum.  I get them from NCTM, other math blogs, and books I read (I inherited/stole this awesome book that you wouldn’t even know is awesome by the title from my former math coach – I didn’t mean to steal it and I even emailed to try to give it back to her once I realized it – it’s “Problem Solving in Mathematics” from the Lane County Mathematics Project).  In any case, this year-long project would involve students choosing some number of these tasks to solve by the end of the year.  Students could present their solutions either in writing, as a video, or as a podcast (or … other ideas?) as long as the product met our problem-solving criteria.  Students would collect these tasks (along with any other final work) in a “Final Draft Math Binder”.  We would celebrate when students produced work ready to go in the binder.

Questions …

–       Alone/together? Should students have to work alone for some number of problems? If they work together, how do I ensure all the students in the group actually thought deeply?  Maybe students can always work together or alone but have to produce individual final products?

–       What do I do once a student figures out a solution so that other students who chose that problem do their own work?  Does it matter if they have to produce their own final product?

–       How many tasks should students solve?  I am hoping next year to have Open Honors (students can choose over the course of the year whether they pursue the honors level of the class).  If so, the number could be different for course vs. honors

–       Should I categorize the problems/require students to try different categories?  Some categories I could have are patterns, shapes within shapes, working backwards (though I wouldn’t call it that), diagramming

–       How do I encourage excitement about the final products and how do we celebrate student work without giving away answers?  One teacher friend suggested I organize questions by term so at the end of a term, we could have a public celebration of work, but then those problems would go away.  Might be a good solution.  Might be annoyingly complicated/arbitrary.

– Where do I find more problems?  Book suggestions?

Idea #2 – Math Moments Blog

I’m not that tech-savvy, but I think it would be cool to have one blog for each of the courses I teach.  Students would then be invited to (and required some number of times to) post …

–       An “ah-ha!” moment, where something clicks that hadn’t before.

–       A this-is-math moment, where they are not at school, but something mathy happens. Woah.

For all of these things, I will eventually (i.e. before the beginning of next year) have criteria and exemplars.  Hopefully funny-but-awesome exemplars that include videos of me and Kogut.

Thoughts?  Ideas, oh people who are smarter than I am?