Graph Theory, Part 1

The last unit for my senior class (Advanced Math Decision Making) is  Networks and Graph Theory. Win for me (and hopefully the students)! My student teachers and I are taking the opportunity to teach a little more like I want to next year (my student teachers are amazing).  Here’s how we’ve done, so far.

Day 1: We started by challenging the students to find a Eulerian path and circuit through the basement (we didn’t use that language). And then took them to the basement.


They explored in groups of ~3, at first with only the criteria for a path/circuit and then with a map. Once students had found the path and explained to a teacher why a circuit doesn’t exist, we sent them upstairs to work on representing the basement diagrammatically and then “fixing” the basement so that it did have a circuit. All the groups finished their diagrams. A few created a circuit in the basement rather trivially by eliminating the middle hallway.

Day 2: We started this day by naming our circuits and paths. Students came up with “S/E” for circuits (because you Start and End in the same place) and “GPS” for paths (because a GPS takes you from one place to another). Students then articulated in writing why no circuit exists in the basement, and gave each other feedback on their reasoning. Finally students were tasked with creating a non-trivial circuits. The methods for doing so were pretty awesome — some groups figured out that you could create a circuit by drawing anything without lifting your pencil. One student started drawing the iPhone (?) dots and making shapes. Once groups created an S/E, they traded with another group and proved that the other group’s graph was an S/E. By the end of the day, we had a collection of S/Es and GPSs, all student generated.

graph theory

Day 3: We sorted 12 graphs into S/E, GPS, and neither. Yay sorts!

graph sort

What’s working

– Students are engaged. One student who has been rather checked out for a few weeks was one of the most determined doing the sort today.

– Students feel ownership. One of the reasons I had students name the circuits/paths was to model what happens when people do mathematics – they need to name things, and they can choose whatever names make sense! Additionally, the students have created as many graphs as teachers. All the graphs on the wall, except one, were created by students.

– Students are making their own sense. They physically experienced the basement. Both the strategies for making circuits mentioned above came from students. Today, a student went to the board to show how she would go around an S/E from the sort and another student interrupted her to tell her to start somewhere else. The original student stuck to her guns and then the other student showed his S/E route, leading to the question of whether you could start anywhere when walking an S/E. (We haven’t finished answering that.)

What I still need to work on

Ohhhh whole-class discussions. How hard you are for me to lead! How to balance argument/excited participation with listening? How to balance desire to go to lunch with listening? Discussions, I aspire to be best buddies with you one day in the near future. Please be open to our future, awesome relationship. Love, Sarah.

Next week … onto the Bridges of Konigsberg and plowing city streets!



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?


Objective: SWBAT identify and eat produce

So, I try to ensure my students work on more than just content math objectives.  Sure, my students calculate the volume of cylinders, write about their process, advocate for their learning needs, and ask for breaks appropriately.  But I also have them master other skills too.

Picture having math class for 68 minutes last period every day.  Many students in my last block class become frustrated throughout the day by a variety of different issues.  During a community meeting at the beginning of the year, the students asked for snacks on Fridays as a celebration for getting through the week as a little reward.  Easy enough to fulfill; I typically have cereals, small candies, fruit cups, and crackers hidden under my desk for hungry children.  Surely I can bust something out once a week for my little ones.  So we did that happily for months.  Occasionally, on other days, a hungry student would ask for another snack.  And I don’t believe that a hungry student can think about whether we are going to use the total or lateral surface area formula so I provide snacks.

And then during a Friday community meeting, my students asked again for snacks, but this time for daily snacks.  I told them I would think about it.  And on the following Monday, I brought a container filled with carrots.  I told them when everyone was working, I had a snack for them.  I unveiled the carrots and bell peppers in a very dramatic fashion.  The students looked concerned.  “Are they cooked?” one little one asked.  “Don’t you have raunch dressing for these sticks?  Or cheese?” another one questioned.  I shook my head and told them I had these carrots and peppers that were full of vitamins to help them think about the math we were going to do today.  Some of them were scared of trying the “sticks” but one student was very eager to get eating.  “Guys! These will help you see in the dark!  And make you feel full fast!”  So some of the students hesitantly ate some carrots.  One student tried a pepper but then immediately spit it out.  I ate the left overs.

Each day, I brought in at least two fruits and veggies for the students to snack on.  We started the routine of introducing that day’s produce.  I would name the selections of the day and talk about the benefits of that particular snack.  Students began to learn names of common veggies.  Broccoli no longer was known as trees.  Spinach lost its nickname of leaves.  Day after day, students tried new veggies.  They would even eat a pepper or two.  And they would try something new when offered.  We still have some impersonations of rabbits when spinach and carrots are paired for the day.  

Weeks go by.  The students chop down on their veggies during work time. When they walk in, they ask what we will have for the day for snack.  Gone are the days that I eat the left overs – the students down them usually before I even get more than one piece.  The day that grapes were introduced might be some of the students favorite day of the year.  Other students in other classes began to ask me why they didn’t get to have veggies in their last period classes.  I told them they could ask their teachers about it.

And then some special days came through our calendar: MCAS celebration and Pi Day.  When some of the older students retested for the MCAS, I had crackers and cheese ready to go for our daily snack something that two months ago they would have gobbled up immediately.  I put out the snacks and some students took some but turned to me and asked “where are the green things?”  I hadn’t prepared any produce for them because of this alternative.  Students were excited for pie on March 14th but immediately asked where their greens were.  Because of what had happened only a week before, I was ready with some veggies too.  A few students wanted only a half slice of pie so they could have more banana peppers and green beans.  When a new student joined our community, he turned his nose up to the snack but one of my students jumped in and said “but we are sharing the good broccoli with you!”  The new kiddo now requests snack as he walks in just like everyone else.

So it’s a routine.  We eat produce.  My students are able to name pretty much anything I bring through and tell me that they go shopping for their own fruit and veggies snacks when they are downtown rather than going to food courts.  During the weekend, I purchase, wash, cut up, and prepare our snacks for the week.  I enjoy our fruits and veggies time and I am glad that it has become a routine.  And I am proud to report mastery levels in nearly all of my students in the objective of identifying and eating produce.



Exhibition (Part I of …)

Kogut and I really should write about exhibition sometime.  But tonight is the night before it goes live, and I’m mostly sitting here feeling incredibly proud of my students.  Just a smattering of examples of why …

– One of my students, who is working on social skills, giving a strong exhibition to a small group of teachers this afternoon.

– My last period class asking each other questions to help each other hit criteria they may have missed during their practice.

– A student who is working on getting to school on time and thus misses math up to 1/2 of the time, keeping up with the project, becoming more engaged because we were going in depth and she can re-orient more quickly, asking thoughtful, genuine questions during practice today.

– A student seeing a video about her topic and exclaiming, “I am going to add some of what I just learned to my presentation!”

– A student, who does not often volunteer to help others, tutoring his groupmates.

– Students informing each other that they will act professional/on time/not giggle/stand tall during Exhibition, no bones about it.

None of this is surprising, but it bears mentioning, and my students are awesome people. I am glad Exhibition gives them an opportunity to shine.


(P.S. No, it’s not perfect and no, not all the presentations will knock your socks off, but that, perhaps, isn’t the point.)

Redo to actually learn

“We do not learn from experience…we learn from reflecting on experience.” -John Dewey

In my land of Standards-Based Grading, a chance to redo ANY assignment is available and encouraged. Even when a term closes, I let students redo their work to improve their year-long grade.  I know that many would argue that this system creates too much work to reassess multiple assignments throughout the unit/term/year and that only some assignments should be allowed to have additional amendments.

I feel that these experiences help a student actual learn and master the material and I tell the students that I don’t have a deadline on learning.  I stumbled upon a recent post about how the most important part of preparing for adulthood is redoing what is needed to be redone and how various educators think about and implement do overs.

I have a template that I give to students so they are able to reflect upon and correct their errors.  While I have been tweaking with the template to make it more “kid-friendly” for many moons, the content has essentially remained the same.  Students are required to talk frankly about and reflect upon the assessment as a whole.  Then they dive into specific questions that they want to try again: they record the problem number, redo the problem, explain their errors, and then create and solve a new similar problem.  Below are parts of the template.


At the beginning of the year, there are days dedicated to learning the template and then providing time for students to work on their own assessment.  As the year moves on, and I fell the pressure to “get through” material, fewer times in classes are given for students to confer with each other and teachers over their assessments.  Instead, they are encouraged to come after school if they want to have some sort guidance throughout their revisions; but then the students lose the time with their peers to debrief and reflect on their assignments.

I am hoping that with a revamping of the curriculum and its structure that there will be more time dedicated and available for students to conference with their peers and teachers to reflect and improve their work.



About three years ago, Kogut and I re-wrote an Algebra 1 project that happens at our school every year. (It’s called exhibition; the end-product includes students presenting their project to approximately 20 students and adults; it’s a long story; we’ll tell you more about it for sure sometime.) Basic background: our Algebra 1 exhibition involves students using linear regressions to analyze real data about worldwide or school issues. My senior math class (Advanced Mathematical Decision Making) exhibition involves a statistical analysis of data about a social issue (linked to the Civics class) and its prevalence in our school.

For several years, I’ve enjoyed the process of giving feedback on these projects. As a rule, I don’t grade the “first draft” of the math work, but rather solely give feedback in relation to the objectives. This feedback-giving barely feels like grading; rather, I get to look at students’ thinking without immediately evaluating it. I get to enjoy the hard work students have done and then offer suggestions for improvement and point out where students’ have skimped on certain parts. Students then have the opportunity to revise before I finally do evaluate their mastery of the objectives.

What I’ve noticed is that students do not miss the grade at all, but rather appreciate that I actually paid close attention to their work. This year, when I gave the seniors back their first drafts, only one person in the entire class asked about a grade. Everyone else (and that student too, once I explained) just started revising their work. The results were almost entirely high quality, and the process felt real, respectful, and meaningful. I look forward to shifting my classroom so that most of the work involves this process of genuine feedback and revision, and to including students as important feedback-givers.