[Explore MTBoS 1] The Border Problem

It’s been a super hectic few months, so Explore MTBoS is exactly what I need to get the blog back in gear. Here we go!

One of my absolutely favorite open-ended problems is The Border Problem, snagged directly from the Boaler & Humphreys’ Connecting Mathematical Ideas. I use the problem as a way to talk with students (and parents) about how we can represent our thinking using numerical expressions, and how being attentive to how we are showing our thinking numerically can help someone else to understand how we are “seeing” or thinking about a particular problem. It’s also super useful as a way to introduce algebraic expressions.

This is a 10×10 grid. Without counting, talking or writing anything down, determine how many squares are in the border.

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Humphreys is super deliberate in how she poses the problem, and it was great to read about the amount of thought that had been put in to thinking through every aspect its framing. I follow her advice and deliberately do NOT print out a copy for each student (since this increases the chance that kids will count squares). Once I’ve given a good 20-30 seconds or so for kids to come up with their idea I ask them to share only their answer with the people at their table (there will always be some kids who will get 40). I then have students write down a numerical expression to show how they thought about it and I pass out smaller 10×10 grids so that they can also show their strategy geometrically. Once students have had a while to familiarize themselves with their strategy and those of their group members I show them a 6×6 grid and ask them to “see” it both using their own strategy and one of two others. I do the same for a 15×15, then ask them what it would be for a much bigger grid (24×24, 231×231, etc.), eventually building to an NxN.

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Getting all the class’ strategies up on the board and discussing them together makes for an awesome conversation. Students want to name the different strategies (2Ls, Cut Corners, etc.), are super willing to give credit to one another for cool ways of looking at the problem that they hadn’t seen, and can’t help but making connections between the different ways of thinking. All good stuff!

To Do List: Part 1

Figured it’d be a good idea to get my thoughts on things I’d like to try next year down. The notes that I keep adding to on my iphone aren’t cutting it, or in existence (I think I’ve accidentally deleted about 10 of them). Here’s the first 6…

1) Get bigger whiteboards – I had students do a lot of work on whiteboards this year and I’m totally sold. The ease with which students can test out ideas is awesome (does make mistakes more fleeting though, no history to analyze later), as is the degree to which kids take ownership over their ideas and personalize them (color-coding, different representations, etc.) Kids need a lot of space though, and so I need to make bigger ones. I also want students to be able to collaborate on a task using the whiteboard (a la Fawn’s classes) Huge thanks here to Frank for his $2 Interactive Whiteboard post. I had them priced out through my school at the end of the year as whiteboards and they were crazy expensive, so going the DIY route seems to be my best option. No Lowe’s or Home Depot in Delhi though so we’ll see what happens. I wonder how much each one will cost in rupees?

2) Plan more effectively for discussions –  For sure I need to read 5 Practices for Orchestrating Productive Mathematics Discussions but I’ve done a fair amount of Anticipating–>Monitoring–>Selecting–>Sequencing–>Connecting in my classes and I think I need to sometimes make discussions more about what-ifs, sequels or mistakes. There have been times when I’ve perfectly sequenced a series of student solution strategies during the summary, and posed solid questions to the group, and the discussion has come off like a bland recap. I think a big assumption here is that students are not interacting with each other’s methods until the “discussion” portion of the lesson (or that the “discussion” portion of the lesson is always happening at the end of class). If we are encouraging students to get all up in each other’s thinking during class then the last practice, Connecting, needs to get way more attention.

3) Continue rethinking HW – Midway through the year I changed my homework structure to a set 8 problems a night (2 practice of current concept, 4 review, 2 extension/challenge which are optional) after hearing about the idea from a session that my wife was in with Steve Leinwand. I call them 2-4-2s. Clever right? Anyways it seemed to work well (The ones I’ve made can be found here). I provided the solutions on the class blog and students were responsible for completing the assignment,  correcting it using the solutions and then coding the problems with a question mark (“I don’t get it”), a star (“I might need to take another looks at this”) or a square (“I can do this all day (son)“). I didn’t grade or collect the homework but we did spend the first few minutes of each class discussing the problems in groups. Students knew to “start with their question marks” and progress up to their squares. If students had all squares then they either shared their method for a particular problem or worked collaboratively on one of the extension problems.

While I feel like 2-4-2s were definitely an upgrade from my previous approach to homework (problem sets based on the current unit) I’d still like to keep rethinking it. Lots of questions are still rattling around in here. For example: Is it really beneficial to provide students with solutions when I assign the HW? Should my students have more choice in their HW? If HW is for practice do all kids need to do HW? Who knows? I’ll keep trying things and see what turns up.

4) More puzzles – They’re fun. They make you smarter. ‘Nuff said. I bought a bunch of the ones I saw at MoMath a few weeks ago, and am psyched to try them out along with arithmagons & kenken/kendoku/calcudoku.

5) More games – Same as above. Games that help to reinforce content/reasoning for sure (I can see the Mistake Game becoming a quick favorite) but also ones that just get kids interacting with one another in a different way and that are just enjoyable. I also have an advisory that I meet with three times a week using the Responsive Classroom/Developmental Designs structure of a morning meeting (or Circle of Power & Respect in the MS), and games feature prominently as a way for students to develop social skills and a sense of community. Ultimate Tic-Tac-Toe is for sure on my list of ones to try.

6) Put more of us in, and out of, the classroom – A humanities teacher on my team had the awesome idea this year to post a picture of each teacher from our 6th grade team from when they were in 6th grade along with their favorite book from that time, or a book that they were currently reading, in the hallway. Though I couldn’t dig up a photo I did contribute my book, and it ended up having a much larger effect than I had anticipated. Kids asked me about it, I discussed another teacher’s book with a group of students, and I even spent my entire lunch discussing a book that a student thought would be their 6th grade book when they became a teacher (in like “a million years”). All in all I liked the way it worked to reinforce the idea that we are a community of learners: always learning and always sharing. I’m for sure going to continue the “what am I reading?” tradition next year along with some other current info about Mr. Hannon so that my students can get a window into my passions and the ways that I’m still growing. I’ve also got a few ideas about how to better use the space around me. Right now I’m thinking one classroom wall for illustrations of the mathematical practices, another for cool mistakes and a hallway board for a rotating spotlight on a “middle school problem-solver”. We’ll see, but I know I’m super interested in continuing to look at the ways in which our learning community constructs, and is constructed by, the physical environment around us.

Blog post #2 in the books. Don’t forget this stuff Jake!

Inaugural Post: SBG Reflections

Thanks to Fawn and Algebrainiac for providing the posts that finally got me off my ass to start blogging and to Bowman for the Angry Birds example that I’ll for sure use this fall. Here goes…


  • Having students provide evidence of an improved understanding before reassessing – This really helped to ensure that students who reassessed did better the second time. Also it sent the message that reassessing was about deepening your understanding of a concept, not just being able to understand your errors so that you can “retake” a test. This evidence could have been generated in class, at home, with me, wherever.
  • Saying “No” to kids who didn’t have this evidence – Again I want students to have a high degree of confidence that they will demonstrate a deeper understanding on the reassessment.
  • Having students assess themselves – After revising their assessments students would complete one of these self-assessment forms in which they reflected upon their thinking and answered the following three questions: 1) What does this thinking prove that I understand? 2) What does this thinking show that I do not yet understand? and 3) What are my next steps in my learning?  We’re a 1-to-1 ipad school and I had students insert pictures from their assessment to support their answers using Remarks. This helped students to acknowledge learning successes, identify areas of need and create a plan for improvement.
  • Allowing students to reassess concepts from earlier quarters – Why should something be off limits in Q3 just because it was taught in Q1? The whole evidence thing helps here, as it means the kid has been doing something additionally to improve. I want to reward that.


  • Google form for reassessment sign-ups – Tried it but it just became one more thing to check. Apparently I wasn’t alone here but I might try and get my hands on the script that Algebrainiac uses that auto-e-mails me when someone has made a request to give it another shot.
  • Offering reassessment and relearning opportunities at the same time – I found it incredibly difficult to work with students to go over concepts while also trying to set up kids with reassessments. As our reassessment policy is department-wide we’re thinking of having all the kids taking reassessments after school to go to one teacher’s room (evidence would have been checked earlier by their classroom teacher). We’ll see how it goes.


  • Providing class time for relearning & reassessment – If I want to send the message that my class is about continuous improvement then I need to put my money where my mouth is and provide opportunities during class for that to happen. A colleague of mine gave me this awesome bingo cage that I may use to randomly choose a standard for in-class reassessment. By scheduling more frequent, smaller assessments on previously taught standards for the whole class I hope to diminish the need for student-initiated reassessments. I’ve also thought about dedicating the last 30 minutes of a class once every few weeks to review/reassessment.
  • Focusing on the Mathematical Practices – Like several of the other bloggers who have done similar posts I’m also interested in making the 8 practices a more central focus of my classroom next year. I’ve already dedicated one whole wall of my classroom next year for us to post student “illustrations” of each of the different practices in action.
  • Activating students as instructional resources for each other – Though students in my class always knew to seek out help from their peers I’d like to create more  formal structures for students who understand topics to provide instruction. Student-led “math meetings” during that 30-minute time maybe or student-created instructional videos using Explain Everything? I’m open to ideas.
  • Self-assessment form – I probably went through 10 different versions of the form linked above and still wasn’t totally happy with it at the end. I want students to have a consistent, easy channel to provide themselves with feedback on assessments and sometimes the process seemed a bit onerous.
  • Communication with parents – Like Fawn I could see requiring a parent’s signature as a nice, and most importantly, easy way to keep parents in the loop about their child’s learning. I’ve also thought about using the comment feature in our online Veracross gradebooks more creatively to provide feedback for parents on a specific assessment.
  • Developing assessment tasks – Like everyone else I’m always looking for better assessment tasks and ways to develop reassessments without sacrificing a significant amount of the time that I should be sleeping.


  • Feedback, not grades – Reading Wiliam’s work has got me thinking critically about the ways students receive feedback on their learning in my class, and I’m eager to continue to try and move towards creating a classroom that revolves more around specific descriptive feedback than grades (be they traditional or standards-based).

Whew. Inaugural post complete! Thanks for reading!