Posted by: rdkpickle | 10.01.2012

problem solving

Well, the year is good and under way. I don’t think the calendar is just having a laugh at my expense – it actually IS October, believe it or not.

September is always a hectic, overwhelming month at work, but things were especially busy for me this year because I was traveling for weddings galore! The year at school started well, though – I felt like the comfortable classroom climate “clicked” a little earlier than usual this year, and I have been impressed at how my students have jumped right into working with each other (both inside and outside of class) and seeking me out for extra help consistently and early when they are struggling. Extra help was chaotic and draining for the first couple of weeks of school, but I kept telling myself that every minute I spent early on meeting my students where they were and helping them rise to my high expectations would pay off dividends for the remainder of the year.

This has already proved to be the case in my two Precalculus classes – after finishing the first unit (a review of fundamental algebra skills that can be challenging for those walking in with a rocky foundation) we spent the next two days with a focus on problem solving. It was actually Molly’s idea that this year after each unit test, we will spend 1-2 days on problem solving – problems that either build off course concepts, or serve as stand-alone explorations of interesting topics. The goal was to help students THINK mathematically and embrace challenge; to encourage them to spend less time doing the procedural part of math (“carry out the plan”) and more time doing the interesting part (“ask the question,” “understand the question,” “come up with a plan,” “extend.”)

So for the first problem solving lesson, we kicked it off by having some fun with Dan’s “bean counting.” I made students give me a quick estimate after watching the first video (before trying to do anything mathematical to solve the problem), and I enjoyed gently ribbing the students who estimated a combined time that was slower than Chris’s 5.6 minute solo time. We did some math. We talked about Polya. They were really interested in what Dan and Chris were listening to.


After exhausting bean counting and the extension (by the way, Dan, the 4 groups that correctly got the extension with no help from me literally SCREAMED and high-fived each other when I played the answer video), I split students into 5 groups, each group tackling one type of modeling problem (drawing a diagram, a number riddle, a rate/time problem, a money problem, and a mixture problem.) They had the remainder of class to work with their group to solve the problem. I tried my best to resist jumping in to help when groups were stuck – encouraging them to keep talking and writing things down, but staying out of the way so that they could have an authentic problem solving experience. This is one of the biggest differences that the confidence from a few years teaching has given me: the ability to discern when students’ struggle is productive struggle, and an arsenal of prodding questions to keep them going when they start spinning their wheels.

For homework each group was tasked with finishing their problem (if they hadn’t been able to do so in class) and preparing to teach the class their problem the next day. I told them they needed to take this seriously – since it was their only homework, I had high expectations about what that teaching would look like.

The next day in class, I gave students about 5 minutes to go ahead and write their work on the whiteboards, including defining their variables, any drawings or diagrams, the set-up of their equation, and all of the algebraic steps to arrive at a solution. I coached them up a little bit on what I expected from them in their presentation of the problem: Be sure each group member is involved in presenting the problem. Start by reading us the problem. Don’t be afraid to tell us some of the things that stumped you at first, or incorrect approaches you tried. Speak slowly and give us time to take notes. Pause and ask us if we have any questions. I also coached the learners: Be respectful of your classmates; raise your hand and ask questions just like you would if it were me teaching – don’t hesitate to ask just because you’re afraid of putting your friends on the spot. It’s okay if they have to ask you to clarify your question – sometimes I do too!

The next 50 minutes were amazing. I mean really, truly, honestly amazing. I sat down at my desk and pretended to be a student and barely had to speak a word as each group taught their classmates. There were plenty of questions and lots of healthy, respectful back-and-forth as students tried to understand each problem. It gave me teacher goosebumps to literally see lightbulbs go off in students’ heads as their classmates explained the ideas in a variety of ways until things finally clicked for everyone.

I’m not going to lie, I actually almost got teary at one point, watching a group with one student who had been having a tough time in class the first few weeks absolutely KILL IT with confident, clear, kick-ass teaching. Blame my sentimental streak on all of the nuptials I’ve witnessed this month, or something. When class ended I ran out of the room barely containing my excitement. Me: “I think I just had the best hour of teaching of my entire career!” Brian (Upper School director): “This doesn’t mean you can retire!”

One last win: I followed up the lesson with a take-home assignment meant to match the types of problems the students taught each other in class. They had the weekend to work on it. Today I woke up to this awesome email with the subject line “Another winning moment in math”:


So that was pretty great. I guess my year has nowhere to go but down. 😉


  1. Your passion is amazing! Your kids are so lucky to have you to inspire them.

    • Thanks for always responding to my posts and making me feel so great! It really means a lot to me. 🙂

  2. […] Oct 2. Rachel Kernodle writes about Bean Counting: "… the 4 groups that correctly got the extension with no help from me […]

  3. I don’t know how I missed it, but I’M DYING!!! This post makes me sooo happy.

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