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Unit Planning: A Focus on Problems

July 25, 2012

When I sit down and focus on planning for this coming school year, I find myself planning for problems rather than planning for sequence, presentation, or activity. I think this is one of the fundamental growths I’ve had between my first-year self and my current teaching self – around this time last summer, I was outlining day-by-day sequences of what I would cover and scripting specific lessons. I had grandiose ideas and felt like I needed a very specific road-map before I could start the year. I was only barely paying attention to the specific problems I was assigning for homework. This philosophy has changed – now I feel like a loosely defined road-map based on groups of skills and a collection of related problems is sufficient to keep me on track for the year. Based on how much I threw away last year, I’m almost afraid to plan a concrete lesson more than a week in the future. Having a collection of meaningful problems to pull from, though, is incredibly soothing to my nerves and is also a better reflection of my teaching style.

In doing this, I’ve been reflecting on the types of problems I assign and this idea I have in my head about scaffolding these problems. Looking back on how I ended my year, I found myself in the following sort of cycle: (1) start a unit introducing a new topic. This may start with some interesting hook or problem or activity to activate prior knowledge, but at some point, it becomes me telling the students a definition or theorem. I model how to solve the basic types of problems. For a while, it is very teacher-centered. Depending on the topic, this can last for a few days – the problems are mostly procedural. Then, (2) introduce some novel situation or contextual problem and start a discussion about it. How is this similar to what we did before? How is it different? What information do we need to solve the problem? 3-Act Problems fit perfectly in this category. After a brief discussion to get them started, they work in groups. I answer questions or throw them back to the group. The problem is rich enough that a discussion takes place – explanations are given, questions are asked, extensions are proposed, connections are made. We spend a whole class period discussing a variation of one type of problem. Sometimes these problems lead to the next piece of procedural knowledge and the process begins all over again (an example I just made up: corresponding angles leading to alternate interior angles). The next day, we tackle a new type of problem. Then, (3) there is some sort of culminating project or activity that may take a few days. I try to be barely involved – instead, I create very detailed instructions that will hopefully leave students self-sufficient. Students are encouraged to work with each other. There is room for creativity and ingenuity. Hopefully both the process and the product are meaningful. Hopefully I’ve scaffolded everything else effectively so this project/task doesn’t overwhelm my students.

This all might be a bit abstract for now, so here are some things that might give a better idea of what I’m talking about:

Procedural Problems: Pretty much anything generated by Kuta Software and anything that can be put on my Wall of Remediation. Also, here are some concrete Parallel Line Problems I might do:

Discussion Problems: Parallel Line Mazes

Building Cities in Geometry

4-Points Quadrilateral Identification

What I think all of these have in common: They are problems that require procedural fluency in order to be successful, but where the answer to the problem is practically meaningless without process to accompany it. These are problems where the meaning is in doing the problem, not in having the correct answer at the end. In fact, for many of these, incorrect answers could also be incredibly interesting. These are problems that could last an entire class period with the right group structure and questions. These are problems that naturally encourage discussion and justification and arguments. These are also problems that I definitely do not want to solve for them – I want them to try it on their own first. Maybe there’s some productive struggle. Maybe they work with their neighbor for a bit. Maybe at some point, we regroup and I ask the group to clarify a few things. I act as more of a facilitator and guide during these problems rather than an explicit model. These are problems that are the gateway for writing in my classroom.

Culminating Task/Projects:

Stained Glass Project

When I taught area, I gave my students meter sticks and had them go outside to measure the surface area of the parking blocks. The prompt was: if the school wants to repaint the parking curbs, how much paint will they need to buy and how much will it cost?

These are hard to come by and I’m still developing them. They’re not perfect and I’m still trying to figure out the best way to incorporate them into my curriculum. But for right now, I think I want at least one mostly hands-off task/project per unit.

How It’s Influenced My Planning: I’m starting to feel that having a collection of these types of problems will help me more effectively plan assessments, activities, and remediation. It all comes down to having problems to pull from. And not just the procedural ones – the really magical ones are the discussion problems. Some of my favorite moments from last year were during class periods that began with some sort of scenario and guiding my students towards finding the answer. This is where my focus has been: collecting problems and searching for meaningful problems to lead discussions and problems that could become a meaningful task or project for my students. I’m hoping this is a better use of my summer time than what I did last year.

Soon to Come: A review of sources I’ve been looking at for these problems. Spoiler alert: Park Math and James Tanton have some amazing problems to pull from.

  1. I’m also wanting to strongly incorporate some form of problem-based learning (my catch-all right now for things like 3Act) into my math classes next year. A resource I learned about at a conference last week was We worked on a sample problem at the time, but I haven’t had a chance to dig into it further yet.

  2. I’m doing lines and angles stuff with my geometry kids and your parallel line activites were perfect to use today. Thanks for sharing!

  3. Julia permalink

    I really like the parallel line maze under “discussion problems”. Do you have the solution key to this? If so, could you please share that with me. I would love to use it in my summer honors geometry course. Thanks!

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