What If Everything I Think About Teaching Is Wrong

That’s what I was often thinking during six fascinating hours in Alex Overwijk and Mary Bourassa’s morning session at Twitter Math Camp.

They have dispensed with traditional units entirely, instead teaching content through activities. An activity could last anywhere from a single class to three weeks. Each major topic in a course comes up within the first few weeks (Al called this “unloading” standards), and is spiraled a number of times before the end. Here’s what the curriculum might look like:
Screenshot 2015-07-30 at 4.46.42 PM
Each column is an activity, and each row is a standard.

Important Notes:
Al and Mary use standards-based grading, which seems like a necessary component. Note the T1, T2, T3, T4 and SE columns — those are assessments, and the standards that are being assessed. They also start the year by co-creating, with students, criteria for great questions, and students decide in groups which questions are most worth answering. They are unafraid as teachers to steer the class toward specific questions, but allow some freedom for students as well. It’s definitely a slow process, building students’ problem posing skills, but a valuable one as well. An activity might start with an image like this:

Look closely. No, more closely. Now what questions do you have?

Al and Mary showed us activities using ropes to illustrate linear equations, stacking cups in all kinds of formations, using a memory game to illustrate quadratic and exponential functions, and using different sized squares to illustrate linear functions, quadratic functions, and the Pythagorean Theorem. They were a ton of fun, and I learned a bit of new math along the way.

A huge part of their success is clearly the quality of activities they have prepared — but through our working group, we are working to build a bank of great activities. Check out what we have so far here.

Why Activities?
An activity-based classroom is ambitious teaching. Making this work requires a pretty significant collection of activities, careful organization to track standards, deliberate messaging to students, and flexibility to deal with the inevitable messiness of activity-based learning.

This approach also has some pretty enormous benefits. It matches what we know about brain science — that in order for learning to be durable, students need practice spaced over time and interleaved between different topics. The way most math classes are structured, practicing skills for a short period of time and then moving on, is practically designed to create the illusion of mastery — knowledge cycles through short-term memory without any incentive or reason to build the neural networks necessary for long-term retention. Activities are also more engaging for students, for obvious reasons. Al first tried this approach 8 years ago because he realized his lowest level 10th grade class wasn’t learning everything. They redesigned the course, with the goal that first it had to be engaging. If students weren’t learning, at least they wouldn’t leave the room hating math. They built this system from that starting point, and have had remarkable success.

One great question from another participant helped drive the point home for me. The participant asked: What if you plan an activity, and it doesn’t work — students aren’t asking the right questions, they aren’t ready for the math, or are focused on other details of the activity besides the relevant standards? My answer — while a failed lesson might look a bit uglier in an activity-based classroom, I teach plenty of failed lessons. Maybe kids are sitting quietly, but more times than I can count this year, a class left my room having learned nothing of substance. I struggle with that. If activity-based teaching has some pretty serious possible benefits, and my students and I might have a bit more fun in the process, it sounds like a good deal. Most of all, if it takes away some of the illusion of learning I get from massed practice and students cramming for a rest, I will be a much better teacher for it.

Where Am I Going From Here?
I’m not going to go full-bore with activity-based teaching this year. I’m starting at a new school, and have lots of things to figure out moving from 8th grade to high school with new curriculum, routines, and lots more. It also feels like a huge risk, and not one that I want to crash and burn with the humans I teach. But I want to incorporate these ideas where I can, and see if I can “test run” them to consider moving forward with this approach. I’ve been thinking about spiraling activities in the context of my Algebra-II class for the fall semester. It’s all juniors, and spans a pretty significant range of content. The topics I’ve been given for the fall are:
Expressions & Equations
Linear Equations
Exponential Functions
Functions (from basic function concepts through inverses and function transformations)
Exponents & Logarithms
Complex Numbers

Two approaches I’m considering, with the caveat that I haven’t started work yet, so I have no idea if these will fly:

No unit on expressions and equations. Instead, I spiral those topics throughout the semester with a variety of activities — I have a few ideas so far, and I’m sure I can build some more. They’re critical skills that my students need, and I think the spiraling approach could be a critical one. And activities focused on expressions and equations will naturally wrap in other standards, and provide a nice change of pace.

No unit on functions. Instead, spiral function concepts throughout the semester. I have a few ideas for this already, including Function Carnival, Graphing Stories, and more, as well as some reading I’ve been doing about how students understand functions and where the idea of a function came from anyway. This is a bit more ambitious, especially with respect to inverses and transformations, but I think if I can successfully wrap functions into everything else we do (with the possible exception of complex numbers) students will have a much broader and deeper understanding of the function concept.

Finally, I’d like to continue my approach to homework from last year. Distributed practice on a variety of topics, but without questions on that day’s learning goal. Students get continuous practice on key concepts throughout the semester, homework is less dependent on how well class goes, and I get a chance to review and address misconceptions from the whole course on a daily basis. It’s small, but I think it makes a big difference.

I’m going to spend some time over the next few weeks puzzling through what each of these might look like, and in particular if I feel like I have the depth and breadth of activities to do them right. My goal is to have a very clear idea of the abstract ideas I am pushing for through spiraled activities, the different perspectives students need to be able to look at them through, and a number of concrete classroom activities that will move students in that direction.

9 thoughts on “What If Everything I Think About Teaching Is Wrong

  1. howardat58

    The article on the development of the function idea is excellent, but appears to limit itself to single value functions. The idea of a mapping is more general, and is exactly a description of what a rigid motion in (or of) the plane actually is. It is always possible to get the output of a process to be single valued, just define the range appropriately. For example the square root function is a mapping of the non negative real numbers to the set of pairs of any real numbers.
    consequently the question “Does this graph represent a function? ” always has the answer “Yes”. One should always specify the domain and the range in the question for “No” to be available. So y = root(x) is a function as described above because for any input x, say x=4, you get the same output today, tomorrow and for ever, the set {2, -2}.
    Just a thought or two !

    1. dkane47 Post author

      This is really interesting. One thing I’ve been thinking about is “simple” and “advanced” conceptions of functions. I like the definition of a function as a relationship that’s reliable as a “simple” definition — it doesn’t capture some ideas, but builds a concrete basis for what is and isn’t a function. Where is the balance between being mathematically rigorous and building concretely on what students already know? You make a good point, but I think I might save that approach until later in the course, after the topic has been spiraled.

  2. Amy Zimmer

    Dylan, It was so fun to meet you! You have such energy and passion! Functions could be a great thread for all the topics. Basically there are four operations, +, – , X and / , so the question is how does that look as we go deeper?

    I like your homework idea. Maybe two or three from the days lesson all the rest spiralling. I like how that would mean the homework wasn’t so dependent on how the day goes.

    Good luck with your new adventure!

    1. dkane47 Post author

      Thanks Amy! Great to meet you as well. The operation approach is an interesting one that has many layers — although the challenge is balancing that approach with the other conceptions of functions that are important.

    1. dkane47 Post author

      I haven’t — just googled it. Looks really interesting, would love to check it out. I think the challenge is that lots of similar resources exist — Geoff Krall’s maps, Illustrative Math’s blueprints which will be released soon, and similar problem-based ideas. Finding the time to put them together in a coherent way, and find tasks that fit together to make this approach work, is a hard problem that I’m pretty interested in.

      1. Teri Owens

        I agree. All the wonderful resources are overwhelming. That is part of why I love those textbooks. The entire (in my case algebra) curriculum is taught with the problem-based style. Then if you have some other activities you love to do you can add them in. Before we started using these books we were “killing ourselves” scouring the Internet to find activities that fit what we needed then making copies (we are not 1 to 1). I had never seen a textbook like it before. Your original post just made me think you might like them.

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