Diversifying My Portfolio

There was the struggle between classroom management and engaging instruction. I invested myself equally in both until the depressing day I realized that my investment in engaging instruction also paid off certain dividends in easier classroom management. I spent that day re-evaluating my assumptions about teaching and re-balancing my investment portfolio.

Then there is the more current struggle between teaching skills (multiplying two exponentials) and teaching concepts (proportional reasoning). I figured, until recently, that in a 120-minute classroom, any time we spent on goofy conceptual digressions was time away from skill instruction we’d have to make up later.

So it’s strange, then, that after a semester of frequent digression, my classes are still on pace with every other Algebra 1 class and my kids set the curve for the semester final exam.

We spent thirty minutes on Friday, for example, investigating Jessica Hagy’s infographic work. I cherrypicked some interesting relationships, covering up Hagy’s graph in each, and asked the students to draw the relationship, also labeling each “direct” or “indirect” variation because, why not.

By the end, we were disputing Hagy’s graphs on technicalities, altering the intercepts ever so slightly to reflect the fact that (eg.) plastic surgeries could, at first, make someone less frightening to children.

After digressing for fully 25% of the period, we got down to the new business of adding and subtracting polynomials. And it struck me as I put an example up on a slide and asked them “what can you do with this?” how little time I spend “teaching” anymore, how these goofy conceptual digressions have trained my kids to look for connections, not just between “plastic surgery” and “frightened children,” but between “old skills” and “new skills.”

I realized, Friday, why we lost dozens of hours in the first semester to goofy conceptual digressions but still outpaced the school.

We didn’t need those hours anymore.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Great post – I have a hard time feeling like I’m seeing the entire forest of “conceptual understanding” that I should be getting kids to see. That makes it hard to decide when a conceptual activity is valuable and leads to lots of trade offs because it is easy to peg skill instruction to the curriculum. I feel like there’s a brick and mortar analogy to be made here. You need to pad all the little skills with some mortar to give them context and prevent them from sliding where they should not.

    I know I need to do more of this, and your results are a nice encouragement that there is value in the bigger picture.

  2. Lately, I’ve reordered the way I do things in class to keep things engaging. Although ‘engaging’ does not necessarily translate to mathematical skills practice, I’ve resorted to wait time less since I’ve got the class with me.

    When “off-task” digression ensues, I often let it go… knowing in the end it works towards building understanding – even if it takes away from class time. Though I’m certain my classes aren’t the best of the school, I know what you’re saying here… And I’m trying to get there myself.

  3. I’m trying to determine, right now, what makes a digression a good digression, and by “good,” I’m specifically meaning “it builds critical thought processes that will also be useful for skill aquisition.”

    “Quick” and “specific” come to mind but I haven’t really put my mind to it enough.

  4. Dude, you’re my freaking hero! It looks like you’re doing an outstanding job getting stuff done to suit the ‘skills’ but also doing some real educating (concepts). You’re use of these ‘funny graphs’ or ‘digital media’ are completely changing the way I look at teaching mathematics. For so long I’ve tried to find ‘good’ applications and all along I’ve missed the boat. Again…you’re my freaking hero!

  5. I’ve been amazed with the proliferation of these funny graph sites how much I study and analyze goofy data.

    The Crappy Graph site, which allows you to design (small d) your own, require a very high conceptual level to create compelling graphs.

    While many find the work of Dan Pink less than stellar, I think his concept of symphony or “big picture” thinking fits well in your example. Math literacy includes the ability to see patterns and find larger implications. I’m sure you might find flaws in this observation but it was certainly my first response to your masterful approach.

  6. I think you bring up excellent points for discussion. How do we, as teachers, build “meaningful” instruction time? If 25% of class time is spent on “digressions” then is it possible to have a 100% effective class with the remaining 75% of the time?

    I think you are hinting that, yes this is very feasible. If you use relevant diversions you can effectively detour students from their social concerns and steer them into Algebra (or English or History, etc.).

    I’m confused about your ending statement though:

    “I realized, Friday, why we lost dozens of hours in the first semester to goofy conceptual digressions but still outpaced the school.

    We didn’t need those hours anymore.”

    If you outpaced the school through your partial digressions, then why do you no longer need those digressions (hours)?

  7. Care to take a stab at an optimal goofy digression:skill practice ratio? The description of your one class suggests 1:3, is that typical? Is it different in your other preps?