I’ve recently been questioning daily learning goals (or objectives, or whatever else you’d like to call them). This is a bit of a straw man argument, because I think I have an answer and an articulation for how to navigate some of these challenges, but I want to put my issues in writing before I try to reformulate my thinking.

**Spiraling**

First, I’ve been captivated by the idea of a fully spiraled curriculum. I wrote more about it here, but the highlights are: Teaching math in concrete units where we focus on one topic for a short period of time, assess, and move on, creates an illusion of learning. Knowledge gets cycled through short term memory without any real incentive for retention and is never revisited. And next year’s math teacher is confused when students don’t know anything. I get a bonus point for throwing in a review day so every student passes the test, but it doesn’t do a ton for real learning — students aren’t synthesizing multiple topics or solidifying understanding over time. Each topic is an island, and a quickly sinking one.

I have two corollaries here. First, interleaving — mixing different topics together — leads to more durable learning, despite being more difficult for students in the moment. So why have just one learning objective? Second, if we’re going to revisit a topic a number of times over the course of the year, students don’t need to master it today. We’ll come back to it and deepen understanding in the future, so why prescribe specific understandings they have to reach that day?

**Formative Assessment
**I spent much of my summer, in particular the three weeks at PCMI, thinking deeply about formative assessment. I’ve been spending more and more class time on tasks designed to see where student thinking is, and building in hinge questions that help me determine where to go next. Teaching Algebra-II and Pre-Calc is particularly conducive to early formative assessment because students come in with such varying levels of knowledge. But I’m very often adjusting my learning goals as class goes on — deciding to push a certain activity until the next day when I can scaffold for a set of misconceptions I just encountered, or moving more quickly through a topic students are confident with. All this means that more of what we do in class has the purpose of exploring student thinking on a topic, rather than trying to get them all to some predetermined place. How does the idea of a concrete, measurable learning goal mesh with honest formative assessment that should, if I use it effectively, change what the most appropriate learning goal is?

**Knowledge Transfer**

I’ve become more generally jaded with daily objectives in that they can mask broader misunderstandings. Much of what I’m doing with all three of my classes right now revolves around exploring and deepening understanding of functions. I can’t break that down into day-sized pieces very easily. Sure, there are linear functions, quadratic functions, function properties function transformation, a wide range of applications, transformations, composition, and more, and each of these has different skills attached to it. But focusing on individual aspects of functions can obscure the broader ideas that underlie everything we’re doing. If I want students to leave class with deep, transferable knowledge of functions, adding up a set of discrete, concrete daily objectives is (I think) unlikely to do the job. Instead, the focus needs to continually come back to big ideas that transfer to new contexts, and those ideas transcend a measurable task I want my students to be able to do. Functions are my example here, but fractions, proportional reasoning, geometric proof and more are similar — it’s easy to get lost in the sauce, and lose the broader purpose for the day’s learning goal.

**Humility**

Finally, as I’ve delved deeper into formative assessment, I am constantly humbled by what my students don’t know. I often thought of formative assessment as how I see what students know about a topic before we start, but I’ve been thinking of it more and more in terms of ongoing assessment that gives me an honest picture of what students understand in the middle of the learning process. Many topics I thought students understood a day or two before I revisit with a new task and find new gaps in understanding. What’s the point of a daily learning goal if, even when students meet that goal that day, they are liable to just forget in a day or two and move on with their lives? Are learning goals just an illusion to make me feel good about myself at the end of a lesson, despite the reality that most knowledge is ephemeral and already slipping away?

I’m really curious for more perspectives on these challenges. I’m trying to put together a new way of looking at my students’ learning goals that takes into account these issues. Stay tuned for that one.

Bowen KerinsCreating a fully spiraled curriculum is super difficult but if anyone can do it, it’s you 🙂

I totally agree that by breaking a topic into too-tiny chunks makes it difficult for students to figure out what works when, or what works generally. One key example for me is graphic: we graph lines, then we graph quadratics … but the parameters and their meanings have mostly changed.

In CME we do “graphing in general” and then apply those general principles to each type of specific graph, so that the most generalizable principles continue to be addressed and sink in.

Good luck out there!!!

dkane47Post authorThanks, Bowen. I’m moving in the direction of a spiraled curriculum for my junior Algebra-II class at the moment, but I think I’m a ways away.

Graphing in CME is a great example of that idea. One of the ways I’m thinking about learning goals is about exactly what you’re talking about — figuring out the most generalizable principles, and acknowledging that those principles in and of themselves are a worthwhile learning goal. It’s not very measurable, but if students are doing mathematical thinking, then I’m happy.

howardat58I just went back to your post on problem solving. Daily learning goals don’t sit well with problem solving, and to me the problem solving is what math is all about. The need for a new method arises naturally out of problems which don’t want to respond to current methods, and so methods are not the most important part. Of course some of them are fascinating in themselves to some students.

dkane47Post authorProblem solving is a good example — but I think students also need some concrete content knowledge to build that problem solving on. Does that synthesis provide a useful place for learning goals?

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