In my last post about the inverted/flipped calculus class, I stressed the importance of **Guided Practice** as a way of structuring students’ pre-class activities and as a means of teaching self-regulated learning behaviors. I mentioned there was one important difference between the way I described Guided Practice and the way I’ve described it before, and it focuses on the learning objectives.

A clear set of learning objectives is at the heart of any successful learning experience, and it’s an essential ingredient for self-regulated learning since self-regulating learners have a clear set of criteria against which to judge their learning progress. And yet, many instructors – myself included in the early years of my career – never map out learning objectives either for themselves or for their students. Or, they do, and they’re so mushy that they can’t be measured – like any so-called objective beginning with the words “understand” or “appreciate”.

Coming up with good learning objectives is something of an art form, and I have a lot of room for improvement in the way I do it. However, I’ve been working with the following workflow for generating learning objectives that works particularly well for my students and fits the ethos of the flipped classroom. Here it is step-by-step.

**STEP ONE**: Comb through the unit you’re going to cover in class and **write down all the things you’d like students to be able to do**, at some point in the near future. Very importantly, use action verbs for these things and avoid anything that cannot be measured. In particular avoid the words *know*, *understand*, and *appreciate*.

For example, here’s the list of objectives that I came up with when I was planning out the unit on the chain rule in the calculus class. These are roughly in the same order in which they appear in the text, and I threw on a couple of additional objectives that address some review items:

- Identify composite functions (that is, functions of the form (y = f(g(x)))) and identify the “inner” and “outer” functions.
- Use the Chain Rule to differentiate a simple composite function, for example a composition of a polynomial and a power function (e.g., f(x) = (x
^{2} + x + 1)^{1/2}). - State the Chain Rule and explain how it works in English.
- Use the Chain Rule to differentiate a composite function involving two functions.
- Use the Chain Rule in combination with other rules from earlier in this chapter.
- Use the Chain Rule to differentiate a composite function in which at least one of the functions in the composite is given as a graph or a table of values.
- Identify situations where the Chain Rule should be used when taking a derivative.
- Use the Chain Rule to differentiate a composite function involving three or more functions.

In the past when I’d taught the chain rule, my only learning objective was something like “Know and use the chain rule”. That’s too vague! There is a lot of nuance in what it means to “know and use” this rule and it’s on me, as the instructor/course designer, to communicate clearly what I intend to assess.

We’re not done with this list.

**STEP TWO**: You’ll immediately see that some of the actions in your list from Step One are more cognitively complicated than others. So step two is: Go back to your list and **reorder the items in it, putting them in order from least complex to most complex**. A handy tool for doing this is Bloom’s Taxonomy, summarized in the pyramid below.

Bloom’s Taxonomy is a standard means of categorizing cognitive tasks by complexity, with the simplest (Knowledge, or “Remembering”) at the bottom and the most complicated (“Creating”) at the top. Go through each of your learning objectives and decide what level of Bloom they most closely correspond to. Then shuffle them around so that the higher up the list you go, the more complex the task is.

Applying this idea to the above list of objectives about the chain rule, I ended up with this ordered list. Here the objectives *start* with the simplest and *end* with the most complex:

- Identify composite functions (that is, functions of the form (y = f(g(x)))) and identify the “inner” and “outer” functions.
- State the Chain Rule and explain how it works in English.
- Identify situations where the Chain Rule should be used when taking a derivative.
- Use the Chain Rule to differentiate a simple composite function, for example a composition of a polynomial and a power function (e.g., ( f(x) = (x
^{2} + x + 1)^{1/2})). - Use the Chain Rule to differentiate a composite function involving two functions.
- Use the Chain Rule to differentiate a composite function involving three or more functions.
- Use the Chain Rule in combination with other rules from earlier in this chapter.
- Use the Chain Rule to differentiate a composite function in which at least one of the functions in the composite is given as a graph or a table of values.

Most of the time, the order of appearance of topics in the textbook mirrors the order of complexity – easier stuff at the beginning, harder stuff at the end – but not always. For example in our book, there are some preliminary examples of the Chain Rule that precede the formal definition, but stating a definition is a less complex task (“Remembering”) than doing an example (“Applying”), so stating the definition appears before any instance of actually performing a computation.

**STEP THREE**: This is a really important step for the flipped classroom. Look at your ordered list of learning objectives and ask: *What is the most complex task that I reasonable expect students to be able to master prior to class, given the resources that they have?* **Find that task and draw a line between it and the ones above it (that are more complex)**. The objectives below the line are your Basic learning objectives, and students will be expected to demonstrate fluency, if not mastery, on those items when they arrive at class. The others are your Advanced learning objectives; students will not be expected to master these before class (although if they do, that’s awesome!) but rather they’ll use the class meeting time and follow-up study to master these over time and with the help of others. Put BOTH sets of learning objectives on the Guided Practice assignment.

In the above example, I decided to split the list at “use the chain rule to differentiate a simple composite function”. I decided that all objectives up to and including that one can, and *should*, be encountered by students prior to class, and students are told on the Guided Practice form that they will be expected to perform all those actions with fluency *when they arrive* – and that *there will be no re-teaching of these concepts in class*. On the other hand, some of these actions are not ones I’d consider appropriate for all students for pre-class work, like working with the chain rule applied to functions that are not formulas. Here’s the final version of that Guided Practice where you can see the learning objective lists in context, including the exercises that students work to practice those learning objective tasks.

Other instructors with different sets of students might choose to set that cutoff line differently. For example, if I had an honors section of calculus, I’d raise the line. If I had a group that was really struggling with earlier derivative techniques, I’d lower it. It’s not raising or lowering the *overall academic standards* of the course, notice. All students in any setting will eventually need to master all of those objectives. It’s just a matter of deciding how much you think students can handle prior to class and how much you think you can handle in class and after class, and setting the cutoff accordingly.

I only recently discovered this idea of the split list of objectives. I wasn’t giving students two lists of objectives, but I noticed that I tended to remix my learning objectives in order of Bloom’s Taxonomy. Going from there to having this split set of objectives was a major leap forward in my implementation of the flipped classroom for several reasons. First, it communicated to students that *they don’t have to learn everything about the new unit prior to class* – just the basics, and I am telling them what those basics are and how to know if they’ve learned them. It also reduces the cognitive load on students and sends the message that they are not “teaching themselves calculus”. Second, it tells students that **the class meeting still matters**, because it’s in that class meeting that we meet the Advanced objectives head-on. Third, it serves as a useful planning device for me, because all I have to do to prep the class meeting is just to design an activity and some formative assessment to address those advanced objectives.

Further down the line, the lists of learning objectives are also a ready-made topic list for timed assessments like tests and the final exam. Want to know what’s on the test? Just take the set union of all the learning objectives we’ve seen up to now.

This is far from a perfect system, but it’s a reliable way to align learning objectives with the actions you want students to perform and the means you want to use to assess them, and it gives students a key ingredient for self-regulated learning: A clear set of criteria that will tell them what they need to know and how to measure whether or not they know it.

*Images: Bloom’s Taxonomy, http://commons.wikimedia.org/wiki/File:Bloom_taxonomy.jpg; main image http://www.flickr.com/photos/fxp/*