I’ve been blabbing more and more about formative assessment the last few months, in particular since my time diving deep into the what and the why at PCMI this summer. I thought it would be a useful exercise to come up with my own definition of formative assessment. Then that exercise turned out to be both harder and more useful than I thought. Here’s where I am now:

**Formative assessment is assessment with the purpose of figuring out what students know and don’t know in time to do something about it.**

Maybe that seems simple and obvious, but there’s plenty I’m leaving out. In Dylan Wiliam’s Embedded Formative Assessment, he identifies five key strategies:

1. Clarifying, sharing, and understanding learning intentions and criteria for success.

2. Engineering effective classroom discussions, activities, and learning tasks that elicit evidence of learning.

3. Providing feedback that moves learning forward.

4. Activating learners as instructional resources for one another.

5. Activating learners as owners of their own learning.

I think Wiliam’s book is great, but I see it more as a guide to great teaching than a book solely focused on formative assessment. While strategies 1, 3, 4, and 5 are important parts of a successful classroom, I have learned more about formative assessment by isolating it as the specific actions I take that elicit evidence of learning, in strategy 2.

This gets me down to a few specific criteria for what effective use of formative assessment tasks look like:

- Tasks focus on the larger concepts and key understandings of a topic
- Tasks are structured to make assessment of understanding efficient for the teacher
- Tasks are structured to minimize bias in evaluating student understandings
- Tasks are structured to minimize false positives

And where they are placed in the learning progression

- Tasks early in an individual lesson allow for a “hinge point” where the teacher makes an instructional decision based on evidence of student understanding
- Tasks early in the study of a topic allow for the teacher to make changes to the structure of the unit based on evidence of student understanding

Building off of this, I end up with two different situations I use formative assessment in.

- I want to know whether students understand a specific standard or skill that is foundational for future learning. This formative assessment is most likely to come in the middle of a lesson, where I can make a decision to spend more time on the skill or move on to a transfer task. I’m more likely to use a short item, possibly multiple choice, to quickly dipstick what students know. Items may be routine or non-routine, depending on the future application of the skill.
- I want to know whether students understand the broader ideas of a concept to gauge what further instruction is necessary. This is more likely to take up a large part of a class period, and is likely more subjective. Tasks require students to transfer their knowledge to a new situation or context.

It’s important to note that the purpose of these tasks is not solely for me to learn what students know — they should also be learning and deepening their knowledge of a topic in the process. But I don’t structure every task deliberately to be used as formative assessment — I pick some to focus my attention on, to discipline myself to avoid the confirmation bias inevitable when I am constantly seeing glimpses of student thinking without necessarily getting the complete picture.

This leads me to an instructional sequence I like:

Let’s say I’m teaching polynomial division. I’m a big fan of the box method.

I introduce it with some examples of polynomial multiplication to contextualize the process, then walk through what the division looks like. Answer questions, and probe with a question for a quick turn and talk — maybe “how do I know if the divisor goes in evenly?” I listen in, and if students feel confident, have them try two on their on. Circulate, and see how they’re doing, and answer questions if necessary. I focus on the end of the circulation specifically on how many students got each question right. Here’s where I make a second decision on whether we need more whole-class instruction, more practice, or if students are ready for a more complex task. Once they’re comfortable with the basic process, I offer up a tougher task — I like this one from Illustrative Mathematics:

Here students have to do some original thinking, and it gives me a much broader, although more complicated, view of how well they understand the connection between polynomial multiplication and division and what the process actually means. I don’t care as much about whether they found the right value of *a* in this problem — I’m looking at their thinking, and have a much more challenging decision to make about what students need to reach proficiency with this standard.

The formative assessment moves form the narrow skills to the big ideas, and gives me chances to do something about student misunderstanding. This is obviously a very neat and tidy example — that sequence would more likely play out over a few days, with plenty of headaches and challenges along the way. But that’s the point — figuring out what those challenges are, and meeting them early while I can do something about it.

howardat58My question is if by this time they haven’t seen the connection between factors of polynomial expressions and roots of polynomial equations then they are missing something. This doesn’t need a reference to the Remainder Theorem at all.