Look on my assessments, ye Mighty, and despair!

Assessment can be a huge, scary thing. Not just unit-sized assessments (which we will focus on here) but every assessment – every question, problem, project, or really anything that tells students and yourself how well they are faring with a particular idea or concept. And then there is that feeling when we put all of this work into constructing a huge, intricate unit assessment (or series of formal/informal formative assessments throughout a unit) only to have it be revealed as far from perfect and crumble apart when students actually get to it (like poor old Ozymandias). In this installment we’ll focus on designing unit assessments, with the conversation continuing in Mathematical Design 1.0 and upcoming blog posts. Let’s get started!

“What if they don’t get it?”

Which route is the fastest? (The above problem from here)

I hear this a lot when teachers look at open-ended questions like the one above. I’ve wondered this question myself with my own assessments. And then comes the landslide of doubt: What if students don’t even try? What if students know how to find the arc length, but they can’t put all of these parts together?

And what typically happens is that I whittle away at that beautiful problem above until I get something like this:


That is a bad problem. In fact, I wouldn’t even call it a problem, honestly. Here’s what this question implicitly tells students, and what they might think when they see this:

  1. “Well, math is a pretty pointless and boring exercise.” This problem is what David Stocker calls “pizza-party math” (very aptly named for this example). Why in the world would anyone care about the length of crust for that piece of pizza? It is a poor excuse of taking a relevant real-world thing and “doing math upon it”, as Stocker might say.
  2. “Alright, I guess I’m going to do something with 8 and 30°. Oh, find the length of this crust, ok.” It already tells students everything they need to know – The radius is 8 inches. This section is 30 degrees. Find the length of this arc.
  3. “Well that was easy. Math is not challenging.” OR “I don’t remember how to find this stuff. Pass.” This for me is the biggest punch in the gut, but is honestly inevitable. Either students know how to find arc length (and I’m sure by the time they take the unit assessment they will have done fifty problems that involve a 30 degree sector) and they are just using that specific skill (this is not a math problem to them, it is math practice), or they don’t remember the formula and therefore excuse themselves entirely from this very obvious attempt to apply that formula.

Instead of asking “what if they don’t get it,” we should instead be asking “why should they get it?” I can think of a whole bunch of reasons why I’d want students to be able to do the first question: It requires me to ask what sort of information is important in this problem; it requires me to recognize arc length as an important measurement; it requires me to then apply this and other mathematical ideas to solve a problem; it requires me reason an appropriate estimation for how many degrees make up that sector with the arc. For the “pizza-party” question, the reason basically boils down to “well I want to make sure that students know how to find arc length.” Ew.

So let’s take a pause. In a moment I’d like you to open up an upcoming unit assessment you’re giving and check through your questions. Ask yourself one thing – “What basic question is at the heart of this problem?” Read between the lines of what is actually written to try and see what the problem is really saying. If your answer to this falls into “How do you (insert specific skill)?” or “what is the definition of (insert math term)?” or “how do I make this assessment take a whole class period?” then we might have a problem with the purpose of this item. Let me demonstrate this reflection process with a particularly bad assessment – mine, from my first semester teaching 8th grade pre-algebra.

Oh no. Let’s look at items 1-5 (although 6-8 are equally bad). What’s the question? “How do we convert between decimals, fractions, and percentages” maybe? There’s that “How do we (insert specific skill)?” red flag. Is converting between these number forms important for this grade? Absolutely, but if I still don’t know my students’ mastery of this skill by the time of the unit assessment, what exactly have I been doing all unit long? Scaffolding is important, but I should have scaffolded this basic skill into a formative assessment (formal or informal) long, long ago. Let me repeat: you don’t have to cram every bit of knowledge and skill explicitly into a unit assessment. A much better plan would be to start from a big mathematical idea and use that to implicitly apply various knowledge and skills. Then you can worry about building those skills throughout the unit, with added incentive that your students won’t just have to mindlessly repeat facts and algorithms, but will have to understand the knowledge and skills well enough to solve a compelling question. Dan Meyer had a great conversation with Daniel Willingham (author of Why Students Don’t Like School) that reinforces this idea with this quote from Willingham’s book:

One way to view schoolwork is as a series of answers. We want students to know Boyle’s law, or three causes of the U.S. Civil War, or why Poe’s raven kept saying, “Nevermore.” Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question. But as the information in this chapter indicates, it’s the question that piques people’s interest. Being told an answer doesn’t do anything for you.

The conversation brings up the difference between posing a question and developing a question. If we’re just “eager to get the answers” instead of curious about how and why we both asking the question in the first place, we’re leading students down a path of thoughtless answer-getting or, worse yet, disinvestment in the math itself. So what are the big mathematical ideas behind converting fractions, decimals, and percentages? Well, there is no single correct answer, but that should be liberating – we can each give it our best shot and come up with totally different, valid answers. And what a difference it makes.

One mathematical idea is that decimals allow us to compare different ratios more easily by using a common base of 10. Percentages help even more with ratios that end up being less than one whole by showing how many hundredths we have rather than how many wholes we have. All of a sudden the underlying question of “how do we convert between decimals, fractions, and percentages” no longer defines the purpose of this concept. Something along the lines of “why do we convert fractions to decimals or percentages?” or “why don’t we always use decimals instead of percentages?” fits much better as an underlying question. And, whoa, I actually want to know the answer to that! Now I can perhaps design a question like…


1) How do the survey results of period 3 and period 4 compare?

And then we can dig deeper into that question if we want specifics, such as…

2) How would decimal form help us better compare these classes? How do these classes compare as decimals? What about as percentages?

3) Do you find it most useful to represent these results as fractions, decimals, percentages, or some other form? Why?

Do students have to use the skill of converting required in the standards? Sure. But the standards now support the assessment rather than wholly define it. This question is providing my students and myself with an incredible amount of information about how we think about decimals, fractions, and percentages, which will serve me not only in converting between these forms, but in applying them to solve diverse problems. Real problems. Not pizza-party problems.

So now try this with your own assessment – try and figure out what sorts of underlying questions your assessment items are really asking. If you don’t like the answer you find, consider the implicit big mathematical ideas to come up with an alternative question that you’d actually want to know the answer to. Because ultimately, we want assessments to be something that students look forward to because of the challenge and accessibility, and something that we look forward to for the feedback opportunities that they provide. And the first step is making questions that do just that.

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