Non-examples aren’t revolutionary to problem design, and would maybe more appropriately fall under problem set design, but I’ve found them to be a critical and incredibly useful way to see where student thinking is and challenge students to push their understanding.
Non-examples are usually routine questions — different from the conceptual thinking that other elements of problem design try to push, non-examples are innocuous but well-placed questions to assess whether a student can differentiate one concept from another. Non-examples require students to attend to precision (MP.6) by discerning between applicable concepts. These are most useful in student practice, after they have been introduced to new material and before they are pushed to deepen their understanding.
While students are practicing area of quadrilaterals, give them a triangle question.
While students are practicing area, give them a perimeter question.
While students are practicing reflections, give them a translation.
While students are practicing the volume of cylinders, give them a rectangular prism question.
Number & Operations:
While students are practicing one operation, give them a word problem requiring a different operation.
While practicing rounding, ask students to write a number in expanded form.
While students are practicing converting between mixed numbers and improper fractions, ask them to round.
While students are practicing simplifying fractions ask them to convert to a decimal.
While students are practicing finding a part of a whole, have them use the part to find the whole.
While students are practicing proportions given as percents, give them a problem using ratios.
While students are practicing finding unit rates, give them a problem requiring a percentage.
While students are practicing finding ratios, give them a problem asking for a probability.
Expressions & Equations:
While students are simplifying exponential expressions (), have them combine like terms with exponents.
While students are solving word problems requiring equations, give them a problem requiring only arithmetic.
While students are practicing square and cube roots, ask them to divide common perfect squares and cubes by other factors.
While students are practicing writing equations from a graph, ask them for a unit rate.
Statistics & Probability:
While students are practicing finding the mean, ask them to find the median.
While students are practicing interpreting measures of variability, ask them to interpret a measure of central tendency.
The idea of a non-example is not to ask random questions interspersed with questions aligned to a daily objective, but to place well-chosen questions at several deliberate places in a lesson that assess whether students can distinguish between similar concepts. If a student answers a non-example correctly, it doesn’t mean they’re necessarily mastering the material. However, if a student struggles on a non-example, and in particular if they apply the skill they’ve been practicing to the wrong concept, it provides valuable information about their fluency with that concept.
This is where choice of non-examples become critical. Random, unrelated questions are unlikely to provoke misconceptions or push students to think critically about the math that they are doing. A non-example should be structurally similar so that a student mindlessly applying a procedure will continue to provide that procedure without pausing to examine the new context. This doesn’t mean trying to trick students — throwing one subtraction question into a mad minute of multiplication — but choosing deliberate questions that push students to be thoughtful and deliberate about the processes they apply to math problems.
At worst, non-examples reveal students with the mindset “the teacher is showing me how to do this thing; I just need to do it over and over to make the teacher happy”. I was floored one time when, in an exercise working with circles, a student encountered a problem asking about a triangle and started to write area equals pi times radius squared, and identified a number as the radius of the circle before I stopped them to think for a moment about what they were working on.
At best, non-examples assess a challenging skill — having enough working memory capacity to think about both the execution of a new concept and the characteristics that allow that concept to be applied, as well as creating a habit in students to attend to precision in applying mathematics to problems they encounter.
Non-examples don’t teach students in and of themselves, and they don’t give valuable information on higher-order thinking, but they can be incredibly useful as an early-stage formative assessment tool to learn whether students are ready to push their understanding to a greater depth.