3. Have you considered if the rule you are teaching expires?
Some rules expire, and if you have considered this, you probably are not wishing content knowledge into students. Rules that expire can be reinforced—sometime inadvertently or unintentionally—which leads students to false and unreliable conclusions.
The following “rules” are often reinforced by words and examples, but they expire.
“When you multiply, the answer always gets bigger…”
– until multiplying by values between 0 and 1 or multiplying with an odd number of negative factors.
“The answer to a subtraction problem always gets smaller…”
– unless it involves subtracting a negative number.
“We never subtract the larger number from the smaller number…”
– except for when the situations indicate that that this is what has to happen.
The key here is to challenge absolutes. When framed as “always” or “never,” look for contradictory examples. Often when we change the type of number with which we are working, the rule no longer applies, making it a rule that expires.
4. Do you frontload academic math vocabulary for a unit?
A “yes” to this question may suggest a wishing of definitions and the mathematical knowledge embedded with them into students. Without context, giving definitions and frontloading academic math vocabulary is unlikely to help a mathematician develop their thinking. A recent NCTM publication reads:
“…we do not learn language through “osmosis” or by solely interacting with fluent or native-like speakers; teachers must explicitly teach it (de Jong & Harper, 2005; de Jong, Harper, & Coady, 2013). However, this does not mean that teachers should focus solely on teaching particular words or phrases in isolation, as this is unproductive for student learning and an inefficient use of instructional time (Gibbons, 2015).
In other words, isolated and frontloaded vocabulary practice may lead to rote recall of definitions, but it does not offer a rich, conceptually driven understanding of the words in real-world or even mathematical contexts. This applies to frontloading of vocabulary for all learners regardless of the languages they speak because all learners are acquiring academic language.
5. Are there students you continually go to because they will have the right answer?
If you have a few students that you call on because they always have an accurate answer, you may be having them wish their answer into the other students.
Try calling on students who have divergent strategies or models to share, creating space for more than one “approved” solution path. To help with this, consider planning lessons by thinking about several possible solution paths. The Thinking Through a Lesson Protocol (Smith, et al, 2008) prompts us to consider all the ways the task can be solved: methods students will use, misconceptions, and errors.
By raising multiple paths, not just the one that is expected (by you or the book), a range of student thinking is validated while also creating opportunities to address misconceptions and errors.
6. Do you call on students until you get the answer you were looking for?
If you answered “yes”, you are probably wishing your way of thinking into all students. When planning lessons, we absolutely should be planning questions that elicit student thinking and reasoning. But if we only have one answer in our heads, the entirety of the math is that one pathway, our pathway, rather than the multiple pathways that offer an opportunity for reasoning!
Every student should have access to the mathematics in our math classrooms, but access does not mean trying to get every student to think the way you do or do it the way the book does it. Nor does it mean that every student will solve the problem the same way. Forcing conventions and procedures before reasoning results in missed opportunity for students to construct their own knowledge. This also limits the opportunities for student to see themselves and their unique assets as vital contributions to the mathematical community in their classroom.