In light of the rapidly approaching school year, there have been a host of articles about how teaching must change. And so I was not terribly surprised to see that National Council of Mathematics Teachers (NCTM) and the National Council of Mathematics Supervisors (NCSM),have jumped on this bandwagon and announced that math teaching must change in their latest report.
An article summarizing NCTM’s report states: “According to the NCTM and NCSM, during the pandemic, the urgency to change the way mathematics is taught has become apparent. According to both agencies, math instruction needs to be more equitable, so it is essential to plan what math classes will look like before returning to school in the coming months.”
Reading through the article, as well as the NCTM/NCSM document itself, other than the fact that online teaching by its nature is different than in-class teaching, it is not apparent how mathematics must be taught differently. In fact, the NCTM/NCSM document’s advice on how math should now be taught is generally the same as it has been for the past three decades. Namely “differentiated instruction”, elimination of ability grouping, full inclusion, and equity for all.
Their pleas for these changes make it seem as if nothing in math education has changed in the past thirty years. If anything, there has been an increase in the practices so recommended. Elimination of ability grouping has been accomplished by so-called differentiated instruction by providing different assignments and expectations for the varying levels of student abilities within the same class. The teaching of procedures and algorithms has given way to “understanding and process”. A disdain for memorization has de-emphasized the learning of multiplication tables. The teaching of standard algorithms is delayed while students learn inefficient and confusing “strategies” that purportedly show the conceptual underpinning behind the standard algorithms.
The document advises that specific teaching practices be implemented in online learning. The document then provides eight practices that the authors of this document believe provide equitable and effective math teaching, and which “provoke students to think.”
Here they are with my commentaries attached:
- Set math goals that focus on learning.
How else are math goals established? The implication, given NCTM’s past history, is that providing instruction for procedures, with worked examples and scaffolding is “inauthentic” and therefore is not focused on learning.
- Implement tasks that promote reasoning and problem-solving.
Most textbooks that were written in previous eras did just that, and did it well.
- Use and link mathematical representations.
By this they mean students should be able to visualize what’s happening by means of pictures. Also, they want students to make “connections” with prior mathematical topics. Robert Craigen, a math professor at University of Manitoba who has been involved in improving K-12 math education says this: “It’s amusing when they speak about “connections” as if this were something different from “isolated facts”. Actually it is the facts that provide connections. Everything else is only the educational analog of a conspiracy theory.”
- Facilitate meaningful problem-solving course.
They want problems to be “relevant”, in the belief that otherwise students have no desire to solve them. Actually, students will want to solve problems for which they have been given effective instruction that allows them to be successful at it.
- Ask questions with a purpose.
This could refer to “intentionality” or “math talk”, or both. Let’s look at “intentionality” first.
Inentionality is the edu-buzzword du jour which has replaced the previous one: “student agency.” From what I can tell from its usage, “intentionality” generally means an overriding goal that strongly colors—and drags along—all other considerations of a lesson. So if the goal is differentiating the lesson to take into account the “variability of all learners”, then any other goals for a particular lesson—say multiplying negative numbers—must be constructed to accommodate weak students and challenge stronger ones.
Math talk: This refers to getting students to talk “like mathematicians” by asking questions such as “Can you convince the rest of us that your answer makes sense?” and “What part of what he said do you understand?” I recently saw an article claiming that “research shows” that students who talk about their math thinking are motivated to learn. In addition, this “math talk” is viewed as a form of formative assessment giving teachers a peek into student thinking and where they need help. “Math talk” is an effective tool only if the instruction they received allows them to make use of it. Otherwise, it is like children dressing up in their parents’ clothes to play “grownups”.
- Develop procedural fluidity that comes from conceptual understanding.
Although they pay lip service to procedural fluency, it is fairly clear that they believe that mastery of the conceptual understanding behind a procedure must always precede the learning of said procedure.
- Support the productive struggle in learning mathematics.
Worked examples with scaffolding are believed to be “inauthentic” and take away from what would otherwise be a productive struggle. Missing from this type of reasoning is that a person who is trying not to drown is not learning how to swim.
- Obtain and use evidence of students’ mathematical thinking.
In other words, students must be able to explain their answers. While this can be done through questioning, it does not take into account that novices (particularly in lower grades) are not as articulate as adults think they should be. Adults have had many years of experience with the topics that novices are trying to learn. “Show your work” now means more than showing the mathematical steps one does to solve the problem. It means justifying every step. Failure to do so, even if a student has correctly solved a problem is viewed as the student failing to “think mathematically” or understand.
I’ll leave it to you to read the NCTM/NCSM document in its entirety. In all fairness, some of their advice is useful. But in my opinion most of it is not.