Developmental Mathematics Revival!

The Rigor Unicorn

How would you define (or describe) “rigor” in college mathematics classes? Can you define or describe “rigor” without using the words “difficult” or “challenging”? I will share a recent definition, and counter with my own definition.

Before anything else, however, we need to recognize the lack of equivalence between rigor and difficult (and between rigor and challenging). The basic problem with those concepts (difficult, challenging) is that they are relational — a specific thing is difficult or challenging based on how that thing interacts with a person or a group of people. Difficult and challenging are relative concepts, not a property of the object being described. I found the calculus of trig functions to be difficult, not because there is any rigor involved — it was difficult for me because of the heavy role of memorization of formulae in that work in the particular class with that particular professor. Other learners find this same work easy.

A recent definition of “rigor” comes from the Dana Center (DC):

We conclude that rigor in mathematics is a set of skills that centers on the communication and use of mathematical language. Specifically, students must be able to communicate their ideas and reasoning with clarity and precision by using the appropriate mathematical symbols and terminology.

See http://dcmathpathways.org/resources/what-is-rigor-in-mathematics-really http://dcmathpathways.org/resources/what-is-rigor-in-mathematics-really

This definition avoids both ‘banned’ words (difficult, challenging), and that is good. This definition focuses on communication of ideas and reasoning, and that seems good. When my department discussed this definition recently at a meeting, the question was naturally raised:

Does rigor exist in the absence of communication?

The problem I have with the “DC” definition of rigor is that it suggests that rigor only exists when there is communication taking place. In other words, rigor does not describe the learning taking place … rigor describes the communication about that learning. Obviously, communication about mathematics is critical to all levels of learning — whether there is lots of rigor or none. I don’t think we can equate rigor with communication. Such an equivalence tempts us to equate rigor with how we measure rigor.

As I think about rigor, I always return to concepts relating to the strength of the learning. I’d rather have an equivalence between rigor and strength, as that makes conceptual sense. The rigor exists even in the absence of communication. Rigor describes the concepts and understanding being developed within the learner, not the object being learned.

My definition:

Rigor in mathematics refers to the accuracy and strength of learning, and specifically to the completeness of the cognitive schema within the learner including appropriate connections between related or dependent ideas.

In some ways, this definition of rigor suggests that “rigor” and “like an expert” might be equivalent concepts. I am suggesting that rigor describes the quality of learning compared to complete and perfect learning. Rigor is not an on-off switch, rather it is a continuum of striving towards the state of being an expert about a set of concepts.

One of the reasons I approach the definition ‘differently’ is that rigor should exist in appropriate ways in every math course — from remedial through basic college through calculus an up to graduate level and research work. Rigor is not a destination, where we can declare “this student has rigor”. Rigor is a quality comparison between the unseen learning and the state of an expert in that particular set of content. When we teach basic linear functions, I seek to develop rigor in which students have qualities of learning like an expert would have, concerning connections and reasoning. When we teach calculus of trig functions, I hope we seek to develop qualities of learning similar to an expert.

I believe the development of rigor is a fundamental ingredient to making mathematics innately easier for the learner. When the knowledge is more complete (like an expert) the use of that knowledge becomes more efficient … and the learning of further mathematics requires less energy (just like an expert). Rigor is the core ingredient in the recipe to make mathematicians from the students who arrive in our classrooms.

Rigor does not start in college algebra, nor in calculus. Rigor is not the same as ‘difficult’. Rigor can exist when there is no communication about the learning. Rigor is the fundamental goal of all learning, at all levels … rigor is a way to measure the quality of learning. Rigor is the goal of developmental mathematics … the goal of quantitative reasoning … the goal of pre-calculus … the goal of calculus … the goal of statistics.