Rigor in the Mathematics Classroom
By Scott Adamson
Scott Adamson
Think about the following statements:
“That math class is rigorous!”
“That college has a rigorous math curriculum!”
“That was a rigorous math lesson/lecture/test today!”
I contend that if these statements were heard by a variety of mathematics educators and if we could see their “thought bubble” containing what they imagine the statements to mean, we would see many different images for the meaning of the idea of “rigor.”
Consider your own thought bubble. What do you imagine when you think about a rigorous math class/program/lesson/etc.?
Unfortunately, there is not a commonly held understanding of the meaning of rigor within the mathematics education community. The University of Texas Charles A. Dana Center reports four commonly held viewpoints on rigor in the mathematics classroom:
Use of logical deductions from stated hypotheses to prove theorems
Adhering to traditionally prescribed content including a long list of topics and concepts
Increased difficulty and more challenging content
Rigor and college algebra may be used interchangeably
We need to ask ourselves, is a rigorous mathematics program one that focuses on proving theorems? One that covers a long list of topics? One that includes the nebulous idea of challenging content? Or one that involves a pathway to college algebra as opposed to a pathway involving a focus on quantitative reasoning and statistics? Are any of these views sufficient or necessary for a rigorous mathematics program?
Consider this problem posed to Introductory Algebra students:
Italian sprinter, Lamont Jacobs, won the 100 meter dash in the 2021 Tokyo Olympics with a time of 9.79 seconds. How fast did Jacobs run?
The problem is intentionally open ended for the purpose of allowing students to respond and then defend their response with mathematical justifications. Before continuing, the reader is encouraged to develop a response to the question!
Below is a typical student response (with hypothetical student comments alongside the written work). Please analyze the response with your image of rigor in mind. Would you classify the response as a “rigorous” response? Why or why not?
Start with the “dirt” formula…d=rt which means r equals d over t.
Plug in the numbers and divide to find that r equals 10.21 m/s.
Use dimensional analysis and cross out units until the desired units remain.
I argue that this does not provide a rigorous response. Imagine that we were able to follow up with this student’s work by asking the following questions:
Explain why the formula d=rt makes sense. What is the mathematical meaning of “dirt formula”?
When you say, “r equals d over t,” what do you mean by “over”?
When you computed 100 meters divided by 9.79 seconds, how do you explain how division makes sense? Consider that division, in this case, says, “how many 9.79 seconds are in 100 meters?” How does this make sense?
When converting to MPH, what is the mathematical rationale for “crossing out the units”?
It is highly likely that most students will be unable to respond to these questions. The inability to respond indicates a lack of rigor in their mathematical understanding. This student has an instrumental understanding but lacks a relational understanding (Skemp, 1978). That is, they may know what to do, but lack an understanding of why they are doing it or how it makes sense.
So...what might we mean when we say “rigor” in the context of learning mathematics? A rigorous understanding based on a rigorous course that is part of a rigorous program with rigorous curriculum is one where students are engaged in (Oehrtman, AMATYC IMPACT, and The Charles A. Dana Center):
Attending to precision, structure and patterns,
Inference, interpretation, reasoning,
Mathematical habits of the mind and ways of thinking, and
Helping students develop their mathematics identity.
Continuing from the above mentioned source, “in a rigorous course, students are asked to: Struggle with real, non-routine problems in context; identify strategies to solve problems; communicate about mathematical ideas with clarity and precision; and justify solutions.”
Let’s return to the 100-meter sprint context. Given that the student audience is one where the mathematical ways of thinking are being developed in an introductory algebra context, a different, possible response from a student follows (with hypothetical student comments alongside the written work):
I want to determine a unit rate…distance traveled in 1 second.
I want to keep the two quantities in proportion with one another.
Creating a correspondence between each of these time and distance segments.
Keeping the quantities in proportion again. I need 3600 copies of 1 second so also 3600 copies of 10.21 meters.
Determining “how many copies” requires a division computation.
Definitions are important. If we define rigor to include ideas like communication, precision, justification, and reasoning, then the second student response would be considered as rigorous. This is not to say that the first response is “bad” and the second response is “good.” The first response contains sound mathematical calculations that result in the correct final answer. However, in the context of teaching and learning in a rigorous context (as defined), the second response is rigorous and demonstrates a well connected, relational understanding of the mathematical ideas needed to solve the problem.
Conclusion
The main takeaway from this blog post is to challenge the reader to consider the meaning of rigor that they hold. For some, rigor is narrowly defined to focus on difficult or challenging computations and procedures. In contrast, consider a definition of rigor that is, perhaps, more robust and extensive. In addition to developing procedural fluency, consider the role the conceptual understanding and application or problem solving may have in mathematics education. Consider how students might be engaged in explaining, justifying, and constructing mathematical arguments. Consider how students might be engaged in communicating their thinking and solution strategies, modeling with mathematics and interpreting results, and critiquing the reasoning of others.
Moving forward, the reader is encouraged to engage in conversations with their mathematics colleagues, departments, and program leaders to create a common definition for the meaning of rigor. Using this definition like a vision statement, consider how this definition may impact pedagogical strategies, curriculum decisions, and assessment practices. That is, in the definition presented here, rigor involves more than just difficult and challenging computational and procedural experiences. Therefore, pedagogical strategies such as active learning are implemented to support rigorous learning of mathematics. Curricula is chosen to support a balance of procedural fluency, conceptual understanding, and application (or problem solving). Formative and summative assessment strategies, which often drive pedagogical and curricular decisions, focus on attainment of rigorous mathematical learning. All students at all levels of mathematics should have access to rigorous, high quality mathematics programs that prepare students for future academic and career choices.
Scott Adamson teaches mathematics to students at Chandler-Gilbert Community College. His primary interest is in developing active learning strategies that promote deep mathematical thinking by students.
References
Charles A. Dana Center (n.d.). What Is Rigor in Mathematics Really?. Retrieved from https://www.utdanacenter.org/sites/default/files/2019-02/what-is-rigor-in-mathematics.pdf . Austin, TX: The Charles A. Dana Center at The University of Texas at Austin.
Skemp, R. (1978). Relational Understanding and Instrumental Understanding. The Arithmetic Teacher, v. 26 no. 3, pp. 9 - 15, National Council of Teachers of Mathematics, Reston, VA