How Do YOU Define the Idea of Rigor in Mathematics Education?
Suppose you heard a colleague say, “Today’s math lesson is very rigorous!” What does it mean to label a lesson as “rigorous?” What would YOUR thoughts be if you heard someone claim that a lesson was rigorous?
Take a moment to think about YOUR meaning for the idea of “rigor” in mathematics.
The University of Texas Charles A. Dana Center reports four commonly held viewpoints on rigor in the mathematics classroom:
- View 1: Use of logical deductions from stated hypotheses to prove theorems
- View 2: Adhering to traditionally prescribed content including a long list of topics and concepts
- View 3: Increased difficulty and more challenging content
- View 4: Rigor and college algebra may be used interchangeably
Let’s unpack each of these views on the context of a college algebra lesson involving quadratic expressions, equations, and functions. As you read, keep in focus YOUR meaning of the idea of rigor.
View 1 - Lesson 1:
The instructor presents a clear and coherent derivation of the quadratic formula. Students carefully copy the instructor’s work from the whiteboard into their notebooks. The instructor then shows two examples of how to use the quadratic formula to solve quadratic equations. Students practice using the quadratic formula to solve quadratic equations.
View 2 - Lesson2:
The instructor shows students how to solve a quadratic equation using synthetic division. Students diligently copy the steps into their notebooks as the instructor describes the process. Additional examples of using synthetic division for solving polynomial equations are shown and then students are given practice problems.
View 3 – Lesson 3:
The instructor shows students how to solve a quadratic equation by factoring. To make the lesson more challenging, the instructor says, “now you try one” and asks students to solve using factoring. Students try unsuccessfully for several minutes and then the instructor solves the problem at the board and students copy the solution.
View 4 – Lesson 4:
An academic advisor is helping a recent high school graduate to decide which math course to take. The student has good grades in her high school math class and so the recommendation is to take college algebra since “you are good at math” and she is told that “college algebra is the prerequisite for other, more rigorous, math courses.”
These examples highlight a common view that rigor in mathematics involves an instrumental understanding of mathematics only - apart from a relational understanding (Skemp, 1978). That is, students are encouraged to focus on what to do (instrumental understanding), but lack an understanding of why they are doing it or how it makes sense (relational understanding). Thompson, Philipp, Thompson, and Boyd (1994) refer to this dichotomy as having a calculational orientation for teaching mathematics versus a conceptual orientation.
“A teacher with a calculational orientation is one whose actions are driven by a fundamental image of mathematics as the application of calculations and procedures for deriving numerical results. This does not mean that such a teacher is focused only on computational procedures. Rather, his is a more inclusive view of mathematics, but still one focused on procedures—computational or otherwise—for “getting answers.””
A teacher with a conceptual orientation is one whose actions are driven by:
- an image of a system of ideas and ways of thinking that she intends the students to develop,
- an image of how these ideas and ways of thinking can develop,
- ideas about features of materials, activities, expositions, and students’ engagement with them that can orient students’ attention in productive ways,
- an expectation and insistence that students be intellectually engaged in tasks and activities.
Return to YOUR meaning of the idea of rigor. Does it include both an instrumental (or calculational) and a relational (or conceptual) orientation? How might we think about rigor in such a way to include both? In this current age, is it important for students to have such a strong focus on an instrumental/calculational orientation? Or, can the procedures and skills that have been the focus of mathematics education be relegated to the ubiquitous technology that students have access to such as Desmos, wolframalpha, and others? In general, 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 Oehrtman, et al. 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.”
What might it look like for all students to experience a high quality, rigorous mathematics lesson? To illustrate one task, situated at the conclusion of a college algebra unit on quadratic functions, consider this 3-Act task and how it might play out in the classroom.
Consider how this meaning of rigor in mathematics education might be evidenced in another lesson focused on the idea of quadratic functions in a college algebra course.
Act 1: The Hook
Students begin the lesson by watching a video from the Pixar Short where Lightning McQueen attempts to make a heroic jump across the mythical Carburetor Canyon. In the video, Lightning McQueen rockets down a ramp and upon hitting the final jump, begins a journey across the canyon. The video stops when McQueen is nearly halfway across.
Next, students are asked, “What do you notice? What do you wonder?” Specifically, students are to focus on mathematical noticing and wondering what comes to their mind.
This approach can be considered rigorous as it encourages students to ask their own questions, to wonder how mathematics can be applied, and to think about what information is needed to explore the questions that they have developed.
This is a departure from the more traditional approach where students are given problems (e.g. homework exercises) that include all the needed data and facts. These facts are then used to answer a specific question that is asked where, likely, students are to plug in the given values into a known formula that is the focus of the lesson. Students begin to believe that mathematics is about plugging the right numbers into the correct formula so that they can compute the right answer.
In this 3-Act framework, students are presented with an open-ended problem situation where they are encouraged to come up with their own questions, to think about the information needed to pursue an answer to their question, and thus to develop effective problem-solving skills.
Act 2: The Mathematical Inquiry
In this case, the class may come to agreement that the primary question to explore is, “Does Lightning McQueen successfully jump across Carburetor Canyon?” In Act 2, students are provided with some of the facts or data that they wondered about in Act 1 which can serve to support their thinking about this primary question.
Let’s focus on the idea of modeling the location of Lightning McQueen during the Carburetor Canyon event. Typically, students will either notice that or wonder if a quadratic function can be used to model the location of the car. This idea is coming from students and they are encouraged to pursue the idea of creating a quadratic model representing the car’s location.
Students may use technology to help them develop a quadratic model in this open-ended task. The following questions often arise.
- Given that the goal is to create a quadratic model, which form of the quadratic (factored, vertex, standard) will you choose? Why?
- You will focus on two quantities, x and y. What do these quantities represent in this situation?
- Convert from one quadratic form to another. What are the advantages of these forms in this context?
- What is the advantage of placing the origin at the end of the ramp?
- Under what conditions is there a successful jump?
- Under what conditions is there an unsuccessful jump?
- Suppose a quadratic model in standard form is created: . What do the different parameters mean in this context?
Students may experience rigor as they work together on the task, share their thinking, and work to create a mathematical model. For example, students first decide which quadratic form to use and present a rationale for why they chose this form. In my experiences, different groups of students have decided to use all three forms. Factored form is advantageous in that the horizontal intercepts are easily obtained. Vertex form is advantageous when students consider symmetry and locate a possible vertex. In both cases, only one more parameter is required. Other students feel comfortable with standard form and, with the use of technology such as Desmos, can manipulate the parameters of this quadratic model using sliders until a satisfactory model is developed.
If interested in how this mathematical model might be developed using Desmos, the reader is encouraged to use this link: https://www.desmos.com/calculator/vemx0igfi0
Note that the rigor comes not from tediously computing but from the meaningful thinking and decision making in which student will engage. The appropriate use of technology allows students to engage in inquiry and discovery. It allows them to apply their understanding of quadratic behavior to a unique situation. The rigor comes as they first make decisions about the quadratic model and, second, interpret the meaning of the parameters of the model. Rigor manifests when students justify and defend their choices and explain their thinking. Rigor is experienced as students make connections to previously studied ideas.
For example, if students are encouraged to interpret the meaning of the parameters of a quadratic model in standard form, , they might first indicate that the parameter is the initial rate of change. Think about the units for this initial rate of change. The quantity x is the horizontal distance, in feet, from the take-off ramp. The quantity y is the vertical distance, in feet, above the hypothetical plateau of the canyon. Therefore, the initial rate of change can be interpreted as follows: For every 1 horizontal foot that the car travels (to the left), its height will increase by 0.679 feet.
Note that the units are “feet per foot.” That is, vertical feet per horizontal foot. What a wonderful context to have students carefully consider the meaning of an initial rate of change!
Act 3: The Resolution to the Problem
Students often ask their instructors, “is this right?” In a situation where computations and procedures are the focus, students often feel the need to find out if their actions lead to a correct result. They feel the need for the teacher to affirm their actions.
In this 3-Act Task framework, students are encouraged to share their work, justify their reasoning, and explain their thinking. The resolution to the problem or the desire to be affirmed about their actions does not come from the teacher confirming that the answer is right. Rather, the resolution comes as the class engages together and fellow students critique the reasoning of others. This activity aligns with the meaning of “rigor” presented earlier where students “struggle with real, non-routine problems in context; identify strategies to solve problems; communicate about mathematical ideas with clarity and precision; and justify solutions.”
In this context, it is possible to create a quadratic model where Lightning McQueen fails in jumping across the canyon. It is also possible to create a model where the jump is successful. Either way, students are encouraged to justify, defend, and explain. They are encouraged to interpret, in context, the parameters of their model. They are encouraged to move between representations and forms in a meaningful way. Furthermore, not every student will experience the task in exactly the same way. However, when sharing their work and then critiquing the work of others, students have a rich and varied experience with the mathematics involved in this task.
One final note: despite the effort to encourage classroom discourse, critiquing the reasoning of others, defending decisions, etc., students will still often ask, “whose model is right?” This presents a wonderful opportunity to discuss the idea of modeling. As George E. Box famously stated, “All models are wrong, but some are useful.” The idea that there is no “right” answer may make some students uncomfortable. This experience can serve to confront the issue of modeling and to help students experience the open-endedness of some mathematical problem-solving experiences.
Conclusion
The main takeaway from this blog post is to challenge the reader to consider the meaning for the idea 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 are 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.
References
American Mathematical Association of Two-Year Colleges. (2018). IMPACT: Improving Mathematical Prowess and College Teaching. Memphis, TN: Author.
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
Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. A. (1994). Calculational and conceptual orientations in teaching mathematics. In A. Coxford (Ed.), 1994 Yearbook of the NCTM (pp. 79-92). Reston, VA: NCTM.
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