By: David Bressoud @dbressoud

I was first introduced to the concept of quantitative reasoning (QR) through Lynn Steen and the 2001 book that he edited, Mathematics and Democracy: The Case for Quantitative Literacy. That fall, Danny Kaplan and I began working with the Macalester faculty to create our own QR program, a process described in my 2009 article “Establishing the Quantitative Thinking Program at Macalester.” I thought I understood its meaning. But an edited volume that appeared this past January, Quantitative Reasoning in Mathematics and Science Education, has both broadened and deepened my understanding of this term.

Steen and the design team he had assembled late in the 20th century described quantitative literacy/reasoning in the first chapter of Mathematics and Democracy:

“Quantitatively literate citizens need to know more than formulas and equations. They need a predisposition to look at the world through mathematical eyes, to see the benefits (and risks) of thinking quantitatively about commonplace issues, and to approach complex problems with confidence in the value of careful reasoning. Quantitative literacy empowers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently. These are skills required to thrive in the modern world.

“… Despite its occasional use as a euphemism for statistics in school curricula, quantitative literacy is not the same as statistics. Neither is it the same as mathematics, nor is it (as some fear) watered-down mathematics. Quantitative literacy is more a habit of mind, an approach to problems that employs and enhances both statistics and mathematics. Unlike statistics, which is primarily about uncertainty, numeracy is often about the logic of certainty. Unlike mathematics, which is primarily about a Platonic realm of abstract structures, numeracy is often anchored in data derived from and attached to the empirical world. Surprisingly to some, this inextricable link to reality makes quantitative reasoning every bit as challenging and rigorous as mathematical reasoning.” (Steen, 2001, p. 2)

Courses in QR have emphasized the interpretation of data as in this introduction to the quantitative reasoning course developed by the Charles A. Dana Center at the University of Texas, Austin:

“Quantitative Reasoning serves students who are focused on developing quantitative literacy skills that will be meaningful for their professional, civic, and personal lives. Such reasoning is a habit of mind, seeking pattern and order when faced with unfamiliar contexts. In this course, an emphasis is placed on the need for data to make good decisions and to understand the dangers inherent in basing decisions on anecdotal evidence rather than on data.” (Dana Center, 2020, p. 1)

See also Shifting Contexts, Stable Core (Tunstall et al, 2019).

While these descriptions include a broad sense of ability to make practical use of the mathematical sciences, the emphasis has been on the interpretation and application of numerical data. But in his introduction to the volume on QR in Science and Mathematics Education, Pat Thompson puts the emphasis on mathematical structures. As he explains it:

“Quantitative reasoning is “an individual’s analysis of a situation into a quantitative structure” (Thompson, 1990, p. 13) such that it entails “the mental actions of an individual conceiving a situation, constructing quantities of his or her conceived situation, and both developing and reasoning about relationships between there constructed quantities” (Moore et al., 2009, p. 3).”

What this means is illustrated through many examples. My understanding is that it entails the ability to take a situation that is inherently but perhaps not explicitly numerical and to build a mathematical model through the recognition of quantitative variables and their connections. His 1990 paper includes the following problem:

“MEA Export is to deliver an oil valve to Costa Rica. The valve’s price is $5000. Freight charges to Costa Rica are $100. Insurance is 1.25% of Costa Rica’s total cost. Costa Rica’s total cost includes the costs of the valve, insurance, and freight. What is Costa Rica’s total cost?” (Thompson, 1990, p. 39)

While this problem asks for a numerical answer, what is really tested is a person’s ability to assign variables, understand their relationships, and build a model—in this case an equation, a system of equations, or a mathematical expression—whose solution or evaluation answers the original question. As we know, solving or evaluating the model is the easy part. Setting it up is where most students struggle.

Pat has pointed out that when faced with this problem, most students and teachers focus on the numbers: $5000, $100, 1.25%. They then try to figure out a way to combine them. Quantitative reasoning first identifies the items that carry numerical value: valve price, freight charges, insurance, and total cost. QR then looks at the relationships: insurance is a percentage of total cost, total cost is the sum of valve price, freight charges, and insurance. One uses this to build an algebraic model. Only then does one look to the numerical values. See the appendix at the end of this column to see how Pat presents a visualization to help teachers see the difference.

This clarifies what is wrong with the way we teach algebra and why teaching it has become controversial. Algebra was created to encode relationships between abstracted quantities and, once encoded, to provide a collection of simple rules for finding solutions. The entire emphasis within algebraic instruction has been on learning the rules for finding solutions. But without the first part, the encoding, the second part is meaningless. Together, they provide a powerful tool for tackling a tremendous variety of problems. The development of algebra as it came to be perfected in the 16th and 17th centuries made possible modern science and technology. But we cheat our students when we ignore the difficulties inherent in the first part, the encoding.

The encoding requires quantitative reasoning. The lack of attention to QR is the reason word problems are so feared (as in Gary Larson’s Hell’s Library).  Word problems like the MEA export problem lie at the heart of what high school and college algebra should be about. They are why algebra is important 

There is a beautiful example of the application of quantitative reasoning in the article by Darío A. González in this volume, “Applying Quantitative and Covariational Reasoning to Think about Systems: The Example of Climate Change.” As he describes it,

“This chapter discusses the role of quantitative reasoning in developing an understanding of the energy budget as a system formed by multiple interacting components in terms of quantities and relationships between them.” (González, 2023, p. 282)

The article is a detailed analysis of students’ thinking as they work through building the elements of a model of the earth’s energy budget. While no derivatives appear explicitly, relationships between rates of change are embedded throughout. In fact, nothing exemplifies QR in this more expansive understanding than the process of building a differential equation to model a given situation.

This takes me back to the first time I taught differential equations. It was in the fall of 1980 while I was a visiting professor at the University of Wisconsin, Madison. This was before computers or calculators were available for use in such a course. It dealt almost entirely with those situations where exact solutions were possible.

(As an aside, I agree with Dick Askey’s statement that exact solutions are important simply because they are so uncommon. But I am also thankful that today technology has made it possible to greatly expand the kinds of dynamical systems one can study and the sorts of questions that can be asked.)

As was typical of differential equations courses of this time, we spent the semester learning how to find exact solutions to a variety of differential equations. But when I constructed the final exam, I thought it would be too boring to ask the students to find exact solutions to problems like those we had solved in class. Instead, I presented a collection of situations that could be modeled by the differential equations we had studied. I asked for solutions. Not surprising in retrospect, few students were successful in arriving at the appropriate differential equation. They were angry with me for testing them on something that had not been taught. Of course, they were right. I was testing both quantitative reasoning and differential equations. How dare I test quantitative reasoning when I had not taught it?

So how does one teach quantitative reasoning? At least part of the answer is the subject of another article from this volume, “Instructional Conventions for Conceptualizing, Graphing and Symbolizing Quantitative Relationships” by Marilyn Carlson, Alan O’Bryan, and Abby Rocha. This piece examines their experiences working with precalculus instructors in their Pathways Project. They wanted these instructors to foster “productive reasoning patterns,” but as they witnessed over many years,

“even when teachers were committed to making their instruction more engaging and meaningful for students, they continued to rely on familiar instructional practices of providing vague explanations and showing students how to find answers.” (Carlson et al., 2023, p. 225)

As they discovered, the key was to raise instructors’ awareness of their vague and procedurally focused explanations; then support them in transitioning to speak with meaning, to provide “conceptually based descriptions when communicating with others about solution approaches [, … descriptions that] are justified with logical and coherent arguments that have a conceptual rather than procedural basis” (Clark et al., 2008, p. 297). The article goes on to provide many examples of what this means in the context of a wide variety of mathematical problems. One shift involves consistently referencing the quantities being represented by symbols, expressions, and graphs———to replace statements like “the graph is going up” with statements like, “the volume of water in the reservoir increases as the number of hours since 10:30 am increases.”

Returning to Lynn Steen’s description of quantitative literacy/reasoning———a predisposition to look at the world through mathematical eyes———we see that his definition encompasses Thompson’s characterization as well as a focus on numerical data. Quantitative reasoning calls for a rich understanding of how the mathematical sciences are manifested across a wide spectrum of human interests where they can be called upon to provide insight. To teach algebra or geometry or calculus or statistics without incorporating quantitative reasoning is to impoverish these pursuits, running the risk of leaving students with a dry and seemingly meaningless shell of technique.

Imbuing quantitative reasoning throughout all instruction in the mathematical sciences at every level is not easy. But it is what everyone who teaches in the mathematical sciences must strive to accomplish.

References

Akar, G.K, Zembat, I.O., Arslan, S., and Thompson, P.W. (Eds.). (2023). Quantitative Reasoning in Mathematics and Science Education. Mathematics Education in the Digital Age. Cham, Switzerland: Springer Nature. https://link.springer.com/book/10.1007/978-3-031-14553-7

Bressoud, D. (2009). Establishing the Quantitative Thinking Program at Macalester. Numeracy 2:1, Article 3. https://digitalcommons.usf.edu/numeracy/vol2/iss1/art3/

Carlson, M.P., O’Bryan, A., Rocha, A. (2022). Instructional Conventions for Conceptualizing, Graphing and Symbolizing Quantitative Relationships. In: Karagöz Akar, G., Zembat, İ.Ö., Arslan, S., Thompson, P.W. (eds) Quantitative Reasoning in Mathematics and Science Education. Mathematics Education in the Digital Era, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-031-14553-7_9

Clark, P. G., Moore, K. C., & Carlson, M. P. (2008). Documenting the emergence of “speaking with meaning” as a sociomathematical norm in professional learning community discourse. The Journal of Mathematical Behavior, 27(4), 297–310.  https://doi.org/10.1016/j.jmathb.2009.01.001

Dana Center. (2020). Quantitative Reasoning, 2nd edition. In-class Activity Instructor Edition. https://dcmathpathways.org/course/quantitative-reasoning

González, D.A. (2022). Applying Quantitative and Covariational Reasoning to Think About Systems: The Example of Climate Change. In: Karagöz Akar, G., Zembat, İ.Ö., Arslan, S., Thompson, P.W. (eds) Quantitative Reasoning in Mathematics and Science Education. Mathematics Education in the Digital Era, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-031-14553-7_11

Moore, K. C., Carlson, M. P., & Oehrtman, M. (2009). The role of quantitative reasoning in solving applied precalculus problems. In Twelfth Annual Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education (SIGMAA on RUME) Conference. North Carolina State University. http://sigmaa.maa.org/rume/crume2009/Moore1_LONG.pdf

Steen, L.A. (Ed.). (2001). Mathematics and Democracy: The Case for Quantitative Literacy. Princeton, NJ: The Woodrow Wilson National Fellowship Foundation. https://www.maa.org/sites/default/files/pdf/QL/MathAndDemocracy.pdf

Thompson, P. W. (1990). A theoretical model of quantity-based reasoning in arithmetic and algebra. https://doi.org/10.13140/RG.2.2.23908.45447

Thompson, P.W. (2022). Quantitative Reasoning as an Educational Lens. In: Karagöz Akar, G., Zembat, İ.Ö., Arslan, S., Thompson, P.W. (eds) Quantitative Reasoning in Mathematics and Science Education. Mathematics Education in the Digital Era, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-031-14553-7_1

Tunstall, L., Karaali, G., and Piercey, V. (Eds.). (2019). Shifting Contexts, Stable Core: Advancing Quantitative Literacy in Higher Education. MAA Notes. Washington, DC: MAA Press. https://www.maa.org/press/ebooks/shifting-contexts-stable-core-advancing-quantitative-literacy-in-higher-education

Appendix

Pat Thompson has offered the following that has helped convey the distinction for teachers with respect to different ways their students think about word problems.

Common student and teacher perspective:

MEA Export is to deliver an oil valve to Costa Rica. The valve’s price is $5000. Freight charges to Costa Rica are $100. Insurance is 1.25% of Costa Rica’s total cost. Costa Rica’s total cost includes the costs of the valve, insurance, and freight. What is Costa Rica’s total cost?

Emphasis on QR:

MEA Export is to deliver an oil valve to Costa Rica. The valve’s price is $5000. Freight charges to Costa Rica are $100. Insurance is 1.25% of Costa Rica’s total cost. Costa Rica’s total cost includes the costs of the valve, insurance, and freight. What is Costa Rica’s total cost?

Download the list of all past Launchings columns, dating back to 2005, with links to each column.