Note: This essay originated from the GLCA Rubrics for Liberal Arts Learning Workshop, March 17-18, 2017.
It was written by the following:
Barbara Andereck, bsandere@owu.edu , Physics and Astronomy, Ohio Wesleyan University
Alice Deckert, adeckert@allegheny.edu, Chemistry, Biochemistry, Allegheny College
Paul Djupe, djupe@denison.edu, Political Science, Denison University
Jeremy Kirby, jkirby@albion.edu, Philosophy, Albion College
Jan Tobochnik, Jan.Tobochnik@kzoo.edu, Physics, Kalamazoo College
William Turner, turnerw@wabash.edu, Mathematics & Computer Science, Wabash College
As we started to talk about our subject, it quickly became apparent that most of our schools were undergoing a transition. Until recently, a Quantitative Literacy requirement generally meant taking a math course or almost any course where students were exposed to numbers. Guided by concerns about what numerate citizens should be able to do with numbers and numerical representations, the emphasis has shifted to literacy and the ability to understand and make arguments with quantitative representations. While numerical competence may arrive through problem sets, assignments that build quantitative literacy are likely to be different, arguably something more.
This is not to dismiss the importance of skill building since the ability to make arguments with quantitative representations hinges on competent modeling, solving, and analyzing, as appropriate. But numerical competence is layered within a set of translations. For numeracy to mean anything, the numerate need to be able to move back and forth between worldly contexts to the quantitative abstractions that may yield understanding – we see that back and forth as acts of translation.
We generally concurred with the AAC&U definition of Quantitative Literacy:
Quantitative Literacy is a habit of mind, competency, and comfort in working with numerical representations. Individuals with strong Quantitative Literacy skills possess the ability to reason and solve quantitative problems in a wide array of authentic contexts and everyday life situations. They understand and can create sophisticated arguments supported by quantitative evidence or models, they can clearly communicate those arguments in a variety of formats, and they can elaborate their implications for those contexts.
However, we recommend emphasizing the importance of translation. Translation takes place from the original context of an issue to quantitative and logical representations, which in turn facilitates interpretation of outcomes and consequences:
Quantitative Literacy is a habit of mind, competency, and comfort in working with numerical representations. Individuals with strong Quantitative Literacy skills possess the ability to reason and solve quantitative problems in a wide array of authentic contexts and everyday life situations and to readily translate written or verbal ideas into mathematical forms and back again. They understand and can create sophisticated arguments supported by quantitative evidence or models, they can clearly communicate those arguments in a variety of formats, and they can elaborate their implications for those contexts. Recognizing formal patterns, they are able to transfer and make use of numerical structures within different contexts.
Ideal Quantitative Literacy-Building Assignment Structures
Here we identify components of assignments that would showcase the complete set of translations to demonstrate quantitative literacy. It is perhaps obvious that not all assignments in Quantitative Literacy courses would include all four components or would ask students to generate all four components. That is, there is considerable value in problem sets, or in essay assignments that ask students to wrestle with one of these components when the others are given. It is not clear, yet, how much of this process needs to be mastered to demonstrate Quantitative Literacy proficiency – that is up to different campuses to decide. Hence, here is where pursuing the learning goal may not have a neat fit with assessment needs. Still, we maintain that assignments that incorporate these four components would build toward Quantitative Literacy proficiency:
- Explain the context of the problem by articulating the premise, hypothesis, and expectations.
- Translate the context of the problem into symbolic or graphical language that can be used to obtain the desired outcome. For example, this may involve representing or organizing quantitative information in a graph or table, or outlining a mathematical model that links what is known about the context to the desired outcome. Students should be able to describe and justify the approach to, or design/model of, any quantitative representations they use.
- Use the symbolic or graphical representation of the problem to obtain the desired outcome by solving, manipulating, or formalizing the quantitative representations to draw appropriate inferences that lead to the desired outcome.
- Reasoning through the outcomes of step 3 and translating the quantitative results back into narrative language, evaluating their importance to the context.
By quantitative representation we mean tables of data, figures, mathematical expressions, equations, animations, etc. By approach/design/model we mean what we will do with whatever data we collect or how we will model phenomena. In some cases this will mean some form of statistical analysis. In some cases it will mean building a mathematical or computational model and testing it either analytically or computationally, using a computer program written by the user or a piece of software written by somebody else.
We have included a set of example assignments from our courses that span a wide range of disciplines and levels in an attempt to make our thinking clear and concrete. Each assignment is annotated to indicate how each of the aspects (1-4) are present either implicitly or explicitly. Note that for introductory-level assignments students are not asked to master all four aspects of the assignment. In some cases they are provided with all but one aspect and asked to discuss or demonstrate proficiency with only that one aspect. In other cases introductory assignments ask students to think about all aspects of the assignment, but provide strong guidance to aid their thinking. More advanced assignments either ask the student to generate solutions that require thinking about all aspects of the assignment with less guidance in the assignment prompt or ask students to generate more sophisticated arguments and manipulations for one or two aspects.
The assignments we have collected also point to disciplinary differences. Clearly, the contexts that are important in different disciplines will vary. But the more substantial differences in quantitative reasoning come in the types of quantitative representations that are helpful to making arguments in the different disciplines. For instance, a political scientist, psychologist, or biologist may rely mostly on statistical evidence to make arguments using numbers. On the other hand, a chemist or physicist would likely produce graphs that show a correlation between two controlled variables and link those correlations through a mathematical model. In this case graphical analysis may be emphasized. The type of quantitative representation used to symbolize the context or problem being studied (aspect 2 of our assignment structure) then informs the type of logical manipulations or ways of inferring conclusions and outcomes from the representations (aspect 3 of our assignment structure). Thus, the most substantive differences between disciplines occur in aspects 2 and 3 of our assignment structure.
Finally, as a group we came to the conclusion that to assess Quantitative Literacy using a rubric may require that the assignment prompt be available to the scorer of that rubric. We maintain that all aspects – context, translation to symbolic representation, manipulation of the symbolic form, and translation back to assess the outcome in light of the original context – are important and necessary components of Quantitative Literacy. However, we recognize that building a capacity to do all of these things in tandem requires very high-level cognitive skills. Thus, any given assignment is not likely to showcase students’ ability equally across all aspects. Many assignment designs provide students with the context and then ask that they demonstrate various levels of skill at translating the context to symbolic form, logically drawing conclusions from the symbolic form and finally assessing the outcome in light of the original context. Without knowing the original context, student work may appear simply as a series of mathematical manipulations, even though those manipulations show a high level of skill in translating a context to mathematical symbolism and then manipulating it to obtain a desired outcome. Or a work product may simply reflect detailed guidance and less the independent Quantitative Literacy proficiency of the student.
Sample Assignments to Build Quantitative Literacy
We conclude by providing examples of assignments from several disciplines that contribute to the development of Quantitative Literacy in one or more of the four components we have discussed. The full assignments can be accessed by the URL links that follow the summary description. The following notes address the skills and the four components of Quantitative Literacy we have identified. We welcome your feedback on any of these sample assignments.
Astronomy 110
Barbara Andereck, bsandere@owu.edu , Physics and Astronomy, Ohio Wesleyan University
This assignment is a group activity related to the moon’s orbit. Work is completed in two parts, with feedback from the instructor between the two. In part 1 students collect information about the times of new and full moon over the course of a year, perform calculations of the duration of waxing and waning phases in Excel, and make a plot. In part 2 they use their data from part 1 to find the eccentricity of the moon’s orbit.
Part 1 of the assignment falls squarely in Step 2 of the Quantitative Literacy process, with significant guidance given to them in the translation process.
Part 2 of the assignment starts in Step 1 of the Quantitative Literacy process, leads them through Step 3 and ends asking them a Step 4 question.
To access the Astronomy 110 assignment, click here.
Introductory Chemistry
Alice Deckert, adeckert@allegheny.edu, Chemistry, Biochemistry, Allegheny College
Exploring Solubility Equilibrium
This assignment describes the Proctor and Gamble “Pur” water purification packets that use the precipitation of Iron(III) hydroxide to remove debris from contaminated water and calcium hypochlorite to sanitize it. Students are provided with the amount of Iron(III) sulfate and calcium hypochlorite in the packets and are led through an analysis of the amount of Iron(III) hydroxide that is formed given the working pH of the packets. They consider the question of whether calcium hydroxide will also precipitate and finally explore the question of whether this works best in cold or warm water.
Although the students are led through the analysis using discrete leading questions, they are asked to translate the context into both chemical and mathematical symbolism. They then must manipulate that symbolic language to obtain numerical answers, and finally they are asked to explain what those numbers tell them about the original context. Thus, all elements of the assignment structure are addressed, but in a very guided way.
To access the Introductory Chemistry Exploring Solubility Equilibrium, click here.
Introduction to American Politics 110
Paul Djupe, djupe@denison.edu, Political Science, Denison University
Students often think of politics as doggedly pursuing an argument, while professors often think of politics as the rules (institutions) that shape behavior. To push this notion while encouraging the first steps in quantitative literacy, students are asked to examine the campaign contribution patterns from the banking industry to senators. I provide them with a dataset with senator ideology, bank donations, and state. Once they find heterogeneity in giving along ideological lines, they are encouraged to look for patterns driven by “power.” In terms of our four-part structure, students are provided the context (part 1) and one operationalization of a concept that they are asked to analyze (part 2). They are then asked to operationalize another concept (power) that would explain the pattern in donations they see in the data (they engage in an informal part 3). They then briefly take the empirical patterns back to the larger set of concerns about political representation.
To access the Introduction to American Politics assignment, click here.
Philosophy
Jeremy Kirby, jkirby@albion.edu, Philosophy, Albion College
This is an exercise in Quantification using Aquinas’ Fifth Way. A number of fallacies can be understood as situations where individuals fail to keep track of that which quantifiers, expressions such as all and some, for example, are thought to modify. Students are first introduced to a prose translation of Aquinas’ argument. They are then introduced to a formal language, mathematical logic, which models the argument, rendering precise the extension of the quantifiers in question. Penultimately, they develop a counter-model that demonstrates how the argumentative structure employed by Aquinas appears unsound. The assignment concludes with an exercise designed to facilitate transference of the concept here learned to another context, namely, the opening argument of Aristotle’s Nichomachean Ethics.
To access the Philosophy assignment, click here.
Introductory Physics
Jan Tobochnik, Jan.Tobochnik@kzoo.edu, Physics, Kalamazoo College; Arthur Cole, Arthur.Cole@kzoo.edu; Physics, Kalamazoo College.
Attached are three activities that we use in our studio physics format, where students work on problems and mini-labs in a lab setting rather than listening to lectures. These activities approximate the assignment structure that was discussed in the Quantitative Literacy dialog.
To access the Introductory Physics assignments, click here.
