Quantitative Reasoning Learning Goals
Developing QR Learning Goals
Bloom's taxonomy categorized and ordered six categories of skills in the cognitive domain. The categories can be thought of as degrees of difficulties. It is a continuum from the simplest behavior, or "Lower Order Thinking Skills," (LOTS) to the most complex ones which are the "Higher Order Thinking Skills" (or HOTS). The three lower skills are remembering, understanding and applying, while the higher order skills are analyzing, evaluating and creating. In the 1990s, Lorin Anderson (see Anderson and Krathwohl 2000), a former student of Bloom, revised the taxonomy by using verbs rather than nouns and by rearranging the order of the categories. This new taxonomy reflects a more active form of thinking. It begins with "remembering" or recalling facts and concepts and ends with "creating" which is bringing together parts or elements to form a whole. The first ones must normally be mastered before the next ones can take place. Just as you cannot understand a concept if you do not first remember it, you cannot apply knowledge and concepts if you do not understand them.
In thinking about quantitative reasoning, Blooms' taxonomy may be helpful in framing more specific learning outcomes as ideally we would like to achieve educational goals in all three domains of learning (e.g., psychomotor, cognitive, and affective). When specifying goals in these areas, it is best to avoid vagueness that can lead to confusion. This can be achieved through the use of action words that provide clear descriptions of what the students can or should be able to do after completing the course or unit in question. Words such as "understand," "appreciate," "think critically" and "demonstrate," which are commonly used in expressing learning goals, do not explain specifically what knowledge or skills are expected of the students. In articulating learning goals, then, it is appropriate to consider whether or not you might be able to assess whether or not the goals have been achieved. If the answer is "no," then you may wish to reformulate your goals. For example, if your goal is "students will understand about changes and percentages," measuring "understand" might be difficult to do. In articulating learning goals, it is best to aim for goals that are neither too broad nor too specific, avoid fuzzy terms, use concrete action works, and focus on the destination/outcome, not the means/process (Suskie 2009).
For examples, if we say that by the end of the course, our students will demonstrate quantitative literacy skills, that would be a vague and broad goal which does not specify how they will demonstrate those skills that they have learned. On the other hand, if we way that students will be able to convert gallons to liters, the goal is too narrow and focused on that particular task of conversion. It would be more appropriate to state that students will make appropriate conversion of units before they compare quantities. This is an indication that the skills they have learned should be transferable to other contexts.
Too vague: Students will demonstrate quantitative literacy skills.
Too specific: Students will be able to convert gallons to liters.
Better: Students will make appropriate conversion of units before they compare quantities.
Broad QR Learning Goals
A variety of organizations have put forward sets of broad QR learning goals that may be helpful in framing more specific QR learning goals for faculty members who are infusing QR throughout the curriculum. For example, the Association of American Colleges and Universities (AAC&U) has put forward a Quantitative Literacy Rubric (Acrobat (PDF) 103kB Aug28 12) that identifies a variety of important skills associated with quantitative literacy. These include: (a) Interpretation, i.e., the ability to explain information presented in mathematical forms [e.g., equations, graphs, diagrams, tables, words], (b) Representation, i.e., the ability to convert relevant information into various mathematical forms [e.g., equations, graphs, diagrams, tables, words], (c) Calculation, (d) Application/Analysis, i.e., the ability to make judgments and draw appropriate conclusions based on the quantitative analysis of data, while recognizing the limits of this analysis, (e) Assumptions, i.e., the ability to make and evaluate important assumptions in estimation, modeling, and data analysis, and (f) Communication, i.e., expressing quantitative evidence in support of the argument or purpose of the work [in terms of what evidence is used and how it is formatted, presented, and contextualized].Many experts in the QR movement have articulated specific learning goals and/or embedded them within their definitions of QL/QR. For example, in the book Mathematics and Democracy, Steen and colleagues (2001: 8-9) outline a comprehensive portrait of several core elements of quantitative literacy including:
- Confidence with Mathematics. Being comfortable with quantitative ideas and at ease in applying quantitative methods. Individuals who are quantitatively confident routinely use mental estimates to quantify, interpret, and check other information. Confidence is the opposite of "math anxiety"; it makes numeracy as natural as ordinary language.
- Cultural Appreciation. Understanding the nature and history of mathematics, its role in scientific inquiry and technological progress, and its importance for comprehending issues in the public realm.
- Interpreting Data. Reasoning with data, reading graphs, drawing inferences, and recognizing sources of error. This perspective differs from traditional mathematics in that data (rather than formulas or relationships) are at the center.
- Logical Thinking. Analyzing evidence, reasoning carefully, understanding arguments, questioning assumptions, detecting fallacies, and evaluating risks. Individuals with such habits of inquiry accept little at face value; they constantly look beneath the surface, demanding appropriate information to get at the essence of issues.
- Making Decisions. Using mathematics to make decisions and solve problems in everyday life. For individuals who have acquired this habit, mathematics is not something done only in mathematics class but a powerful tool for living, as useful and ingrained as reading and speaking.
- Mathematics in Context. Using mathematical tools in specific settings where the context provides meaning. Notation, problem-solving strategies, and performance standards all depend on the specific context.
- Number Sense. Having accurate intuition about the meaning of numbers, confidence in estimation, and common sense in employing numbers as a measure of things. Practical Skills. Knowing how to solve quantitative problems that a person is likely to encounter at home or at work. Individuals who possess these skills are adept at using elementary mathematics in a wide variety of common situations.
- Prerequisite Knowledge. Having the ability to use a wide range of algebraic, geometric, and statistical tools that are required in many fields of postsecondary education.
- Symbol Sense. Being comfortable using algebraic symbols and at ease in reading and interpreting them, and exhibiting good sense about the syntax and grammar of mathematical symbols.
Even while QL/QR is distinct from traditional mathematics, many mathematicians and mathematical organizations have played a leading role in the QL/QR movement. According to the Mathematical Association of America (MAA) (1998), "The foremost objective of both liberal and professional types of higher education should be to produce well-educated, enlightened citizens, who can reason cogently, communicate clearly, solve problems, and lead satisfying, productive lives." They further argue, "A quantitatively literate college graduate should be able to:
- Interpret mathematical models such as formulas, graphs, tables, and schematics, and draw inferences from them.
- Represent mathematical information symbolically, visually, numerically, and verbally.
- Use arithmetical, algebraic, geometric and statistical methods to solve problems.
- Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results.
- Recognize that mathematical and statistical methods have limits" (MAA 1998).1
The National Council of Teachers of Mathematics (NCTM) has also put forward a comprehensive set of learning goals for pre-K through grade 12 education that focuses on five different areas including (1) numbers & operations, (2) algebra, (3) geometry, (4) measurement, and (5) data analysis and probability. Data analysis and probability is the area that is perhaps most closed linked to QR/QL, and for this area NCTM argues that instructional programs from pre-kindergarten through grade 12 should enable all students to:
- formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them;
- select and use appropriate statistical methods to analyze data;
- develop and evaluate inferences and predictions that are based on data; and
- understand and apply basic concepts of probability.
Similarly, the participants in the American Statistical Association's Guidelines for Assessment and Instruction in Statistics Education (GAISE) project produced two reports focusing on recommendations for statistics education in pre-K-12 years and introductory statistics courses at the college level. In the pre-K-12 report, Franklin and colleagues (2007: 1) write, "Our lives are governed by numbers. Every high-school graduate should be able to use sound statistical reasoning to intelligently cope with the requirements of citizenship, employment, and family and to be prepared for a healthy, happy, and productive life." The GAISE framework is intended to complement the NCTM standards.
In the GAISE College Report, Alliaga and colleagues (2010) defines statistical literacy as "understanding the basic language of statistics (e.g., knowing what statistical terms and symbols mean and being able to read statistical graphs) and fundamental ideas of statistics." This report also presents goals for students in an introductory statistics class. These goals range from believing and understanding why data beats anecdotes and association is not causation to recognizing common sources of bias in surveys and experiments. Of course, some of the variation among the learning goals of the MAA and the AAC&U and the ASA has to do with differences in concepts (QL vs. statistical literacy, for example).
Moreover, Ganter (2006: 13) contends that a full discussion of quantitative literacy must address the three dimensions of QR2: (1) elements of QL, such as confidence with mathematics, cultural appreciation of the role of data in society, interpreting data, logical thinking, making decisions, mathematics in contexts, number sense, practical skills, prerequisite knowledge, and symbol sense; (2) areas of life that utilize QL, such as citizenship, culture, education, professions, personal finance, personal health, management, and work; and (3) skills needed for QL, such as arithmetic, data, computers, modeling, statistics, chance, and critical reasoning.
Disciplinary or Course Specific QR Learning Goals
Of course, all these components of QR indicated above are relatively broad and it will be important for faculty who are infusing QR throughout the curriculum to operationalize more specific QR learning goals that can be tackled within particular courses. For example, Milo Schield (2010) argues that students in the social sciences should have a firm understanding of (1) the distinction between association and causation as encountered in ordinary English; (2) the influence of confounders in observational studies (including influence on statistical significance); (3) how counts, measures and associations are influenced by the way things are defined, grouped and presented; and (4) the ability to read and interpret rates and percentages as presented in tables and graphs. A faculty member in history may, on the other hand, be interested in helping students to distinguish between absolute numbers and rates in explaining historical trends (a course specific learning goal). Indeed, when QR is infused throughout the curriculum the kinds of QR learning goals that a faculty member chooses to focus on may vary considerably across courses and disciplines.
1Many schools have also articulated specific QR learning goals, and these often overlap with those of the Quantitative Literacy Committee of the MAA. For example, Frostburg State University students are expected to interpret mathematical models and be able to draw inferences from them; communicate mathematical information symbolically, visually, numerically, and verbally; use arithmetic, algebraic, geometric, or statistical methods to solve problems; and estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results (Limbaugh 2005: 3).
2Kirst (2003: 107) defines mathematical literacy as "the basic skills of arithmetic, algebra, and geometry that historically have formed the core of school mathematics." In contrast, he defines quantitative literacy as "reasoning with data in their natural contexts, especially in situations that citizens encounter in judging public issues (e.g., pollution, taxes) or private decisions (e.g., cell phone plans)."
Aliaga, Martha, George Cobb, Carolyn Cuff, Joan Garfield (Chair), Rob Gould, Robin Lock, Tom Moore, Allan Rossman, Bob Stephenson, Jessica Utts, Paul Velleman, and Jeff Witmer. 2010. Guidelines for Assessment and Instruction in Statistics Education: College Report. Alexandria, VA: American Statistical Association.
Anderson, Lorin W. and David R. Krathwohl. (Editors). 2000. A Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom's Taxonomy of Educational Objectives, Abridged Edition. Pearson.
Bloom, Benjamin S. 1984. Taxonomy of Educational Objectives Book 1: Cognitive Domain, 2nd edition. New York: Longman.
Franklin, Christine, Gary Kader, Denise Newborn, Jerry Moreno, Roxy Peck, Mike Perry, Richard Scheaffer. 2007. Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K-12 Curriculum Framework. Alexandria, Virginia: American Statistical Association.
Ganter, Susan L. 2006. '"Issues, Politics and Activities in the Movement for Quantitative Literacy." In Current Practices in Quantitative Literacy, edited by Rick Gillman. Washington, DC: Mathematical Association of America. Pp. 11-15.
Kirst, Michael W. 2003. "Articulation and Mathematical Literacy: Political and Policy Issues." In Quantitative Literacy: Why Numeracy Matters for Schools and Colleges, edited by Bernard L Madison and Lynn Arthur Steen. Princeton: NJ: The National Council on Education and the Disciplines. Pp. 107-120.
Limbaugh, Jim. 2005. "Frostburg State University: Basic Proficiencies in Quantitative Reasoning." Report on behalf of the faculty planning group on quantitative reasoning.
Mathematical Association of America. 1998. Quantitative Reasoning for College Graduates: A Complement to the Standards. Mathematical Association of America.
National Council of Teachers of Mathematics (NCTM). N.d. "Standards."
Schield, Milo. 2010. Personal Communication (email from Milo Schield to Esther Wilder).
Steen, Lynn Arthur and the Quantitative Literacy Design Team. 2001. "The Case for Quantitative Literacy." Pp 1-12 in Mathematics and Democracy: The Case for Quantitative Literacy, edited by Lynn Arthur Steen. Princeton, NJ: National Council on Education and the Disciplines. Pp. 1-22.
Suskie, Linda. 2009. Assessing Student Learning: A Common Sense Guide (2nd ed.). San Francisco: Jossey-Bass.