A Report to the CUPM
Quantitative Literacy and Service Course Subcommittees
On the curriculum for students in non-mathematically intensive disciplines
The vast majority of students attending college today are studying in disciplines that are not mathematically intensive. In fact, the majority of students in most college mathematics departments are majoring in disciplines that are not mathematically intensive. However, these students, as college-educated individuals, still deserve at least a minimal competency in quantitative reasoning to go along with their competencies in writing and speaking. The foundation for the acquisition of such competencies should begin early in the student's college career and be reinforced throughout the student's undergraduate program.
In today's world quantitative reasoning is essential for evaluating concerns in every sector of life--the community, the state, and the nation. The analysis of problem settings and problem solving is expected of our leaders, but it often involves quantitative components. Further, such situations are also a part of personal decision-making. In fact, the primary reason for colleges and universities to require that these students take some mathematics should be to address the students' need to be educated at a level in quantitative reasoning commensurate with a college educated mind, and that are expected of individuals in today's highly quantitative society. A secondary reason for requiring some mathematics for these students is to provide them with more appreciation for the role of mathematics and quantitative skills in society.
Mathematics departments must seriously consider the needs of these students because of the the skills that they are expected to have to be successful in our society, the attitudes many have towards mathematics, number of students involved, and their lost potential (and possibly that of their children) as students mathematics, science, engineering and technology.
This report will speak to the needs of those students majoring in disciplines that are not mathematically intensive, with the one exception of students majoring in elementary education. The latter have very special mathematical needs and those needs should be discussed in a separate report.
The students whose needs are described in this document are those who currently take only one mathematics course in college. Their rationale might be to satisfy a minimum competency in mathematics or to satisfy a prerequisite for study in general education science courses and applied statistics course taught outside the mathematics department. A few may actually take a general mathematics course to learn how a mathematician or statistician thinks or develops his/her ideas.
The MAA's Quantitative Reasoning Report of 1995 establishes mathematical and quantitative goals for these students. That report justified the following five goals:
1. Interpret mathematical models such as formulas, graphs, tables, and schematics, and draw inferences from them.
2. Represent mathematical information symbolically, visually, numerically, and verbally.
3. Use arithmetical, algebraic, geometric and statistical methods to solve problems.
4. Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results.
5. Recognize that mathematical and statistical methods have limits.
The report "Mathematics and Democracy: The Case for Quantitative Literacy" just published by the National Council on Education and the Disciplines re-iterates the importance of these goals for all of our students.
In addition, the NCED report clearly draws the distinction between mathematical study which asks students to climb the ladder of abstraction, to rise above context, and is about general principles and quantitative literacy which asks students to stay in context, to see context through a quantitative lens, and to use simple mathematics in complex situations.
This distinction informs the belief of this committee that most students have been ill-served by the traditional college algebra course which was - and fundamentally continues to be - designed to prepare students to eventually take calculus. The quantitative literacy needs of students who do not intend to, nor are expected to, study calculus are ill-served by courses with that goal in mind. In particular, pre-calculus courses are designed with the assumption that they are not terminal mathematics courses. Further, the pre-calculus courses frequently omit a wide range of topics and forms of quantitative reasoning that will be confronted by a college-educated individual.
Current controversy concerning what should or should not be labelled "college algebra" does little to answer the real issues at hand--namely, what should be preparatory today as knowledge and technical skills for the study of calculus, and how should this prerequisite material be taught and intertwined with the quantitative literacy needs for those students
intending to take calculus.
Therefore the most significant recommendation of this report is
Recommendation One: Each mathematics department should encourage and support their college or university in establishing a quantitative literacy program for all students. This program should pay particular attention to the needs of students not studying in mathematically intensive fields. A typical program may consist of a foundation course offered by the mathematics department (covering topics to be identified below), and subsequent courses with quantitative components offered by many disciplines.
In fact, because many students who begin a pre-calculus program of study eventually do not actually take calculus, these programs must also pay attention to the quantitative literacy needs of the students, while not losing sight of their primary goal. The topics from algebra, trigonometry and elementary study of functions should be carefully delineated and taught in new ways, including significant use of concepts in "word problem" situations to contextualize the learning. The next significant recommendation of this report is therefore
Recommendation Two: A new preparatory course, or courses, should be developed for those intending to, or expected to, take calculus. Careful attention should be paid to the course titles given to this course. In particular, "college algebra" should not be used as a catch-all title for all pre-calculus courses.
This committee further notes that many students entering college are not prepared to begin working towards either of these curriculum goals (calculus or quantitative literacy) without significant assistance. Therefore, we have a final recommendation.
Recommendation Three: Institutions should develop a well-structured developmental mathematics program to prepare students for collegiate level work in both mathematical and quantitative reasoning.
While much of the work in these developmental courses will cover topics included in pre-algebra and algebra, careful attention should be given to the fact that the primary goal of these courses is to help the students be successful in wide variety of settings, particularly in applied contexts arising in general education courses outside of the mathematics department. This suggests that the general pedagogical structure of such courses should emphasis quantitative literacy rather than computational proficiency.
The following additional recommendations are organized around the working group topics proposed by the CUPM. They speak to issues of concern for students majoring in non-mathematically intensive disciplines and who may only take one course from the mathematics department.
Mathematical Ideas and Themes
A1 - To achieve some level of quantitative literacy, all students at a college or university should have exposure and experience with a diverse set of mathematical ideas. There is some consensus that this set of ideas should include, at a minimum, elementary probability and descriptive statistics, estimation and approximation, interest and finance, developing and utilizing linear and exponential models, and problem-solving.
A5 - For students in disciplines that are not mathematically intensive it is of less importance that they understand that mathematics is "an engaging field with contemporary open questions" than it is to understand that mathematics can actually be used to solve the problems that they currently face, and will have to face in their lives. This does not imply, however, that mathematics should be taught as a bag of prescribed tools for solving standard, previously described problems.
B1 - All students should have some mastery of elementary facts and skills. These should include the usual arithmetic skills, algebraic skills at the level of at least one year of high school algebra, geometric skills and facts involving length, area, and volume for standard figures, and statistical skills consisting of elementary probability, presentation of data, and interpretation of statistical concepts and tests.
Students should know what a function is and be able to provide description of functions in multiple representations and to provide descriptors of functions as increasing, decreasing, increasing faster than, etc. Students should understand the idea of linear relationships and an understanding of when relationships are not linear, and those for which a linear model will suffice as an approximation. Approximation should be more general than arithmetic and include notions such as geometrically approximating lengths by line segments, approximating areas by inscribing rectangles or circumscribing rectangles, approximating volumes by packing in cubes, etc. Further, students should understand the idea of optimization of a function.
B2 - All students should acquire the rudiments of problem solving. This means that students must develop an internalized resource of mathematical ideas and facts, along with knowledge of problem solving strategies. Further, they should have sufficient experiences working with such knowledge so as to develop the connections they need between the mathematical facts and skills and the strategies to apply in order to set up a problem plan for solution.
Complexity of problem solving should also be something they understand. They should realize the difference between a one-step problem and a problem that which can be viewed as a multistep situation where many subproblems occur.
Students should experience problem solving in a wide variety of settings without the technicalities of work that a major in a field would experience. For example, geometric settings abound in our every day life, as do many items related to the environment, personal business, etc. In other words, working quantitatively should come naturally to students through their involvement in the many different areas that impact their lives as individuals and citizens.
B3 - Students in non-mathematically intensive fields may have intuitive skills of visualization on which to build understanding. These skills could include understanding the impact of color on a presentation; the emotional impact of the positioning of actors on a stage; or the marketing impact of the arrangement of a window display for a retail store. Mathematics instructors should identify and capitalize on these skills to develop better visualization skills in quantitative contexts.
Proofs and Arguments
A2 - As stated in the general goals above, all students should be able to think analytically and critically about information that they receive and about problems they must solve. In particular, students must be able to formulate problems in a real-world context, solve them, and interpret them back to this real-world context.
A3 - While students studying in non-mathematically intensive disciplines do not need to be experts at proof constructions, they should be able to verbally describe the inductive and deductive reasoning processes; should be able to distinguish between instances of inductive and deductive argument; and should be able to produce simple inductive and deductive arguments, as well as counterexamples. They should be able to identify logical fallacies and be able to utilize the techniques of survey analysis to correctly interpret data (in particular, sort out overlapping responses when a set of questions have been asked of the same individual).
Relationship to other Disciplines
A4 - Students should be able to build elementary mathematical models of problems from other disciplines. Students should be able to use elementary mathematical models when enrolled in courses from other departments. That is, knowledge from the mathematics classroom needs to be transferable to other contexts - this is at the heart of quantitative literacy.
F2 - Each department should work with faculty in other disciplines to establish course prerequisites that are based on requisite student background rather than on course titles. In particular, a blanket use of "college algebra" as a prerequisite doesn't guarantee any requisite student knowledge, unless this link is very clearly articulated. Mathematics departments should work with other disciplines to develop courses for the general student population that provide these students with the necessary skills for success in other general education courses.
F3 - It will generally take more than one course for students to achieve the five quantitative literacy goals identified above, so departments must work with other disciplines to develop quantitative experiences in courses outside of the mathematics department that build and extend the work begun in the student's first year experience.
Use and understanding of technology
A6 - Students who are not majoring in mathematics intensive fields will be expected to use technology in their workplace and homes. Therefore they should be able to demonstrate the ability to use the graphics features of a calculator to obtain information and should be able to demonstrate the ability to use computer based tools, such as spreadsheets and CAS, to organize and manipulate data and formulas. The former competency presupposes that the students can use the calculator functions of their electronic calculator.
F4 - Each department must provide a diverse set of technological supports for instruction by its faculty. To clearly demonstrate and teach the idea of using the right tool for solving a problem, instructors must have the right tools available. The quantity of these tools, and the mix that they are held in will depend on the student population (both size and quality), and the institution's budget and mission.
A7 - All students should have the ability to read mathematical literature at least to the level that it is presented in publications for the general public, for example as contained in the Science News weekly. They should be able to write mathematically using both mathematical
notation and non-mathematical language with a reasonable level of proficiency, as well as be able to communicate mathematics orally.
B4 - All students should be encouraged to use quantitative reasoning as one way of looking at situations throughout their undergraduate curriculum. Thus, it is essential that faculty in the mathematics department be supportive of, and encourage, their colleagues in less mathematically intensive disciplines to be accepting, and to encourage, intensive projects which involve a quantitative approach whenever this is reasonable.
D2 - Efforts should be made to provide students the opportunity to communicate mathematics within the context of their major area of study. This may involve reporting on the statistical outcomes of surveys or studies; explicitly using an inductive or a deductive argument to present or critic a view on a social, political, or economic issue; or solve a problem in budgeting, theater set design, or office staffing.
Assessment and Evaluation
E1 - For institutions without quantitative literacy programs in place, establishing them may be an evolutionary process dependent on institutional priorities, staff development, and student needs. Ongoing assessment of the program provides a mechanism in this process for maintaining an institutional memory, for focusing on both long range and short term goals, and for identifying the next critical step in program development.
E2 - Because the students described in this report are from disciplines that are not mathematically intensive, it is essential that mathematics faculty involve colleagues from these other disciplines throughout the entire curriculum design, implementation, and evaluation process. Faculty from other disciplines can best help mathematics faculty to identify the quantitative needs of their students and to provide feedback about whether the courses offered by the mathematics department service this need effectively.
F1 - Because of institutional admission policies and institutional goals, mathematics departments must ultimately develop individualized quantitative reasoning programs. Generally, these programs will have three levels: a developmental mathematics course(s) for students unprepared for collegiate level mathematics; a foundations course, typically offered in the mathematics department; and a secondary quantitative experience (building on ideas taught in the foundations course, but not necessarily offered by the mathematics department) that allows students to add depth to the mathematical understanding.
Recruitment, retention, and satisfaction
D1 - It is essential that departments provide space for students to work together both formally in the classroom, but also informally outside of class. For students outside of mathematically intensive disciplines this means that they have access to spaces provided to majors, but more importantly that they are encouraged to meet and to study mathematics together in some reasonable context - whether that be the library, the student union or the residence hall.
With some encouragement, these students make very effective use of various communication methods available through the internet. This may actually provide an additional means to help the students with the writing skills.
D3 - One idea for recognizing the quantitative work of a student in a non-mathematical discipline can be through an award recognizing "the best presentation of quantitative information" in a written project. Candidate projects can be submitted by the faculty sponsor of the work; this would also provide significant encouragement to those faculty who support quantitative literacy.
D4 - The group of students being discussed in this report contain hidden potential for the mathematics, science, engineering, and technology community. Teaching these students should not be delegated to part-time faculty or graduate students. Full-time faculty are uniquely positioned to identify students who have buried mathematical potential and to encourage this potential over a period of several semesters as it blossoms.
F5 - Not only do regular faculty need to understand the extent (or lack thereof) and nature of students personal and mathematical backgrounds in these quantitative reasoning courses, but they should ensure that part-time faculty and graduate students are aware of it as well. Further, regular faculty should be responsible for helping part-time faculty and graduate students develop a wide repetoire of teaching techniques and approaches necessary to respond to the varied learning styles of students - particular those who are in non-mathematically intensive fields as their learning styles can vary greatly from that of students drawn to SMET.
G1 - While we may only convert a few of the students enrolled in our quantitative reasoning courses into mathematics majors or majors in other mathematically intensive disciplines, a positive - successful - experience in this course will begin to change the public environment from which we draw our majors. Students with these positive experiences will support decisions by younger siblings and by their own children to study in mathematically intensive fields. They will be the individuals who will demand change and improvement in our public education systems.
G2 - Since minority students and women are not present in reasonable proportions in the SMET curriculum, they are disproportionately among the students taking one or no mathematics courses. That is, many of them are among the students described in this report. As suggested elsewhere in this report, mathematics departments should take the instruction of quantitative literacy courses and general education courses seriously as they provide a significant opportunity to attract talented individuals to SMET from among groups currently underrepresented.
G3 - References to successful programs at a variety of institutions can be found in the MAA's Quantitative Literacy Report of 1995, in the current report "Mathematics and Democracy: The Case for Quantitative Literacy", and at the website http://www.stolaf.edu/other/ql/ql.html.