In the same paper Resnick goes on to say "...we may do well to conceive of mathematics education less as an instructional process (in the traditional sense of teaching specific, well-defined skills or items of knowledge), than as a socialization process. In this conception, people develop points of view and behavior patterns associated with gender roles, ethnic and familial cultures, and other socially defined traits. When we describe the processes by which children are socialized into these cultural patterns of thought, affect, and action, we describe long-term patterns of interaction and engagement in a social environment, not a series of lessons in how to behave or what to say on particular occasions. If we want students to treat mathematics as an ill-structured discipline-- making sense of it, arguing about it, and creating it, rather than merely doing it according to prescribed rules--we will have to socialize them as much as to instruct them. This means that we cannot expect any brief or encapsulated program on problem solving to do the job. Instead we must seek the kind of long-term engagement in mathematical thinking that the concept of socialization implies.''
In this spirit quantitative literacy for college students is not something gained by taking one specific course in the curriculum, by learning some specific mathematical content, or by developing a particular level of computational facility. Becoming quantitatively literate must not be thought of as acquiring a certain set of concepts, facts, and procedures (through a remedial course, if necessary) followed by completion of a good course in problem solving. Further, becoming quantitatively literate is not the consequence of an excellent survey course (or introductory statistics sequence) which does no more than engage the student in doing interesting application problems. Rather, a student becomes quantitatively literate through a broad program aimed at developing capabilities in thought, analysis, and perspective--through a program aimed at developing the ingredients expressed in Part II of this report in such a manner that the student will have formed attitudes and habits of thought which provide certain "long-term patterns of interaction and engagement.''
Four-year colleges and universities ordinarily dictate standards for acceptance. For example, two or more years of college preparatory high school mathematics may be required for admission, whether the institution be public or private. Clearly, any program for the development of quantitative literacy must take into account the level of mathematical attainment at which the student enters and seeks a smooth transition for the student. Further, in order to maintain a quantitative literacy program of high quality, four-year colleges and universities should expect transfer students to have made the same amount of progress toward quantitative literacy as native students. Performance on a good placement test (covering computational facility, mathematical reasoning, and problem solving) should be immensely helpful to advisors in determining where a student enters the quantitative literacy program.
When the degree of quantitative literacy expected for all college graduates was described earlier, it was described as more than a matter of remediation. However, one or more remediation stages may be needed in a quantitative literacy program.
"All right,'' we might say, "if people are so enthusiastic about English, they should take English. Why doesn't the English department teach them how to write?'' The English department can't do the whole job. Writing is so complex an activity, so closely tied to a person's intellectual development, that it must be nurtured and practiced over all the years of a student's schooling and in every curricular area.
The Composition class can give students some transferable skills: the concept that effective writing is focused and well organized, strategies for structuring forms like comparison or argument, principles of clear prose style, and conventions of grammar and punctuation. Like every class, however, the composition class is a community, with its own set of expectations, types of writing, and unique values. When asked, "What is the goal of your course?'' faculty members usually give a discipline-specific answer--"to make my students think like economists,'' or "To teach my students the basic questions, methods, and values of psychology.'' Instructors in those fields must show students how to apply composition skills and how to carry on the common types of thinking and writing in the individual discipline.
Just as the complexity of the writing task is so great that the English department should not be expected to assume responsibility for the entire job of its development for the student, so also the complexity of the task of a student's becoming quantitatively literate requires the commitment of more than the department of mathematics. Instructors in other fields must show students how to apply quantitative reasoning to gain disciplinary knowledge and understanding.
However, in assuming a leadership role for the quantitative literacy program, the mathematics department will see to it that there is a means within the institution for students to secure a foundation in their quest for quantitative literacy . For example, depending on the student's entering computational facility level and intended major, the student may be directed to one of a set of courses which seek to advance the ingredients of quantitative literacy described in Part II of this report. For many mathematics departments this may well mean the establishment of at least one new course whose thrust is quantitative literacy as well as the reexamination (and possible alteration) of other courses in the department and college to make them better at fostering quantitative literacy goals. The foundation experience should aim to develop the student's capability to DO quantitative reasoning -- not just to see it done! But the foundation experience should not stand alone!!!
The aspects of intellectual competency termed "control,'' "beliefs,'' and "practices'' in our earlier discussion of the dynamics of quantitative literacy ordinarily involve changing attitudes and habits. And new attitudes and habits take time to acquire. Consequently, the foundation for the quantitative literacy program should be obtained early in the baccalaureate career to give the student the growing time essential to the program's success. If students are to acquire quantitative literacy as a base for life-long learning and see connections between mathematics and other disciplines, they must really experience such learning and connections.
Experiencing such learning means not suppressing the use of quantitative reasoning in science and social science general education courses the student takes for the degree. It means encouraging students to carry out projects involving quantitative reasoning in courses outside the mathematics and natural science departments. Also, since quantitative literacy can be expected to be more strongly developed if it is encountered in settings students consider relevant to their interests, the curricula for all program majors should in some way include the experience of quantitative reasoning in courses at the junior and senior level. Thus, throughout their undergraduate careers students should be faced with reinforcement and strengthening of the type of thinking encountered in the foundation.
Those curriculum committees in the college and university which set the general education program for all students must be made aware of the "why" of quantitative literacy . That awareness can lead to the formation of a network of faculty, staff, and administrators who can develop a plan for a program and engender the good will, hard work, and cooperation needed to bring the program into being. In addition, a focused program should involve the establishment of explicit targets of accomplishment for the student which can be meaningfully assessed.
In a successful quantitative literacy program faculty members will have a role as coaches who again and again model quantitative reasoning and who critique student performance so that it may be recognized as good when it is and be improved or further developed when it is not so good. The work of these faculty coaches will be enhanced by the presence of a supportive learning environment for quantitative literacy on the whole campus.
Across the campus, faculty (especially those serving as advisers) should act as cheerleaders for students acquiring quantitative literacy by exuding the value of the experience. Supportive learning environments counteract anxious and phobic responses to quantitative reasoning situations (see S. Tobias and C. Weissbroad, "Anxiety and math: an update,'' Harvard Educational Review (1980), and Anne Wescott Dodd, "Insights from a Math Phobic,'' Math. Teacher (1992). Establishment of a mathematics clinic paralleling the writing clinic could be of immerse value too! (In fact, verbalizing or writing questions arising in a quantitative literacy setting may help a student develop multiple communication skills.)
The development of courses and course materials is normally a demanding activity, but for a foundation course for a quantitative literacy program, it is especially demanding. Writing in September 1990 about the development of their course first offered in the 1987-88 academic year at the University of Chicago, J. Cowan, S. Kurtz, and R. Thisted explained:
When we set out to develop this course, we expected to do research, to write books, to write programs that supported the books we were going to write, and in short to erect a pedagogical monument to mathematical thought within the liberal arts out of nothing but our own will. Such hubris!
Of course, lofty goals like this don't just die; and we weren't being hypocritical in holding them. We persevered in this program; we wrote essays (not quite yet books); we wrote programs (but without all the polish we would have liked); we've taught (and continue to teach) a good course that is getting better. The problem with this approach is that it has not been efficient.
...Our major objective has been to produce an exportable course. ...We've had considerable discussion about what "exporting'' means; but we're barely to the point of getting our colleagues to teach the course here, let alone getting the course taught at other universities.
...We still intend to produce the materials to export the course. Indeed, we have been working and continue to work. There is no doubt that we underestimated the effort necessary to get the job done; but our resolve to do it is intact.
(From "A course in the Mathematical Sciences,'' in S. Golberg (ed.), The New Liberal Arts Program; A 1990 Report)
Course materials for a foundation course must place emphasis on students' doing reasoning, rather than merely being exposed to it. Materials should capture student interest, which may well mean that they are dated in nature (of a "throw away'' character). Topics studied should have genuine application (there are plenty of mathematical topics with both utility and beauty, so beauty need not be sacrificed). Exercises and problems generally must be better problems or exercises, not just harder exercises or problems than those which have appeared frequently in mathematics courses aimed at increasing mathematical competency. The "natural'' use of hand calculators and appropriate computer software should be involved. And materials must tackle inappropriate "beliefs'' that students may carry, such as "to do mathematics is to calculate answers.'' Further, materials should be usable for teaching methods different from the lecture and listen mode.
Producing projects or laboratory exercises or computer programs for courses may take considerable time and energy. For example, in the University of Chicago development experience the faculty wrote the following about a computer program they wrote to simulate the propagation of a plague: "This program required about a month of evenings to write, and provides a 10 minute demonstration for the class together with a single homework assignment.''
One source for materials is the Consortium for Mathematics and its Applications (COMAP), Inc. Some of its UMAP modules have the characteristics of the course materials sought for a foundation course. Although they are written for grade levels 8-12, the American Statistical Association's Quantitative Literacy Series presents materials of the nature we are suggesting.
In order for a high-quality and focused quantitative literacy program to be established at a college, the college needs to support faculty who are willing to set up the program--which may mean developing new courses and course materials in the mathematics department or elsewhere in the institution. Further, attending development workshops, perhaps in periods when classes are not in session, may help faculty to be prepared to participate in the quantitative literacy program at all levels. The common set of goals for the institution's program and the means for their attainment need to be understood well by all those faculty who provide the program. In particular, all faculty should know the base the foundation course seeks to establish, so the follow-up components can reinforce and firmly build on that base. In order to be effective the quantitative literacy program must be "bought into'' by the faculty in a college as a whole and the curriculum must reflect the belief that quantitative reasoning belongs in courses outside departments of mathematics.
Students entering college with mathematical deficiencies have presumably had opportunities to learn the mathematics, and for them those opportunities did not work. Therefore, the college remedial course should not be a mere rehash, and certainly not an accelerated one, of the traditional secondary or even elementary course. Courses that cover the same old ground in much the same old way tend to be just as uninspiring and unintelligible for these students as the originals, and therefore even less likely to succeed. Students should be able to find even remedial courses fresh, interesting, and significant.
For such remedial work the "fresh, interesting, and significant'' approach we advocate is to study the mathematics in context. As expressed in the report "Reaching for Quantitative Literacy'' (in Heeding the Call for Change: Suggestions for Curricular Action, (1992)), "The key is to have the context relate to student interest, daily life, and likely work settings." Or as also stated there, teaching approaches should provide relevance to the student's life as the student perceives it. As anyone who reads college student newspapers knows, students can be interested in problems they may genuinely face as citizens in our communities, states, and the nation.
For nonremedial quantitative literacy courses both to capture student interest and to enable students to make connections with other disciplines, a wide variety of issues should be discussed in problem settings. Citizens who are leaders in their communities will need to respond to issues of personal finance, the environment, personal health, politics, the local schools, and the like. General education courses in the sciences and social sciences commonly raise issues which may be discussed in quantitative terms--a good quantitative literacy program will encourage them to do so.
But what approaches can be used to stimulate student learning of mathematics in context? The report Everybody Counts notes that educational research as to how students learn points to the need for teachers to view their roles as broader than they may have in the past. On page 58 and following we find:
...Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding...
All students engage in a great deal of invention as they learn mathematics; they impose their own interpretation on what is presented to create a theory that makes sense to them. Students do not learn simply a subset of what they have been shown. Instead, they use new information to modify their prior beliefs. As a consequence, each student's knowledge of mathematics is uniquely personal.
Evidence that students construct a hierarchy of understanding through processes of assimilation and accommodation with prior belief is not new; hints can be found in the work of Piaget over fifty years ago. Insights from contemporary cognitive science help confirm these earlier observations by establishing a theoretical framework based on evidence from many fields of study.
No teaching can be effective if it does not respond to students' prior ideas. Teachers need to listen as much as they need to speak. They need to resist the temptation to control classroom ideas so that students can gain a sense of ownership over what they are learning. Doing this requires genuine give-and-take in the mathematics classroom, both among students and between students and teachers.
Teachers' roles should include those of consultant, moderator, and interlocutor, not just presenter and authority. Classroom activities must encourage students to express their approaches, both orally and in writing. Students must engage mathematics as a human activity; they must learn to work cooperatively in small teams to solve problems as well as to argue convincingly for their approach amid conflicting ideas and strategies.
...As students begin to take responsibility for their own work, they will learn how to learn as well as what to learn.
Other research that is especially pertinent to the goals of quantitative literacy are studies on how to teach problem solving. In an essay "Teaching Mathematical Problem Solving: Insights from Teachers and Tutors'' (1989)), R. Shavelson, N. Webb, C. Stasz, and D. McArthur studied expert teachers and tutors of the process and compiled what they believe to be some important features of successfully teaching problem solving. These features include:
While acknowledging that this list is incomplete, the essay goes on to observe that the features do not appear individually in teachers and tutors, but they collectively become well integrated in the working of excellent teachers and tutors. Taken as a whole, the features suggest strategies for teaching which are more interactive in character than the traditional lecture and listen mode.
There are methods for teaching mathematics courses aimed at quantitative literacy which have the added benefit of blending well with succeeding components in a quantitative literacy program. We advocate the following examples: establishing collaborative learning situations, utilizing a wide variety of writing assignments, studying significant mathematical models, conducting explorations using calculators or computers, and employing team projects.
A description of the nature of collaborative learning situations appears in the report of the Sigma Xi workshop cited earlier.
During the workshop, Jack Lochhead demonstrated collaborative learning by involving all participants, working in groups of two, in the solution of simple problems. It became immediately evident that the work could be structured in such a way that each participant has the experience of formulating and articulating questions, and the experience of formulating and articulating responses to questions. The ground rules were very simple. One of the pair assumed the roles of reading and answering questions, talking aloud throughout; the other assumed the role of asking questions. With the next problem the roles reversed. What a wonderful way for individuals to explore problem solving and at the same time explore their own mental constructs and confront misconceptions in a friendly environment.
Uri Treisman, reports success with students considered to be at risk in entry-level calculus by combining work in small groups with traditional lectures. The students register for the regular calculus course and also for a scheduled laboratory that meets for a two hour session twice a week. These laboratories are devoted to the investigation of problem solving with the students working in small groups under the supervision of a competent mathematician who understands how to select appropriate problems and assist small groups in their investigations of problem solving. This extension of time-on-task, working in small groups under supervision, enables students to have experience in doing mathematics and to recoup in part the deficit in problem solving experienced during the precollege years. The essential characteristics of collaborative learning are scheduled periods of working in small groups with other students on common problems in a supportive environment (p.12).
A second method for teaching for quantitative literacy is to employ writing. A writing assignment may be simple: asking students to explain problem results, or asking them to develop lecture notes for a class period. More complex assignments are reporting on a team project or a discovery laboratory experience or making a critical analysis of a media presentation of quantitative information. Asking a student to write an explanation of problem results demands that the student clarify his/her thinking about the problem and how it can be solved. Further, it enables the faculty member reading the student's paper to respond more pointedly to the student's thought processes and hence be more helpful. Two excellent sources for help with implementing writing as a teaching tool are the MAA Notes Number 16, Using Writing to Teach Mathematics (edited by Andrew Sterrett), and the collection of essays Writing to Learn Mathematics and Sciences edited by Paul Connolly and Teresa Vilardi and published in 1989 by Teachers College Press of Columbia University. A review of these sources and descriptions of the quantitative reasoning courses developed with the help of the Sloan Foundation reveals substantial linkage between the use of writing as a means to quantitative literacy and the other methods suggested above. For example, all three of the courses mentioned earlier, which the Sloan Foundation helped develop, require students to write papers in response to questions about significant mathematical models.
Many students will encounter quantitative reasoning in their major programs of study through consideration of significant mathematical models. As noted in the MAA report A Call for Change (p. 5): "There are three ways in which models are used in attempts to solve problems originating in the world around us: (i) information is derived from mathematical models that others have built and used (weather reports, governmental studies, stock market projections); (ii) models that others have derived are used to analyze real-world situations; and (iii) people derive their own mathematical models of the situations either from known quantitative relationships or from collected data.'' All three of these ways may be brought into a course on quantitative methods by making a careful choice of models to consider. At the same time the choice can be made so as to study mathematics which is entry-level for college students.
Some mathematical models may be set up as computer-based exploratory environments in which the student may change parameters and recognize relationships in a model by "playing'' with it on the computer. Such explorations may increase students' mathematical power. As stated in A Call for Change (pp. 6-7), "Greater accessibility to mathematical strategies and representations provided by a technologically rich environment opens a new array of real-world problems to mathematical solutions. Calculators and computers allow students to explore mathematical ideas from several different perspectives, resulting in a deeper mathematical understanding. Through the regular use of calculators and computers in collegiate mathematics courses, students learn more mathematics and can more rapidly apply that understanding in problem solving.'' In particular, calculators and computers may be used "to pose problems, explore patterns, test conjectures, conduct simulations, and organize and represent data'' (ibid., p. 7). But also, perhaps in a more mundane view, employing calculators and computers may "force'' students to come to grips with the order of operations involved in elementary computations such as those involved in determining the standard deviation of a data set. In the latter instance the use of technology may create a need to know something the student has not learned previously, but which the student is now willing to pursue seriously.
Computer-based or calculator-based explorations need not appear in courses as "exercises'' but may be solidly integrated into student team projects. Projects assigned to student teams may involve the exploration of a model simulated by a computer program or the carrying out of activities more routine in nature, such as conducting a poll. The key to their development is that they involve mathematical ideas in context. Such projects would be expected to result in written, and perhaps oral, presentation of results obtained.
The methods for teaching mathematics courses aimed at quantitative literacy discussed here may be used in various proportions within a course and may be used in combination with each other as the paragraph on joint projects suggests. These methods not only can mesh well with the later components of a quantitative literacy program, and incorporate important features of teaching problem solving, but they also may provide the needed care which a successful transition from students' earlier experiences with mathematics may require, as well as a means for attacking the false beliefs about mathematics or the anxious or phobic responses to mathematics that some students exhibit.
Indeed, many myths about mathematics and the ability to do it have been propagated in our society. Activity-based learning methods cut through the alienation some feel towards mathematics and reach out to those who have until now seen mathematics as a "solitary'' or "purely intellectual'' endeavor. Cooperative learning where students encounter problems they must discuss with other students helps breed success for those who may not have previously known great success in quantitative reasoning situations -- thus changing students' perceptions of their own ability. Further, using team projects and studying a variety of mathematical models may lead to realizations that mathematics is not only for the few, nor is it a numbing experience (in fact, it could become enjoyable!). Working with a team of students may lead an individual student to a sense of community in a class (a desirable trait for liking a class, as pointed out in S. Tobias, They're Not Dumb, They're Different, p. 22) and also to the recognition that there may be more than one way to do a problem. Cooperative learning creates an atmosphere of acceptance and openness for those who may have known mathematics anxiety or phobia in the past, helping develop at least a "neutral'' attitude towards mathematics, if not a positive one. And using calculators and/or computers for explorations may present mathematics as not just computation. In general, those methods suggested above should also counteract the types of feelings expressed by students such as Eric towards the teaching style of his physics class as reported in Tobias (ibid; p. 21):
I still get the feeling that unlike a humanities course, here the professor is the keeper of the information, the one who knows all the answers. This does little to propagate discussion or dissent. The professor does examples the "right way'' and we are to mimic this as accurately as possible. Our opinions are not valued, especially since there is only one right answer, and at this level, usually only one (right) way to get it.
Undergoing transition from the type of teaching method Eric encountered to the types suggested above for courses aimed at quantitative literacy demands that students be given special attention. If students are to become confident in their ability to analyze, discuss, and use quantitative information, they must encounter sufficient practice and success in using quantitative information. Students must be expected to do quantitative reasoning--reasoning which is substantially different from completing routine problems. Where previous student experience has been largely routine in character, students may need to be "phased in'' to teaching methods which present greater and different demands on them. Explaining to students why they are being asked to do certain activities often garners both greater acceptance of the activity by the students and more effort in carrying out that activity. Further identifiable goals need to be articulated to students, and students need to be advised on how to study, what are reasonable expectations for themselves, and how to use the available resources (including faculty office time).
Finally, the ways used to evaluate mathematics courses aimed at quantitative literacy should reflect the course goals and the teaching methods used. For example, evaluation components may include written assignments, participation in discussions, and examinations. And examination questions may involve routine questions along with discussion and open-ended questions; in short, students should be expected to respond to "why'' questions and not just "how'' questions and to show that they know how to raise pertinent questions.
Two excellent sources for additional ideas and elaboration of some of these ideas are the essay "Teaching Statistics'' in the MAA volume Heeding the Call for Change and the MAA publication A Sourcebook for College Mathematics Teaching.
A focused quantitative literacy program must not be viewed as one more "hurdle'' for underrepresented groups in our culture. In particular, while the program must be carefully developed, it must also be sensitively executed. Periodic assessment of student attitudes and needs may make desirable adjustments possible and ensure the cooperative spirit of students.
Ingrained beliefs die hard. So women and ethnic group members who have long been told, often in subtle ways, that they are not capable or are less capable of doing mathematics may need repeated encouragement to do the hard work on which success depends. Students who are labeled "at risk'' may find themselves fighting anxious and phobic responses and be in need of special care. The formation of study groups in the execution of teaching strategies should be done in full awareness of the special needs groups of students may have.
Probably the most crucial time in the quantitative literacy program for those who come from underrepresented groups is the foundation experience. Thus, critical to success in the program is the advice resulting in the initial placement of the student as well as the help and encouragement received in that initial experience.
While it is acknowledged that the establishment of a quantitative literacy program at any institution is likely to take considerable energy on the part of the mathematics department, the climate is ripe to move on such a program now!
The desire to improve opportunities for certain groups to participate more fully in our society, the actions that the government and the mathematics community are taking to improve the standards of school mathematics, and the pressures on colleges and universities for accountability regarding undergraduate education all provide avenues for change in core programs for all baccalaureate students. The accountability pressures alone can lead to a natural reexamination of the extent to which graduates are quantitatively literate, but the establishment of a program of writing across the curriculum also provides an ideal time to advance a program of quantitative literacy. The latter is true because then the parallel between mathematics across the curriculum and writing across the curriculum can be more easily grasped. Also it is easier for a faculty and others to institute changes of the magnitude suggested in a package than to try to adopt one major change followed by another.
As noted indirectly in the discussion of mathematics across the curriculum, the establishment of a quantitative literacy program will take a network of dedicated faculty and others. Users of mathematics on campus should be actively involved in setting the program. While the mathematics department should be expected to bring leadership to the establishment of a program and its standards, and play a prominent ongoing role in its execution, a quantitative literacy program can normally be expected to be overseen by a general education committee of the college. To be effective, such a committee must accept responsibility for advocating quantitative literacy for all students as well as being a support group for those faculty striving in the front lines for student development.
A quantitative literacy program can most easily be implemented in stages and should be expected to take time to be fully executed. Having made a plan, a committee may first introduce foundations experiences along with appropriate placement processes for entry into those experiences. Later the continuation experiences can be added to the program or defined for the program. What is important is to GET STARTED on a quantitative literacy program, and, from a practical standpoint, to realize that the program can be phased in.
It should be noted that it is not being suggested that the concept be implemented in an artificial way--e.g. by asking art majors to take irrelevant mathematics courses in their junior and senior years. Rather some students may be taking general education courses which have a substantial quantitative component (presupposing the foundation experience) during their junior and senior years in college. Or some students may be completing research methods courses in disciplines outside mathematics, or doing substantial projects in their coursework which involve quantitative reasoning.
Some computer science courses may be acceptable quantitative reasoning experiences, while others will do little to foster the goals of Part II. But computer literacy and quantitative literacy are not the same. Generally computer literacy aims to enable a student to use a personal computer effectively in word processing and provides no experience in quantitative reasoning.
The starting point for any program in quantitative literacy will ordinarily be the college entrance requirements. These requirements and an appropriate related placement test established on both the goals of quantitative literacy and background needs for courses for various majors and degree programs should determine the foundation experience to which the student is directed. In many colleges there will be multiple tracks in the quantitative literacy program, but each will normally have the following components:
The components will take on a variety of expressions. To clarify our vision we offer the following hypothetical examples of possible programs:
Example 1. Students are admitted with three or four years of specified college preparatory high school mathematics courses and sufficiently high ACT or SAT scores. Guided by a testing program and student interest, each student takes general education courses that require quantitative reasoning projects at least one of which employs statistical methods and all of which involve mathematical models. Students individually contract for additional experiences--some take courses while others do some research or write a junior paper and a senior thesis.
Example 2. Students admitted to the college have three or four years of specified college preparatory high school mathematics courses and sufficiently high ACT or SAT scores combined with high class rankings at graduation. A testing program and student interest determine whether students take an interdisciplinary course in quantitative reasoning, a precalculus course, or a calculus course--these are to be (part of) the foundation experience and are to be formulated and taught in such a way as to foster the goals in Part II. In each subsequent academic year each student is required to submit in at least one course a paper which includes a substantial quantitative component furthering attainment of the goals of quantitative literacy (as in Part II).
Example 3. A college has an entrance requirement of two years of college preparatory high school mathematics including one year of algebra. A placement test and student interest direct students into one of five courses termed remedial, competency, mathematics for social science or business, precalculus, and calculus. The course chosen is (part of) the foundation experience for the student and is to be formulated and taught in such a way as to foster the goals in Part II. Before the junior year, students are subsequently to take one course designated "Q" (see below) in the social science or science section of the college's general education program. In their junior and senior years students must take two additional upper division "Q" courses in different semesters. A "Q" course must have a substantial quantitative component in some form which builds on the foundation experience and furthers the quantitative literacy goals (in Part II).
Example 4. The college has no explicit entrance requirement in mathematics (presupposes only high school graduation mathematical experience). Admission to the foundation quantitative literacy courses presupposes knowledge of intermediate algebra. Two explicit remedial courses are taught according to the philosophy espoused in Part II--one aims at the student who has not studied mathematics for a long period of time, while the other is intended for those who have studied mathematics more recently, but have not yet attained the level of completion of an intermediate algebra course. One or both remedial courses which emphasize collaborative learning and project assignments are prescribed as prerequisite for the foundation experience for the quantitative literacy program for those students who have not yet mastered intermediate algebra. For those with intermediate algebra as background, a testing program provides guidance for placement into one of six courses termed remedial, quantitative reasoning I, finite mathematics, mathematics for elementary teachers, precalculus, and calculus. All six courses are formulated and taught in such a way as to foster the goals in Part II. In their junior and senior years students must complete two additional courses from a quantitative reasoning listing of courses.
However the program is formulated, targets for student accomplishment through the foundation experience should be set, and the entire program should be directed towards the goals of Part II. In any case, the program should have an assessment component which can be used to improve the program.