Mathematics courses in high school, at least up to pre-calculus, definitely play a gateway function for students going to college. At the University of Colorado, for example, students who are not able and prepared to do well in calculus right away simply have to abandon any hope of certain majors in the sciences. Moreover, they are completely shut out of a career in engineering. This handicap seems to affect a disproportionate number of students from public schools--including many students of color--where they have not been groomed for higher education by strong advice to stay on track in mathematics from middle school and through high school.

So mathematics certainly does serve as a gateway to certain careers.. What I don't know is whether calculus is absolutely necessary in the pursuit of certain careers that now require it, or whether it plays just a filter role to eliminate students. If this is true, something should be done about it. On the other hand, because mathematics is a rigorous and challenging course of study that trains the mind, I require of my own children that they take the highest level of mathematics possible all through their high schools years--regardless of what they wish to pursue as careers in college and later in life. But I also know that for a variety of reasons (e.g., lack of interest, self-motivation, parental advice, peer support, or good counselling) even in our middle-class suburban school district, many students do not take mathematics every year of high school. In fact many stop after algebra in the 9th or l0th grade.

I have read, however, that Asian-American students as a group tend to take more mathematics classes consistently through high school, which may in turn help explain why these students are more highly represented in the sciences and engineering both in college and subsequent careers. Our inability to keep so many American students interested in mathematics throughout high school is clearly a serious problem, if for no other reason than the need to keep open wide options of future careers.

Question

I am not sure how to define or measure the quantitative skills and knowledge we need to function as adults. With powerful technology so easily accesible, the skills of simple computation that we all used to need seem no longer so critical. The term "numeracy" is not well known, but the term "quantitative literacy" seems to have some appeal. I take it to mean an understanding of the uses of computation and measurement, with some ability to use the associated skills. Minimally, QL requires an ability to understand graphs, tables and charts--and not to be intimidated or put off by them. Although students pick up some QL understanding and skills in precollegiate mathematics classes, QL education should go well beyond formal mathematics classes. We need to find a way to apply what is learned in formal mathematics courses to everyday "real life" situations, circumstances, scenarios and case studies. Question 3 I don't think the problem is a case of either/or--that is, either formal mathematics or "context-rich QL." Rather, it requires re-thinking our expectations as educators as to what quantitative skills and knowledge we hope students going through our educational system should acquire in order to be a functionally literate adult and productive citizen.

Periodically the media picks up yet another report on how low is our mathematics achievement as a nation and how poorly we compare to countries such as Taiwan, Singapore, Japan, Czech Republic, Poland, etc. I don't really know what these stories mean except that, in comparison with other countries that ourperform us on these tests, we don't expect all our students to take a lot of mathemtics in high schools. So while our average scores may not be impressive, I bet our best students who do take a lot of mathematics do as well as the best from anywhere.

I am concerned, whether we talk about formal mathematics or quantitative literacy, that weak schools, poor families, and minority communities seem to produce fewer students with strong mathematics background, thereby creating an obstacle which blocks students from pursuing--or even imagining--certain careers that are creative and rewarding both intellectually and financially. For me, that creates a critical social problem because it widens the existing gaps between classes and races.

You note that technology makes many traditional quantitative skills seem "no longer critical." Many high school students draw the same conclusion--which may explain why so many drop out of mathematics as soon as they are allowed to. If advanced high school mathematics is no longer necessary for life and work, but only critical for those going into the hard sciences, what persuasive reasons can be given to those who do not aspire to be a scientist or engineer?

One of the chief problems for students who opt out of mathematics too early is that they foreclose certain careers. If a student doesn't continue in mathematics through calculus or pre-calculus while still in high school--perhaps for lack of early interest in pursuing a career in science or engineering--that student will need to do a lot of catching up if he or she discovers later on that indeed, science and engineering are interesting after all. So I would never discourage students from taking mathematics every year of high school. I agree, however, that because of the ready accessibility of inexpensive and powerful calculators, many students probably don't see the need to learn computation skills, and certainly not to hone those skills to perfection. Nevertheless, I believe that all students should take a lot of mathematics in high school no matter what their eventual career and life goals may be because mathematics teaches us to think in a systematic and disciplined way. I think these skills can be transferred to other areas of study and work.

Although calculus has been--and still is--an important prerequisite for students going into the "hard" sciences, for many other fields (e.g., medicine, business, public policy) statistics and computing are much more important than calculus. Does this suggest that we should give students more choice earlier in their academic careers--perhaps after grades 8-9 or so? Might this convince them to stay in mathematics longer?

I notice that high schools are introducing new mathematics sequences and courses, notably statistics and computing. I think these are all good options, and that the calculus sequence should certainly not be the only one available to those students who want to continue with mathematics. Having such options will probably encourage more students to stay with a mathematics or quantitative skills sequence. (Here, as elsewhere, I use "mathematics" as a shorthand to include all types of quantitative reasoning and skills courses.)

Jerry Bentley,University of Hawaii As the social sciences (including history) have come to rely more on quantitative data, statistical methods have became a routine part of undergraduate programs. To what degree do history and the social science courses in high school reflect the increased quantification of these disciplines? In particular, do these courses do much to enhance students' nascent QL skills? If not, do you think that they should or could do so?

Actually, there has been a fairly sharp and clearly detectible move away from quantitative analysis in professional historical scholarship. For some time now, the move has definitely been toward qualitative and especially cultural analysis. But that doesn't mean that quantitative concerns should not be a part of history education. To the contrary, introducing quantitative literacy across the curriculum would make a welcome complement to writing across the curriculum. My suspicion is that high school courses make little effort to deal with quantitative issues, although my only real basis for this judgment are the reviews I prepared for publishers of two high school textbooks in world history. My suspicion is that if you tried to smuggle quantitative issues into history by way, for example, of economic history, students would complain that this is a course in history, not economics, so why do we have to count? My response would be that all disciplines and fields overlap to some greater or lesser extent with others, and if you want a reasonable understanding of the world and the way it works, you have to take a lot of considerations into account.

As you know, many critics have faulted higher education for failure to ensure that all graduates are well-equipped in basic knowledge and skills. How important is quantitative literacy in the priorities of colleges and universities today? Is there any consensus on the nature and level of QL that a college degree should represent? Do colleges know how numerate their graduates are? Do they care?

The University of Hawaii requires all undergraduates to complete a course in either logic or mathematical reasoning, but I suspect that there are lots of institutions with no such requirement. My own sense is that our requirement is just about right. There are a very few people who simply cannot deal with numbers, just as there are a very few who cannot learn a foreign language. For those very few people, there needs to be an alternative to a mathematics or foreign language requirement. For mathematics, logic is an obvious alternative.

Most of the commissions that have weighed in on college and university curricula seem to believe that some kind of quantitative coursework is desirable for all undergraduates, but I don't have a sense how many institutions make mathematics a core requirement. Colleges certainly have no idea how numerate their students are. Basically they don't care, or if they do they don't have the time, money, or motivation to develop tests that would reveal the level of numeracy among their students.

Perhaps gratuitously, I'll add that a large part of the problem with mathematics at the college level are the mathematics departments. For twelve years I've served on our core curriculum committee, and I have always been a very strong supporter of the math-or-logic requirement. But mathematics is probably the biggest source of student complaint about the core curriculum. The mathematics faculty seems to think that some sort of minor league professional mathematics is the appropriate thing for undergraduates completing core requirements. My own sense is that something on the order of mathematical thinking--and I hope you don't ask me for a definition of that!--is a lot more pertinent for most undergraduates.

Many people talk as if QL skills are somehow different than what is taught in school mathematics. Do the terms "mathematical literacy" and "quantitative literacy" mean different things to you? Are there any significant differences between what tends to be taught in school mathematics and what you would expect of a numerate citizen?

Well, this question is asking precisely what I hoped you would not ask! Indeed there are ways in which quantitative literacy would be different from a mastery of mathematics at either the beginning or advanced level. Maybe it has partly to do with allergies and phobias. Some people-- numerophobes--are allergic to mathematics. QL would enable such folks to deal with the kinds of quantitative issues we all face every day, even if they couldn't make their way very far through set theory or topology. QL might even make it more difficult for politicians, public officials, lobbyists, public relations types, and others with particular narrow interests to manipulate opinion and flim-flam the general public. That obviously happens quite frequently. Because of their math allergy or numerophobia, many people don't bother to think about the logic of their arguments. So indeed, some kind of preparation that would help people realize that numbers are not scary--that would help them engage quantitative data rather than breeze over it with glazed eyes--would be most appropriate for citizenship in these times.

Why do colleges have QL requirements if they don't care how numerate their graduates are? Are they so wedded to an "input" system of controls that they disregard outputs? Is this true for other elements in liberal education, or is it peculiar to QL? Might it be because they don't really know what they mean by QL?

Actually, I'm not sure that many colleges do have QL requirements. Mostly, I would guess, they have math and logic requirements, which are not necessarily the same as QL requirements. Personally, I wouldn't doubt that all faculty and administrators care deeply that their graduates are numerate, but it is a very difficult thing to prescribe meaningful requirements for large numbers of people. It is almost impossible to guarantee that a respectable percentage of students will graduate with the desired outcome of almost any course at all. Moreover, the curriculum is contested ground, since all departments and disciplines want bodies to justify their existence (and growth). The required core curriculum is both a political and an educational compromise that sometimes works well but often completely flops. This much is true for all fields and disciplines. When it comes to quantitative courses, the problem hinted at by your previous question also comes into play. For most people, quantification brings mathematics to mind and there is little awareness that there might be something such as QL distinguishable from mathematics.

2. Some research studies have shown that the study of logic does little to make people think logically. [Mathematicians are a good example: they know formal logic quite throughly, but are hardly more logical in argument than other faculty.] Much the same can be said of courses in statistics and algebra. Some claim that if students see logic and statistics *used* in context (e.g., in a history or econonomics course) they are more likely to be able to use it later. Might this suggest that QL is better taught in courses like history (even if it requires smuggling it in) and economics rather than in algebra, statistics, or logic?

Interesting idea here, on the order of writing across the curriculum. But basically I don't think it would work. For my own part, I might occasionally deal with some quantitative issues in history courses, but QL cannot be the focus of my courses, and I couldn't deal with quantitative issues in any kind of systematic way. Some students might well see the significance of quantitative reasoning better if it were in some real-world context, but I don't believe you could do quantification across the curriculum systematically enough to ensure that many students would graduate with respectable QL. In addition, a lot of instructors themselves are not at a QL level that would enable them to deal effectively with quantitative issues. A more realistic approach might be for instructors of courses in mathematics, logic, and statistics to frame their lessons with real-world examples.

Michele Forman, As the social sciences (including history) have come to rely more on quantitative data, statistical methods have became a routine part of undergraduate programs. To what degree do history and the social science courses in high school reflect the increased quantification of these disciplines? In particular, do these courses do much to enhance students' nascent QL skills? If not, do you think that they should or could do so?

Before responding, I need to note that the social sciences definitely do not include history. History is in the humanities. The social sciences are predominantly a late 19th and early 20th century development that applies "scientific" tools to human society. The two disciplines occasionally overlap, but generally use very different lenses to view the world. Names are very touchy issues to many in the field (but not to me). I don't teach social science. I'm an historian, so I can only answer in terms of history teaching and learning.

Good history courses in high school strongly reflect increased quantification, especially through social history that often relies on demographic or other quantifiable data. Sometimes our work does increase (or at least emphasize) the importance of what students know and can do relating to these quantification skills. For example, when examing historical evidence in the form of a graph, I might ask my students to explain how the graph would appear different (while containing the same information) if we were to collapse the y axis by half. We can discuss how this might affect superficial interpretation of the evidence. And what study of Islam would be complete without Islamic art and tesselations? (Escher was inspired by the Al-hambra in Spain.) I definitely think that history teachers should consciously integrate numeracy skills into teaching because (a) interdisciplinary teaching is important and effective for all of us, and (b) we need these tools as historians. After all, even a timeline involves quantification.

Many people talk as if QL skills are somehow different from what is taught in school mathematics. Do the terms "mathematical literacy" and "quantitative literacy" mean different things to you? Are there any significant differences between what tends to be taught in school mathematics and what you would expect of a numerate citizen?

I certainly have a limited understanding of these terms, but I think of "quantitative literacy" as being a subset of "mathematical literacy"; quantitative literacy is similar to (or the same as) "numeracy". I see mathematical literacy as going beyond working with numbers or arithmetic. I believe our more capable students are well prepared as numerate citizens, but our weaker students are not. Students who fulfill their mathematics requirements with "general math" or business courses learn very useful skills, for example, but miss out on learning how to reason abstractly.

I understand mathematics to be another language, and like all languages, it allows us to interpret our universes differently. Because mathematics increases possibilities for interpreting evidence, it helps us see the world from different vantages and reason better and more creatively . I would certainly hope that all students are exposed to mathematics beyond arithmetic.

For a host of different reasons, many students arrive in college not fluent in elementary (high school level) algebra, geometry, and statistics and are therefore unprepared for the quantitative demands of courses in the natural and social sciences. Often colleges respond by providing supplementary (non-credit) QL programs that help students remediate these deficiencies. Question: Should more than that be required for graduation from college? In other words, should numeracy for a college graduate represent something beyond entrance-level expectations? If so, what should it be?

I endorse raising graduation standards. While certainly not a mathematics teacher, I nonetheless believe that the vast majority of our students are significantly more capable than we have thought them to be--in history, mathematics, and other disciplines. As for college, numeracy must involve more than entry-level expectations, especially since these are so weak at many colleges. I am puzzled that our students recognize the importance of mathematics education (a student of mine once wrote in a paper, "Without math, you can't have a life.") but many avoid taking challenging mathematics courses. Only college entrance requirements drive many of my best students to take higher level courses in mathematics.

Pam Paulson Despite strong links between mathematics and some aspects of the fine arts (e.g., musical scales and rhythm, perspective drawing, digital image enhancement, computer-based choreography), most people still think that quantitative literacy is primarily important for the natural and social sciences. In your view, how important is QL for students interested in the arts? What kind of QL is most important for them?

Basic quantitative literacy sneaks into many aspects of the arts and is key to realizing expression of what is in the mind's eye. Students seem to have a new appreciation for numeracy when it is connected to the art form they work in, especially when it is necessary to the application of their ideas. Measurement comes into play in every art form. Fractions are important in technical theater, music, visual arts, and dance. New computer programs have encouraged many new ways to connect the arts and mathematics. Any time you are mounting an arts production you are involved in basic mathematical computations for ordering materials and supplies, dealing with budgets, sewing costumes, angles for sets and lighting, etc. Artists definitely need basic mathematical literacy.

Mathematics is a subject that students love to hate. In school, many drop out as soon as it becomes an elective; in college many take mathematics only if it is required and even then often put it off as long as possible. Do you have any advice for mathematics teachers and curriculum developers about how mathematics can be made more attractive to students who like it least?

Based on our experience at the arts high school, we have found that when mathematics is taught as an extension of the arts, students find it relevant and important. They see the connections applied as they create and perform their work. Students have made tesselations in drawing, worked with fractals in dance, created model set designs, etc. As long as students can see first hand the relevance to their interests and goals, they perceive mathematics as an important area of study. Unfortunately many arts teachers also feel uncomfortable with mathematics, and may not do as much with it as they might.

Do the terms "mathematical literacy" and "quantitative literacy" mean different things to you? Are there any significant differences between what tends to be taught in school mathematics and what you would expect of a numerate adult?

There does seem to be a difference between what is taught in school and what most adults in the arts seem to need to know, although some of the new integrated mathematics programs appear to be helping bridge this gap. The basic literacy skills of computation are essential. Reading and understanding data and graphs is also important, as well as basic geometry. Finding and creating patterns is a very important part of the arts. Problem solving is also critical.

Although there are many links between mathematics and the fine arts, most of them are invisible to the public. How do we get students and parents to recognize this link--as they now recognize the link between mathematics and science? Will high school counselors ever tell students that mathematics will be a big help to a career in the arts?

The kinds of mathematics used most varies by arts area, so one of the important considerations in making links visible is to be clear about the kinds of mathematics being used. For example, theater and visual arts focus more on geometry, while music relies more on algebra. However, because computers can calculate things like sound waves and tuning frequencies, most students don't learn the mathematics: they just learn how to operate the electronic equipment. Media arts also relies heavily on computer use, for example to calculate such things as focal length. In dance the connections often revolve around using mathematical models as structures for choreography. There is also the use of planes and just plain arithmetic. Budget and cost considerations cross all of the arts areas.

Once the types of distinctions between arts areas are more visible, it is easier to talk with students and teachers about their relevance. But then we run into the next stumbling block: although describing these distinctions and connections may make sense in a verbal conversation, I don't think parents will understand these links based only on verbal descriptions. It will probably also take some application on their part. The parents will have to actually put some of the concepts into action. By starting this kind of description for parents when their children are in the early grades, they will have a better chance of seeing the links. For example, parents can easily understand patterns in an activity, but not necessarily just from a verbal description. Having mathematics/arts examples used in interdisciplinary contexts will help.

We have found that few counselors ever tell students about opportunities to major in the arts in college because most don't believe there are worthwhile careers in the arts. Of course this depends on the individual and their background, but I think it is a long way off for counselors to realize themselves that there are career opportunites in the arts and that there are connections with learning in other subject areas like mathematics.

Your list of quantitative skills needed by adults and artists omits the one subject that is dominant in high school mathematics classes, namely algebra. Do you think we need a different kind of high school mathematics to make it appealing and useful to arts students?

Indeed, algebra is the one area of mathematics it is hard to grasp in many of the arts areas. In some cases where it is important the electronic equipment programmed with the algebra does all the work, so students only need to operate the equipment. In my opinion, the integrated approach works best for students in the arts. Students have a very hard time connecting the segmented parts of mathematics by themselves. They really need help to "add it all up." At the Center we are using a new book called Discovering Geometry: An Inductive Approach that is filled with color images and art works including, for example, Islamic art as examples of mathematics. It also deals with optical art and perspective and relates these concepts to the arts. Coming in the back door has been successful with many students. First they see the beautiful line drawings, for example, and then after they are intrigued, they ask, "how can I learn this" and "where is the compass?"

Peggy Skinner Many people talk as if QL skills are somehow different from what is taught in school mathematics. Do the terms "mathematical literacy" and "quantitative literacy" mean different things to you? Are there any significant differences between what tends to be taught in school mathematics and what you would expect of a numerate citizen?

Is QL different from mathematical literacy? Are the teachings different from the uses? I think there is a difference. Aside from simple calculations, the most important concept for individuals to understand is proportion. In real life, individuals are constantly asked to think about issues that relate to percentage or doubling or halving something, for example, to calculate how to double a recipe, to determine an 8% sales tax, or to understand a risk factor for disease. It is interesting that of all the calculations that students use, the easiest for them is the calculation of their percentage score on a test. Here they create their own need to know. Students are taught these kinds of skills in lower and middle school, but not necessarily in upper school unless they take a statistics and probability class. (Our school offers that class to those who they define as nearly mathematical illiterate because they did not succeed in algebra and move along smoothly to pre-calculus and calculus). I define a subject-literate individual as one who knows and can use information. A numerate citizen actually uses very little of their mathematical skills.

Although everyone says that quantitative skills are essential for success in science, the science most students see in high school uses very rudimentary QL methods--typically just those of the middle school curriculum (e.g., formulas, graphs, averages). Can the teaching of QL be embedded in science courses, or is it better left to the mathematics teachers?

I certainly have to teach basic QL skills. Even though we have an excellent mathematics department, the students cannot always make the transition from one class to another, especiallly if the timing is not quite right. Science teachers spend a good deal of time teaching basic mathematical skills. I teach percent change many different ways in grade 9-10 biology. Every time I present it during the year (about 4 times), I leave the formula on the board but the students still work hard. Science teachers typically have more ways to demonstrate how a principle is used than math teachers do. Moreover, they have the equipment needed give each concept a conceptual framework.

As you know, mathematics is often called a "critical filter" that blocks students with weak school education from rewarding careers. Some fault the pristine formalism of mathematics which poses seemingly insurmountable hurdles for many students; many now argue to replace formalism with a practical, context-rich quantitative literacy. Just how important is it for all students to master formal mathematics? Is context-rich QL a more reasonable expectation?

This is a hard question that hits philosophical issues. Should you track students? Should only some of the students take a road that includes mathematical literacy? It is clear that using it will make students follow a traditional series of classes. When one filters and then places students, those that survive will continue to take additional classes. Many of those students will be well served by the rigor and skills that they encounter in preparation for future work in engineering, mathematics, or physical sciences. Those that do not take that road do not need to master formal mathematics. Most individuals find that life uses only simple algebra or the interpretation of graphs--probably no more than 8th grade mathematics.

Senta Raizen, The National Center for Improving Science Education, Washington, DC. Many people talk as if QL skills are somehow different from what is taught in school mathematics. Do the terms "mathematical literacy" and "quantitative literacy" mean different things to you? Are there any significant differences between what tends to be taught in school mathematics and what you would expect of a numerate citizen?

Quantitative literacy is different from what is traditionally taught in school mathematics, although not necessarily what would be taught if the NCTM standards were more widely implemented. For example, I expect people who have QL skills to be able to think in orders of magnitude when appropriate, make reasonable estimates of various quantities (including linear, area, and volume measures), find shortcuts to arithmetical operations to do in their heads, do ratio/proportional reasoning and arrive at good approximations, and have developed some spatial and directional sense. I also think people should be able to relate various types of graphical information to their numerical expressions, taking account of scales (linear, logarithmic and other curves), smoothing of data, and the rationales behind various projections. They also need to understand the kind of information that is carried on consumer labels (e.g., food, appliances). I don't believe traditional school mathematics teaches these competencies.

As you know, mathematics is often called a "critical filter" that blocks students with weak school education from rewarding careers. Some now urge that students learn algebra at earlier ages, while others fault the pristine formalism of mathematics for posing seemingly insurmountable hurdles for many students. Just how important is it for all students to master formal mathematics? Might a context-rich experience in applications of mathematics be a more reasonable expectation for all students?

It is important for all students to learn quantitative literacy skills that will serve them the rest of their lives. More and more, information is going to be provided in visual terms, so this too should be taken into account in thinking of the mathematical needs of all students. So I do believe in an applications- and context-rich approach to mathematics--but only in part. Many students are turned on by quite abstract mathematical puzzles, even if they don't have any immediate practical applications. This is also true, I believe, of some areas of mathematics that have completely dropped out of the curriculum, such as construction geometry. Altogether, the US curriculum is weak in geometry, an area of mathematics that has high aesthetic appeal for many students.

I do believe, therefore, that all students should have some exposure to formal mathematics after exposure to various other types of mathematical experiences. It seems to me the argument about algebra, however, has some aspects of illogic about it: if students who now take algebra do well in their subsequent schooling, then (so the argument goes) if all students took algebra, they would all do well in their subsequent schooling. In other words, this popular argument would have us believe that if A (an unknown factor) leads to B (taking algebra) and C (doing well in school), then B by itself leads to C.

Mathematics is a subject that students love to hate. In school, many drop out as soon as it becomes an elective; in college many take mathematics only if it is required and even then often put it off as long as possible. What should change--the subject (perhaps from mathematics to QL) or the approach? Can an emphasis on QL help resolve this age-old dilemma?

I'm not sure what the distinction is between the subject and the approach to it. QL has to be based in good understanding of a number of important mathematical concepts, although possibly not in all that are necessary for mathematical literacy. Also, as I said, some formal mathematics should make up part of the path toward QL. At some point it would be interesting to explore the distinction between QL and ML, that is, what underpins QL versus what underpins ML, and what the differences in expected competencies really are. If we agree that everybody needs QL, but ML is variable depending on your profession (software engineer, physicist, financial analyst, social scientist, biostatistician), then perhaps we can better define what aspects of ML extend beyond or are different from QL.

You say, as do many well educated people, that many students are "turned on by quite abstract mathematical puzzles" and that "all students should have some exposure to formal mathematics." But what about the many students who are *not* turned on by abstract mathematics? Why should all students be expected to learn formal algebra, for instance, in addition to the broadly useful QL skills you describe?

I don't mean to suggest that all students should be forced to take whole courses in formal mathematics such as algebra or calculus. What I do mean is that as students, with the help of their teachers, find "shortcuts" to mathematical operations based on regularities, they should get assistance in making those explicit and in learning some of the underlying structures that make the shortcuts work. In this way, formal mathematics enables further strides in mathematical thinking.

As you point out, school mathematics has not traditionally emphasized the kinds of things you describe as important for QL. But is QL really the responsibility of mathematics teachers, or might students learn more QL if it were embedded throughout the curriculum with all teachers taking responsibilty?

You raise a very good point. I guess the real question is: What will be harder, getting mathematics teachers to go beyond the traditional curriculum, or getting all the other teachers to teach what they will consider the mathematics teacher's responsibility? I'll offer a quick anecdote: I was observing a 7th-grade science class doing a lab on conservation of mass, which involved weighing some water in a beaker on a triple-beam balance. The particular student I was observing had done everything correctly. However, when it came to the weighing, she couldn't figure out whether she had .147 grams, 1.47 grams, 14.7 grams (the right amount) or 147 grams. I told the teacher afterwards that this student was having problems with the decimal system and place value, whereupon the teacher shot back: "That's the math teacher's job!" (Of course, this is also a good example of the student having no sense of quantities!) Well, there you have the dilemma in a nutshell.

Richard Millman As you know, many critics have faulted higher education for failure to ensure that all graduates are well-equipped in basic knowledge and skills. How important is quantitative literacy in the priorities of colleges and universities today? Is there any consensus on the nature and level of QL that a college degree should represent? Do colleges know how numerate their graduates are? Do they care?

All colleges have some kind of QL requirement but it is nearly always a mathematics requirment and many times not well thought out. I am reminded of Uri Treisman's comment in response to precalculus being required of all students that a final course in a subject should never be "pre-" anything. Part of the issue is that our institutions are so different. My view of the correct procedure is to first ask what is your mission and purpose and what are the educational goals you derive from them? Thus there would be many different types of QL requirements.

Most institutions have no idea how numerate their graduates are. The liberal arts colleges may care, but it is not a very widespread concern on campus. Most often, numeracy is delegated to the mathematics department where it is viewed as mathematics requirement rather than as QL.

Many people use almost interchangeably terms such as "quantitative literacy," "mathematical literacy," "quantitative reasoning," and "numeracy," the latter especially in British commonwealth countries. What's your view about the relation of mathematics to QL (or QR)? If QL is different from mathematics, and if mathematicians and mathematics teachers are responsible for teaching mathematics, then who is responsible for teaching QL? As you view QL, is it something that mathematicians and mathematics teachers are particularly qualified to teach?

Each campus should have a QL committee with representatives of departments who could teach the subject (quanitative social scientists, mathematics, computer science, etc.) plus some "laypeople.". But again: what you want as an educational goal for your students is the crucial first step.

Some colleges have introduced "mathematics-across-the-curriculum" as a strategy to infuse QL skills in students who (sometimes for good reason) resist taking mathematics courses. Is it reasonable (and effective) to think of QL like writing--as a responsibility of the whole faculty and as something that should be emphasized to some degree in every course? If only science faculty stress QL, will the average student take it seriously? On the other hand, does anyone other than science faculty really care?

In an ideal world, quantitative literacy should be across the curriculum. Unfortunately, people outside the sciences (plus a few social scientists) are phobic about mathematics or QL and would hate to teach it. (Look at what happens when mathematicians are asked to put writing in their courses. Of course, I like it personally but there are many others who refuse.) I am afraid that "mathematics across the curriculum" is doomed by exactly what we are trying to combat: mathematical or quantitative phobia. Having said all that, when Knox makes a curriculum change (which we are starting now), I will propose mathematics (or QL) across the curriculum.

June K. Phillips, Dean, College of Arts and Humanities, Weber State University, Ogden, Utah A common pattern on many campuses is to require a foreign language of B.A. students and mathematical (or quantitative) skills of B.S. candidates. Other institutions embed similar choices--mathematics or foreign language--in their general education requirements. Does this alternative make educational sense? Is quantitative literacy analogous to foreign language literacy?

Our general education is in two sections: core requirements for all students regardless of degree sought and degree-specific requirements. Core requirements include composition, American institutions, computer literacy and quantitative literacy. The latter can be fulfilled by achieving a score of at least 65 on the COMPASS algebra exam, a score of at least 3 on the AP Calculus or AP statistics exam, or by taking a 3- credit mathematics course (Contemporary Mathematics, Statistics, College Algebra, Pre-calculus, or any higher-level mathematics course). The B.A. or B.M. degrees require, in adddition, two years of foreign language (or equivalent). B.S. students must take six credit hours over and above the general education requirements that emphasize scientific inquiry. Nearly one hunderd courses are available to fulfill this requiremement, including seven offered by the mathematics department.

Thus we do not see FL/QL as alternatives in general education although FL/SL (scientific inquiry) do define degrees. Defining the B.S. this way is new and came about during the semester conversion process we went through recently. I think the faculty didn't like the B.S. being called a "default degree," so now it has some substance. One consequence is that we're having fewer English majors with a B.S.--and that's fine with me. They'd rather fulfill the FL requirement for the B.A. than take more mathematics or science.

I might mention that we have compatible requirements throughout the state system for the general education component. This was achieved during semester conversion in order to make transfer of general education more uniform. We have an agreement that if general education is completed in one state institution, the other accepts the whole general education package. For a host of different reasons, many college students are not fluent in elementary (high school level) algebra, geometry, and statistics and are therefore unprepared for the quantitative demands of many courses in the natural and social sciences. Often colleges respond by providing supplementary (non-credit) QL programs that help students remediate these deficiencies. Question: Should more than that be required for graduation? In other words, should numeracy for a college graduate represent something beyond entrance-level expectations? If so, what should it be?

We offer two remedial courses in mathematics for those not prepared for the quantitiative literacy courses: Pre-Algebra and Elementary Algebra. Special fees are paid to support those courses and credits do not count toward graduation. The courses should help students succeed in the QL courses which they still must take and pass. These are not substitutes for meeting the requirements.

Some colleges have introduced "mathematics-across-the-curriculum" as a strategy to infuse QL skills in students who (sometimes for good reason) resist taking mathematics courses. Is it reasonable (and effective) to think of QL like writing--as a responsibility of the whole faculty and as something that should be emphasized to some degree in every course? If only science faculty stress QL, will the average student take it seriously? On the other hand, does anyone other than science faculty really care?

We don't have a mathematics across the curriculum program and I wouldn't expect us to do so. Frankly, our mathematics department is the least connective across disciplines and we have had to hire people with good high school backgrounds as instructors to meet the needs of the developmental and QL courses. Our regular mathematics professors are in a world of their own. As we get more students into the new B.S. track, my guess is that most students will meet the "scientific literacy requirements" through course in natural sciences, business, and measurement courses offered in social sciences and not in mathematics. Often one hears that QL issues become confused (or incoherent) when they are externally prescribed through state-wide articulation agreements that may not match the situations of individual campuses. What's your sense of how well Utah's state-wide articulation in working?

For transfer students there are certainly advantages to the articulation agreemeent among state institutions. What we have agreed to is a 'parity' arrangement whereby if students complete the general education package at one university, it is considered to be "locked in" and can be transfered as a block to the other institution. (This is similar to what one does with an Associate's degree.) If a student has not completed the entire general education block, then we apply parity: students who did 75% of general education at one place, are considered to have done 75% at the other. The trick then is looking at what was done and what was not.

This system was imposed by the Regents through the back door of semester conversion. They brought together state-wide discipline-based teams with representatives from each campus to coordinate their course numbering systems, etc. While they claimed they did not want "cookie cutter" institutions, that is exactly what they did want. By working with disciplinary groups to seek conformity, they in fact created practically identical general education packages without ever looking at general education as a whole. Once the science, foreign language, mathematics and other faculties agreed to what they wanted, you ended up with a general education package that had never been debated as a whole or passed through an individiual faculty senate as a single construct.

The only institution that held out was Utah State. They had recently gone through a general education revision that included upper division general education courses, which reflects a trend at a number of schools who have revisited general education in the last decade. Admirably in my opinion, they held fast with keeping some general education at the upper division and will require transfer students to take those courses. At Weber, we lost our requirement to have a required literature course which had been part of our package. The English department didn't fight for it (they just gave up) and are now realizing how it has affected their enrollments and the balance with writing in the department. Formerly, most faculty taught almost equal shares of literature and composition. Now their loads are 3/4 composition.

My concern is that we do not have a thoughtfully created, widely debated general education on campus. We might have ended up in the same place since when push comes to shove, most debates on the academic issues associated with general education break down into a type of distribution requirement. But now, as a result of articulation issues, we have a distribution system based on convenience of transfer above all else!

Angelo Collins Many people talk as if QL skills are somehow different from what is taught in school mathematics. Do the terms "mathematical literacy" and "quantitative literacy" mean different things to you? Are there any significant differences between what tends to be taught in school mathematics and what you would expect of a numerate citizen?

It appears to me that mathematics has two foci. One is a way of thinking and reasoning that allows a person to solve some types of problems. The other is a set of tools (arithmetic, algebra, geometry, trigonometry, calculus) that enable, support, and provide context for these problems. To me, the use of the tools is mathematical literacy, while the tools and thinking together constitute quantitative literacy.

My experiences with contemporary school mathematics is limited, but my impression is that there are schools in which quantitative literacy is emphasized more than the use of the tools and the reasoning that these tools support. However, there are also schools where the use of the tools without a purpose or context is emphasized. In some places the use of tools is a criteria for entrance into advanced courses and therefore the skills of using these tools becomes a limiting factor for students.

I guess I would prefer a salesclerk who could estimate or calculate a discount over one who knows calculus but does not have these skills.

Schools are under pressure these days both to prepare students for college (SAT, AP, etc) and to meet the needs of the modern high- performance workplace. These different goals emphasize rather different mathematical areas (e.g., the former, advanced algebra and formal methods; the latter, data analysis and contextualized problems). Can teachers do it all? Can students learn everything they must learn? How would you sort out what is essential QL from what is optional?

This question highlights the ambiguity of the American public about the purposes for schooling. We seem to expect our schools to satisfy different purposes for schooling: the liberal ideal that represents certain types of knowledge because this is what all educated persons know, the needs ideal which focuses on being able to function in the work place, and the student-centered ideal which focuses on the development of persons. There is no way that schools can teach all that we expect citizens to know for all three purposes.

Although it has become a cliche, I would suggest less is more. Focus on teaching fewer mathematical concepts, but choose those that have the greatest mathematical power and current and predictable social impact. Then teach these concepts with reasoning about their development and utility.

Is it better for the skills that are central to quantitative literacy to be taught primarily by mathematics teachers or to make QL a responsibility of all teachers in all subjects, embedded in every course? Even if the latter might be ideal, is it practical? Are teachers QL? Will they willingly and eagerly push their students in this direction?

I may be old fashioned, but I believe that at the middle and high school level quantitative literacy should be taught by those who have a depth of understanding of mathematics (which is prerequisite to quantitative literacy), and that is the mathematics teachers. Teachers cannot be expert in all disciplines. What is distributed across the curriculum is easily overlooked.

Elizabeth Stage your vision of the quantitative abilities of a numerate citizen? Is there a core that absolutely everyone should know and be able to do? I would define the core in terms of the kinds of tasks that citizens should be able to complete successfully. As a voter, that would include understanding the quantitative arguments made in voter information pamphlets like the one issued by the secretary of state in California, or by politicians about sich things as the federal budget, the deficit, the debt, or the balance of payments. The issue is not arithmetic, but compound interest, extrapolations, and underlying models. For example, one recent voter initiative dealt with complex projections of growth by tying the schools budget increases to the lesser of the increases in the school age population, the total population, the state's cost of living, and the national consumer price index.

As a consumer, the core would include number sense, estimation ability (to recognize when a scanner is broken, or that something was entered twice, or with a misplaced decimal), and the ability to figure out personal finances such as whether it's worth taking out a loan to pay off your credit cards. To stay healthy, consumers also need to be able to read and interpret nutritional labels, and to figure out what information is relevant for oneÕs own diet and health choices.

As a patient, the core would include interpreting statistics to decide whether to have surgery and to be able to formulate appropriate questions about medical data (For example, for such and such surgery there's generally an 80% success rate. However, given your age and physical condition, the likelihood is somewhat lower. There's a 5% mortality rate from the surgery, and a 20% chance of serious postoperative infection.)

As an employee, apart from job-specific skills, thereÕs an increasing need to be able to develop spreadsheet models and Òwhat ifÓ manipulations, as well as an ability to decipher quantitative information, interpret it, and communicate it to others in forms (e.g., graphs, charts, tables, formulas) that make sense in context.

I could go on enumerating, but you get the idea. The core includes mainly arithmetic and middle school concepts, emphasizing a facility with estimation, manipulation using technology, or mental arithmetic, all of it in context, all of it sense-making. This program has very little overlap with traditional school or undergraduate mathematics.

I am ambivalent about including Òenough mathematics to have opportunities to learn new things.Ó On one hand, thatÕs the time to learn it, in the context of nursing or studying science or whatever. But on the other hand, as a matter of equity, what I have said above looks suspiciously like Òcheckbook mathÓ and you know thatÕs not what I mean. The problem formulation (what do I pay attention to?), implementation (what model makes sense?), and conclusion (does the answer make sense?) ought to be enough to get people into the door of the courses that they need, but it isnÕt.

Just how important is it for all students to master formal mathematics? Might a context-rich experience in applications of mathematics be a more reasonable expectation?

More reasonable to whom and for which students? It is no more important for all students to master formal mathematics than it is to be able to recite ÒArma virumque cano,Ó or to know which fork to use at a formal table setting. And it is no less important. I learned those things, and many more (e.g. ÒDonÕt wear white shoes after Labor Day or before Memorial Day,Ó) and they have served me extraordinarily well. As long as such symbols are held in high esteem, access to them and success in them are the right of all students.

If we can get society (and particularly some mathematicians in California, I assume that you arenÕt going to quote me verbatim, but you know exactly which ones I mean) to recognize that an elegant proof is not superior to an elegant aria or an elegant three point shot in basketball, then we cannot replace the traditional standards with ones that make more sense. Yes, I think that every student should be pushed and supported to achieve excellence at the highest levels in some spheres; call that mastering formalism. And I think that every student should be given opportunities to learn well things that are useful and important as citizens, voters, consumers, etc.

Of course I think that content-rich experience in applications of mathematics is a better goal for all students, but until itÕs widely valued, then it sells some students short. In my own experience, content-rich applications (studying physical chemistry) actually got me interested in mathematics per se. IÕd skated through calculus, but was jarred into interest in math when a chemistry professor said, at the end of my freshman year in college, ÒIÕm not happy with this course being based on differential equations. Does anyone want to audit Linear Algebra with me next year and see if thatÕs a better model for the chemistry weÕre trying to describe?Ó All the tumblers fell into place for me. I did audit the course (I never thought to take it for credit, as I was simply interested in learning the stuff) and he did rewrite the course and the book for the course. Whew! But I had already passed Calculus III, so I could afford to learn mathematics to answer a question that was meaningful to me. (Of course you can say that this was an unusual professor at a liberal arts college who had time to revise courses and involve undergraduates and that I was an unusual freshman. Yes, but when I taught middle school mathematics I found that even the conventionally best students were far more engaged in meaningful problems, where the generalizations and abstractions and formalisms were their own, derived from the contextualized and concrete.)

We have a political problem here, which in some places, as you know, has become partisan and very mean-spirited. I console my friends whoÕve been on the front lines in California that the reason that we have the problem is because we have had some success, that we have taken all students to heart and actually made a dent in the social order. I truly believe that we are at a crossroads, that we may have to wait another generation if we donÕt navigate these shoals successfully, but that we have to create a social and political climate where your innocent question isnÕt so loaded and the answers to it arenÕt so guarded. The good news and the bad news is that ten years ago I would have answered the question, ÒFormalism, no. Context-rich, yes. Next question.Ó Many critics have faulted higher education for failure to ensure that all graduates are well-equipped in basic knowledge and skills. How important is quantitative literacy in the priorities of colleges and universities today? Is there any consensus on the nature and level of QL that a college degree should represent? Do colleges know how numerate their graduates are? Do they care?

IÕm tempted to say that I have no standing to answer this question, but I think that higher education is in even worse shape than K-12, to the extent that the degree or diploma is meaningless except as a measure of persistence. ItÕs hard to imagine a major in which some quantitative literacy isnÕt necessary, in context, but the colleges that I know about that keep track do so entirely out of context, so math is another filter and a bad representation of math, at that. The California State University ELM exam (I think that stands for Elementary Level Mathematics, but am not sure) is a particularly pernicious test of out- of-context, who cares stuff. So, I donÕt think that thereÕs consensus and in the current political climate IÕd be afraid to see what that consensus would look like. (The UC/CSU Math Diagnostic Test is another favorite, though I have heard that it now has constructed response items and doesnÕt classify students on the basis of their ability to add fractions with unlike denominators, which was the pivotal item for getting placed into calculus.) Many public universities (e.g., CUNY) have come under attack for the extent of remedial work offered to undergraduate students. Indeed, on many campuses the entire QL effort is devoted to helping students overcome deficiencies in preparation for college. Should numeracy represent more than remediation? If so, how much more?

This goes to the definition of numeracy. If itÕs the traditional one, then the colleges have a huge task at getting students to be numerate in context. The business about remedial work is a budget matter, I think. Who pays for the failure of K-12? Why donÕt we send them to community college, which is cheaper, etc.

This question reminds me of something that I may have told you and is somewhat tangential, but itÕs a story that I like. Smith College (my alma mater, from the previous story) has a Five College calculus reform model, spearheaded by Jim Callahan, one of the professors of mathematics from whom I learned mathematics but didnÕt take courses. He told me that they had had to develop a one-semester course for students who entered with 5Õs on the BC calculus AP exam which helped them to connect the calculus that they had studied to the contexts in which Smith freshman had learned it--economic, social, etc. I loved it! He was discouraged about other faculty being willing to take up the reform ideas, but I thought that the need for such a course was wonderful.

IÕm obviously running out of steam, but you said that you wanted a 7-10 day turn on these questions, so I think that I should sign off for tonight. I hope that I have given you enough grist for the mill of another round! Or, I need feedback about how useful or not useful my replies so far have been.

============ Doug Bennett 1. As you know, many critics have faulted higher education for failure to ensure that all graduates are well-equipped in basic knowledge and skills. How important is quantitative literacy in the priorities of colleges and universities today? Is there any consensus on the nature and level of QL that a college degree should represent? Do colleges know how numerate their graduates are? Do they care? 2. Some colleges have introduced "mathematics-across-the-curriculum" as a strategy to infuse QL skills in students who (sometimes for good reason) resist taking mathematics courses. Is it reasonable (and effective) to think of QL like writing--as a responsibility of the whole faculty and as something that should be emphasized to some degree in every course? If only science faculty stress QL, will the average student take it seriously? On the other hand, does anyone other than science faculty really care? 3. For a host of different reasons, many college students are not fluent in elementary (high school level) algebra, geometry, and statistics and are therefore unprepared for the quantitative demands of many courses in the natural and social sciences. Often colleges respond by providing supplementary (non-credit) QL programs that help students remediate these deficiencies. Question: Should more than that be required for graduation? In other words, should numeracy for a college graduate represent something beyond entrance-level expectations? If so, what should it be? ================= =============== Varieties of Ql: Here is a sample (in no special order) of distinct varieties of QL, each arguably serving some legitimate societal goal, each found somewhere among the variety of current programs, but none with an obvious claim to being the correct QL: Algebra for All. The current slogan of Riley and the import of many of the new high-stakes state tests: require every graduate to pass a test (rather like parts of the SAT I math test) that requires modest ability in reading and interpreting formulas, understanding graphs, and solving very simple equations. Quantitative Reasoning. The substance of George Cobb's WNC essay, and the name of many college degree requirements. Emphasizes broad synthesis of logical, visual, verbal, and computational thinking. Manipulative algebra is incidental to this goal. Problem Solving. The longest-lasting NCTM mantra, often interpreted in a very broad sense that moves war beyond the boundaries of traditional mathematics (e.g., where to site a new shopping center in a community). In this perspective, the problem and its possible solutions are paramount; particular mathematical skills involved in the solution are secondary. Preparation for AP. Traditional mathematical skills, well practiced, embedded in core theory and embroidered with template-style problems that echo major classics of mathematics exams of the past. The criteria for topics is their utility in (traditional) calculus. The dominant ideal of the suburban Volvo crowd. Computer Mathematics. A common desire of employers--for skills in solving (and presenting the solutions of) quantitative problems using standard computer packages such as spreadsheets, simulation programs, and other mathematical computer models. Usually ignored by mathematics teachers, but taught (often poorly) in business courses. [A bank manager once said, comparing computer-based methods with those taught by mathematics teachers: "I would fire anyone I found doing their calculations by hand."] SCANS Skills. A rather popular innovation in some charter schools (including many of those focused on the fine arts), the SCANS skills cover broad categories required of employees, entrepreneurs, and community leaders: acquiring information, allocating resources, working with others, improving systems, and working with technology. QL is embedded in (but not easily separable from) the SCANS skills. Considerable emphasis on communication skills not found in other QL programs. Functional Mathematics. The outcome of the "Beyond Eighth Grade" project that Susan and I did with Tom Bailey for NCRVE. A full NCTM-like three year curriculum that gives priority to the kinds of mathematical topics and skills needed by ordinary people in life and work. Includes topics not ordinarily studied (financial mathematics, planning and scheduling) and postpones topics needed only for specialized college work. Civic Literacy. The quantitative skills necessary to make wise decisions about public matters, whether as an elected official or just as a single voter. Understanding the need for data; abilities to sort through conflicting claims; skepticism about the reliability or significance of data; recognition of the limits of computer models. Instrumental Mathematics. John Dossey's definition of QL as the ability to interpret and apply aspects of mathematics to understand, predict, and control "relevant factors in a variety of contexts." More a schema than a definition, this approach focuses on comprehensive mastery of both procedural and conceptual understanding over six aspects and four levels. Parental Literacy. Appropriate understandings to enable parents to help children learn quantitative methods as they grow up, including recognition of how quantitative skills emerge (and can be encouraged) in young children. Involves attitudes as well as skills, dispositions as well as abilities. Cultural Literacy. The goal of many mathematicians for the common man-- to recognize the contributions of mathematics just as they appreciate the accomplishments of writers, musicians, and artists. Reflected in the many "Math for Poets" courses on college campuses, and in the popularizations of mathematics by writers such as Ivars Peterson. Rarely thought about in the K-12 system. Language of Science. Akin to the "prepare for AP" focus but tied more closely to traditional scientific applications and (in recent years) broadened to include statistics and combinatorics that are increasingly important in the life sciences. Focuses more on advanced skills than than the NCTM Standards which are in other respects, very similar. NCTM Standards. A vigorous, comprehensive curriculum in mathematics that takes three years for the average student and would leave them well prepared for further study in any career. In effect, a proposed high level alternative to the K-8 core that is now the de facto low standard for schools and employers. Stresses mathematics at the expense of less commonly recognized aspects of QL. Quantitative Practice. The substance of Peter Denning's essay in WNC, stressing the importance of practices which can be learned but which cannot be accurately described. The burden shifts away from "literacy," book learning, and classroom instruction to apprenticeship environments in which mathematics is used and learned by use but perhaps never explicitly exhibited in words and symbols. Mathematical Modeling. The substance of Henry Pollak's essay in WNC; a cycle of interaction between real-world issues and abstract reasoning akin to the process of hypothesis-building and testing in science. Here QL includes all aspects of the cycle, from mathematicizing the problem to analyzing the mathematics, from collecting data to verifying (or refuting) predictions of the model. ================= Ted fiske 1. For the last ten years, national and international studies of literacy have distinguished between verbal, quantitative, and document literacy -- the latter being about comprehending data presented in charts, maps and other graphic forms. I wonder what you as a writer infer (or assume) about the literacy skills of your readers. Are people more likely to understand a complex idea if it is presented in verbal, quantitative, or document (graphical) form? Are there any widespread "illiteracies" that frustrate your effectiveness as a writer? 2. As you know, many critics have faulted higher education for failure to ensure that all graduates are well-equipped in basic knowledge and skills. Based on your experience as an observer and analyst of higher education, how important do you think quantitative literacy is in the priorities of colleges and universities today? Is there any consensus on the nature and level of QL that a college degree should represent? 3. Many public universities (e.g., CUNY) have come under attack for the extent of remedial work offered to undergraduate students. Indeed, on many campuses the entire QL effort is devoted to helping students overcome deficiencies in preparation for college. Should numeracy represent more than remediation? If so, how much more? ======================= To: CJShroll@aol.com 1. It often seems that people representing the world of work tend to harbor a different conception of what's important for high school graduates than do those of us who work in higher education. What's your vision of the nature and level of quantitative literacy that you would like every high school graduate to possess? Does this vision differ in any significant way from what colleges expect? 2. Is quantative literacy the same as mathematical literacy? Are there aspect of one that are not really essential for the other? Are they both equally important for students preparing for the modern world of work? 3. National and international studies of literacy currently distinguish between verbal, quantitative, and document literacy --the latter being about comprehending data presented in charts, maps and other graphic forms. I wonder what you have observed about the literacy skills of both new and experienced workers. Are employees more likely to understand a complex idea if it is presented in verbal, quantitative, or document (graphical) form? Are there any widespread "illiteracies" that frustrate employers who are trying to achieve high-performance standards? To: "suzanne m. wilson" Subject: Re: Quantitative Literacy 1. Do the terms "mathematical literacy" and "quantitative literacy" mean different things to you? Are there any significant differences between what tends to be taught in school mathematics and what you would expect of a numerate adult? 2. As you know, mathematics is often called a "critical filter" that blocks students with weak school education from rewarding careers. Some fault the pristine formalism of mathematics which poses seemingly insurmountable hurdles for many students; many now argue to replace formalism with a practical, context-rich quantitative literacy. Just how important is it for all students to master formal mathematics? Is context-rich QL a more reasonable expectation? 3. The QL skills of high school graduates and college students covers the full spectrum of school mathematics from arithmetic to advanced algebra. Consequently, colleges generally do not set very rigorous QL standards for graduation. Indeed, college graduation requirements for QL are generally lower than those set forth by NCTM as the goal for all high school graduates. Should colleges stiffen their graduation requirements? Should they require college-level QL for all students? Would this imply raising (or imposing) rigorous QL entrance requirements? 4. Mathematics is a subject that students love to hate. In school, many drop out as soon as it becomes an elective; in college many take mathematics only if it is required and even then often put it off as long as possible. What should change--the subject (perhaps from mathematics to QL) or the approach? Can an emphasis on QL help resolve this age-old dilemma? Evelyn: The University of Colorado has a quantitative skills requirement for graduation, which can be accomplished with individualized "math modules;" I am not sure if any "outcomes assessment" has been done with this requirement. If you are intersted, I can pursue this further with our office of institutional research. If you want to see those courses you can do so in the catalogue on our web page: www.weber.edu.

Peggy Skinner: So, with that introduction, let me give a few thoughts to you. It is interesting that I recently was on a hiring committee for Bush to add an upper school math teacher. My questions in every interview included a basic one about the nature of a quantitative literate individual. What does a student need to be literate in math? I asked. (only one person out of 8 had thought about that issue before coming to the interview). I followed that a series of questions about conversations that they had had with science teachers to attempt to make math usable outside of the math classroom. We do a national search for our positions at Bush and interviewed about 8 of the best candidates. Only one of the 8 addressed the question of literacy. With reference to your specific questions: