* Interviews on the subject of quantitative literacy (QL)--its
importance for society, its role in education, and its impact on economic
and social institutions. Comments, corrections, and additions are welcome
by e-mail to**Lynn A. Steen.* (QL Home Page)

Jerry Bentley, University of Hawaii Robert Bernhardt, East Carolina Univ. Angelo Collins, Vanderbilt University Margaret Cozzens, U. Colorado, Denver Ted Fiske, Education Writer Michele Forman, Middlebury High School Rick Gillman, Valparaiso University Aimee Guidera, Nat'l Alliance of Business Evelyn Hu-DeHart, U. Colorado, Boulder |
Richard Millman, Knox College Pamela Paulson, Minn Ctr for Arts Educ. June Phillips, Weber State Univ. Senta Raizen, Nat'l Ctr Impr. Science Educ. C.J. Shroll, Nat'l Coal. for Adv. Mfg Peggy Skinner, The Bush School Elizabeth Stage, New Standards Project Suzanne Wilson, Michigan State Univ. |

§ *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
sciences in high school reflect the increased quantification of these
disciplines? In particular, do these courses do much to enhance students'
nascent QL skills?*

Actually, there has been a fairly sharp and clearly detectable 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 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--for example, by way 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, so if you want a reasonable understanding of the world and the way it works, you have to take into account a lot of different considerations.

§ *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 of 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 their students' actual level of numeracy.

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 expectation 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") seem 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 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? Or 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 mathematics or logic requirements, which are not necessarily the same as QL requirements. Personally, I wouldn't doubt that most 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.

§ * 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 thoroughly, 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 economics course)
they are more likely to be able to use it later. Might this suggest that
QL is better taught in courses like economics or history (even if it
requires smuggling it in) 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 a 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. (Top)

§ *What is an appropriate QL requirement for college
graduates? How should it differ from high-standards (e.g., NCTM-like) QL
expectations of high school graduates?*

Appropriate QL requirements of college graduates was a contested topic when I served on the QL Subcommittee of CUPM, as well as how college QL requirements should differ from high school QL requirements. We never settled the argument, and I strongly suspect that it cannot be settled to everyone's satisfaction. Even if you could decide this issue at some moment in time, the requirements would change with time and have to be redefined every five years or so. We are shooting at both an elusive and a moving target. Think, for example, of how much the definitions of computer literacy have changed over the years.

To me, QL is the amount of quantitative knowledge that a graduate needs to be a functioning and contributing member of contemporary society. For high school graduates, I would guess that this means the mathematical and statistical skills needed to read a newspaper intelligently. For college graduates, one would require deeper knowledge.

In college, a new wrinkle arises. One can ask what constitutes a quantitatively literate major in English, psychology, history, biology, etc. This is the approach that the MAA QL Subcommittee took: let each department define its own level of QL for its own majors. Then the question of how to attain those goals arose. Our committee settled on some foundation courses, probably taught by the Mathematics Department, followed by requirements within (or related to) the major. This structure is very similar to writing-across-the-curriculum programs.

I like this approach, since it speads the responsibilty for QL throughout the institution. But that is also its weakness, since one has trouble interesting many faculty in other departments in this idea. And the poor mathematics department is stuck trying to create and staff the foundation courses--which generally requires upper level faculty because these courses are not straight-forward to teach. It is a tough sell.

§ *To what extent is QL part of mathematics and to what
extent does it differ? Whose responsibility is it?*

To me QL is *all* mathematics, but many others strongly disagree. If
you use the structure indicated above, then foundation courses belong to
the mathematics department, unless an institution has a strong engineering
program that wants to to take on some of this duty. (That's politically
unlikely; they may do it for their own majors only.) Then the
follow-up courses are each department's responsibility.

Of course, certain mathematics or statistics courses could be used (or created) to service several departments; these types of forces are always at work in an institution. I would recommend avoiding the question of whether QL is mathematics or not altogether, because even if you could answer it, it would not help you very much. Certainly mathematics and statistics have strong roles to play in any viable QL program. That is enough. (Top)

§ *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 certain 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--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.

§ * Most people seem to equate QL with a functional ability
to use mathematics and ML with a familiarity with mathematical tools. But
you say just the reverse: "To me, the use of the tools is mathematical
literacy, while the tools and thinking together constitute quantitative
literacy." Could you elaborate on this point?*

It is just a question of terminology; in fact I originally had the names and descriptions reversed. It seems to me that some see being able to reach correct answers to problems as one goal for mathematics education while others see being able to think, reason and argue using quantitative ideas as a different purpose for mathematics.

§ *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*that focuses on being able to
function in the work place, and the *student-centered ideal*that
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 cliché, 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?*

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.

§ *I'm sure you are right that it is easy to overlook things
that are distributed across the curriculum. But "writing across the
curriculum" has enjoyed considerable success because nearly everyone
values writing enough so that it is not easily overlooked. Might not the
same happen with QL?*

I am not convinced that writing across the curriculum has had the impact that its proponents claim. (Top)

§ * How important is quantitative literacy in the priorities
of colleges and universities today? *

Quantitative literacy, especially the area that includes data analysis, is so important that it is becoming a required course related to each major, much like writing relative to one's major came to be stressed a decade ago. We now have a component of our core that requires each student to have these QL skills in addition to a university-wide mathematics requirement.

§ *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)?*

I personally view quantitative literacy as different from mathematical literacy. As I see it, QL includes working with data (not just numbers) in meaningful ways and includes an analysis component that most mathematics courses neither contain nor care about. Statistics is a big piece of QL but not the whole of it. Many people can teach it, but not all mathematicians, nor all of any other discipline. There are, however, many competent economists, psychologists, biologists, etc., who can teach QL courses.

§ *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 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?*

I believe that QL should be like writing as a responsibility of faculty in many disciplines. The social sciences seem to care more than the natural sciences, but in this day and age virtually everyone now believes QL is important.

§ *Many public universities 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?*

QL should not represent just the remedying of deficiencies, any more than writing across the curriculum represents only a response to deficiency. The goal is writing (and QL) relative to ones major. QL should have its own distinctive character. Done right, QL should help people work with numbers--but that should not be its only goal.

§ *I gather that in your view QL, being focused on
interpretation and analysis of data, is rather different from ML
(mathematical literacy). Yet mathematics remains the critical filter for
entrance to college and to many careers. Do the demands of dealing with
data suggest that perhaps QL would be a more effective entrance screen
than mathematics?*

Although many students bring more QL to college now than in years past, I think mathematics is still the right entrance requirement, partly because QL is closely related to majors (in the social sciences) that high school students won't understand. Actually QL is probably a better screen for entrance to jobs--especially those entered right out of college (either a two- or four-year college). (Top)

§ *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?*

I do think readers understand complex ideas better if there are, in newspaper parlance, "graphics." Finding ways to illustrate ideas in a visual way is thus a much more important part of writing than it used to be. Indeed, it has gotten so that people expect more of this. We live in a culture where we are surrounded by quality graphics, and there has been a sort of revolution of rising expectations on such matters. Just ask any college admissions officer saddled with a poorly designed viewbook.

I also think people have gotten more sophisticated about certain basic elements of quantitative literacy, such as margins of error in polls. The media have been pretty good about educating people on this, so it's become part of everyday life. A good parallel might be that boys always used to learn the concept of percentage from baseball batting averages.

As far as more sophisticated statistics are concerned, I'm not sure that people know as much as they should. I never took a statistics course myself, and I regret that I did not.

§ *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?*

Lots of colleges now build something called "quantitative literacy" into their core programs, though what they mean by it varies all over the map. I think all curricular planners now pay at least lip service to equipping students with quantitative skills. That's a big change from twenty or thirty years ago. But I'm sure most students still graduate without any of the statistical skills that I wish I had.

One thing you might want to ask some experts about is the impact of "digital" thinking. School mathematics has always been organized so as to build up to the calculus as the ultimate goal -- which is to say that it looks at change and progression and how numbers relate to each other. Discrete mathematics, on the other hand, which is the mathematics underlying computers, isolates numbers and approaches quantitative thinking very differently. Because computers are so much a part of our culture, there are people who think that school mathematics will eventually have to be reorganized to emphasize discrete mathematics.

§ *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?*

Indeed, more attention should be paid to numeracy in remedial work, but I don't know how to define what's enough. Remedial education at the college level is a huge issue that mainly has to do with legislatures getting tired of paying different levels of education to teach the same subject matter.

§ *You are not the only one who suggests that statistics and
discrete mathematics perhaps deserve greater attention than calculus. But
as you know, the engine of AP Calculus, driven by strong political forces
in many states, leaves high school mathematics teachers with little choice
about where to aim their curriculum. Any thoughts about what might be
done to restore better balance?*

That's obviously a long political fight. You're talking not just about retraining teachers but about textbooks at all levels. And I have no idea what the politics is within the National Council of Teachers of Mathematics.

My suggestions would be that change should start at the top--with college-level discrete mathematics courses that then get bumped down to the high school level. Perhaps it's an area where the College Board could exercise some leadership. How about another AP math course along these lines? (Top)

§ *As the social studies have come to rely more on
quantitative data, statistical methods have became a routine part of
undergraduate programs. To what degree do history and social studies 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?*

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?
*

Although I have a limited understanding of these terms, I do think of "quantitative literacy" as being a subset of "mathematical literacy." 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 a 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 unprepared for the quantitative demands of college courses. 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?*

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 although students recognize the importance of mathematics education (a student of mine once wrote in a paper, "Without math, you can't have a life"), many avoid taking challenging mathematics courses. Only college entrance requirements drive many of my best students to take higher level courses in mathematics.

§ *As you say, many people do seem to equate QL with
arithmetic, and ML with something more. Am I correct in interpreting your
responses to mean that students need the "something more" to really
succeed in the study of history? I'm fascinated by your hints of
increased possibilities for interpreting evidence, and I'd welcome some
elaboration on this point.*

Students definitely need something more to succeed in history. Not only to make it possible to interpret evidence, but to engage in higher order reasoning. I'm not suggesting that all students should master symbolic logic, but certainly the discipline of mathematics expands reasoning abilities. I often explain to students that a historical analysis has a lot in common with a geometric proof, for example.

§ *Lots of people--especially mathematics teachers--are
puzzled by the anomaly of people believing mathematics is important yet in
large measure avoiding courses that promise significant, rigorous
education in mathematics (or statistics). You must have some insight, as
a historian, into why people's actions may not conform to their beliefs.
Can you think of any explanations that might plausibly apply to
mathematics education?*

No great insight here. But I often hear students expressing fear of mathematics, especially at higher levels, and a deeply embedded feeling that they are not "good at mathematics." Very few students say this about other subjects, especially the humanities. In the humanities (including history), most students seem to feel that they can succeed, given enough time and support. They may not enjoy the study of the humanities, but they believe they can be successful. Many approach mathematics, however, as something based on innate ability. I don't know where this comes from. (Perhaps from elementary school teachers, some of whom might feel insecure in teaching mathematics?) (Top)

§ *What is an appropriate QL requirement for college
graduates? How should it differ from high-standards (e.g., NCTM-like) QL
expectations of high school graduates?*

Let me begin by saying that I believe it is the fundamental responsibility of K-12 education to prepare students to be functional citizens of this country. At 18, a person can vote, pay taxes, join the army, marry, incur personal debt. Preparing a young person to do these things cannot be a college activity because not only is it too late, but not everyone goes to college. I do believe that the NCTM standards begin to move towards providing the quantitative components of this requirement although I think that statistical ideas are extremely shortchanged.

What does QL mean at a University then? It seems obvious to me that it fits naturally into a liberal education as much as literature and writing and science do: its purpose is not so much functional (read this as life skills) as it is to provide depth, integration, context, analysis, and creative impulses.

I realize that this is probably a minority opinion, but if QL at college simply means to teach skills that a student needs to cope with the world, we are reinforcing a very poor image of our secondary schools and the state of our nation.

I teach an intuitive calculus course for our liberal arts students. I tell them up front that they will not learn anything "useful" in this course. I do tell them that they will leave with some understanding of what their friends, roommates, and significant others are doing in their engineering and science courses. I've received feedback from students who have said that it finally feels good not to feel like an idiot while sitting at a Burger King with two engineering friends. This is quantitative literacy at a university level.

§ *To what extent is QL part of mathematics and to what
extent does it differ? Whose responsibility is it?*

QL is to mathematics as writing is to English. Clearly, the skills generally attributed to QL can be learned in mathematics courses. (Of course, often they aren't and this is part of the problem.) But they can also be learned elsewhere. I have had students forcefully argue that since they learn critical analysis skills in writing courses, why should a mathematics course (e.g., a traditional high school geometry course) be required to do this?

So just as writing has justifiably moved across the curriculum, so should
QL. And this means beyond the obvious science departments. There should
be extensive discussions on campuses of what it means and how to do QL in
literature, communication, speech, foreign language, and theology courses.
Until QL saturates a university like writing has, not only will it be
unevenly gained by students, but it will also perpetuate the myth of C.P.
Snow's two cultures--particularly since, as Jerry King wrote in *The Art
of Mathematics*(Plenum, 1992), QL forms the bridge between the two.
(Top)

§ *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, or from what is emerging in
state standards?*

The business and education communities continue to speak past each other regarding the skills and knowledge required for success in the world beyond high school. While there are many notable efforts to bridge the gap between the worlds (SCANS, Skill Standards, employers sitting on academic standards-setting committees, countless surveys of employer hiring requirements), there is still a lack of clarity of what (and why) high school graduates should know and be able to do.

That is not to say that there isn't growing consensus on the need for standards and other exit requirements, but rather that the break-down occurs in the application. For example, educators have latched on to the SCANS competencies as "this is what business says it wants" and have created curricula to address such things as "team work." Nevertheless, business still finds that they are not able to hire the folks they need.

We believe this breakdown comes from educators interpreting what business says it needs, but not having a full understanding of the application and contextualization of those needs. Employers have failed to clearly articulate specific examples and applications (rather than ambiguous categories of needs) of the skills and knowledge they require of entry level workers. Murnane and Levy did a great service in their research by spelling out the "new basic skills" and defined ninth grade proficiency in mathematics (your quantitative literacy category) as a prerequisite for any solid career.

Section II of our *Formula for
Success*

One of the most often reinforced messages we hear is that the expectations of higher education and the entry-level workplace are more similar than different. As part of our "Making Academics Count" campaign to encourage employers to ask for school records, we are exploring ways to make the transcript of a student's school record more meaningful.

Recently we convened a group of human resource directors from companies of varied size and different industries along with admissions directors from two- and four-year colleges to discuss what they look for in applicants and how that information could best be communicated on student records. There was incredible consensus in the group on the need for student records to display both academic achievement--critical thinking skills, communication skills, and mastery of basic academic skills--and, in addition, what we have traditionally called employability skills (e.g., SCANS).

Interestingly, this group recommended that we stop labeling SCANS-like competencies as "employability" skills since that title conveys the inaccurate impression that these skills apply only to students who are not going on to college. Both employers and educators also wanted to have the opportunity to learn how students apply their learning, both inside and outside the classroom. The common theme that emerged was the need for a communication tool which was demand-driven and competency based.

At about the same time we convened teams of educators and employers who had taken part in teacher "externship" programs to begin to capture changes in how and what teachers teach based on their own first hand exposure to the application of knowledge and skills in the workplace. In addition to preparing a report on this meeting, we intend also to set up a web site to document examples of applications of academic topics that teachers themselves developed.

§ * 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?*

I discussed your QL questions with my colleague Karen Larsen who has led much of our work in the math/science skills arena. We really struggled with this particular question: Karen doesn't see a difference between quantitative and mathematical literacy, while I see the former being broader than the latter. Karen posits that mathematical literacy also encompasses how mathematics connects to other academic subjects and the arts. Contextualization helps students see this linkage in a way not offered by pure manipulation of numbers.

We both agree that the difference in what is taught and what is necessary to be a numerate citizen is application and contextualization of knowledge. (This is the core issue in our response to your first question as well.) The employer community absolutely supports the higher standards for all students, but also believes that there needs to be greater expectation placed on students for understanding when and how to apply the knowledge students acquire (apart from passing the high stakes assessment!). NAB and BCER are working to try to support efforts to infuse workplace examples into classrooms, to get employers themselves into classrooms and teachers and students into workplaces so the "ah ha!" can occur: "So that's why I need to understand the Pythagorean theorem!"

§ *Mathematics in the world of work is often embedded so
thoroughly in particular settings that neither supervisors nor their
employees readily recognize it as the "mathematics" they were taught in
school. Does the study of formal school mathematics prepare students to
cope with the quantitative demands of work, or might a different
context-rich QL curriculum work better?*

All stakeholders in our education system (that is, everyone in society) would be better served if students' omnipresent question "why do I need to learn this?" could be answered with a rich, accurate reply rather than with "because it is on the test." Your suggestion of a context-rich QL curriculum is vital to this pursuit, and employers must work with educators to ensure that the curriculum continues to change and keep up with the changing context of the workplace.

Our effort to document actual workplace examples of application has led us to believe that teachers are the lynch pin to the contextualization effort. We had tried interviewing and visiting several firms to get them to explain or document the academic applications used in their companies. They all had a very difficult time doing so. They simply couldn't break down their tasks into discrete academic skills and knowledge. However, teachers who have been in the workplace are able to say, "ah ha...that is an application of x." That then has an ensuing impact on how the teacher returns to the classroom and teaches that concept. We are hoping to expand this work. If you know of anyone else who is interested in working on this, please let us know! (Top)

§ *Mathematics is often called a "critical filter" that
blocks students with weak school education from rewarding careers. Should
it be the most important gatekeeper for higher education?*

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 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 all 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 suburban middle-class school district, many students do not take mathematics every year of high school. In fact many stop after algebra in the 9th or 10th 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.

§ *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?*

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 accessible, 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--at least 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.

§ *Some fault the pristine formalism of mathematics which
poses seemingly insurmountable hurdles for many students; many now
advocate 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?*

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 our mathematics is achievement as a nation and how poorly we compare to countries such as Taiwan, Singapore, Japan, and Poland. I don't really know what these stories mean except that, in comparison with other countries that outperform us on these tests, we don't expect all our students to take a lot of mathematics in high school. 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.

Whether we talk about formal mathematics or quantitative literacy, I am concerned that weak schools, poor families, and minority communities seem to produce fewer students with strong mathematics background, thereby creating an obstacle that 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? 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. (Top)

§ *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 requirement 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 about the institution's mission and purpose and what educational goals they derive from them. This procedure would yield 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.

§ * 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 (quantitative social scientists, mathematics, computer science, etc.) plus some "lay people." But again: what the institution wants as an educational goal for its 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? *

In an ideal world, quantitative literacy should be taught 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 plan to propose mathematics (or QL) across the curriculum.

§ *Granted, curricular requirements should be based on
educational goals (and college presidents have a duty to remind faculties
of that logic). However, with respect to QL the educational goals of
liberal arts colleges do not differ all that much, and other four-year
degrees tend to enhance (rather than diminish) these liberal arts goals.
So is it really that unreasonable to expect consensus on the nature and
level of QL required for a liberal arts college degree?*

It may be that liberal arts colleges have similiar missions but most
can't agree on what a liberal arts education is. Moreover, the quality,
enthusiasms, and desires of students are very different at different
places. For example, compare institutions in the top group of the *U.S.
News and World Report*ratings with those in the bottom group, or
national with regional liberal arts colleges. A requirement that would be
very reasonable at one place would be impossible at another. If all
adopted similar requirements, the marketing of some colleges would be
deeply affected.

Most top liberal arts colleges require a second language. Yet if that requirement were implemented in the third or fourth tier of national liberal arts colleges, I believe there would be a drop in many institutions' enrollments. On the other hand, there are no requirements of any kind at Grinnell (except freshmen seminar) yet Grinnell is a very fine school.

This reality doesn't mean that we shouldn't try to come up with a definition of quantitative literacy which would work at national liberal arts college (of the first two tiers). I would like to see that happen and would be happy to participate in a workshop or have some of our Knox faculty be there.

§ *You seem to suggest that on many campuses educational
goals that imply a degree of QL for all students coexist uneasily with a
mathophobic faculty who hesitate to implement meaningful QL requirements.
But isn't this explanation perhaps too facile? In other areas faculty
freely introduce requirements (e.g., foreign languages) that exceed their
own comfort zone. Might it be that resistant faculty really do not
believe that QL is all that important?*

In graduate education we hear that the language requirement is still present because it was there when today's faculty received their degrees--even in these days when there is so much scientific literature in English. Second languages for undergraduates are supported for the reasons given above. But faculty remember the mathematics courses that they went through in the 60s or 70s (and maybe 80s) that were focused either on substantial stuff for engineers or on math for poets fluff that wasn't substantial.

I believe things would be different if we had imaginative courses
linking, for example, mathematics and literature or mathematics and art
(e.g., *Godel, Escher, Bach* or a more accessible version).
Have you ever read Borges story "The Aleph?" Some of his other stories
also make heavy use of mathematical metaphor. One even uses the Dupin
Indicatrix from differential geometry!

What I am saying, prompted by your good question, is that we need to have a substantial course which shows intellectual vitality of mathematics from an interdisciplinary perspective. Knowlege is multifaceted and so issues need to be looked at from a variety of viewpoints. Moreover, liberal arts colleges intentionally prepare students for the intellectual life--what is sometimes called the "life of the mind." These two conocepts, if included in such a course, would convince faculty across the campus that quantitative literacy was a fine part of a liberal arts college. (Top)

§ *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?*

Quantitative literacy sneaks into many aspects of the arts and is a key to realizing expression of what is in the mind's eye. Students gain 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 high school 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 music 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 opportunities 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 important cases electronic equipment that is 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." For
example, 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?" (Top)

§ *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?*

At Weber, general education involves two components: 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). In addition, the B.A. or B.M. degrees require two years of foreign language (or equivalent), while B.S. students must take six credit hours (over and above the general education requirements) that emphasize scientific inquiry. Nearly one hundred courses are available to fulfill this requirement, 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
unprepared for the quantitative demands of college courses. Institutions
often 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?*

We offer two remedial courses in mathematics for those not prepared for the quantitative 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 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?*

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. 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 requirement through course in natural sciences, business, and 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 agreement 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 such things as course descriptions and numbering systems. 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 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, we ended up with a general education package that had never been debated as a whole or passed through an individual 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 well 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! (Top)

§ * 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 very 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. Can an emphasis on QL help resolve this age-old
dilemma?*

QL has to be based on good understanding of a number of important mathematical concepts, although possibly not on 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 that 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?
*

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 responsibility?*

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. (Top)

§ *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?*

Business wants new employees from the educational system who can do mathematics accurately, within benchmark time periods, and frequently with the use of a calculator. All businesses need people who can perform four-function mathematics, and many need those who can perform seven-function mathematics--including both numeric quantities and currency calculations.

Usually when I hear educators talk, particularly from higher education, I hear that they also want people who have taken mathematics. However, there is a decided difference between the standards and performance level required to pass a mathematics class at the secondary level and the standards and performance level required to meet the needs of employers. The standards to pass a mathematics class are much lower than the standards to perform acceptably in the world of work.

The second important aspect of mathematics or quantitative literacy I imagine is included in the broad area of "problem solving." In an academic setting this usually means the story problems at the end of the chapter. In the world of work it means dealing with real, unpredictable, and unorganized situations where the first task is to organize the information and only then calculate to find an answer. Here too it is very possible for students to be academically successful without being very proficient at problem solving. In many cases teachers skip the story problems, sometimes because they come at the end of the chapter, but often because they are so inane (but that is another discussion). In the world of work, organizing the information is the most important aspect. The mechanical calculations are now often done with computers or calculators or electronic cash registers.

Mathematics classes would be more successful and more helpful to both students and businesses if they started with the story problems. First help students learn how to organize their thinking and then help them learn whatever mechanical calculations are needed to solve the problem. Problems presented must be current and practical--not about trains leaving New York and St. Louis. This is not about the teachers, but about the system of preparing teachers and the role they are expected to play in the educational process. They should not be "the sage on the stage" but "the guide on the side."

The expectations and requirements of business and college do differ. However, obviously, they shouldn't. Engineers and doctors still need to add and subtract the same as clerks and production workers.

§ *Is quantitative 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?*

I do not know if quantitative literacy and mathematical literacy are the same. Even after looking up the words in the dictionary the terms were not sufficiently clear. This confusion reveals a common problem in terminology and communication. My advice to business and to education when trying to communicate is, whenever possible, to provide examples. Perhaps my own education was faulty, but even as a student and teacher of mathematics, I found terms like quantitative and mathematical literacy not sufficiently clear, specific, and measurable to be discernible.

Indeed, as I became engaged with your questions, I found them perplexing. After stepping back from them for a bit, I realized the confusion is due to semantics, which is often the basis for differences (real or perceived) between business and education when it comes to a discussion of literacy.

In my work over the past several years trying to bridge the gap between education and business, I have often identified the problem as one more of communication than of substance. That is not to say that there are not significant gaps between the skills desired by an employer and the skills that a prospective employee may have. But for me the question centers on the whether employers are able to communicate their skill needs clearly and accurately to educators, and then whether educators are able--or more importantly, willing--to discern what employers want. Often educators listen politely and then do what they know how to do rather than really considering how to change.

The last and most important aspect of the relationship between education and business is one of measurement. In order for the relationship to improve, we must find common measurement for what we mean when we say words like "literacy" and "skills" and even "education." Both education and business are guided by measurement; unfortunately at this time we are often measuring different things or measuring the same thing differently. This is not to point a finger of blame at educators (or at business for that matter), but rather to call attention to a "systems" problem. A credo of many in the field of system and process improvement is: "We are not here to fix the blame, but to fix the system."

§ *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?*

The use of charts, graphs, maps, etc. are absolutely essential in the world of work and therefore should be essential in the world of education. Charts, graphs, diagrams, etc., are the evolving language of business, made necessary because of the significant problems associated with communication based only on words, written or oral. The ideal situation is when someone has the ability to communicate in many different ways including charts, graphs, words, symbols, and more.

Employees are most likely to understand a complex idea when it is presented in different ways. This is true for all people--in life as well as in work. Employers are often frustrated that many people coming to them have only learned to communicate with words. If we look a the world of work we see rapidly growing use of charts, graphs, system diagrams, process maps, and quality system documentation (e.g. ISO 9000).

The common description of "high-performance" includes the ability of employees to work together by communicating effectively. This can only be accomplished with the use of multiple communication methods including charts and graphs and the like. In today's workplace we see a great deal of diversity--cultural, regional, ethnic, nationality, etc. To imagine that all these people are using the same words and same syntax leads to disappointment and frustration.(Top)

§ *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 to calculate is 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.

I recently was on a hiring committee at my school to add an upper school mathematics teacher. In every interview I asked a basic question about what skills a citizen needs to be quantitatively literate. Only one person out of eight had even thought about that issue before coming to the interview. I followed with a series of questions about conversations that candidates had had with science teachers to attempt to make mathematics usable outside of the mathematics classroom. Few had done much of that either.

§ *I'm astonished (though I probably shouldn't be) that so
few of your candidates for the mathematics position had thought at all
about the QL needs of citizens. Do have any idea why this is so? Are the
teacher preparation programs that bad? Or do all your prospective
teachers think only about filtering students for success in
calculus?*

Since we interview experienced teachers, we had several who were veteran public school people--a few with thirty years in the system. Their responses were something along the line of not having time for communication with other departments, the need to keep mathematics isolated (not integrated), and the importance of keeping students on track for studying calculus. I asked the question to about eight finalists and really felt that none addressed the question. Even our own mathematics teachers (all excellent) don't seem to have the vocabulary to answer a question about quantitative literacy. I do think that our teacher preparation programs or other kinds of preservice training need attention.

§ *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, especially 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 have to work very hard to comprehend it. Science teachers typically have more ways to demonstrate how a principle is used than mathematics 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. (Top)

§ *What is 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 such 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 one's 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, there's an increasing need to be able to develop spreadsheet models and "what if" 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 vision of QL has very little overlap with traditional school or undergraduate mathematics.

I am ambivalent about including in the core enough mathematics to have opportunities to learn new things. On one hand, that's 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 "checkbook math," but you know that's not what I mean. 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 unfortunately it isn't.

§ *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 "Arma virumque cano," 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. "Don't wear white shoes after Labor Day or before Memorial Day") 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.

Unless we can get society (and particularly some mathematicians in California) 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. I do think that every student should be pushed and supported to achieve excellence at the highest levels in some spheres; call that mastering formalism, if you will. But I also think that every student should be given opportunities to learn well things that are useful and important as citizens, voters, and consumers.

Of course, content-rich experiences in applications of mathematics is a better goal for all students. But until it's widely valued, then it sells some students short. In my own experience, content-rich applications (studying physical chemistry) actually got me interested in mathematics. I'd skated through calculus, but was jarred into interest in mathematics when a chemistry professor invited a few students to join him in auditing linear algebra to see if it offered a "better model for the chemistry we're trying to describe." Subsequently, 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.

Unfortunately, we have a political problem here which in some places has become partisan and very mean-spirited. I console my friends who've been on the front lines in California by saying that the reason we have this 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 don't navigate these shoals successfully. We have to create a social and political climate where your innocent question isn't so loaded and the answers to it aren't so guarded. The good news, and the bad news, is that ten years ago I would have answered the question, "Formalism, no. Context-rich, yes. Next question."

§ *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?*

I'm tempted to say that I have no standing to answer this question, but in fact I think that higher education is in even worse shape than K-12--particularly to the extent that the degree or diploma is meaningless except as a measure of persistence. It's hard to imagine a major in which some quantitative literacy isn't necessary, in context, but the colleges that I know about that keep track of QL do so entirely out of context. Thus mathematics (and a bad representation of mathematics, at that) becomes yet another filter.

The California State University Entry Level Mathematics (ELM) exam is a particularly unsatisfying test of out-of-context, who-cares stuff. The UC/CSU Math Diagnostic Test is another favorite, though I have heard that it now has constructed response items and doesn't classify students on the basis of their ability to add fractions with unlike denominators, which used to be the pivotal item for getting placed into calculus. In short, I don't think that there is any consensus. In the current political climate I'd be afraid to see what a consensus would look like.

§ *Many public universities 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 it's 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 don't we send them to community college, which is cheaper?

This reminds me of a related story. A while ago, Smith College had to develop a one-semester course for students who entered with 5's on the AP BC calculus exam to help them connect the calculus that they had studied to the contexts in which Smith freshman had learned it--economic, social, etc. I loved it! I thought that the need for such a course made a wonderful point about context-free learning.(Top)

§ *Do the terms "mathematical literacy" and "quantitative
literacy" mean different things to you?*

In many ways these terms do mean different things to me, although I don't think they have to. As soon as one calls something "mathematics," it seems that mathematicians feel like they have the right to stipulate what that is. And while I think they have the right to participate in the discussion, I don't think they have the right to claim pride of place--at least not if the term is supposed to be more general than a discipline-specific idea.

Mathematics is a discipline, which means it has a discourse, community norms and expectations, and the like. One question that gets raised when you used the term "mathematical literacy" is: What is the relationship between that term and those discipline-specific features? If I am mathematically literate, can I participate in the discipline's discourse? Do I understand its norms and rules and expectations and vocabulary? As a term, quantitative literacy might get one out of that quandary. But I really don't know.

I think a lot of what I learned in statistics courses (e.g., principles of sampling and measurement, theories of population and distribution) might reasonably fall under the category of "quantitative literacy." I am not sure they are part of mathematical literacy.

The term "literacy" also makes things complicated. It can mean the ability to read and write, but sometimes what we want it to mean is the capacity to reason in certain ways. Sometimes, for me, the confusion entails not knowing what people presume "literacy" means.

§ *Are there any significant differences between what tends
to be taught in school mathematics and what you would expect of a numerate
adult?*

Absolutely. I actually don't think that much school mathematics enables adults to be numerate. Whether it is because of the content or the teaching, school teaches people to not reason, instead of to reason. I remember as a child scribbling in the margins of my elementary school assignments questions about alternative solution paths for problems. (I was really curious about mathematics.) But my teacher, like most I fear, told me there was only one way to solve the problems.

For reasons that are peculiar to mathematics, but also to education more generally, schools do not typically teach people to think. Even the language of mathematics can discourage thinking, inquiry, and reasoning. I asked a question in a mathematics class at Stanford once, and the professor responded that the answer was "obvious." Another time a mathematics professor responded in class to a question by suggesting that the solution was "trivial." For someone with no knowledge of what those terms meant within mathematics, I was silenced by these professors instead of encouraged to discuss the ideas.

A numerate adult can do many things: Read the newspaper well, interpret research and scholarship, understand checking accounts and budgets, etc. But among the most important things I would wish for students--and the adults they grow up to be--would be an ease with numbers, with mental mathematics at the grocery store, with square footage problems when building a house, with solving a problem using mathematical rules or principles. Students do not leave school with such ease.

§ *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?*

I see it as a two-part problem: More people ought to go into the field as mathematicians, embrace the formalism and understand its power and beauty. And everyone else ought to have the mathematical or quantitative capacity to reason. To address both parts of the problem, one can't drop the formalism.

Furthermore, I don't think we know enough to understand the role that learning formalism plays in the development of quantitative literacy. For instance, I acquired my knowledge of many of the principles of statistics not through conceptual discussions, but through "crunching numbers." One learns to set a volleyball well, by setting a volleyball thousands of times. There is no other way to do it. I don't think we know--in mathematics--what role the learning of formalism plays in developing conceptual understanding.

I think we often forget that there are at least two concerns running alongside one another. Learning requires motivation. Although motivation is sometimes inspired by forces beyond our control, sometimes we can actually increase the probability that someone will be motivated to learn. In schools, this often takes the form of making things "relevant" or "interesting." Part of the call for "practical, context-rich" stuff is a concern for motivation.

Related to, but separate from, motivation, is the concern for what people learn. The discussions are often so muddled that I haven't yet heard a good argument one way or the other about how to resolve the "what people should learn" (after they become motivated). Could one imagine a context-rich, formalism-free K-12 curriculum that would still get enough people inspired to join the ranks of mathematicians? I can't. As a child growing up, I loved mathematics for the formalisms, for the lack of narrative, for the systems of axioms and theorems, and for the ways in which one was asked to reason. That was its appeal to me. However, it is unclear to me whether--if one could solve the motivation problem--all students ought to have the chance to learn and think in those ways.

§ *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?*

Colleges could do that, but then they would also have to take responsibility for the fact that they have "educated" the K-12 teachers who do not know the mathematics it would take to meet those higher standards. I would rather see higher education set standards for its graduates first and create requirements and courses (in an experimental way) that were designed to enhance students' capacity to reason mathematically and quantitatively. I would be interested in ways that a college could do this so as not to punish the individual student, but rather focus the effort on the institution.

What measures of success might there be for the college or university that didn't put the responsibility simply on the shoulders of the student, but rather on the professors and the department? Mathematics and statistics departments are infamous for blaming the student: "If you did the homework, if you had a mathematical mind, if your other teachers had taught you what you need to know, ... ." I would like to see a reform that focused on the institution's and the professoriate's responsibility. Eventually, one might then move to requirements for individuals, but I don't think starting with those requirements leads to institutional change. One can still simply blame the students rather than the university curriculum or its teachers.

§ *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 don't think the content really has to change, at its core. But the teaching does. At the elementary level (and probably K-12 in general), teachers don't have a flexible knowledge that allows them to inspire and encourage students to think. At the university level, teachers might have the knowledge, but many of them lack an empathy for what it takes to learn. And sometimes they lack a sort of pedagogical flexibility that would enable the use of a broader set of instructional strategies.

The content might need to change in the sense of designing a curriculum that teaches the traditional central ideas, but in ways designed to make them interesting and appealing to learn.(Top)

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