What is Most Worth Knowing in Mathematics? Towards a Relevant Secondary School Mathematics Curriculum for All Students by John Knudson-Martin, Teacher Licensure Coordinator, Science and Mathematics Education Department,Oregon State University, Corvallis, Oregon
Introduction
L
IKE MOST IMPORTANT QUESTIONS,
“What is most worth knowing in mathematics?” cannot be answered without a context. The context for this question involves the needs of the persons who are learning the mathematics. The answer to this question, therefore, is different for a priest, lawyer, carpenter, nurse, farm worker, or geologist. Similarly, designing a relevant secondary mathematics curriculum depends on the needs and aspirations of the secondary school students. High school students who aspire to different careers require a math curriculum that supports their varied career goals. In the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics (NCTM), 2000), NCTM agrees with this assertion and proposes that, “To ensure that students will have a wide range of career and educational choices, the secondary school mathematics program must be broad and deep” (pg. 287). They further propose that, “All students are expected to study mathematics each of the four years that they are enrolled in high school, whether they plan to pursue the further study of mathematics, to enter the workforce, or to pursue other postsecondary education.” (pg. 288) These proposals point to a rigorous mathematics curriculum that supports the varied career paths that high school students pursue when they leave school. However, the curriculum taught in most U. S. high schools, containing the traditional Algebra 1, Geometry, Algebra 2, Pre-Calculus, Calculus sequence, is primarily designed to support students pursuing the study of science and engineering. Certainly, this curriculum is important for a subset of the students. But, not all students lean toward these disciplines; yet, all students need a strong mathematics foundation. The world simply relies on mathematics in a broad range of disciplines. The questions then become: ■ How well does this traditional curriculum support the future mathematical needs of all high
The Oregon Mathematics Teacher ● October/November, 2008
school students? ■ How did this curriculum evolve into what it is today? ■ Are there better ways to organize the high school mathematics curriculum to better serve the diversity of students? The demographics of relevance: Who is served by the traditional high school mathematics curriculum? Each year, the U. S. Department of Education compiles data on the major areas of study of college graduates. Table 1 (see next page) presents these data for the 2003-4 academic year, grouped by majors requiring and not requiring calculus (Department of Education, 2005). These data are organized according to the Classification of Instructional Programs categories. This classification system was designed to standardize the reporting of areas of college study so that comparisons between institutions, states, and countries can be made. Most of these fields of study are easily sorted by whether or not they require calculus (e.g. Engineering, Psychology). Three fields of study, however, required further examination. These fields were Health profession and related clinical sciences, Multidisciplinary/interdisciplinary studies, and Transportation. The undergraduate majors included within the Health professions category are nursing and health technician-related majors. A survey of university catalogs showed that uniformly calculus was not required for these majors. The Multidisciplinary studies category included a variety of integrated majors (e.g., religious studies, mathematics and computer science). These majors were assessed individually and included in the table as Multidisciplinary majors requiring and not requiring calculus. The Transportation category also included a variety of majors, the majors were again assessed individually, and included in the table as Transportation majors requiring and not requiring calculus. From Table 1, 41.1% of the college graduates ma-
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jored in a field of study requiring calculus. Among these students, 53.4% majored in business. A survey of mathematics requirements among major universities showed that business majors are usually required to take only one course in calculus. This course is designed as a survey of principles of differential and integral calculus. Thus, only the math, science, and engineering majors students pursue a substantial study of calculus. In 2004, these majors made up 19.1% of the total college graduates (Department of Education , 2005). Clearly, a high school mathematics curriculum that prepares students to study calculus is serving a minority of college graduates. When considering all high school graduates, the percentage of students served by the traditional curriculum becomes smaller. According to Department of Education statistics, 67% of high school graduates matriculate to college (Department of Education, 2003a). Among those students who do go to college, approximately 50% graduate (Department of Education, 2003b). These results show that only 14% of high school graduates go on to complete a field of study that requires calculus. Reducing this information to the math, science and engineering majors only, just 6% of high school graduates complete a college major requiring the substantial use of calculus. The conclusion from these numbers is clear – for the large majority of students in high school, preparing for the study of calculus is largely irrelevant. The path to the current curriculum Twice in the last half century, calls from the federal government were made for an increase in the rigor of the mathematics curriculum in our secondary schools. First, in response to the launch of Sputnik in 1957, U.S. leaders called for more scientists to be trained to help win the arms and
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Table 1. Number and Percentage of College graduates in 2004 in majors requiring and not requiring Calculus. Fields with majors that require Calculus
No. of graduates
Agriculture and natural resources
22,835
Architecture and related services Biological and biomedical sciences Construction trades Engineering Engineering technologies Mathematics and statistics Mechanics and repair technologies Military technologies Physical sciences and science technologies Precision production Computer and information sciences Business Multi/interdisciplinary studies requiring Calculus Transportation majors requiring calculus Subtotal: Percentage of total: Fields with Majors that do not use Calculus Area, ethnic, cultural, and gender studies Communications, journalism, and related programs Communications technologies Education English language and literature/letters Family and consumer sciences Foreign languages, literatures, and linguistics Legal professions and studies Liberal arts and sciences, general studies, and humanities Library science Parks, recreation, leisure and fitness studies Philosophy and religious studies Psychology Public administration and social service professions Security and protective services Social sciences and history Theology and religious vocations Visual and performing arts Health professions and related clinical sciences Multi/interdisciplinary studies not requiring Calculus Transportation majors not requiring Calculus
8,838 61,509 119 63,558 14,391 13,327 159 10 17,983 61 59,488 307,149 5,249 1,026 575,702 41.1% No. of graduates 7,181 70,968 2,034 106,278 53,984 19,172 17,754 2,841 42,106
Subtotal:
823,660
Percentage of Total: Total:
72 22,164 11,152 82,098 20,552 28,175 150,357 8,126 77,181 73,934 23,913 3,618
58.9% 1,399,362
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Most Worth Knowing
space race with the Soviets (Bruner, 1963). When, in the 1980s, international tests showed that the mathematics ability of students in U. S. schools were lagging behind the ability of students in other developed countries (National Center for Educational Statistics, 1985), education leaders again called for an increased focus on science and mathematics in our schools (United States Department of Education, 1983).
Fundamental Strand Algebra I (8th Grade) Geometry Algebra II Pre-Calculus
College Algebra/Pre-Statistics
Calculus
Statistics
Calculus-based College Preparatory Strand
Social Science College Preparatory Strand
Each time education and political leaders called for more focus on science and mathematics, the amount of mathematics required of our students has increased. In many of today’s schools Algebra I, previously the typical college preparatory 9th grade mathematics class, has moved to the middle school. Many seventh and eighth graders now commonly take geometry and some complete Algebra II before reaching high school. Pre-calculus and calculus courses have become the major college preparatory courses for juniors and seniors in high school. Yet, most juniors and seniors, even the college bound students, are not likely to ever use the content of these classes. Most do not plan to study advanced mathematics in college nor do they plan to pursue careers where calculus is required. The laudable goal of preparing students to fill the country’s scientific and technological career needs has largely succeeded in transforming the secondary school curriculum to what it is today. However, the unintended consequence of this effort has been to create a curriculum that does not support the mathematical needs of the majority of students graduating from high school. Now is the time to consider how to revise the secondary mathematics curriculum to continue to prepare budding scientists while also providing a relevant and rigorous mathematics education to the majority of students. A dual-strand college-preparatory curriculum Revising the secondary school mathematics curriculum to support all students planning to pursue college study can be accomplished by offering course choices for juniors and seniors in high school. The
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Figure 1. Dual strand college-preparatory high school mathematics curriculum
mathematics taught in Algebra I, Geometry, and Algebra II cover concepts and skills important for all students. These courses make up the Fundamental Strand of the proposed curriculum and should be taken by all students. The content in these courses has many real world applications, helps students learn reasoning and problem solving skills, and prepares students for the study of both calculus and other advanced mathematics. It seems reasonable, however, to split the high school mathematics curriculum into two strands after Algebra II (see Figure 1). Calculus-based college-preparatory strand A Calculus-Based College Preparatory Strand does not need to be created – it already exists in the courses and curriculum now taught in our high schools. This curriculum needs to be liberated - released from the constraint of having to be a curriculum for every student. Without such constraints, teachers will more effectively and efficiently serve the needs of students pursuing the study of science and engineering with more direct attention to the needs of calculus bound students. Social sciences college-preparatory strand Over the last 50 years, much thought and effort has gone into organizing and refining a secondary school mathematics curriculum that supports the study of science. Seemingly, students who pursue non-science majors in college are assumed to be prepared for their college study by taking a portion of the curriculum designed for preparing the science majors. This direction is not effective because it over-educates these students in a mathematics they will Continued...
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not use in their future careers and college study. A strand of high school mathematics that is designed to support the mathematical needs of this majority of college bound students needs to focus these students on topics relevant to the future pursuits rather than topics of science and engineering. The study of statistics, which has relevance to students majoring in the social sciences, is an appropriate capstone course for this strand. To prepare students for the study of statistics they need to take a course that covers probability, functions, problem solving, and mathematical modeling. This curriculum is similar to what is now offered in many college and universities and called College Algebra (e.g., Small, 2004). At the college level this curriculum is meant to provide undergraduates with the mathematical knowledge they need as college trained professionals. At the high school level, this curriculum would serve a similar role while preparing students for a rigorous course in statistics. Combining the Social Sciences College-Preparatory Strand with the Algebra 1, Geometry and Algebra 2 of the Fundamental Strand, provides students not planning to pursue careers in science and engineering with the rigorous, four-year curriculum the National Council of Teachers of Mathematics (NCTM, 2000) has recommended. What is now needed is for the College Algebra/Pre-Statistics course to be designed, for colleges and universities to recognize it along with the Social Sciences College-Preparatory Strand as complete and appropriate preparation for college study, and for high schools to put this curriculum in place. A multi-strand curriculum and de-tracking The Multi-Strand Curriculum proposed here may call up images of tracking curricula that in many schools have denied access to college preparatory opportunities for entire groups of students (Oakes, 2005). This proposal is not tracking and it supports the college aspirations of more students than does a traditional, calculus-based mathematics curriculum. In the proposed Multi-Strand curriculum, students take the same mathematics courses through the 10th grade and then choose a college preparatory mathematics strand that matches their career plans. This model is consistent with the philosophy of detracking outlined by Oakes (2005) and similar to the detracked curriculum implemented in the Rockville Centre School District in New York where students are taught a single college-preparatory curriculum through the 10th grade and then allowed to choose
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among different college-preparatory options in their junior and senior years (C. C. Burris, Principal, Southside High School, personal communication, March 24, 2008). Conclusion Jerome Bruner’s The Process of Education (1963) was instrumental in changing the course of curriculum design in this country. In his introduction, Bruner invoked the words of Benjamin Franklin to frame how a school curriculum should be organized. Franklin wrote in Proposals Relating to the Education of Youth in Pensilvania (sic) (1749), “As to their studies, it would be well if they could be taught every Thing that is useful, and every Thing that is ornamental: But Art is long, and their Time is short. It is therefore propos’d that they learn those Things that are likely to be most useful and most ornamental. Regard being had to the several Professions for which they are intended.”
Franklin’s words seem no less true today than they were at the founding of public education in Philadelphia and Bruner was right to frame curriculum design with Franklin’s words. Art is indeed long these days and students have a limited time to consume the knowledge that prepares them for college and careers. Franklin’s words, “Regard being had to the several Professions for which they are intended” remind us that secondary mathematics education should be a relevant pursuit that supports the careers to which students aspire. And Franklin’s inclusion of the most ornamental suggest a link to the grand mathematical theorems and concepts that have a place in the school curriculum. The current curriculum, with its emphasis on engineering/scientific mathematics, has served needs of some students and undoubtedly increased the production of scientists from the colleges. It is now time to consider what mathematics is most worth knowing for all of the students attending society’s schools and heed the recommendations of the NCTM to revise the mathematics curriculum to provide a rigorous and relevant four-year curriculum for all.
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References Bruner, J. S. (1963). The Process of Education. Cambridge, MA: Harvard University Press. Franklin, B. (1749). Proposals Relating to the Education of Youth in Pensilvania. Retrieved from University of Pennsylvania electronic archives at: http://www.archives.upenn.edu/primdocs/1749proposals.html. Miller, J. (1990). Whatever Happened to the New Math? American Heritage, 41(8: 76-82. National Center for Educational Statistics, (1985). Report on the Second International Assessment of Education Study. Washington, D. C.: United States Department of Education. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA.: Author. Oakes, J. (2005). Keeping Track: How Schools Structure Inequality (2nd ed.). New Haven, NJ: Yale University Press. Schoenfeld, A. H. (2002). Making Mathematics Work for All Children Issues of Standards, Testing, and Equity. Educational Researcher, 31(1), 13-25. Small, D. (2004). (5th ed.). Boston: Mac Graw-Hill. United States Department of Education, (1983). A Nation at Risk, the Imperative for Educational Reform. Washington, D. C.: Author. United States Department of Education, Institute of Educational Sciences. (2003a). “Graduation and postsecondary participation rates of recent high school students, by selected high school characteristics: 1999 -2000.” [Data file]. Available from National Institute for Educational Statistics Website, http://nces.ed.gov/programs/digest/ d03/tables/dt187.asp. United States Department of Education, Institute of Educational Sciences. (2003b). “Average Graduation and Transfer-out Rates for Full-time, First-time Students in Title IV Institutions in 1997, Initially Enrolled in 4-year, by Sector and State: 2003” [Data file]. Available from National Institute for Educational Statistics Website, http:// nces.ed.gov/das/library/tables_listings/show_nedrc.asp?rt=p&tableID=2490. United States Department of Education, Institute of Educational Sciences. (2005). “Bachelor’s Degrees Conferred by Title IV Institutions, by Race/Ethnicity, Field of Study, and Gender: United States, Academic Year 2003-04” [Data file]. Available from National Institute for Educational Statistics Website, http://nces.ed.gov/das/library/ tables_listings/show_nedrc.asp?rt=p&tableID=2077.
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