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Humanistic Mathematics Project: General Education and the Role of Axiomatic Thinking and Proof
What Happened Historically to Form the Current System
General Education is a concept and system used in the U.S. colleges and universities, according to which students have proposed a set of courses that provide broad overviews of many different academic disciplines. The focus of General Education is on providing the basic or introductory knowledge in languages, mathematics, sciences, and the arts and humanities. This system began to develop in the 1880s when James McCosh proposed a distribution system at Princeton University to change the focus on specific courses (Katz 2). Courses in Princeton University became less narrow, and obligatory and disciplinary general courses became the priority for educating students (Katz 2). The development of the General Education program was caused by the necessity to address the changes in society.
World War I changed the political situation in the United States. Columbia University reacted to the changes while proposing broad liberal arts courses aimed to educate undergraduates with a focus on the highly structured program (Katz 3). World War II brought about significant changes in higher education associated with changing social demands. After World War II, Harvard developed the Harvard Red Book in which the principles of General Education in the United States were discussed. The main principle stated in the book pointed at the necessity to provide undergraduates with the vision or taste of different disciplines in a broad manner (Katz 3). These ideas were also associated with the need to absorb the new influx of students coming from a more varied educational background after World War II, when returning soldiers began to enter colleges actively (Josephson par. 5). Thus, in 1949, Harvard chose to propose three to five courses according to the General Education program to provide students with the cultural foundations for life in American society (Josephson par. 5). The political and social realities of the 1950s and 1960s supported the idea that the U.S. society needed general education in contrast to highly specialized education.
In the 1970s, general-education courses were revised to build the curriculum around a range of abstract concepts like moral reasoning , quantitative reasoning , and social analysis (Katz 3). Even though the list of disciplines included in the General Education programs changed over time, mathematics was a key component of different General Education programs adopted in the U.S. higher schools. According to Brint and the group of researchers, such approaches as the core distribution areas , traditional liberal arts , cultures and ethics , and civic/utilitarian models were followed in the U.S. colleges and universities during the 20th-21st centuries (Brint et al. 605). Although various models represented the basic idea of General Education, different higher schools chose their approach to developing the general-education mathematics course appropriate for the college requirements.
Principles of General Education influenced the higher schools vision of educating students regarding the fundamental knowledge and basic disciplines. The California Community College system also follows principles of General Education and proposes the Intersegmental General Education Transfer Curriculum (IGETC) for California community college students oriented to transfer to the University of California and California State University ( California Community Colleges Chancellors Office ). Students can transfer after completing the three-unit mathematics course associated with Mathematical Concepts and Quantitative Reasoning Area ( California Community Colleges Chancellors Office ). From this perspective, the General Education programs created the base for many college programs, including the courses developed for the California Community College system.
What Is the Current System
The California Community College system proposes the majority of obligatory General Education courses in English and mathematics for sophomores and several General Education courses for seniors. Thus, the General Education courses in California colleges are divided into several subject areas, including courses in English, mathematics, science, and the arts and humanities ( California Community Colleges Chancellors Office ). Besides, the current California Community College system implements the principles of General Education while proposing the IGETC for California community college students and providing the one-semester undergraduate mathematics course for all students, regardless of the major ( California Community Colleges Chancellors Office ). The IGETC can be discussed as more appropriate for students who specialize in non-science subject areas because of the broadness of proposed mathematics and science courses. However, despite the effectiveness of the General Education programs, there are still challenges and barriers associated with completing the mathematics requirements for transferring within the California Community College system and to the University of California or California State University.
The problem is in the fact that the General Education mathematics courses developed for the California Community College system should address the issue of applicants and students decreasing knowledge rates and skill levels. Thus, about 50% of California community college sophomores are discussed as having the below college-level knowledge in mathematics (California Community Colleges 25). California community colleges respond to this issue while proposing the General Education mathematics courses that include the topics learned in high school and while developing math workshops to refresh students knowledge ( California Community Colleges Chancellors Office ). As a result, more than 70% of Black and Latino students become placed in the lowest General Education mathematics courses to improve their academic performance, as it is found with the help of regular assessments (California Community Colleges 12). Therefore, it is possible to state that the General Education mathematics courses should be more adapted to address not only disparities in knowledge and skills but also ethnic differences.
Current mathematics courses proposed according to the General Education program and the IGETC cannot be discussed as effective enough to meet the needs of the California students. Despite proposing the IGETC, the majority of students have only a 10% probability of attempting transfer-level mathematics (California Community Colleges 27). Therefore, a significant proportion of students starts their college education in need of additional basic skills education in English, math, or both (California Community Colleges 27). In this case, the current system followed in the California community colleges can be discussed as only partially addressing the principles of General Education and the needs of students in the state. More attention should be paid to developing the improved variant of the mathematics course effective to meet general and transferring requirements.
Axiomatic Thinking and Proof
Although the focus on axiomatic thinking and proof is typical for upper-level mathematics courses, these topics should be included in the improved one-semester mathematics course for undergraduates specializing in a non-science major in California colleges. The reason is that axiomatic thinking and proof are the main steps to developing students critical thinking and problem-solving approaches. Several specific module topics can be proposed to be integrated into the course.
The introductory topic is the history of axiomatic thinking and proof. While focusing on learning classical approaches to studying mathematics and resolving math problems, it is necessary to discuss the Greeks axiomatic approach and unique methods of proof (Knapp 1). This first step involves contextualizing the concepts of mathematics according to historical, philosophical, and cultural aspects.
The second topic can discuss the principles of the axiomatic system and the approach to developing theorems. This step is based on focusing on the axiomatic system as the fundament for developing theorems in mathematics. Students should know what aspects are selected by mathematicians to develop and prove their theorems (Knapp 2). Besides, students receive the opportunity to refer to elementary mathematical discoveries necessary for the further study of mathematics.
The third topic includes the focus on the methods of proof and abstract mathematical structures. Studying the logic with references to axiomatic systems and theorems, students develop their abilities to think logically and draw inferences. Furthermore, it is a necessary condition to develop basic mathematical thinking with references to such abstract mathematical structures as axioms, theorems, or classes. Proving is one of the basic skills for students regardless of the subject area. Skills in proving are necessary to explain, persuade, synthesize the gained knowledge, and generate new knowledge (Knapp 1). Developing their skills in axiomatic thinking and proving, students can read and understand the mathematicians and researchers conclusions and focus on logical implications. The proposed topics can be discussed as effective to address the students needs in developing their abilities to prove and think critically.
Works Cited
Brint, Steven, Kristopher Proctor, Scott Patrick Murphy, Lori Turk-Bicakci, and Robert Hanneman. General Education Models: Continuity and Change in the U.S. Undergraduate Curriculum, 1975 2000 . The Journal of Higher Education 80.6 (2009): 605-642. Print.
California Community Colleges. System Strategic Plan. 2013. PDF file. Web.
California Community Colleges Chancellors Office. 2014. Web.
Josephson, Edward. Before the Core: The History of General Education at Harvard. 1978. Web.
Katz, Stanley. Liberal Education on the Ropes. 2005. PDF file. Web.
Knapp, Jessica. Learning to Prove in Order to Prove to Learn. 2005. PDF file. Web.
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