Class of 1996! Class of 2000! Many of us aspire to be part of a “Class.” We work our way through elementary school, middle school, and finally high school. A significant amount of attention is always placed on mathematics education and how to improve student achievement. Publications such as the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics stress the importance of reforming not only the curriculum standards but also the teaching principle. The NCTM calls for teachers’ knowledge and understanding of mathematics, students as learners, and pedagogical strategies.
Upon high school graduation, some enter the work force right away, and others enroll in some form of post-secondary education – vocational schools, community colleges, liberal arts universities, comprehensive universities, or research universities. There is a push in college mathematics, as well, to improve the deficit of student success, mathematical enrollment, and the production of mathematicians, scientists, and engineers. As a student transitions from high school to college, how different is their learning development? What sudden or drastic changes in their cognitive psyche could there possibly be? The answer is not much, if any, so the solution that has been presented is to implement the teaching principle of the NCTM.
The teacher, however, is only one-third of the picture. The mathematics education environment consists of the student, the teacher, and the curriculum being taught. In relational terms, the student and his or her success are highly influenced by the teacher and curriculum, although not completely dependent. The teacher and curriculum can in a sense be altered or “controlled” to vary the student’s success. At the college level, the student has a greater amount of freedom in choosing the curriculum path and instructor he or she will take. The focus here is on two-year colleges or community colleges, because they play a significant role in the realm of post-secondary education. Approximately 44 percent of undergraduates in the U.S. are enrolled in two-year colleges, 25 percent of which enroll in mathematics or statistics courses. “Nearly 10 percent of U.S. students who receive a doctorate in the mathematical sciences began their studies in a two-year college” (National Research Council, 1991, p.4). As demonstrated by these numbers, it is essential to address mathematics education in the two-year college.
The courses that a college student undertakes and hopefully succeeds in will determine which degree he or she will receive. Ideally, these courses become progressively more in-depth and focused. The work “should be intriguing with a level of challenge that invites speculation” (NCTM, 2000, p.19). I was at one point in my undergraduate career an electrical engineer major. The first semester track for a bachelor’s degree included two electrical engineering introductory courses, one of which past students had labeled as the “weed-out” course. Essentially, this implied that the course curriculum was, for the first year student, very complex and rigorous so as to “weed-out” the not-up-to-par students. Through countless hours of studying and perseverance, I successfully completed the course. Regardless, I did not appreciate the pressure or the stress it placed upon me. If this was what the remaining years of my college life were going to resemble, I decided I would much rather pursue a degree of equal caliber but one that also allowed for exploration of my own interests at my own pace and that did not set out to fail me but encourage me. I, of course, was faced with resistance when I had to speak with the Dean of Engineering. He claimed that as a female minority, I had successful a future ahead of me marked by multiple employment opportunities and quickly increasing salaries; however, I was still lacking the exposure to the inspirations and motivational interests that would keep me working hard and towards that goal, i.e. I was not intrigued.
Engineering is a field that continually seeks to produce more minority and female engineers. “The number of individuals in their mid-twenties will decline until the end of this century, making it increasingly difficult to sustain present production of new engineers and scientists” (NRC, 1991, p.21). Considering this, it is important not only in the recruitment phase of getting such individuals started in these programs but also to support them and motivate them, maintaining their curiosity whilst challenging them. Mathematics and other science fields need to be concerned with this issue as well. Students not focusing on a mathematics or science field will still require some mathematics for graduation and for their daily lives, so their experience in the mathematics classroom should be just as meaningful if not more.
Only a small fraction of our population – consisting of primarily white males – complete a mathematics education that matches their potential and interests. The result is an appalling waste of human potential, denying to individuals opportunity for productive careers and to the nation the resources for economic strength (NRC, 1991, p. 20).
The Mathematics Achievement Partnership (Achieve) acknowledges the mathematics deficiency of U.S. students and asserts that student need to be more prepared mathematically not only in college but also in the work force. They emphasize that “U.S. employers cannot find enough skilled workers: A recent survey of more than one thousand employers found that more than one third of job applicants are turned away because they lack mathematical and verbal skills needed for the jobs” (Achieve, 2001, “Raising the Bar” section).
Every student at every level of education, especially post-secondary, has his or her own aspirations for the future, motivations, and values. The level of importance and interest in his or her education also varies. Individuals who apply to college enroll for numerous reasons. It is, for certain students, a family tradition or expectation to further one’s education beyond high school. Others’ enrollment is self-motivated. Of the students who do enroll, regardless of their reasons, nearly half apply to two-year colleges.
For some, geographic accessibility, low tuition and fees, or the open admission policy are key reasons… Other students are attracted by small class size, the focus on teaching and learning, and the availability of faculty office hours. Still other students need the preparatory work offered through developmental courses… Students who have attended college previously [may attend a] community college to gain necessary training for a change in career (Wood, p.101-102).
A number of students enroll in courses offered at community colleges for their personal interests, such as software training, self-defense or finance management. The majority, however, enroll with the intentions of obtaining their associates degree or transferring to a four-year college. The variation in these students’ goals requires a catering of their learning track. Students interested in a mathematics-intensive program will require a different path and curriculum focus than students in a non-mathematics-intensive program. Among those focused on a mathematical profession, some will enter into the field of education, and they too will require a variant educational track. What mathematics is taught in the classroom for this array of college students is an important issue that is shared with the classrooms of grades K-12, as demonstrated by the curriculum material created by the Core-Plus Mathematics Project, Contemporary Mathematics in Context: A Unified Approach.
The Core-Plus project, a high school reform mathematics program, includes a “three-year core curriculum for all students, plus a flexible fourth-year course that continues the preparation of students for college mathematics” (Core-Plus, 2001, “Introduction” section). Course four consists of ten units or chapters from which the instructor can choose to assign his or her students, depending on their post-secondary educational goals. In this manner, the student is provided the option not only to continue his or her mathematics education but also to explore and learn the mathematics that will suitably serve him or her in the future. The material gives the student a rationale for learning the mathematics and the motivation to learn through problems that interest him or her. A student that is more enthused and immersed in the subject matter will have a greater desire to take additional courses in that area. This is an important aspect of the curriculum, because if a student sees a purpose in the mathematics and has a curiosity for it, he or she is more likely than not to consider pursuing a career in the science and technology field. The American Mathematical Association of Two-Year Colleges (AMATYC) affirms, “The curriculum should provide the connectedness that is inherent in mathematics. Connectedness gives mathematics its power, establishes its truth, and reveals its beauty” (1993, p.8).
Similarly, the two-year college mathematics curriculum should address the retention and success of its students, based not only on their grades but also the students’ enthusiasm. A student, through hard work and dedication, can achieve academic success, but if this student lacks the interest or fails to see its significance, he or she will refrain from continuing on with let alone begin his or her studies in the field. Mathematics courses offered by the two-year college, especially the introductory courses,
should encourage many students to continue their study of Science, Mathematics, Engineering, and Technology, serving as “pumps” rather than “filters”… Course design [should accommodate] a diverse group of student who exhibit differing educational backgrounds, experiences, interests, aspirations, and learning styles. Students’ experiences with mathematics in introductory level courses are pivotal and influential in determining career choices (Wood, p.103).
Students opting to major in a mathematics-intensive program at the two-year college or as a transfer student at a four-year college will require an array of mathematics courses beyond calculus, including but not limited to differential equations, discrete mathematics, probability, and statistics. These courses should be transferable to four-year colleges so that the student is not discouraged in having to repeat the curriculum. Admittedly,
it is indeed difficult for the two-year college curriculum to meet the needs of many transfer institutions that do not agree with each other on the preferred curriculum for their native students… Students not completing an associate degree who choose to transfer courses piecemeal may find transfer more problematic, as not all courses they have completed at the two-year college will necessarily be accepted for credit towards graduation at the four-year institution (Wood, p.105).
Two-year college courses should also be of the same caliber and as comprehensive as the core courses offered at four-year colleges, paving the way for a smoother transition from the two-year college. There are a number of students who, after transferring from a two-year college, find the transition to be quite difficult. Various factors contribute to this, such as increased class size, less accessibility to faculty, or increased distractions (e.g. more student organizations to get involved in). Sometimes, however, the major source of such a problematical transition is the increased intensity of the course and increased expectations. Therefore, two-year college “faculty must demand quality performance from mathematics-intensive majors, who will become the mathematicians, scientists, engineers, and economists of the future” (Wood, p.102). Returning to the issue of recruiting students into mathematics-based careers, especially females and minorities, Wood also notes “two-year colleges, with their significant minority population, are prime sources for the recruitment and preparation of a diverse work force of the mathematicians, scientists, and engineers of tomorrow” (p.103).
“Undergraduate mathematics provides a powerful platform for careers in many fields” (NRC, 1989, p.53). For the individuals pursuing studies in fields other that science, mathematics, engineering, or technology, mathematics should still be a “powerful tool not an overwhelming barrier that students must surmount to enter their chosen disciplines” (NRC, 1991, p. 25). These students, especially, should be aware of the diversified applications and presence of mathematics.
While two-year college and lower-division mathematics students are preparing for a multitude of future occupations, there exists a common core of mathematical experiences, viewpoints, concepts, and skills that should be learned by all students… Mathematics provides a language for the sciences; plays a role in art, music, and literature; is applied by economists; is used in business and manufacturing; and has had an impact on history… The reform of mathematics education in the first two years of college speaks of building mathematical power as a basic goal. In every topic and in every course, students should be discovering the usefulness of mathematics as a means to deal with the world around us. The initial courses available at college can be thought of as general education mathematics, which provides sound mathematics instruction and the incorporation of knowledge from other disciplines (Crossroads, 1995, “Pedagogy” section).
Although not all students enrolled in a two-year college will pursue an actual degree, they should still a sense of purpose in mathematics and the skills to cope self-assuredly with today’s society. On a day-to-day basis, people encounter probability/statistical polls and graphs, financial reports, computers, etc. Our daily lives, alone, require a sound understanding of basic mathematics, so regardless of a student’s purpose for taking a course(s) in mathematics, the curriculum should cater to that purpose. In the end, however, each student, math-focused and non-math-focused, should
develop critical habits of mind – to distinguish
evidence from anecdote, to recognize nonsense, to understand chance, and to
value proof… Citizens who are bombarded daily with conflicting quantitative
information need to be aware of both the power and the limitations of
mathematics (NRC, 1989, p.8-9).
A critical aspect of getting students to proactively interact with one another and with the instructor is to address their source of motivation or lack of. Students will willingly explore the material presented to them if they possess a personal interest or are alternatively motivated, e.g. to make a good grade. According to attribution theorists such as Bernard Weiner, students seek “to understand the world around them to determine the causes of success and failure. Among the most prevalent inferred causes of success and failure are ability, effort, task ease or difficulty, luck, mood, and help or hindrance from others” (Graham, 1996, p.71). The motivations of a community college student are more apparent than those of a K-12 student, because he or she is voluntarily enrolling into school. The college student has an explicit goal of earning an associates degree, transferring to a four-year college, or expanding his or her knowledge for career or personal purposes. Although the students may be motivated overall, the teacher must still draw the student into the specific mathematics course they are taking and keep them actively involved.
According the NCTM Principles and Standards of Mathematics Education, the most effective classroom teaching is demonstrated in an environment in which the teacher is a director, moderator, facilitator, and intellectual coach. The underlying theory of this approach is constructivism.
Constructivist learning is based on students' active participation in problem-solving and critical thinking regarding a learning activity, which they find relevant and engaging. They are “constructing” their own knowledge by testing ideas and approaches based on their prior knowledge and experience, applying these to a new situation, and integrating the new knowledge gained with pre-existing intellectual constructs (Briner, 1999).
NCTM recommends that teachers implement this style of instruction and that they understand the students’ level of learning - what challenges are the students likely to encounter in learning these ideas, how can the ideas be represented to teach them effectively, and how can their understanding can be assessed (NCTM, 2000, p.17). This knowledge of teaching, or pedagogy, can just as well be applied to the two-year college mathematics courses, especially for those students in lower-level mathematics. In fact, the AMATYC stated in their Guidelines for Mathematics Departments at Two-Year Colleges (1993) that “faculty should implement the various recommendations of [the NCTM standards]” (p.5). These particular students have a greater difficulty grasping the mathematics; otherwise, they would have completed the lower-level course work in high school. Various factors, such as past failures or no sense of real-life connections, contribute to these students’ disinterest or lack of motivation. One myth that they are likely to fall victim to is that “success in mathematics depends more on innate ability than on hard work” (NRC, 1991, p10). Therefore, it is essential to construct a representation of mathematics that is logical and well connected. The Mathematics Achievement Partnership (2001) also agrees “students will benefit from the chance to replace mind-numbing [work] with interesting and relevant mathematics” (“Supporting Teachers” section). In addition, providing the students with the opportunity to interact with their instructor and with one another to develop their mathematics knowledge will foster a more positive temperament towards the field of mathematics. Students that are more mathematically self-motivated can also gain from such a learning atmosphere through more thought-provoking course work. Presuming these students are more mathematically developed, they will not require as much teacher encouragement; however, the instructor will still serve as a moderator and springboard for ideas.
Much of the
ineffective teaching strategies that occur in college mathematics classrooms
are brought about by tradition and insufficient training. Once college students themselves, many
college instructors are more likely than not to have been taught through
traditional lectures. Of course, the
tendency is then to teach as one was taught.
“Only infrequently does the education of our future college teachers
provide models of appropriate instructional techniques as well as intellectually
challenging opportunities to address issues of how mathematics is taught and
learned” (NRC, 1991, p.29). For this reason, it is still important for faculty to be
“familiar with the discoveries of [educational] psychology as applied to mathematics
education… and with [the] findings in the psychology of learning and thinking”
(AMATYC, 1993, p.5).
Educational psychology courses are a requirement for educators of grades
K-12. Cognitively, there is no
significant difference between 11th and 12th grade and
the first two years of college.
Overall, the most effective modes of learning apply across the board of
adulthood. At age 18 or 35, an
individual is equally influenced and stimulated by new information and by its
discovery, but because having a student successfully and meaningfully complete
grades K-12 is so critical in our society and is seen as such a substantial
achievement, more attention is placed on the learning and thinking development
for that period. Not everyone who
completes K-12 will continue on to a post-secondary education, but for the
number of students who do, it is equally influential that they are taught in an
environment just as engaging and conducive to learning as they were in
K-12. “Student achievement at each
grade level correlate[s] positively with the quality of the teachers who taught
those students.” (NRC, 2001, p.47).
The teaching principle proposed in Professional Standards for Teaching Mathematics, can be and should be applied to the faculty of two-year colleges to nurture a more effective teacher and classroom. One aspect that may not seem as critical in terms of reform in the two-year college is the mathematical education/knowledge of the instructors, because most, if not all, two-year colleges require that their mathematics instructors have a bachelor’s degree in mathematics and a master’s degree in a similar field. Deep and detailed knowledge about mathematics, however, does not signify equal amount of knowledge in relaying that information effectively and successfully. It is a suggestion and an expectation that instructors comprehend not only the factual information but also the “the important ideas that are central to their [students’ goals], the challenges students are likely to encounter, how ideas can be represented to teach them effectively, and how students’ understanding can be assessed” (NCTM, 2000, p.17). The most influential factor in each of these implications is time. These pedagogical issues – experience in teaching, making curricular judgments, mathematical expectations, communication skills, responding to students’ questions and familiarity with technology – take considerable time to develop.
Teacher instruction is especially important for the students who are pursuing a career in the education field. Even if a student has not declared their major in education, the teacher should be aware that his or her students could potentially be future teachers.
The quality of introductory mathematics and science courses at two-year colleges will influence the knowledge and skills these future teachers take to their own classrooms… Mathematics faculty should model the pedagogy that student will later use in their classrooms and incorporate the recommendations of the NCTM Principles and Standards for School Mathematics (Wood, p.104).
“It is rare to find mathematics courses taken by prospective teachers that pay equal attention to strong mathematical content, innovative curricular materials, and awareness of what research reveals about how [students] learn mathematics” (NRC, 1991, p.28-29).
An important point to maintain in the mathematics classroom is the engagement of the students into the material. Without the students’ focus and interest, a teacher’s efforts are essentially futile. Generating student involvement is the backbone of constructivist learning; the students are proactively and cooperatively working to construct and comprehend the mathematics. Teachers need to listen as much as they speak. Numerous strategies are available to the teacher for maintaining and elevating student participation and attention. They include the teacher’s own disposition of mathematics, the use of realistic problems, multiple approaches to problems, and technology.
First and foremost, a teacher must enjoy his or her role as teacher. Students feed off of the instructor’s energy and enthusiasm. “Students’ understanding of mathematics, their ability to use it to solve problems, and their confidence in, and disposition toward, mathematics are all shaped by the teaching they encounter in [the classroom]” (NCTM, 2000, p.16-17). If a teacher does not display interest or confidence, how can he or she expect the students to exert any themselves? It is the responsibility of the teacher to spark the curiosity of the students and to then use that curiosity to get them involved in the discovery of the mathematics. As stated by the Nation Research Council (2001), “the [effective] teacher appreciates multiple perspectives and conveys to learners how knowledge is developed from the vantage point of the knower” (pp.58).
It was discussed previously that the classroom mathematics be tied into real life. This is crucial to all students, those pursuing mathematic-intensive degree and those who are not. “Well chosen tasks can pique students’ curiosity and draw them into mathematics… [They] should be intriguing, with a level of challenge that invites speculation and hard work” (NCTM, 2000, p.18-19). The importance of the material is better portrayed and better received in the context of its real life applications. What should a student care about minimizing a quadratic function? Put into the context of finding the production level that minimizes the average cost per unit, however, differentiation suddenly has an applicable appeal. A student with entrepreneurial interests would be more responsive to such a problem as opposed to being given numerous functions to differentiate and minimize or maximize. Connecting the mathematics with real experiences places relevance on the learning and will ideally appeal to the student.
Mathematics must not be presented as isolated rules and procedures. Students must have the opportunity to observe the interrelatedness of scientific and mathematical investigation and see first-hand how it connects to their lives. Students who understand the role that mathematics has played in their cultures and the contributions of their cultures to mathematics are more likely to persevere in their study of the discipline (Crossroads, 1995, “Pedagogy” section).
Creating this connectedness with real life is even more important to the two-year college student, because by enrolling himself or herself into the college, he or she is in fact preparing himself or herself for real life.
In coordination with students’ application of mathematics in real life scenarios, the problems should be solved using multiple approaches. Solving a system of equations, for instance, can be solved through standard algebraic manipulation, which is the “direct” method. Not every student may realize this a resolution method, but that is not to say they will not construct an appropriate solution. Various students may choose to use graphs, estimation, or computers. The AMATYC advocate that instructors
model the use of multiple approaches – numerical, graphical, symbolic, and verbal – to help student learn a variety of techniques for solving problems… and provide rich opportunities for students to explore complex problems… This will motivate students to go beyond the mastery of basic operations to a real understanding of how to use mathematics, the meaning of the answers, and how to interpret them (Crossroads, 1995, “Pedagogy” section).
Allowing each student to take on problems from different perspectives creates an environment of discussion and collaboration, because the students argue their approaches and respond to others’ arguments (NRC, 1989, p.61). The acquired skills of articulate oral and written communication are indispensable to the college student who will soon be entering the work force, where most individuals are required to communicate their work’s progress and status with their colleagues and supervisors by means of oral reports and written presentations. Even outside of the professional and inside the social context, it is just as important for students to be able to make and recognize a valid argument. For instance, reading up on polls or conducted surveys or watching political debates on television, students should be able decipher on their own whether or not they agree with or support the stated claims.
The use of technology is one of the multiple approaches students and teachers can take advantage of in the classroom. Technology incorporates but is not limited to calculators, computers, audio and visual media, and mathematical software. As time goes on, technology advances with increasing capabilities, improved sophistication, faster speeds, greater portability, and easier accessibility. The technology of today and that of tomorrow does and will allow for enhanced and more profound exploration than any pencil and paper can. With the quick manipulation of software such as Geometer’s Sketchpad and Mathematica, students can conjecture, test, and prove with greater ease and less frustration, and with less frustration students are more likely to advance in their explorations and constructions. Teachers should
use dynamic computer software to aid students in learning mathematics concepts and model the appropriate us of technology as tools to solve mathematical problems… Emphasis should be placed on the use of high-quality, flexible tools that enhance learning and tools that they are like to encounter in future work (Crossroads, 1995, “Pedagogy” section).
Extremely advanced software such as Mathematica can be used in numerous contexts of mathematics, from plotting points on a Cartesian plane to illustrating the three-dimensional parameterizations of a curve created by rotating spheres. The level of use is up to the user – the instructor and the student. To be able to utilize such technology, the teachers must be knowledgeable and comfortable with it, and that requires furthering the teacher’s own educational development. Aside from knowing how to use the technology, the instructor must also know how and when to meaningfully and effectively incorporate it into the curriculum. He or she must take caution to not excessively use the technology and in a sense reverse the initial intention of using it. “The use of technology within the instructional process should not require more time. In fact… it should provide the time that is needed to implement the needed reforms in mathematics education” (Crossroads, 1995, “Pedagogy” section).
Learning the essentials of the technology is a facet of teaching that takes place outside of the classroom. It is a part of enhancing the teacher’s professional development, which leads to better teaching. The Mathematics Achievement Partnership (2001) addressed this for the middle grades, stating that by working with the state teachers “will receive opportunities for sustained lessons that provide a deep and rich understanding of the mathematics that students should learn” (“Supporting Teachers” section). Professional development does not necessarily imply receiving formal instruction; it also integrates evaluating one’s self and working together with other teachers. “Opportunities to reflect on and refine instructional practice – during class and outside class – are crucial… Collaborating with colleagues regularly to observe, analyze, and discuss teaching and students’ thinking is a powerful, yet neglected, form of professional development” (NCTM, 2000, p.19).
Tending to their professional development and through hands-on experience, teachers can improve on their pedagogical knowledge, acquiring new and shaping old strategies. “The manner in which students learn is inseparable from the content. Research regarding how students learn mandates development of new pedagogical methods and implementation of proven teaching techniques” (Crossroads, 1995, “Pedagogy” section). In tandem with the appropriate curriculum – a curriculum that not only educates but also entices and demonstrates significance – teachers can provide a nurturing intellectual environment for his or her students. Such an environment is beneficial at all levels of education in all domains, both formal and informal. Students in such an interactive and positively reinforced setting can develop only in a good way. In the realm of mathematics, they acquire the analytical and problem-solving skills required for day-to-day living, which is why it is important to provide such an environment both in grades K-12 and at the college level. “The demands of [society] require that all students become empowered citizens capable of critical thinking” (Crossroads, 1995, “Foreword” section).
Cleary, there is a purpose in providing creative intellectual stimulation in the classroom, but it is just as significant outside the classroom. Take into account the Internet and television, such as an Internet site about Pearl Harbor and a television documentary on the Buena Vista Social Club. Should either be a mere regurgitation of factual tidbits – dates, names – who would want to read through it or sit through it, other than someone looking for just the dates and name? As learners, people want be educated to not only of the facts but also of the stories behind them, the implications, and the what-ifs. People in turn develop a desire to question and conjecture, and without that the progression of our world would be at a stand still.
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