The objective of this paper was to offer a critical analysis of a college-level math course. The position was taken that the said course’s learning objective could be generally defined as a behaviorist approach. Therefore, in view of the prevailing scientific literature, the following analysis offers effective pedagogical strategies that can benefit the overall construction of the class in question. The successive discussion is organized by the following interrelated constituents: content, objectives, teaching methods, and assessment.
The human mind can only know what the human mind has made"
~ Giambattista Vico, 1710 ~
Is the general and common fear of taking math classes an inherited trait or a developed phobia due to passive and ineffective teaching methods? According to Lochhead (1992, p. 543), mathematics is the only privileged school discipline that the majority of educated adults voluntarily claim to be incompetent. He posits that the teachings of constructivism releases or dismantles the mental blockage of math and leads to a counter-construction, inferring that everyone can learn. Constructivism not only changes the learner’s cognitive concepts of the nature of math awareness but also convinces learners to construct math by themselves and for themselves. (p. 544). Although the constructivist learning framework is widely applied throughout K-12 programs, including special aid programs, (Mercer & Jordan, 1994), a nationwide quantitative study by Goubeaud & Wenfan (2004) revealed that teacher educators seem to be more constructivist in instruction than higher education faculty. Accordingly, the aim of this paper is to critically look at a single case of instruction method of a college course on the basis of peer reviewed literature. The course in question was taught at CUNY John Jay College of Criminal Justice in Fall 2006.
As a mandatory prerequisite class for undergraduate students, the above-mentioned course, entitled Modern Math, displayed a systematic treatment of the foundation of college algebra. Topics covered were complex numbers, systems of linear equations and inequalities, various forms of functions, such as increasing or decreasing, quadratic, polynomial or exponential; as well as the theory of equations, logarithms and related applications.
Content of the Course
The course in question covered basic foundations for algebraic understanding of problem solving. Starting with a review of fundamental knowledge, the course materials were unambiguously textbook sources with the tendency to gradually build on previous covered knowledge. A linear progression of topics, starting with rudimentary elements, i.e., properties of real numbers or fractions, reinforced with problem solving tasks and rigid calculations, were systematically covered. The structure of the course did not allow the student to construct the acquired knowledge in a new dimension but gave the student an early and short taste of success by initially presenting the course materials in a simplified manner while the given formulas were expected to be memorized. Accordingly, the content and structure of the course in question can be seen as representing B. F. Skinner’s type R conditioning of his influential doctrine of operant conditioning. Since “any response that is followed by a reinforcing stimulus tends to be repeated” (Hergenhahn & Olsen, 2001, p. 77), the initial introduction of simple materials of the described course not only resulted in the somehow learning of the mathematical material but also reinforced repeating behavior towards forthcoming more complex materials. The stimulus-response theory might help the student to quickly retrieve the memorized material under test conditions, but does it lead to a fundamental understanding of the underlying meanings of mathematical concepts? Cobb, Wood, and Yackel (1992) argued that rote memorization of course materials leads to passive learning. They suggested that active learning was fostered by teaching more mathematical concepts and less theories and recipes (p. 16). Consequently, the course would benefit if it stressed mathematical thinking and not mathematical memorization because the student would be able to apply the acquired understanding to more complex and realistic problems outside the box or classroom environment. Furthermore, it is advised as a teaching strategy for the course in question that the learner be initially immersed into a complex topic. The presented material should be interesting and pragmatic enough to not only motivate the student to construct own ways of solving problems but should result in an augmenting discovery of possibilities. Thus, the recipe (Cobb, et al, 1992) should be in the student’s head to craft a comprehensive frame of thinking.
Objectives of the Course
The objectives of the course were to acquire mathematical skills of models that were strictly guided from the required textbook. Rather than stressing logical thinking and understanding of mathematical concepts, the course’s objective was the development of computational skills. For instance, the course in question neglected to use or introduce methods of proofs and independent mathematical reasoning. The nature of algebraic complex numbers was explained and demonstrated by means of second-hand data and quantified models. In a nutshell, the objectives were confined to textbook problems without emphasizing real life problems or solutions. To some extent the objectives of the course were enunciated by providing the means and formulas to solve the asked problems. For example, constant repetition of elements of numbers in class seemed to be a central goal for skill acquisition. Inasmuch as the course was a prerequisite for higher levels of math classes, and aimed to prepare the students to the next level, the said course as a whole was constrained to textbook problem solving without an emphasis on mathematical thinking. In contrast to it’s behaviorist nature, a constructivist approach of teaching would be desirable for a learner-driven outcome.
Teaching Methods of the Course
The analyzed course’s teaching method primarily consisted of knowledge transmission to a broad passive audience through lecture and textbook applications in the old fashioned “expository format” (Inch, 2002, p. 111). Emphasis was on rote memory learning of mathematical rules and formulas. For instance, formulas for bionomic calculations were handed out, but the derivation or formation of the distributed formulas were not addressed, hence method of proofs was absent. The sole demand was to learn the formulas by heart and be able to retrieve upon testing conditions. As a result, the said course can be characterized as a traditional math instruction with behaviorist drill and problem-solving tasks. Since the turn of the century, traditional teaching based on the framework of behaviorism, such as the course in question, is being replaced by inquiry-based teaching, facilitating a constructivist framework of learning. Advocates of the constructivist-teaching paradigm (Draper, 2002), recommend a more student-centered math classroom that “deemphasizes rote memorization of isolated skills and facts and emphasizes problem solving and communication” (p. 523). According to Larochelle & Bednarz, (1998) a constructivist classroom is rich in conversation. By conversing, the teacher infers the learning level and preparation of the student and coaches the communication so that the learner is able to construct meaning, understanding, and knowledge. Teachers who embrace constructivism reject the transmission model of teaching (Richardson, 1997).
Although the course in question solely relied on secondary data sets to solve problems transmitted by the lecturer (transmission model of teaching), the constructivist classroom adopts raw data as primary source. It further implements interactive materials to enhance experiential learning for the students (Jaramillo, 1997; Inch, 2002). Accordingly, the questioned course should use course materials that are gathered by the students for examination and interpretation, i.e., in place of presenting overused word problems such as, Betsy is six times as old as her brother Bob when the train leaves Boston. It is advantageous, for the learning effect of the course in question, if raw data are collected from newspapers or if connections can be seen between algebraic numbers and street signs and parked cars or even political polls, among others. The doctrine of constructivism postulates that the students teach themselves rather than passively consume lecture notes and calculations on the board. Constructivism engages the learner to internalize teaching concepts in a new light, empowering the learning effect. The role of the teacher transforms into a coach who guides the topics with proximity to the learners. The students work together in small groups to solve problems and engage in inquiry in order to construct knowledge out of experience (Cobb et al., 1992).
Assessment of the Course
Besides class participation, weekly homework assignments were collected and graded. Furthermore, the assessment of the course consisted of one mandatory final exam and three quizzes to test the learning of factual information by means of a multiple-choice format. The lowest quiz score was dropped. This latter technique finds its root within negative reinforcement in operant conditioning. Subsequently, the removal of the lowest grade (aversive stimulus) increases the behavior to score better in future quizzes (desired behavior).
"'Knowing' mathematics is 'doing' mathematics" (National Council of Teachers of Mathematics, 1989, p. 7, in Draper, 2002) catches the constructivist framework. In order to assess the students’ knowledge is to immerse them actively in the process of teaching by discovery learning. Correspondingly, an additional beneficiary assessment in the present course discussion can be measured if a student goes to the board and works on a given problem by explaining each step. In case the student feels lost at the board, another student could assist so that they resolve the algebraic problem as a team. This way the teacher is assured that the students understand the mathematical concepts in comparison to factual reception of course material and, at the same time, comes to see what kind of comprehensive difficulties the students might have.
Relating to the constructivist theory of learning, the learner is active and continuously constructs and reconstructs conception of phenomena. The learning is not assessed with separate examination at the end of the course, but assessment methods are integrated into the learning process itself (Tynjala, 1998). The objective of the assessment is to encourage the learning process resulting in the discovery of qualitative changes in the student’s knowledge base. As a result, the course in question would benefit from an assessment method that stresses the application or performance that displays “development of metacognition and critical thinking” (p. 176) in an authentic and constructive way.
Even though comparative studies that assess the learning outcomes of students in primary schools or colleges do not yield statistical significant differences between traditional versus constructivist teaching methods (Chung, 2004; Tynjala, 1998; also see Anghileri, 1989; Fosnot, 2001), the effectiveness of constructivist teaching as higher-order learning is supported.
Despite the fact that the title of the course in question anticipated a modern framework for mathematical discoveries, the teaching method was old fashioned and truthful to its behaviorist forerunners. Especially in higher education, the teaching of constructivism should be applied to immerse the learner into critical thinking because concept-driven learning prepares the college student for professional life. In contrast to rote memorization of factual knowledge that is lost later on due to lack of substantial meaning and connection, the higher-order learning facilitates a long-term base of knowledge.
Moreover, constructivism not only rejects the idea that students come to class with no built-in mental content (tabula rosa) but functions as a therapeutic means to release the perception of incompetence. Class participation in active discovery and confrontation with real life ideas is increased in the process of constructing fear-free learning.
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