Mathematics as a Discipline Has a Very Important Place - Essay Example

Does more calculus improve learning in intermediate micro and macroeconomic theory Introduction Mathematics as a discipline has a very important place in the school curriculum all over the world. It also enjoys close relationship to almost every other discipline taught in the school. The relationship between mathematics and economics is as good as is between mathematics and physics. The physics has its own mathematics called as mathematical physics, whereas the economics has its favorite as calculus, hence more appropriately the relation ship between calculus and economics. This is evident by the fact that in USA most undergraduate programme in economics has a calculus prerequisite. Does learning calculus really improve learning micro and macroeconomics theory The question seems important to improve the student performance in economics theory. This will also help in understanding why the student with poor calculus understanding finds it hard to grasp the concepts in micro and macroeconomic theory. The calculus itself is a very vast topic categorized in to two parts the differential and integral calculus. Which of these two parts share close proximity with economics theory The undergraduate economics prerequisite should be specifically for differential or integral or for both In the following paragraphs I intend to address this question of significance of calculus learning for the intermediate students learning micro and macroeconomic theory by replicating the article, "Does more calculus improve student learning in intermediate micro and macroeconomic theory", Co-authored by J.S. Butler, T. Aldrich Finegan and John J. Siegfried from the department of economics, university of Vanderbilt, Nashville, TN 37235, USA, published in the journal of applied econometrics , mar/apr 1998; 13,2. The students records of nearly all the Vanderbilt university undergraduates who entered the college of Arts and Science as freshmen during the August 1983 -1986 and studied Microeconomic theory (micro -2) or intermediate Macroeconomic theory (macro -2)or both, constituted the data for this study. The student who took at least one calculus course and both semesters of principals (macro -1, micro -1) of economics at Vanderbilt before taking either intermediate theory course formed the sample for the study. The independent variable here is the calculus course taken by the students. Depending on the variety of calculus course available the key independent variables were seven namely Math 170, Math 171A-B, Math 172 A-B, Math 221 A-B and Math 222. The dependent variable is performance in micro-2 and macro -2. A two stage estimation procedure was used to cope with the problem of selection bias. In the first stage ordered probit model was used to predict the highest level of calculus attained by each student prior to taking each intermediate theory course. In the second stage the grades in Micro-2 or Macro-2 (the dependent variable)is regressed on the actual level of the calculus attained (independent variable). This can be written in simple words as the grades earned in the calculus course are used to predict the grade in the economic course. The ability and motivation of the students became the control variables. The ordinary least square estimation is used as there are twelve categories of grades depicting cardinal measure of performance. The ordered probit was appropriate for the purpose of this study as the various calculus courses can be ordered in accordance with the difficulty of the course material covered, whereas the difference in the courses can't be measured cardinallly. Individuals are the units of abservation. Table one Calculus courses taken by Vanderbilt undergraduates: descriptions and ranking in the ordered probit Course (credit hours) Description Math 170 (4) Analytic Geometry and calculus: A basic course in rudiments of analytic geometry and differential and integral calculus, with emphasis on application. Designed for students who do not plan to further study the calculus. No background in trigonometry is required or used in the course. Math 171 A-B (3-3) Analytic Geometry and calculus: A two semester in basic calculus recommended for most students. 171 A: functions, limit, differentiation of algebric functions, application of differentiation, introduction to integration. 171B differentiation and integration of transcendent functions, methods of integration. Math 172 A-B (4-4) Analytic Geometry and calculus: a more intensive two semester course in basic calculus designed for students interested in majoring in math or in math oriented science. 172 A: functions, limit, differentiation of algebraic functions, applications including extreme problems, areas, volumes, centriods; 172B: differentiation and integration of transcendent functions, applications, methods of integration, coordinate geometry, polar coordinates, infinity. Math 221 A-B (3-3) Intermediate Calculus: A two semester sequenced earmarked for students who have completed 171A-B. analytical geometry, polar coordinates, infinite series, vectors, parametric equations, vector analysis, partial differentiations and multiple integration. Prerequisite 171B Math 222 (3) Intermediate Calculus: A one semester course for students who have completed 172 A-B; content equivalent to 221 B; indeterminate forms, solid analytical geometry, vectors in three space, partial derivatives, multiple integrals; Prerequisite 172B Ordered probit ranking,bottom to top ( cumulative credit hours shown in brackets) Number of students who completed stated calculus course before taking: Micro 2 Macro 2 Math 170(4) Calculus Survey 164 128 Math 171A(3) Calculus I 49 30 Math 172A(4) CalculusI for math major 11 9 Math 171(6) Calculus II 228 202 Math 172B(8) CalculusII for math majors 42 39 Math 221 A(9) CalculusIII 31 29 Math 221Bor 222(12 or 11) CalculusIV 84 57 total 609 490 Math 221Bor 222 are considered equivalent courses. Students who take 221B completed 12 hrs of Calculus(171A-B followed by 221A-B), while those taking 222 complete 11 hrs (172A-B followed by 222). The variables include students' SAT math score, the number of years of advance math, physics and chemistry (each considered separately)taken in high school, his or her expected major at the time of application for admission, sex, whether the student had achieved a high score on the College Board Foreign Language achievement test to satisfy the Foreign Language option of the Vanderbilt's general education requirement. Hypothesis: the students with more calculus (predicted or actual) and higher grades in their last completed calculus should have done better in intermediate theory. Results: Regarding ordered probits predicting the level of calculus attined by sample students: the students with higher SAT score, two years of advanced mathematics in high school and expected to major in natural science took more calculus as compared to those with low score, less mathematics in high school, expected to major in humanities or social sciences other than economics. One year of high school physics was also found to be significantly associated with more calculus whereas one year of high school chemistry was not. Amount of calculus seemed to unaffected by the score on the College Board Foreign Language achievement test. Female students took more calculus then males. Regarding second stage regression: a second semester of Calculus definitely leads to better grade in micro theory but same can't be said for macro theory. For micro-2 the students exposure to reference group ( 170 or 171 A or 172 A) has little influence. Completing either two semester sequence leads to large and highly significant (at 1% level) improvement in micro-2. The first semester grades were found to be highly relevant to improvement in micro-2 as compared to the second semester Calculus. Regarding instructor's grade deflator in intermediate theory: The student who took a section of theory form an instructor with an ex post deflator of plus one could expect to receive a course grade higher by almost one letter then would have been received from an average instructor with a deflator near zero. Regarding grades in economic principles: doing better in micro principles leads to better performance in micro-2. Like wise in macro principles and macro-2. strong significant relationship was also found in converse pairs i.e. micro-1 and macro-2 and vice versa. Regarding freshmen GPA: this was found to be highly significant predictor of grades for both the theory courses. Regarding SAT score: SAT score plays only a minor role in explaining performance in intermediate theory. Regarding class size: no significant relationship was found. Regarding sex: the women performed better in micro theory and same as men in macro theory. Conclusion: A second semester of calculus is associated with a full letter grade improvement in intermediate micro performance. How well students do in calculus seems related to their performance in intermediate micro if they take only one semester of calculus; but if they take more calculus, their performance in mathematics courses is unrelated to their success in intermediate micro theory. How well students do in calculus at any level is unrelated to their success in intermediate macro theory. The greater success in intermediate micro theory associated with second semester of calculus is independent of the selection bias caused by students with greater quantitative aptitude simultaneously enrolling in more calculus courses and doing better in intermediate economic theory. Read More