www.EducationNews.org online publication, December 27, 2010
Important Mathematics Deficiencies in United States Grade School Teaching Practices
Richard Pan Massachusetts Institute of Technology alumnus
[email protected]
Dedicated to the author's younger students.
Abstract In the present pedagogical note, a list of six (6) important examples of deficiencies common in U.S. grade school mathematics teaching practices are described, which may contribute to a significant barrier to modern mathematics thinking. The six topics are an inconsistent introduction to signed arithmetic, non-existent teaching of the integer fractions, i.e., the "rational reals," in relation to the real number line, lack of a clear introduction to formal logic, inappropriate names for cornerstone theorems such as the Division Algorithm Theorem, longstanding ambiguity in usage of the words "function" versus "map," and last, teaching of the Fundamental Theorem of Calculus without adequate knowledge of the existence of the Riemann integral. These six examples, while not selected for perfect characterization of K-12 mathematics education in the United States, span the middle school-to-high school curricula of great interest. The author's goal here is to invite pedagogical debate, thereby promoting better teaching of these topics for our nation’s education-minded community. Introduction An important question may arise: why is there so great a gap between United States grade school mathematics and pure mathematics defined by university curriculum? The United States of America's federal government's recent attention to education, shown by sharp funding jumps in the 2009 Recovery and Reinvestment Act and in the new United States Budget (Klein, 2009), demonstrate the Obama administration's commitment to higher standards in basic education. Criticisms of weak pedagogical practices in United States grade school classrooms in the fundamental subject of mathematics are justified. Significant Pedagogical Deficiencies in Grade School Mathematics Teaching Six examples of United States grade school mathematics teaching deficiencies or gaps are presented here as instances of how weaknesses in mathematics thinking are entwined with teaching itself: brute emphasis on computation instead of ideas, words, and logic, whether a flaw of ambiguity is accepted within the mathematics community, and further, a weaker teaching style than the subject needs. Although the examples listed in Table I are not intended to characterize nor to be comprehensive by any means, they may provide new insight into the breadth of United States mathematics teaching controversies. The author presents them here so that all teachers, parents, and students may publish on these important controversies, allowing complete freedom of mathematics pedagogy discussion. Table I. Pedagogical Flaws in United States Grade School Mathematics Teaching 1 Poor signed arithmetic teaching Non-existent teaching of the rational reals 2 as a subset of all real numbers 3 Weak, casual introduction to logic Naming of important theorems which 4 obscure proof and motivation 5 Inconsistent definitions of function v. map Fundamental Theorem of Calculus teaching 6 that neglects existence of Riemann integral
First, the subject of negative numbers may often be given an inconsistent introduction. All school children should learn how to count the natural numbers {1,2,3,...} and how to do integer arithmetic, addition, subtraction, multiplication, and division. However, it is common among middle school students to be uncertain "why a negative integer times a positive integer is a negative integer but a negative integer times a negative integer must be a positive integer" based on such counting. These results can be proved as pure counting theorems, where repeated removal of negative integers is shown the counting logic equivalent of repeated addition of positive integers (Pan, 2009). The pedagogical decision made in Connected Mathematics 2 series' text Accentuate the Negative (Lappan et al., 2009) to drop a clear visual red and black chip board model and instead offer a complicated mechanics explanation based on physical time multiplied by directed running rates, either positive going to the right or negative going to the left along a number line, to give net physical displacement, either positive or negative, is most questionable. Pedagogy does make a difference here. This is an example of how abandoning the abstract, intrinsic logic of counting as an introduction to pure mathematics, for a confusing time and velocity model from the subject of physics, is a great mistake. Second, it may be surprising to many - parents of school-aged children and established professionals alike - that the current United States grade school math curriculum spends upwards of several years on integer fraction arithmetic, i.e., addition, subtraction, multiplication, and division, then with a minimum of justification or statement of assumption, extends properties such as commutativity and associativity learned from integers only, to any real numbers! That is astonishing! Indeed, how does a grade school student even know integer fractions, e.g., rational reals, satisfy basic properties such as commutativity of multiplication or distributivity of multiplication over addition? Further, how do students know it for any real, such as π or 3 ? Must it be true for example, that π + 2π = (1+2)π = 3π ? Based on current textbook thinking such as that of the Connected Mathematics 2 series (Lappan et al., 2009), grade school students are asked to take it on faith. It is asserted here without demonstration that commutativity of addition and multiplication, associativity of addition and multiplication, distributivity of multiplication over addition, and existence of additive inverses and existence of non-zero multiplicative inverses, all can be demonstrated or proved in an abstract sense, for integer fractions and further, using standard theorems from elementary mathematical analysis, these theorems for integer fractions can be extended to any real number, such as π or 3 . What's disturbing about this current grade school text is not that it uses these theorems without proof, it is that they do not give them any statement at all. Students who do not bother asking a teacher how it is known distributivity must be true for integer fractions may never learn such a result is a real number theorem which requires what is called mathematics proof. Instead, they are left wondering “Why must distributivity hold for integer fractions and for reals too?” In the current scenario of United States grade school math education, such questions are often left unstated and thus unrealized, stagnant from grade seven until university study when a theory of calculus or algebra class is taken, at least five to seven years later! How great an educational loss this is, for many students never take advanced mathematics classes! A brief statement to all middle school and all high school students that distributivity of multiplication over addition can be proven for integer fractions and for all reals is not so great a teaching burden and helps students position this knowledge in their thoughts, while alerting them to much more depth from the real number line. Rational numbers are very notable due to the importance of counting
arithmetic in science, technology, and computers. Abstract theorems which at first appear irrelevant to business or medicine, turn out to occupy central roles in statistics and engineering! Mathematics teachers owe their students a much clearer warning. Third, the United States' current grade school math curriculum does not encourage simple yet important notions of elementary logic, despite considerable opportunity. Euclidean geometry, most often taught to tenth graders, is a natural chance to teach logical necessity versus sufficiency (given logical statements p and q, p q states q is necessary for p or the logical equivalent, p is sufficient for q), which may have nothing to do with causality from the scientific world (p → q describes event p which "causes," "leads to," "precedes," or "is causally necessary for" event q). Although introductory logic may seem cumbersome and tied to command of grammatical English, basic logic concepts such as the negative, the converse, the contrapositive, and truth tables too may sharpen and aid any student's thinking. Language itself can generate examples which turn non-trivial and interesting in the blink of an eye: "If two sides of a flat Euclidean triangle have equal length, then the corresponding opposite angles must have equal measure” has a logical converse, “If two angles of a flat Euclidean triangle have equal measure, then the corresponding opposite sides must have equal length,” which in this case is also true though converses are not true in general. Pure logic may suggest new science as well: "significant history of smoking S plus exposure to drug X increases likelihood of cancer Y" suggests heavy smokers who never get Y were not exposed to X at sufficient doses although other hidden conditions S' may no longer hold. Fourth, it is a surprise a leading high school mathematics text such as Holliday et al.'s Glencoe Algebra 2 (2003) gives a confusing name, the "Remainder Theorem," for the statement If a polynomial f(x) is divided by x-a, the remainder is the constant f(a), and Dividend equals quotient times divisor plus remainder. f(x) = q(x) • (z-a) + f(a) where q(x) is a polynomial of degree one less than the degree of f(x). (pg. 365). This text deserves praise for citing "the division algorithm" when discussing "Dividing Polynomials" (p.233) though they give no statement of the division algorithm. The theorem stated above as the "Remainder Theorem" for polynomials with real or complex coefficients is actually known in mathematics as a special case of the Division Algorithm Theorem (Rotman, pgs. 239 & 241) which may proven as an important generalization of the Euclidean Division Algorithm for integers. Middle school students should be accomplished at long-hand division of integers and high school students revisit the idea of the Euclidean Division Algorithm when they practice long-hand division of polynomials. High school may be an apt time for students to know the true origin of important theorems and the correct name here, Division Algorithm Theorem, helps makes this possible. In contrast, the name "Remainder Theorem" may be misleading for suggesting an incorrect relation with the Chinese Remainder Theorem for integers, which itself has a polynomial generalization (Rotman, p. 288), lacking any immediate relation to the Division Algorithm Theorem.
The above example shows established mathematics texts such as the Glencoe Algebra 2 may offer confusing or inconsistent names of basic theorems. Students who depend on consistent teaching will bring imprecise thinking with them to jobs and university study. More detailed and better references at the end of each chapter in prominent texts ought to improve students’ mathematical base of knowledge. Fifth, an important stumbling block to new college mathematics students fresh out of United States high schools, is a long-standing ambiguity in use of the words "function" versus "map." Although U.S. high schools may intentionally avoid use of the mathematics word "map," they are supposed to provide a careful definition of the word "function," namely, a set of ordered pairs satisfying the condition that no two distinct ordered pairs share the same first element (with different second elements). What may confuse freshmen is the common practice among university texts to call a map (which precisely speaking is defined as the aforementioned function together with a domain set and a range set) a function, an established mathematics ambiguity which makes the phrase "surjective function" confusingly redundant. Pan (2008a) describes the map versus function ambiguity, noting an important distinction among inverses: an injective function, defined above, necessarily has an inverse while an injective map must also be surjective in order to have an inverse map. Superior teaching of mathematics advocates a consistent approach to the topic at the boundary between high school math and university mathematics, as described by Pan (2008a). Sixth, The Advanced Placement Calculus website's primary teaching resource on the Fundamental Theorem of Calculus (FTC) gives a statement of the FTC: For the purposes of this paper, we refer to the Fundamental Theorem by its parts: Antiderivative part of the FTC: d x If f is continuous on an open interval containing a , then f (t )dt f ( x) dx a Evaluation part of the FTC: If f is continuous on [a, b] , and F is any antiderivative of f , then
b
a
f (t )dt F (b) F (a) .
(p. 98, L. Townsley, 2006, open citation)
The most general pedagogical criticism of this FTC theorem is the lack of a clear statement of existence of the Riemann integral. In both antiderivative part and in evaluation part, the above FTC makes no clear statement the Riemann integral must exist for functions continuous on an interval or at a point (Townsley, 2006). Instead, the above FTC antiderivative statement tacitly assumes existence of the integral and combines it with a differentiation operation, which likely is confusing for novice analysis students, when they later consider integration of arbitrary functions discontinuous at more than a countable number of points. Students may appreciate an alternative pedagogical approach of first proving or disproving existence of the Riemann integral for any arbitrary integrand function f. They should learn Riemann integrability is a standard mathematical criterion for which any function can be tested, without regard to differentiability of the resulting integral. Examples of non-integrable functions, i.e. functions for which the Riemann integral does not exist, such as the Dirichlet
function (Bartle and Sherbert, pg. 143) and integrable, discontinuous functions, an example of which is the Thomae function (Bartle and Sherbert, pg. 143), are given direct proof of nonintegrability or integrability by the student, in addition to stock examples of continuous functions. Once existence of the Riemann integral is clear, then students may feel greater confidence examining topics of continuity of the Riemann integral and differentiability of the Riemann integral. A lengthy though clear exposition of this traditional approach of proving existence of the Riemann integral, using U(P,f)-L(P,f) < Ɛ arguments, free of assumptions of integrand continuity, before introducing the FTC, is found in Ross (1980, chp. VI), which ought to be better known at the United States high school level! America’s growing reliance on The College Board’s Advanced Placement program as the backbone of United States high school education (Pan, 2008b), and the Advanced Placement Calculus teachers’ capacity for didactic instruction of what was considered sole province of “university theory of calculus teaching,” justifies careful examination of the official resources prepared by The College Board, for instance, L. Townsley’s Proving the Fundamental Theorem of Calculus (2006), D.M. Bressoud’s Why Do We Name the Integral for Someone Who Lived in the Mid-Nineteenth Century? (2006) and recommended texts. These documents may suggest gaps in pedagogical emphasis and lack of sufficient formal proof requirements and adequate teaching examples and counterexamples. The present author and his teaching colleagues may wish to argue outstanding texts already in mainstream American college mathematics curriculum, Bartle and Sherbert (1992) and Ross (1980) for instance, instead of Stewart (1999) currently used for Advanced Placement Calculus, may better meet the needs of United States high school and middle school students. Conclusion Specific pedagogical deficiencies are identified in the United States grade school mathematics curriculum, spanning middle school negative integers and the Fundamental Theorem of Calculus from Advanced Placement Calculus. Although the list is not comprehensive, the topics may be considered motivation for education reform. They are also offered to any teacher concerned with standards of mathematics knowledge as opportunities for teaching improvements and education publications. Acknowledgements The author wishes to thank Boston Public Schools for substitute teaching employment which gave him in-depth experience with grade school students starting formal classroom study. The author acknowledges as well the most generous donation of a complete set of the 2009 Connected Mathematics 2 texts from Mr. Jack Ahearn of Pearson Prentice Hall. References Bartle, RG and Sherbert, DR (1992) 0471510009. John Wiley & Sons, Inc.
Introduction to Real Analysis (2nd ed.).
ISBN
Bressoud, DM (2006) Why Do We Name the Integral for Someone Who Lived in the MidNineteenth Century? AP® Calculus, 2006-2007 Professional Development Workshop Materials,
Special Focus: The Fundamental Theorem of Calculus, pgs. 105- 111. Internet resource document: http://apcentral.collegeboard.com/apc/public/repository/Calculus_SF_Theorem.pdf, December 26, 2010. Holliday, B, Marks, D, Cuevas, GJ, Casey, RM, Moore-Harris, B, Day, R, et al. (2003) Algebra 2. ISBN 0078279992. Glencoe/McGraw-Hill. Lappan, G, Fey, JT, Fitzgerald, WM, Friel, SN, and Phillips, ED (2009) Grade Seven volume Connected Mathematics 2 (CMP2) Series. ISBN 9780133661194. Prentice Hall®, Pearson Education, Inc. Klein, A (2009) Obama Budget Choices Scrutinized. Education Week 28(32), 1, 17. Pan, R (2008a) Important Ideas In The High School-to-University Mathematics Transition: Functions, Maps, and Inverses. Education and Science Essays, Pan, R. (2010), ISBN 978-0-55795278-6, Lulu.com. First published at EducationNews.org Internet publication December 22, 2008. Archive: http://www.ednews.org/categories/Commentaries-and-Reports. Pan, R (2008b) United States High School Mathematics Tests for Top 1-10% of Students Show Surprises, I. Advanced Placement Calculus Test Registration Rises Though Failing Students Increase. Education and Science Essays, Pan, R (2010), ISBN 978-0-557-95278-6, Lulu.com. First published at EducationNews.org Internet publication, December 29, 2008. Archive: http://www.ednews.org/categories/Commentaries-and-Reports/. Pan, R (2009) Signed Arithmetic of Integers: The Case for Consistency of Counting Proofs. Mathematics Teaching-Research Journal (MTRJ) Online, Vol. 3, Num. 4, pgs. 46-63. Ross, KA (1980) Elementary Analysis: The Theory of Calculus. ISBN 038790450-X. SpringerVerlag. Rotman, JJ (2000) Prentice-Hall, Inc.
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