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Toh-Lim Siew Yee Hwa Chong Institution Effects of Using History of Calculus on Attitudes and Achievement of Junior College Students in Singapore

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Overview Rationales Aims Teaching Package on History of Calculus Methodology Results Discussion

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Rationales Numerous studies advocate the use of history of mathematics to improve students’ learning outcomes such as attitudes and achievement (see Fasanelli and Fauvel (2006) for review). Comprehensive reports: Fasanelli and Fauvel (2006) Tzanaki (2008)

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Rationales Numerous qualitative studies Lack of quantitative studies Only 3 studies done in Singapore which produce inconclusive results due to their poor experimental designs (Lim & Chapman, in press)

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Aims To encourage curriculum planners and teachers in Singapore to incorporate history of mathematics in the curriculum by convincing them about the benefits of using history of mathematics through a quasi-experiment. To develop a teaching package on the history of calculus that can be used by teachers.

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Teaching Package on History of Calculus In this study, history of calculus includes (1)the use of anecdotes and biographies of mathematicians (Bidwell, 1993; Higgins, 1944; Wilson & Chauvot, 2000), (2)the discussion of historical motivations for the development of content (Katz, 1993b), and (3)the use of original materials from historical sources (Arcavi & Bruckheimer, 2000; Jahnke et al., 2000).

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Teaching Package on History of Calculus: Use of Anecdotes and Biographies “War” between Newton and Leibniz Leibniz1674 Leibniz – published work in 1674 Newton1683 Newton – published work in 1683 Totally different notations and symbols

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Notations NotationAuthorYear of Publish Sir Isaac Newton (British) 1670s (Not published immediately) dx, dy, andGottfried Wilhelm Leibniz (German) 1675 f'(x), f''(x) etcJoseph Louis Lagrange (Italian) 1797 fx, f’x 1806 f(x), f'(x), f''(x), f'''(x)

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Teaching Package on History of Calculus In this study, history of calculus includes (1)the use of anecdotes and biographies of mathematicians (Bidwell, 1993; Higgins, 1944; Wilson & Chauvot, 2000), (2)the discussion of historical motivations for the development of content (Katz, 1993b), and (3)the use of original materials from historical sources (Arcavi & Bruckheimer, 2000; Jahnke et al., 2000).

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Teaching Package on History of Calculus: Historical Motivations Although some mathematical concepts appear to be disconnected from real-life in modern times, history enables students to understand the need for the development of these concepts. Aim of using historical motivation: To make students appreciate the value of mathematics by letting them see the motivations behind the creation of knowledge by mathematicians, which are mostly due to real-life problems in the past (Burton, 1998).

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Teaching Package on History of Calculus: Historical Motivations Sir Isaac Newton used calculus to solve many physics problems such as the problem of planetary motion, shape of the surface of a rotating fluid etc. – recorded in Principia Mathematica

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Teaching Package on History of Calculus: Historical Motivations Gottfried Leibniz developed calculus to find area under curves

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Teaching Package on History of Calculus: Historical Motivations How much wine is in the bottle? - Green, 1971

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Sum of area of n rectangles from x = a to x = b 0 x y a b ……….. y1y1 y2y2 ynyn y3y3 Using areas of rectangles to approximate area under a curve

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Teaching Package on History of Calculus: Historical Motivations Archimedes – Method of Exhaustion

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It’s easy to find the areas of polygons such as squares, rectangles and triangles. How can I find the area of a circle? Archimedes 287B.C. – 212 B.C.

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Uses of Circles

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Teaching Package on History of Calculus In this study, history of calculus includes (1)the use of anecdotes and biographies of mathematicians (Bidwell, 1993; Higgins, 1944; Wilson & Chauvot, 2000), (2)the discussion of historical motivations for the development of content (Katz, 1993b), and (3)the use of original materials from historical sources (Arcavi & Bruckheimer, 2000; Jahnke et al., 2000).

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Teaching Package on History of Calculus: Use of Original Materials from Historical Sources Willebrord Snellius Dutch astronomer and mathematician

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Teaching Package on History of Calculus: Use of Original Materials from Historical Sources Fermat’s Principle: Light follows the path of least time. Pierre de Fermat French Lawyer

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Snell’s Law x d v1v1 v2v2 a b

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Light follows the path of least time.

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Snell’s Law d x a b d - x

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Snell’s Law Light follows the path of least time.

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Importance of Snell’s Law

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Land Problems Euclid Greek Mathematician 365 B.C. – 275 B.C. Prove that a square has the greatest area among all rectangles with the same perimeter.

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y y Euclid’s Solution to Solving Max/Min Prob – Without Calculus Let y = v + x 2x + 2u + 2y = Perimeter of the rectangle (A + C) = Perimeter of the square (B + C) = 2x + 2v + 2y So u = v (since 2x + 2u + 2y = 2x + 2v + 2y) So Area A = ux < vy = Area B, (since x < y) So Area (A + C) < Area (B + C) So the Area of an arbitrary rectangle < the Area of a square u v x u A B C y v y x Prove that a square has the greatest area among all rectangles with the same perimeter.

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Fermat’s Solution to Solving Max/Min Prob – An Introduction to Calculus x y Area, A = xy (1) Perimeter, P = 2x + 2y Sub (2) into (1): A = Prove that a square has the greatest area among all rectangles with the same perimeter.

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Other Sources of History of Math Textbooks: Burton (2003)2003 Eves (1990)1990 Webpages: ebresources.html ebresources.html

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Other Sources of History of Math Youtube Search for “history of mathematics” BBC & The Open University “The Story of Mathematics”

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31 Methodology Duration: 22 one-hour tutorial sessions over 4 months Topics: Techniques and applications of differentiation Techniques and applications of integration Participants: 17 year-old junior college Year 1 students

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32 Experimental (History of mathematics) group 2 classes (51 participants) Experimental (History of mathematics) group 2 classes (51 participants) Control (No history of mathematics) group 2 classes (52 participants) Control (No history of mathematics) group 2 classes (52 participants) Methodology

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33 Key: O r : Achievement pretest r, for r = 1, 2 and 3. X : Treatment (history of mathematics) P r : Achievement posttest on calculus topic r, for r = 1, 2 and 3. A 1 : Attitudes pretest A 2 : Attitudes posttest Methodology

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34 Methodology - Instruments Attitudes Toward Mathematics Attitudes Toward Mathematics Inventory (ATMI) (Tapia & Marsh, 2004) Modified Academic Motivation Scale (AMS) (Lim & Chapman, 2011)

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35 Attitudes Toward Mathematics Inventory (ATMI) Measures general attitudes toward mathematics: Enjoyment General motivation Self-confidence Value

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High Self-determination Level High Autonomy High Sense of Control Low Self-determination Level Low Autonomy Low Sense of Control Intrinsic motivation Amotivation Extrinsic motivation External Regulation Introjection Identification Non-valuing Incompetence External rewards or punishment Internal rewards or punishment Personal importance Valuing Interest Enjoyment Inherent satisfaction Self-determination Continuum (adapted from Ryan and Deci, 1985) Amotivation: I can't see why I study mathematics and frankly, I couldn't care less. External Regulation: Rewards such as good grades or punishments by parents Introjection: Because of the fact that when I do well in mathematics, I feel important. Identification: Because I believe that mathematics will improve my work competence. Intrinsic Motivation: For the pleasure I experience when I discover new things in mathematics that I have never learnt before. Academic Motivation Scale (AMS)

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37 Methodology - Instruments Achievement in Mathematics 3 sets of pre and post calculus tests modified from past years G.C.E ‘A’ level questions. Content validated by a setter of the GCE ‘A’ level 9740 H2 mathematics paper from UCLES.

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38 MANCOVA (for attitudes) and ANCOVA (for achievement) using SPSS 19 Independent variable: History of mathematics Dependent variables: Posttest scores of attitudes and achievement Covariates: Pretest scores of attitudes and achievement Data Analysis

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Control (n = 52) Experimental (n = 51) FactorsMeanSDMeanSD Attitudes Tests ATMI Enjoyment Motivation Self-confidence Value Modified AMS Amotivation Intrinsic motivation Identification Introjection External regulation Achievement Tests Test 1 (Techniques of Differentiation) Test 2 (Applications of Differentiation) Test 3 (Integration)

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40 Results – Attitudes Toward Mathematics Inventory (ATMI) Experimental group performed better in all domains of ATMI (i.e., enjoyment, motivation, self- confidence, value). Statistically significant result on the combined dependent variables (F(4, 94) = 2.70, p = 0.035, partial ƞ 2 = 0.103) Value of math – Experimental group perform significantly better than control group at a Bonferroni adjusted alpha level of (F(1, 97) = 6.75, p = 0.011, partial ƞ 2 = 0.065)

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41 Results – Modified Academic Motivation Scale (AMS) Experimental group performed better in all domains of attitudes except for amotivation. Experimental group performed better than the control group on the combined dependent variables (F(5, 90) = 2.31, p = 0.051, partial ƞ 2 = 0.031). Marginally insignificant.

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42 Results – Modified Academic Motivation Scale (AMS) Experimental group performed significantly better than control group in intrinsic motivation (F(1, 94) = 4.94, p = 0.029, partial ƞ 2 = 0.050), and introjection (F(1, 94) = 7.07, p = 0.009, partial ƞ 2 = 0.070), at a Bonferroni adjusted alpha level of 0.01.

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43 Results – Achievement ANCOVA conducted on each of the three achievement tests, with pre-test scores on the same topic as covariates. Experimental group performed significantly better than control group in Test 1 (F(1, 100) = 9.72, p = 0.002, partial ƞ 2 = 0.089), and Test 3 (F(1, 100) = 15.78, p = 0.001, partial ƞ 2 = 0.136).

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Qualitative Data?

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Students’ Comments "Mrs Toh is really a very awesome teacher. She is dedicated and very passionate about mathematics. Her enthusiasm influences us to love maths as well! Moreover, she is always willing to go the extra mile to explain the derivation instead of just asking us to accept the definition. I feel very fortunate to be in her class. ^_^ Thank you for everything Mrs Toh. ^_^“ – Leong Yuan Yuh, 10S72

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Students’ Comments "Haha, I'd like more of real life application of the math topics and explanation of formulas (like what it actually means, rather than using it as a fixed thing which has no meaning).” – anonymous, 10S74 "I like learning about the Histories of math.” – Chen Sijia, 10S74

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47 Discussion Results show that the use of history of mathematics in classrooms can improve certain aspects of students’ attitudes toward mathematics, particularly in value, intrinsic motivation and introjection. Studies (e.g. Vallerand et al. (1993) and Gottfried (1982)) have shown that these aspects of attitudes are positively related to good students’ learning outcomes such as academic achievement, low anxiety and low dropout rate from school.

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48 Discussion The experimental group performed better than the control group in all three achievement post- tests. The results are statistically significant for the first and third test. Hence the use of history of mathematics in classrooms should be strongly encouraged. Teachers’ training institutions may also want to consider conducting courses on history of mathematics for teachers to equip them with the skills and resources to use history of mathematics in their lessons.

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49 Limitations Experiment only involves the teaching of calculus. The participants of this study come from only one junior college in Singapore.

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Toh-Lim Siew Yee Hwa Chong Institution Questions?

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