1Toh-Lim Siew Yee Hwa Chong Institution firstname.lastname@example.org Effects of Using History of Calculus on Attitudes and Achievement of Junior College Students in SingaporeToh-Lim Siew YeeHwa Chong Institution
2Overview Rationales Aims Teaching Package on History of Calculus MethodologyResultsDiscussion
3RationalesNumerous 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)
4Rationales 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)
5AimsTo 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.
6Teaching Package on History of Calculus In this study, history of calculus includesthe use of anecdotes and biographies of mathematicians (Bidwell, 1993; Higgins, 1944; Wilson & Chauvot, 2000),the discussion of historical motivations for the development of content (Katz, 1993b), andthe use of original materials from historical sources (Arcavi & Bruckheimer, 2000; Jahnke et al., 2000).
7Teaching Package on History of Calculus: Use of Anecdotes and Biographies “War” between Newton and LeibnizLeibniz – published work in 1674Newton – published work in 1683Totally different notations and symbols
8Notations Notation Author Year of Publish Sir Isaac Newton (British) (Not published immediately)dx, dy, andGottfried Wilhelm Leibniz (German)1675f'(x), f''(x) etcJoseph Louis Lagrange (Italian)1797 fx, f’x1806 f(x), f'(x), f''(x), f'''(x)
9Teaching Package on History of Calculus In this study, history of calculus includesthe use of anecdotes and biographies of mathematicians (Bidwell, 1993; Higgins, 1944; Wilson & Chauvot, 2000),the discussion of historical motivations for the development of content (Katz, 1993b), andthe use of original materials from historical sources (Arcavi & Bruckheimer, 2000; Jahnke et al., 2000).
10Teaching 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).
11Teaching 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
12Teaching Package on History of Calculus: Historical Motivations Gottfried Leibniz developed calculus to find area under curves
13Teaching Package on History of Calculus: Historical Motivations How much wine is in the bottle?- Green, 1971
14Using areas of rectangles to approximate area under a curve y………..y1y2yny3abxSum of area of n rectangles from x = a to x = b
15Teaching Package on History of Calculus: Historical Motivations Archimedes – Method of Exhaustion
16How can I find the area of a circle? It’s easy to find the areas of polygons such as squares, rectangles and triangles.How can I find the area of a circle?Archimedes287B.C. – 212 B.C.
18Teaching Package on History of Calculus In this study, history of calculus includesthe use of anecdotes and biographies of mathematicians (Bidwell, 1993; Higgins, 1944; Wilson & Chauvot, 2000),the discussion of historical motivations for the development of content (Katz, 1993b), andthe use of original materials from historical sources (Arcavi & Bruckheimer, 2000; Jahnke et al., 2000).
19Dutch astronomer and mathematician Teaching Package on History of Calculus: Use of Original Materials from Historical SourcesWillebrord SnelliusDutch astronomer and mathematician
20Teaching Package on History of Calculus: Use of Original Materials from Historical Sources Fermat’s Principle:Light follows the path of least time.Pierre de FermatFrench Lawyer
26Land ProblemsProve that a square has the greatest area among all rectangles with the same perimeter.EuclidGreek Mathematician365 B.C. – 275 B.C.
27Euclid’s Solution to Solving Max/Min Prob – Without Calculus Prove that a square has the greatest area among all rectangles with the same perimeter.yyuvBCvyxuAxLet y = v + x2x + 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
28Fermat’s Solution to Solving Max/Min Prob – An Introduction to Calculus Prove that a square has the greatest area among all rectangles with the same perimeter.Area, A = xy (1)Perimeter, P = 2x + 2yxySub (2) into (1): A =
29Other Sources of History of Math Textbooks:Burton (2003)Eves (1990)Webpages:
30Other Sources of History of Math YoutubeSearch for “history of mathematics”BBC & The Open University“The Story of Mathematics”
3117 year-old junior college Year 1 students MethodologyParticipants:17 year-old junior college Year 1 studentsTopics:Techniques and applications of differentiationTechniques and applications of integrationDuration: 22 one-hour tutorial sessions over 4 months31
32Methodology Experimental (History of mathematics) group 2 classes (51 participants)Control(No history of mathematics) group2 classes (52 participants)32
33Methodology Key: Or: Achievement pretest r, for r = 1, 2 and 3. ControlO1P1O2P2O3P3ExperimentalXAttitudesA1A2Key:Or: Achievement pretest r, for r = 1, 2 and 3.X: Treatment (history of mathematics)Pr: Achievement posttest on calculus topic r, for r = 1, 2 and 3.A1: Attitudes pretestA2: Attitudes posttest33
36Academic Motivation Scale (AMS) Self-determination Continuum (adapted from Ryan and Deci, 1985)Low Self-determination LevelLow AutonomyLow Sense of ControlHigh Self-determination LevelHigh AutonomyHigh Sense of ControlAmotivationExtrinsic motivationIntrinsic motivationExternalRegulationIntrojectionIdentificationIntrinsic Motivation:For the pleasure I experience when I discover new things in mathematics that I have never learnt before.External Regulation:Rewards such as good grades or punishments by parentsIdentification:Because I believe that mathematics will improve my work competence.Introjection:Because of the fact that when I do well in mathematics, I feel important.Amotivation:I can't see why I study mathematics and frankly, I couldn't care less.Non-valuingIncompetenceExternal rewards or punishmentInternal rewards or punishmentPersonal importanceValuingInterestEnjoymentInherent satisfaction
37Methodology - Instruments Achievement in Mathematics3 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.37
38MANCOVA (for attitudes) and ANCOVA (for achievement) using SPSS 19 Data AnalysisMANCOVA (for attitudes) and ANCOVA (for achievement) using SPSS 19Independent variable: History of mathematicsDependent variables: Posttest scores of attitudes and achievementCovariates: Pretest scores of attitudes and achievement38
39Control(n = 52)Experimental(n = 51)FactorsMeanSDAttitudes TestsATMIEnjoyment3.310.803.400.75Motivation3.100.863.13Self-confidence3.540.713.710.84Value3.650.593.880.45Modified AMSAmotivation1.820.891.65Intrinsic motivation3.000.853.28Identification3.200.723.420.73Introjection2.723.140.99External regulation2.810.672.90Achievement TestsTest 1(Techniques of Differentiation)68.4618.9377.1612.18Test 2(Applications of Differentiation)53.0117.9062.5514.69Test 3(Integration)40.7719.5756.2119.70
40Results – 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 ƞ2 = 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 ƞ2 = 0.065)40
41Results – 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 ƞ2 = 0.031).Marginally insignificant.41
42Results – Modified Academic Motivation Scale (AMS) Experimental group performed significantly better than control group inintrinsic motivation (F(1, 94) = 4.94, p = 0.029, partial ƞ2 = 0.050), andintrojection (F(1, 94) = 7.07, p = 0.009, partial ƞ2 = 0.070),at a Bonferroni adjusted alpha level of 0.01.42
43Results – AchievementANCOVA 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 inTest 1 (F(1, 100) = 9.72, p = 0.002, partial ƞ2 = 0.089), andTest 3 (F(1, 100) = 15.78, p = 0.001, partial ƞ2 = 0.136).43
45Students’ 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
46Students’ 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
47DiscussionResults 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.47
48DiscussionThe 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.48
49Experiment only involves the teaching of calculus. LimitationsExperiment only involves the teaching of calculus.The participants of this study come from only one junior college in Singapore.49