Transcript Effects of Using History of Calculus on Attitudes and Achievement of Junior College Students in Singapore Toh-Lim Siew Yee Hwa Chong Institution [email protected].

Effects of Using History of Calculus
on Attitudes and Achievement of
Junior College Students in Singapore
Toh-Lim Siew Yee
Hwa Chong Institution
[email protected]
Overview






Rationales
Aims
Teaching Package on History of Calculus
Methodology
Results
Discussion
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)
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)
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.
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).
Teaching Package on History of Calculus:
Use of Anecdotes and Biographies
 “War” between Newton and Leibniz
 Leibniz – published work in 1674
 Newton – published work in 1683
 Totally different notations and symbols
Notations
Notation
x, x
dx, dy, and
dy
dx
Author
Year of Publish
Sir Isaac Newton
(British)
1670s
(Not published immediately)
Gottfried Wilhelm
Leibniz (German)
1675
f'(x), f''(x) etc Joseph Louis
Lagrange (Italian)
1797 fx, f’x
1806 f(x), f'(x), f''(x), f'''(x)
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).
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).
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
Teaching Package on History of Calculus:
Historical Motivations
 Gottfried Leibniz developed calculus to find area
under curves
Teaching Package on History of Calculus:
Historical Motivations
How much wine is in the bottle?
- Green, 1971
Using areas of rectangles to approximate area under a curve
y
y  f(x)
y1
y2
y3
 A1  A2  A3
0
a δx
δx
yn
δx
………..
 An
δx b
x
Sum of area of n rectangles from x = a to x = b
  A1   A2   A3  ...   An
 y1 x  y2 x  y3 x  ...  yn x
Area under the curve = lim (y1  y2  ...  yn ) x
 x 0
b
By fundamental thm of calculus, area under the curve =  y dx
b
 y dx = lim (y1  y2  ...  yn ) x
a
 x 0
a
Teaching Package on History of Calculus:
Historical Motivations
 Archimedes – Method of Exhaustion
http://media.texample.net/tikz/examples/PDF/archimedess-approximation-of-pi.pdf
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.
Uses of Circles
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).
Teaching Package on History of Calculus:
Use of Original Materials from Historical Sources
2
1
sin 2
 constant
sin 1
Willebrord Snellius
Dutch astronomer and
mathematician
1580 - 1626
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
1607 - 1665
Snell’s Law
b2  (d  x)2
2
b
v2
1
v1
a x
2
2
a
x
d
a x
b  (d  x)
t

v1
v2
2
2
2
2
sin 2 v2

sin 1 v1
sin 2 v2

sin 1 v1
Snell’s Law
Light follows the path of least time.
a x
b  (d  x)
t

v1
v2
2
2
2
2
1
2
dt
1 2
2

(a  x ) (2 x)
dx 2v1

1
2
1 2
2

(b  (d  x) ) (2)(d  x)(1)
2v2

sin 2 v2

sin 1 v1
Snell’s Law
  1 
dt  1  
x
dx
  
  

1
2
dx  v1   a2  x 2   v2   b2  (d  x2
)
sin
sin
2
b2  (d  x)2
b
d-x
a2  x2  1
a
x
d




Snell’s Law
Light follows the path of least time.
dt sin1 sin2


dx
v1
v2
sin1 sin2

v1
v2
sin 2 v2

sin 1 v1
0
sin 2 v2

sin 1 v1
Importance of Snell’s Law
Land Problems
Prove that a square has the greatest area among all
rectangles with the same perimeter.
Euclid
Greek Mathematician
365 B.C. – 275 B.C.
Euclid’s Solution to Solving Max/Min Prob
– Without Calculus
Prove that a square has the greatest area among all
rectangles with the same perimeter.
y
v
B
y
u
x
A
u
v
x
C
y
y
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
Fermat’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 + 2y
x
P  2x
y
y
   (2)
2
P  2x  Px 2

Sub (2) into (1): A = x 
 x
 2  2
dA P
P
  2x  0  x 
dx 2
4
d2 A
P
 2  0  max area at x 
2
dx
4
Other Sources of History of Math
 Textbooks:
 Burton (2003)
 Eves (1990)
 Webpages:
 http://www-history.mcs.st-and.ac.uk/
 http://aleph0.clarku.edu/~djoyce/mathhist/nonw
ebresources.html
Other Sources of History of Math
 Youtube
 Search for “history of mathematics”
 BBC & The Open University
 “The Story of Mathematics”
Methodology
Participants:
 17 year-old junior college Year 1 students
Topics:
 Techniques and applications of differentiation
 Techniques and applications of integration
Duration: 22 one-hour tutorial sessions over 4 months
31
Methodology
Experimental
Control
(History of mathematics) group
(No history of mathematics) group
2 classes (51 participants)
2 classes (52 participants)
32
Methodology
Achievement
Attitudes
Control
O1
Experimental
O1
Control
A1
Experimental
A1
X
P1
O2
P1
O2
X
P2
O3
P2
O3
P3
X
P3
A2
X
X
X
A2
Key:
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 pretest
A2: Attitudes posttest
33
Methodology - Instruments
Attitudes Toward Mathematics
 Attitudes Toward Mathematics Inventory (ATMI)
(Tapia & Marsh, 2004)
 Modified Academic Motivation Scale (AMS) (Lim &
Chapman, 2011)
34
Attitudes Toward Mathematics Inventory (ATMI)
Measures general attitudes toward mathematics:




Enjoyment
General motivation
Self-confidence
Value
35
Academic Motivation Scale (AMS)
Self-determination Continuum
(adapted from Ryan and Deci, 1985)
Low Self-determination Level
High Self-determination Level
Low Autonomy
High Autonomy
Low Sense of Control
High Sense of Control
Amotivation
Extrinsic motivation
External
Regulation
Introjection
Intrinsic motivation
Identification
Amotivation:
Intrinsic
Motivation:
External
Introjection:
Regulation:
Identification:
Interest
External
Internal rewards Personal
Non-valuing
IBecause
can't
why
Ifact
study
mathematics
andin
For
thesee
pleasure
Igood
experience
when
Iwill
Rewards
ofbelieve
the
that
grades
when
orI do
well
Isuch
that
mathematics
Enjoyment
rewards
or as
or
punishment
importance
Incompetence
discover
things
in mathematics
that I
Inherent
frankly,
I new
couldn't
care
less.
punishment
Valuing
punishments
mathematics,
by
I feel
parents
important.
improve
my
work
competence.
satisfaction
have never learnt before.
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.
37
Data Analysis
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
38
Factors
Control
(n = 52)
Mean
SD
Experimental
(n = 51)
Mean
SD
Attitudes Tests
Enjoyment
Motivation
ATMI
Self-confidence
Value
Amotivation
Intrinsic motivation
Modified
Identification
AMS
Introjection
External regulation
Achievement Tests
Test 1
(Techniques of Differentiation)
Test 2
(Applications of Differentiation)
Test 3
(Integration)
3.31
3.10
3.54
3.65
1.82
3.00
3.20
2.72
2.81
0.80
0.86
0.71
0.59
0.89
0.85
0.72
0.85
0.67
3.40
3.13
3.71
3.88
1.65
3.28
3.42
3.14
2.90
0.75
0.75
0.84
0.45
0.84
0.75
0.73
0.99
0.99
68.46
18.93
77.16
12.18
53.01
17.90
62.55
14.69
40.77
19.57
56.21
19.70
Results – Attitudes Toward Mathematics Inventory
(ATMI)
 Experimental group performed better in all
domains of ATMI (i.e., enjoyment, motivation, selfconfidence, 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 0.0125 (F(1, 97) =
6.75, p = 0.011, partial ƞ2 = 0.065)
40
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
ƞ2 = 0.031).
Marginally insignificant.
41
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 ƞ2 = 0.050), and
 introjection (F(1, 94) = 7.07, p = 0.009, partial
ƞ2 = 0.070),
at a Bonferroni adjusted alpha level of 0.01.
42
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
ƞ2 = 0.089), and
 Test 3 (F(1, 100) = 15.78, p = 0.001, partial
ƞ2 = 0.136).
43
Qualitative Data?
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
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
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.
47
Discussion
• The experimental group performed better than
the control group in all three achievement posttests. 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
Limitations
 Experiment only involves the teaching of calculus.
 The participants of this study come from only one
junior college in Singapore.
49
Questions?
Toh-Lim Siew Yee
Hwa Chong Institution
[email protected]