int. j. math. educ. sci. technol., 2002
vol. 33, no. 6, 801±818
What e€ ect does the introduction of graphics calculators
have on the performance of boys and girls in assessment
in tertiary entrance calculus?
PATRICIA A. FORSTER*
Faculty of Community Services, Education and Social Sciences, Edith Cowan University,
2 Bradford Street, Mt Lawley, Western Australia 6050
e-mail: [email protected]
and
UTE MUELLER
School of Engineering and Mathematics, Edith Cowan University, 100 Joondalup Drive,
Joondalup, Western Australia 6027
e-mail: [email protected]
(Received 4 June 2001 )
The paper reports an investigation of gender-related e€ ects in the Western
Australian Calculus Tertiary Entrance Examinations for the period 1995±2000,
three years before and three years after graphics calculators were introduced.
Consideration is given to students’ total examination scores, scores on questions
grouped by curriculum component and students’ actual use of graphics
calculators on two questions from the year 2000 examination. Results show
that over the six-year period the performance of girlsÐon the examination as a
whole and in most curriculum componentsÐwas superior to that of boys at the
lower end of the achievement scale, while boys recorded the best performance at
the top end of the scale. The results are partially explained by the participation
rate in calculus for girls being lower than that for boys. Where superior
performance is recorded for girls it is frequently attributable to competence
with analytic methods. Superior performance favouring boys typically occurred
on questions where diagram played a role in the solution.
1. Introduction
In 1998, graphics calculators became required equipment for Western Australian Tertiary Entrance Examinations (TEE) in mathematics and science subjects. Since that time we have investigated extensively the extent and nature of
students’ use of the calculators in the Calculus examination [1±3]. We have also
systematically explored and described the characteristics of examination questions
and the nature of changes between questions set prior to and since the introduction
of the graphics calculator [4]. In particular, we considered the opportunity for
graphics calculator use, the role of a diagram in the solution process, whether
or not a well-known algorithm was called upon and the curriculum component to which each question belonged. The ®ve main curriculum components
* The author to whom correspondence should be addressed.
International Journal of Mathematical Education in Science and Technology
ISSN 0020±739X print/ISSN 1464±5211 online # 2002 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
802
P. A. Forster and U. Mueller
in the Calculus TEE are Functions and Limits, Theory and Techniques of
Calculus, Applications of Calculus, Vector Calculus, and Complex Numbers. The
classi®cation of questions was a critical evaluation of our own practices as
examiners, for one or both of us were on the examining panel for Calculus in the
years 1996±2000.
For the 1998 examination, we carried out a preliminary analysis of students’
total scores by gender and compared the results with examination outcomes for
1995±1997 [3]. There was no statistically signi®cant di€ erence between the mean
total examination scores of females and males in any of the years. The 1998
examination contained one question that attracted widespread calculator use for
generating a graph. A comparison of scores for it with scores for similar questions
from 1995±1997 gave no strong indication that the introduction of the calculators
had been discriminating in its e€ ect. For other questions in the 1998 paper,
including on Function and Limits, Complex Numbers and Applications, work
shown in scripts indicated that the minority of candidates (as low as 25% for one
question) chose to use the tool where its use was an option. With the 1998 Theory
and Techniques questions there was very limited scope for using the calculator,
and no opportunity in the item on Vector Calculus.
Thus, issues to be investigated with the introduction of graphics calculators to
the Calculus TEE, in relation to gender, are opportunities to use the technology,
the extent to which it is used by girls and by boys, the nature of its use, and the
distribution of scores for girls and boys opting for graphics calculator approaches
and for those who choose traditional analytic approaches. In this paper we explore
these issues for the period 1995±2000. Speci®cally, we consider students’ total
examination scores for the six years, scores as per the ®ve main curriculum
components, and scores by gender and use of graphics calculators for two
questions on Applications of Calculus from the year 2000 examination.
2. Background
2.1. The literature
Graphics calculators appeared in 1985. The introduction to schools was
gradual but accelerated upon their inclusion in public examinations. The calculators were allowed for the United States Advanced Placement Calculus in 1993
and the `A’ level examinations in the UK in 1994. In Australia, the calculators
were introduced in examinations for the Victorian Certi®cate of Education (VCE)
in 1997, in the Western Australian TEE in 1998, and are allowed in school-based
assessment for university entry in Queensland. The introduction of the calculators
to schools was accompanied by a research programme but perusal of journals
indicates that relatively little attention has been paid to issues of gender. A
repeated ®nding, however, is superior performance by both males and females
in groups using graphics calculators, compared to groups without their use [5±7].
Other outcomes are as follows.
Ruthven [5] recorded that boys scored better overall than girls in a group using
the calculators (in a high-school setting) but girls’ performance was superior to
that of boys on items requiring symbolic answers for given graphs. The better
performance of girls was attributed to the availability of graphical checks. Checks
potentially decrease anxiety, which is particularly relevant to girls for they exhibit
less con®dence than boys under conditions of uncertainty (Fenema, cited in [5]).
E€ ect of graphics calculators on performance of boys/girls in calculus
803
On items where graphs were supplied and interpretation was required, but no
equation, Ruthven found there were no signi®cant di€ erences in performance
between calculator and non-calculator groups, and no gender-related e€ ects within
the calculator group.
Boers and Jones [8], in the context of undergraduate Calculus where graphics
calculator usage was encouraged in the course, recorded di€ erences in overall
examination performance favouring females. Superior performance by females was
due mainly to success in algebraic questions. In questions that required the
construction of graphs, females only marginally outperformed males; a greater
(although not statistically signi®cant) percentage of boys and girls chose the
calculator option over pure algebraic, or mixed `algebraic and calculator’ approaches; and the percentage of boys who chose a pure calculator approach was
greater than the percentage of girls. In particular, for an item that required the
integration of algebraic answers (from early parts of a question) with the sketch of a
graph, the percentage of males who were successful was greater than the percentage of females.
Smith and Shotsberger [6], in the context of a college (undergraduate) algebra
examination with graphics calculators included, also reported that the percentage
of males using only a graphical approach in some questions was greater than the
percentage of females. Moreover, a greater percentage of females than males used
mixed graphical and algebraic approaches, partly for checking one method against
the other.
Dunham [9] found that students’ performance in a `graphics-calculator-enhanced college precalculus course’ improved signi®cantly for both sexes with
respect to visual items. Furthermore, in a pretest males showed greater competence in visual items but this gap closed in the post-test (cited in [8]). Nimmons
[7], using a comparative study design in a college algebra setting, found that on
visual items both male and female students in graphics calculator groups scored
signi®cantly higher than students in groups without the technology. Females in the
calculator groups showed greater gains than males in visualization skills and in the
level of understanding, which re¯ects Dunham’s ®ndings. However, Cassity’s [10]
investigation of the relation of the variable `mathematical performance at the
conceptual level’ to the variables spatial visualization, mathematical con®dence,
utilization of the graphics calculator in the classroom, basic algebra ability and
gender, reported no statistically signi®cant results relating to gender. Lawton’s
[11] study showed an overall improvement of mathematics performance with the
calculators but no signi®cant gender-speci®c di€ erences in performance.
In addition to performance, student attitudes and con®dence have been subject
to inquiry with regard to gender and again results are mixed. In brief, Dunham [9]
recorded signi®cantly higher positive correlations between task-speci®c performance and con®dence for females than for males at the end of the graphics calculator
enhanced course. In contrast, Lawton [11] found no signi®cant correlation
between mathematical performance and mathematical con®dence for males and
females. Almeqdadi [12] investigated student attitudes towards graphics calculators in a university calculus course and identi®ed signi®cant gender-related
di€ erences favouring males, no signi®cant di€ erences between the low and high
achievers, and no signi®cant interaction between gender, students’ achievement
level and students’ attitudes towards the calculators. Merckling [13] recorded that
use of the calculator in a high-school setting had a positive e€ ect on students’
804
P. A. Forster and U. Mueller
attitudes to learning mathematics, and that the result was particularly true for
males. Dunham’s [9] interviews with students suggested that low-con®dence
females and males relied on the calculator more than on algebraic approaches;
and high-con®dence females were more likely to use an algebraic approach, while
high-con®dence males were most likely to mix the methods. Boers and Jones [14]
observed that low-achieving students relied more on the calculator than other
students in examinations, and that this did not pay o€ . Nimmons [7] found that
overall retention in graphics calculator groups and the retention of females in
particular was greater than in non-calculator groups; and the calculator activities
seemed to bring about lasting positive changes in female students’ dispositions to
mathematics.
Thus, in regard to gender-related e€ ects, it is established that the overall
performance and understanding of both boys and girls can be enhanced by
inclusion of the calculators. Other aspects which are particularly relevant to our
inquiry include di€ erences in the extent to which algebraic and calculator methods
are used, uptake of the calculator for checking, visualization skills and use of the
technology according to ability level. To date, research ®ndings on these items are
inconsistent and have commonly been based on small sample sizes or given little
attention in the literature.
2.2. Data collection and analysis
The Curriculum Council of Western Australia who administers the TEE
supplied us with the marks for each candidate for each examination question
and with summary statistics for the population, for the years 1995±2000. The
summary statistics for 1995±1997 were based on total numbers of examination
candidates, and for 1998±2000 on scores of students who attempted the questions.
We retained the two approaches in our analysis by gender. In all cases, two-sample
t-tests were carried out to test the statistical signi®cance of di€ erences in mean
scores of boys and girls.
Markers of the Calculus TEE in 2000, who were all experienced in teaching
Calculus, were also a direct source of data. We asked them to record on a proforma,
for a minimum of twenty students each, whether students had used graphics
calculator or analytical methods in a question on rectilinear motion, and the part
marks scored. Data were collected for 588 of the total 1886 scripts. Examination
scripts from a school are spread among several bundles and bundles are allocated
randomly to markers. Each script is marked by two markers, and di€ erences
reconciled. The ®rst author, as a marker, collected similar data from 214 scripts for
a question on exponential growth. Inferring students’ use of graphics calculators
from their written answers is a subjective exercise so the results relating to the
rectilinear motion and exponential growth questions need to be read with this in
mind.
We draw on our previous classi®cation of characteristics of questions for the
1996±1999 examinations [4]. The classi®cation was extended for the present
inquiry to include the questions from the 1995 and 2000 examinations. The
characteristics we refer to were adapted from Senk, Beckman and Thompson
[15] and are:
(1) curriculum component;
E€ ect of graphics calculators on performance of boys/girls in calculus
805
(2) the role of a diagram in the solution (whether students were asked to
interpret a diagram, make one, or a diagram could have assisted the
solution); and
(3) the role of graphics calculators (questions were calculator active, where
there was a de®nite advantage in using the tool; neutral, where the question
could reasonably be answered with or without the tool; or calculator
inactive).
We each carried out the classi®cations separately and negotiated our di€ erences of
opinion to reach agreement. Further, when we refer to use of a graphics calculator
in the analysis we are referring to use of capabilities over and above those of a
scienti®c calculator.
2.3. Examination conditions
The Calculus TEE is a state-wide examination. It comprises one three-hour
paper. Total marks are 180. All questions are compulsory. Graphics calculators are
required equipment, and non-symbolic calculators and the Hewlett Packard
HP38G with limited symbolic capabilities are allowed. Students can take four
A4 pages of notes into the examination (two sheets written on both sides),
which was introduced to address the issue of di€ erent storage capacities of the
calculators.
3. Results
3.1. Participation and total examination scores
Table 1 shows the numbers of girl and boy candidates and the mean total raw
examination scores for girls and boys.
The ®gures in table 1 indicate that about twice as many boys as girls participate
in the Calculus TEE and that during the period from 1995 to 2000 the number of
male candidates decreased slightly (3% for the period), while the number of female
candidates decreased substantially (22%). Curriculum changes for 1995±2000
which could explain the declining candidature, in particular the greater decrease
for girls, are:
(1) increased use of technology in the teaching and learning of mathematics;
(2) the removal of the Calculus as a prerequisite for some Western Australian
university courses, with universities o€ ering bridging courses instead;
(3) the change of the university entry requirement from six subjects with
satisfactory performance to four.
That the retention of girls has been a€ ected more than the retention of boys is
most likely related to students’ career aspirations. There are three TEE
Number of girls
Girls’ mean mark
Boys’ mean mark
Number of boys
1995
1996
1997
1998
1999
2000
706.00
103.45
104.04
1387.00
660.00
104.97
104.19
1264.00
618.00
120.29
119.49
1269.00
562.00
99.01
98.55
1320.00
577.00
106.08
104.62
1358.00
549.00
102.37
97.82
1337.00
Table 1. Candidature and total raw examination scores for 1995±2000 Calculus TEE.
806
P. A. Forster and U. Mueller
5th Percentile
8.0%
95th Percentile
6.0%
4.0%
2.0%
0.0%
-2.0%
-4.0%
1995
Figure 1.
1996
1997
1998
1999
2000
Percentage di€ erence in scores between female and male candidates at the 5th
and 95th percentiles.
mathematics subjects in Western Australia. We note that only the candidature in
Calculus has decreased, whereas the number of students studying Discrete
Mathematics has increased and the number of students in Applicable Mathematics
has remained stable. Of the three subjects, Discrete Mathematics requires the least
strong mathematics background and Calculus the strongest.
The ®gures in table 1 show also that the mean total raw examination scores in
Calculus for girls were greater than the mean scores for boys in the ®ve years from
1996 to 2000, with the result from 2000 being statistically signi®cant at the 98%
level of signi®cance. The year 2000 examination is the only one for which the
di€ erence in mean scores (4.6 or 2.6%) was substantial (see table 1). Throughout
the time period, at the lower end of the achievement scale girls obtained higher
scores than boys, while boys consistently outperformed girls at the upper end. The
performance gap between boys and girls at the 5th and 95th percentiles is shown in
®gure 1. This gap is much wider at the bottom end (5th percentile) than at the top
end, with percentage di€ erences in favour of girls at the bottom end twice as large
as those favouring boys at the top end. The percentage di€ erence between scores of
girls and boys at the 5th percentile ranges from 3.9 to 6.6% and at the 95th
percentile between 0.6 and 2.8%. In addition, progressive shifts occurred in the
percentiles above which performance of boys is superior to that of girls (30th
percentile in 1995, 40th in 1996, 65th in 1997, 70th in 1998, 65th in 1999 and 90th
in 2000). These outcomes are explained in part by the lower participation in TEE
Calculus by girls, where one aspect is that fewer girls at the lower end of ability
take the subject.
We sought also to explain the signi®cantly better mean performance of girls in
2000 in terms of the extent to which use of graphics calculators was possible in
examination questions. However, there was no clear link. For 1998±2000 when
graphics calculators were allowed, the percentages of part-questions which were
technology active or neutral (1998, 27%; 1999, 39%; and 2000, 36%).
Next we investigated the call on visualization as measured by the percentages of
part-questions in which there was a role for a diagram in the solution. The ®gures,
E€ ect of graphics calculators on performance of boys/girls in calculus
807
(1995, 32%; 1996, 28%; 1997, 24%; 1998, 47%; 1999, 45%; and 2000, 48%), did not
explain the di€ erences in total scores for boys and girls in the year 2000 examination. The increase in the role of diagrams between 1997 and 1998, upon the
calculators ®rst being introduced, would have been when an e€ ect on scores could
have been expected.
3.2. Results by curriculum component
In this section we report mean scores on questions grouped by predominant
curriculum component. We classi®ed each question on the 1995±2000 examinations as relating to one curriculum component. Where parts of a single question
related to di€ erent curriculum components, the question was classi®ed as belonging to the component which attracted most marks. This decision arose because we
had marks for whole questions but not part-questions. Cumulative distributions
by gender for the curriculum components Theory and Techniques of Calculus,
Complex Numbers, Function and Limits and Applications of Calculus for the
years 1995 and 2000 are shown in ®gure 2. They are representative of trends in the
distributions for the six years 1995±2000.
In reporting the results, we start with Theory and Techniques of Calculus, the
component having the strongest trend in regard to scores by gender, and ®nish
with Vector Calculus, the component having the least consistent results by gender.
In all six years, di€ erences between girls’ and boys’ scores in questions relating
to Theory and Techniques favoured girls (see table 2). In ®ve of these years the
di€ erences were statistically signi®cant. The greatest absolute di€ erences in mean
marks was 1.50 in the year 2000 (0.8% on the examination). With Theory and
Techniques questions, the use of graphics calculators and diagrams can assist the
solution, for example, a graph can be used to help with establishing where a
function is di€ erentiable. However, for questions in this component there is
typically a minimal call or no call on the use of graphics calculators and diagrams
(see table 2). Thus, Theory and Techniques questions have mainly required
analytic methods, both before and since the introduction of graphics calculators,
and girls have regularly done better on them, which is consistent with the ®ndings
of Boers and Jones [8]. Plots of the cumulative distribution of percentage marks for
boys and girls for all years 1995±2000 highlight the consistency and show that the
trend of superior performance by girls persists at all levels of achievement (see
®gure 2 for the 1995 and 2000 distributions).
Complex Numbers is the component with the next most consistent outcomes in
relation to gender after Theory and Techniques. In ®ve of the six years from 1995±
1995
Maximum score
Girls’ mean mark
Boys’ mean mark
¤
26.00
¤
17.98
¤
17.32
Part-questions with a role for
Graphics calculators
Ð
Diagram
17%
¤
95% con®dence level,
¤¤
1996
¤¤
1997
¤¤
1998
¤¤
1999
2000
¤¤
35.00
¤¤
24.33
¤¤
23.07
36.00
¤¤
29.80
¤¤
28.83
28.00
¤¤
20.34
¤¤
18.91
29.00
19.14
18.73
25.00
¤¤
19.06
¤¤
17.56
Ð
11%
Ð
8%
0%
0%
22%
11%
0%
0%
99% con®dence level
Table 2. Results for the population on Theory and Techniques of Calculus questions.
808
P. A. Forster and U. Mueller
Distribution TTC 1995
1
Distribution TTC 2000
1
0.8
0.8
0.6
0.6
0.4
0.4
F
0.2
F
0.2
M
0
M
0
0
0.2
0.4
0.6
0.8
1
Distribution CN 1995
1
0
0.8
0.6
0.6
0.4
0.4
F
0.2
0.4
0.6
0.8
F
0.2
M
0
1
Distribution CN 2000
1
0.8
0.2
M
0
0
0.2
0.4
0.6
0.8
1
Distribution F&L 1995
1
0
0.8
0.6
0.6
0.4
0.4
F
0.2
0.4
0.6
0.8
F
0.2
M
0
1
Distribution F&L 2000
1
0.8
0.2
M
0
0
0.2
0.4
0.6
0.8
1
Distribution AC 1995
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Distribution AC 2000
1
0.8
0.6
0.4
M
0
0.2
0.4
0.6
0.8
F
0.2
F
M
0
1
0
0.2
0.4
0.6
0.8
1
Figure 2. Cumulative distribution of % raw scores in Theory and Techniques of Calculus
(TTC), Complex Numbers (CN), Functions and Limits (F & L) and Applications of
Calculus (AC).
E€ ect of graphics calculators on performance of boys/girls in calculus
1995
Maximum score
Girls’ mean mark
Boys’ mean mark
1996
1997
¤
1998
1999
¤
809
2000
¤¤
35.00
14.94
14.59
41.00
23.76
23.54
38.00
¤
22.45
¤
21.54
42.00
21.38
21.4
34.00
¤
16.42
¤
15.6
36.00
¤¤
17.79
¤¤
16.42
Part-questions with a role for
Graphics calculators
Ð
Diagram
56%
Ð
46%
Ð
38%
27%
55%
30%
70%
22%
56%
¤
95% con®dence level,
¤¤
99% con®dence level
Table 3. Results for the population on Complex Number questions.
2000, di€ erences favoured girls and for three of these years the di€ erences were
statistically signi®cant (see table 3). The greatest absolute di€ erence in mean
scores was 1.37 in 2000 (0.76% on the examination).
The introduction of graphics calculators led to noticeable changes in the types
of questions. As indicated in table 3, there has been a role for diagram in the
majority of complex number questions and this has principally involved the
construction and interpretation of Argand diagramsÐdrawn without the assistance
of a calculator, which is re¯ected in the relatively low ®gures relating to calculator
use. With the introduction of the graphics calculators, the number of questions
related to arithmetic and to conversion from algebraic to polar representation
(or the reverse) decreased substantially. This resulted in lower average scores in
1998±2000 compared to 1995±1997. Girls mean percentage scores out of the
maximum possible scores for this component (see table 3) were 51%, 48% and
49% for 1998±2000 and 43%, 58% and 59% for 1995±1997. Boys’ scores showed the
same trend. The aggregated ®gures in table 3, however, do not imply performance
by gender is linked to use of diagrams or to opportunities for calculator use.
However, a question-by-question analysis of the 29 questions, over the six years,
indicated that girls performed signi®cantly better than boys on nine questions.
They variously required interpretation of a diagram, the construction of a diagram,
optional use of a diagram, or included no role for a diagram; of these, one in 1999
was technology neutral in that it allowed a graphical solution to an equation.
Boys performed signi®cantly better than girls on three questions. One of these,
in 1998, required students to make a diagram and was technology neutral (the
diagram could have been produced on the calculator); in the other two (in 1998 and
1999), use of an Argand diagram, generated without the technology, could have
signi®cantly assisted the solution. Thus, for Complex Numbers, use/non-use of a
diagram is a stronger predictor of signi®cance than opportunities to use technology. Where signi®cant di€ erences in performance occurred, all di€ erences on
questions that did not involve a diagram (i.e. required analytic methods or
computation) favoured girls. All di€ erences that favoured boys were on questions
that allowed use of a diagram.
With regard to the distribution of scores, there was virtually no di€ erence in
performance between boys and girls at the upper end of the performance scale in
any of years 1995±2000 (see ®gure 2 for the 1995 and 2000 plots). The trend of
superior performance by girls was evident at lower levels of achievement, and was
pronounced and extended highest up the scale for 2000, which again is explained
by the declining participation in calculus by girls.
810
P. A. Forster and U. Mueller
1995
Maximum score
Girls’ mean mark
Boys’ mean mark
1996
1997
1998
1999
¤¤
2000
¤
15.00
8.84
8.68
35.00
22.16
22.48
27.00
19.94
20.15
26.00
14.08
13.85
33.00
¤¤
19.02
¤¤
18.13
28.00
¤
19.33
¤
18.74
Part-questions with a role for
Graphics calculators
Ð
Diagram
33%
Ð
36%
Ð
45%
67%
67%
50%
50%
36%
45%
¤
95% con®dence level,
Table 4.
¤¤
99% con®dence level
Results for the population on Functions and Limits questions.
Di€ erences between girls’ and boys’ scores on Functions and Limits questions
favoured girls in four of the six years (see table 4) and were signi®cant in 1999 and
2000, after the introduction of the calculators, but the greatest absolute di€ erence
in mean scores was small (0.5% on the 1999 examination). However, a number of
aspects of the questions from the curriculum area Functions and Limits are of
interest. In 1998, the year in which the calculators were ®rst allowed, there was
ample opportunity for calculator use and this was accompanied by an increased
role for diagrams in the solution process (see table 4). For example, graphs of
functions produced on the calculator could be used to establish or check limiting
values, continuity and di€ erentiability. Checking has been suggested as being
particularly bene®cial to girls [5, 6], but the results by gender in table 4 lend little
support for this. The opportunity to use diagrams and graphics calculators
progressively decreased in 1999 and 2000, and at the same time the requirement
to use algebra increased. Hence, the signi®cant di€ erences in scores that favoured
girls in those years might again indicate superior performance by girls on
analytical methods. However, the cumulative distributions by gender for 1999
and 2000 show that girls at the lower and middle levels of the achievement
scale bene®ted most from the change in emphasis (see ®gure 2 for the 2000
plot). The distributions for years up to and including 1998 indicate minimal
di€ erence in achievement across the scale of performance (see ®gure 2 for the 1995
plot). Lower con®dence/lower achieving students have been observed to choose
calculator approaches when they were an option, more frequently than algebraic
methods [9, 14] and there were still opportunities in 1999 and 2000 where this was
possible. Thus, it is not clear to what girls’ superior results in Functions and
Limits can be attributed. A question-by-question analysis did not illuminate the
issue. Actual use of the calculators by gender in this component needs to be
ascertained.
In Applications of Calculus, di€ erences in performance favoured boys in four
out of the six years, were signi®cant in two of those years (1995 and 1998), and
substantial to the extent of 1% on the 1995 examination (see table 5). However,
girls have closed the achievement gap at virtually all levels: while the cumulative
distribution of percentage scores for 1995±1998 indicated superior performance by
boys for the major part of the scale, in 1999 the plots for boys and girls were
virtually identical and in 2000 girls recorded marginally superior performance for
nearly the whole scale (see ®gure 2 for the 1995 and 2000 plots).
E€ ect of graphics calculators on performance of boys/girls in calculus
1995
Maximum score
Girls’ mean mark
Boys’ mean mark
89.00
¤
53.27
¤
55.10
Part-questions with a role for
Graphics calculators
Ð
Diagram
23%
¤
¤
1996
1997
1998
¤
811
1999
2000
59.00
27.89
27.68
61.00
37.31
38.27
73.0
¤
36.7
¤
38.4
72.0
46.5
46.6
77.0
39.7
38.6
Ð
22%
Ð
25%
30%
55%
45%
45%
48%
57%
95% con®dence level
Table 5.
Results for the population for Applications of Calculus questions.
Since the introduction of the calculators, opportunities to use them in Applications and the role of diagrams have been relatively high, but the ®gures in table 5
do not suggest any link between students’ performance by gender and these
question characteristics. So, we conducted a question-by-question analysis for
the six years to explain the di€ erences in performance. The style of the question
depends on the Application, so we grouped the questions by Application for the
analysis.
In questions on simple harmonic motion and rectilinear motion, boys’ mean
performance was better than that of girls in ®ve of the six years from 1995 to 2000.
All signi®cant di€ erences (recorded in two years for simple harmonic motion and
in ®ve years for rectilinear motion) favoured boys and occurred on questions where
there was a role for a diagram. In most instances, the opportunity or requirement
to use a diagram occurred in several parts of the questions. Sometimes the diagram
was provided and at other times it could be generated on a calculator. The
requirement to interpret a diagram was more consistent with signi®cant di€ erences
than was using the technology.
The questions on rectilinear motion where signi®cant di€ erences occurred
were relatively hard questions, as measured by students’ percentage scores. They
called on high-level interpretation, in particular where velocity and acceleration
were involved, or when a solution required interpretation of a given context [2].
However, that signi®cant di€ erences occurred on hard questions and that these
favoured boys was not consistent for the other Application questions.
For questions on volumes of solids of revolution, di€ erences in performance
favoured girls in four years and were signi®cant in those years. The role diagrams
played was minor. Graphics calculators could be used to evaluate the volume
integrals after their derivation. The other applications (including area under a
curve, optimization, related rates and incremental change, continuous growth and
decay) did not show consistently superior performance by one gender group and
did not have signi®cant di€ erences favouring one gender.
There is typically only one question on Vector Calculus in the Calculus TEE.
There was little opportunity in the questions for use of a diagram for the
years 1995 to 1999 or for the use of a graphics calculator in 1998 and 1999,
indicating the call was mainly on analytic methods. However, unlike for Theory
and Techniques, these characteristics were not associated with consistently better
performance by girls: in fact when di€ erences were signi®cant they favoured boys
(see table 6).
812
P. A. Forster and U. Mueller
1995
Maximum score
Girls’ mean mark
Boys’ mean mark
15.00
8.42
8.35
Part-questions with a role for
Graphics calculators
Ð
Diagram
17%
¤
95% con®dence level,
¤¤
1996
¤¤
1997
1998
1999
¤¤
2000
10.00
¤¤
6.84
¤¤
7.42
18.00
10.79
10.71
11.00
6.46
6.39
12.00
¤¤
5.43
¤¤
6.23
14.00
6.62
6.79
Ð
0%
Ð
0%
0%
0%
25%
25%
50%
50%
99% con®dence level
Table 6. Results for the population on Vector Calculus questions.
3.3. Analysis of the year 2000 question on rectilinear motion
In the following two sections we report the extent and nature of students’
actual calculator usage as ascertained by perusal of students’ examination scripts
instead of relying on ®gures relating to possible calculator usage. First, we discuss
a question on rectilinear motion and then consider a question on continuous
growth. Both questions are from the year 2000 Calculus TEE.
The question on rectilinear motion was attempted by 541 of the 549 girl
candidates and 1310 of 1337 boy candidates. The mean score for girls was 6.05 and
for boys was 6.67 out of the possible 11 marks. The di€ erence in mean was
statistically signi®cant at the 99% level. Data were collected for 588 examination
scripts. A chi-square test of goodness of ®t of the distribution of marks within the
sample showed the sample to be representative of the results for the entire
population at the 90% con®dence level. Technology usage was inferred when
students provided an answer without including any written solution steps. The
question is as follows
A hot air balloon begins a 60 minute ¯ight by rising upwards from the side of a
hill. Its vertical velocity v (metres per minute) is given by:
v ˆ t…t ¡ 35†…t ¡ 60†=1400
where t is the time from the start in minutes.
(a) What is the maximum upward velocity?
(b) While the balloon is ascending, when is its acceleration the greatest?
(c) When does the balloon reach its maximum altitude and how far above its
starting point is it at that time?
(d) Does the balloon land above or below its original elevation? Explain your
reasoning.
Sample data are summarized in table 7. Data collected for part (d) are not included
in the summary because the use of the calculators was di cult to discern.
Part (a) could be solved by graphing the velocity function with a graphics
calculator and using its in-built capabilities to `jump’ to the maximum turning
point of the graph. An analytic approach required expansion and subsequent
di€ erentiation of the expression or di€ erentiation using the product rule, and was
more time consuming and potentially more error prone than the graphical
approach. Since justi®cation of the answer was not required, the graphical solution
would have been the more practical approach to take.
E€ ect of graphics calculators on performance of boys/girls in calculus
Females
Answer only
Calculus working
Males
Answer only
Calculus working
a
813
Part a
Part b
Part c
Number Average
studentsa markb
Number Average
students a markb
Number Average
students a markb
¤¤
89
69
1.70
¤¤
1.17
262
160
1.74
¤¤
1.46
¤¤
¤¤
36
113
1.08
¤¤
0.75
136
268
1.43
¤¤
0.97
¤¤
¤¤
27
118
2.63
¤¤
2.27
107
276
3.15
¤¤
2.23
¤¤
Sample size: 161 females, 427 males
maximum 2, 18b maximum 2, 18c maximum 4
99% con®dence level
b
18a
¤¤
Table 7. Graphics calculator usage by gender for the 2000 rectilinear motion question.
Part (b) could also be solved by relying entirely on in-built calculator functions.
After entering the velocity function into the graphics calculator, the interval
[0, 35] over which the balloon was ascending could be determined visually
and the derivative (acceleration) could be plotted without calculating the
symbolic form. The global maximum acceleration was needed and the time at
which it occurred …t ˆ 0† could read from the graph. Alternatively, the interval
[0, 35] could be determined by inspection, from the velocity function; and the
problem could be treated as a constrained optimization problem, requiring
investigation of all interior critical points and endpoints. The graphical approach
had the advantage of directly focusing attention on the endpoints. For parts (a)
and (b), sample data indicated that boys who chose to use their calculators were
the group who obtained the highest scores on average, followed by girls who
used their calculators, then boys and, last, girls who used analytic approaches (see
table 7).
Part (c) built on part (b) as it required the use of the interval on which the
velocity was positive. Once the integral
of the velocity function was set up it could
„ 35
be evaluated on the calculator … 0 v dt ˆ 216:927 m†. The analytic approach was
more cumbersome and error prone. The order of achievement was the same as for
parts (a) and (b) except that, with analytic approaches, the mean score for girls was
greater than the mean score for boys (see table 7). On all parts (a) to (c) of the
question a clear majority of each sex chose to use numerical integration in
preference to evaluating the integral over the velocity function exactly and a
greater percentage of boys than girls chose to use it. This pattern of choosing/not
choosing to use the technology matches the pattern recorded by Boers and
Jones [8].
For part (d), the solution involved observing that the area enclosed by the
velocity function
and the horizontal axis was greater for [0, 35] than for [35, 60]; or
„ 60
evaluating 35 v dt (by calculator or
„ 60by hand) and comparing the result with the
answer for part (c); or evaluating 0 v dt and noting that it was positive. For an
analytic solution the integral expression from part (c) could be used. In the sample,
part (d) was answered by 130 girls (mean score 1.93, out of a maximum of 3) and
by 342 boys (mean score 2.15). We note that other aspects of this problem were
814
P. A. Forster and U. Mueller
Question 13a-d
Girls
Boys
a
Question 13e
Number
studentsa
Average
markb
459
1329
6.29
¤¤
5.94
¤¤
Question 13f
Number
students a
Average
markb
Number
studentsa
Average
markb
492
1133
1.91
1.98
497
1152
0.83
¤
0.87
¤
number in the cohort girls ˆ 549, boys ˆ 1337
maximum 10, 13e maximum 3, 13f maximum 1
99% con®dence level
b
13a-d
¤¤
Table 8.
Performance for the population on the 2000 question on continuous growth.
the need to make sense of the context of a hot air balloon in mathematical terms,
and the possibility of mixed calculator/analytic approachesÐthese were coded as
`Calculus working’ (see table 7).
3.4. Analysis of the year 2000 question on continuous growth
The year 2000 question on continuous growth was attempted by 549 girls
(mean score 8.75) and 1329 boys (mean score 8.38). The di€ erences between
boys’ and girls’ performance were statistically signi®cant at the 95% level and
girls achieved greater mean scores than boys. The question is given below, and
table 8 shows performance by gender by part-question. The part-mark data were
recorded for this question, at our request, on the central marks collection forms for
the examination and provided to us by the Western Australian Curriculum
Council.
The size of P…t† of a population of bacteria in a culture at time t minutes is
modelled by the equation
dP=dt ˆ P ¡ P2 =1000
…1†
(a) For which values of P is the growth rate dP=dt zero?
(b) For which value of P is the growth rate greatest?
¡
(c) Show by di€ erentiating that P ˆ 1000=…1 ‡ Ce t † satis®es equation (1), for
any value of the constant C.
(d) Find C, given that at time t ˆ 0 the size of the population is 100.
(e) Sketch a graph of P as a function of t.
(f) What is the limiting size of the population as t ! 1?
Part (a), worth 1 mark, was solved easily by factorization. It could have been
solved graphically, but less e ciently, on the calculator. Part (b), worth 2 marks,
was easily solved by calculating the derivative by hand, equating it zero and solving
the equation by hand to obtain the maximum. Part (c), worth 5 marks, was a high
level question and required sophisticated algebra. Part (d), worth 2 marks,
required relatively simple algebra. Thus, the aggregated marks for parts (a) to
(d) were associated in large part with algebraic competence.
Part (e) required the generation of a graph (see ®gure 3). The given function is
not part of the TEE Calculus and its unfamiliarity to students would have forced
E€ ect of graphics calculators on performance of boys/girls in calculus
f(x)
1000
900
800
700
600
500
400
300
200
100
(500, 2·19) point of inflection
5
Figure 3.
Girlsa
Boysa
815
10
15
20
x
¡
The graph of P ˆ 1000=…1 ‡ 9e t †.
Adequate
range (e)
Correct
y intercept (3)
Correct
shape (e)
Correct
limit (f)
45
63
53
67
51
64
75
90
a
Number in the sample girls ˆ 57, boys ˆ 157; number who attempted (e) girls ˆ 47,
boys ˆ 128; number who attempted (f) girls ˆ 48, boys ˆ 131.
Table 9. Sample data on % of students with correct answers to (e) and (f) when
attempting them.
calculator generation of the graph. The process involved prediction of the limit to
in®nity in order to obtain the range for an adequate screen display, thus algebraic
thinking was integral to the graphing (unless a calculator with an autoscale facility
was used). Then, when drawing the graph, students needed to be aware of the
horizontal asymptote and draw it in or at least ensure that their graph did not drift
above f …x† ˆ 100. The graph did not need to be interpreted, except for part (f ), but
the limit required in (f ) was ideally obtained before the graph was produced.
The 214 scripts marked by the ®rst author showed little if any working for the
graphing, which indicates that the graphs were usually obtained using a graphics
calculator. The sample data are summarized in table 9. They indicate that girls had
more problems than boys in constructing the graph and in establishing the limit.
About one third of girls and boys who identi®ed the limit did not obtain an
adequate range, indicating that they failed to make the connection between the two
quantities.
Boys achieved marginally better results on the graph for part (e), and
statistically signi®cant but not substantially better results for the limit (see table
8). Thus, for this question, again girls were stronger with analytic methods and
boys showed superior performance with the part that involved a diagram. The
mathematics in some parts of this question was challenging, but the context of
bacterial growth did not in itself result in complexity, unlike the motion of the hot
air balloon in the previous question.
816
P. A. Forster and U. Mueller
4. Conclusions
There are a number of variables relevant to boys’ and girls’ di€ erential
performance in the Calculus TEE. The most prominent is the much smaller
number of girls choosing to study Calculus at the upper secondary level. This can
explain why overall examination performance for girls at the lower end of
achievement of the cohort is better than that for boys, and why girls’ mean scores
were consistently higher than boys’ mean scores over the ®ve years from 1996 to
2000. However, boys continue to outperform girls at the top end of the scale.
Secondly, in questions that required analytic solution methods, for example in
questions from Theory and Techniques and Complex Numbers, analysis indicates
that girls regularly scored better than boys. Signi®cant di€ erences in performance
favouring boys occurred only on questions where diagrams played a role in the
solution, for example in questions on Complex Numbers and Applications. Finegrained analysis of students’ actual calculator usage in two questions from the 2000
Calculus TEE supports these generalizations. Furthermore, results on the Complex Number and Applications components indicated the variable `role of diagram’
is a stronger predictor of outcomes by gender than `opportunities to use graphics
calculators’. However, girls performed signi®cantly better than boys on some
questions where a diagram played a role; and in some instances, diagrammatic or
graphical approaches are only feasible if the calculator is used.
The di€ erences in performance over the six years 1995±2000 were substantial
to the extent of 2.6% on the 2000 examination, the only year in which di€ erences in
overall performance were statistically signi®cant. The greatest source of di€ erence
in that year was the component Theory and Techniques (0.8% in the mean
favouring girls) and this was greater than the e€ ects relating to other components
in all six years, except in 1995 Applications contributed 1% which favoured boys.
After Functions and Limits, Applications has allowed most opportunity for the use
of graphics calculators. While data on actual calculator usage by gender are not
available for Functions and Limits, sample data for the rectilinear motion
application in 2000 indicated that the majority of boys and girls chose to use the
technology when it was an option and scored better than students not choosing it;
and a greater percentage of boys than girls chose it and scored better than the girls.
There are important time and accuracy advantages arising from choosing appropriately to use the calculator in the context of the Calculus TEE. Complexity
introduced by context was another aspect of the questions on rectilinear motion.
In conclusion, our study revealed trends in overall performance and in some
curriculum components in the Calculus, and ®ne-grained analysis highlighted the
trends. Anomalies are that increased opportunity for visual methods upon introduction of the calculators has not been accompanied by higher mean total scores in
the examination for boys; and that Vector Calculus questions have called mainly
on analytic methods, yet have not shown a pattern of superior performance by
girls. Further investigation into actual calculator usage by gender, at di€ erent
levels of achievement, and taking into account the level of di culty of questions
are directions for future research which might further explain the discrepancies.
Because of the anomalies and the unequal sizes of the boys’ and girls’ groups,
we are hesitant to make broad generalizations as a result of the study, nor do we
claim our ®ndings are applicable across all mathematics subjects. However, if
con®rmed in other contexts, the trends that we have identi®ed have important
implications for the equitable setting of examinations which screen students’ entry
E€ ect of graphics calculators on performance of boys/girls in calculus
817
into university courses, where entry for some courses is highly competitive. A mix
of question types that overall are not likely to favour one sex over the other would
be recommended. Implications for preparing students for such examinations
where graphics calculators are included are that girls might need to be encouraged
to embrace the use of the technology and the visual approaches that it allows, and
competence with analytic methods might need to be emphasized to boys.
Acknowledgements
We sincerely thank the markers who contributed to our inquiry and the
Curriculum Council of Western Australia for their support and permission to
copy the two questions from the year 2000 Calculus TEE paper. Copyright for
these questions belongs to the Curriculum Council. Comments in this paper are
not to be taken to represent views of the Curriculum Council.
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