Calculus for Kids
Andrew Fluck, Christopher K.H. Chin, Dev Ranmuthugala, Irene Penesis
CALCULUS FOR KIDS
Andrew Fluck, Christopher K.H. Chin, Dev Ranmuthugala, Irene Penesis
University of Tasmania and Australian Maritime College
Abstract
The ‘Calculus for Kids’ project builds on ‘Calculus in Primary’ which was conducted with
final year primary school and first year secondary school students aged between 10-12
years old in four Australian states. Classroom teachers were trained to provide instruction
in the use of MAPLE mathematics software. They taught their students in 1:1 laptop
classrooms (and one computer laboratory) to use MAPLE to solve real-world problems
using integral calculus. After eleven lessons, the students took a version of the first year
engineering degree calculus examination, where they gained an average Distinction grade.
The project was significant because it showed very young students could achieve at much
higher levels when using computer technology. The resulting discussion can examine the
ethics of withholding such support in mainstream classrooms and what we mean by
‘knowing calculus’. Both questions are reviewed in this presentation.
‘Calculus for Kids’ was designed to extend the project to include lessons in which similarlyaged students devise mathematical models to describe real-world activities. With funding
from the Australian Research Council (linkage project LP130101088) the project has been
extended to more states and schools over a longer term of engagement.
Introduction
Educators have long held high hopes for the use of computers in schooling. These hopes have included
access to advanced ideas at a younger age. Seymour Papert suggested this would be possible in the
area of mathematics:
Many topics that were unteachably abstract in the context of pencil and paper technologies
will be considered as appropriate for children in the context of a digital technology that
makes the previously formal become concrete. (Papert, 2000).
The implications are that the right choice of software can provide students with opportunities for
practice and rapid feedback in a motivating environment or have higher order cognitive goals
(Wenglinksy, 1998). The Calculus for Kids project was created to take advantage of this potential. It
also provides a chance to address a looming shortage of mathematics skills in Australia.
Universities have reported falling numbers of enrolments in tertiary mathematics, but in New South
Wales at least, pre-tertiary mathematics completions with studies involving calculus have remained
fairly constant since 2001 (MANSW, 2014). In the USA, SimCalc (Kaput Center, 2014) has been used
to assist 12 to 14 year old students develop conceptual understanding of key concepts in Calculus
(Rochelle & Kaput, 1996). This provided the impetus to re-think the curriculum by considering that
the “most effective way of promoting learning is to embed basic skills instruction within more
complex tasks” using computer technology (Roschelle, Pea, Hoadley, Gordon & Means, 2000).
Further studies with SimCalc revealed student-level effect sizes of .63, .50, and .56 (Roschelle,
Shechtman, Tatar, Hegedus, Hopkins, Empson, Knudsen, & Gallagher, 2010), all above the ‘hinge
point’ of .4 (Hattie, 2009). An effect size of 1 is typically associated with advancing learners'
achievement by one year, or improving the rate of learning by 50%, and corresponds to one standard
deviation. Effect sizes above .4 are unusual in educational research. This demonstrates the correct use
of software has significant potential in addressing the shortage of these STEM (science, technology,
engineering and mathematics) skills.
Australia’s international ranking for school mathematics is declining. In the international PISA studies
it was ranked 8th in 2003, 9th in 2006 (DEEWR 2008) and 15th in 2010 (PISA 2010), placing
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Calculus for Kids
Andrew Fluck, Christopher K.H. Chin, Dev Ranmuthugala, Irene Penesis
Australia below several Asian countries and New Zealand. In his 2011 report, Gonski recommended
the need for additional investment (Gonski et al., 2011). “Australia’s weak performance in reading and
mathematics…illustrates a serious cause for concern and suggests significant education reform is
needed…”
The obvious disjunction between ICT use in the world of work and the less consequential uses in
schools can be seen as a reason why students feel disenfranchised and disengaged with education. The
responses of some education systems can seem quite dramatic. For instance, the US state of Indiana no
longer requires cursive writing to be taught because of the perceived importance of keyboards
(Loughlin, 2011).
Therefore, the lack of STEM skills appears to be an important problem which could well be addressed
by the correct choice of software and an appropriate pedagogical approach.
Literature
Expectations computers will transform learning
There has been speculation that children can learn concepts, including mathematics, at younger ages if
they have free access to computers, are as ICT-literate as they are reading-literate, and are unfettered
by traditional age-related achievement. A first demonstration using the acquisition of systems concepts
(Resnick, 1998) has hinted at the validity of this idea. Following the 2007 Australian Federal election,
the first item on the new cabinet’s agenda was computers in schools. An interview with Mark Pesce
(2007) highlighted ICT in schools as a Trojan horse to force teachers to think about not only their
training but also changing the curriculum. Principals interviewed on the same program expressed their
expectations that the $1.2 billion initiative (Rudd, Smith & Conroy, 2007) would provide a real vision
and move the system forward. The Australian Government committed to a Digital Education
Revolution with a focus on schooling in Years 9–12, where students nationwide were to be provided
with computer access throughout every school day.
Disjunction between ICT in school and at work
We witness on a daily basis the stark difference between traditional calculus instruction in schools and
practical applications by professional engineers. Traditionally students have been taught how to
integrate a function from first principles using a series of rules or patterns they memorise. This helps
students to understand how to integrate a new function in the future. However, as the catalogue of
function integrals grows, the use of poorly-memorised results using ‘first principles’ can impede
practical calculation. Therefore professional engineers use a variety of specialist software to ‘crunch
the numbers’. One might argue that reliance on computing equipment in engineering is analogous to
the widening use of word processors in lieu of pens in newspaper offices; there is certainly a
discussion to be had about the way these technologies redefine the underlying skills or their
acquisition. The crucial transformational role of ICT in schooling is underlined by the need to
introduce ICTs as an integral component of broader curricular reforms that are changing not only how
learning occurs but also what is learned – identified for the Australian government in Making Better
Connections (Downes, Fluck et al., 2002).
Some educational institutions are realising the importance of bringing engineering applications of
calculus into the mathematics classroom. Horowitz and Ebrahimpour (2002) described the use of
Matlab software at Penn State University (USA) to solve optimisation problems and predict the effect
of drag forces. Tang, Ram and Shah (2005) used multimedia instructional materials and the Maple
software (Maplesoft, 2014) to work on the inventory control problem and do curve fitting. The use of
computers to better match the activity of professional engineers was a feature in both these
approaches.
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Calculus for Kids
Andrew Fluck, Christopher K.H. Chin, Dev Ranmuthugala, Irene Penesis
Some major reports into the efficacy of ICT as a support for student attainment in numeracy and
literacy showed it can be limited when used in an inappropriate curriculum (e.g. Robertson & Fluck,
2006 and Dynarski et al., 2007, the former carried out for the Australian Research Council). Parr
(2000) showed ICT is about as effective as other methods for improving education, such as decreasing
class sizes, when assessed using non-ICT based tests. Therefore we argue the impact of ICT as a
general capability to assist the teaching of conventional subjects is likely to be severely limiting when
assessed using conventional methods.
Method
The aim of the project was to show that advanced mathematical concepts can be taught to and
understood by primary school students, who can then use these new skills to solve real-world
problems usually only attempted by senior secondary students. This was achieved using specialist
computer software and the development of an integrated ICT system, curriculum, and associated
teacher training. The trial was conducted using a sample of students from 5 schools. The project aimed
to:
 introduce techniques to assist children to learn and better understand concepts such as
areas and volumes;
 develop improved ways of teaching children to understand the structure of mathematical
equations through optimising the mechanisms they use to symbolically represent those
equations;
 create learning modules for integral calculus and differential equations accessible by
primary students;
 enable students to collaborate in problem-solving activities during the learning process.
Our main conceptual tools were Rogers’ theory of innovation adoption, the non-template problemsolving method of Allen (2001), a realistic mathematics education approach (Gravemeijer, et al, 1999),
and a methodology we developed on the transformational use of ICT in school education which
combines professional software tools with multimedia instructional materials.
The basic design for the research was an intervention method using a cyclic approach, consisting of
four stages: produce/modify procedures and material, train the teachers, intervention in schools, and
assess results. We recruited schools for the project from four Australian states, ensuring a wide range
of ICSEA scores (see Table 1). The local facilitators were chosen by each participating school and
attended a one-day training session at the University of Tasmania. The preference was for situations
where students were allocated a laptop for the duration of the program; however, due to the
unavailability of equipment, a class in one school used desktop computers in a laboratory.
In each teaching package we provided materials (see Figure 1) for 12 one-hour modules to be taught
over six weeks with links to worked examples in MAPLE worksheets and extension activities for
students to autonomously consolidate learning. The material consisted of real-life situations presented
through high-visual-impact media that students can understand and apply the mathematical techniques
to solve them. Our challenge was to provide teachers with the confidence that they could master and
convey the material. This included grounding in the operational aspects of the mathematics in the
training day, followed by the other calculus topics in subsequent sessions. Ethics approval was gained
to use a purely post-test method since we understood virtually no student would have learned calculus
beforehand. The post-test contained 14 application questions taken from the first year engineering
calculus examination, with one affective question: “What is calculus good for”? Students were
allowed to use the MAPLE software as they undertook the post-test.
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Calculus for Kids
Andrew Fluck, Christopher K.H. Chin, Dev Ranmuthugala, Irene Penesis
Figure 1 Example of interactive learning materials
The highly visual teaching format used full-colour cues, both stationary and animated, to create a
delivery tool that suited the learning styles of both girls and boys, in a topic that is currently skewed
towards males both in terms of school-curriculum and career choice. Groups of upper-primary-school
children from five schools followed the learning module during one term. The initial lessons focused
on the use of MAPLE, to ensure students could use this tool to solve familiar problems. Once the
operational aspects of the software had been mastered, the concept of integral calculus was introduced
in the fifth lesson. By the eighth lesson, students were quite happy to put together a combination of
definite integrals (as per Figure 1). They went on to solve problems using integral calculus involving
real-world applications, such as the quantity of wood chips required to fill a curved garden bed and
how much paint will be needed for a decorated theatre wall.
Results and Discussion
Our small proof-of-concept pilot project addressed the major issue of STEM skill shortages faced by
many Australian universities. To our surprise, the students scored highly on the university-level posttest, gaining Distinction and Credit grades. Table 1 provides a breakdown of the results by school and
gender.
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Calculus for Kids
Andrew Fluck, Christopher K.H. Chin, Dev Ranmuthugala, Irene Penesis
Table 1 Student demographics, location, school advantage, and overall performance by
gender in post-test questions
9
9
17
12
7
n
11
10
8
11
7
73
89
56
67
86
17
8
13
23
7
n
27
23
16
23
11
Standard
Deviation
74
87
73
70
90
Scores - combined
Mean (%)
n
16
13
8
12
4
Standard
Deviation
ICSEA
*
1004
1118
984
959
887
Scores - Males
Mean (%)
Location
Urban
Urban
Rural
Rural
Rural
Standard
Deviation
State
NSW
QLD
VIC
TAS
TAS
Mean (%)
Scores - Females
75
88
58
63
90
18
9
27
26
10
* Index of Community Socio-Educational Advantage (ICSEA): The mean ICSEA value is 1000 with a
standard deviation of 100. Values below the mean indicate schools with fewer advantages.
The results were astonishing, since young children demonstrated competence of what is currently
university-level integral calculus (although the engineering undergraduates have to solve the problems
without a computer). In just one narrow topic, 2D integral calculus, it was possible to harness
children’s passion for engaging with computers, with skill acquisition beyond the project expectations,
both in terms of visual conceptualisation and the understanding of a set of mathematical processes
(Fluck, et al, 2011).
The Australian Curriculum (ACARA, 2014) offers a number of content descriptors for the study of
integral calculus.
 Senior Secondary | Mathematics | Mathematical Methods | Unit 3 (e.g. ACMMM115)
 Senior Secondary | Mathematics | Specialist Mathematics | Unit 4 (e.g. (ACMSM124).
These units are normally studied in Year 12, whereas the students in this project were in Years 6 or 7.
Therefore the students were using computers to learn and demonstrate achievement at a level five
years in advance of their chronological age. An effect size of 1.0 is associated with advancing learner’s
achievement by one year (Didau, 2014), so this project might reasonably claim an effect size of 5.0.
This is well beyond Hattie’s (2009) hinge point of 0.4 as a measure of medium educational impact.
The project has led to numerous journal papers, a book chapter, an outstanding paper award at an
international conference, positive feedback from the local community and the local press,
commendation from the State Education Minister and media interviews. This success story is featured
in the Maplesoft “User case studies” (Chin, et al, 2012) and was presented at the Australasian
Association for Engineering Education (AAEE) conference in 2011 (Penesis, et al, 2011), where it was
commended by engineering peers.
Each participating school received a community report suitable for the school newsletter. The success
of one school (where students were selected from several classes and used a computer laboratory) was
congratulated by the Tasmanian Minister for Education who stated in the local press: “This is an
outstanding result, given that integral calculus is a branch of mathematics which has widespread
applications in science, economics and engineering”. It was also outstanding in that this school had the
lowest index of community socio-educational advantage.
This project demonstrated that primary school students, some of whom were as young as 10 years of
age, can handle integral calculus when equipped with computer tools. With time on their hands after
finishing the project early, students in one school went on to experiment with features of the software
which were not taught. They discovered a wizard for calculating volumes of revolution, and used this
to design goblets. Figure 2 illustrates one such exploration, showing how the students were familiar
with the mathematical notation yet playful in their activity, a good omen for future curriculum
transformation.
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Calculus for Kids
Andrew Fluck, Christopher K.H. Chin, Dev Ranmuthugala, Irene Penesis
Figure 2 Goblet design in progress using the volume of revolution wizard in MAPLE
Conclusion
A 2014 Australian Research Council grant will enable this project to go into more schools and track its
impact on NAPLAN numeracy results. In addition, it will allow students to look at a greater range of
real-world problems and construct their own functions to model them. Two additional lessons have
been inserted into the learning materials on parabolic functions. The main reason for this was to
counter criticism that the students in the project were ‘merely pressing buttons’. The lessons on
parabolas will provide students the opportunity to tailor a function to fit a real-world situation, then
use the tools of integral calculus to solve the problem. Such a criticism brings into stark relief what we
intend when we say a student ‘understands calculus’ or has mastered the subject. If passing an
examination in the topic at an academically advanced level does not signify such understanding, then
what does? Or do we always mean this understanding must always be demonstrated without the help
of electronic equipment?
This cuts to the core of what might be entailed in the re-design of curricula through the use of
computers. If our social decision is that calculus comprehension must be demonstrated without
electronics (even four-function calculators), then it would be strange if authors were asked to give up
their word processors.
As the Calculus for Kids project proceeds, we will be keeping a close eye on ways this activity
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Calculus for Kids
Andrew Fluck, Christopher K.H. Chin, Dev Ranmuthugala, Irene Penesis
impacts upon general numeracy, hopefully through generating mathematical understanding and delight
in successful achievement. Our other eye will be on the future of existing curricula, wondering what
other projects can contribute in a meaningful way to transformation and re-design using computer
tools.
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