
Using Multimedia for the Teaching of Decimal
Notation
Gary Asp, Dianne Chambers, Nick Scott, Kaye
Stacey & Vicki Steinle
University of Melbourne
Asp, G., Chambers, D., Scott, N., Stacey, K., & Steinle, V. (1997). Using Multimedia for the Teaching of Decimal Notation. In Clarke, D., Clarkson, P., Gronn, D., Horne, M., MacKinlay, M., & McDonough, A. (Eds.), Mathematics  Imagine the Possibilities. Proceedings of the Thirtyfourth Annual Conference of the Mathematical Association of Victoria. (pp. 6067) Melbourne: Mathematical Association of Victoria.
In this article, we will describe how we are using
multimedia to integrate practical experience of children in
schools and findings from current research into the
universitybased component of the teacher education program
at the University of Melbourne. In order to teach a
mathematics topic in a manner that is likely to result in
the successful learning of the topic by children, the
teacher must have a thorough knowledge of its mathematical
features and of the ways in which children approach it.
Fortunately, recent research in mathematics education has
focussed on children's thinking and on the way in which they
understand key concepts in important topics of mathematics.
Not only has this research produced a valuable and growing
body of knowledge concerning children's thinking about
mathematics topics, it has also confirmed how important it
is that teachers have an excellent understanding of this
knowledge to successfully teach these topics. It is vital
that students in teacher education courses acquire this
knowledge. Multimedia provides us with new opportunities to
include the study of children's thinking and practical
experiences with children in schools in universitybased
components of our teacher education program. Multimedia also
gives us new opportunities for making a very wide range of
teaching materials and ideas for lessons accessible to our
student teachers. In this article, we will outline the
components of a resource on the teaching and learning of
decimals that we are currently constructing for use in our
teacher education program at the University of Melbourne,
and suggest some possible ways it might be used with teacher
education students. The resource is being developed for use
on the World Wide Web using a Web browser such as Netscape
Navigator. Our long term aim is to extend the use of the
resource to teachers working in schools.
Why decimal notation?
Our multimedia resource will collect together a large
array of information about teaching decimal notation. This
topic has been chosen because it is an important aspect of
numeracy and because many students in school have difficulty
with it. Our society uses metric measures, so that decimal
notation is widely used. An understanding of decimals is
also critical in using a calculator. For example, to
apportion a weekly cost or amount of 100 units equally among
7 days involves dividing 100 by 7. A calculator display will
give 14.285714. We have evidence that many people do not
understand how large the decimal part of such a number is.
Students learn to interpret the size of decimal numbers from
about Year 4 onwards. About 25% of Year 4 students learn
it quite quickly, however, about 25% of Year 10 students
have not yet mastered this important concept (Moloney and
Stacey, 1995)
There are a variety of common errors. Many primary
students are misled by the length of the decimal part. They
think that 14.285714 is larger than 14.3 because 285714 is
larger than 3. Most students learn that this is not the case
by the early years of secondary school. However, many
secondary students will think that 14.285714 is smaller than
14.2. They reason that 0.2 is in tenths, whereas 0.285714 is
in millionths (or some other tiny part) and they have
difficulty coordinating the number of parts with the size
of the parts. Since tenths are larger than millionths, they
frequently conclude that 14.2 is larger. Alternatively they
may think about 1/285714 and similarly reach the wrong
conclusion. There are many other difficulties as well. For
example, some students (especially older students) rely very
strongly on analogies with money. They may reason that
14.285714 is larger than 14.2 because the first is like $14
and 28 cents and the second is like $14 and 20 cents.
However, these students are unable to decide whether 0.4502
is larger or smaller than 0.45, because they see them both
as 45 cents. These students have no real idea of place value
in the decimal region to back up a useful analogy.
The misconceptions and partial understandings that are
described above are useful for teachers because they
highlight particular aspects of decimal notation that
students need to learn: the place value properties are
critical, the coordination of number of parts with the size
of parts that is an obstacle to understanding fractions is
again a difficulty, and students need to be able to
reunitise (thinking of 2 tenths as 20 hundredths, etc). Our
research (Moloney and Stacey, 1995, 1997) shows that these
difficulties are widespread. Furthermore, even students who
can reliably decide which of two decimal numbers is the
larger often have very little understanding of place value,
because they are applying in rote fashion the "adding zeroes
until they are equal length, then treating as whole numbers"
rule. Teacher education students need mastery of the ideas
behind decimal notation. They need to know the ideas that
children will find hard, they need good teaching strategies
for working on these ideas and they need to be able to
diagnose children's difficulties. Our multimedia resource is
designed to help them learn about these important issues. If
it proves to be successful in our teaching, the resource
will be extended to cover other topics, such as whole
number, fractions and choice of arithmetic operations (that
is initially those most closely related to decimal
numbers).
How the resource will be used
Our project is to make a technically and intellectually
ambitious teacher resource module called Learning About
Decimal Numbers initially for the first year primary
mathematics education subject at the University of
Melbourne. We aim to strengthen the theoretical and
practical knowledge of our students regarding the teaching
of decimal numbers by giving them opportunities to study
children's work, to see and hear children explain their
ideas about decimal numbers, to interact with tests which
will help them understand and identify misconceptions, and
to learn about effective teaching methodologies. By using
multimedia, the module combines, and allows links between, a
range of learning experiences and related teaching resources
that would not otherwise be possible. This includes audio
and video of children working with decimal numbers and
images of their written work which illustrate children's
ideas, interactive tests and games to diagnose
misconceptions, textbased information, links to teaching
resources and opportunities for electronic discussion.
In Semester 2 1997 a prototype of the resource is being trailled with
students in Mathematics1 (485129) and their evaluation and comments will
be used for further development of the resource. By 1998 the resource
will be an integrated whole and used with prospective secondary teachers
as well. The resource is intended to enrich lectures, tutorials and assignments
and to allow maximum benefit to be gained during teaching rounds by assisting
students in developing an organised theoretical approach to the discipline.
The resource will be used:
 in lectures as a source of uptodate Australian
illustrative material
 as part of the initial arithmetic competence test, a
hurdle requirement for all preservice primary
teachers
 in tutorials and for assignments, so that students
can interact with 'virtual children' (this parallels a
science 'dry lab') so that they can study children's
thinking and draw on a convenient database to suggest
teaching strategies
 by teacher education students on teaching rounds as a
support to their teaching and for collecting data to be
analysed later at university.
In addition, it is expected that many teacher education
students will want to revisit the online text material,
tests and games, and teaching ideas and resources in their
own time in order to consolidate and extend their
understanding of ideas introduced in the lectures and
tutorials. In the long term, we hope the resource will also
be available to practising teachers across Australia.
Components of the resource
Text based information
The resource will provide students with ready access to a
variety of written information, including some which will
relate closely to the material in lectures and other
material for background reading. The textbased information
will outline the history of decimal notation, the main ideas
(mathematical, psychological and pedagogical) and give
useful links and references to printed and online material
for further information. This information will be available
on the university intranet, along with the rest of the
resource, but will also be made available in Portable
Document Format (PDF) so that students can print and access
these text materials in a workbook format.
Profiles of individual children
A feature of the resource will be the profiles of
children, showing the ways in which individuals think and
learn about decimal numbers. These case studies are now
being constructed from material gathered in an associated
research project (Improving learning outcomes in numeracy:
Building rich descriptions of children's thinking into a
computerbased curriculum delivery system funded by the
Australian Research Council). Each case study will contain a
child's annotated written work; analysed test results; audio
of the child explaining his or her ideas; video of the child
demonstrating how he or she works with materials (with
possibly a backup library of videotapes) and embedded
questions that will facilitate teacher education students in
identifying the thinking patterns illustrated. In the
future, it may be possible to show the progress that a child
makes after certain teaching activities. Additional case
studies will be incorporated as the research project
progresses. An example of a screen in the pilot version of
the resource is shown in Figure 1. This gives a portion of
an interview transcript with a child. This interview is very
closely based on a real interview, but it has been 'tidied
up' somewhat to remove extraneous information and irrelevant
or unclear comments. All information identifying the child
who originally gave the interview has also been disguised or
removed and a child model used for photography. In this
interview, Susan was asked to select from the cards marked
0, 1, 2, ... 9 to complete the number which started 3 point
blank blank. Her choice of 3.10 as the smallest possible
number is explained in Figure 1: she believes that 1 ten and
0 hundreds is smaller than 3.01 because this contains 1
hundred.
A teacher education student reading the interview would
be able to see that Susan has not just carelessly given the
incorrect labels for the tenths and hundredths columns; she
is actively trying to incorporate the relationships that
exist in integers (hundreds are bigger than tens) in her
interpretation of decimal numbers.
Figure 1: An excerpt from an interview with
'Susan'.
Online tests
We have developed a brief test that can be given and
marked by computer which makes a reasonably accurate
diagnosis of a person's ideas about decimals. Further tests
to assess other aspects will be developed later. In Figure
2, a copy of part of a written version of the current test
as completed by a child is shown. The instructions for the
test are to circle the larger number of each pair. This
screen appears in the pilot version of the resource when
teacher education students are studying students'
misconceptions and partial knowledge of decimals. They can
see the answers this child has given and they can also see
the diagnosis. In order to help them understand the thinking
that has led to the child's answers, "postit" notes are
available on certain items. By clicking on these, teacher
education students can obtain an explanation of why Susan
chose the answer she did. For example, the answers on the
test as a whole and the explanations she gave when
interviewed, indicated that Susan generally chose longer
decimals as larger, because she thought the place value
columns from the left after the decimal point went tens,
hundreds, thousands etc. The postit note explains that in
this instance she probably chose 1.538 as larger than 1.541
as 8000 is bigger than 1000. The online testing and
diagnosis will be used:
 to test our teacher education students' own knowledge
and provide detailed feedback to each individually.
Students can use the resource in their own time to
remediate basic weaknesses identified.
 to teach our students about children's thinking. For
example, a tutorial exercise for our students is to
answer the questions in the way that a child who had a
particular misconception would. The computer feedback
tells them if they have successfully imitated the
problem.
 to allow students to collect data from children when
they are working in schools, which can be analysed in
various ways during tutorials.
 eventually to allow teachers in schools to run the
tests with their children and receive immediate detailed
feedback.
Figure 2: Choices that 'Susan' made on a decimal
comparison test, with explanations.
Teaching ideas
A thorough understanding of the way children think about
a mathematical topic and a thorough understanding of the
mathematical demands of a topic provides student teachers
with a strong basis on which to choose teaching activities
for individuals, groups and classes. Because it is in a
multimedia format, the resource will be able to contain or
point to a large range of teaching ideas, eventually indexed
and crossreferenced from the parts of the resource that
describe children's difficulties. In addition, the range of
teaching materials will be able to be extended from
textbased ones. Some components will be:
 electronic teaching games available for downloading
(These are being written and tested as part of our
research project)
 links to internet sites offering lesson plans, reviews of published
material etc (currently there are a number of useful US curriculum sites
but the range of Australian sites is constantly growing and more teaching
material is being provided e.g. from EdNA and the Victorian Department
of Education)
 lesson plans from published sources where copyright
release can be obtained
 a discussion forum, and 'Frequently Asked Questions'
file
 some written materials about wellknown teaching
methodologies and research results.
In tutorials, students will study the profiles of
children and of whole classes and select appropriate
teaching strategies from the resources provided.
Future directions
Multimedia opens up new possibilities in teaching at all
levels. In teacher education, a number of resources are now
becoming available, such as "Learning from Teaching"
(Mousley and Sullivan) and "The C&D PD CD " (Chance and
Data Professional Development CD, Watson & Moritz, 1997)
which help bridge the gap between the university program and
experience with children. Although not a substitute for
experience with children, resources such as these have
distinct advantages as the scenarios presented have been
distilled from research data and clearly illustrate
representative case studies found in all classrooms.
Generally, interactions with children are witnessed by only
one or two adults. They are hard to capture in real life and
consequently hard to analyse with any reliability. Our
resource takes advantage of the careful analysis of selected
interactions with children, to build awareness of
welldocumented, widespread features of children's thinking.
In the classroom, teachers often have limited time to
observe individual children's work and would therefore be
greatly assisted if they were able to quickly spot patterns
in children's thinking. To do this, they must be very
familiar with what they might expect to see. They need to
gain this awareness in ways which supplement and build on
their real life experiences. Mathematics education research
is providing knowledge concerning things teachers can expect
to see as children learn mathematics, and our resource and
the others noted above are working to bring this knowledge
to prospective and practising teachers. In this way advances
made in mathematics education research are being translated
into practice.
Multimedia is now providing new opportunities for ongoing
professional development of practising teachers. Our
eventual aim is to make our webbased resource available for
practising teachers all over Australia and beyond, so that
they can consult it on a regular basis, for assistance with
individual students (possibly online diagnosis with
suggested teaching strategies) and for suggestions for
teaching. Hence, teachers will be able to update their
professional knowledge in their own time and relate this
knowledge to problems that they face in their own
classrooms. In this way, we hope to contribute to improving
the outcomes for numeracy of Australian children.
References
Moloney, K. & Stacey, K. (1995). Understanding
Decimals. Australian Mathematics Teacher, 52(1), 4
&endash; 8.
Moloney, K. & Stacey, K. (1997). Changes with Age in
Students' Conceptions of Decimal Notation. Mathematics
Education Research Journal. 9 (1), 25 &endash; 38
Mousley, J.,Sullivan, P, Mousley, P. (1996) Learning
About Teaching. Adelaide: Australian Association of
Mathematics Teachers
Watson, J. and Moritz, J.(1997) The C&D PD CD:
Professional development in chance and data in the
technological age. In Scott, N. & Hollingsworth, H.
(Eds.) Mathematics creating the future. (pp 442 
450) Adelaide: Australian Association of Mathematics
Teachers.
