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Using Multimedia for the Teaching of Decimal Notation

Gary Asp, Dianne Chambers, Nick Scott, Kaye Stacey & Vicki Steinle

University of Melbourne

 

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 university-based 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 university-based 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 Grade 4 onwards. About 25% of grade 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 co-ordinating 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 co-ordination 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 re-unitise (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, text-based information, links to teaching resources and opportunities for electronic discussion. In Semester 2 1997 a prototype of the resource is being trialed with students in Mathematics1 (485-129) 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 up-to-date Australian illustrative material
• as part of the initial arithmetic competence test, a hurdle requirement for all pre-service 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 on-line 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 text-based information will outline the history of decimal notation, the main ideas (mathematical, psychological and pedagogical) and give useful links and references to printed and on-line material for further information. This information will be available on the university intra-net , 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 computer-based 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 back-up 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, "post-it" 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 post-it note explains that in this instance she probably chose 1.538 as larger than 1.541 as 8000 is bigger than 1000. The on-line 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 cross-referenced 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 text-based 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 eg 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 well-known 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 well-documented, 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 web-based 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 on-line diagnosis with suggested teaching strategies) and for suggestions for teaching. Hence, teachers will be able to up-date 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.