4.14 MATHEMATICS AND INFORMATION AND COMMUNICATION TECHNOLOGIES
The National Council for Educational Technology (NCET) have identified six major ways in which information and communication technologies (ICTs) can provide opportunities for students learning mathematics.
These include
- learning from feedback: provided by the computer in a fast, reliable, non-judgemental and impartial manner
- observing patterns: based on a computer's ability to produce many examples in a short time
- seeing connections: between formulae, tables and graphs
- working with dynamic images: allows students to manipulate geometrical diagrams
- exploring data: students can interpret and analyse real data in a variety of representations
- "teaching" the computer by means of an algorithm:
- encourages the student to express their commands unambiguously and in the correct order.
This section gives a brief overview of some of the work that is currently being undertaken in Irish mathematics classrooms with the aid of various software packages. It is by no means an exhaustive list. It concentrates on packages that are currently available to most schools and on some of the initiatives that are being run as part of the Schools Integration Project (SIP) under Schools IT 2000. More information on these projects can be obtained from the National Centre for Technology in Education (NCTE).
Many Education Centres have been offering training to teachers of mathematics on the software mentioned below and are a source of further information for interested teachers.
Aset of ICT Guidelines to be issued to secondary schools will include a discussion of the issues, challenges and opportunities offered by ICT for teaching and learning.
DYNAMIC GEOMETRY SOFTWARE
In recent years a number of software packages which are collectively described as dynamic geometry packages have been developed. These include Geometer's Sketchpad, Cabri-Géomètre and JavaSketchPad. More information and samples can be viewed on the Internet by searching for the phrase "dynamic geometry". These packages are having a significant impact on the teaching and learning of geometry in very many countries. They have the potential to provide students with a new and fundamentally different geometry learning experience. They can be used in a variety of ways from a versatile demonstration aid for the teacher to a powerful tool for student-centred investigative learning.
The Geometer's Sketchpad is a visualisation tool which emphasises spatial reasoning and logical abstraction. It allows for discussion, co-operative learning, and improved communication skills in mathematics. Students can discover properties and test conjectures by dragging parts of their constructions. As constructed relationships remain valid while one drags, geometry becomes dynamic. Sketchpad is a dynamic geometry environment. Translations, rotations and reflections can be performed to analyse the effects of motion in the plane. Much useful work in coordinate geometry can be achieved by getting students to construct points and lines and asking the program to find the mid-point and slope and hence the equation of lines. Interactive scripting records and generalises the steps of a construction so that it can be repeated on another figure or played again and again to create more complex figures. Teachers and students can add comments, thus making scripts an ideal medium for communicating mathematically.
Geometer's Sketchpad is also the focus of a Schools Integration Project (or SIP) under the auspices of the NCTE. There are four schools involved and the project commenced in the 1999/2000 school year. The current target group is students of Junior Certificate mathematics. The synthetic geometry section of the revised syllabus is the main focus of the project. Preliminary work concentrates on constructing line segments and half-lines. To date, lesson plans have been developed in the area of Geometry theorems relating to angles, parallelograms, circles and triangles. Additional work has also been undertaken in the areas of Coordinate Geometry and Transformation Geometry. The lesson plans for all of these topics will be disseminated via Scoilnet (http://www.scoilnet.ie) at the conclusion of the project. A sample lesson plan idea on how the teaching of the bisector of an angle and incircle construction might be approached is provided in Geometry lesson idea 3 to give mathematics teachers an indication of the possibilities which the Geometer's Sketchpad offers. The package is available in Education Centres, from where further information can be obtained.
COMPUTER ALGEBRA SOFTWARE
Computer algebra systems, such as Maple and Mathematica, have been significantly changing the manner in which many mathematicians, scientists and engineers go about their professional work. These packages are not designed for mathematics education per se, but rather as powerful tools to aid the professional in his or her work. Nevertheless, it is useful for mathematics teachers to become familiar with such software for several reasons. First, many of the students who pass through our care in schools will actually continue their mathematical experiences at third level and beyond and will thus be exposed to this type of software. Secondly, mathematics teachers are often looking for ways to maintain their levels of enthusiasm, and this kind of software offers opportunities for them to pursue their mathematical interests in new ways. Finally, despite the fact that these packages were not designed as educational ones, they still offer significant potential for use as such. One such package which has received exposure in Irish schools recently is MathView (now renamed LiveMath).
LiveMath is the focus of one of the innovative SIP projects under the auspices of the NCTE. There are four schools involved and the project commenced in the 1999/2000 school year. Although the current target group is Transition Year students there are many possibilities for teachers and students of Junior Certificate Mathematics. LiveMath is a computer algebra package which allows students to interact with the computer and hence improve their understanding of mathematics. The package also has graphic abilities which are noteworthy. The SIP project has exploited many of these features. To date, lesson plans have been developed in each of the following topic areas:
Algebra, Co-ordinate Geometry (Line and Circle), Functions and Graphs.
The lesson plans for these topics will be disseminated via Scoilnet (http://www.scoilnet.ie) at the conclusion of the project. A sample lesson plan idea on Algebra is provided in Algebra lesson idea 2 to give mathematics teachers an indication of what is involved. The package is available in Education Centres, from which further information can be obtained. Mathematics and Science with MathView is a book by a practising Irish second level mathematics teacher which describes additional ways of using LiveMath; a reference to this book is made in the Resources section. LiveMath is now web-enabled and by visiting the website (http://www.livemath.com) it is possible to download a plug-in and interact with lesson plans from other users.
MATHEMATICAL SYMBOLS FOR EXAMINATION PAPERS AND OTHER DOCUMENTS
One of the ways in which computers are being used by teachers of all subjects is in the production of notes and examination papers. One problem for the mathematics teacher in this regard is a perceived difficulty with mathematical type-setting. There are a number of packages which can be used by mathematics teachers to overcome this problem. These include MathType, LaTe? and Equation Editor. Equation Editor is an easy-to-use package which comes free with Microsoft Word, and which is sufficiently powerful to meet all of the needs of the mathematics teacher in regard to the production of notes, articles for journals, examination papers and solutions and so forth. It is possible over a couple of years to build up a number of examination papers using the computer and it then becomes very easy to produce examination papers subsequently by combining and making adjustments to parts of old ones.
To verify that equation editor has been loaded from the Microsoft Word disk on a PC, within Word click on Insert, select Object, and search the list for Microsoft Equation. If it does not appear as an option, you need to load it from the Microsoft Word disk supplied by your dealer.
SPREADSHEETS
Spreadsheets are one of the most commonly available types of application, designed primarily for a wide range of business uses. Almost all modern computers come with at least one spreadsheet application as standard "bundled" software, so that this is one tool which is available to any teacher in a school with computers. Spreadsheets lend themselves to being used very effectively in mathematics learning, and their use forms part of the mathematics curriculum in many countries. Even within a didactic model of teaching, they can be used to free the student from tedious calculations that may stand in the way of conceptual understanding in standard numerical procedures. In a more investigative and student-centred learning environment, their power and ease of use provide a facility to tackle material which would be difficult or impossible to approach in this way without a computer. Furthermore, the use of spreadsheets provides a solid aid to the understanding of some basic algebraic concepts (such as variables and functions).
An example of how a spreadsheet can be used in the context of mathematics teaching is provided in Functions and graphs lesson idea 1.
CALCULATORS
The issue of calculators has been addressed in the context of changes in content (see section 3.3). A set of guidelines on calculators is being produced, in which a pedagogical rationale for their use is outlined. The purchase of suitable machines for schools is also discussed. It is worth noting again that the Department of Education and Science has set up a research project to monitor numeracy related skills (with and without calculators) over the period of introduction of the revised courses.