Understanding Why Maths Education Must Continue To Change

by Mark R. O'Brien

For those who read a lot of educational material, the heading will be familiar. Developed from the title of an article by Renate & Geoffrey Caine¹ I've changed it slightly to reflect a specific interest in maths education and to also reflect the fact that here in Australia, maths education has been changing. The developments instigated  in the 80's with an increased focus on problem solving, investigations, activity oriented teaching and the incorporation of programs like JIM and DIME materials were significant developments and put us at the forefront of maths education. This was perhaps reflected in our sound results in the Third International Math and Science Study (TIMSS)². For many reasons this development seems to have stalled somewhat at a time when there is a great need for continued change. In this article I will present some of the arguments for continued change.

Curriculum Framework

No longer do we have a syllabus of prescribed learning laid out in fine detail for all to follow carefully. The Curriculum Framework³ is a document that allow teachers to teach using whatever materials and methods are appropriate for specific students. It is a document that allows for the restructuring of courses. It provides a framework that is more flexible than any previous syllabi to allow for the changes we need to make now and in the future. It is a document that reflects changes in how the educational world believes learning takes place, an area that I will expand on later in this article.

We can look to the Curriculum Framework³ for guidance on pedagogy (pp 33-37) and for the overarching outcomes of our education system but more importantly we need to look at it as permission to make the changes necessary to improve all students learning, to connect better with them and the world they are growing into.

The underlying philosophy of the Curriculum Framework will soon be spread officially to our year 11 and 12 students making it the standard for education throughout all our schooling.

Developments In Education and Learning

Recent research into teaching and learning has developed new theories and practices that are different from those expounded when most teachers did their training. Currently, there is a trend to use brain research and theories about how people learn. As a result of this teachers have more knowledge available to them in this area.  If we are to maximise the learning opportunities for students it is necessary for us to incorporate this knowledge into our teaching and learning practices.

Learning Styles:

Theory on Learning Styles is based upon research that shows that because of a number of differing factors that contribute to whom we are, individuals perceive and process information differently. We have both concrete and abstract perceivers and active and reflective processors. The implications of this are that "Teachers should design their instruction methods ... using various combinations of experience, reflection, conceptualisation and experimentation⁴."

Brain Theory:

Studies in neuroscience have revealed a great deal of new information about how the brain works and this knowledge can be used to enhance learning. The implications of this information for educators are that we should "organise a curriculum around real experiences and integrated, whole ideas⁵" and "focus on instruction that promotes complex thinking and the growth of the brain⁵".

Proponents of brain-based learning have used the results of neuroscience to develop a core set of principles to guide education. Two significant outcomes of this are that "Teachers must design learning around student interests and make learning contextual⁶" and "Teachers must immerse learners in complex, interactive experiences that are both rich and real⁶."

Multiple Intelligences:

Howard Gardner's work on multiple intelligences⁷ suggests that we need to use a wider range of learning opportunities with our students to ensure that we make the most of the different types of intelligences that various students have. The simplest example of this is to use diagrams as well as language when working with concepts. A lot more work continues to be done on how we can best take advantage, as teachers, of what Gardner is talking about. Mark Wahl⁸ has looked specifically at its effect on maths education and this type of information can help us to understand what changes we can make. The simple fact is that the understanding and knowledge of multiple intelligences must be allowed to play its part in the evolution of our teaching methods.

Constructivism:

Support for the philosophy that says that "we construct our own understanding of the world we live in⁹" is steadily growing as the way we must structure our teaching. One area of support for using a constructivist approach in mathematics teaching was from the NCTM, searching for solutions to the problems highlighted by the US results in the TIMSS study. Constructivist principles would guide us to ensure that "the learning process focusses on primary concepts, not isolated facts⁹" and tells us that "The purpose of learning is for an individual to construct his or her own meaning, not just memorise the right answers and regurgitate someone else's meaning.⁹" In this way we can develop a felt sense or felt meaning of concepts.

The Learning and Teaching Principles in our Curriculum Framework reflect these knew 'knowledges'. If we neglect this research evidence, the way we teach may not maximise the learning potential of our students. Not surprisingly, many experienced teachers have already developed some of this knowledge simply by reflecting on their experiences with students in their classrooms. The research often supports and enhances their "felt sense" of what teaching at its best looks like.

Changes in Our World

Changes in our world are precipitating major changes in education. A fifteen minute browse through the book "The Learning Revolution¹⁰" is enough to convince even the biggest sceptics that we can no longer continue as we have done for many years. Changes must be made and they need to be made quickly to cope with the rapid changes our world is now undergoing.

It is difficult to summarise the myriad of ideas and huge quantity of information presented in this book but suffice to say the book "spells out the shape of things to come and the alternative models we can use to usher in an age of potential plenty¹⁰".

In my introduction I mentioned the work of Renate & Geoffrey Caine¹ . The Caines have presented material on the need for changes in our education systems. To summarise part of this material I present here the changes they believe need to be made in the underlying beliefs of our education system as we move from the Industrial Age to the Information Age.

Industrial Age (what we perhaps believe or at least what our education system was originally based around);

"Only experts create knowledge"

"Teachers deliver knowledge in the form of information"

"Children are graded on how much of the information they have stored"

Information Age (underlying assumptions suggested by the Caines that can guide us in the new century);

"Dynamical knowledge (the sort of knowledge that is naturally and spontaneously invoked in authentic interactions in the real world) requires individual meaning making based upon multiple sources of information."

"The role of educators is to facilitate the making of dynamical knowledge."

"Dynamical knowledge is revealed through real world performance."

If these thoughts are unnerving then perhaps another comment by the Caines can reassure us and stimulate us; "the process of moving from one model of schooling to another that is as yet unknown is causing both chaos and confusion as well as immense opportunity and new possibilities¹".

Changes In Mathematics

The advent of graphics calculators has already caused changes in the mathematics we teach and how we teach it. Content that was for years part of our courses suddenly becomes unnecessary. Certainly, a lot of the content in past algebra courses is under question. Understandings of concepts are reached using a different set of learning experiences.

Spreadsheets and computers still cry out for much greater integration into our mathematics courses if we are to reflect the knowledge necessary in the information age. In spite of the difficulties caused by practicalities, they remain as tools of which we must make more use. Perhaps a change in how we structure maths education may simplify their implementation.

As Computer Assisted Design and Drafting (CADD) packages become the means of operating in design and construction we must make changes to our geometry content that are more significant than simply renaming it and regrouping content. Maybe "view coordinate geometry¹¹" and "object coordinate geometry¹¹" are more important than say, triangular pyramid.

Do Global Positioning Satellite (GPS) systems mean that we should look closely at what is taught in our trigonometry courses, with a view to better preparing students?

These factors and others must cause us to question what and how we teach. We have made the right start by moving away from content and towards processes, knowledge that is transferable to new situations. We need to continue to look at our courses in terms of facilitating the making of "dynamical knowledge", in the area of mathematics.

Outcomes-based Education¹²

In an outcomes-based education we are required to focus on the desired outcomes being achieved by our students. Mathematics teaching has already been on the right track. A more detailed look at the four requirements of an outcomes-based education may help us to see what changes we still need to make:

Clarity of Focus - Asks us to understand clearly what outcomes we are aiming to achieve and to inform students of what these outcomes are - although the outcomes may now have changed (or changed from objectives to outcomes) many maths teachers have worked towards specific goals and informed students of these goals in recent years.

Expanded Opportunity - To keep the opportunity for students to achieve outcomes when they are ready open, we will need to change the structure of our teaching. Lessons and resources will need to be more flexible, able to cater to students working at their own rate. Changes in Maths in Practice and Modelling With Maths have attempted to cater for this requirement.

High Expectations - We need to work towards all students achieving outcomes at their highest potential and then moving them on to even higher goals. We need to reduce the comparisons between students and instead develop realistic goals for each individual. It is here that the streaming commonly used in maths classes has come under fire although I am not sure that an argument can't be mounted which shows that certain ways of streaming can improve the achievement of outcomes.

Designing Down - Maths teachers are very good at working back from the learning goals to a series of sequenced lessons, what we do need to realise is that the learning goals have changed. Designing down from the overarching outcomes and the student outcome statements will result in changes in what and how we teach.

Most of us are still working towards properly understanding what is meant by an outcomes-based education system and what its implications are for our classrooms. One thing that it surely does however is call for our courses to be more reactive to students' competence and understandings rather than follow prescribed sequences of content.

At the recent Swan Curriculum Conference on Curriculum Innovation Di Kerr stated that an outcomes approach should "focus on what students know and understand" and Sue Willis stated that the "teachers job is to provide the inputs that allow students to learn". Neither of these things is really different from what we should always have been doing.

Summary

The average secondary maths teacher out there is about 46 years old. It's a long time since we went to college and a lot of us haven't completed any formal education since. We have seen the coming and going of many "new" things in education. Some have been improvements, some were just politically expedient. In the face of continuing change these are some of the options we can take:

• Change nothing and hope it goes away - realising of course that some of these things won't go away - knowledge about how we learn will only increase, the information age isn't reversible ...
• Change nothing because we are already doing our job perfectly - realising of course that our job may not be the same one we were doing a few years ago, our students may not be the same, the world they are going to live in is different ...
• Change professions - realising of course that the information age has caused changes in most professions, and that the education system would then lose a huge resource of experience and knowledge ...
• Start making changes - accept the need to continue changing, work out what is required to make the changes, set yourself up for your future years in teaching by making your own teaching knowledge more "dynamical", look for opportunities to learn new knowledge ...If you choose this option can I make a few suggestions as a fellow teacher and as a parent of the children I hope you will be teaching - don't accept more work, we already work very hard! - demand change rather than additions - cut something old before you add something new - demand resources that support the changes you want to make.

Unfortunately with previous changes in education we were never provided with the underlying reasons for change and we couldn't access the facts required to decide for ourselves if the change was right or wrong, good or bad, necessary or unnecessary. This time I believe that the information about why mathematics education must continue to change is available to us. If you choose the last option above a good place to start is to look at some of this information.

I've learnt a lot of what is in this article using the Internet. As I've worked towards developing new materials I've often been able to base decisions on the huge amounts of knowledge easily accessible to me in the comfort of my own home. You'll notice that I have included footnote numbers throughout this article. If you wish to access further information about any of these people or articles I've included web addresses wherever possible in the 'bibliography'. Probably the single most useful site I have found is the New Horizons site - http://www.newhorizons.org