Stephen ARNOLD
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## Technology and Mathematics: Impact upon Practice, Curriculum and Pedagogy

Stephen Arnold

I begin with several observations that define my position: these are based upon research, wisdom of practice and my own experience. You are free to accept or reject as you please, but you will know where I am coming from.

 First, mathematics itself: not a body of knowledge so much as a process, a way of approaching the world: in the words of a former syllabus document, "Mathematics is a search for patterns and relationships". When we observe systematically, draw links and connections and, importantly, draw abstractions from these observations, we are doing mathematics. Fields as diverse as geometry, statistics, algebra, measurement and number theory - all are unified, not by their content but by the process that defines mathematics. Mathematics as Content... Geometry Algebra Calculus Statistics Measurement Logic Music Number Theory Arithmetic... OR Mathematics as Process A way of interacting with the world and yet independent of the world? A systematic search for patterns and relationships using a variety of tools and both formal and informal methods

Thirty years ago, I was a new teacher, charged with bringing my school into the computer age! Since then, technology has impacted upon almost every aspect of our lives and culture, and mathematics has been no exception.

Statistics provides a nice example: the study of statistics used to be dominated by the intricate and tedious computations that defined what it was seen to be. Certainly, the end result was always to make sense of data, to draw conclusions and to make inferences, but to those involved in the process (and anyone looking in from the outside) statistics was about calculation.

No longer is this true: the advent of technology has transformed this vital and increasingly important branch of mathematics from a focus upon computation to a focus upon interpretation. And this transformation has occurred at every level: from the practice to the teaching and learniing.

Statistics might be, perhaps, the most dramatic example of mathematical practice transformed by technology, but it is by no means isolated. From the nature of proof itself to a fundamental shift from the continuous to the discrete, mathematics has been rejuventated and renewed by new tools, new techniques and new domains of study - all defined by technology.

Next, my position on teaching and learning.

After thirty years of classroom research and practice involving the use of technology within the domain of mathematics, what have we learned:

• about good teaching and learning, and

• about the teaching and leaarning of mathematics in particular?

We have learned that students learn best when:

• they are actively engaged in constructing meaning about content that is relevant, worthwhile, integrated and connected to their world.

Further, we know that they learn mathematics best when:

• they are active participants in their learning, not passive spectators,

• They learn mathematics as integrated and meaningful, not disjoint and arbitrary, and

• They learn mathematics within the context of challenging and interesting applications.

And the role of technology?

Technology in mathematics learning plays two major roles:

• as a tool for representation, and

• as a tool for manipulation and scaffolding.

Good technology supports students in building skills and concepts by offering multiple pathways for viewing and approaching worthwhile tasks.

Technology offers new tools for students to engage with their mathematics, ideally within contexts that offer meaning and challenge.

Good use of technology makes possible what could not be done previously, and our students are our firmest critics in this regard: they will not tolerate poor and unecessary use of technology.

But these are powerful tools and, in careless or untrained hands, the more powerful the tool, the greater the damage that can be done.

While tools such as graph plotters sit comfortably alongside what we have always done, other tools confront and challenge our comfort zones: I am thinking of computer algebra tools in particular.

Algebra has long served as the gatekeeper, the way to sort those who would succeed from the rest. But in an age increasingly reliant upon the sills and understandings that we, as teachers of mathematics, can provide, can we afford to continue to discriminate, to deny access to so many?

But won't they lose their skills if they use such tools?

And what will be left to teach them and, more importantly, what will be left in the exams?

The same questions that were asked previously, every time our status quo has been threatened - and the answers are new questions: what skills are important in today's world? Certainly not the same skills as were needed twenty, thirty, fifty years ago.

And what questions will be left to ask? Worthwhile questions rather than trivial questions!

We need to remember that this is not the world we grew up in: when I was in high school, only 20-30% of the school population continued to the end of compulsory schooling; it is now around 90%.

We need to remember, too, that we were the few - we were the chosen. Those who did not succeed as we did fell by the wayside and vanished into the "lower grades" - our memories are very selective concerning what was "the norm" in our day - those basic skills that we claim have been lost in recent decades through the dangers of technology and new pedagogies were mastered in reality only by very few.

We ARE seeing changes in curriculum: the high point of mathematical study in school used to be the Calculus - all our efforts were directed towards preparing students to this end: through the primary years to be ready to start algebra, up to year 10 culminating in equation solving and coordinate geometry - ready for that pinnacle of achievement that is the Calculus.

But is the study of the Calculus still the greatest need for our students in today's world? I have come to believe that we might be better served preparing our students with a strong statistical background if they are to make sense of the data and information that bombards them from every source, and to be well prepared for the great variety of fields now dominated by data and its interpretation.

We need to be teaching a mathematics that is more experimental than theoretical, that draws on data from many sources: from measurement, from the internet, even from the world around them through data logging - and to use this data to lead them to the beauty of functions and, for some, eventually, the analytical study of the Calculus.

And the best tools to use? Those that most empower our students, offering new ways to visualise and manipulate their mathematics, not simply doing their mathematics for them but scaffolding and supporting their learning.

Why do I like to use technology in my mathematics teaching?

It helps my students to be better learners...

It scaffolds their learning, allowing them to see more and to reach further than would be possible unassisted.

Good technology extends and enhances their mathematical abilities, potentially offering a more level playing field for all.

It is inherently motivating, giving them more control over both their mathematics and the ways that they may learn it.

It encourages them to ask more questions about their mathematics, and offers insight into the true nature and potential of mathematical thinking and knowledge.

Good technology also helps me to be a better teacher...

It offers better ways of teaching, new roads to greater understanding than was previously possible.

It encourages me to talk less and to listen more... students and teacher tend to become co-learners.

It makes my students' thinking public, helping me to better understand their strengths and weaknesses, and to better evaluate the quality of my own teaching and of their learning.

It frequently renews my own wonder of maths, helping me to think less like a mathematics teacher and more like a mathematician.

Why do I love using technology in my mathematics classroom?

Because, like life, mathematics was not meant to be a spectator sport.