2. Common Ground, Common Themes
Mathematics in the past has been described as the "Queen of the Sciences". I would argue that this is a misnomer, but mathematics certainly shares much with physics, chemistry, biology, the earth sciences and other branches of Natural History. Certainly, scientists use mathematics on a daily basis as a fundamental tool by which data may be generated (often through measurement), then collated, organised, analysed and interpreted. While free to wander imaginary domains of the mind, many mathematicians choose to make this real world of ours their principal playground, seeking to explain and understand all that lies around and within us, so better to appreciate what lies beyond. The line between scientist and applied mathematician is often hazy.
Interestingly, it may be when we teach these disciplines that one important aspect of commonality is made obvious. Both domains of knowledge are today firmly grounded on constructivist principles of learning, perhaps more so than other areas of curriculum. Few who have studied and read widely concerning student learning would today deny that they learn best when provided with opportunities to construct their own knowledge, to negotiate meaning, both socially and individually, to engage actively with their peers and with that which is to be learned and understood. Knowledge which has been pre-digested by teacher or text is a very poor substitute for that which is argued about, wrestled with and made their own. Significant research over the past two decades in both science and mathematics has left little doubt on this matter – even to the extent that these principles are reaching those last bastions to fall to educational research: the classrooms!!
Technology offers another strong area of commonality. Mathematics has been described as a "tool-based activity" but no more so than science. Both have always relied upon extensions of the human form, both physical and cognitive to assist in their development and in their teaching. Each year that passes, this common dependence upon Information and Communications Technologies becomes ever more established. In fact, data loggers and graphic calculators, perhaps even more than computers, appear to offer much to both domains. The physical, personal link that these tools provide between students and their world is fast becoming critically important in the teaching of both subjects and fast blurring the lines between what may be considered mathematics and what science.
The new Learning Objects being developed in collaboration between Federal and State Governments and New Zealand also speak directly to this question of integration. Almost one hundred Stage 1 Learning Objects have been released to schools and teachers in the ACT, across the domains of Literacy for students at risk, mathematics and science. One of the key features of these Objects lies in their integrated nature, crossing boundaries readily between the domains of knowledge. It is possible that the compartmentalised labelling into Literacy, Mathematics and Science may even serve as a distraction for teachers and students, who need to recognise the arbitrary nature of these boundaries.
Far Out Lenses!Perhaps in this place we find the heart of arguments for integration: are we reaching a place where the lines drawn between the two domains are being erased by the new technologies? If so, then might we be approaching a place – at least in the middle school years – where integration may bring real advantages to the teaching and learning of both disciplines? If the primary purpose of mathematics in schools is to produce numerate citizens, then certain essential skills may be recognised: student mathematical thinking needs to be flexible, confident and transferable, skills best achieved through an integrated approach!
Numeracy is to number as literacy is to literature - not necessarily closely tied at all! To be numerate is to use mathematics effectively to meet the general demands of life at home, in paid work, and for participation in community and civic life. [AAMT, 1997]
If numeracy is a functional skill for life, then why have schools often been so spectacularly unsuccessful in developing numerate students?
Two reasons stand out:
Mathematics has been taught and continues to be taught in a largely decontextualised way, as a set of isolated skills, practised independently and rarely connected to life in authentic ways.
School mathematics has been far too "algorithm-dependent", teaching formal methods as the preferred (and often the only acceptable) way to approach mathematical situations. This has been at the expense of an emphasis upon skills of estimation and mental computation.
What might be done?
Encourage and teach multiple strategies, both formal and informal, with an emphasis upon mental computation and flexible approaches to problems.
Wherever possible, situate those problems within authentic contexts, in this way encouraging students to develop mathematical knowledge and skills which are transferable and applicable.
Interestingly, these have been a particular focus of the Learning Objects released in the first round!
After many years of exploring the use of new technologies in my own teaching and learning, I have one crystal of wisdom which I am confident to offer: learning to use the new tools is really about learning to ask new questions. The new technologies of learning frequently serve to make our old questions trivial, and our old methods, at best unnecessary and, at worst, a distraction from the real learning we would strive for. So what is the role of new questions in our consideration of integration?
For comments & suggestions, please e-mail Steve Arnold.
