The Changing Shape of Mathematics Education

Andy Kemp - Head of Mathematics at Taunton School, Taunton.

The majority of content in the current Mathematics curriculum (in the UK and most of the rest of the world) has remained mostly unchanged for hundreds of years. Its basis and focus was designed around the concept of equipping students with the skills necessary for life. In the period before the late twentieth century, this clearly included a heavy emphasis on calculating skills: addition, subtraction, multiplication and division. However, in the current technologically rich society in which we now find ourselves in the UK, these skills no longer have the importance they once did. Ask the average student (or teacher) to add up more than two or three numbers and rather than reach for pen and paper they reach for a calculator. 

Does this show a weakness in their understanding? Or are they simply making use of the most appropriate tool for the job? The argument against the use of calculators has always been based on access; what will people do when they don’t have one? With the widespread use of mobile phones, everyone now has access in their pockets but this still presents the question as to whether it is appropriate or not. In order to resolve this issue and decide whether the use of technology in mathematics is something we want to encourage, or shun as Gardiner (2001) would have us do, we must first decide exactly what it is we mean when we talk about mathematics. 

What is Mathematics? 

For many, Mathematics is defined by their experience in school. Therefore, Mathematics is the act of calculating, whether this is carrying our numerical calculations like finding the area of a circle; or algebraic calculations such as what is the root of this equation?


These skills certainly form a part of mathematics, but is this it? Is mathematics really just a huge collection of skills and tricks? Buchberger expresses the views of many students when he says: 

“Above all, the students ask: ‘Will we ever need this? Who of us, having graduated, will ever differentiate or integrate a function again? Not even extracting a root will really occur later, and if it does, we simply use the pocket calculator! How many of us have, in later life, been assigned problems from mathematics classes (“Two trains travelling from A to B with speed u, ...”)?’” (Buchberger 2002 p.3) 

For many, the focus of our current mathematics curriculum is based around teaching students a collection of skills that for the most part, they will never use again once they leave school. This rather dismal view of the subject is challenged by Conrad Wolfram (2008) describes mathematics as being made up of four parts:
  • translating a problem to a mathematical form (i.e. framing the question) 
  • deciding what result is required mathematically 
  • doing the calculation (i.e. moving from the mathematical beginning point to the mathematical end point) 
  • interpreting and validating the result. 
The majority of time in schools is spent on the third part of this process ‘doing the calculation’ and of the four parts, ironically, this is the only part that can be more accurately and more quickly carried out by using technology! 

Does technology trivialise school mathematics? 

Basic numerical calculators for the most part trivialise many of the skills which form the focus of the primary curriculum. Effective use of a calculator enables a student to add, subtract, multiply and divide numbers of any size (within the capabilities of the calculator). The calculator doesn’t remove the need for students to have an appreciation for the concepts of arithmetic, however it does raise the question whether it is good use of time to spend months or even years teaching students how to add whole numbers, then negatives, fractions and decimals, and then later on algebraic objects, vectors, matrices and complex numbers... All of these use the same ‘concept’ which most students can understand (even if they cannot accurate carryout) by the age of 5 or 6, the time is spent teaching students the different tricks required to apply the same concept to each of these different mathematical objects. 

However the mechanical task of adding together any of these different objects can be handled more effectively (and more accurately) by the use of technology. The same can be said for many areas of mathematics. The current A-level syllabus has at its pinnacle the topic of Calculus (Integration and Differentiation). Around half of the pure content is related in some way to this topic, but the emphasis is very much on teaching students a collection of tricks for how to integrate or differentiate functions (substitution, parts, chain rule, product rule, quotient rules etc.). Many, if not most of these questions are made trivial with the availability of technology such as CAS (Computer Algebra Systems) which are capable of carrying out symbolic algebraic manipulations such as solving equations, simplifying expressions, factorising and expanding expressions, symbolic integration and differentiation, and many other techniques. 

This does not suggest that the concepts of calculus are unimportant, as they are crucial for many areas of the physical sciences, and economics. What it does allude to is that with the effective use of technology, it would be possible to rationalise the amount of time spent in schools focusing on the ‘calculating’ part of the mathematics process outlined above allowing for more time to be spent on Wolfram’s other aspects of mathematical knowledge. 

The knowledge economy 

Politicians frequently tell us that that Mathematics is essential in modern society. Adrian Smith’s report into the state of Mathematics says: 

“Mathematical concepts, models and techniques are also key to many vital areas of the knowledge economy, including the finance and ICT industries. Mathematics is crucially important, too, for the employment opportunities and achievements of individual citizens.” (Smith, 2004 p.v) 

But when we talk about Mathematics being of central importance in the knowledge economy, do we really mean that students need to be good at carrying out calculations? I suggest that this is a wrongheaded view. When mathematics is mentioned as being central to the knowledge economy, I believe people are really talking about the other three areas we mentioned that make up mathematics: 
  • translating a problem to a mathematical form (i.e. framing the question) 
  • deciding what result is required mathematically 
  • interpreting and validating the result.
These are the skills which are essential in an knowledge based economy. The ability to take a problem and turn it into something mathematical, carry out the calculations using technology and then interpret the results. 

For example, an area where the mathematical focus is on these three other parts of the mathematical process is in the field of statistical research. The mathematics involved in carrying out quantitative research is immense and complicated. However, what does the researcher need to know? They need to know how to frame their research as a mathematical question, know which mathematical tests and tools are appropriate and then how to interpret these results. What they do not need is to be able to manually calculate the standard deviation or chi-squared of a set of data as this is much better and more appropriate handled by a software package such as SPSS. 

The example of statistical research and the training given to students in this field provides a possible model for the way we teach mathematics in schools and branching out to the other forms of mathematical knowledge such as translating the problem into a mathematical form, deciding what mathematics is appropriate to the question and interpreting these results. This would allow teachers to bring balance back to mathematical understanding and allow the current fascination with the mechanical calculation skills to be reduced accordingly. 

The use of technology may appear to advocate ‘cheating’ but the real issue is that many of the teachers are limited by their own training. Many teachers learnt to do all these skills by hand and to move from this conceptual space to one where technology is embraced is hard but not as impossible as they may think. Since the advent of scientific calculators we no longer teach algorithms for finding the square root of a number but students are still able to understand what a square root is. If we are able to use technology to support conceptual understanding in one way, why would it be any different with the various other topics that make up school mathematics? 

The future? 

Whilst there are arguments against the use of technology in the Mathematics classroom the potential is brilliantly illustrated by Dan Kennedy when discussing the use of Graphical Calculators in Mathematics: 

Look around you in the tree of Mathematics today, and you will see some new kids playing around in the branches. They’re exploring parts of the tree that have not seen this kind of action in centuries, and they didn’t even climb the trunk to get there. You know how they got there? They cheated: they used a ladder. They climbed directly into the branches using a prosthetic extension of their brains known in the Ed Biz as technology. They got up there with graphing calculators. You can argue all you want about whether they deserve to be there, and about whether or not they might fall, but that won’t change the fact that they are there, straddled alongside the best trunk-climbers in the tree – and most of them are glad to be there. Now I ask you: Is that beautiful, or is that bad?” (Kennedy 1995) 

In my opinion, this is a positive vision of the potential of technology to change the way we teach mathematics in school. No longer do we need to be bound by the historical route through the subject that demands a full grasp of the intricacies of the skills of arithmetic before a student can tackle algebra. No longer does a student need to have a firm grasp of all the techniques of working with algebra before they can explore calculus. By using technology, students can move freely around the ‘tree of mathematics’; a 10 year old student is easily capable of understanding the concept of integration in terms of the area under a curve, so why not let them use technology to look at problems to do with curved areas? 

The problem so far has been that technology has been perceived to be an adjunct to the mathematics classroom. Even in excellent examples/case studies of teachers using technology to help their students, technology is seen as a ‘bolt on’ rather than a fundamental part of the pedagogy to help improve progression and understanding. A substantive mathematics education will remove the old forms of thinking about the subject and move towards addressing the four parts of mathematics we identified. This change has the potential to completely revolutionise school mathematics, allowing us to structure the mathematics curriculum in terms of conceptual difficulty rather than as it is at present in terms of its computational difficulty. These changes would finally enable us to provide students with a curriculum that will help them acquire the skills they will need to take their place in the knowledge economy. 

References: 

Buchberger, B. (2002) Computer Algebra: The End of Mathematics? ACM SIGSAM Bulletin, Vo136, No. 1, March 2002. pp. 3-9. 

Gardiner, T. (2001). Education or CAStration. MicroMath 17.1 pp 6-8. 

Smith, A., 2004, Making Mathematics Count: Report of the Inquiry into Post-14 Mathematics Education. [online] Department for Education and Skills Publications. Available at: http://www.mathsinquiry.org.uk/report/MathsInquiryFinalReport.pdf [Accessed 22nd August 2010] 

Wolfram, C. (2008). Computer Algebra Systems in the Mathematics Curriculum. [online] MEI supported by Texas Instruments. Available at: http://www.mei.org.uk/files/pdf/MEA_CAS_Report_v1a.pdf [Accessed 22nd August 2010] 

Mathematics technology comes in many forms, from the basic four-function calculator and Scientific calculators already common in the classroom; through graph plotters and dynamic geometry packages; all the way to Computer Algebra Systems (handheld forms like the TI-Nspire CAS, and software like Mathematica) and even so called knowledge engines like WolframAlpha.
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