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Digital technology in the classroom is often talked about. Some agree that it has an important place in modern education, the classroom and students’ lives. Others worry that it is removing the need to be able to carry out mathematical processes, can potentially allow ‘cheating’ and can distract from learning. Untangling the story behind all of this is truly a challenge, and my recent visit to the Geogebra Global Gathering in Linz, Austria highlighted to me the varying perspectives that exist across several dimensions.
Many overlapping questions spring to mind when considering the use of digital technology at any level of education:
In this blog I’m going to consider the place of digital technology in the maths classroom. Merrilyn Goos et al (2003) neatly describe the relationship we may have with digital technology as master, servant, partner, or extension of self, as below:
Technology as master: this describes the situation when learners or teachers are dependent on technology. These users are unable to evaluate the accuracy of any output generated. For example, students inputting data in a table in a graphical calculator and using an inbuilt function, such as finding the mean, without understanding the way in which it is calculated. The calculator acts like an impenetrable black box, with outputs taken as true and valid without evaluation. I see an analogy to a teacher unfalteringly using the answers in the back of a textbook, even in the face of every student getting a different answer for a particular question.
Technology as servant: this describes when technology is used as a replacement for pen and paper – a faster, potentially more reliable way of carrying out calculations. Using technology in this way doesn’t change the nature of classroom activities, other than potentially making patterns and relationships quicker to identify. I would argue this is very much how a calculator is used in a typical maths exam at the end of secondary school by many students: they use it to carry out calculations that they could achieve (with difficulty) using pen and paper, and they calculate, evaluate and write down the answers. For teachers, technology may serve them through their use of one of the myriad of online maths packages which track and analyse student responses and which can – in some cases – identify errors and misconceptions. Teachers’ time is saved by the auto-marking and they can receive an overview analysis of their classes. In both these examples it’s worth considering what is lost through the automation of the process: does carrying out calculations or marking by hand offer the opportunity to deepen understanding of both a mathematical process and our understanding of our learners? What balance is needed? How much practice ‘by hand’ in each case is needed? Does the attainment of the learner affect your answer? What about the experience of the teacher?
Technology as partner: with technology as a partner, mathematical ideas are investigated and explored from a different perspective. Potentially, for example, this may be through dynamic models, parallel multiple representations, or considering geometrical solutions to justify algebraic ideas. Often, with technology as a partner, relationships can be investigated, parameters altered, limiting or boundary cases discovered. This idea was clearly exemplified to me by Tom Button (MEI) at a recent workshop about technology in maths at the University of Nottingham.
His question was: What is the locus of the midpoint of the intersections of the parabola y=x2+4x+3 and the line y=3x+k as k varies?
Algebraically I may choose to solve this as follows;
The x-co-ordinates of the two intersections (assuming they exist) are the solutions of x2+4x+3=3x+k, or 0 = x2+x+3-k
Hence their co-ordinates are;
The midpoint of these is:
producing the line x=-½. Some additional work identifies the values for k for which the parabola and line intersect and hence the values of y for which the locus exists.
What happens if we consider plotting the parabola and straight line using a dynamic graphing tool, for example Geogebra? Allowing k to be a slider, identifying the intersections and tracing their mid-point produces a pleasing animation:
Firstly, I can see the solution: a vertical line for particular values of y. Now considering the boundary case, an obvious place at which something interesting is happening is when the line first intersects the parabola. This happens when they cross just once, in other words when the line is a tangent to the curve, so when both have the same gradient.
y=3x + k, gradient is fixed at 3
y=x2+4x+3 has gradient dy/dx = 2x + 4, and 2x + 4 = 3 when x = - ½ and y = 5/4
Therefore, the locus is the line x = - ½ for y>= 5/4
Thinking about the problem dynamically gives me an alternative entrance to the solution and, in this case, I would argue, relies less on algebraic manipulation and more on deep understanding of the mathematics.
Technology as an extension of self: technology is seamlessly integrated into the classroom. This is a level I am yet to witness or read about, yet still search for. Is it a unicorn? Have you had any experience of situations where technology is integrated into learning activities and learning itself?
Whilst at the Geogebra gathering I could identify certainly the first three of these relationships when listening to presenters and talking to delegates. The products and services being developed aimed at enabling different relationships, discussed the wants and needs of their audiences and the aspirations of those making decisions about the future of education.
I would ask developers, curriculum designers, resource designers and others working in this area to consider: which of these relationships they are encouraging? Why do they believe that is valuable and how might it support the learners, teachers or other stakeholders?
What are your aspirations for your learners (or yourself) when you use digital technology in the classroom?
Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2003) Perspectives on technology mediated learning in secondary school mathematics classrooms. Journal of Mathematical Behavior, 22, 73–89.
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