Interoperable interactive geometry for Europe
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Das Muster: Technology on Demand/The TECHNOLOGY ON DEMAND Pattern


Problem / Challenges / Motivation

Students solving math problems should learn when to use which software in which context. They should be able to use the software whenever they think it will be useful. Students should experience the usefulness of technology solving mathematical problems and reflect on it.


  • To solve math problems with IT is not always necessary so there has to be created a demand for technology to solve the problems.
  • Students have no or low abilities in using related software in itself so they avoid using IT as long as possible.
  • Exploring mathematical assumptions using technology requires i.e. systematically approaches of varying specific parameters and keeping track of the changes which are not normally known to students from school.


Math problems are selected where the use of technology is necessary. There are several ways to achieve this: (1) Problems must be of sufficient high complexity. (2) Problems must induce operations which would be too much work or too monotonous work to be done by hand (for example, always the same calculations with different data). (3) Representations or visualizations of data which can’t be created efficiently without the software are needed to solve the problems.

Students get instructions how to use the specific software (s. section 2). Math problems should contain hints on which software is appropriate. These hints should be as open as possible. For example, they should not mention specific software packages (as Open Office Calc), but types of software (as spreadsheet calculation programs). Ideally, alternatives are given in a way that students can reflect on the advantages and disadvantages of different tools in the specific context. In addition, students must have access to computer tools whenever they need it.

The phrasing of the problems includes questions, tasks and hints which guide the exploring using the technology without suppressing the possibility to follow own ideas and other paths.


The learning of software usage is most effective in contexts where it is necessary to use the software. The need to use a software tool should come before the instruction not the other way around. This is called just-in-time learning or learning-on-demand ((2),(3),(4)).

Computer applications are cognitive tools when they support people’s thinking. “Cognitive tools refer to technologies, tangible or intangible, that enhance the cognitive powers of human beings during thinking, problem solving, and learning.” ((5) p. 693). Cognitive tools allow for creating useful representations, they help to explore a given situation in microworlds, they support deep thinking about content, or they just take away routine jobs from the learner to free his or her cognitive resources (6).

Cognitive tools in the context of learning mathematics can be spreadsheet calculation programs, dynamic geometry systems, computer algebra systems, or simply handheld calculators. The problems have to be posed so that the software i.e. allows to explore an assumption or to falsify the obvious first idea to a solution. If the software can be simply used to avoid thinking then the problems has to be changed.


A typical geometry problem inducing the need for technology is the following:

Given the instruction of inversing points with respect to an inversion circle (without giving away the mathematical term), explore the following questions:

  • What is the inverse of a line?
  • What is the inverse of a circle?
  • What happens if the circle is moved?
  • You can use a dynamic geometry system for exploration.
A similar example in the field of algebra (cf. (1)): ~~Make conjectures of several unit fractions concerning their decimal representation.

What kind of decimal do you get?~~

If it is not a terminating decimal: How long are the periods and the delays of the periods? Make conjectures on the base of your data. Which properties determinate the kind of decimal? Which properties determinate the length of the period and the delay? Test your hypotheses with other unit fractions. Hints/Techniques: You can use the spreadsheets made available in our LMS. Which of the unit fractions are good indicators for your conjectures?


  1. Bescherer, C.: LoDiC – Learning on Demand in Computing. In: Proceedings of 8th IFIP World Conference on Computers in Education 2005, Cape Town, 4.–7. July (2005).
  2. Bescherer, C., Spannagel, C., & Müller, W.: Pattern for Introductory Mathematics Tutorials. To appear in the proceedings of the EuroPLOP 2008 conference (in press).
  3. Eisenberg, M. & Fischer, G.: Symposium: learning on demand. In: Proceedings of the Fifteenth Annual Conference of the Cognitive Science, pp. 180–186, Lawrence Erlbaum Associates, Hillsdale, NJ (1993).
  4. Jonassen, D. H., & Reeves, T. C. Learning with technology: Using computers as cognitive tools. In D. H. Jonassen (Ed.), Handbook of research for educational communications and technology, 693-719. New York: Macmillan (1996)
  5. Salomon, G.: No distribution without individuals’ cognition: a dynamic interactional view. In: G. Salomon (ed.), Distributed cognitions. Psychological and educational considerations, pp. 111–138. Cambridge University Press, New York (1993).
  6. Spannagel, C., Girwidz, R., Löthe, H., Zendler, A. & Schroeder, U.: Animated Demonstrations and Training Wheels Interfaces in a Complex Learning Environment. Interacting with Computers, 20(1), 97–111 (2008).