## ICT Integration: Justifying the task

## Mathematical justification

The use of ‘shadow reckoning’ to determine the height of an unknown object based on simultaneous measurements of the length of its shadow and that cast by an object of known height is a classic textbook question found in almost all textbooks on applied geometry. The technique apparently dates back to the 6th century BC when the Ancient Greek philosopher, Thales of Miletu, is said to have used this method to measure the height of the Great Pyramids (Swetz, 1994).

The selection of various incarnations of this problem depicted in the header graphic and the image below suggest not much has changed since then.

## Problem Solving?

In all of these classic examples, the information is presented to the student in the same format; a side-on view clearly depicting a pair of similar triangles.

Whilst this may not immediately seem like an issue, this formulation of the question precludes students from engaging in any true problem solving. In mathematics, the crux of problem solving is identifying or creating techniques which can be used obtain the desired outcome. Once a method has been determined, everything else is merely mechanical. Yet, in the classic problems (see again, the header picture above) students are presented with a ‘problem’ scenario overlaid with two geometrical figures, a pair of similar triangles. Combine this with the fact that the question is likely found in the imaginatively titled “Problem Solving” section of the “Similar Triangles” chapter, it would take a remarkable student to ignore all the prompts for an appropriate method to solve the problem.

Using Google Earth as an alternative environment for this problem will not address the latter issue. That is, most students will realise that an exercise placed at the end of a week of lessons regarding similar triangles will likely be related to similar triangles. Yet the question of *which* triangles is left unanswered and students are required to formulate the real world problem into a mathematical problem for themselves, an importnat skill.

Further, this objection goes to the core of geometry as an abstract form of thinking and reasoning. The geometer deals with triangles, not made of steel, brick or wood, but of imagination. Lines have zero thickness, points are of zero size, and triangles emerge between any three points the geometer care to imagine. The visual representation of this problem given to students (a horizontal cross section, does not require imagination, or abstraction, the triangles are on the page and not in the mind. It is linking the abstractions of geometry to solid, real-world situations which we should celebrate as problem solving, not turning the crank on a pre-prepared exercise.

## Learning in Geometry

Models of learning in geometry are largely based on the concept of van Hiele levels, loosely analogous to Piaget’s stages of cognitive developments. The van Hieles, were contemporaries and colleagues of Piaget and their work is similarly, a foundation for constructivist approaches to learning in the area of geometry. Pierre van Hiele (1999) has criticised teaching of geometry in schools as it is structured and presented in a way baed on axioms, theorems and other such generalisations. This approach necessarily assumes that students are operating at a formal deductive level (c.f. Piaget’s formal operations stage) yet many students lack foundational understandings about geometry that is achive through play, exploration and interaction with real-world geometric objects (Crowley, 1987). I would argue that although exploring geometrical objects using computer software lacks the tangibility of blocks or cardboard shapes, yet students’ use of computers, mobile phones and iPods has made the manipulation of objects through their depiction on a screen an everyday, ‘real-world’ experience. This task based in ‘Google Earth’ will give students an opportunity to explore real-world geometry.

**Crowley, M. L. (1987)**. The van Hiele Model of the Development of Geometric Thought. In M.Montgomery Lindquist (Ed.) Learning and Teaching Gemretry, K12, 1987 Yearbook of the National Council of Teachers of Mathematics, (pp.116). Reston, Va.: National Council of Teachers of Mathematics.

**Swetz, F. J. (1994)**. *Learning activities from the history of mathematics*. Portland, ME: J. Weston Walch

**van Hiele, P.M. (1999).** “Developing Geometric Thinking through Activities That Begin with Play.” Teaching Children Mathematics, 6, 310–316.

Again, very thorough. I hope others can learn from your approach!