A Euclidean point is at some place~it has a position, and that is all you can say about it. A Turtle is at some place~it, too, has a position~but it also faces some direction~its heading. In this, the Turtle is like a person – I am here and I am facing north – or an animal or a boat. (Mindstorms, p.55)
Papert’s presentation of Turtle Geometry as a ‘computational geometry’ (55), highlights his inventiveness for combining ‘humanist’ mathematical content through LOGO programming language. Interestingly, he shows that how Turtle Geometry would only ‘fit’ with children’s learning through three principles: the continuity principle, the power principle and the principle of cultural resonance.
The ‘continuity principle’ is met is by giving a Euclidean geometric point or position ‘its heading’ (55). In doing so, Turtle Geometry makes mathematics ‘like a person – I am here, and I am facing north’ (55). Consequently, children can identify with the Turtle and are thus able to bring their knowledge about their bodies and how they move into the work of learning formal geometry’ (55).
The offering of the Turtle to children in two physically different entities (floor and light Turtle) allows them to see how three essential properties of position, heading and ability to obey of ‘Turtle Talk’ gives rise to the power principle and principle of cultural resonance. Crucially, it shows how entities are mathematically the same – isomorphic – while simultaneously showing a materially different appearance (56). As a result, Papert demonstrates that the sense of power this brings to children linking body knowledge and movement in affecting the creation of both Turtle Talk (programming) and graphically representing the Turtle’s motion. He calls this heady stuff, as children realise their thoughts through syntonic learning (63).
Such a powerful concept means that the ‘effect of work with Turtle Geometry on some components of school math is primarily relational and affective (68). The impact of making mathematical knowledge more ‘concrete’ through builds children’s cultural syntonicity, deepen mathematical concepts such as ‘navigation’ and ‘variable’ (69), and most exciting of all, the concept of recursion, the trick of setting up a never-ending process (71), like learning itself.