Interested primarily in genetic epistemology, [the philosophy of how humans gain and develop knowledge] Piaget’s work on mathematical development stemmed more from a wider interest in logic than a desire to discover how children are numerate. However, we know that key to Piaget's theory is that children think differently to adults. This is [in Piaget terms] because they simply haven’t yet had the range of experiences needed to think in an adult way.
This, I believe, is a key point that is of great use if you need to get your head around the constructivism theory on schema's.......
Piaget contended that human thoughts are organised into schemas, which are built upon and retrieved as cognitive development progresses. The child remains very much at the centre of this process, an active learner, constructing knowledge as they constantly revisit what they already know as they experience more and more of their world.
In mathematical terms, it’s easy to see how schema is important, as the child builds on basic numerical concepts; also the idea that logic develops from concrete to abstract is key in mathematical understanding.
The critics of Piaget [eg.Donaldson, Borke, Bruner] question the validity of many of his experiments because they felt that the children didn’t understand what it was they were being asked. The questions simply did not make ‘human sense.’ Further experiments were developed so that children could contextualise what was being asked and the success rate was much higher with children much younger than Piaget had written.....more children could internalise knowledge [central to cognitive development according to Piaget] when they understood the scenario of what was being asked...by way of example – Egocentric or not – being able to decentre from personal viewpoint - the mountains test vrs the cross, villain and the policeman test. (see Donaldson, 1979).
Piaget has become like a very comfortable old friend during the last couple of years and I still find his work interesting and the work of the people who looked deeper into his work and disputed its validity. One cannot deny his [continuing] influence in many international education systems. Piaget and his impact on mathematical teaching was the serving of the day – an [albeit very] ‘pro-piagian’ video and a slightly less bias power point, made two hours pass in a blink..... enjoyable stuff.