Piaget believed that children were born with no cognitive understanding of mathematics, or “numerosity” – the ability to understand small quantities (Marmasse, Bletsas and Marti 2000). However, more recent research has shown that children of just a few months old understand very small quantities, distinguishing between 2 and 3 items, but not between 4 and 6, showing their understanding is not fully formed (Starkley et al 1990).This ability proves that infants are born with an innate number sense. 

There are many different number systems, outlined in the article by Marmasse, Bletsas and Marti (2000), dating back to prehistoric times. When writing was invented, there became symbols which represented this ability to count. These symbols we call tags. The easier these tags are, the easier it is for children to learn to count. For example in China, 15 is spoken as “ten five” and in French, 92 is spoken as “four twenty twelve” (as in 4 times 20 and then add on 12). This system is helpful when it comes to teaching children place value.

Marmasse, Bletsas and Marti’s (2000) article also describes the 5 principles of counting, outlined by Gelman and Galistel (1978). It is these principles that young children must grasp in order to count accurately:

• One to One
  • one counting tag for each item
    • One, Two, Three NOT One, Two, Two.
• Stable Order
  • counting tag order must be repeated and consistent (not as below)
    • Counting 3 items – One, Two, Four
    • Counting 4 items – One, Four, Three, Five
• Cardinal
  • last tag represents the cumulative amount of items
    • If counting the apples, you know that once you have counted 10, that the ten applies to the total of the apples, not the specific last apple you counted.
• Abstraction
  • anything can be counted
    • from age 2/3 children can count groups of mixed items
• Order Irrelevance
  • doesn’t matter where you start counting from (left, right, top, bottom)
    • This skill doesn’t emerge until age 4/5

Early counting involves using manipulatives i.e. fingers or toys. This then progresses onto verbal counting, removing these prompts with children counting in their heads, and eventually being able to recall facts from memory. For example, we all know that 3+7 = 10 without actually having to count on 3 from 7.

The part of the article that particularly resonated with me was where they discussed different teaching approaches. Marmasse, Bletsas and Marti discuss the two different teaching approaches as “traditional” and “constructivist”. These two approaches are much the same as Skemp’s instumental and relational teaching (as discussed in my previous blog available at https://blogs.glowscotland.org.uk/glowblogs/myeportfolioekma1/2016/09/23/316/).

It is clear that, while Piaget was partially correct in saying children are not born with any mathematical ability, numerosity is both innate and developed through learning experiences. There are some principles of basic mathematics that cannot be developed until around age 4/5, however, Piaget’s belief that it is not until age 8 that children develop a mature number sense has been clearly disproved by theorists such as Starkey. This is an important factor to consider in the classroom, and I will take this forward in teaching maths in school. Children are more capable than they are given credit for in mathematics, and I intend to use this knowledge and try more complex lessons and mathematical discoveries in future.

References

Marmasse, N., Bletsas, A. and Marti, S. (2000) Numerical Mechanisms and Children’s Concept of Numbers. Available at http://web.media.mit.edu/~stefanm/society/som_final.html (accessed on 1/12/16)

Starkey, P., Spelke, E.S., and Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36: 97-127.