Model pupil: Children have an innate grasp of certain concepts
What happens in children’s early lives is important in all aspects of development. There is a worldwide move to address the needs of young children systematically, based on current research. The Harvard Centre for the Developing Child, for example, publishes short, usable papers for caregivers online. These working papers are freely available and can be downloaded from open access websites.
Short papers like these provide valuable information about the early years. They focus on nutrition and health, and on the physical, emotional and cognitive development of infants, toddlers and pre-schoolers. They also show how “toxic stress” impacts children’s future development in all respects.
When children suffer too much toxic stress, which comes from lack of care, their emotional development and their natural ability to learn may be impaired. If they are neglected, do not feel safe and do not receive constant, adequate care and real time interaction, they are affected in many ways.
One of the areas of a young child’s development, where stress and lack of care may have a serious impact, is in the learning of mathematics. In the instructional film Taximaths: How children make their world mathematical, created at the Centre for Educational Practice Research at the University of Johannesburg’s Soweto campus, children’s development of number concepts from four to eight years is dramatised. The wording of this title intends to show that children do, indeed, make their lives mathematical by way of their experience, by learning from teachers and learning elsewhere, for example on apps. Yet the basis upon which they develop their knowledge of numbers and other aspects of mathematics is innate.
Children enter the world with the capacity to distinguish, approximately, between more and fewer objects in small quantities. Infants have an innate ability to estimate magnitude and space. They can distinguish between one, two, or three objects. They can see the difference between 13 and five sweets, for example, by approximation.
This inborn trait to see the difference between many sweets and few sweets is developed and expanded by learning from their environment. The experiments that came to the conclusion of “differentiation of many and few” were conducted, of course, without the infants’ use of oral language. Much of children’s learning about the mathematical world around them happens when they have not yet acquired the ability to speak. They see and hear people talk about mathematical things and observe maths in the physical and the social world around them. They see numbers, shapes, positions, dimensions, size, quantity, distance, magnitude, weight and speed, and hear people talk about these phenomena. When they have learned to speak, they love talking about who runs the fastest and who has the most (or the least) of just about anything.
Once children learn to use language, first by showing their understanding when others speak to them, and then by gradually starting to use language themselves, they begin to give names to their intuitive knowledge of maths as they learn these from people interacting with them. They will speak about a big tree and a small tree, about being in front of someone in a queue, or behind a person. They will talk about going forwards, backwards and sideways, and to climb up and down, as well as turning left and right. They happily group things that look/sound/feel the same together and estimate size and number, comparing, for example, the number of chocolate crunchies they got at a birthday table. They judge distances and quantity; for instance, one glass contains more milkshake than another.
Pre-school siblings do this a lot when it comes not only to edible treats, but also size and number of whatever the parents or caregivers “hand out”. Equal distribution is important to young children. They will also quickly tell you when a line is not straight or too long or too short for a purpose. They quite comfortably use much of the language that is needed to learn the full spectrum of early mathematics.
But then a small “disaster” strikes. Formal teaching of “counting” sneaks into the scenario. Parents and crèche aunties and teachers at ECD (early childhood development) centres (and even some grade R teachers) teach them the number words and drill them in oral choir fashion. This in itself is not harmful, for now they are learning at least the language sounds of individual number words. For young children phonological sensitivity is a good thing. It means that their “working memories” have taken up the sounds of the number words and will have them stored for future use.
But, for some reason, many adults seem to think that if a child can remember this list of words, such a child now also understands the concepts that the words represent. Big mistake — probably so big that it may be seen as one of the causes for South African children’s huge gap in early maths competence. Knowing the list of words in the counting sequence has little to do with understanding the individual concept that each of these words represents.
One can test a four-year-old who can “count” to 10 or further. Put about 15 objects on a table and ask the child to take one. Then ask her to take two. Then ask her to take five. Perhaps she will be able to count out five objects one by one. Do it with a few other numbers. If she can competently count out, with one word corresponding with one object, she is only at the beginning of the trajectory of learning to develop number concepts. It can take up to six months for this same child to learn what the “sixness” of six is conceptually. Counting is only one skill that develops, along with others, in the craft of learning maths in the pre-school years. The heart of early maths learning is actually inductive reasoning, and not counting itself.
Once children know the sequence of the words, they slowly learn that one number (representing a whole set) follows another — that four comes after three and that three comes before four. These position words (in any of the local languages) require much practice. They are some of the language precursors for learning in mathematics.
We found in intensive testing of 900 grade one children at the beginning of the school year in 2014 and 2015, that the Gauteng sample of kids tested (in an in-depth individual interview test that was later standardised) showed some disturbing results. Conceptually these children were not ready for the grade one curriculum.
Our school curriculum expects children to have formed strong concepts of small numbers when they enter grade one. If they have not, the speed at which the curriculum is delivered will make it difficult for them to cope. Pre-schools and grade R can do a lot to prepare children. But that preparation does not mean counting up and down to one hundred or more. Counting is only one part of building maths knowledge foundations.
In new research, with a different test, we are currently assessing 400 kids and it is evident, so far, that grade one learners start school with an immense conceptual backlog and enter the grade one curriculum with a sizeable competence vacuum.
Professor Lara Ragpot is an educational psychologist in the department of childhood education, and a research affiliate in the Child Cognition Lab (SARChI Chair), University of Johannesburg, Soweto campus.
Prof Elizabeth Henning is NRF/DST South Africa Research Chair (SARChI): Learning Language, Science and Mathematics in the Primary School, University of Johannesburg, Soweto campus