Continuing her series on human tendencies, Barbara Isaacs explores exactness, orientation and order…
Montessori believed that children were born with a ‘mathematical mind’, which she defined as a natural tendency for exactness, orientation and order, usually manifested in older children as capacity for logical, systematic thinking. In her writing in The Absorbent Mind she compares this genetic gift to the warp on a loom, into which experiences and learning are woven as the child develops and has opportunities to use shapes, pair and match, sort and use principles of one-to-one correspondence, order, sequence and make patterns.
Current research confirms young children’s early capacity to recognise that two objects are different from one object as early as the first year of life. The activities of everyday living and sensorial materials in the Montessori classroom explore these mathematical qualities and provide a strong foundation for later systematic learning of arithmetic, geometry and algebra.
However, even before young children are introduced to these areas of learning, we can observe their natural instinct to organise – the capacity is usually demonstrated by schematic behaviours when they repeatedly use trajectory, enveloping or enclosure to express their natural tendency for repetition and also organisation in a particular way. Elinor Goldschmied based the heuristic bag activities on toddlers’ natural tendency to compare, match and grade objects. Fitting paper tubes into each other, or posting shapes through appropriate openings not only enhances their eye-hand coordination but also their capacity to note similarities and differences. During the second year of life these experiences are often accompanied by adults’ comments that encourage what is happening – such as, “Try it again”, “Take your time”, “Yes, you have done it!” – and also explain: “Yes, the cube fits into the square”, “Twist the piece until it fits”, “You have too many, take some out”, or perhaps ‘”Which comes next?”, “Which one is different?” or “Are they the same size?”. This is a perfect example of how mathematical language accompanies the child’s experiences from early on, alongside the encouragement toddlers receive from their parents and carers.
As children settle in a Montessori nursery they are introduced to the cylinder blocks, which give them opportunities to experience one-to-one correspondence as they learn to find the right sockets, first by trial and error and later by visual discrimination. The experience of the cylinder fitting snugly into only one of the holes highlights that they belong together. The child will have similar experiences highlighting one-to-one correspondence later, when on his/her third birthday he/she may find that the three candles on the cake match with the numeral three written in icing or on a birthday card.
In a Montessori setting, young children are encouraged to hold the knobs of the cylinders with their thumb and first two fingers, as they are trying to fit the cylinders into the sockets. The focus on the handling of the cylinders enhances their pincer grip and supports the ability to hold a pencil. The movements modelled by the adults are always careful, deliberate and precise – paying attention to details and handling of the resources – maximizing levels of concentration through the careful manipulation. As children continue to use the cylinders, the adults will introduce language such as big and small, wide and narrow, tall and short; thus through first-hand experiences they scaffold the child’s understanding of the cylinders and build the foundation for later learning.
Alongside Piaget, Montessori understood intuitively the importance of real experiences as an essential foundation for the development of conceptual frameworks. To this day she is admired for her golden bead materials, which demonstrate perfectly her deep understanding of how children learn to understand quantities and mathematical language whilst manipulating objects. The golden beads are an excellent example of her thinking. They are introduced to children who have a secure understanding of numbers to 10. Using the units, tens, hundreds and thousands as represented by the beads in their bars of tens, squares of hundreds and cubes of thousands, children have an opportunity to experience and compare one unit (one bead) with the feel and weight of the one thousand cube with its 1,000 individual beads. What is more, they will at a later date, also have the opportunity to actually count the 1,000 beads – organised in 100 ten-bead bars. The activity clearly demonstrates to the young child conservation and reversibility of number, enhancing the mathematical mind by weaving experiences of the organisation of the decimal system into the warp of the mathematical mind and scaffolding the child’s developing abstract thinking.
Barbara Isaacs is the academic director of Montessori Centre International.