Number is, and always has been, an essential aspect of mathematical education. Almost everything can be reduced to numbers in some form, if only counting. "How many ...?" is a question that is regularly asked by children or their parents, with the inevitable counting that ensues. The (cardinal) counting system, based as it is on the simple concept of +1, is a part of the "poetry" of growing up. Parents count to and with their children, as simple counting, or within nursery rhymes or songs; one, two, three, four, five, once I caught a fish alive.
Sometimes, it is extended to the ordinal form, as in first, second, third.
However, counting, although a baseline element of number, is not the be all and end all of maths, but it is one aspect to which children regress, if they are less secure in developmental stages.
On entering my first Primary classroom, the maths resources were based on matching and grouping cards, similar to the cotton reel picture below, with many different variations on real objects and clip art pictures, which then had to be matched, as 1:1 correspondence with some kind of counting material, as in picture 2, with the reels removed.
This exploration and "replacement" of one element with another, is an essential step on the road to more abstract interpretation and manipulation of ideas that have their base in the real world.
Before moving on, try to estimate how many reels there were, either from the picture or the counters. I wonder how many readers will have automatically counted them?
Sometimes, it is extended to the ordinal form, as in first, second, third.
However, counting, although a baseline element of number, is not the be all and end all of maths, but it is one aspect to which children regress, if they are less secure in developmental stages.
On entering my first Primary classroom, the maths resources were based on matching and grouping cards, similar to the cotton reel picture below, with many different variations on real objects and clip art pictures, which then had to be matched, as 1:1 correspondence with some kind of counting material, as in picture 2, with the reels removed.
This exploration and "replacement" of one element with another, is an essential step on the road to more abstract interpretation and manipulation of ideas that have their base in the real world.
Before moving on, try to estimate how many reels there were, either from the picture or the counters. I wonder how many readers will have automatically counted them?
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Grouping the counters into groups of two, three, four, etc allows insights into the broader properties of the number being explored. In this case, 7 groups of 2, 4 groups of 3 and 2 extra, 3 groups of 4 and 2 extra. The properties demonstrate an equality which is fundamental to conservation of number.
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Eventually, grouping into tens was reached, beginning then to demonstrate the relative values of a collection of ten and the unitary elements. Once you can "read" the image, the group of ten equates to the number 1 in 10, as long as the learner understands the need for the 0 to signify an "empty set". So far, we have explored 1:1 correspondence through matching, grouping and set theories.
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Having established a baseline understanding, the learner has to accommodate the idea that numbers can be manipulated, through what we call the "four rules (or functions in the language of Zoltan Dienes) of number", addition, subtraction, multiplication and division.
Essentially the learner has to understand that they are effecting a change of state, where the number that you started with is acted upon by an "instruction".
Addition, already encountered in the counting system, essentially entails the understanding that the original number is being enlarged by a specific amount. this can entail counting on from the "conserved" number, not from 1. If a child regresses to counting from 1 they have not secured conservation of number.
Linking the counting element to some kind of number based visual system, like Dienes or Numicon, adds another visual, but this can lead to visual representation, diagrammatic modelling, which can then be transposed to the number system.
"Greater than, less than, difference between", are all underlying concepts that are explored early in a child's maths experience. Difference between does not need to imply subtraction, but often is presented as such, which can introduce unnecessary complications and potentially remove effective methodologies.
Essentially the learner has to understand that they are effecting a change of state, where the number that you started with is acted upon by an "instruction".
Addition, already encountered in the counting system, essentially entails the understanding that the original number is being enlarged by a specific amount. this can entail counting on from the "conserved" number, not from 1. If a child regresses to counting from 1 they have not secured conservation of number.
Linking the counting element to some kind of number based visual system, like Dienes or Numicon, adds another visual, but this can lead to visual representation, diagrammatic modelling, which can then be transposed to the number system.
"Greater than, less than, difference between", are all underlying concepts that are explored early in a child's maths experience. Difference between does not need to imply subtraction, but often is presented as such, which can introduce unnecessary complications and potentially remove effective methodologies.
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I have not mentioned number lines or 100 squares, or any other systematised supports. These need to be introduced with care, if they are not to add any layers of confusion, that can then cause misconceptions that have to be unpicked. In the attempt to "simplify", especially for struggling learners, it is all too easy to add complexities that confuse, rather than support.
In my early classrooms, where "maths resources" may have been picked up during a walk at the beach or the woods, a trip to a charity shop, or some other means of sourcing artefacts or counting supports, improvisation was needed.
Any form of counting material enabled exploration of arrays, underpinning the commutative law , that 3*4=4*3
Equally, we sometimes explored square numbers as in the picture below.
The principle that maths is everywhere is an important one. It should not exist in the purity of the maths lesson, but needs to demonstrate that what is learned in maths has application across many areas of life.
In my early classrooms, where "maths resources" may have been picked up during a walk at the beach or the woods, a trip to a charity shop, or some other means of sourcing artefacts or counting supports, improvisation was needed.
Any form of counting material enabled exploration of arrays, underpinning the commutative law , that 3*4=4*3
Equally, we sometimes explored square numbers as in the picture below.
The principle that maths is everywhere is an important one. It should not exist in the purity of the maths lesson, but needs to demonstrate that what is learned in maths has application across many areas of life.
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