Now that I am about to post this, it feels a little too serious for such a hot day. Be warned this is going to be a long post about the challenge of mathematics language, teaching abstract mathematics topics and a uniquely Montessorian ages 3-6 years mathematics material- the fractions circles.
This last term, I was teaching KS2 relationship between fractions, decimals and percentages in a non- Montessori classroom. In my opinion, this is as abstract as primary mathematics gets. The main difficulties the children had was in using the language of these abstract ideas, such as ‘2/3 of these children came to school by bus’. Unfortunately I may not get very far from the beginning when we reviewed ‘Fractions’ because a rather thorough understanding of fractions was so vital that at every step of the way, different fractions-related obstacles found different children. – Most of these examples are of me and small groups of 2-6 children so we were quite able to not leave any child behind. Having said that, helping all children achieve rather thorough understanding can be very easily done if we are patient with the concrete and visual stage.
First, we reviewed fractions. Fractions are often overrepresented as a geometric shape cut up into parts. Not even real shapes by the way, just drawings of circles and rectangles coloured in…This nonetheless is all very well since you and I mostly survived it, that is until later, when algebraic elements are introduced and we try to help children understand a fractions statement that looks like this:
A fraction of a number: This is an example of fractions throwing the learner off in a most unnecessary way by diving straight in (or by introducing half of a number and then a quarter of a number as a half of a half, which does not at all help the learner when a fraction has an odd denomination. Teaching a quarter as a half of a half does not even aid the furthering of understanding in any way. It is completely useless unless it is discovered by the learner himself.) It also reveals how poorly prepared these children are when they get here in year 5.
1. So what do we do? We must allow an extensive amount of time with concrete experiences of the use of ‘a fraction of’ language. If a child knows a third of a circle/cake/pizza means I cut it up into 3, he must now be aided to appreciate that a third of a quantity (number) means I split that quantity (number) into 3 parts and I want one out of that 3 parts.
3. We can then begin emphasising reading a fractions statement and doing what it asks you to. So, instead of asking the child for a fraction of the quantity, the child now reads the instructions and provides the amount.
Up till this point, all your child needs to know is how to read numbers and count well, which means you can do this at home with a year 2-4 child, who has been introduced to fractions.- Even if all he can eg. name 2/3 and thinks it means 2 coloured in out of 3 because that is how he was taught in school…. (can you imagine one day if they swapped it to the not- coloured in bits?!) If your child has been introduced to multiplying and dividing, then once he is familiar with no. 3, he may soon discover that ‘splitting’ is like sharing which is dividing. Again, this really works best if the child discovers it and they usually do. If they do not, telling it to them unfortunately will not really aid them in remembering any of this.
All of the above, by the way, is my Montessori- based invention. It is not at all how fractions is taught in a Montessori Elementary School because I was not trained in (Montessori) Elementary. I was however, trained in primary teaching which did not at all tell you how to teach fractions. – They assumed we knew!
In a Montessori Children’s House, fractions is introduced from about 3.5 to 4.5 years, once a child is confidently counting reading and writing numbers up till 9 or 10. There is a hierarchy of materials introduced beforehand to prepare your child for this. ‘Preparation’ is vital in an authentic Montessori environment and a directress decides whether child is ready for certain materials by observing how engaged he was with the prior materials and by following your child’s interest. By interest, we do not mean James likes cars and singing. – we mean, this tells us he is interested in movement, sounds and language. Montessori preparation for mathematics is definitely worth a post.
This is a typical set of fractions materials in a Children’s House. Children are first introduced to the handling of it ie. how to carry and move it. Then they are shown how to remove pieces and put them back. This is quite a challenge as you can imagine and will require of the child a great deal of concentration, perseverance, interest, problem- solving and visual- discrimination skills. Following that, the names of halves, thirds and so on are introduced a few at a time. Then, the fraction numeral representation is shown. And then adding, taking away and so on. Of course in the meantime, other mathematics materials are also introduced so that by the time operations (+, – etc) with fractions are introduced, operations in general have been introduced. By this time, the child is at least 5 years old.
Next up: a less dry post about summer holiday mathematics for 2-4 year olds!