# Opening up the world of fractions

Kids know that after one comes two and after two comes three. This is how numbers are taught to them and it is the first numerical logic they encounter (1, 2, 3, 4, 5, …). However, this logic works only in the realm of natural numbers, where indeed after one comes two. In the realm of real numbers one cannot say what comes after one, since you can always find a number that is closer to number one. This, however, is rarely taught to kids early enough or well enough. It is taught in high school, but many of the students never truly realise the vast difference in the logic of numbers. This is the topic of my mothers PhD Kaarina Merenluoto) doctoral dissertation  (intro here in Finnish, f.ex. one of several articles in English). So the school is unable to guide kids through the conceptual change needed for them to realise what real numbers are and how vastly different the logic is. The job, then, is up to parents as schools focus more on how to calculate with real numbers, such as fractions. Indeed, even though students rarely realise this, they enter the realm of real numbers as soon as they start to calculate with fractions.

So many kids learn the rules of calculating with fractions but never really understand what fractions are.  And if kids don’t understand them, the rules are just memorised, falsely in some cases, and never really learned. This leads to difficulties when learning more advanced math. So opening up the world of fractions well and with time is important.

I suggest  doing this only after the kid has a solid understanding of natural numbers where the numbers go 1, 2, 3, … But once the child can count and play with these numbers, then introducing fractions works quite well.

First, as always, comes the part where the adult introduces the concepts in his/her language and speech. Here it is done most easily by saying things like a half of the bread (f.ex. as in please eat at least…) or third, two thirds etc. and connecting these with real life phenomena as bread, cake, juice etc. A child might not understand what 1/2 is but every one of them gets what an adult means if they say that the child can have a half of a cookie. So including these mathematical expressions in the everyday conversations at home helps the kids once it is time to move onto more abstract versions of the same concepts.

Another important aspect is to connect these expressions with their mathematical equivalents with the same time and care as what is spent on the natural numbers. This is not something that a parent has to start with, but it is something that needs doing at some point, at latest when fractions are introduced in school. Schools tend to spend way too little time on this. There are less books for fractions than natural numbers, way less. Maybe I’ll write one and add it to this website. This can, however, also be done with food. First a parent needs to write the mathematical expressions of the fractions he/she wants to use on pieces of card board – make them into cards. Next one can utilise any kinds of foods, cucumber sticks, carrot sticks, bread, cookies etc. The thing to remember is to compare. Have the “whole” always there, so it is possible to compare the smaller parts to that whole. For instance, take four cucumber sticks, cut one into three equal parts. Cut another into four equal parts and a third into two equal parts and leave the last whole. Then combine one part of each with the cards made earlier. First do this with the child and then the child can have a go herself/himself. It can be a game with the prize being all those edibles the child has connected with each other in the right way. Another way to do this is to cut up the stick into 2 or more pieces and have the child guess if one piece is 1/2, 1/3 or even a smaller number.

A part from food, play with “fraction-cakes” (murtokakku in Finnish). You can buy them online or cut them up from coloured cardboard as many schools in Finland do. There are all sorts of things one can do with these. One favourite of mine is to take, lets say, one half (1/2) and see how many one eights (1/8) fits into it. My daughter loves to try to build a whole with pieces from different sets (lets say 1/2, 1/4 and two 1/8). Just playing with what they mean and how they relate to each other.

Fraction-cakes help introduce kids to what the numerical expressions of fractions really mean. Can be cut from cardboard.

A third aspect of introduction is basically asking the kid what is the next number after one (or any other natural number) and when they say two (or what ever comes next), first tell them and next remind them that there is always another number in between. This is best done only after the kid is familiar with fractions. Also playing with a measuring tape helps teach this. The measuring tape is a very useful tool in learning math and comes useful also in addition and subtraction. There you can show that there is something in between 1 and 2.

Fourth, I recommend using physical objects like these fraction-cakes when they actually calculate with fractions. It helps kids to have a tactile and very concrete tools they can use to understand the math. I’ve used these with several kids when substituting, also for kids with special education needs, and they really help with grasping what the numbers mean.

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