# Teaching “and a half” early, possibly before general proper fractions

Fractions are a well-recognised issue in maths learning, with all sorts of complex issues involved. One particular aspect of this is difficulty recognising fractions as numbers which describe the size of things, especially fractions greater than 1.

I once saw a talk or read a paper (which I am unable to now find) suggesting that we can help to deal with this right from the start by introducing children to mixed numbers with halves and quarters (such as $1 \frac12$) before we introduce them to general proper fractions (such as $\frac37$). The argument was that if children can recognise half as an amount, then they can easily recognise one and a half and two and a half as amounts too, and even add them together consistently. This would mean they could understand these simple mixed numbers before learning about other fractions, and then, when they do learn about other fractions, all the mixed numbers can be inroduced at the time.

My question is twofold:

• Does anyone have a reference for this idea?
• Has anyone actually tried it in practice who can describe how it worked (or did not work)?
• I don't have a reference for you, but I think it is important to distinguish the verbal formulation "three and a half" from the written formulation "$3 \frac{1}{2}$". Just as children can understand the notion of "twelve" way before they understand how to write it as $12$, children are generally prepared to understand the concept of "half" well before they understand the relationship between the $1$ and $2$ in the expression $1/2$. – mweiss Aug 12 '14 at 18:00

I have tried this with my own son (whom my wife and I are homeschooling), and at least for him, the approach seems to be working. What we have basically done is introduced fraction concepts as they are naturally needed and discussed them like we might discuss any other number.

Notice the lack of symbols in my description of our conversation. This is because we have not yet introduced any symbols when thinking about fractions. My hypothesis is that introducing symbols based on division to represent fractions is pretty meaningless until children have a better understanding of division, so hence my delay in that respect.

• There's a flaw in your hypothesis about fraction symbols. Certainly explaining fraction notation as division before students have a notion of division would make little sense. But fraction notation still conveys meaning as a part-whole relationship, which is an essential understanding of fractions. It can certainly be explained in these terms. While a fraction can be reinterpreted as a division problem, the notation is not there to represent a division problem. – JPBurke Aug 14 '14 at 12:53
• (I'm not saying there's anything wrong with your approach, just that your hypothesis overlooks other mathematically valid ways of interpreting the fraction notation.) – JPBurke Aug 14 '14 at 13:04
• Worthy point @JPBurke. To be fair, David's son seems to have a decent understanding of the part/whole relationship for at least thirds. – DavidButlerUofA Aug 26 '14 at 17:28
• @DavidWees I'm not a teacher and I don't have any kids. Now that it's 5 years after you wrote this answer, I'm wondering if your son now would be able to figure out the answer to $\frac{1}{2} + \frac{1}{3}$ as a number written as fraction with both the numerator and denominator positive integers. I don't think I struggled with fractions so I don't know what it's like for people who did or how to teach them. Do you think it's worth introducing to new kids like your son was halves first then introducing sixths then thirds by showing that $\frac{2}{6}$ has the property that it can be expressed – Timothy Oct 10 '19 at 19:03
• as $1 \div 3$? Maybe after that, they could be given the problem of figuring out $\frac{1}{5} + \frac{1}{6}$ without first introducing them to thirtieths. Then they might think until they realize that they can both be expressed as a fraction whose denominator is 30. – Timothy Oct 10 '19 at 19:06

I have found the following resource online:

"Fractions: pikelets and lamingtons", State of New South Wales Department of Education and Training, 2003 PDF online, accessed 29/3/2014 (Note that according to the acknowledgements, most of it was written by Peter Gould, though he is not listed as the author.)

This document contains activities designed to support children's learning of fractions, as well as discussion of how children learn fractions and some review of literature. The activities described begin with children's natural language of "half" and "quarter", and then move directly to sharing numbers of pikelets and lamingtons between children in such a way that they must be cut into halves and quarters. So basically, the concept of "and a half" is introduced before other types of fractions.

I can't find further references to this idea in other literature yet, but I will add them to the answer if I do.

The following teaching sequence is described on page 61:

Students’ informal notions of partitioning, sharing and measuring provide a starting point for developing the fraction concept. Drawing on the research on developing the fraction concept and the identified problem of representing fractions as a/b, the following sequence is used to develop the fraction concept.

• Subdividing continuous quantities into halves and quarters using the names halves and quarters and identifying the sub-units.
• Sharing of numbers of continuous models. First using the halving strategy (e.g. Share 6 pikelets between 2 people, share 3 pikelets between 2 people, share 3 pikelets between 4 people) then coordinating the partitioning with the number of sharers.
• Recordings using sharing diagrams for the continuous model (typically circles or rectangles) showing the partitioning.
• Traditional recording of fractions used as “environmental print” linked to sharing diagrams. If a student choses to write one-quarter as 1/4 it is accepted as a common shorthand form.
• Comparison of units by re-dividing a continuous quantity, such as a paper streamer, leading to representing equivalent fractions (e.g. fraction wall).
• Identifying fractions as numbers and locations on a number line (comparison of location, Which is larger, 1/3 or 1/2?)
• Addition and subtraction of fractions using representations of equivalent fractions as subdivisions of the same unit of length ( 2/3 + 1/4). For example, addition or subtraction using the “fraction wall”.
• Symbolic operations with fractions based on students’ coherent and stable meanings for fractions that may be expressed symbolically. It is important to build meaning for fractions arising from the process of division. In this way halves, quarters, thirds and fifths arise from problems of sharing rather than being defined as abstract numerical quantities.

The teaching sequence described in the document builds fraction ideas on the foundation of equal sharing, and starts by using children's natural language of "half" and "quarter"begins with the idea of sharing things equally among people (pikelets and lamingtons actually), introducing the idea of sharing three pikelets among two people in only the second activity

• After that, maybe they could be challenged to compute the value of $\frac{1}{3}$. They might decide to consider themselves not knowing what the value of $\frac{1}{3}$ is until after they compute it using halves and quaters and so on. They might multiply $\frac{1}{2}$ by 3 to get $1\frac{1}{2}$. Then they might multiply $\frac{1}{4}$ by 3 to get $\frac{3}{4}$. Then they might multiply $\frac{3}{8}$ by 3 to get $1\frac{1}{8}$. Then they might come up with constructing the rest of the real numbers by Dedekind cuts of the dyadic rational numbers, the numbers that can be gotten by starting from an – Timothy Oct 10 '19 at 19:13
• integer and then dividing it by 2 as many times as you want. – Timothy Oct 10 '19 at 19:13