When you ask people what topics in maths they like and dislike, fractions tend to appear in the dislike column – often vehemently. Recent polling found that maths teachers think fractions are important for students entering High School. I teach in a career course in maths for people who have missed out on maths on the way through school, and a large component of that is fractions.

I love fractions. I prefer them to decimals because they are more exact. One seventh is so much neater than its decimal equivalent. I like adding fractions and multiplying fractions and drawing pictures of them. But that is nerd-speak. I probably should have deleted this paragraph.

I previously wrote about what makes decimals difficult and many of the same things are true of fractions. (It was exactly a year ago!)

- It is difficult to work out relative size for fractions – the bigger the denominator, the smaller the value
- There are infinitely many expressions for the same fraction
- When you multiply by a fraction between zero and one, the result is smaller than what you started with – which is different to whole number multiplication.
- Adding and subtracting fractions requires transforming fractions to a common denominator, but multiplying and dividing doesn’t.
- Multiplying fractions is probably the easiest operation – except for mixed fractions.
- Dividing by fractions is just silly – apart from in algebra, who ever divides by fractions?

Jo Boaler and many other maths educators encourage visual representations of fractions. Often textbooks use circles to show fractions. Circles are pretty and remind us of cookies and pizza (a staple of fraction examples.) Circle representation suffers from the same problem as pie charts, in that it can be difficult to tell the relative size of a fractional value. For example, which is bigger, 3/5 or 5/7? For this reason bar models, also called tape diagrams, are useful for expressing fractions. So long as the bars are the same length, you can compare different fractions. I love bar models, and have a supply of graphics for all sensible fraction amounts.

A customary way to teach fractions is to start with equivalent fractions, getting people to convert between, say thirds and sixths. I have recently seen students complete these exercises accurately, and yet when faced with adding thirds and sixths, they do not transfer their skills from the “equivalent fraction” page. It is as if the two things were unrelated.

It is particularly interesting teaching adults, as many already have entrenched ideas on how to do fractions, some of which work, but have no meaning to them, and many of which are mixed up together. Though my knowledge of “Nix the Tricks” makes me want to avoid the whole “cross multiplication” method, it works for many of the students. When I try to get them to understand what they are doing, they express impatience, and often end up more confused. So in the interests of getting a correct answer, they do use “cross multiplication”. To be fair, that is what I use. It works.

I do find myself wondering why we bother with adding and subtracting fractions. These days calculators can perform fraction operations in an almost miraculous way. Prior to calculators, it was important for some people to know how to calculate with fractions, but is it now? Maybe it should wait until we get to algebra, where it really is needed.

On the other hand, it has been found that “Early proficiency with fractions uniquely predicts success in more advanced mathematics.” Hugues Lortie-Forgues, Jing Tian and Robert S. Siegler This statement needs some exploration of the role of causation, prediction, lurking factors etc. Is it that proficiency in fractions actually encourages success in later maths, or is it that people who understand fractions in the way they were taught at school also understand advanced maths better, or is it that people who find fractions difficult are put off mathematics?

I asked my friends of all mathematical persuasions if and when they use fractions. Many, but not all said they tended to think more in terms of percentages. Common uses of fractions were in cooking, craft, distance travelled when running or flying and time to complete a task. Americans tend to use fractions more as they have yet to embrace the metric system and use cups rather than scales for measuring when baking. None of the examples my friends gave involved operations on two fractions such as adding or multiplying but rather they were used as ways to state an amount, or to find the fraction of a whole number.

One area where fractions are particularly relevant for me is in probability. Probability is an area where fractions are particularly suitable for expressing likelihood and also for calculating complex probabilities. Decimals just don’t work as well for me.

As usual I have no answers, but one useful thought from my reading:

“Interventions that focus on rational number magnitudes appear to be especially effective in helping children learn fraction and decimal arithmetic.”

So maybe the takeaway lesson is to spend plenty of time ordering and comparing fractions before attempting any other operations.

What do you use fractions for?

Why do you think fractions are so feared and disliked?

How important are fractions, really?

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## 6 Comments

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These are some facts about fractions that are not explained carefully at school. Thanks for sharing by the way…

Great post! Thanks for sharing.

Thank you for sharing your insights, it is quite interesting to see why working with fractions is challenging for students.

I love how you mentioned that before focusing on operations students should first visualise fractions, identify their equivalence and understand their magnitude. I agree that understanding fractions involves first realising how ‘big’ or ‘small’ a fraction is and realising that they can be multiple expressions for the same fraction (Bailey, Hansen & Jordan, 2016). I think just drawing on your focus on visual learning experiences, focusing on ‘visual maths’ is also very important in facilitating this understanding, for example not only using examples such as pizzas and cookies, as they can develop misconceptions that fractions only ‘work’ with circular pictures or graphs. I definitely agree that using bar graphs helps to focus students on understanding fractions as an overall concept by seeing equivalence, however I think to extend students understanding further introducing hands on materials such as playdough, which they can use to physically placed or overlayed on top of another helps students to physically understand and recognise which fraction is larger or smaller or the the same comparatively and why. This can lead to students really investigating and problem solving ‘why’ certain fractions are equivalent, making connections such as 3, 6, 9 and 12 are all divisible by 3 and so they can make equivalent fractions easily.

Also, following on from your strategy of using the bar model to show equivalence, a number line strategy can also be effective in getting students to realise that fractions are within the whole numbers and visually identifying where they are placed in relation to each other. This strategy also enables students to identify where mixed numerals are placed and to better understand ‘why’; thereby moving beyond just identifying if the numbers in the denominators are bigger or smaller, but making more critical and reasoning based decisions (Wong, 2013).

I love how you have really broken down students thinking process and made it clear why students find the topic of fractions so frustrating and irrelevant, I guess as mathematics educators it will be our role to show students how fractions are used in the world around us, maybe bringing to class cookbooks or science books to see how fractions are utilised in our daily lives, thereby making their learning experience much more relevant and meaningful.

Wong, M. (2013). Identifying fractions on a number line. Australian Primary Mathematics Classroom, 18(3), 13–18.

Bailey, H. (2017). The codevelopment of children’s fraction arithmetic skill and fraction magnitude understanding. Journal of Educational Psychology, 109(4), 509-519.

Thank you for your great contribution. I’m currently exploring bar models further for teaching multiplication, and they are really versatile.