*By guest blogger* David Robson

Cast your mind back to your teenage maths lessons. Without a calculator, would you have been able to estimate the answer to the following sum?

12/13 + 7/8

Don’t worry about giving the precise number; just say whether it lies closest to 1, 2, 19, or 21*.

By the end of middle school, most American pupils have been studying fractions for a few years; these questions should be embarrassingly easy. But when Robert Siegler, a psychologist at Carnegie Mellon University posed the problem to a group of 8th graders (13 to 14 year olds), he found that they performed little better than if they’d simply guessed – with just 27 per cent choosing the right answer. In another test, around 50 per cent of 8^{th} graders failed the arguably easier task of putting a series of simple fractions into size order, from smallest to biggest.

As if this dismal performance wasn’t depressing enough, it comes after decades of educational reforms. American educational psychologists first identified that teens often struggled to understand fractions in 1978, and since then various government commissions and teacher committees have attempted to address the issue, including thousands of educational studies and the introduction of widely used textbooks aimed specifically at deepening children’s understanding of fractions and decimals. These efforts may have amounted to billions of dollars’ worth of spending. Yet Siegler found next-to-no improvement in performance over the 40-year period.

Interestingly, students from other cultures – most notably, East Asian countries – do not face these problems. Siegler and his colleague Hugues Lortie-Forgues have explored the various potential explanations in their recent article for *Current Directions in Psychological Science*, including research that hints at a tantalizing link between language and mathematical reasoning.

As they point out, a basic understanding of fractions and decimals is essential for any further mathematical or science education – so finding a way to remove this stumbling block could have serious consequences for the pupils’ future careers.

The pair first looked at the “inherent” sources of difficulty in these kind of fraction problems – the cognitive hurdles that every child must pass to arrive at the correct answer. The absolute size of the numerators and denominators in a fraction can be intuitively misleading. For example it takes some time and brainpower to work out that 5/11 is smaller than 2/3, since we need to discount the fact that 5 is bigger than 2, and 11 bigger than 3. As a result, even estimating the rough magnitude of a fraction becomes a burden, and that, in turn, could prevent you from checking whether your calculations are on track.

And there’s no getting away from the fact that the rules of fraction arithmetic can be a headache – you need to remember to convert the numbers to a common denominator when adding or subtracting, but not when multiplying, for instance. (1/3 + 1/2 becomes 2/6 + 3/6 = 5/6, for instance.) As an adult these may not seem like huge cognitive leaps, but as a child the reasons can seem opaque and confusing.

Yet children in other cultures do not stumble over these hurdles – suggesting that the inherent sources of difficulty cannot be the whole explanation for why US kids struggle so much. One study from 2015 compared the performance of 6^{th} and 8^{th} graders in the US and China on a test of fraction arithmetic. By the 6^{th} grade, the Chinese students already reached about 90 per cent accuracy, compared to 32 per cent in the US. Even after two extra years of education, the US students only scored 60 per cent, lagging considerably behind the Chinese. (The same is apparently true for many East Asian cultures, including Taiwan, Japan and South Korea.)

For this reason, Siegler and Lortie-Forgues also examined the “culturally-contingent sources of difficulty” – which, to my mind, are far more revealing.

One possibility is that US teachers simply aren’t as good at maths as teachers in other countries, with various studies suggesting that the majority struggle with a conceptual understanding of fractions in particular. For instance, Siegler found that a majority of trainee teachers incorrectly predicted that multiplying two fractions less than one would result in an answer greater than one. Chinese, teachers, by contrast, are less likely to make these mistakes. Nor will the students find the right answers in their textbooks: Siegler and Lortie-Forgues argue that US maths books offer fewer examples and questions for students to practice – particularly on the trickier concepts, like division – compared to sample texts from East Asia.

But perhaps the most intriguing explanation involves the language we speak. East Asian languages phrase fractions in a more literal way: the Korean term for 1/3, *sam bun ui il*, translates as “of three parts, one” – compared to the English “one third”. This small difference means that even very young Korean children, in the 1^{st} grade ( 5-7 year olds), can intuitively match a fraction to a picture – immediately elevating their mathematical understanding.

Together, these different factors may contribute to a kind of vicious cycle in the North American educational system. Students struggle to understand the basic mathematical concepts, and then, if they become teachers, they are less well-equipped to cultivate the necessary thinking in the next generations.

From a British perspective, I’d have been interested to read more about the cultural differences in the West. How do British and European schools compare to the US and East Asian schools? My guess (based on my own education) is that our students would certainly share some – if not all – of the difficulties faced by the American children. If nothing else, our shared language could be one barrier.

The big question, of course, is whether psychology can suggest ways to cultivate a deeper understanding of mathematics. There is some hope that it might: for instance, one study found that simply changing the way we label fractions in English – so that it is closer to the Korean phrasing “of three parts, one” – seems to boost American children’s understanding. But in general, strong evidence for effective interventions is so far lacking.

That may change soon. Siegler and Lortie-Forgues say they are planning to test a new method that will attempt to build the children’s understand on stronger foundations, filling in many of the conceptual gaps that the students (and teachers) are currently missing. Will they succeed, where so many other initiatives have failed? Watch this space.

—Hard Lessons: Why Rational Number Arithmetic Is So Difficult for So Many People

*12/13 + 7/8 = 187/104, or 1.8, meaning that 2 is the correct answer

**Post written by David Robson**** (@d_a_robson) for the BPS Research Digest. **David is a freelance writer based in London, UK. He is currently writing a book, *The Intelligence Trap*, for Hodder & Stoughton (UK)/WW Norton (USA).

I was having a problem with fractals until my mother did not show me fractals on apple. She divided it first on two parts, explaining what 1/2 means. After that, she divided apple on quarters, explaining what 1/4 means. I’ve ate these “fractals”, and remembered them on very simple, cheap and tasteful way.

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While language plays a part, the teaching method impacts the understanding of fractions as well.

In fact, in Singapore, the language of instruction for mathematics is English. However, children from ages 9 and above start working on questions that require “putting a series of simple fractions into size order”, in both ascending and descending orders.

Perhaps apart from looking at language, it would be worthwhile to look at how fractions are taught to the children too.

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I found the solution at the end interesting because is was needlessly complex if you have number sense in addition to computational ability. Each of the two fractions is about 1 so the total is close to 2. Similarly 5/11 is clearly less than 2/3 since 5.5/11 is half and 5/11 is less than that.

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Most people are having trouble with fraction. That is why learning the fundamentals of fractions and knowing how to do the basic mathematical operation on it is a lesson that we should never miss and a must to learn. But sometimes doing it manually may result in a wrong answer. That is why I used a tool to confirm that the computation is correct. For checking if the calculation of fractions is correct, I used an app which shows the correct answer and display the step by step solution. It is actually a fraction calculator from http://www.fractioncalc.com. It is very useful in solving fraction and a very good tool to be used both by teacher and student for reference if their answer is correct and if their solution is right.

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