# Why is learning fraction arithmetic so difficult?

I have recently attended two events, both concentrating upon the teaching of fractions in schools.

As part of the Shanghai teacher exchange programme, I recently observed a teacher from Shanghai teach a mathematics lesson in a local secondary school. Much has been said for and against the merits of these exchanges, so I wanted to approach the observation with an open mind. I was keen to see what good practice we, as UK mathematics teachers, can take from the partnership.

The lesson itself focused on comparing two fractions. After a brief explanation, students were given a handful of carefully chosen questions. All questions were linked, served purpose and correct mathematical terminology, such as co-prime, was insisted upon and praised. The teacher used careful scaffolding, so that subtle teaching points were teased out. The pace of the lesson was much slower than I was used to but contained a number of new ideas and approaches. Students were asked to decide whether two numbers are co-prime and then determine how the highest common factor and lowest common multiple are calculated in three situations; where the two numbers are co-prime, where the smaller number was a factor of the larger and when the two numbers had a common factor but the smaller number was not a factor of the larger.

The lesson moved on to comparing fractions, firstly by considering whether the numerators are the same, or could easily be made the same. Students then considered examples where the denominators are the same, or could easily be made the same. There was a clear link between the work finding lowest common multiples and comparing fractions, with the same numbers being used in each section of the lesson. At each stage, students were challenged to think mathematically, considering a range of methods to determine the most efficient.

In the review after the lesson we learnt more about the Shanghai approach; the stable, objective-led curriculum and the consistent approach to topics by teachers achieved by lessons being planned and reviewed by the department. This means that when an experienced member of staff moves school, their knowledge and labours are retained. Lesson observations are frequent but concentrate upon the mathematical content of the lesson, not the performance of the teacher. The lesson is then refined and improved, “there are no outstanding teachers, just outstanding lessons”. Differentiation is achieved not by acceleration, but by delving deeper into the topic presenting students with richer, more challenging questions on the same topic.

"there are no outstanding teachers, just outstanding lessons"

A few days later I attended a lecture at the University of York by Hugo Lortie-Forgues entitled ‘Why is learning fraction arithmetic so difficult?’ He looked at why students struggle to master fraction and decimal arithmetic and suggests that a student’s ability in the topic of fractions is a strong predictor of later mathematical achievement.

As part of his research, Hugo asked 12 and 14 years olds the question: “What is the answer to the closest whole number of the sum 12/13 + 7/8.” The response options were 1, 2, 19, 21, and “I don’t know”. In a 1978 study 24% chose the correct answer. In 2015 it was 27%. How would your classes do?

In a second batch of questions, students were asked to solve a series of fraction arithmetic problems. Forty six per cent of the problems were correctly answered, with only 25% able to correctly answer fraction division questions. In China, the same questions were asked to students of the same age, with an average score of 92%. Incidentally, when primary school teachers were asked the same questions only 55% answered the fraction division question correctly.

The research concludes “relative to Western countries, East-Asian countries have highly knowledgeable teachers (Ma, 1999) and place a large emphasis on students solving difficult mathematics problems (Son & Senk, 2010). Moreover, East Asian students come to the task of learning rational number arithmetic with better knowledge of whole number arithmetic (Cai, 1995) and better knowledge of fraction magnitudes (Bailey et al., 2015).”

There are many aspects of the teaching of mathematics in Shanghai we are not in a position to replicate, but it has been interesting to experience their approach and consider what aspects I can adopt to change my teaching for the better.

Research taken from 'Why is learning fraction and decimal arithmetic so difficult?'

## AlisonLH

I was interested to read your article and can relate it to what I experienced when I was fortunate enough to visit Shanghai in 2014. I observed a lesson on co-ordinates, with 55 primary aged children In the class. The lesson begun with the teacher showing the children a map of the classroom and asked them to describe positions using the co-ordinate grid. This was then replicated with a map of the school and then repeated with a map of the area the school was in. All of us observing commented on how slow the pace of the lesson was and why was there so much repetition? It was not moving learning on but doing something the children seemed to have grasped at the start. However, the teacher afterwards explained how the children needed the different contexts to transfer the skills for a deeper understanding. This was something that I considered carefully in my own teaching and that of teaching in my school....simplicity and repetition, as well as time spent on a concept before moving on too quickly. But how could this be managed within an overloaded curriculum?

Having now spent a year exploring this, the benefits are now starting to reveal themselves. For example, Last term I taught a unit on decimals. In the past I would have probably whipped through place value, ordering, comparing and rounding in 2 lessons and at the time the pupils would have, in my opinion, understood it. However, having spent a week and a half on decimals, my class are actually now making much faster progress in areas of maths where they need to use decimals, and the amount of time previously spent is now proving very beneficial. Thus I am spending less time having to go over basic skills concepts again and have more time for applying maths at a much greater depth.