Ever-envious of Singapore’s much-heralded success in teaching maths, politicians are keen to see its methodology arriving in UK classrooms.

Education minister Elizabeth Truss explained some of the background to the government’s current proposals for teaching maths in a recent speech.

###### This article was originally published on The Conversation. By Steve Chinn*, University of Derby*

She mentioned the term “mastery” and enthusiastically welcomed Singapore Maths, a series of textbooks following the “mastery model” by Marshall Cavendish Education, that will be published in the UK from 2015 by Oxford University Press.

One might be tempted to assume Singapore Maths might have something to do with the Ministry of Education in Singapore. I am a huge admirer of the education system in Singapore and have even done some consultancy work for their ministry, but I doubt that the title reflects their direct involvement.

## Learning for mastery

[pullquote]A learning goal has to be broken down into a number of small learning objectives.[/pullquote]The mastery method has been around in educational circles for a while. The term “learning for mastery” was introduced by American educational psychologist Benjamin Bloom in 1968. His idea was that a learning goal has to be broken down into a number of small learning objectives.

This is a methodology that predates computers, but it is often so protracted it needs computer power to be practical. It also relates to precision teaching, pioneered by another American psychologist Ogden Lindsley, again where a goal is broken down into miniscule progressive steps.

So in a maths lesson, a goal for a student might be to: “carry out whole number addition”. One objective that would contribute to this goal could be to “add two three digit whole numbers with carrying in the tens”. In 1983, Robert Ashlock and his colleagues went further, breaking down addition into 23 objectives and subtraction into 24 objectives.

## Not for everybody

I would argue that learning in this way might handicap understanding because the process can be so slow that learners forget the early stages when, and if, they reach the later stages.

Such methods are often prescribed for children who are having difficulty in learning maths. But they are usually inappropriate, particularly if it is the only methodology. It is inherent in the detailed nature of the structure that children who are lagging behind will not catch up by sole use of this methodology. The emphasis, for all learners, should be understanding maths concepts, which will then support memory.

There are other concerns about an over-emphasis on mastery, especially when it is closely linked to behavioural methods of teaching. The level of mastery has to be defined. If, as the word implies, it is a 100% performance, then many children will never achieve that level. If progression to the next topic is denied until mastery is achieved, then too many children will not progress.

All pupils learn differently, and so it may not be possible to establish a strict hierarchy in the different components of arithmetic. In fact, Ann Dowker at Oxford has noted a child may perform well at a difficult task while performing poorly at an apparently easier task. By limiting progression to an inappropriate hierarchy of steps, many children may be denied success in maths.

[pullquote]An ex-student of mine, who was very dyslexic, never mastered recall of all his times tables.[/pullquote]On an anecdotal note, an ex-student of mine, who was very dyslexic, never mastered recall of all his times tables. He did, however, achieve a degree in maths. When I asked him about times table knowledge in the third year of his degree, he assured me that such knowledge was not a huge component of his programme.

## Model students?

In her speech, Truss said that, “The mastery model of learning places the emphasis on understanding core concepts.” Actually mastery is not often about understanding concepts, but instead is about what Bloom’s Taxonomy called “knowledge-remember” – remembering knowledge, not about understanding and higher levels of cognitive ability

I have concerns about exactly what the minister means by “core concepts”. A pre-school child might have mastered the accurate recitation of the core numbers, but they may not have acquired any underlying sense of number.

The mastery model, as with all models, will work for some children, but not for all. There is no data available on which profiles of children respond best to this teaching model. However, thirty years of experience of teaching children who have difficulties with maths tells me that it will rarely be appropriate for that population.

*Steve Chinn does not work for, consult to, own shares in or receive funding from any company or organisation that would benefit from this article, and has no relevant affiliations.*

This article was originally published on The Conversation.

Read the original article.

### A response by David Thomas (@DMThomas90) – A Maths Teacher in West London.

**Mastery Learning in Maths: Actually Explained!**

“Mastery learning is the belief that students should master a skill before moving on to learn a new one. In contrast to the classic spiral curriculum, where students raced between topics without properly learning any of them, a mastery curriculum gives students the space to learn a skill, understand it conceptually, and practise until it’s automatic.

This approach matters because of its effect on working memory. Students who have mastered previous skills have their working memory freed to learn new ones, while students who haven’t get bogged down in the basics and don’t have the working memory space to learn something new.

There are some important subtleties of definition that Steve Chinn picks up on. What it means to have mastered a topic must be clearly defined from the outset, or confusion will ensue. As understanding improves when students develop their conceptual map of maths and draw links between topics, we know that mastery early in school will not mean perfection. For me, mastery means two things:

- The student can demonstrate or explain the concept orally, concretely, visually and abstractly.
- The student can apply the concept automatically, so that it is not dominating their working memory.

Chinn does not engage with these fundamentals of mastery learning.

His first criticism is that mastery learning will not help children catch up, and that they should instead be taught with an emphasis “on understanding maths concepts”. Given that Singapore Maths and its mastery model is renowned for its focus on developing understanding, this seems like an odd criticism. Conceptual understanding is at the heart of mastery learning, especially of Singapore Maths and its concrete-pictorial-abstract model of learning mathematical concepts.

His second criticism is that mastery learning is flawed because the ordering of skills for teaching is imperfect. This is true – there is no universally accepted hierarchy of all skills. This does not detract from the obvious fact that some skills are dependent on others, and that these dependencies are important for the order in which we teach. Adding fractions requires a knowledge of lowest common multiples, which requires a knowledge of times tables. We may disagree on whether we should teach names of shapes or bar charts first in the gap between them, but we know they have to come in that order.

The next criticism is that mastery learning is flawed because some people, for unknown reasons, appear to learn things differently. Even if we accept this argument, I cannot see where it leads. Is the implication that we therefore don’t need to care about the order in which we teach topics, and should pull them from a hat? If order doesn’t matter for some people, why deprive the others of being taught in a logical sequence?

It is particularly dangerous to support such arguments with anecdotal success stories like the dyslexic maths student whose times table recall was not perfect. Anecdotes do not a policy make. This anecdote seems compelling precisely because it is so rare, and it is so rare because it is an exception to a large body of well established research. This student succeeded in spite of imperfect times tables, not because of them. That they succeeded against the odds is not a reason for us to stack the odds against everybody else.”

Images Source: Mathematics Mastery – one year on via Ark Schools – Click here to read more information about their story.

You need to Login or Register to bookmark/favorite this content.

There is, as ever, complexity in these ideas around mastery not least between the use of the language – to master a subject and the proper noun Mastery Maths that is being used for this particular approach – as Chin points out this is emphasised by Singapore Maths and its connection with the renowned education system in the city state of Singapore.

Thomas also makes some reasonable points about the importance of recognising the precursor skills before trying to develop these. This is, of course, not new – the various attempts at curriculum have always tried to do this and Maths (above many other subjects) has a more hierarchal nature which allows this, though we may find the number of steps that Ashlock defined as rather too many.

My concern is that there is a sense again of the rather simplistic “what works” mentality that has pervaded a certain sort of thinking in the last few years – we can see this in the phonics debate where a simple system (SSP) was decided to be right for “all children” – Mastery maths will be the same, I fear, and we should be aware of the differences in culture between the areas in the East where this system is applied and the cultural in the West.

However I would like also to pick up on Thomas’ rather dismissive statement about the spiral curriculum which at best comes from ignorance or at worst a deliberate inaccurate ridicule. The spiral curriculum from Bruner’s work builds on the, “the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development” (Bruner, 1960:3 [The process of education]) and that, “‘A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them’ (ibid.: 13).” – which rather supports the Mastery idea more than students “raced between topics without properly learning any of them” as Thomas caricatures.

There is also at the moment a sort of love affair with the textbook and I feel that the pushing of the Mastery method has something to do with this – it is true that many of the high performing (by the PISA measure – leaving aside at the moment the questions about the accuracy of PISA a ruler) use textbooks but so do many of the lowest performing by the same measure and there is also the equation of a textbook as being a lump of dead tree as opposed to electronic or other measures.

By all means let teachers of Maths look at the Mastery system and explore if they think this is useful for their pupils in their teaching situations but let’s please (please, please) stop having politicians deciding that they know what is the best for the curriculum – as David Bell said recently we need to stop the, “ridiculous situation of MPs with no teaching experience setting the curriculum” (Independent, Jan 9th, 2015)