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 pyschologist 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.
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.
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.