# Singapore Maths

Teachers folding paper to solve a problem at a Maths no Problem seminar in 2012

## Students can under perform in mathematics because they find it boring or they can't remember all the rules.

### The Singapore method of teaching mathematics develops pupils' mathematical ability and confidence without having to resort to memorising procedures to pass tests - making mathematics more engaging and interesting.

Ofsted, the National Centre for Teaching Mathematics (NCETM), the Department for Education, and the National Curriculum Review Committee have all emphasised the pedagogy and heuristics used by Singapore. This method is now being used successfully in the UK by the Ark academies, the Harris Federation, Primary Advantage as well as numerous state, free, and independent schools.

### Features of Singapore Maths:

- Emphasis on problem solving and comprehension, allowing students to relate what they learn and to connect knowledge
- Careful scaffolding of core competencies of :
- visualisation, as a platform for comprehension
- mental strategies, to develop decision making abilities
- pattern recognition, to support the ability to make connections and generalise

- Emphasis on the foundations for learning and not on the content itself so students learn to think mathematically as opposed to merely reciting formulas or procedures.

#### Rationale of Singapore Maths

Borrowing heavily from the Cockroft report (1982) the emphasis of the programmes is all on problem solving.

In all the material you will find that the teaching focuses on the use of three core competencies: Visualisation, Finding Patterns, and Mental Strategies.

The Singapore method of teaching mathematics is based on research from a variety of sources. The work of educational psychologist Jerome Bruner, Richard Skemp's work on relational and instrumental understanding, and the work of Zoltan Dienes on systematic variation.

#### Jerome Bruner

Bruner studied how children learned: he coined the term "scaffolding" to describe how children often build on the information they have already mastered.

In his research on the development of children (1966), Bruner proposed three modes of representation: enactive representation (concrete or action-based), iconic representation (pictorial or image-based), and symbolic representation (abstract or language-based). Rather than neatly delineated stages, the modes of representation are integrated and only loosely sequential as they "translate" into each other. Symbolic representation remains the ultimate mode, for it "is clearly the most mysterious of the three."

In accordance with this understanding of learning, Bruner proposed the spiral curriculum: a teaching approach in which each subject or skill area is revisited in intervals at a more sophisticated level each time. Using this technique of a spiral curriculum, material is presented in a logical sequence. Initially it is enacted with concrete materials, later it is represented by models and then by abstract notation.

This learning theory is the basis for the Concrete -> Pictorial -> Abstract approach which is prevalent in the materials available at Maths No Problem.

#### Richard Skemp

In 1976, Skemp writes about instrumental and relational learning in his paper named:

"Relational Understanding and Instrumental Understanding"

Richard R. Skemp

Department of Education, University of Warwick. First published in *Mathematics Teaching* 77, 20–26, (1976)

Skemp distinguishes between the ability to perform a procedure (instrumental) and the ability to explain the procedure (relational). He argues that these are two different methods of learning - relational and instrumental. In reading Skemp's article it is clear that relational understanding is necessary if children are to progress beyond seeing mathematics as a set of arbitrary rules.

#### Zoltan Dienes

Based on Dienes' ideas (Dienes, 1960), systematic variation is used throughout this method of teaching. The idea is that you vary the lesson through a series of examples that deal with the same problem/topic. It is employed in several ways, including mathematical variability -where the learning of one particular mathematical concept is varied and perceptual variability -where the mathematical concept is the same, but the students are presented with different ways to perceive a problem. The idea of multiple embodiment is to use different ways to to represent the same concept. The Singapore maths books present this in a systematic way to ensure students comprehend what they are learning.

#### Singapore Maths' success

The maths programme is now being used by home educators, tutors, and schools around the globe. It has been extremely successful in producing a consistently high level of maths comprehension which has been substantiated by international studies such as TIMSS and PIRLS.

One of the findings in the TIMMS report is that most countries see a drop in comprehension in year 8 while Singapore students continue to excel which indicates a strong foundational base. As well as having a greater amount of high achievers, a significantly higher percentage of pupils in Singapore attain intermediate level than in the UK showing that the challenged learners benefit the most and retain their level.

There are some heuristics that appear in these series that cannot be found in UK textbooks, but in essence the key components are: Teaching to Mastery, the extensive use of the concrete-pictorial-abstract approach, and extensive use of the bar model method for visualisation. The emphasis is on comprehension and not on rote learning. Children are taught mental strategies to solve problems rather than learning to solve them through repetitive computational procedures.

There is a great deal of information on the Singapore Ministry of Education website, which you can find here.

As well, there is quite a substantial comparison made by the latest research paper from the Review Committee for the new national curriculum, which can be found here.

#### About the books

We deal with all the major publishers in Singapore and distribute all the programmes, including books that can not be found on this website.

Each year of the Singapore Maths school books comes in two sections: A and B. Within each of these sections, there is one textbook and one or more workbooks. These are not alternative programmes, but they are meant to be used one after the other. Students should progress onto B only once A has been completed. The Primary Mathematics 3rd Edition books are the original "Singapore Math" series while the My Pals are Here Second Edition is a more up to date series. All the books are in English and they use the metric measurement system. Math in Focus is a US adaptation of the My Pals are Here series. English speaking students should have no trouble with the content.

#### About the maths programme

Singapore Maths has produced a world-class level of achievement for years. Singapore students scored first in the past three "Trends in International Mathematics and Science Studies" (TIMSS). These studies are conducted by the International Association for Evaluation of Educational Achievement (IEA). IEA is an independent, international cooperative of national research institutions and governmental research agencies based in Boston, USA. Singapore's 4th and 8th grade students scored top place for Mathematics in 1995, 1999, 2003 and 2007.