The teaching of mathematics for all the primary years is carried out using a Singapore-style syllabus. The national mathematical curriculum of Singapore has been acknowledged internationally as the most successful method to acquire mathematical understanding and skills. It has been ranking (see International Test Rankings TIMMS and PISA) as the number one most successful course of primary study worldwide for the past three decades. Due to the exceptional results it achieves the method has been imitated widely throughout the west.
Singaporean-style methods have been successful because they include a systematic teaching of the mathematical concepts required to perform numerical operations. Children acquire a deep understanding of mathematics. Key features of the approach include the CPA (Concrete, Pictorial, Abstract) progression, number bonds, bar modelling, and mental math. These ensure that students learn to think mathematically and rely on the depth of knowledge gained in previous lessons to progress to harder material.
The first half of each daily lesson will be dedicated to progressing through the maths textbooks. This will ensure the regular practice required to acquire mathematical literacy (both understanding concepts and applied skills). The second half of each lesson goes beyond textbook study and aims to give each student a rich and positive personal relationship to maths. In line with the ‘Classical Learning’ philosophy that characterises our school, this section is dedicated to revealing to students the beauty of mathematics: the simple, clear and awe-inspiring descriptions of truth that mathematics is uniquely capable of. Furthermore we employ methods to make the maths class engaging and fun so that children will desire of their own accord to understand the subject matter.
| Monday | Tuesday | Wednesday | Thursday | |
|---|---|---|---|---|
| 9:00 – 10:10 | Textbook progression/maths games | Textbook progression/memory work | Textbook progression/maths history/philosophy of concepts | Textbook progression/memory work |
The Monday class gives time for children to make free use of educational mathematical games. Primary 1 – 3 children will have opportunity to manipulate a choice of tactile teaching aids (‘manipulatives’) such as: Montessori-style wooden toys; Dienes base ten set; measuring cups and liquids; Legos; relevant puzzles; fraction blocks; snap cubes; abacus (traditional and simplified) and more. The older students in primary 4-7 can choose to play from a selection of board games including: ticket to ride; prime climb; mathematical card games; number scrabble; Go; Achi; Tsuro; Dara; Lost Cities and more.
On Tuesdays and Thursdays children will use part of the class to memorise basic mathematical facts through music. They will use song to memorise the times tables; squares; cubes; metric measurements; significant laws such as associative, commutative, distributive, identity and so on. In this section children will use the joy of music to memorise items that are above their level. This will act as foundational for further learning - when they come to need these they will already have learnt them by heart and thus will then be ready to apply them with ease.
As part of this section we will use games to revise the content memorised. An example of a game is where students separate into teams and players take turns answering questions. Anyone with a correct answer takes a turn shooting at a target winning points for their team. Another example is using a game of musical chairs to decide which type of question each students will answer. These games make the classroom more kinetic, introduce playful competitiveness, team spirit and group motivation to the learning experience. This approach is called Gamification (gamifying learning objectives). It has been used very successfully by learning systems such as the Acton Academy schools.
The Wednesday class gives time to learning about important mathematicians of the past; great mathematical discoveries; beautiful solutions that have marked the history of mathematics (such the Pythagorean theorem); and introduction to fun unsolved puzzles. If you tell a child that a prize of one million dollars has been offered to anyone to who can prove the Goldbach conjecture, they sit up and try their hardest to understand what prime numbers are.
Additionally students study basic questions from the philosophy of maths. They discuss questions such as what constitutes proof and in what ways maths represent reality. The Wednesday class takes the form of a conversation and gives learners a wider context for seeing what the field of mathematics contributes to our world. It gives them experience of the purposes of mathematics and in this way it grounds their study of numeracy skills. In conversation with the children ideas and histories are presented and children are encouraged to explore these through dialogue.
Our Maths Teaching Methodology
In our attempt to enrich the mathematical learning of our students Saint Andrew’s Church School has brought together the best elements from a choice of pedagogies. We can briefly summarise what has been said already as such:
We use the Singaporean method of teaching primary maths. This offers learners the best foundation in understanding and practicing mathematical ideas and hence in numerical literacy. We place this within a classical education ethos, where students are invited to see mathematics as one of the ways people have sought truth and beauty in creation. In line with the classical method we emphasise learning basic bits of mathematical knowledge by heart using enjoyable songs.
We have also adopted what is best about the Montessori teaching methods: we give children the opportunity to explore the concepts learnt at their own pace using a range of manipulatives (educational toys that you can manipulate with your senses). We also employ the teaching strategy known as Gamification which aims to increase learner engagement in the classroom by incorporating game elements into the educational environment.
Two more points need to be added to complete the presentation of our school’s unique methodology for mathematical learning:
1. We use a type of Socratic questioning in the teaching of new material. The idea is that children (and all people in general) understand the answer better after they have understood the question. When we approach new data with a question in mind we are far more able to absorb and make use of the data. Asking questions achieves two necessary processes: it focuses our attention and provides us with a framework within which we can order and make sense of data.
A simple way to understand the power of questions to motivate learning is to consider the scheme of an engaging movie or novel. The first and longest part of films is dedicated to involving the viewer in the problem. Here is the problem that needs to be solved (an alien ant species is coming to earth), here is why it matters (they will take over, eat all our agriculture and then eat the people too. We will all die). By this point, the viewer has understood why the hero needs to take action to address this issue. The viewer is then able to appreciate all the actions that our hero takes to finally press this one button and shut down this one machine.
We use questions in the same way at key turning points to introduce our learners to new ‘solutions’. An example of this comes after children have understood addition. We ask: what if I want to add 3 and 3 and 3 and 3 many many times. What would be a more efficient way to write/compute this. After the children have had an opportunity to answer this question as a group, they are ready to understand the essence of addition and multiplication. You can also ask: what is the opposite of an operation and allow children to derive subtraction from addition, and division from their knowledge of multiplication.
At later stages you can ask students: what happens if you want to divide things that don’t divide neatly? What happens if you apply addition or multiplication to things that are not numbers? You discover sets and Boolean logic. You allow children to discover as a team before you present to them mathematical processes for what they really are: solutions to a dilemma.
A great way for children to understand the process of mathematical discovery is to give them opportunities to experience what happens when you break the rules – you discover new branches of mathematics. After children have a solid foundation and understand how the rules work, they can see the joy of breaking them. An example of a rule is that you can add any two numbers but you can only subtract an equal or smaller number from a larger one. What happens if you break this rule? You open up the concept of negative numbers.
This Socratic way of teaching achieves two of our purposes. Children are engaged to focus more keenly on new material, but also, they come to see how everything in mathematics is connected. They understand the generalization element that makes patterns, that in turn make mathematics beautiful. Children can deeply understand that addition is a generalization of counting, multiplication is a sort of addition, exponentiation is repeated multiplication and so on.
2. Finally let us point out an important element of gamifying learning in the classroom. Games make the learning less individualistic. They harness the child’s desire to be part of a community and to contribute to it with their labour. Each child can improve the learning community with their own achievement. The example of the relay race games used for revision illustrates this well. Each team is composed of students of different abilities (ages). The first children are given a number and they need to add something to it. When the task is completed they pass it to their teammates who then perform a more complicated operation (multiplication for example). At the end of the series the last player in the team hands the result to the teacher. The team with the correct answer produced the fastest wins the race. Another example is finding the solution to a word problem in small teams (rather than as individuals) and then presenting your answer to the class on the blackboard.
In line with our emphasis of using the motivating element of community in our maths classes we run an annual ‘Maths sports day’. Children make teams and compete in a range of fun maths team games the way they would in a sports day. Where possible we aim to run this maths fun day together with another school.
The tables that follow list the particular competencies for each year group: