Head of Junior School
Higher Order Mathematical Thinking
I recently attended day one of a three-day workshop being run by The Association of Independent Schools of WA (AISWA). The workshops are focusing at assisting school to develop a whole school approach to mental computation.
The leader of the workshop, Dr Paul Swan, is an award-winning author, acclaimed speaker and workshop presenter, and developer of games and materials to support students to learn mathematics. He is an Honorary Life Member of the Mathematical Association of WA and an Honorary Fellow of the Australian College of Educational Leaders (ACEL). He was awarded his PhD for his work identifying the computational choices of upper primary and lower secondary students.
In the Junior School, we have been engaged on this process for a number of years. We have been developing the boys' counting skills, which involves pattern work, multiples and sequencing. We have also been addressing the boys' basic number skills and working to consolidate their understanding of these key facts. We have been targeting efficient mental strategies for solving number and word problems. The work we are currently engaged in will assist us to bring our previous work together and to further develop the boys' mental mathematics skills.
In our session, Dr Swan shared with us the link between reading comprehension and problem solving. In Year 1 and Year 2, we spend a significant amount of time developing the boys' number skills and knowledge. In Year 3 they move to word problems. This then involves more reading comprehension. Issues with comprehension will result in issues with Mathematics. The link between reading comprehension and success in school is even more clear.
In the Junior School, we will continue to work to develop our boys' ability to work mentally in an efficient manner. Key to this is their understanding of the basic facts for addition, subtraction, multiplication and division. These total 390 facts. We want our boys to be able to recall these facts in three seconds or less. We are not asking for rote learning, but the development of certain understandings that will ensure they know the concepts and can recall them to solve a question or fact.
There are 390 number facts:
• 100 addition facts from 0 + 0 = 0 through 9 + 9 = 18
• 100 subtraction facts from 0 - 0 = 0 through 18 - 9 = 9
• 100 multiplication facts from 0 x 0 = 0 through 9 x 9 = 81
• 90 division facts from 1÷1=1 through 81÷9=9 (There are only 90 division facts because division by zero is not possible, thus the only divisors are 1, 2, 3, ..., 9.)
This is a lot to remember. We aim to develop the boys' understanding of the commutative property of addition and subtraction (2+5 = 7, 5+2 = 7, 3x4 = 12, 4x3 = 12), the addition and subtraction property of 0, place value through partitioning (125 = 100+20+5), halving and doubling, near doubles, building to 10, counting on from the larger number and more.
As we have developed a boy's understanding of key concepts in mathematics, the task of learning 390 facts will become far less daunting. Of the 100 addition facts, 64 can be solved quickly by counting on from the larger number, 32 are simply the understanding of the commutative property (2+3 = 3+2). The remaining 25 involve near doubles, doubles, building to 10 and bridging 10.
Multiplication is much the same. Moving up from 0 x 0 to 10 x 10, there are 100 facts. The multiplication property of 0 reduces the total to 80. The multiplication property of 1 reduces the total to 64. Once the boys understanding the communicative property (3x4 = 4x3), the facts to learn drop to 32. These include square numbers, multiples of two, five, three and nine.
Similar understandings will reduce the number of facts to learn. Strategies such as factoring, doubling and halving, number lines, related facts, using know facts, distributive property and arrays will assist boys in their consolidating their number skills.
The work we will be engaging in this year will address the key understandings and provide the boys with a range of strategies to apply to more complex number problems. A key area of our work will be to develop strong knowledge of the key basic number facts that impact learning across the subject.
Mr John Stewart
Head of Junior School
