Charlotte Wilkinson in her latest Newsletter challenges us as teachers to consider the broader picture rather than the narrow on; "Learn your Tables in Primary School and all well be well for Maths at Secondary!"
Those of you over 40 (and those younger as well) need to reflect on your achievement and attitude to mathematics while at school and before University.
How many of your friends hated Maths?
How many of your friends still say they cant do maths?
How many of you were encouraged to take other subjects instead of Maths at Secondary School ((Just one young lady in my 6th form maths class in 1960)
And now we are being told to make sure students memorize their tables in the first months of schooling.
Charlotte is advocating for understand the tables and their relationships and not to learn 100'2 of isolated facts. We have all seen the despair on children's faces when they dont get "10 out of 10 " of they can remember something that has been asked.
Picture this: I am a visiting Consultant/Advisor of Mathematics in a Central Auckland School's Year 6 class:
"Please Sir can you help me I have forgotten what 6 x 4 is?"
"Have you any counters in your room?" "Yes!" and rushes to get them
"What do you notice with the counters?" "THank you Sir the answer is 10!"
This of course led to a discussion about what the signs + and x meant and how display with counters
Wilkie Way September
Basic Facts or Algebra?
Many teachers view primary mathematics as computation, learning the basic facts. Our new curriculum with its wording of memorising basic facts quite early in a students mathematical journey is adding weight to this view.
The Ministry of Education are pushing the need to as turn out pupils who know their basic addition and subtraction facts and know their multiplication tables. The perception being that armed with these crucial mathematical skills children will be successful in secondary school mathematics.
Parents also set this as an expectation on primary schools and most primary school
assessments support this view of primary mathematics.
While knowledge of addition, subtraction, multiplication and division facts
are essential to mathematical success they are only part of the story.
Recent results from the UK (who implemented a compulsory multiplication test at year 4 many year
ago) have recorded the lowest pass rate at GCSE maths (year 11) since records began. Mathematical
knowledge and looking good on narrow primary assessments does not guarantee success at later maths.
Consider the early addition experiences presented to young children.
I have three red teddies and two blue teddies. How many teddies altogether?
This will often be recorded as 3 and 2 makes 5 and 3 + 2 = 5. The use of signs at this stage implies a
meaning on the equals sign as “makes”. Teachers in their teaching continue to develop the understanding of equals as “makes” by the language they use. They may use the word “makes” or “is” or even use the word “equals” or “is the same as”, however children will construct meaning from what they already know.
Learners will construct the meaning of the equals sign to be “makes” from the context in which it is used.
The focus of children’s learning is on the result of the combination of two or more numbers.
Some teachers introduce signs as a way of avoiding the issue of children being unable to read words and
the signs are far easier for children to learn to write than words. By using signs children can be quickly
propelled into carry out mathematical activities involving the use of worksheets and pencils which allows the teachers to keep children occupied on tasks independent of the teacher.
Some teachers may themselves believe in the importance of teaching equality and that the equal’s sign
indicates a relationship between 3 + 2 and 5 in the equation 3 + 2 = 5. They will often use scales or a
number beam balance to convey the idea of equality but while a number sentence is only portrayed as
3 + 2 = 5 the understanding of the meaning of the equals sign as “makes” seems to pervade children’s
thinking for years to come.
Children’s first generalization regarding equations is constructed incorrectly with the assistance of their
teachers. They practice pages of “sums” all reading “number + number = total” or “number – number =
what’s left”. Occasionally they may be presented with the equation in reverse 5 = 3 + 2. For children who still write letters and numerals back to front this is just the same equation presented back to front, 3 and 2 still make 5. The focus is on the result of joining 3 and 2 and not on the relationship between 3 and 2 and 5. Their learned expectations of an equation result in children’s responses to equations presented as 4 + ___ = 7 as completing the missing number as 11 instead of 3. How often do you see this in a primary school classroom? Children see the + which tells them the operation is addition, they see two numbers, 4 and 7 and an equal’s sign which means what do these two numbers make?
Plenty of research exist identifying huge issues with children’s understanding of the equals sign and even more importantly teachers lack of understanding regarding the children’s misconceptions.
Research evidence would suggest delaying the introduction of the equals sign until it can be introduced ina way that encourages an understanding of a relationship between two quantities rather than as a signal o perform an operation. The equals sign should not be seen as “what’s the answer”.
However - our curriculum expects students to be using the addition and subtraction operational symbols
and the equals symbol in their first year. (and multiplication and division symbols in year 2)
The junior teaching programme for mathematics needs to include a focus on the relationships and
patterns inherent in numbers and through studying these relationships and patterns children will become
competent in knowing their basic addition and subtraction facts and then their multiplication and division facts. Basic facts practice must also be included as children need time to learn these essential facts in order to use them as tools for working with higher mathematics relationships and patterns. Completing equations is not practice for recall.
Teachers need to transform their current arithmetic activities and word problems from problems
with a single numerical answer to opportunities for discovering patterns and making conjectures or
generalizations about mathematical facts and relationships and justifying them. (Blanton & Kaput 2003)
For example: A year one class working on facts to five. The basic facts they need to learn are 0 + 5, 1 +
4, 2 + 3, 3 + 2, 4 + 1 and 5 + 0. You may question whether 1 + 4 is the same or different to 4 + 1. To five
year olds they are different as they will compare by direct observation of the numbers, they have not yet
generalized that addition is commutative and the result is unchanged by reversing the numbers in the
equation. Nor will they see the relationship between 4 + 1 and 3 + 2.
(The curriculum expects students to have memorised these facts by the end of 6 months - this does not
mean recording the equations.)
When introducing this work I would start with looking at number five and begin with the partitioning rather than the joining of two sets. How many ways can you break up five objects into two groups? From this the children will discover there are only three ways, 5 and 0 or 0 and 5, 4 and 1 or 1 and 4, and 2 and 3 or 3 and 2. (There are many songs and poems around the number 5 that make good contexts)
Model the use of the addition sign and use the equals sign to record 4 + 1 = 1 + 4. In this way you
are using the equals sign to show equality and promoting the mathematical property of addition being
commutative. (Expected year 1 knowledge)
While learners seem to experience little difficulty with operational signs like addition or subtraction the
equals sign needs to be used in a variety of contexts to promote the understanding of equality. Children
can be presented with the equation 4 + 1 = 3 + 2. Questions to promote thinking about the relationship
between the two sides of the equation need to be asked which focus the children on the changes that
have occurred. These questions should also be supported by concrete evidence. A line of double sided
counters showing four yellow and one red could be changed by turning one counter to red to show three
yellow counters and two red counters. Ask the children if the number of counters altogether has changed.
Emphasize 4 + 1 is equal to 3 + 2, supported by the recording. How have the numbers changed? The
counters assist children to describe the change, four yellow counters is now one less, there are three
yellow counters and the red counters have one more so one red counter is now two red counters. Initially children may not have the language to explain the change. Part of the teacher role is to develop the children’s language to explain the changes. Language underpins all mathematical thinking.
Children must understand that equality is a relationship that expresses the idea that two mathematical
expressions hold the same value. If they can use relationships in mathematics then mathematical
reasoning involved in finding the result to 5 + 6 they can say “ I don’t know what 5 + 6 is but I know 5 + 5 is 10 and one more is 11.” Using mathematical relationships will enable children to recall basic addition and subtraction facts. If the right answer is what children (and teachers) believe is the most importantaspect of mathematics then methods employed will stay in the safe option. Counting will always give me the right answer, thinking about relationships is not required and therefore will not be used.
The relational patterns begun with numbers up to ten continue with all numbers. If children
have been able to generalize the relationships with small numbers then using them with bigger
numbers is not as difficult as trying to undo misconceptions learnt in their first year of school mathematics
