The Wilkie Way:September Newsletter TABLES OR ALGEBRA?

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

AIMS: Activities in Mathematics and Science

 In the 1980's there was a move by many educators to provide contexts for Mathematics Learning, this including looking at the Maths within Science as well as stand alone Maths.

At the same time there was a push for more Girls to be involved in these  subjects as traditionally they were considered for boys alone!!!!

A great deal of movement was begun in California out of Lawrence Hall of Science Berkeley where EQUALS and Family Math were born.  At the same time happening in Fresno Pacific University AIMS was born

I was fortunate to visit AIMS in Fresno (1987) and through their generosity came away with many practical resources for sharing in New Zealand. I also attended a Week Long Workshop in Perth facilitated by AIMS Tutors.

It has changed over the past 40 years but I am glad that it is still doing some great Professional Development for Teachers.  In fact they have just turned 40.

Elementary School Teachers, check out their latest Newsletter at AIMS 40th Newsletter you may then wish to join their Mailing List.

Congratulations AIMS on reaching 40, just a little short of half my time on this earth. 

 

 

 

Capturing Penguins- Ordered Pairs

 I came across this activity many moons ago so thought it was timely to resurrect as another way to introduce or practise using Ordered Pairs.

Personally hope you are still encouraging children/students to play "Battle Ships" and similar games that I can remember playing, some 70 years ago. 

It does require a little preparation but I hope the outcomes are worth it  And it could spark interest in finding out about Penguins, especially the Hoiho as it is very endangered. 

 

Mathematical "Trick" with a Calculator

 I have to admit I am not a great fan of too much Technology with children, as I believe students learn best by doing, talking about what they have been doing and then writing(in maths language).

DO

SAY

WRITE 

It is difficult for me to understand how students can learn to Measure if all they have is a Tablet in their hand. Measurement is about Stepping, Holding, Weighing(Balance scale) etc.

That said her is a Fun Calculator Activity for you to try on your students or children.   After repeating it a number of times you may wish to ask them to see if they can work out "why it works."

Enjoy.  

Three Gazinta

 Often when encouraging students to learn/memorise tables we do not always look at some of the maths "hidden" in the numbers.

I can remember seeing the "Lights go on" when exploring with students various Divisibility Rules.  I still use the Divisible by Four rule for working out which years are Leap Years, or when the Olympic Games are scheduled.

Three Gazinta could be the springboard for further investigations as well as a context for simple additions and division by 3.  Tables with a Purpose! 

 

The Mathematics of Kaleidoscopes

 With the New Zealand Mathematics Curriculum of the 1990's each Achievement Objective was prefaced by words such as; "Through problems and investigations students will achieve.."

This was a time when those involved in Mathematics Education Teaching and Learning also encouraged "Contexts for Studying Mathemathics" so that the maths skills and ideas had real life ideas to explore and learn the mathematics.

On Thursday I visited an Oral Surgeon, and before he filled y mouth with Instruments he asked me what I used to do. This of course opened a quick discussion about "How Children Learn Maths and My Philosophy"  When I mentioned Contextual Learning he said that he did Orienteering with his daughter! Wow what a great physical as well as mathematical activity. Geometry and Measurement in CONTEXT! I then offered to send him some repeatable maths activities for the family to do together, otherwise he was going to go online to find Maths Pages. I encouraged the games as they would be learning together, they are fun and they would be "talking mathematicss" 

This then got me thinking about other activities that students and families could become involved with. The first is KALEIDOSCOPES. I have never come across anyone who doesn't like at the wonderful patterns made by a Kaleidoscope.  But how many of us have explored the mathematics?  The attached PDF might be the start of a great exploration ending up with making your own kaleidoschope in the future.  Kelston Intermediate back in the 80's had a Metal Work Teacher who had a unit about the making of a Kaleidoscope but no real exploration of the mathematics that help make the patterns.

 

Enjoy your dip into the wonderful world of Kaleidoscopes (Acknowledgement to Creative Publications-Dale Seymour-for Kaleidoscope Math by Joe Kennedy and Diane Thomas)

 

 

 

Beware The Science of Maths

I am on Charlotte Wilkinson's mailing list, as I respect her research and writings, and have her permission to share her thoughts and writings. 

Her latest epistle (April Newsletter) https://www.wilkieway.co.nz/blog/post/163475/april-newsletter-2026/ asks us to Question and Reflect on Teaching Approaches we may have thrust on us or encouraged to implement as its "best for the students"  see her final paragraph.

Beware The Science of Maths

Riding on the wave of the popular “Science of Reading” there is now a movement calling itself the Science of Maths.

Information for this newsletter is taken from The Science of Maths Reconsidered: A critical examination of foundational claims. by Kate Raymond (University of Oklahoma USA) and Melissa Gunter (Central Connecticut State University USA)

While there is much common ground between the “Science of Maths” (SOM) and current research in maths education most arguments made by SOM are based on scant evidence.

Areas of agreement: There is a need for high quality instruction, large scale research of instructional practices, and clear goals and direction for students.

SOM conclude that these goals can only be achieved through the use of direct instruction, they fail to demonstrate that inquiry, discovery, or other student-centred approaches cannot accomplish the same goals.

SOM claims it is a myth that students should not be exposed to procedural instruction until they have demonstrated adequate conceptual understanding.
This is a long standing debate within maths education - when in fact effective mathematics teaching focuses on the development of BOTH conceptual understanding and procedural fluency. Conceptual knowledge and procedural knowledge work in tandem and are often intertwined. To use an algorithm well, students have to have a strong foundation in understanding of numbers and place value. They need a strong foundation in understanding of what it means to add, subtract, multiply or divide before introducing an algorithm.

SOM claims it is a myth that inquiry learning is the best approach.
The argument given is very thin in that the view of inquiry based learning is interpreted as an approach that offers no support or guidance to students. Inquiry with support and scaffolding for student success is of benefit to students. There is little evidence to suggest that inquiry methods with support and scaffolding are inferior to explicit instruction methods.

Further myths claimed by SOM include:

• Teaching algorithms is harmful
• Productive Struggle is important
• Growth mindset increases achievement
• Executive training function is important
• Timed assessments cause maths anxiety

The emergence of SOM as a contempory contributor to the discourse of mathematics education should be treated with caution especially when picked up by politicians, policy makers and the media. Their inclination to support SOM is probably because of its focus on procedural fluency (which is easily measured) rather than sense making, reasoning, or problem solving for which is harder to gather “hard data” as evidence of this occurs over time and in application outside of the school setting. (Becoming numerate!)

We should focus on the common ground:
Timed assessments: - be wary of timed assessments when used ineffectively in providing useless data, creating high stakes assessment practices, used to compare students, or used as a means of withholding something, e.g. morning tea break.
Explicit instruction: defined as “an instructional design and delivery approach characterized as unambiguous, structured, systematic and scaffolded.” This approach can equally be applied to inquiry, problem based learning or other student-centred approaches. However to add to the definition should be “responsive and flexible to individual learning needs.”

It is imperative that as educators we do our due diligence with all new ideas, examining each critically and always ask ourselves why?

We need to make sure we go beyond asking ourselves;
What do I need to teach? (the curriculum)
How am I going to teach it (which resource am I going to use?)

2 ©Copyright N C Wilkinsons Ltd 2026 All rights reserved.

I would like to add a phrase, "Courses for Horses" in other words there is no one instruction that will fit the needs of all students. We as Teachers/Educators need to be aware of this and adjust our teaching strategies to meet the needs of all students in our classrooms.

Consider the learning styles or the students: 

Primary Learning Styles (VARK Model)

• Visual (Spatial): Prefer maps, diagrams, graphs, charts, and patterns to understand information.
• Auditory (Aural): Learn best through listening, discussions, lectures, and speaking.
• Reading/Writing (Verbal): Consume information best through text-based input and output, such as reading notes and writing.
• Kinesthetic (Physical): Learn through experience, hands-on practice, simulations, and movement.

 "I can remember being asked to demonstrate how I would teach simple line graphs to a Year 10 group of girls in a private girls school.

I explained to the class that we were going to start the "maths lesson" outside on the tennis courst, where I had prepared a number line including negative numbers.

Once outside I asked some of the girls to stand on a number on the number line and then proceeded to give them an equation such as:  y = 2x + 3

The students standing on the various numberline points had to look at the number between their feet and do the calculation:   2(-2) +3 =-1,  2(0) + 3 = 3,  2(3) + 3 = 9   etc.

Once the calculations had been completed with assistance from others especially for the negative numbers I instructed.   "When I say "go" if your total is negative step that many steps backwards, if positive  then step those many steps forward.   Go

What have we made?  A straight line!

After a number of these activities including a quadratic equation or two we then went inside and discussed what we found.   This included drawing the graphs of the various equations.

At the debrief, with the schools teachers I was asked, 

"Did you notice the girl at the back right who answer, or tried to answer all the questions? 

"Yes, I did"

"She has never answer a question in maths before"

"Lets disregard the fact that I am a Male teacher in an all girls school, and consider if she is a Visual/Kinesthetic Learner?"

This is just one example how a changed teaching style can empower and involve a student who may have been "turned off" for most of the time.

One size(approach) does not fit all