Making Maths Meaningful!
With
Myra Dunley-Owen
and
February 2015 Edition
Dear School Leaders, Educators, and Learners
Welcome to the second newsletter of 2015! As mentioned last month, we’re focusing on fresh
approaches to the teaching and learning of Mathematics in 2015. I encourage you to keep trying
different approaches to Maths! We can all benefit from trying out new approaches. I’ve been
particularly impressed with our new cohort of thirty Grade 1 teachers (part of JUMP – the Junior Uplands
Maths Programme – also sponsored by The Momentum Fund) who are now using a skipping rope in
their classrooms to practice counting forwards and backwards. A wonderful numeracy exercise that has
the added benefit of expending the boundless energy of Grade 1 learners!
I hope you enjoy reading about the following topics in this newsletter:
 Assessment practices – to highlight the ‘taught and caught’ part of Mathematics
 Number sense – to ensure the foundations are properly laid for Mathematics
 Monthly competitions – to engage your mind
 Pi Day on 14 March – to celebrate with your learners
 Maths Teacher Feature – to highlight the amazing Mr BN Mashigo at Khutsalani Secondary
Please let me know if you have any comments, ideas or suggestions about Mathematics in your
schools. I would love to hear from you!
Mathematically yours,
Myra
Myra Dunley-Owen
Maths Specialist
Uplands Outreach
[email protected]
The beauty of Maths
MATHEMATICS
Did
If people do not believe that
Mathematics is simple, it is only
because they do not realise how
complicated life is.
- John von Neumann
You
Know?
Fibonacci was educated in North
Africa while his father was a
diplomat in Algeria.
Want to know more?
Google
Fibonacci and the sunflower!
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A Fresh Approach to Assessment – as a Tool of Diagnosis!
The first term of 2015 is rapidly drawing to a close, and we should by now have been assessing our
learners to ensure that effective learning has taken place. Assessment provides useful information and
must be used optimally to inform effective teaching and promote purposeful learning. We must also
remember that an effective teacher does not only have good mathematics content knowledge, but also
has knowledge of how to teach mathematics, and how learners learn mathematics. These three
qualities develop mathematical proficiency in teachers.
Also, time needs to be provided for teachers to continue their professional development, sharing with
one another about common problems and working together to develop their teaching proficiency.
Diagnostic reports of the Annual National Assessment (ANA) results and the NSC (matric) have been
printed and teachers are attending workshops on the common errors identified and discussing how to
improve results in 2015. The new project “1 + 4” for all Grade 8 and 9 teachers is underway with all the
re-planning and new time tables to accommodate these teachers being out of the classroom every
Monday. However, I wonder if this is targeting the right teachers? If we analyse the maths results of
each grade we can see clearly that the results decrease as we move through the grades:
Using Maths to assess Maths performance!
From Grade 1 to Grade 4 there is a
decrease of 31%, and if we move from
Grade 6 to Grade 9 we should also expect a
decrease of this order if the pattern
continues.
This would take us to 12%, which is very
close to the 11% that was actually achieved
in the 2014 ANA!
Furthermore we all know that learning maths is accumulative so that if learners only know about 40% of
the basics in Grade 6, it is expected that the results in the Senior Phase will be considerably less. Why
must we then target only the Grade 8 and 9 teachers – the problem begins much earlier than this!
Yes, these weekly workshops will improve the ANA results at the end of 2015 – but only because the
teachers have ‘trained’ the learners in the different topics – but I doubt if there will be conceptual
understanding. This will not help the learners who take mathematics in the FET phase, nor will it
improve the matric results. And the poor teachers in the FET phase will still have to try and teach
abstract concepts with learners not knowing the basic concepts needed. However, I know our maths
teachers, and they will continue to do their best by giving extra classes daily, over weekends and also
during their holidays. Should we not treat the “cause” of the poor results and not simply “band-aid” the
symptoms along the way? Matric matters – but primary school matters more. This is the key reason why
Uplands Outreach is now working with Grade 1 teachers, thanks to support from The Momentum Fund.
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The basic concept focus for Foundation Phase Mathematics
Long-term studies indicate the following: beginning first graders (those just starting out in school) that
1. understand numbers,
2. understand the quantities those numbers represent, and
3. understand simple arithmetic
will have more success in learning mathematics through to the end of primary school, while other
studies suggest the effect actually lasts throughout the rest of their lives.
Maths is critical for success in many fields, and South Africa is simply not doing a great job of teaching
maths. Studies find that once students fall behind, it's very difficult to correct misconceptions and to
bring them up to speed. Results also showed that understanding numbers and quantity is a necessary
foundation for success as the student progresses to more complex math topics. This reinforces the idea
that maths knowledge is incremental, and without a good foundation, a student won't do well because
the maths gets more complex as one proceeds through school.
So what do we mean by ‘number sense?’ Howden (1989) describes number sense as a “good intuition
about numbers and their relationships. It develops gradually as a result of exploring numbers,
visualising them in a variety of contexts, and relating them in ways that are not limited by traditional
algorithms.” As learners work with numbers, they gradually develop flexibility in thinking about numbers,
which is a feature of number sense.
: It is essential that, from the beginning, teachers differentiate between
numbers, numerals, digits and cardinal and ordinal numbers.
 is developed - it cannot be taught
 is helped by good, suitable activities that
must be provided by the educator
 Levels of the same age will be at different
levels of understanding
 Learners cannot and should not be forced to
operate at a level of understanding they
have not yet reached. Forcing a learner is
bad for their development.
 Learners must be allowed to function at
levels they feel comfortable with, and should
use the understanding and tools they have.
NUMBER CONCEPT
Number concept simply means the understanding of number. All of the next
abilities develop over time. This includes:






the ability to count
the ability to understand number symbols and what each symbol
really represents
a feeling for the size of numbers
the ability to use numbers sensibly to solve problems
the ability to estimate an answer to a calculation, and to judge
whether a calculated answer makes sense
knowledge of the vocabulary such as more, less, first, second, …..
last, halve, double
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How do we develop this “number concept?” By learning to count!
The young child learns to count by counting objects that are very similar. In the beginning, they see the
differences between objects and do not want to, even cannot, count together different objects or
objects with different colours or shapes.
Counting does not mean reciting number names. Learners must learn to count a collection of objects
correctly, and realise that the last name in the counting sequence gives the number of objects.
This process takes time.
Young learners go through different phases before they reach this understanding. This is perfectly
natural and cannot be forced. What is needed for this development to take place is plenty of counting
practice with real objects. During this time learners learn to recognise and write number symbols.
The levels of number concept start with the child who can already count
out correctly, and can write the number symbol for a number.
Level 1: Counting ALL with models
4 + 2
e.g.
1
2
3
4
5
6
 the child has to create each number before using it
 the process has to occur repeatedly
 the child has to construct the numbers repeatedly by counting from the beginning before
acquiring a feeling of “how much” a number is
Level 2: Counting on

an abstract feeling for “muchness” of number is called numerosity of number - this is an abstract
concept – the child will now be at level 2
For example:
 Counting all without models
4 + 2
Counting from the beginning until answer is reached
1, 2, 3, 4, ..., 5, 6
 Counting on from first
4 + 2
Counting forward from the first number in the problem
4, ..
/ /
5, 6
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Level 3: Decomposing – using number facts
 A child can decompose or change numbers in a given task in order to perform one or more different
but equivalent tasks which give same answer e.g. add 25 to 38 - some think of 25 as 23 + 2, then add
2 to 38 giving 40.
 Others think of 25 as 20 + 5, and 38 as 30 + 8. Then they add like this:
20 + 30  50 + 8  58 + 5  63
 When dividing 51 suckers among 3 children it is futile decomposing into 50 + 1 - parts like 30 + 12
+ 9 are more useful
 Different children find different things easy - DO NOT INSTRUCT CHILDREN OR SUGGEST HOW
TO DECOMPOSE NUMBERS - the aim is they make the calculation easier and more convenient for
doing the calculation
General remarks on the levels….
 A child may count on to solve 7 + 8 , but count all to solve 25 + 17 - means child has not
acquired numerosity of large numbers - reverts to level 1 to make sense of big numbers
 A child may use level 3 method to solve 25 + 2 8 (20 + 30  50 + 13  63) but may resort to
drawing a picture and making marks to solve 19  3 (a level 1 method) - this indicates child is
unfamiliar with the division operation
Level 3 does not imply decomposing into number of tens and number of units e.g. 54 is not thought of
as “five tens and four units” – this used to be taught –but it is a limiting and rigid conceptualisation
seldom used in mental operations and it makes division very difficult - the natural way is to decompose
into fifty and four
Remember:
 Levels of number concept cannot be
forced.
 the teacher has an important role to aid
each child’s development of number
concept by giving activities that allow
them opportunities to move to higher
number ranges and higher levels - but
should not demand methods and skills
that the child is not ready for.
Sources/References:
Notes on Early Arithmetic - Hanlie Murray (ex
University of Stellenbosch)
Developing Number Sense - Sue Southwood, Rose
Spannenberg & John Stoker
Elementary and Middle School Mathematics: Teaching
Developmentally - John A. Van de Wall
Department of Basic Education – Policy documents
5
Have any of you
Yes ma’am, isn’t
heard of the
it that one that
Cartesian plane?
crashed last month?
cras
A true story recounted by a teacher from
a Gauteng school
MONTHLY COMPETITION TIME!
Prizes
There are
of a R100 CNA voucher for the person that emails the correct answers to
each category to Beauty Mashego at [email protected] first. Terms and conditions
apply. Please include your name, school and cell number in your email.
MONTHLY COMPETITION for Foundation Phase
Find the number I stand for.
 I am a 2-digit number
 I can be counted in 4’s
 I am less than 80
 My ten’s digit is 3 more
than my unit’s digit
MONTHLY COMPETITION for GET Phase
Put each of the digits
0, 1, 2, 3, ……… , 9
in the circles alongside so
that the sums of the numbers
at the corners of each shaded
triangle are the same.
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MONTHLY COMPETITION for FET Phase
3
1
The isosceles right-angled triangle
shown has a vertex at the centre
12
4
of the square. What is the area of
the common quadrilateral?
4
ANSWERS TO NEWSLETTER 6 COMPETITIONS
FOUNDATION PHASE
1. 11 squares
2. 20 triangles
GET PHASE
1. 6
2. 6
3. 4 and 5 (or any 2 numbers
less than 6)
FET PHASE
Answer: R6 million
PI DAY IS MARCH 14 – can you see why?
Quick Facts
Pi Day is annually observed on March 14 (3.14) in honour of the mathematical constant pi (π).
Pi Day is the unofficial holiday that celebrates the mathematical constant pi (π) on March 14 in the
month/day date format because the digits in this date correspond with the first three digits of π (3.14).
It has become an international observance that is celebrated live and online and also celebrates Albert
Einstein’s birthday.
Symbol for Pi
What is Pi (π)?
Pi (π) is the mathematical constant that has been known for almost 4000 years. Its value is the ratio of
any circle’s circumference to its diameter in Euclidean space or the ratio of a circle’s area to the square
of its radius. The value of pi is approximately equal to 3.14159265 ….. , but it is an irrational number
and its decimal representation never ends or repeats.
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Some Pi snippets
 is a Greek letter spelt out as pi, and is the symbol used to represent a particular number.
It is an irrational number and the first 50 decimal places of its value are
3,141 592 653 793 238 462 643 383 279 502 884 197 169 399 375 10 ......
 relates the radius or diameter of a circle to its area and to its circumference.
 has a very long history, but it was not given the symbol and name that we use today until 1706 by
the Welsh mathematician William Jones. However, not much notice was taken of the idea until it was
published by the more famous Swiss mathematician Leonhard Euler in 1737. The ancient Egyptians (c.
2000 BC) knew of the diameter/area relationship of a circle. They recorded that the area of a circle was
found by taking eight-ninths of the diameter and squaring it. This gives a value of about 3,16 for . The
Babylonians, at about the same period, has a stated value of 3,125 which they used for their work on
the circle. All of the these estimates were produced by careful measuring and observation.
No matter how big or how small a circle is, if you divide its circumference by its diameter, you always get
the same number. It is known as pi – or by the symbol  - a name given to it by the ancient Greeks.
It is an irrational number usually rounded off to 3,14.
In 2011 a supercomputer in Tokyo calculated

to a billion decimal places.
What do people do on Pi Day?
There are many activities that celebrate Pi Day such as
games, creating some type of pi ambiance, eating “pi” foods, converting things into pi, making strange
mathematical activities like having a contest to see who knows the most digits of pi. Many people
celebrate Pi Day by eating pie and discussing the relevance of π. Many teachers will use this date to
engage students in activities related to pi by singing songs about pi and developing pi projects.
Mathematicians, teachers, maths students of all ages and other enthusiasts celebrate the number with
pi recitations, pie-baking, pie-eating contests and math-related activities.
The First Pi Day
The Pi Day celebration was founded by Larry Shaw and it was first held in San
Francisco in 1988. The celebrations began with the public and museum staff marching around a
circular space, and then eating fruit pies. The museum has since then added pizza to its menu and the
programme has grown to include activities such as creating Pi puns, Pi-related activities, and many
other activities that involve Pi.
What use is Pi?
Pi is incredibly useful to scientists, engineers and designers. Anything
circular and anything that moves in circles involves pi. Without pi, people wouldn’t be able to build cars,
understand how the planets move, or work out how many baked beans fit into a can!
Sources: www.exploratorium.edu/pi; Theoni Pappas, The Music of Reason
8
Maths Teacher Feature:
Mr B.N. Mashigo
My Career Path: Teaching was not my dream when I was a child. I
wanted to become a doctor, but because of financial constraints and
the scarcity of bursaries in those years, I could not take that path
after completing my matric. After matric I was teaching Maths, and
the following year I went to college to do my teaching diploma. Since
then I never thought I will quit teaching - because the love of
teaching developed and grew. That’s how I became and stayed a
teacher. After completing my training I joined Khutsalani Secondary
School. I’ve taught at this school up until today. I joined UNISA and
studied my BA degree. I also joined Uplands Outreach where I
managed to get my ACE (Advanced Certificate in Education) in Maths
through the University of Johannesburg.
My children: I have one child who is at Wits University studying a B. Commerce. He will be doing his
second year this year.
My favourite Maths topic: My favourite topics are trigonometry and geometry. A learner must first
think before attempting to answer any question.
My favourite colour: I do not have a specific colour. All colours are the same to me. I only match the
different colours when I am wearing clothes.
My favourite food: I don’t have a specific food. Any food that comes in front of me I eat. I choose
between traditional food like morogo and meat. Morogo is better than meat.
My favourite way to pass the time: I like to watch TV and my favorite programmes are sports and
religion programmes. I like gospel songs. I also enjoy reading.
My favourite part of T4E: I have been deeply involved with Uplands Outreach since it was established.
The T4E (Teachers for Excellence Maths Programme) helped me a lot by developing my knowledge of
Maths, and a little bit of knowledge in computers. By participating in the T4E programme, we managed
to be provided with laptops, data projectors, and printers which we are using at our schools. The
support materials I received during the T4E helped a lot during my teaching last year. I managed to
produce good results last year because of the programme.
Concluding thoughts:
Why is Maths important in every day life?
We need to be able to do the following:
 Count and do calculations using
numbers
 Know about money and how to do
calculations
 Measure and to do calculations
 Recognise shapes and know some of
their properties
 Understand and be able to use fractions
and percentages
 Interpret data and graphs
 Recognise and use patterns to make
predictions
Thank you for reading this newsletter. Remember, we want to hear from you! Please send us any
questions, comments or suggestions to [email protected] or [email protected].
~ End of February 2015 Newsletter ~
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