3. BIG ideas covered this week
My prior knowledge of pre-number concepts included what children have learned or their
‘general knowledge’ before understanding numbers and numeration. I didn’t particularly
know what pre-number involved however, my recent understanding tell me that the skills
and Early concepts that define Pre-number concepts includes:
Determining attributes,
Matching by attributes,
Sorting by attributes,
Comparing attributes,
Ordering attributes
Patterning
These 6 components form the basis of all later study of number relationships and
numeration used in everyday life; whether it be defining items by the common attribute
such as colours, sizes, shapes, differences, similarities or patterns
4. Concepts, Skills & Strategies
Each pre-number concept has its own concepts, skills and strategies. The example I
will use is SORTING BY ATTRIBUTE
Concept- A child being able to take a large group of items (counters, teddy bears,
attribute blocks) and group them into smaller amounts by separating, describing the
defining the specified attribute (i.e. colour, size, shape, length etc.)
Skills- has a clear idea of how they will group by attributes. How the child will
choose to sort the attribute blocks, how the child will organise the piles
Strategies- Putting the concepts and skills into action. How the child goes about
sorting the attributes. The process of action- moving the attribute blocks into a pile/
container/ sorting chart.
7. Teaching strategy and resources that assist
children to understand a key
mathematical concept
Patterning for Year 1:
Activity 1
Allocate student a one- or two-digit number (e.g., 6,8,10,12 etc.)
and a set of objects (MAB blocks, counters, bears).
Ask the children to show and record their number in as many
different ways as possible and using the objects provided - in
words, in diagrams, in numbers and using the objects.
Students use cubes or plastic squares to build a staircase
pattern. They draw, count and record the number of squares
used in the pattern. Ask questions about the pattern the
students have made.
How many squares make up the first shape in the pattern? the
second shape? the fifth shape?
How many squares were added to the first shape to make the
second shape? the second shape to make the third shape?
8. Misconception
How would you avoid or remediate this misconception?
Students in year one might not be able to identify a growing pattern in the form of
but rather confuse the situation by adding extra on to ‘fill the gaps’.
I would try to break the pattern down to the core starting with counting how many are in each
section of the pattern instead of setting it in a pictorial way … example- 1, 2, 3…? What’s next?
From there, gradually and slowly I would use concrete counting materials such as small plastic
bears, do set out the patterns in linear formations or groups, adding one on each time making
sure the child is understanding and following. I would then try to relate it back to the pattern
above asking if they she any relationships and similarities between what’s happening with the
concrete materials and the pattern. I would then explain that when you look at patterns like these
you are best to count the number in each section first then try and find something that is the
same about it- whether it be counting by 1’s, 2’s ,5’s etc.
9. ACARA Links & Scootle Resources
Patterning:
FOUNDATION YEAR
Strand: Number and Algebra
Sub-strand: Patterns and algebra ACMNA005
Content Description: Sort and classify familiar objects
and explain the basis for these classifications. Copy,
continue and create patterns with objects and
drawings
Elaborations: observing natural patterns in the world
around us, creating and describing patterns using
materials, sounds, movements or drawings
Site2see:
http://www.resources.det.nsw.edu.au/Resource/Acce
ss/37f3cd25-eeb2-4be3-87e0-e83f98c717b7/1
In the ‘Early stage 1 and Stage 1’ part of the website
there are some basic and age appropriate activities
for children in foundation year to have a go of which
copying, extending and creating new patterns.
Comparing and Ordering: FOUNDATION
YEAR
Strand: Measurement and Geometry
Sub strand: Using units of measurement / ACMMG007
Content Description: Compare and order duration of
events using everyday language of time
Elaborations: knowing and identifying the days of the
week and linking specific days to familiar events,
sequencing familiar events in time order
Skwirk: http://skwirk.com.au/esa/Ordering_Events.html
This link requires children to recall basic events of a
day from a short video clip. The children are then
required to order them correctly (digitally). This is a
good link to begin to give children the idea of ordering.
(Australia Curriculum, Assessment and Reporting Authority, 2016)
10. Resources & Ideas
For Young children creating basic knowledge is important. By
providing mats such as these children will be able to sort
coloured bears or other small counting items into colour or
size etc.
Again by providing colored bears and a temple for children
they are able to pattern. Using templates (as pictured)
children are able to Build, Copy ,Create, Extend a pattern
and Insert the missing element.
Comparing- Providing scales in a Foundation Year setting
(and older years) is a great resource which allows children to
experiment with objects- creating the appropriate language
such as more than, less than, heavier, lighter etc.
By unintentionally teaching ordering to children they will begin
to pick-up sequence. Whether it be the school routine, home
routine or activities that surround sequencing. Activities such
as the one below are frequently seen in Foundation year
settings (and younger).
11. Textbook Notes
Number sense includes:
Understanding number concepts and operations of
numbers
Development of strategies for dealing with numbers and
operations
Accurately and effectively mentally compute to detect errors
and recognize expected results
Decide flexible ways to decipher the correct mathematical
judgments
An expectation that numbers are useful and that work with
numbers is meaningful and makes sense
• Appropriate modeling, posing process questions,
encouraging thinking and creating a classroom environment
which supports number sense assists in developing number
sense
Number sense is developed in early childhood and
primary settings for a lifelong process which is
continually build upon due to 3 stages
1. Pre-number and informal number
Classification
Patterning
2. Early number development
Conservation
Comparisons
One-to-one correspondence
3. Number development and counting
Place value
Skip counting
Counting back and forth
Oral and written cardinal and ordinal number
recognition
12. PATTERNS:
1. Mathematics is the study of patterns. Problem-solving skills are required when creating, constructing and describing
patterns and this is a very important part of mathematics learning. Patterns combine several attributes: geometric,
relational, physical and affective attributes which help to develop recognition skills i.e. shape, symmetry, sequence,
function, color, size, texture, number, like, happiness.
2. Patterns help children develop number sense, ordering, counting, sequencing. They are also helpful in developing
thinking strategies. Encourage children to “think out loud” as they search for patterns and ask them to explain what
they did. Encourage them to share their thinking.
3. Children need opportunities to create their own patterns. These can give insight into their mathematical thinking.
Copying a pattern requires children to choose the same pieces and arrange them in the same order, or finding the
next one to continue and complete a pattern. Extending a visual pattern gives children the opportunity to explore
different mathematical ideas.
GROUP RECOGNITION:
1. Subitising: The skill to “instantly see how many” in a group. It is an important skill to develop. Sight recognition of
quantities up to 5 or 6 is important. It is much faster to recognize the number in a small group than counting each
individual member of that group.
2. Children who can name small groups demonstrate a knowledge of early order relations. They understand that 3 is
more than 2, and that 3 is 1 less than 4. From here, they will develop more sophisticated counting skills. It also
accelerates the development of addition and subtraction.
3. Sight recognition. As children grow older, their ability to recognize quantities will improve. Sight recognition is
evidenced by children’s skills in reading the number of dots on a domino. Certain arrangements are more easily
recognized. Children usually find rectangular arrangements easiest to follow. The grouping of objects in common
patterns is also effective in the counting process. Research supports the notion that subitising is a prerequisite for
counting.
13. COMPARISONS AND ONE-TO-ONE CORRESPONDENCE:
1. Comparison of quantities is another important part of learning to count and the development of number awareness. When making
comparisons, children must be able to discriminate between important and irrelevant attributes.
2. Children need to become familiar with descriptions of relationships such as “more than”, “less than”, “as many as”
3. Ordering is involved when comparisons are made among several different things. Children will realize that 4 is the number between 5
and 3. Graphs can also be used and can be particularly helpful for children in the early development of numbers.
COUNTING:
1. Rational counting is a goal for all young children. A ‘rational counter’ says each number name and realizes that the last number counted
represents the number in the group.
2. Four important principles of counting include: Each object to be counted must be assigned one and only one number name, the number-
name list must be used in a fixed order every time, the order in which the objects are counted does not matter, the last number name
gives the total of the objects in the group.
3. Rote counting: A child using rote counting knows some number names but these names are not in correct counting sequence.
Rational counting: Rational counters exhibit all four counting principles. A Child gives a correct number name as object are counted, uses
one-to-one correspondence and is able to answer the question about the number of objects being counted.
4. Counting on: A child gives correct number names as counting proceeds, and can start at any number and begin counting. Counting-on
practice leads children to discover valuable patterns and is also essential for developing addition.
Counting back: Many children find it difficult to count backwards. It helps children establish sequences and relate each number to
another in a different way. A number line can be used for counting on or backwards from any starting point.
5. Skip counting: The child gives correct names but instead of counting by 1’s, counts by 2’s, 5’s or other values. This method provides
counting practice, early recognition of even and odd numbers, and readiness for multiplication and division.
6. Counting change is a very important skill. Most of the difficulties associated with counting change can be traced to weaknesses in
counting, particularly skip counting. Teachers should encourage accurate and rapid skip counting.
(Reys, et al., 2012)
15. BIG ideas covered this week.
I didn’t have very much prior knowledge on number
sense or numerations at all before learning about its
concept. I had never really thought about the
underlying significance if the way numbers are order,
why place value was important and the concept of
operations. This is known as Number knowledge, which
organised by formal ideas (numeration and place
value) and informal ideas (number sense).
16. Concepts, Skills & Strategies
Number Sense
Concepts: Number sense essentially refers to a student’s “fluidity and flexibility with numbers,” (Gersten & Chard, 2001). He/She has
sense of what numbers mean, understands their relationship to one another, is able to perform mental math, understands symbolic
representations, and can use those numbers in real world situations
Skills: solve problems, spot un/reasonable answers, understand how numbers can be taken apart and put together in different ways,
see connections among operations, figure mentally, and make reasonable estimates
Strategies:
Model different methods for computing: record a number of different approaches to solving a problem
Ask students regularly to calculate mentally: Mental math encourages students to build on their knowledge about numbers and
numerical relationships.
Have class discussions about strategies for computing: Having high quality classroom discussions about strategies help students to
develop their own thinking while providing them the opportunity to critically evaluate their classmates’ approaches.
Make estimation an integral part of computing: Most of the math that we do every day— reading the time, looking for house numbers,
how much paint to buy, how much money you have in your wallet, which line to get in at the grocery store relies not only on mental
math but estimations.
Question students about how they reason numerically: Asking students about their reasoning—both when they make mistakes AND
when they arrive at the correct answer—communicates to them that you value their ideas, that math is about reasoning, and, most
importantly, that math should make sense to them. Exploring reasoning is also extremely important for the teacher as a formative
assessment tool. It helps her understand each student’s strengths and weaknesses, content knowledge, reasoning strategies and
misconceptions.
17. Numeration
Concepts: naming, writing, reading, interpreting and
processing numbers
Skills: using numbers in mathematics. Identifying a numeral
with a written word, understand the value of a number and
correctly using it in the appropriate form
Strategies: teaching place value, actively using numbers in
every-day conversations to bring familiarity to children from
very young ages.
Mental computation
Concepts: emphasises the mental processes used to
achieve the answer
Skills: using previously learnt strategies to work out and
obtain exact or approximate answers mentally
Strategies: dependant on the question the following
strategies can be put into place to assist with problem
solving (as many as needed can be used per question):
Count on
Count up
Count back
Doubles/ halving
Multiples of ten
Remaining factors
Place Value
Concepts:
The value of a digit in the context of a number.
(Ones, tens, hundreds, thousands, ten thousands,
hundred thousands, millions etc.)
Children begin with rote counting, 1,2,3,4... From there
they get to two digit numbers, 11, 12, 13 and to three
digit numbers 100, 101, 102... To a child the 1 in 1, 10
and 100 often mean the same thing. However, in place
value, a 1 is one, a 10 is 1 group of ten, 100, is ten, tens
or 1 group of 100. Therefore, the difficulty is
understanding the place of the specific number and
knowing that the placement changes the value of the
digit.
Skills: being able correctly identify the value of each number
Strategies:
Base 10, 5,2,3
MAB blocks
Expanders
18. Language Model
YEAR 2&3
Student language-
“Count the number of TENS and ONES in
the following pictures”
“Circle the odd numbers and square the
even numbers”
“Write the value of the underlined digit”-
“54=50”
“Write the following sentences in number
form” “2 ONES, 6 TENS”
Concrete counters
MAB Blocks
Work sheets which display pictures of
houses, numbers etc. which can be used
to assist with mathematical questions like
Number Sense.
Materials language-
“Write the following sentences in number
form” “4 ONES, 7 TENS 3 HUNRDEDS =
374”
“ Circle the greater number of the two”
“652 or 1067”
MAB blocks
Pictures of MAB blocks
19. Mathematic Language-
“ Colour in the provided squares required to
represent the following numbers”
“Use a number expanded to investigate the different
values each number holds.”
“Continue this pattern “sixty-
three, sixty-five, sixty-seven, _______”
Number expander
Worksheets
Numeric patterns
Symbolic Language-
“Write the following sum in correct place value form”
“Write down the value of the digits shown below”
222, 963, 1372, 22
“Write each number in expanded notation”5,123,
380, 1,360
20. Teaching Strategy &
Resources
Mental Computation- Year 3
To teach mental computation further, teachers would first assess the student’s basic addition and subtraction ability.
Given that it was of standard, the teacher could start by asking simple addition and subtraction problems and have
the children work them out mentally (5+7, 8+4 etc.), stopping and asking the children how they got to that specific
answer (whether the answer be right or not). Moving onto double addition number children will be required to
implement strategies to assist.
Strategy: Nearest whole number.
Ask questions such as 10-3, 20+13, 40+25
Ask the children how they worked that out… by adding the tens first, then the ones
Given that all children are grasping that concept bring in other double-digit numbers to see how they manage
the sum, if they are finding the nearest whole number.
This would require a lesson on counting on/back for number up to 3 then adding or subtracting the count
on/back off.
These questions could be 32+58, 68+21,45+99
If children are having difficulties or aren’t developing as planned writing down the workings of this strategy
create a visual to work off when mentally computing.
21. Misconceptions
How would you avoid or remediate this misconception?
A misconception is not being able to identify the meaning of place value in
a number.
For example a teacher sets out 24 popsicle sticks and asks a student
what purpose the 2 or 4 holds- meaning 2 collections of 10 sticks (20)
plus an extra 4.
If a child isn’t able to answer this correctly it show that they are not
recognising numbers as place value but rather ‘face value’- when seeing a
‘2’ and the ‘4’ children will assume it equals 2 sticks and 4 sticks
respectively, when it really stands for 20 and 4.
To avoid this misconception much emphasis needs to be placed on
helping children to gain a solid platform understanding of Tens and
Ones. This can be done by completing and further explaining problems
such as above by using manipulates to move into groups of Tens and
Ones.
22. ACARA Links & Scootle Resources
Year 1
Strand: Number and Algebra
Sub-strand: Number and place value ACMNA014
Content Description: Count collections to 100 by partitioning
numbers using place value
Elaborations: understanding partitioning of numbers and the
importance of grouping in tens, understanding two-digit
numbers as comprised of tens and ones/units
Scootle:
http://www.scootle.edu.au/ec/viewing/S6182/index.html - I
found this to be a wonderfully informative demonstration and
explanation of place value
https://content.echalk.co.uk/esa/Maths/units/units.html -
‘Digital whiteboard’ would be a great resource when showing
a class the value that each column holds
Year 3
Strand: Number and Algebra
Sub-strand: Number and place value ACMNA055
Content Description: Recall addition facts for single-digit numbers
and related subtraction facts to develop increasingly efficient
mental strategies for computation
Elaborations:
Recognising that certain single-digit number combinations always
result in the same answer for addition and subtraction, and using
this knowledge for addition and subtraction of larger numbers
Combining knowledge of addition and subtraction facts and
partitioning to aid computation (for example 57 + 19 = 57 + 20 – 1)
Scootle:
http://www.scootle.edu.au/ec/viewing/S6194/index.html - great
strategies and methods of teaching mental computation, visual
representations and information
https://www.scootle.edu.auh- Take away online activity which is
able to be manipulated to assist with educational discussions
about finding whole number or adding to the nearest 10 when
learning to mentally compute.
(Australia Curriculum, Assessment and Reporting Authority, 2016)
23. Resources & Ideas
Laminated place value mats are an
accessible and cheap classroom
resource that students can write on,
place concrete materials on and
develop their skills.
Mathematical Place value dice
rolling game- provides visuals and
concrete resource opportunities for
children’s assessment
http://www.topmarks.co.uk/place-
value/place-value-charts
Digital resource child can be switched
from teacher mode to student mode. It
allows children to practice adding the
values of numbers together.
24. Textbook Notes
CHAPTER 8
EXTENDING NUMBER SENSE: PLACE VALUE
1. Children must have a clear understanding of our number system if
they are going to be mathematically literate. Place value is critical to
this understanding and is one of the cornerstones of our number
system.
2. The number system that we use is called the Hindu-Arabic system. It
originated in India by the Hindus and was transmitted to Europe by the
Arabs, but many different countries and cultures have contributed to it’s
development.
3. There are 4 important characteristics of the Hindu-Arabic system:
Place value….the position of a digit represents it’s value
Base of ten….The system has ten digits and ten is the value that
determines a new collection
Use of zero…..this allows us to represent symbolically the absence of
something.
Additive property….Numbers can be written in expanded notation and
summed with respect to place value.
Once children understand these characteristics, the development of their
number sense will improve.
4. Nature of place value: Place value develops from many various
experience, such as counting and mental computation. 2 key ideas relating to
place value:
Explicit grouping or trading rules are defined and consistently followed
The position of a digit determines the number being represented.
5, Children develop their skills with the aid of various models (i.e. base-
ten blocks, abacus).
Place value mats serve as a visual reminder of quantities. Children are
able to compose and decompose numbers and begin to recognize
equivalent representations.
Understanding place value is essential to counting and facilitates
operating with larger numbers.
25. CHAPTER 10
Solving Mathematical problems
1. Children need to learn a variety of methods of problem solving and learn when to use each.
Students need to recognize that an appropriate method is needed to find a solution, decide on
which strategy to use to solve the problem, be aware that estimation can be used, check the
reasonableness of the result which is an important part of problem solving.
2. Mental computation can deepen children's’ understanding of how numbers work and increase
fluency in mathematics. Mental computation is an important real-life skill that can be used in all
aspects of life. More than 80% of all mathematical computations in daily life involve mental
computation and estimation.
3. Research has shown that the use of calculators can improve a students skill with paper and
pencil, both in basic operations and in problem solving. To use a calculation well, students need
to enter and interpret the information correctly and know about it’s functions.
4. Estimation is a valuable process that produces answers that are close enough to allow for
good decisions without performing exact computations. Before problem solving, students can
use estimation to get a general sense of what to expect. During the problem, estimation is used
to check if the computation is moving in the right direction. After solving the problem, students
can use estimation to reflect on their answer and decide if it makes sense.
(Reys, et al., 2012)
27. BIG ideas covered this week
Something that I didn’t know or hadn’t thought about
was how the 6 early concepts and beginning processes
have such big influence of pre-algebra, and that
patterning was considered to be algebraic. A repeating
pattern has a core element that applies geometric and
number patterns to assist in translating pre-algebra.
Pre-algebra explains the purpose of patterns and
sequencing in math, problem solving to decide
outcomes. This has all become clearer to me, as
previously I had never thought that deeply about the
working of algebra and the repetition and translation or
an equation.
28. Concepts, Skills & Strategies
Concept: how numbers or objects are ordered to make a pattern.
This requires students to use all 6 pre-number concepts to use
number reasoning to evolve and discover the relationship between
geometrical shapes and numbers.
Skills: being able to identify the elements, describe what is being
repeated, repeat and copy, grow and extend, replace missing
elements and translate number and geometrical patterns.
Strategies: by problem solving the function- the relationship
between number and pattern, students will be able to work out
answers. This can be set out in a table to display the information
that you know and provides a space for the answer to which you
have to discover using function.
29. Language Model
Student Language: Pattern, repeating, growing, missing, replacing, changing
Can you repeat this pattern?
Can you copy this pattern?
Continue the following pattern by drawing the shapes in the provided space
Identify the pattern of following numbers- 2,4,6,8,10,12… what’s happening to the numbers?
When looking at this scale, do you think it’s even? (One with 6 stones the other with 2). What do we need to do
to the side with 2, to make it even?
Resources:
Laminated pictures of scales
Scales- concrete material
Concrete shapes, counters, MAB
Pictures of familiar objects
Symbols (#,*,4)
Materials Language: Pattern, repeating, growing, missing, replacing, changing
Identify and fill in the missing pattern of the following numbers. 3,6,9,12,15,18, ___, 24,____,30.
continue the following pattern
Resources:
Laminated pictures of scales
Scales- concrete material
Concrete counters- bears, stones etc.
Mathematics Language: Relationship, function, formula, expressions, equations, variables, equality, inequality
What is the relationship of the following numbers? 1,4,7,10,13,16,19,22?
Jessica is baking a cake. The recipe calls for 9 cups of flour. She already put in
3 cups. How many more cups does she need to add?
There were 9 roses in the vase. Jessica cut some more roses from her flower
Garden. There are now 15 roses in the vase. How many roses did she cut? Create a formula and equation to find the
answer.
Joan bought a soft drink for 3 dollars and 5 candy bars. She spent
A total of 23 dollars. How much did each candy bar cost? Display a formula and answer to show the answer.
Resources:
Work sheets with working space, pictures and diagrams
Concrete counters to assist with workings
Symbolic language:
2 + __ = 5
__ x 4 = 20
2x + 5 =11
2x > 5
2x + 5 ≠ 2x + 3
30. Teaching strategy and resources that assist
children to understand a key
mathematical concept
Variables to represent an unknown.
x + 5 = 9
To get children to understand how to work out unknown
variables, concrete materials and their own individual
laminated working which they can write on and use
manipulative to assist them working out the answers.
I would pose the question to child… what plus 5 make 9? How
many more counters do you need to make 9?
I would show them the counting on strategy… 5, 6,7,8,9 =4
And would show them how to subtract to find the answer. 9-
5=4
Therefore 4+5=9.
Once all children are on similar understandings we would work
on more difficult numbers.
31. Misconception
How would you avoid or remediate this misconception?
Misconception: Students not being
able to identify the relationship or
function of the pattern. Children might
struggle to grasp the concept that as
the number of cows increase by 1 the
amount of fences required increases
by 2. But rather see it as a more
confusing equation, which they need to
solve. In order to remediate this
misconception I would use concrete
materials to model what is happening
in each step, highlighting to them that
it only increases by 2 each time.
32. ACARA Links & Scootle Resources
Foundation Year
Strand: Number and Algebra
Sub strand: Patterns and algebra ACMNA005
Content Description: Sort and classify familiar objects and
explain the basis for these classifications. Copy, continue and
create patterns with objects and drawings
Elaborations:
Observing natural patterns in the world around us
Creating and describing patterns using materials, sounds,
movements or drawings
Scootle
https://www.scootle.edu.au/ec/viewMetadata.action- Site2See
link- has some great online activities (which could be evolved
into concrete classroom materials) about patterning, copying,
ordering etc. which sets a basis for Foundation children
https://www.scootle.edu.au/ec/viewMetadata.action?id=L10h - a
visual and audio digital activity with assits children in
comprehending creation and repetition of patterns by creating a
pattern with sound and listening it back to ensure it follows
pattern.
Year 1
Strand: Number and Algebra
Sub-strand: Patterns and algebra ACMNA018
Content Description: Investigate and describe number
patterns formed by skip counting and patterns with objects
Elaborations:
Using place-value patterns beyond the teens to generalise
the number sequence and predict the next number
Investigating patterns in the number system, such as the
occurrence of a particular digit in the numbers to 100
Scootle:
https://www.scootle.edu.au/ec/view- great resource that
teaches patterning, numbers before and after, jumps of 10
etc.
https://www.scootle.edu.au/ec/viewMetadata.action?idh -
Site2see- Digital activity in Stage 2 and Stage 3 is a great
resource which requires children to think and work out the
numerical pattens in order to unlock a safe.
(Australia Curriculum, Assessment and Reporting Authority, 2016)
33. Resources & Ideas
Having lots of numbered dice as a classroom
resource will assist children in creating
visuals to balance out their equation.
Having scales, which are available for
student interaction, is an effective way to
reiterate how to balance numbers out. Having
two different variables will assist in balancing
the scales to achieve equality. Also the bears
which are photographed in the picture are a
fantastic resource to display patterning,
repetition and attributes.
34. Textbook Notes
CHAPTER 15 Algebraic thinking
Children need a solid foundation in arithmetic before they begin to study algebra. By building on their understanding of the number system, students learn to describe relationships
and formulate generalizations. Students learn to apply their number and algebra skills to conduct investigations, problem solving and communicate their reasoning.
Algebra is a study of patterns and relationships. It is a way of thinking and provides strategies for analyzing representations, modeling situations, generalizing ideas and justifying
statements. Algebra is an art. Children gain a better understanding of the underlying structure and properties of mathematics. Algebra is a language that uses carefully defined terms
and symbols. Algebra is a tool and children need to learn to use it in a practical and meaningful way.
Problems, patterns and relations are each an essential part of primary school mathematics. All of these have connections to algebraic thinking. Routine and no routine problems
provide good opportunities for developing algebraic concepts and thinking. Patterns are an important part of mathematics because they help children organize their world and
understand mathematics. A repeating pattern has the core element that it is repeated over and over. This simple pattern raises many questions about how children work with patterns.
As children become familiar with repeating patterns, they can extend other patterns or make their own.
Two types of relation that can help children build algebraic concepts and thinking: Properties of numbers and functions. Use number properties to promote algebraic thinking
(commutative, associative, distributive and identity). A function is a way of expressing that relation.
When helping children to learn to think algebraically, use algebraic terms and symbols as you would if you were teaching a foreign language. You should weave the terms and
symbols naturally into your discussion. Children often use a mixture of symbols and words when they begin to express mathematical situations..
Young children understand sharing equally and balancing a scale, but they can be baffled by the equal sign. Many children think that the equal sign means “get an answer”. You can
help children grasp the meaning of the equal sign by introducing it in an activity in the classroom. Help children use the inequality symbols (< and >) to describe larger or small
quantities.
There are 3 different used of Variable. The most common in primary schools is a Placeholder i.e. 3 + a = 7. The letter or use of a box is a placeholder for the number that makes
the sentence true. Variables are also used in Generalizations – any number subtracted from itself is equal to zero. Variables are also used in formulas and in functions. Functions
may be treated as patterns. By making tables and discussing the patterns, you can help students find the general rule (the function) that relates the 2 quantities.
(Reys, et al., 2012)
36. BIG ideas covered this week
Measurements take into account length, area, time,
liquid, mass, which is essential in everyday living.
Something which I had never thought about is the fact
that it links a persons knowledge of numbers to spatial
situations like measuring volumes, areas, lengths and
angles along with everyday problem solving situation
which we unconsciously perform.
Every subject throughout a child’s schooling life will
involve an aspect of measurement whether it be
geographically measuring how many kilometres an
island is, or measuring how much paint is required for
an art lesson.
37. Concepts, Skills & Strategies
Concept:
Children must understand the attribute they are measuring, an understanding
of the concept under investigation is essential before decision-making. This is
essential before a child moves any further. Example: the concept of volume. A
child must know that the volume of an object refers to the amount of space,
which is take up by a three-dimensional object. This might refer to how much
water can be held inside a jug. This is measured as mm3, cm3,m3 etc.
Skill:
Using their knowledge about concepts to determine or discover ways to find
the volume of a shape.
Strategy:
This could be by sight- looking for a bigger or smaller volume, weight- heavier
or lighter or mathematically problem solving.
38. Language Model
Children’s language:
Length
The table was very long but not very wide.
Mary has two brothers. One is very tall and the other one is short.
Perimeter
The farmer built a fence to keep his sheep in the paddock.
Area
What shape is an orange? Round or square?
The farmhouse was very large but the barn was small.
Mass
Tip some water into an empty glass and it will soon become full.
The elephant is a very big animal, and a mouse is so little.
Capacity/ Volume
Which of the two cups is full/ empty?
Time:
If today is Monday, what day is it tomorrow?
Mum will be here soon and Dad will arrive later.
Angle:
Circle the corners on the following shapes
Draw a star on all of the right angled corners
Money:
How much does each of the following items cost?
Order the following objects from most to least expensive
Temperature: Warm, hot, cold, freezing
What kind of weather is it today?
Do you think the North Pole is hot or cold?
39. Materials Language:
Length-
can you build a fence that is 6 blocks long
Perimeter-
can you build square fences to keep the cows in?
How many blocks did it take you to make the fence?
Area-
how many blocks have you used to cover the floor
for your cubby house?
Mass-
which is heavier? Rocks or nails?
Capacity/ Volume-
how many spoonful’s of rice will it take to full up that
cup?
Time:
how many handclaps can you do before the ball land
on the ground?
Angle:
If this angle were on a clock, what time would it be?
Describe the angle. Is it a full turn, half turn or a
quarter turn?
Money
If 2 ice-blocks cost $2 each, how much money would
I have to pay?
If I have $1, how many lollies could I buy if they were
10c each?
Temperature:
What kind of clothes do you need when it’s hot or
cold?
What would you wear to the snow?
Do you think this girl is going to a hot or cold place?
Why?
40. Mathematics Language:
Length-
Can you measure a piece of ribbon that is 6cm long
If I want 5 pieces of 12cm ribbon, how much will I
have altogether?
Perimeter- Find the perimeter
Area- Find the area of this shape
Mass- how many kg of bananas did you buy?
Capacity/ Volume- which has more L? a bottle of
milk or a can of lemonade?
Time: Draw the hands on these clocks
Angle:
- How many degrees are a right angle, acute angle,
obtuse angle, straight and reflect angle? Order them
from biggest to smallest.
Money:
I have lots of 10c and 20c coins. Can you make $2
using only those coins?
Show me 3 different ways to make $3, using only
10c, 20c and 50c coins
Temperature:
Do you think that 2° is hot or cold?
What is the temperature today? Look at a
thermometer
If it is 30° today, what type of clothes would you be
wearing?
41. Symbolic Language:
Length-
- Measure the following shape to find the total
length of the highlighted wall.
Perimeter
Find the missing side
Area- find the area of the following circles
Mass- convert the following amounts
Capacity/ Volume- Compare the capacity.
Which has more? And by how much more?
Time: what is the difference between 5:36pm
to 10:55am the next day?
Angle:
work out the missing angle.
Money:
If I have $16 and needed to buy 4 apples
worth $2 each, how much change would I
have left over?
I have $10, but I need $45 to by a dress. How
much more money do I need? $45= $10+
$X
I want to buy a hat, which costs $70. I earn $
17.50 an hour. How many hours would I need
to work before I could buy the hat? $17.50 ×
X=$70
Temperature:
if it was -2° how much hotter would it need to
get to reach 4°? -2° + x= 4
42. Teaching strategy and resources that assist
children to understand a key
mathematical concept
Step 1: Identify the attribute (concept)
Mass
Draw on children’s personal experiences about common objects, using known language such as heavier and lighter. This could be rocks, canned food, and toys. Show them
example of 2 of the same cups with different amounts of sand in them- discuss which one is heavier/lighter, and why?
Step 2: Choose an appropriate unit of measurement for the attribute being measured
Using a scale, children can use arbitrary units to determine weight of multiple blocks being placed on either end.
By counting the blocks, we could say that 6 wooden blocks is lighter than 10 wooden blocks. Or 4 wooden blocks is lighter than a 1 cup of sand.
Moving onto standardized units, children could use electronic scales to determine whether 90grams of sand is more or less heavy than 40grams of sand. This is an introduction to
the language of Mass
Step 3: Measure the object using the chosen unit
As in step 2, applying number concepts by counting the mass or items being weighed, comparing the weights, ordering cups of sand from lightest to heaviest etc.
Step 4: Report the number of units
Doing a classroom experiment, children would investigate some provided objects to weigh to determine their mass. Children would be encouraged to draw pictures of what they see
and write mathematical sentences such as 20grams of play dough is 30grams less than 50grams of play dough.
43. Misconception
How would you avoid or remediate this misconception?
A misconception would be that children confuse the
appropriate units of measurement for each attribute. An
example might be that a child confuses the units for
volume/capacity (ml, L etc.) with Mass (g, kg etc.). a
way to avoid this confusion would be to make sure it
clear is to children that there is a big difference
between to two. Poster and indications should be
placed on classroom walls and the two subjects should
be taught a different times to ensure that children have
a solid grasp on one concept before they move onto
another.
44. ACARA Links & Scootle Resources
Foundation Year
Strand: Measurement and Geometry
Sub-strand: Using units of measurement ACMMG006
Content Description: Use direct and indirect comparisons to decide which is longer,
heavier or holds more, and explain reasoning in everyday language
Elaborations:
Comparing objects directly, by placing one object against another to determine
which is longer or by pouring from one container into the other to see which one
holds more
Using suitable language associated with measurement attributes, such as ‘tall’
and ‘taller’, ‘heavy’ and ‘heavier’, ‘holds more’ and ‘holds less’
Scootle:
https://www.scootle.edu.au/ec/ceh- Teachers Resource which provides a page dedicated
to comparing familiar objects related to weight and time. It has information, videos for
teachers and children and lesson ideas.
https://www.scootle.edu.au/ec/viewMetadata.ah- Site2See- great resouce which has
worksheets and digital activities about measurement, length.
Foundation Year
Strand: Measurement and Geometry
Sub-strand: Using units of measurement ACMMG007
Content Description: Compare and order duration of events using everyday language of time
Elaborations:
Knowing and identifying the days of the week and linking specific days to familiar events
Sequencing familiar events in time order
Foundation Year
Strand: Measurement and Geometry
Sub-strand: Using units of measurement ACMMG008
Content Description: Connect days of the week to familiar events and actions
Elaborations:
Choosing events and actions that make connections with students’ everyday family routines
scootle
https://www.scootle.edu.au/ec/viewMetadata.actionh- digital game resource, which teaches children the different types of clocks,
AM and PM, as well as teaching them how to tell the time.
https://www.scootle.edu.au/ec/viewMetadata.actionh - an interactive video and game, which gives children an idea about routine
and timing. The events, which most children would do during a school day with children’s language, used throughout. The children
are required to watch the video and recall what happens
(Australia Curriculum, Assessment and Reporting Authority, 2016)
45. Resources & Ideas
Time: interactive resource which teachers could display
on an interactive white board to show how time is read
and counted through changing minutes.
http://www.visnos.com/demos/clock
The following link has heaps of great resources which
can be purchased for the classroom to display weight,
volume and length
http://www.hand2mind.com/category/science/measureme
nt/357
46. Textbook Notes
Measurement is the main topic of curriculum
mathematics which is used most frequently in
students everyday life
Measurement is one of the 3 content strands
in the Australian Curriculum
Directly linked to other areas of mathematics
such as multiplication, division, addition and
subtraction as well as art, geography and
SOSE.
Children develop estimation skills, problem
solving
Measurement process involves:
1. Identify the attribute being measured
2. Measure with informal units
3. Measure with formal units
4. Apply the measurement to a real life context
Very young children grasp the concept of
length well as it is said to be the most easily
perceived
Capacity is considered the attribute that
describes 3D shapes
Mass is the amount of a substance- Weight is
the pull of gravity on that substance
Area is the amount of surface within a plane
shape or region within a boundary
To help children understand an attribute it is
suggested the teacher compare 2 objects on
that attribute: perceptually, directly and
indirectly
ACARA stated that the attributes of
measurements is studied in the primary years
are length, area, volume, capacity, mass,
temperature, time and angle
Language is crucial to develop at all stages as
a process of estimation
(Reys, et al., 2012)
48. BIG ideas covered this week
Geometry is an organized, logical and coherent system
for the study of shape, space and measurement.
Geometry involves the study of lines 1D, planes 2D and
solid 3D shapes. It is one of the earliest forms of
mathematics and was used for navigation pyramids,
Roman aqueducts. It is essential to everyday life and is
central to mathematical processing. It is used for
abstraction, generalization, consistency, patterning,
reasoning, and also helps many mathematics ideas to
be “pictured” such as multiplication, area of circles and
other shapes.
49. Concepts, Skills & Strategies
Concepts: understanding that the different shapes all have different names and attributes to go with it.
Skills:
Skill 1: Visualising- understanding the shape, recognizing the shape from a picture or concrete display, being able to describe
the attributes etc.
Skill 2: Communication- children should be able to talk and write about the attributes of a shape or pattern, describing
similarities and differences, using geometric language and describing spatial relationships
Skill 3: Drawing and modeling- drawing the shape to scale and model
Skill 4: Thinking and reasoning- classifying, analysing and seeking similarities and differences, reasoning and synthesising
Skill 5: Applying geometric concepts & knowledge- to real life experiences. Link shapes to objects and function
Strategies:
Effective exposure to the correct language And materials
Response to visuals and touch- being able to get a sensory exposure to shapes helps with defining the attributes
50. Language Model
Children’s language:
How many corners does a door have?
How many sides does a box have?
How many faces on a tin can?
Does a ball have any corners?
Resources: Familiar concrete materials- ball, can, box
Materials language:
How many sides does a square have?
Do squares and rectangles have the same amount of
sides and corners? What’s different about them?
Describe a circle
Resources: Attribute blocks, pictures of shapes
Mathematics language: Vertices, faces, perimeter,
circumference, area, volume, scalene, equilateral, isosceles,
quadrilateral,
Name some quadrilaterals
Work out the perimeter of a rectangle?
Work out the circumference of a sphere
Draw and describe the difference between scalene,
equilateral, isosceles triangles
Resources- pictures, attribute blocks, worksheets
Symbolic Language:
Find the area of a rectangle that is 7cm long and 5 cm
wide
Find A
51. Teaching strategy and resources that assist
children to understand a key
mathematical concept
Using Skill 3 of the 5 basic skills of geometry, I would
provide my students with plasticine or play dough for
them to mould and construct shapes in order for them
to gain their own understanding about the attributes of
each shape. They would them be required to write
down exactly what they think and how they would
describe the shape they have made.
52. Misconception
How would you avoid or remediate this misconception?
A child not having being exposed to extended terms for
shapes early enough and struggles with the
terminology and ability to classify shapes into
categories such as parallelogram, oblong, congruent
and symmetry and struggle to keep up with the
terminology that the teacher is using. If I had a child in
my class who struggled with this, I would make a
conscious effort to use those words when describing
the attributes to them individually.
53. ACARA Links & Scootle Resources
Foundation Year
Strand: Measurement and Geometry
Sub-strand: Shape ACMMG009
Content Description: Sort, describe
and name familiar two-dimensional
shapes and three-dimensional objects
in the environment
Elaborations: Sorting and describing
squares, circles, triangles, rectangles,
spheres and cubes
Scootle:
Video to help children become
aware of the shapes around them.
https://www.scootle.edu.au/ec/viewMetadata.action?id=M020186&q=&topic=&start=0&sort=alignment&contentsource=&contentprovider=&resourcetype=&v=text&showLomCommercialResources=false&field=title&fiel
d=text.all&field=topic&learningarea=%22Mathematics%22&contenttype=all&contenttype=%22Interactive%20resource%22&contenttype=%22Collection%22&contenttype=%22Image%22&contenttype=%22Moving%20i
mage%22&contenttype=%22Sound%22&contenttype=%22Assessment%20resource%22&contenttype=%22Teacher%20guide%22&contenttype=%22Dataset%22&contenttype=%22Text%22&contenttype=%22StillIma
ge%22&contenttype=%22MovingImage%22&contenttype=%22StillImage%22&contenttype=%22MovingImage%22&commResContentType=all&commResContentType=%22App%20(mobile)%22&commResContentTy
pe=%22Audio%22&commResContentType=%22Book%20(electronic)%22&commResContentType=%22Book%20(printed)%22&commResContentType=%22Digital%20item%22&commResContentType=%22Learning
%20object%22&commResContentType=%22Other%22&commResContentType=%22Printed%20item%22&commResContentType=%22Software%22&commResContentType=%22Teacher%20resource%22&commR
esContentType=%22Video%22&userlevel=(0)&kc=any&lom=true&scot=true&follow=true&topiccounts=true&rows=20&suggestedResources=M019797,R10709,M019176,M019177,M009190,M013921,M019368,M0198
01,M019798,S4970,M009195&accContentId=ACMMG009&fromSearch=true
Site2see- Teachers resource to assist
with teaching the basics of geometry.
There are activities provided to help
with 2D and 3D shapes.
https://www.scootle.edu.au/ec/viewMetadata.action?id=M016317&q=&topic=&start=0&sort=alignment&contentsource=&contentprovider=&resourcetype=&v=text&showLomCommercialResources=false&field=title&fiel
d=text.all&field=topic&learningarea=%22Mathematics%22&contenttype=all&contenttype=%22Interactive%20resource%22&contenttype=%22Collection%22&contenttype=%22Image%22&contenttype=%22Moving%20i
mage%22&contenttype=%22Sound%22&contenttype=%22Assessment%20resource%22&contenttype=%22Teacher%20guide%22&contenttype=%22Dataset%22&contenttype=%22Text%22&contenttype=%22StillIma
ge%22&contenttype=%22MovingImage%22&contenttype=%22StillImage%22&contenttype=%22MovingImage%22&commResContentType=all&commResContentType=%22App%20(mobile)%22&commResContentTy
pe=%22Audio%22&commResContentType=%22Book%20(electronic)%22&commResContentType=%22Book%20(printed)%22&commResContentType=%22Digital%20item%22&commResContentType=%22Learning
%20object%22&commResContentType=%22Other%22&commResContentType=%22Printed%20item%22&commResContentType=%22Software%22&commResContentType=%22Teacher%20resource%22&commR
esContentType=%22Video%22&userlevel=(0)&kc=any&lom=true&scot=true&follow=true&topiccounts=true&rows=20&suggestedResources=M009190,M019368,M019177,R10709,S4970,M013921,M019798,M019797,
M009195,M019176,M019801&accContentId=ACMMG009&fromSearch=true
(Australia Curriculum, Assessment and Reporting Authority, 2016)
54. Resources & Ideas
3D geometric shapes that unfold to
the net of the shape. Helps children
to understand that within 3D shapes
there are a seris of 2D shapes that
create them.
http://www.teaching.com.au/product?
KEY_ITEM=LER0921&KEY_ALIAS=
LER0921#
Geometric flip chart-
http://www.teaching.com.au/product?
KEY_ITEM=NX8600&KEY_ALIAS=N
X8600
55. Textbook Notes
Geometry has been neglected in schools
Geometry, along with measurement is one of the 3 essential content strands of the Australian Curriculum
More than preferred students only focus on the ability to recognize the shape and name, rather than their ability to learn
the defining attributes that make the shape what it is.
Need to help children differentiate between a ball- which is hollow and a sphere which sometimes is not
Children need to be able to describe properties of 3D objects to explain how 2 or more objects are alike to different
geometrically
Making 3D shape nets with an effective teaching strategy
Geometry helps to represent the space in which live, describe out location ,movement and relationships between objects
and space
Transformation and symmetry can be used to describe our world and to problem solve
Although geometry is learnt though visual, pictorial and concrete materials, children also need to be able to mentally view
shapes and net.
(Reys, et al., 2012)
56. Bibliography
Bibliography
Australia Curriculum, Assessment and Reporting Authority.
(2016). Mathematics. Retrieved 5 29, 2016, from Australian
Curriculum: www.australiancurriculum.edu.au
Reys, R. E., Lindquist, M. M., Lambdin, D. V., Smith, N. L.,
Rogers, A., Falle, J., et al. (2012). Helping children learn
mathematics. Brisbane, QLD, Australia: John Wiley & Sons
Australia Ltd.
(Reys, et al., 2012) (Australia Curriculum, Assessment and
Reporting Authority, 2016)
