Week 12: Geometry

The Big Idea:

Geometry is known as one of the earliest forms of mathematics. It is an organised, logical and coherent system which involve the study of shapes, space and measurement. It is the study of 1D (lines), 2D ( planes) and 3D ( solids) shapes. Teaching geometry is central to mathematical processing and understanding as well as this it provides us with real life problem solving opportunities.


Concepts, Skills and Strategy:

Concepts- an organised, logical and coherent system for the study of shape, space and measurement.

Skills- the skills needed in geometry are communicating, drawing, naming, the ability to recognise similarities and differences, reading and navigating maps, visualising, modelling, thinking and reasoning.

Strategy- useful strategies for geometry include: sorting and classifying shapes, modelling, creating a net of a shape, drawing and building shapes with play dough, clay or any other material.

Misconceptions:

There is one common misconception when it comes to geometry and that is the understanding that a piece of paper can be mistaken for a 2D shape when in fact it is known to be a 3D shape because of it's depth and thickness. Furthermore, a paper is not a rectangle although it can be described as having a "rectangular shape" it is a " prism".

Resources:

According to the Measurement and Geometry section in ACARA Geometry is found in the Foundation Year and is described as " sorting and describing squares, circles, triangles, rectangles, spheres and cubes" along with this students will need to be able use spatial reasoning to visual 2D and 3D shapes as stated in the elaboration. This falls under the Scootle resource as ACMMG009.

Sourced Teaching Strategy:

( Jamieson-Proctor, R. 2019 Powerpoint presentation )

Textbook Concepts, Skills and Strategy:

In Primary mathematics it is important to understand the attributes and relationships of different geometrical objects. Throughout their early childhood and through to their primary years, students continue to describe and sort different objects. In doing so it will help them develop their thinking stage at Level 1 of Hiele's theory and move onto Level 2 of his theory. Children need to be able to recognise geometric shapes as objects used in the real world, they should also be able to list the names of common shapes including: triangle, square, rectangle, circle. As well as this they should also be familiar with other words that are associated with the shape such as a centre, radius, diarmeter  or circumference. Models play a key role in geometry, face models can easily be made with construction paper and edge models can be made with pipe cleaners or toothpicks that is connected with clay or tape ( Reys, R. 2012).


References:

Australia, E. (2018). Search - Scootle. Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMMG009

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406597&chapterid=190039

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?year=11751&year=11752&strand=Number+and+Algebra&strand=Measurement+and+Geometry&strand=Statistics+and+Probability&capability=ignore&capability=Literacy&capability=Numeracy&capability=Information+and+Communication+Technology+%28ICT%29+Capability&capability=Critical+and+Creative+Thinking&capability=Personal+and+Social+Capability&capability=Ethical+Understanding&capability=Intercultural+Understanding&priority=ignore&priority=Aboriginal+and+Torres+Strait+Islander+Histories+and+Cultures&priority=Asia+and+Australia’s+Engagement+with+Asia&priority=Sustainability&elaborations=true&elaborations=false&scotterms=false&isFirstPageLoad=false

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 517,518,521,537,538). Milton, QLD: John Wiley & Sons.

Week 11: Measurement

The Big Idea:

This weeks focus was on the concept of Measurement, which begins in the Foundation Year. Knowing measurements are essential to everyday life such as knowing money or time. There are clear links to number knowledge to spatial situations such as different measurements including: volume, area, length and angles. Measurement is crucial in problem solving and it also connects mathematics to with different curriculum areas (english, arts, science). There are 4 main steps in the sequence of measurement:

1. Identity the Attribute - introducing and understanding the concept, this involves use of children language and non standard traits to measure ( pens, phones, blocks- counting units). 
2. Appropriate unit of measurement for attribute being measured- firstly, the use of non-standard units for measurement and secondly, using standardise units of measurements ( rules cm/m or protractors) 
3. Measure the object using the chosen unit- apply the number concept, matching the unit of measurement to the object through counting, comparing, ordering or sequencing.  
4. Report the number of units- represent the data with numbers, graphs or pictures. 

Concepts, Skills and Strategy:

Concepts- Knowledge of counting is essential in measurement as they will need to know numbers and how to count before they attempt to measure. Counting unit measurements such as km,m,cm,mm, l as well as any other useful units of measurement.

Skills- A few basic skills needed when learning measurement besides knowledge of number and counting is knowing what is countable and what is not countable. Being able to record the count as well as write the numerals is also important as it is a part of the 4 main steps outlined above. The proper use of equipment such as scales, weights, measuring tape, ruler as well as other measuring devices is also crucial.

Strategy- In measurement is it important to have the knowledge of multiplication and the times tables. Another useful strategy in measurement is the knowledge of all formulas ( eg. perimeter = L xW)

Misconceptions:

A few misconceptions when it comes to measurement for children is that all measurements should be the same if the piece of paper is the same. However, this is not the case as there are different types of paper with different lengths therefore it can not be assumed at all pieces of paper have the same length and width. Making a direct comparison is also a misconception as items of different sizes does not necessarily mean that the item will be heavier, it may be the opposite and weigh lighter even though it is a bigger book.


Resources:

According to ACARA, under the heading Measurement and Geometry for the Foundation Year students need to be able to compare both objects directly against one another using metric units to measure and estimate. As well as this it states that the suitable language used during the Foundation Year are: heavy, heavier, holds more, holds less, tall and taller are just a few examples given by ACARA as to the children's language that is used when describing the measurement of objects. Furthermore, comparing and ordering durations of events is also useful in the Foundation Year students will need to know and identify the days of the week on the calendar using sequencing to familiarise themselves with time order.  


Sourced Teaching Strategy:


( Youtube: https://www.youtube.com/watch?v=q8o7n-A0SC0)


Textbook Concept, Skills and Strategy:

The key ideas in measurement which are outlined in the textbook is that students start to explore the relationship between area and perimeter. Apply the concept and skills learnt in the pattern and relationship to find the result and lastly to create reasoning by explaining the possible solutions. According to the textbook measurement is can also be known as the notion for all concepts- number, pattern, geometry and data. Measurement can be brought into other subjects as it overlaps into many subjects. Measurement is known as a process in which a object is given a number to be associated with. As measurement is a process used in the " real world" it is important for children to measure objects outside of classroom materials, they will also need to develop their estimation skills as well as further develop their number sense ( Reys, R. 2012). Overall, measurement is crucial in their mathematical understanding as it is brought further into the context of the real world and it is for this reason and others that it should be taught in the Foundation Year.

References: 

Introduction to Nonstandard Measurement for Kids: Using Paper Clips to Measure. (2015). Retrieved from https://www.youtube.com/watch?v=q8o7n-A0SC0

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406594&chapterid=190032

Mathematics. (2019). Retrieved from https://australiancurriculum.edu.au/f-10-curriculum/mathematics/?year=11751&year=11752&strand=Number+and+Algebra&strand=Measurement+and+Geometry&capability=ignore&capability=Literacy&capability=Numeracy&capability=Information+and+Communication+Technology+%28ICT%29+Capability&capability=Critical+and+Creative+Thinking&capability=Personal+and+Social+Capability&capability=Ethical+Understanding&capability=Intercultural+Understanding&priority=ignore&priority=Aboriginal+and+Torres+Strait+Islander+Histories+and+Cultures&priority=Asia+and+Australia’s+Engagement+with+Asia&priority=Sustainability&elaborations=true&elaborations=false&scotterms=false&isFirstPageLoad=false

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 558,583,607). Milton, Qld: Wiley.

Week 10: Early Algebra part 2

The Big Idea:

This week further reinforced our understanding in the importance of Algebra. The main concept similar to last week is Algebra is a statement of a relationship between two variables. Algebra begins in the Foundation Year where children are taught to create and explore new ideas and key processes. There are two different types of patterns which relate to Algebra being: Geometrical Pattern and the Number Pattern, both of which begin in the Foundation Year and are carried through to Primary and Secondary school. It is important to note the difference between Formula and Formulae while a Formula is an equation a Formulae however is a statement of relationships between two variables.

Concepts, Skills and Strategies:

Concepts- It is the process of building and describing a pattern as well as this the concept of algebra also includes pattern, sequencing and generalisation which simply stands for a statement of relationship as mentioned previously. Further concepts also include Equivalence and Equation where the sign "=" stands for a point of balance, where one side has to be equal to the other for a state of balance to be met.

Skills- There are a number of skills used to help understand and work through algebraic equations. Skills such as: recognising patterns, growing a pattern, the process of repeating and describing a pattern.

Strategies- The use of materials or other manipulative can help with the process of solving an equation. Materials such as MAB blocks or balancing scales are useful in visualising the equation.

( ACU LEO: http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36561)

Misconceptions:

As algebra is a complex concept to grasp there are many misconceptions that can occur. A common misconception among children is the difference between an Expression and Equation. An expression does not have a statement of equality or equal sign although it is a part of an algebraic equation, an example would be 7n + 3x-1 ( this would be a form of an expression). An equation however, has an equal sign a form of an equation would be 4x +5=2x+13.

Resources:

Stemming from last weeks post ACARA's description of Algebra in the Foundation Year under the subheading "Patterns and Algebra" can be defined as classifying familiar objects and explain the classifications. Furthermore, specific references to numeracy should be provided with understanding of patterns and the relationship within these patterns.

Sourced Teaching Strategy:

(Jamieson-Proctor, R. ( 2019) Powerpoint presentation ) 

Textbook Concepts, Skills and Strategies:

Following last week, the textbook outlines Algebra and Algebraic thinking in Chapter 15 when it states that it is: a study of patterns and relationship, a way of thinking, it is characterised by order and consistency, used to define terms and symbols and it is a tool. Primary students learn to develop algebraic thinking through modelling, generalising and justifying by explaining their reasoning or thinking behind their answers. Algebra is an essential part of the curriculum in the Primary years, as it builds on their understanding of Problems, Patterns and Relations.

References: 

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406589&chapterid=190027

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?strand=Number+and+Algebra&strand=Measurement+and+Geometry&strand=Statistics+and+Probability&capability=ignore&priority=ignore&year=11751&elaborations=true

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 485,487,488). Milton, Qld: Wiley.

Search - Scootle. (2018). Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA005

Week 9: Early Algebra

The Big Idea: 

The concept of algebra is introduced in the foundation years through recognising patterns, shapes and symbols, being able to build and describe the patterns created. At this stage in early algebra children may familiarise themselves with the words: Isolating, getting on its own and strip it down as it is the terms used in the Children's language stage when describing algebra. Algebra is a statement between two variables. An algebraic equation needs to be consistent all the way through. Simply put, algebra is the ability to understand patterns and order.


Concepts, Skills and Strategy:

Concepts- It is multiplicative, using bases 2 and 10. Although it is often Base 2. Algebra is referred to as an abstract number understanding, it consist of repetitive patterns, relationship equality and equilibrium ( balance). In algebra letters take the place of numbers and are represented as such these are often based on a repeating pattern. All patterns in algebra have to be consistent for it to be defined as a pattern.

Skill- The importance of pattern and order in Mathematics is vital in the understanding of algebra. A skill needed is to be able to recognise patterns and words ( letters) relating to the pattern, creating basic sentence problem to find a missing element or to grow a pattern. For example:

3XS+1=N.S

3 X number of sticks + 1= Number of sticks


Strategy- Useful strategies used in Early Algebra can include summing, multiplication as it is key to algebra, knowledge of patterns and sequences, the ability to problem solve independently and understanding symbols.

Misconceptions:

It is easy for misconceptions to occur especially when learning algebra. It is automatically presumed that algebra will make mathematics more complex as it combines both numbers and letters as well as different operations to find the total. As well as this a very early misconception that occurs when beginning to learn algebra is that students' may confuse the letter " X" with the variable " x". Finally,  another misconception combining non-like terms in an equation can further confuse a student for example : 5x + 4= 9x. This may confuse them as mentioned previously at this stage as they are not yet familiar with " x" being used as a variable.


Resources:

In ACARA under the Foundation Year Mathematics section subheading " Patterns and Algebra" the content description discussed how early references to algebra should be taught in the classroom with specific references to numeracy to be able to recognise and use patterns and relationship by observing natural patterns in the world around us and describe these patterns through material, sound, movement and drawings.

Sourced Teaching Strategy:

( Pintrest: https://www.pinterest.com.au/pin/157626055681170752/ )

Textbook Concept, Skills and Strategy: 

The main concepts in algebra which the textbook discusses is that algebra is the study of patterns and relationships, it is a way of thinking, it is characterised by order and internal consistency, it is known as a language that is carefully designed to define terms and symbols and lastly algebra is a tool. The skills that will be essential in the early years when students start to develop algebraic thinking and being to progress are the abilities to understand patterns in numbers, using language and symbols appropriately. Finally being able to work out problems, understand patterns and relations are all critical in early and primary school mathematics. A general strategy that applies to algebra is to number the terms, this can help students' to recognise and generalise a pattern ( Reys, R. 2012)  

References:

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406589&chapterid=190027

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?strand=Number+and+Algebra&strand=Measurement+and+Geometry&strand=Statistics+and+Probability&capability=ignore&priority=ignore&year=11751&elaborations=true

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 485,487,488). Milton, Qld: Wiley.

Search - Scootle. (2018). Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA005

Week 8: Number and Place Value part 2.

The Big Idea: 

Following on from last week, this weeks main focus was on numbers. Specifically, number sense and numeration. Number sense includes the fluidity and flexibility of numbers whereas numeration is the process of reading, writing and building numbers. In the Foundation Year students learn single digit numbers, numbers consisting of 0 to 10. As they progress through to Year One they have more knowledge on double digits this include 10-99. When they reach Year 2 it becomes more complex as students are learning triple digit numbers, numbers consisting of 100 to 199. In Year 6 students mainly operate in Base 10 using the place value houses,

Concepts, Skills and Strategy:

Concept- Place Value can be defined as the mother of all concepts. Where the number is situated informs us about the value of that number. Large number concept need to be taught at an early age as it is the basis to algebra which all students will learn as they proceed through school, if they are unable to comprehend the concept it will be difficult for them to grasp further content. It is important for students to understand that the numbers in the place value system are group differently according to their base system. The base number is the largest digit which is one less than the base number. For example:

Base 10 = 9
Base 5 = 4


Skill- A high level of number knowledge and understanding, knowledge of basic number facts and a general knowledge of operations: addition, subtraction, multiplication and division are the basic skills needed in order to understand place value and number.

Strategy- Mental computation can be used when teaching place value. Knowledge of order of operations and estimation as well as rounding are useful when trying to visual numbers. Other strategies includes: using a calculator with numbers beyond 10, the count on strategy, double up and back and build up and down.

Misconceptions: 

Misconceptions generally occur when double digits are taught. If the student has not understood the concept properly they may begin to build misconceptions when it comes to larger numbers. A common misconception is that the number 10,11, 12 is not known to be a " teen" number when in fact teen numbers start immediately when there is a double digit. Furthermore, although the number eleven and twelve may be written as 11 (one - one)  and 12 (one two ) it is important for the student not to pronounce it as it is written. It is crucial for students to understand and recognise the consistency with number names such as thirteen, fourteen, fifteen all the way to  nineteen. Reading back to front can assist students who might have difficulty reading number to understand and learn better a number such as 18 for example is read back to front the number eight comes before the one (also known as teen). Lastly, a numeral expander can help easily overcome misconceptions when a number such as 304 is written it is said to be three hundred and four, children commonly mistake it and write 3004 as they know how the number 300 and the number 4 so they combine both numbers together assuming they are correct in doing so. Hence why the numeral expander is a useful tool as it spreads out the numbers into place value houses making it easier for children to understand. 

Resources: 

Similar to last week The Australian Curriculum ( ACARA) under the heading Patterns and Algebra students are to " investigate and describe number patterns formed by skip counting and patterns with objects". Moreover, they are to use place value patterns beyond the teen numbers to help generalise the number sequence and predict the upcoming number. In the numeracy strand it is crucial for the students to recognise and use the patterns and relationship. 

Sourced Teaching Strategy: 

(Blogger.com:  http://www.promotingsuccessprintablesblog.com/2014/10/place-value.html)

Textbook, Concepts, Skills and Strategy: 

The C/S/S this week is similar to the C/S/S mentioned in last weeks blog post in that the position of the digit represents its value and that the two main ideas for promoting number sense includes : Explicit grouping or trading rules are defined and consistently followed and The position of the digit determines the number being represented. All prior knowledge of all operations are needed in order to gain a deeper understanding of number and place value.  

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406579&chapterid=190019

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?strand=Number+and+Algebra&strand=Measurement+and+Geometry&strand=Statistics+and+Probability&capability=ignore&priority=ignore&year=11751&elaborations=true

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 219,220,227,237). Milton, Qld: Wiley.

Search - Scootle. (2018). Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA001

Week 7: Number and Place Value

The Big Idea:

Students in Year Six should be able to understand decimal numbers and numbers up to one million. There are two sections in which number knowledge is broken into these include Formal and Informal. Formal is taught in primary school it combines reading, writing and understanding of numbers. Informal is when no mathematics skills are required, most informal math skills are based on everyday activities and requires no mental computation or basic calculations. Mental computation refers to ones understanding of operations the process surrounding the operation, it requires estimation and basic knowledge of number facts. Place value is the value of the number that is represented based on the position. 

Concepts, Skills and Strategy:

Concept-  The concepts in Place Value is that the place holder is always zero. It is multiplicative meaning it is associated with the mathematical operation of multiplication. The lines of symmetry pass through the ones house, the decimal point separates a whole number from a fraction.

Skill- The skills needed to place numbers correctly into Place Value columns are the ability to understand number, know basic number facts and have well rounded knowledge of the concept of operations.

Strategy- Obtain information and process accurately and efficiently. Use the 7 thinking strategies (count back, double/ half, use 10) : addition and subtraction as well as ( double 2x, 4x, 8x, counting x5 and x10, build up x3 and x6 and build down x9): multiplication and division.

Misconceptions:

The common misconception when it comes to Place Value is the inability to understand the value of the number when numbers are regrouped or renamed in an addition or subtraction problem. A misconception regarding place value in addition is that when students are given the number 35 + 26 their answers may equal 511, as they understand that 5+6= 11 and 3+2=5 therefore they add both columns separately and combine both numbers together getting the incorrect total of 511. This can happen when students do not yet understand the concept of Place Value. 

Resources:

The Australian Curriculum ( ACARA) under the heading Number and Algebra subheading Number and Place value it states that they have to " Establish an understanding of the language and processes of counting by naming numbers in sequences, initially to and from 20." Furthermore, it is critical in estimating and calculating whole numbers, recognising and using patterns and relationships,  understanding use of number context. It develops their fluency with forwards and backwards counting and their understanding that numbers are said in a particular order and the pattern in which it is said. 

Sourced Teaching Strategy: 

 

( Blogger.com: http://twocandoit.blogspot.com/2012/05/place-value-and-telling-time.html ) 

Textbook Concept, Skills and Strategy: 

The position of the digit represents its value the understanding of Place Value develops through counting and mental computation. Promoting number sense and resting on 2 main ideas which include: Explicit grouping or trading rules define and consistently followed and The position of a digit determines the number being represented. Prior knowledge of number and basic addition and subtraction, the ability to focus on the leading number is an important to number sense and estimation ( Reys. R, 2012).

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406579&chapterid=190017

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406579&chapterid=190018

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?strand=Number+and+Algebra&strand=Measurement+and+Geometry&strand=Statistics+and+Probability&capability=ignore&priority=ignore&year=11751&elaborations=true

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 219,220, 227). Milton, Qld: Wiley.

Search - Scootle. (2018). Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA001

Week 6: Pre-number

The Big Idea:

Pre-number begins at a very young age. It combines both the symbolic and children's language stage. It requires the students to know how to sort, order and recognise patterns. The more significant attributes students associate with learning numbers is by the shape, size and number. Pre-number refers to the beginning processes of a child's understanding to numbers. The order in which it occurs are as follows: Pre-number, early number, subitising ( remembering patterns), cardinal ( eg. how many people are there on the desk), ordinal ( 1st,2nd,3rd place) and nominal ( Maning something eg. student number).

Concepts, Skills and Strategies:

Concept- Pre-number concepts are referred as the foundation to mathematics and may not include numbers. The concepts include: Matching, Sorting, Comparing and Patterning. Matching may be as simple as recognising mum and dad, children are able to do this from a very early age as they match them based on their attributes therefore being able to distinctly recognise their parents from other parents. The Caldwell pattern was designed to teach children how to visualise numbers and patterns to develop their understanding of early number. This pattern helps children recognise number patterns quickly and effectively introducing the concept of subtising. Creative activities such as role playing (Mrs. Tens and Mrs. Ones) can also be useful in introducing the concept of place value further developing their understanding.

Skills-  The basic skills children will need in order to understand Pre-number will be: Learning to count ( numbers 0-10), sequencing, shape, size as well as matching, sorting, comparing and ordering mentioned above.

Strategy-  Useful strategies when introducing Pre-number are one to one correspondence where the children point to the object as they count, stable order ( getting the number order correct), order irrelevance ( no matter which way the order or pattern is the number of objects are still the same).By far the most common strategy used is by finger counting.

Misconceptions:

Students' might not fully understand that counting is a strategy to determine " how many" and the last number counted determines " how many". They may also avoid counting objects which have already been counted. 

Resources: 

In the Australian Curriculum, ACARA under the heading Foundation Year in subheading Number and Algebra it states that students will learn how to " connect number names, numerals and quantities, including zero, initially up to 10 and then beyond.

Sourced Teaching Strategy: 

  

( Caldwell Pattern, YouTube: https://www.youtube.com/watch?v=Xaxy8ylxuzo)

Textbook Concepts, Skills and Strategy:

Developing early number concepts in children is vital in ensuring that they are learning both ordinal, nominal and cardinal numbers. Number cards would be useful for children when learning about ordinal numbers as it can help students solve a problem and clarifying the notion. Nominal numbers are used when providing a label or classifying something such as postcode or license plate, these numbers provide information but does not use the cardinal or ordinal aspects ( Reys.R, 2012). Cardinal numbers are the natural numbers used to identify the cardinality ( size) of a set.  

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406574&chapterid=190012

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406574&chapterid=190011

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?strand=Number+and+Algebra&strand=Measurement+and+Geometry&strand=Statistics+and+Probability&capability=ignore&priority=ignore&year=11751&elaborations=true

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 181,195,196,207). Milton, Qld: Wiley.

Search - Scootle. (2018). Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA001

Week 5: Division

The Big Idea:

The Big Ideas taken from this week is that both division and multiplication both stem from the concept of equal groups. Multiplication requires two equal groups added together to find the total. Whereas, division is the opposite as it requires separating the total into equal groups. Finally, both multiplication and subtraction are linked to division through the use of quotation stories.

Concepts, Skills and Strategy: 

Concept- To teach children how to distribute a certain number of items or objects equally and fairly among them. In order for children to understand the concept of division they have to know how to share. The separation process consist of two processes being Partition and Quotition. Partition is the idea of sharing a large number to see how many there are in each group. Quotition is the process of repeated subtraction where small quantities are subtracted from a larger amount.

Examples 

Partition - Lucy has 6 lollies to share with his 3 friends. How many lollies can each of Lucy's friends receive?

Answer: 6 divided by 3 equal 2

Quotition - There are 12 goody bags and I want to give 1 bag each to my friends. How many friends can I give goody bags to?

Answer: 12 divided by 1 equals 12


Skills- The skills required in order to do division are Multiplication and Subtraction.


Strategy- The strategy needed in division are knowing the multiples very well, being able to do multiplication. Other strategies that can be used in division include Use doubles, Properties to zeros, Counting strategy, Build up or down strategy and using real-life scenarios can also help when demonstrating division.

Misconceptions: 

The misconceptions when it comes to division are that children don't often use the language of division to describe the process. They do not understand that groups and sets need to be subtracted in order to solve the problem. Another misconception is that when dividing the number 0 does not need to be added, this is false as the number should always be recorded.  The language " Goes into" changes what we want students to know about division, where the total is shared out into equal groups. Unlike the other operations, there are several symbols to represent division including: ÷ / - .


Resources:

The Australian Curriculum ( ACARA), states under Minimum Standards-Numeracy that students in years 3 and onwards should be able to apply the appropriate strategies to solve problems which require multiple steps. Using mental multiplication demonstrating by using the skills of doubling and halving as well as simple multiplication and division.

Sourced Teaching Strategy:

(https://www.pinterest.com/mariehdutoit/long-division/)

Textbook Concepts, Skills and Strategy:

Similar to multiplication, the concept of division is developed through concrete, pictorial and symbolic representations. There are two situations that are considered these include: Measurement (Repeated Subtraction) and Partition (Sharing). Fluency in basic skills such as addition, subtraction and multiplication are needed for years 3, 5 and 7. Some strategies used are concrete materials or counting specifically finger counting ( Reys. R, 2012).

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406570&chapterid=190005

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406570&chapterid=190004

NAP - Minimum standards - numeracy. (2016). Retrieved from https://www.nap.edu.au/naplan/numeracy/minimum-standards

Numeracy Learning Continuum. Retrieved from https://www.australiancurriculum.edu.au/media/1077/general-capabilities-numeracy-learning-continuum.pdf

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 267,276). Milton, Qld: Wiley.

Week 4: Multiplication

The Big Idea: 

Multiplication was the focus this week. Unlike subtraction, there is only one type of Multiplication. However, even though there is only one form of Multiplication there are 4 main strategies for Multiplication that students need to be aware of. Student's will need to be able to demonstrate the algorithm using materials such as place value mats or MAB blocks.

Concepts, Skills and Strategy:

Concept- To understand the Multiplication students need to be familiar with Addition as multiplication is a form of repeated addition. It is important to introduce The Set model when beginning to teach multiplication as it allows the students to gain a visual making it the easiest way for students to understand multiplication. The concept of multiplication is simple, as it is a form of repeated addition all parts are added to equal each other, it is the repeated addition of equal groups or sets. An example would be how :

4+4+4=12

same as

3 x 4 =12

4 Ways to Picture Multiplication

1. The Set: 2 acorns in 3 plates. How many altogether?
2. The Array: Length of a rectangle is 5cm and has a width of 3cm. What is the area? 
3. The Measurement/ Length Model: 4 ribbons each 2cm. How many meters?
4. The Combination Model: 3 coloured shirts, 2 coloured pants. How many outfits can I make? 

Skill- To be able to learn multiplication, a skill needed is to understand the basics of addition. Materials such as counters, MAB blocks and paddle pop sticks can help students learn as the ability to visual problems is a skill that is often required for children to understand. During the Children's Language Stage it may be referred to as plates of, sets of or groups of. However, as they progress and develop high order thinking skills the use of language will also change to a more mathematical way of thinking and use of language. Knowing that they have gained the skills required to move from the Children's Language Stage to the Maths Language stage having the knowledge and ability to know that Multiplied by and By both mean the same thing. As they move onto the Symbolic Language Stage they will also realise that the symbol for multiplication is represented by " x" and that multiplication is the number of groups x how many in the group they will have the skills and knowledge to do more complex multiplications as they progress. 

Strategy- There are 4 strategies for Multiplication 

Counting Strategy- 5 and 10, using hands to count for 5 and both hands for 10. MAB blocks can also be used for 10. Multiplication grid can also be used 

Real World Strategy- Used for 0 and 1. 1 involved group, row, stack, set ( example: 1 flock of 10 sheep). Writing a number sentence and identifying a pattern. 0 used for real world situations ( example: 6 nest no egg. How many eggs did I get?) 

Use Double- Used for numbers 2,4 and 8. Also known as the mental computation strategy knowing the basic double facts for addition. 2 is concept of multiplication as repeated addition. 4 uses the double double strategy and 8 uses the double x 3 strategy ( example: 6x6=12 12x12=24)

Build Up/ Down- Build up only used for 3 and 6 whereas the Build down is used for the number 9. In order to use the Build Down strategy for the number 9 the 10s fact has to be known. 

Misconception: 

There are 3 misconceptions when it comes to Multiplication. These are assuming multiplication will always equal a larger value, they have been taught it is repeated addition so generally they assume the result will be a larger number. Secondly, they have to multiply in the order given which is false for example 5 x13x2, if the student is. to familiar with the 2 digit multiplication they should start with 5 x2 then it will make it easy for them to multiply two of the two digit numbers together. Lastly, the incorrect use of the Order of Operations always assuming multiplication is before division students should perform in the method of which the equation appears in.

Resources:

According to the Australian Curriculum ( ACARA) students in years 3 and upwards should be able to recall multiplication facts of two, three, five and ten and to be able to relate that to division. They should also be able to represent and solve problems that involve multiplication using both mental and written strategies.

Sourced Teaching Strategy:

                    (https://www.homeschoolmath.net/teaching/md/multiplication_algorithm.php)


Textbook Concept, Skills and Strategy: 

The concept of multiplication is developed through concrete, pictorial and symbolic representations. Children need to know the basic of all four operations but mainly addition as well as multiplication facts, using visuals or stories can help children understand what is being asked. The " think addition" strategy proves useful when it comes to multiplication. There are many strategies when it comes to multiplication the Commutativity strategy which is a key strategy in learning multiplication facts, the Skip counting strategy which is effective with the multiples of 2,5 and 10 and the Repeated addition strategy which extends on the Skip counting strategy to factors that are less than 5 ( Reys. R, 2012).

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406568&chapterid=189998

NAP - Minimum standards - numeracy. (2016). Retrieved from https://www.nap.edu.au/naplan/numeracy/minimum-standards

Numeracy Learning Continuum. Retrieved from https://www.australiancurriculum.edu.au/media/1077/general-capabilities-numeracy-learning-continuum.pdf

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 287, 357, 361). Milton, Qld: Wiley.

Week 3: Subtraction

The Big Idea: 

This weeks focus was on Subtraction. Students will need to know the concept of Addition very well in order to understand Subtraction as they are the inverse of each other. It is important not to use the word minus and only refer to take-away and can not be related to other types of Subtraction. There are 3 different types of Subtraction.

Concepts, Skills and Strategy: 

Concept- Understanding the concept of Subtraction the total always needs to be known, it is important to know one part of the total in order to find the missing part. Creating a Subtraction story can also help students with their understanding of the topic, in a subtraction story the total is always on top.

Skill- When trading starting in the ones column this can be done by using materials such as MAB blocks rounding up numbers and counting by moving up the ability to subtract without a calculator.

Strategy- As stated previously a subtraction story demonstrates 3 types of subtraction which students need to be aware of and these are: take-away, difference or comparison and missing addend.

Misconception :

A big misconception is that big numbers are not always at the top in the case of the same number such as: 8-8 or 8-9 which is an impossible calculation. When teaching negative numbers children may get confused an example would be -8 +9=?. Furthermore, children will not be able to describe how they completed the equation or why they did.

Resources:

According to the Australian Curriculum ( ACARA) under NAPLAN under the Calculating section. It states that children in Years 3 and onward should be able to recall basic facts with smaller numbers and use them in both addition and subtraction. Using number facts to up 20, to assist in the students use of adding and subtracting from two digit numbers. Furthermore, students should be able to use both mathematical functions to solve simple problems and link terms to the appropriate operation to be able to recognise problems that only require one operation.("NAP - Minimum standards - numeracy", 2016).


Sourced Teaching Strategy:

                            (Pintrest- https://www.pinterest.com.au/pin/184577284701548376/?lp=true)


Textbook Concepts, Skills and Strategy:

According to the text, subtraction facts that were under 10 were to be taught to children 6 years of age and subtraction facts that were above 10 to be taught to children 7 years old. Borrowing or carrying of numbers were started to be taught to children ages 8 and above. The concept of subtraction are developed through concrete, pictorial and abstract representations, a visual representation using counters, blocks or a number line can be used to demonstrate both addition and subtraction. Using pictures and a number line can help students visualise and demonstrate the idea of subtraction, it is equally as important to note that subtraction problems can be easily thought of as missing addend problems as the same strategy of "think addition" is used. In some instances both operations ( addition and subtraction) can be taught together, as mentioned the " think addition" strategy is vital when learning and understanding subtraction. Children should be encouraged to recognise, think about and use the relationship between both operations in order to find the answer (Reys. R, 2012).

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406565&chapterid=189992

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 286,351). Milton, Qld: Wiley.

Year 3 Mathematics- Work Sample Portfolio. (2011). Retrieved from http://docs.acara.edu.au/curriculum/worksamples/AC_Worksample_Mathematics_3.pdf

Year 3 Work Sample Portfolio. (2014). Retrieved from http://docs.acara.edu.au/curriculum/worksamples/Year_3_Mathematics_Portfolio_Satisfactory.pdf

Week 2: Addition

The Big Idea:

This week the main focus was on Addition  and the importance of informing students that Addition does not necessarily mean that the total is going to be greater than as this is a misconception which will be elaborated on further on in this blog. It is also important to note the use of language at a Materials Language Stage  as the students have yet to gain a clear understanding of the Mathematic Language and Symbols used to express a form of equation. Hence why, avoiding the words " add" and " equals" at a Materials Language Stage is crucial. Simply put, Addition is joining groups to find a total.

Concepts, Skills and Strategy:

The C/S/S when it comes to the process of Addition are as followed:

Concept- The use of objects when explaining the concept of Addition, by joining numbers together. Developing a story-board or games with different themes (eg. counters, mats, MAB blocks) as well as learning through song by using well-known simple songs that children easily identify with to keep them engaged in the topic providing them with the opportunity to discuss the topic.

Skill- Knowledge of the concept and the ability to follow through. A skill is knowing what to do with the concept and how to do it properly. An ability to perform a simple " sum" is considered a skill. Such as 3+8= 11.

Strategy- Method used to complete a task effectively. A strategy which has proven to be ineffective when teaching children is the " finger counting" strategy as it is a distraction for children. The 3 main strategies used for Addition are the : Count-on strategy used for numbers ( 1-3) beyond that this method is proved to be ineffective, Double-up strategy ( used with larger numbers and numbers which are the same) and Use 10 strategy or Build Up strategy ( used for adding numbers that are close to the number 10. An example would be : 5+5= 10 or 10+0=10)

Misconceptions: 

As mentioned briefly, a misconception when it comes to Addition is that the total is always going to be a greater number. However, this is not the case. Children should not be taught that by adding two numbers together the total is going to be greater the reason for doing so is that as they mature and learn more about different uses and different forms of equations they will come in contact with negative numbers and how a negative number + a positive number will equal a negative number and a positive + a positive equals a positive same way as a negative + a negative will equal a positive. That is why installing the idea of Addition will always be equal to a greater number is a false assumption and should not be taught to students that way. 

Resources:

According to the QCAA and Australian Curriculum ( ACARA) under the section titled Number and Algebra students should establish an understanding of language and the process of counting numbers in sequences (Qcaa.qld.edu.au, 2014). Students should be taught how to connect and join numerals and quantities which also includes the number zero to ten. The goal by the end of the Foundation Year in terms of Addition is that the students should be able to analyse and read numbers up to ten and knowing the simple process of addition. ACARA states that in the Foundation level students learning Number and Algebra should be able to:

Understand- Connect names, numerals and quantity by addition
Fluency- Counting numbers in sequence or patterns
Problem Solving- Using materials to come up with a solution for the problem by using familiar counting sequences
Reasoning- Creating patterns and explanation of the process
( Australiancurriculum.edu.au, n.d)

Sourced Teaching Strategy:

( Amsi.org.au, 2011)

Mentioned previously, the use of counters is a very effective method when it comes to teaching Addition is it gives children a visual representation of the process and how it is presented. 

Textbook Concepts, Skills and Strategy: 

This text summarises the Big Idea related to C/S/S in Addition. Before start schooling it is generally assumed that all children will have the basic idea of Pre-number these will help build an understanding of the early number concept and the foundations of basic skills needed later on in their Mathematic journey ( Reys, R. 2012). The ability for a child to know how to count correctly is a skill however, in some cases a child may know how to count and the process involved but are unable to establish the last number is an indication of how many they are. It is encourage for students to organise their objects for a clearer understanding, this will also reduce the amount of errors a child makes such as accidentally double counting. Therefore, they will gradually improve on their skills as they are able to sort out the numbers efficiently and without hesitation. Furthermore, some of the key strategies  mentioned in the textbook that are used in counting are the Count-On strategy which was mentioned previously and the Count-back strategy. Both of theses strategies are useful when it comes to teaching the children Addition and the basic principles of Counting. This strategy is critical when developing Addition. It teaches children to recognise different numbers and how the starting number differs from the next number ( Reys, R. 2012).

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406552&chapterid=189986

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 281,285,343). Milton, Qld: Wiley.

Search - Scootle. (2018). Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA002

Week 1: The Language Model

The Big Idea:

This weeks tutorial mainly focused on two key ideas how Mathematics is a language and should be taught as such and the effective structure of the Language Model for teaching mathematics in the classroom. There are four main stages of the Mathematical language these include:
1. Children's Language Stage
2. Materials Language Stage
3. Mathematics Language Stage
4. Symbolic Language Stage

It is crucial to follow these stages through in order for the students to develop a clear and concise understanding of the concept to the Mathematical language.

Concepts, Skills and Strategy:

No mathematical C/S/S were discussed this week. However, the four stages of the Mathematical Language Stages were presented and discussed in detail.

Stage 1: Children's Language:  Mathematics is explained to students at a level in which they are able to understand and interpret the basic meaning and structure of the language. An example would be the word " addition" which means to add however, a teacher may use a different more simpler term for the children such as the word " join". This will help the students gage the concept of the word. Placed in a sentence this can be " Martha and Susie joined Jessica for lunch".

Stage 2: Materials Language:  Similar to the Children's Language stage, the Materials Language stage is simply a process of replacing words with objects and being able to visualise produce verbal cues. This can be done by replacing what was said previously with a more visual piece of stimuli such as counters or blocks.

Stage 3: Mathematics Language:  It begins to explain and elaborate on the proper use of the mathematic terminology such as using the word " subtract" instead of " take- away".

Stage 4: Symbolic Language:  This final stage is where symbols are introduced to students as a means to further simplifying the language. Instead of writing the word " add" the symbol " +" should be written to explain what is meant.

Misconceptions:

A misconception I have known in mathematics is that it was taught in a different style and structured differently which until recently I have learnt is not true as it is a universal language and is taught the same or similarly around the world as well as this mathematics should be taught the same as learning a language. At a younger age I have always believed myself to be not good at mathematics and one has to be " really smart" to be able to obtain the information and understanding in mathematics. However, I now realise just like with learning a new skill or sport it takes time to learn, understand and perfect those mathematical skills.

Resources:

The tool I have chosen for the assignment is a Blog Website by the name of " BLOGGER". I have chosen this form media because not only have I used it multiple times and am familiar with the settings but, I believe Blogger is an effective and simple tool to use in the classroom as it is very straightforward and will not confuse the children. As well as this, Blogger has very bright and colour themes this would intrigue the students and make them concentrate on what is being presented. This media platform can be extremely useful in the classroom when it comes to presenting ideas for topics or sharing information gathered. The URL to this blog is ( https://clarissatchia.blogspot.com )
Sourced Teaching Strategy:

( Pintrest: https://www.pinterest.com.au/pin/251146116706710818/)

The picture above shows an excellent teaching resources that we can use in the classroom with kindergarten and primary students. The Number Mat is a combination of being able to spell out the number  " S-E-V-E-N" as well as write and count the number using different materials such as the frogs and bears shown above.

Textbook Concept, Skills and Strategy:

There are multiple Big Ideas introduced in the text. However, one in particular that stood out is What determines the Mathematics being taught: Needs of the child. Catering, adapting and changes in thinking help influence a child's ideas and approach to mathematics. Some of the key concepts that were understood and implemented in the 1920's is that of a child's understanding on subtraction, that facts under 10 were focused on children between the age of 6 and facts over 10 were aimed towards children at the age of 7. The more complex form of subtraction borrowing and carrying a number were aimed towards children 8 and above ( Reys, R. 2012). In mathematics a child needs to be able to  understand simple concepts such as making connections to problems and applying a solution or develop new ways to solve the problem. Fluency is also critical in a child's understanding of mathematics as it helps them develop the appropriate skills required to form a procedure with accuracy, efficiency and flexibility. Problem solving is also critical for children to be able to comprehend and develop as it requires them to be able to make their own choices and decisions formulating their own problems as well as model and investigating the issue, which will lead to communicating their solutions effectively. Lastly, students need to be able to provide a logical reasoning for their solutions such as proof of analysing, evaluating and justifying their answers.