Week Eight: Measurement

Synthesise the big ideas

• There are four main steps that are involved in the teaching sequence for measurement:
  - Identifying the attribute/concept
  - Choose an appropriate unit of measurement for the attribute being measured
  - Measure the object using the chosen unit
  - Report the number of units (Jamieson-Proctor, 2016a) 
• Standardised and non-standard units can be used measurement (Jamieson-Proctor, 2016a) 
• Measurement can help children to learn about other topics in mathematics (Reys et al., 2012)
• Estimation is the mental process of arriving at a measurement without the aid of measuring instruments  (Reys, Lindquist, Lambdin, Smith,, Rogers, Falle, Frid, & Bennett, 2012)
• Formulas for measurements:
  - Area of rectangles and parallelograms: A = b x a
  - Area of a triangle: A = 1/2 (a x b)
  - Area of a trapezium: A = 1/2 [a x (b + B)]
  - Volume: V = l x w x h  (Reys et al., 2012, pp. 417-418) 
• "Measurement is a process by which number is assigned to an attribute of an object or event" (Reys et al., 2012, p. 405)
• There are four steps to the measurement process:
  - Identify the attribute being measured and compare objects and events
  - Measure with informal units
  - Measure with standard units
  - Apply the measurement to real-life concepts (Reys et al., 2012, p. 405)
• Different types of measurement include:
  - Length: the distance between 2 points
  - Perimeter: the distance around a region
  - Circumference: the distance around a circle
  - Capacity: how much a 3D container can hold
  - Mass: the amount of a substance
  - Weight: the pull of gravity on that substance
  - Area: the amount of surface within a plane shape or region within a boundary
  - Volume: how much space a 3D object takes up
  - Time, temperature, speed and density (Reys et al., 2012, pp. 406-411)

How have the big ideas changed your understanding of the topic?

• Although I had a basic understanding of the ideas presented this week I was unsure about how effectively to teach them. I also lacked an understanding, until this week, that non-standardised units were encouraged in teaching measurement before using standardised units 

Demonstrate your understanding of the mathematical concept and related skill and strategies children need to assimilate and be able to use, that are related to the topic of measurement

• Many different concepts this week were covered including; measurement, money, time, volume, area, length, angle, perimeter, weight, mass, capacity, temperature, 
• Length is the distance between 2 points. This concept is one that children learn early on (Reys et al., 2012)
• The skills of length:
  - Identify the attribute being measured and compare objects and events
  - Measure with informal units
  - Measure with standard units
  - Apply the measurement to real-life concepts (Reys et al., 2012, p. 405)
• Teaching strategies for length:
  - Maths picture books
  - Tapes, cubes, rulers, trundle wheel, string, paperclips, pencils, straws (Jamieson-Proctor, 2016a; Reys et al., 2012)

Language model for concept

Figure 1.34: Language model for length
Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238196&chapterid=36568

Describe/demonstrate a specific teaching strategy and appropriate resource/s that could be used to assist children to understand a key mathematical concept related to measurement 

Different teaching strategies that are appropriate to assist children to understand the concept of length include:

• Units of length
• Measurement song
• Standardised - centimetres 
• Non-standardised - paper clips 

Describe/demonstrate a specific misconception children might have in relation to measurement

• A specific misconception that students might have in relation to the concept of length is the that if an object, such as a belt, changes shape, than the length changes. This misconception shows that children do not understand that units remain the same (Reys et al., 2012) 
• A way to remediate this specific misconception is to have children measure the same object in different shapes but explain that it is still the same length 

Provide appropriate URL links to the ACARA year, strand, substrand, content description, elaborations and Scootle resources for the earliest mention of measurement 

• Measurement can be seen in the Foundation year, ACMMG006, Measurement and Geometry strand, Using units of measurement sub-strand (ACARA, 2016)
• The content descriptors for ACMMG006 is "use direct and indirect comparisons to decide which is longer, heavier or holds more, and explain reasoning in everyday language" (ACARA, 2016, p. 1) 
• The elaborations for ACMMG006 are:
  - "comparing objects directly, by placing one object against another to determine which is longer or 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'" (ACARA, 2016, pp. 2-3)
• The following are some Scootle resources that could be used to aid in the teaching of ACMMG006: 

Figure 1.35: Scootle Resource One
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMMG006&userlevel=(0)

Figure 1.36: Scootle Resource Two
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMMG006&userlevel=(0)

Figure 1.37: Scootle Resource Three
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMMG006&userlevel=(0)

Figure 1.38: Scootle Resource Four
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMMG006&userlevel=(0)

Figure 1.39 Scootle Resource Five
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMMG006&userlevel=(0)

Provide appropriate links to resources and ideas you have sourced personally to assist students to develop concepts, skills and/or strategies related to measurement 

Resources for students to understand the concept of length:

• Units of length
• Measurement song

Resources for students to understand the skills associated with length: 

• Measuring using standardised units and non-standardised units if I were to use this video I would pause it before the man started explaining the non-metric system of measuring and explain how we measure, using centimetres, millimetres, metres...

Resources for students to understand the teaching strategies in length:

• These videos are examples of measuring in standardised and non-standardised unit:
  - Standardised - centimetres 
  - Non-standardised - paper clips 

Provide a concise synthesis of the textbook chapter/s related to measurement

Chapter 17 - Measurement 

• Measurement provides many application to everyday life and can be used to help learn about other topics of mathematics (Reys et al., 2012)
• Children must develop estimation skills with measurement (Reys et al., 2012) 
• "Measurement is a process by which number is assigned to an attribute of an object or event" (Reys et al., 2012, p. 405)
• There are four steps to the measurement process:
  - Identify the attribute being measured and compare objects and events
  - Measure with informal units
  - Measure with standard units
  - Apply the measurement to real-life concepts (Reys et al., 2012, p. 405)
• Different types of measurement include:
  - Length: the distance between 2 points
  - Perimeter: the distance around a region
  - Circumference: the distance around a circle
  - Capacity: how much a 3D container can hold
  - Mass: the amount of a substance
  - Weight: the pull of gravity on that substance
  - Area: the amount of surface within a plane shape or region within a boundary
  - Volume: how much space a 3D object takes up
  - Time, temperature, speed and density (Reys et al., 2012, pp. 406-411)
• A unit must always remain constant (Reys et al., 2012)
• Informal units include:
  - digits, inches, hands, hand spans, feet, straws, handprints, envelopes, tiles, spoonfuls, cups, bottles, marbles, table tennis balls, rocks, buttons, fruit, coins (Reys et al., 2012, pp. 413-414) 

Figure 1.??: Standardised units (Reys et al., 2012, p. 415)

• Formulas for measurements:
  - Area of rectangles and parallelograms: A = b x a
  - Area of a triangle: A = 1/2 (a x b)
  - Area of a trapezium: A = 1/2 [a x (b + B)]
  - Volume: V = l x w x h  (Reys et al., 2012, pp. 417-418) 
• To be able to change from one unit to another, children must know the equivalence or relation between 2 units  (Reys et al., 2012)
• Estimation is the mental process of arriving at a measurement without the aid of measuring instruments  (Reys et al., 2012)

References 

Australian Curriculum Assessment and Reporting Authority. (2016). Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1

Education Services Australia. (2016). Scootle: Mathematics. Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA005&userlevel=(0)

Iken Edu. (2011). Maths for kids: Units of length. Retrieved from https://www.youtube.com/watch?v=Ll21j2r8gDc

Jamieson-Proctor, R. (2016a). EDMA202/262 mathematics learning and teaching 1: Week 8 Part 1. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36561

Jamieson-Proctor, R. (2016b). EDMA202/262 mathematics learning and teaching 1: Week 8 Part 2. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36562

Jamieson-Proctor, R. (2016c). EDMA202/262 mathematics learning and teaching 1: Week 8 Part 3. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36563

Matholia Channel. (2013). Measuring length in centimetres. Retrieved from https://www.youtube.com/watch?v=tuBLuIW1U70

Murdock, L. (2014). Math monsters standard and nonstandard measurement. Retrieved from https://www.youtube.com/watch?v=OetmuNyzblQ

Numberock. (2015). Metric system song for kids: Measurement music video. Retrieved from https://www.youtube.com/watch?v=djTNUp4XIRo  

Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn mathematics. Queensland, Australia: John Wiley & Sons Australia, Ltd.

Teaching Without Frills. (2015). Introduction to non-standard measurements for kids: Using paper clips to measure. Retrieved from https://www.youtube.com/watch?v=q8o7n-A0SC0

Week Seven: Early Algebra

Synthesise the big ideas

• Skills that apply to both geometry and number patterns include:
  - Recognising the pattern
  - Describing the pattern
  - Repeating the pattern
  - Growing the pattern
  - Replacing missing elements of the pattern
  - Translating the pattern (Jamieson-Proctor, 2016a)
• "Use a pronemtal to symbolise the step number so that you are not restricted to any one value" (Jamieson-Proctor, 2016a, p. 15) 
• When finding a number pattern in a linear sequence the key is to look for a relationships between the numbers (Jamieson-Proctor, 2016b, p. 3)
• Different number theories include:
  - Odds and evens
  - Prime and composite
  - Multiples and factors
  - Positive and negative
  - Whole and fraction (part/whole)
  - Place value (Jamieson-Proctor, 2016b, p. 9) 
• There are three different concepts related to algebra:
  - Patterns and functions
  - Equivalence and equations
  - Patterns, sequences and generalisations (Jamieson-Proctor, 2016c)

Figure 1.28: Terms and Symbols of Algebra
Jamieson-Proctor, R. (2016c). EDMA202/262 mathematics learning and teaching: Week 7 Part 3. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36563

• "An algebraic equation is a statement of a relationship" (Jamieson-Proctor, 2016c, p. 9)
• A repeating pattern has a core element that has been repeated over and over (Reys et al., 2012) 
• Growing patterns are when each successive term changes by the same amount of the preceding term (Reys et al., 2012)
• Functions are a way of expressing a relation (Reys et al., 2012)

How have the big ideas changed your understanding of the topic?

• Algebra has always been a favourite maths concept of mine, and it is an area that I was good at in high school. However I did not realise, before this week, that algebra began so early in school. I was under the idea that algebra begun in the later primary years and early high school

Demonstrate your understanding of the mathematical concept and related skill and strategies children need to assimilate and be able to use, that are related to the topic of early algebra

• The concept for this week was algebra. Algebra is "...a study of patterns and relationships, [and] a language that uses carefully defined terms and symbols" (Reys et al., 2012, p. 351)
• The skill in regards to the concept of algebra include:
  - Recognising the pattern
  - Describing the pattern
  - Repeating the pattern
  - Growing the pattern
  - Replacing missing elements of the pattern
  - Translating the pattern (Jamieson-Proctor, 2016a)
• The thinking strategies for the concept of algebra include:
  - concrete materials to map out the equation

Language model for concept

Figure 1.29: Language model for algebra Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36561

Describe/demonstrate a specific teaching strategy and appropriate resource/s that could be used to assist children to understand a key mathematical concept related to early algebra

• Different teaching strategies to assist children in understanding a key mathematical concept related to early algebra include the following videos:
  - Algebra video
  - Algebra basics

Describe/demonstrate a specific misconception children might have in relation to early algebra 

• There is a misconception among children, in relation to pre-algebra, that algebra is maths that uses letters instead of numbers (Jamieson-Proctor, 2016c)
• A way to re-mediate this specific misconception is to ensure that children understand that "Letters are only a symptom of algebra just like a runny nose is a symptom of a cold. Primary students in Years F-7 need to create patterns and describe the relationship between the steps in the pattern" (Jamieson-Proctor, 2016c, p. 18)

Provide appropriate URL links to the ACARA year, strand, substrand, content description, elaborations and Scootle resources for the earliest mention of early algebra

• Algebra can first be seen in the Foundation year, ACMNA005, Number and Algebra strand, patterns and Algebra sub-strand (ACARA, 2016)
• The content descriptor for ACMNA005 is "sort and classify familiar objects and explain the basis for these classifications. Copy, continue and create patterns with objects and drawings" (ACARA, 2016, p. 1)
• The elaborations for ACMNA005 are:
  - "observing natural patterns in the world around us;
  - creating and describing patterns using materials, sounds, movements or drawings" (ACARA,
     2016 , pp. 2-3).
• Scootle resources to support the teaching of ACMNA005 include: 

Figure 1.30: Scootle Resource One
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA005&userlevel=(0)

Figure 1.31: Scootle Resource Two
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA005&userlevel=(0)

Figure 1.32: Scootle Resource Three
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA005&userlevel=(0)

Provide appropriate links to resources and ideas you have sourced personally to assist students to develop concepts, skills and/or strategies related to early algebra 

Resources for students to understand the concept of pre-algebra:

• Algebra video
• Algebra basics

Resources for students to understand the skills in algebra: 

• Equation playground
• Pattern song
• Patterns

Resources for students to understand the teaching strategies in algebra: 

• Concrete materials such as those shown in the lecture can help children understand the teaching/thinking strategies behind algebra 

Figure 1.33: Jamieson-Proctor, R. (2016c). EDMA202/262 mathematics learning and teaching: Week 7 Part 3. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36563

Provide a concise synthesis of the textbook chapter/s related to early algebra

Chapter 15 - Algebraic Thinking 

• Algebra is a study of patterns and relationships (Reys et al., 2012)
• The teaching of algebra in primary school should build on ideas that are an essential part of the curriculum (Reys et al., 2012)
• Both routine and non-routine types of problem provide good opportunities for developing algebraic concepts of thinking (Reys et al., 2012) 
• A repeating pattern has a core element that has been repeated over and over (Reys et al., 2012) 
• Growing patterns are when each successive term changes by the same amount of the preceding term (Reys et al., 2012)
• Functions are a way of expressing a relation (Reys et al., 2012)
• There are four different properties of numbers:
  - Commutative
  - Associative
  - Distributive
  - Identity (Reys et al., 2012)
• Children can learn the language and symbols associated with algebra as they are learning about numbers (Reys et al., 2012)
• There are 3 different uses of variables:
  - Place-holder
  - Generalisations
  - Formulas and functions (Reys et al., 2012)
• Modelling helps students to organise their thinking (Reys et al., 2012) 
• Justifying allows students to explain their thinking (Reys et al., 2012)

References 

Australian Curriculum Assessment and Reporting Authority. (2016). Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1

Education Services Australia. (2016). Scootle: Mathematics. Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA005&userlevel=(0)

Funza Academy. (2014). Algebra made easy: Math concepts for kids. Retrieved from https://www.youtube.com/watch?v=OU87O69sTLM

Jacknjellify. (2008). X finds out his value. Retrieved from https://www.youtube.com/watch?v=J2TYyUftI8k

Jamieson-Proctor, R. (2016a). EDMA202/262 mathematics learning and teaching 1: Week 7 Part 1. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36561

Jamieson-Proctor, R. (2016b). EDMA202/262 mathematics learning and teaching 1: Week 7 Part 2. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36562

Jamieson-Proctor, R. (2016c). EDMA202/262 mathematics learning and teaching 1: Week 7 Part 3. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1238194&chapterid=36563

Maths Online. (2013). Kindergarten lesson: Shape patterns. Retrieved from https://www.youtube.com/watch?v=wHzjLsTadVk

Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn mathematics. Queensland, Australia: John Wiley & Sons Australia, Ltd.

Scratch Garden. (2014). The patterns practice song: Scratch garden. Retrieved from https://www.youtube.com/watch?v=MBjjxSx45-Q

Zaaxa.com. (2014). Algebra basics. Retrieved from https://www.youtube.com/watch?v=YVJAdfE-L68

Week Six: Number & Place Value

Synthesise the big ideas

• Number knowledge consists of:
  - Formal ideas related to numeration and place value
  - Informal ideas that we call number sense (Jamieson-Proctor, 2016a) 
• Mental computation is a numeration skill that allows people to calculate mentally (Jamieson-Proctor, 2016a) 
• Mental computation develops from:
  - Number sense
  - Exploring a range of useful mental strategies
  - Knowledge of multiple facts (Jamieson-Proctor, 2016a)
• To compute mentally students must:
  - Decide what operation to perform
  - Select a strategy for carrying out the operation
  - Perform the operation(s)
  - Make sense of the answer (Jamieson-Proctor, 2016a)
• The decimal point separates the whole from the fraction part of the number (Jamieson-Proctor, 2016a)
• There are three types of numbers in the world:
  - Cardinal numbers
  - Ordinal numbers
  - Nominal numbers (Jamieson-Proctor, 2016a) 
• There are seven types of number subsets:
  - Prime and composite numbers
  - Square and cubic numbers
  - Triangular numbers
  - Odd and even numbers
  - Pascal's triangle
  - Exponents, integers and real numbers
  - Directed number (Jamieson-Proctor, 2016a)
• Place value helps to:
  - Form a picture
  - Help to calculate
  - Help to estimate
  - Help to learn new numbers (Dietz, 2016) 
• We use the Hindu-Arabic numeration system. This system has four important characteristics:
  1. Place value
  2. Base of ten
  3. Use of zero
  4. Additive property (Reys, Lindquist, Lambdin, Smith, Rogers, Falle, Frid & Bennett, 2012)
• Two types of materials help young children develop place value:
  1. Ungrouped materials
  2. Pre-grouped materials (Reys et al., 2012)
• Australian curriculum documents recommend the use of calculators and computers in school (Reys et al., 2012)
• Number theory provides an avenue to extend and practise mathematical skills (Reys et al., 2012)

How have the big ideas changed your understanding of the topic?

• Prior to this I had very little knowledge of mental computation because it has never exactly been my strong suit 
• This week also changed my understanding about place value and the use of calculators. Prior to this week I didn't know that the curriculum placed importance on children using calculators in maths. Whenever I was taught in school calculators were frowned upon

Demonstrate your understanding of the mathematical concept and related skill and strategies children need to assimilate and be able to use, that are related to the topic of number and place value

• There were many different concepts covered this week, including number, numeration, number sense and place value. 
• The concept of place value base 10 "...means that any number can be represented using only 10 digits (0-9)" (Reys et al., 2012, p. 169). In place value "the position of a digit represents its value"  (Reys et al., 2012, p. 168)
• The skill of place value base 10 include:
  - trading
  - regrouping
  - renaming (Reys et al., 2012)
• The thinking strategies of place value base 10 is to make place value mats and use concrete materials to understand 

Language model for concept

Figure 1.21: Jamieson-Proctor, R. (2016). Language model for place value. Retrieved from http://leo.acu.edu.au/mod/forum/discuss.php?d=403262

Describe/demonstrate a specific teaching strategy and appropriate resource/s that could be used to assist children to understand a key mathematical concept related to number and place value

• A teaching strategy for place value is to use different concrete materials in teaching place value. Concrete materials can include beans, straws, counters, MAB and many others. 
• Other resources include:
  - Place value concept
  - Place value song

Describe/demonstrate a specific misconception children might have in relation to number and place value 

• A common misconception about number and place value is that children have an understanding of the size of numbers greater than 100, even if they can count that high (Reys et al., 2012)
• A way to re-mediate this misconception is to have children have hands on practice with concrete materials, which display the size of numbers greater than 100
• This can be done by using resources such as beans, counters, MAB blocks. Beans and counters can be glued on to cardboard to show the size of numbers

Provide appropriate URL links to the ACARA year, strand, substrand, content description, elaborations and Scootle resources for the earliest mention of number and place value

• Place value can first be seen in Year 1, ACMNA014, Number and Algebra Strand, Number and Place Value substrand
• The content description for ACMNA014 is "count collections to 100 by partitioning numbers using place value" (ACARA, 2016). 
• The elaborations for ACMNA014 are:
  - "understanding partitioning of numbers and the importance of grouping in tens;
  - understanding two-digit numbers as comprised of tens and ones/units" (ACARA, 2016) 
• Attached are three Scootle resources which aid in the teaching of ACMNA014:

Figure 1.22: Scootle Resource One
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA014&userlevel=(1)

Figure 1.23: Scootle Resource Two
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA014&userlevel=(1)

Figure 1.24: Scootle Resource Three
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA014&userlevel=(1)

Provide appropriate links to resources and ideas you have sourced personally to assist students to develop concepts, skills and/or strategies related to number and place value

Resources for students to understand the concept of place value:

• Place value concept
• Place value song

Resources for students to understand the skills related to the concept of place value:

• Place value song
• Place value song

Resources for students to understand the teaching strategies related to the concept of place value: 

• Number expander - can also be made in paper by students

Figure 1.25: Hands On. (2015). Place value. Retrieved from http://www.handson.co.uk/primary-resources/maths-numeracy/place-value.html

• MAB blocks

Figure 1.26: Alexander, A. (2014). Math: Place value with base ten blocks. Retrieved from http://wes1stgradeparentblog.blogspot.com.au/2014/09/math-place-value-with-base-ten-blocks.html

• Base 10 sticks - can be made by students and can be made by many different materials

Figure 1.27: The Crafty Classroom.com. (2012). Base ten sticks. Retrieved from https://au.pinterest.com/pin/78672324710128926/?from_navigate=true

Provide a concise synthesis of the textbook chapter/s related to number and place value
Chapter 8 - Extending number sense: Place value

• We use the Hindu-Arabic numeration system. This system has four important characteristics:
  1. Place value
  2. Base of ten
  3. Use of zero
  4. Additive property (Reys et al., 2012)
• Development of place value promotes number sense and rests on 2 key ideas:
  1. Explicit grouping or trading rules are defined and consistently followed
  2.  The position of a digit determines the number being represented (Reys et al., 2012)
• Place value (in the Hindu-Arabic number system) means that any number can be represented using only 10 digits, 0-9 (Reys et al., 2012)
• Two types of materials help young children develop place value:
  1. Ungrouped materials
  2. Pre-grouped materials (Reys et al., 2012)
• Place value models may be either proportional or not proportional (Reys et al., 2012)
• Later errors in computation can often be traced back to a lack of understanding of place value 
• In developing place value and establishing number names it is far better to skip beyond the teens and start with the larger numbers (Reys et al., 2012)
• Counting suggests many patterns (Reys et al., 2012)
• The calculator can be successfully used to illustrate regrouping with very large numbers (Reys et al., 2012) 
• Practice in skip counting helps decrease bumps in place value learning (Reys et al., 2012)

Chapter 10 - Solving mathematical problems with mental and written strategies, calculators and estimation 

• Australian curriculum documents recommend the use of calculators and computers in school (Reys et al., 2012)
• Finding the balance between mental, written and calculator (or computer) methods is an essential aspect of the Australian curriculum (Reys et al., 2012)
• Children should be encouraged to always try mental computation before using paper and pencil or a calculator  (Reys et al., 2012)
• Estimation is a valuable process that produces answers that are close enough to allow for good decisions without performing elaborate or exact computations (Reys et al., 2012)
• There are three different points at which to perform the estimation process:
  1. Before solving a problem
  2. During the problem
  3. After solving the problem 
• Using compatible numbers is often helpful in estimation 

Chapter 14 - Extending students with number theory 

• Number theory allows the opportunity to extend and connect mathematical ideas met in other chapters (Reys et al., 2012) 
• Number theory provides an avenue to extend and practise mathematical skills (Reys et al., 2012)
• Classifying numbers as odd or even is one of the first number theory topics that children encounter (Reys et al., 2012)
• Children begin to learn about factors and multiples when learning about multiplication and division (Reys et al., 2012) 
• Prime numbers are numbers which are greater than 1 that has only 2 factors, 1 and itself (Reys et al., 2012) 
• A number with more than 2 factors is known as a composite number
  - Every composite number can be expressed through prime factorisation (Reys et al., 2012) 
• A number is divisible if there is no remainder (Reys et al., 2012) 
• Polygonal numbers include square and triangle numbers (Reys et al., 2012)

References 

Alexander, A. (2014). Math: Place value with base ten blocks. Retrieved from http://wes1stgradeparentblog.blogspot.com.au/2014/09/math-place-value-with-base-ten-blocks.html

Australian Curriculum Assessment and Reporting Authority. (2016). Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#level1

Dietz, J. (2016). EDMA202/262 Tutorial Week 6. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1237883&chapterid=32751

Education Services Australia. (2016). Scootle: Mathematics. Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA014&userlevel=(1)

Hands On. (2015). Place value. Retrieved from http://www.handson.co.uk/primary-resources/maths-numeracy/place-value.html

Jamieson-Proctor, R. (2016a). EDMA202/262 mathematics learning and teaching 1: Week 6 Part 1. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1237883&chapterid=32746

Jamieson-Proctor, R. (2016b). Language model for place value. Retrieved from http://leo.acu.edu.au/mod/forum/discuss.php?d=403262

Math & Learning Videos 4 Kids. (2015). Place value first grade: Tens and ones. Retrieved from https://www.youtube.com/watch?v=1F3AycEDksY

Mr. Peters’ Classroom. (2015). Place value song: Rude by MAGIC! Parody. Retrieved from https://www.youtube.com/watch?v=gsvrhKka1nc

Mr. R’s Songs for Teaching. (2011). Place value math song: Ones, tens, hundreds. Retrieved from https://www.youtube.com/watch?v=5W47G-h7myY

Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn mathematics. Queensland, Australia: John Wiley & Sons Australia, Ltd.

Teacher Tipster. (2010). Teacher tipster: Place value song. Retrieved from https://www.youtube.com/watch?v=ATgnG0M3S3Q

The Crafty Classroom.com. (2012). Base ten sticks. Retrieved from https://au.pinterest.com/pin/78672324710128926/?from_navigate=true

Week Five - Pre/Early Number

Synthesise the big ideas

• Pre-number is the early concept and beginning processes which include:
  - determining attributes
  - matching by attributes
  - sorting by attributes
  - comparing attributes
  - ordering attributes, and;
  - patterning (Jamieson-Proctor, 2016)
• There are four different ways that patterns might be used in developing mathematical ideas:
  - copying a pattern
  - finding the next one
  - extending a pattern
  - making their own patterns (Reys et al., 2012).
•  There are three different ways early number is development:
  - conservation
  - group recognition
  - comparisons and one-to-one correspondence (Reys et al., 2012). 
• There are four different counting principles:
  1. when counting objects each object is assigned only one number name
  2. stable order rule
  3. order irrelevance rule
  4. the number name which was assigned to the last object is how many objects there was (Reys
      et al., 2012)
• There are three different counting strategies:
  - counting on
  - counting back
  - skip counting (Reys et al., 2012) 
• Calculators are a helpful tool in demonstrating counting practice and getting over children's hesitation when reaching certain numbers, such as the next decade or century (Reys et al., 2012) 
• There are five counting principles:
  - one-to-one correspondence
  - stable order
  - cardinal principle
  - abstraction
  - order irrelevance (Jamieson-Proctor, 2016) 
• There are three different types of numbers:
  - cardinal
  - ordinal
  - nominal (Jamieson-Proctor, 2016) 
• Attribute blocks, which were used in the tutorial, are helpful in teaching pre-number 

How have the big ideas changed your understanding of the topic?

• Prior to this week I was unaware that pre-number understanding involves attributes 
• I was also unaware that there was three different types of numbers; cardinal, ordinal and nominal 

Demonstrate your understanding of the mathematical concept and related skill and strategies children need to assimilate and be able to use, that are related to the topic of pre/early number 

• There were many concepts this week including, pre-number, patterning, ordering, sorting, comparing, matching, early number, counting and subitizing. 
• Pre-number is a concept that "...leads eventually to meaningful counting skills and number sense" (Reys et al., 2012, p. 143).  
• The concept of pre-number is done using the following skills:
  - determining attributes
  - matching by attributes
  - sorting by attributes
  - comparing attributes
  - ordering attributes and;
  - patterning 
• The strategies for pre-number are:
  - matching by having all objects on a sorting mat
  - sorting by placing objects in different cups, circles or lines for the different attributes

Language model for pre number

Figure 1.16: Language model for the concept of pre-number

Describe/demonstrate a specific teaching strategy and appropriate resource/s that could be used to assist children to understand a key mathematical concept related to pre/early number 
When teaching children the concept of pre-number, attached are some resources that could be used to assist:

• Concept of pre-number: Taller and Shorter 
• Concept of pre-number: Bigger and Smaller
• Concept of pre-number: Heavy and Light
• Pre-number Concepts

Describe/demonstrate a specific misconception children might have in relation to pre/early number 

• A common misconception, in relation to the concept of pre-number, is that pre-number work is usually done before children do anything with numbers in school (Reys et al., 2012). This is a misconception for children and adults. 
• A way to re-mediate this is to make it known what the topic being taught is and the concept that it relates from 

Provide appropriate URL links to the ACARA year, strand, substrand, content description, elaborations and Scootle resources for the earliest mention of pre/early number 

• Pre-number can first be seen in the Foundation Year, ACMNA289, Number and Algebra strand, Number and Place Value sub-strand. 
• The content description for ACMNA289 is "compare, order and make correspondences between collections, initially to 20, and explain reasoning" (ACARA, 2016). 
• The elaborations for ACMNA289 are:
  - "comparing and ordering items of like and unlike characteristics using the words 'more', 'less',
     'same', 'not the same as' and giving reasons for these answers;
  -  understanding and using terms such as 'first' and 'second' to indicate ordinal position in a
     sequence;
  - using objects which are personally and culturally relevant to students" (ACARA, 2016)
• Attached are images of two Scootle resources which aid the teaching of ACMNA289

Figure 1.17: Scootle Resource One
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA289&userlevel=(0) 

Figure 1.18: Scootle Resource Two
Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA289&userlevel=(0) 

Provide appropriate links to resources and ideas you have sourced personally to assist students to develop concepts, skills and/or strategies related to pre/early number 

Resources for students to understand the concept of pre-number:

• Concept of pre-number: Taller and Shorter 
• Concept of pre-number: Bigger and Smaller
• Concept of pre-number: Heavy and Light
• Pre-number Concepts

Resources for students to understand the skills related to the concept of pre-number: 

• Pre-number skills - this is a game that students could play using attribute blocks to development their pre-number skills 
• Attribute blocks - these are a resource that allow children to classify based on different attributes, such as shape, colour, thickness and size. 

Figure 1.19: Discount School Supplies. (2016). Attribute blocks: Set of 60. Retrieved from http://www6.discountschoolsupply.com/Product/ProductDetail.aspx?product=31378&Category=6353

Resources for students to understand the strategy of pre-number: 

• Sorting mat 

Figure 1.20: Teachers Pay Teachers. (2013). Free printable colour sorting mat. Retrieved from https://au.pinterest.com/edumom01/sorting-by-shape-color-size/

• Cups for sorting different attributes 

Figure 1.21: Osborn, S. (2016). Button sorting cups. Retrieved from http://aboutfamilycrafts.com/button-sorting-cups/

Provide a concise synthesis of the textbook chapter/s related to pre/early number

Chapter Seven:

• There are two pre-number concepts:
  1. classification
  2. patterns (Reys et al., 2012)
• There are three different ways to develop early number:
  1. conservation
  2. group recognition
  3. comparisons and one-to-one correspondence (Reys et al., 2012)
• Patterns facilitate the counting process (Reys et al., 2012)
• There are four different counting principles
  1. when counting objects each object is assigned only one number name
  2. stable order rule
  3. order irrelevance rule
  4. the number name which was assigned to the last object is how many objects there was (Reys
      et al., 2012)
• There are two different counting stages:
  1. Rote counting where you know some number names but not necessarily the proper sequence
  2. Rational counting where the child gives a correct number name when objects are counted in
      succession (Reys et al., 2012)
• There are three different counting strategies:
  1. counting on
  2. counting back
  3. skip counting 
• "Cardinal numbers...answers the question 'how many?'" (Reys et al., 2012, p. 158)
• "Ordinal number...answers the question 'which one?'" (Reys et al., 2012, p. 158)
• "Nominal numbers provide essential information for identification but do not necessarily use the ordinal or cardinal aspects of the number" (Reys et al., 2012, p. 159)

References 

Ashlar Kidz. (2015). Maths::concept of heavy and light – i. Retrieved from https://www.youtube.com/watch?v=nqU0yQzwxOo

Ashlar Kidz. (2015). Pre-number concept of bigger & smaller – ii. Retrieved from https://www.youtube.com/watch?v=5dv30Akzybs

Ashlar Kidz. (2015). Pre-number concept of taller & shorter – ii. Retrieved from https://www.youtube.com/watch?v=NAYNiyNnxc0

Australian Curriculum Assessment and Reporting Authority. (2016). Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#cdcode=ACMNA289&level=F

Cusack, B. (2013). EDX1280 foundation pre number skills. Retrieved from https://www.youtube.com/watch?v=DiVjbuarRsg

Discount School Supplies. (2016). Attribute blocks: Set of 60. Retrieved from http://www6.discountschoolsupply.com/Product/ProductDetail.aspx?product=31378&Category=6353

Education Services Australia. (2016). Scootle: Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1

Jamieson-Proctor, R. (2016). EDMA202/262 mathematics learning and teaching 1: Week 5 Part 1. Retrieved from http://leo.acu.edu.au/mod/book/view.php?id=1180683&chapterid=28372

Osborn, S. (2016). Button sorting cups. Retrieved from http://aboutfamilycrafts.com/button-sorting-cups/

Pioneers Education. (2013). Learn maths – Class 1 – Pre-number concepts. Retrieved from https://www.youtube.com/watch?v=5HlIC90m-Hc

Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn mathematics. Queensland, Australia: John Wiley & Sons Australia, Ltd.

Teachers Pay Teachers. (2013). Free printable colour sorting mat. Retrieved from https://au.pinterest.com/edumom01/sorting-by-shape-color-size/