Friday, 15 April 2016

WEEK 6 - PLACE VALUE

THE BIG IDEAS
Face value and place value are two very different things. Face value is the value of the symbol represented whilst place value is the value of the place the symbol is in, for example in 34 the face value of the 3 is 3, however the place value is 30. This is a concept that may not be fully understood by all children, leading to situations where students are unable to think abstractly to understand the quantity of a place value as the symbolic method of understanding (face value) is much more commonly use to express large numbers. 

    Helping children learn mathematics (2012) discusses this on page 179 where it states: 
    "Research reports that many children lack an understanding of the relative sizes of numbers greater than 100. This results of many factors - one of which may be the lack of opportunity to model large numbers, which helps children develop a visual awareness of the relative sizes of numbers"
    This means, as a teacher I need to provide my students with many opportunities to picture and develop a conceptual understanding for large numbers to relate the face value and the place value to a physical quantity. This can be done through activities or videos. 

    Video: What is face value and place value? 
    Video: Example of a video that demonstrates the size of large numbers - This video could be used with older students when looking at space

    THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES
    There are 7 concepts that fall under place value. These concepts include:


    1.         Place holder – 0
    2.         The base is the multiplier – 10 in a Base 10 system
    3.         The PV system is symmetrical around the ones place
    4.         The decimal point separates the whole from the fraction part of the number 
      5.
    5.         The number of digits required equals the base number – 10 digits in Base 10, 5 digits in Base 5, 2 digits in Base 2
    6.         The largest digit is 1 less than the base number – 9 in Base 10, 4 in Base 5, 1 in Base 2, M-1 in Base M
    7.         When operating on numbers, trading happens when the base number is reached – 6 + 4 = ? Need to trade 10 ones for 1 ten 
    For this section, only 'Place Holder' will be explored further. 

    Concept 
    For place holder, the concept is that all place values must contain a digit and if it is empty, it is filled with a 0 . This doesn't contain value, however it adds to the value of the other digits in the number. This can be introduced in a video (such schoolhouse rock - my hero, zero below) or through experimenting with a place value mat or similar resource. 

     Number cups is a cheap and easily constructed resource that can assist with this concept. By starting with one cup and focusing on a number (for example five) the students can watch the place value of the 5 change by adding more cups with zeros (5, 50, 500, etc)

     Video: Schoolhouse rock - My hero, zero
    Link: Number Cups
    Video: How to make number cups


    Skill
    The ability to change the value of a number using place holders and read the value of a number that contains place holders. This skill can be practiced using resources such as number cups (as mentioned above), apps (such as Kids Maths Place Value), number expanders and other activities that can be done in class. 


    Strategies 
    The strategies for place value includes the 'Big 7' mental computation strategies and turn around facts explored in weeks 1 to 4. These included:
    Addition and subtraction strategies:
    1.Count on/back for 0,1, 2, 3
    2.Doubles/halving for numbers that are the same (4+4, 6+6, 234+234)
    3.Use 10 for 8 and 9
    Multiplication and Division strategies:
    4.Double (x2); Double, Double (x4); Double, Double, Double (x8)
    5.Counting for x5 and x10
    6.Real world for x0 and x1

    7.Build up for x3 and x6 and build down for x9

    8. Turn around facts for x7

     These strategies are important because mental computation plays a large role in the number sense we use as a part of place value. 

    THE LANGUAGE MODEL
    The language model is used to display the relationship between the visual, verbal and symbolic elements of mathematics and forms 'stages' of learning. 


    Student language - During student language, all language used should be familiar to the child and accompanied with familiar objects. 
    Materials language - At this stage, there is still no introduction of mathematical terms. The language is very similar to student language however the visuals used have become more abstract. 

    Mathematics language - Moving away from 'stories', this stage introduces mathematical terms. 

    Symbolic language - This stage introduces symbols. 

    THE LANGUAGE MODEL FOR PLACE VALUE




    THE MISCONCEPTION

    A child may not understand the use of a place holder as you don't say the zero when saying the number. In this case, the child would write five hundred and two as 52 because the only numbers spoken is the five and the two. A child in this position does not yet understand place value (or the 'houses') and therefore needs more work with a place value mat. 

    THE ACARA LINK
    Place value is first introduced in year one.
    Strand: Number and Algebra
    Substrand: Number and place value
    Code: ACMNA014
    Content descriptions: Count collections to 100 by partitioning numbers using place value
    Elaborations: 
    • Understanding partitioning of numbers and the importance of grouping in tens
    • Understanding two-digit numbers as comprised of tens and ones/units
    Scootle resource ideas:
    1. eChalk: Hundreds, tens and units is a virtual place value mat that can be used when introducing the use of MAB blocks at the materials language stage
    2. Importance of Zero is a quick 30 second clip that has a song that explains zeros importance as a place holder, this song could be taught to the children when learning about place holders. 

    1. The place value hop mat teachers children place value with a new 'hopscotch' like twist. This helps children learn how to say large numbers as well as provides a section for concrete materials to be involved. This mat also assists with the concept of place holders. 
    2. This childrens' language place value activity introduces the concepts of a place value mat without entering materials language.  


    THE TEXTBOOK SUMMARY
    • Place Value is first mention in year one of the Australian Curriculum
    • The number system we use is called the Hindu-Arabic system, it was primarily invented in India by the Hindus and transmitted to Europe by the Arabs, but many countries and cultures contributed to its development
    • The Hindu-Arabic system has 4 important characteristics: the position of the digit represents its value, it is a base ten system, a symbol for zero exists and allows us to represent symbolically the absence of something and numbers can be written in expanded notation and summed with respect to place value. 
    • Practice in skip counting helps decrease bumps in the place value road. 
    • Reading and writing numbers are symbolic activities and should follow much modelling and talking about numbers. This reccommendation is based on research that highlight the dangers of introducing children to symbolic numbers too soon. A sustained development of number sense should accompany reading and writing numbers. This ensures that the symbols the students are writing and reading are meaningful to them (Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al.,2012).
    THE REFERENCES
    ACU,. (2016). Learning Environment Online. Leo.acu.edu.au. Retrieved 3 March 2016, from http://leo.acu.edu.au/course/view.php?id=18458

    Australia, E. (2016). Home - Scootle. Scootle.edu.au. Retrieved 3 April 2016, from https://www.scootle.edu.au/ec/p/home

    Australian government,. (2016). Home - The Australian Curriculum v8.1.Australiancurriculum.edu.au. Retrieved 3 April 2016, from http://www.australiancurriculum.edu.au/

    Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al. (2012). Helping children learn mathematics. Milton, QLD: John Wiley & Sons.

    YouTube. (2016). Youtube.com. Retrieved 3 April 2016, from https://www.youtube.com/

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    Friday, 8 April 2016

    WEEK 5 - PRE-NUMBER AND EARLY NUMBER

    THE BIG IDEAS
    One of the important points this week was the five principles that are involved with counting. These include:


    1. One to one correspondence (one number for each object)
    2. Stable order (the correct order of numbers e.g. one, two, three)
    3. Cardinal principle (the last number counted tell us how many there are altogether)
    4. Abstraction (There are things that can be counted such as the number of blocks in the toy box, and other things that can't like the number of sand grains on the beach)
    5. Order irrelevance (no matter the order the objects are counted the number will always be the same unless more is added or removed) 

    Helping children learn mathematics (2012) discusses the intricate natures of counting on page 150 where it states: 
    "What is counting? It is a surprisingly intricate process by which children call number values by name. A close look at the counting process shows that finding how many objects are present involves two distinct actions. A child must say the number-name series starting with one, and point to a different object as each number name is spoken." 
    These principles are important to note as all principles must be demonstrated by a child before they can be considered as knowing how to count. This means for me as a teacher, interviewing children in my foundation class at the beginning of the year is necessary to fully understand the stages of development the children are currently situated and gain insight into how to further assist this development. 

    This may be an assessment piece in a school setting, however in a prior to school setting, this could be a checklist to be revisited over an extended period of time to demonstrate the child's progress. 

    VIDEO: Counting interview This video is a recording of a child that demonstrates the cardinal principle, stable order and is in the beginning stages of one to one correspondence. This interview does not address abstraction or order irrelevance, however additional questions may be added to include these aspects. 


    THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES
    Focusing on pre-number, there is a group of six concepts and related skills that fall under this group. These concepts include:

    1. Determining attributes
    2. Matching by attributes
    3. Sorting by attributes
    4. Comparing attributes
    5. Ordering attributes
    6. Patterning
    For this section, only 'sorting by attributes' will be dissected and explored further. 


    Concept 
    For sorting by attributes, the concept is that we are taking a large group of objects and splitting them up into smaller groups based on one or more attributes. The concept of sorting can be taught using discussions starting with the questions "Can can we make smaller groups?". Children begin to recognise that objects can be catagorised, this can begin simply by showing the child objects that can be catagorised by strong and obvious differences and slowly move into other groups of objects that require more thinking. 

    VIDEO: Class recognising different attributes they can sort by In this video the children are demonstrating the concept of sorting by explaining what attribute the shapes have been sorted by.  

    Skill
    The ability to move objects into groups based on an attribute. This skill can be practiced using any objects that share one or more attribute. 
    VIDEO: Child sorting rocks by what it reminds him of
    VIDEO: Child sorting keys in various groups

    During these videos, the children are seen grouping various objects differently. This is practicing the skill of sorting. 

    Strategies 
    In sorting, the thinking strategy used is planning how to complete the task. Strategies used may be a sorting chart, separating into groups (as shown in the videos in the 'skill' section) or placing each group in a different cup or container. 











     THE LANGUAGE MODEL
    The language model is used to display the relationship between the visual, verbal and symbolic elements of mathematics and forms 'stages' of learning. 


    Student language - During student language, all language used should be familiar to the child and accompanied with familiar objects. This may vary depending on the 'story'. For example, 'Three bird families were sitting in a tree and they decided to go home to their nests. Can you help them go to the right nest?'. In this situation, an image of a tree, three separate nests and groups of three different birds would be used. 
    Materials language - At this stage, there is still no introduction of mathematical terms. The language is very similar to student language however the visuals used have become more abstract. For example, 'Can we separate these counters into groups? What makes each counter in this group the same?'

    Mathematics language - Moving away from 'stories', this stage introduces mathematical terms. For example, 'How can we sort these objects? What attributes can we sort by?'

    Symbolic language - Pre-number and early number does not use symbolic language or symbols. 





    THE LANGUAGE MODEL FOR SORTING




    THE MISCONCEPTION

    A misconception children may have with sorting is misinterpreting the attribute being sorted. This is particularly prominent when sorting more than one attribute (an example of this is in the link below). This can be remedied by using a table for a more visual representation of what attribute is being sorted.   


    THE ACARA LINK
    Sorting is first introduced in the foundation year. 
    Strand: Number and Algebra
    Substrand: Patterns and algebra
    Code: ACMNA005
    Content descriptions: Sort and classify familiar objects and explain the basis for these classifications. Copy, continue and create patterns with objects and drawings
    Elaborations: 
    • Observing natural patterns in the world around us
    • Creating and describing patterns using materials, sounds, movements or drawings
    Scootle resource ideas:
    1. Kitchen Stacker: Sort and Label is an online game that allows students to sort a large number of different objects by a variety of attributes. 
    2. Clean up time is a video that sorts toys by a number of attributes (colour, type and shape). This could be used as an introduction to a lesson as it ends with the open-ended question 'Can you think of anything we can sort?'
    3.  Letter Detective: Letter Case allows students to sort upper case an lower case letters

    1. 'Sorting Game' by My First App is an app designed for the ipad used to introduce sorting using one attribute during pre-number.
    2. ABCYa Educational Games: Rainbow fuzzbugs counting, sorting and comparing is an online game that introduces sorting as well as names of different attributes (such as largest and smallest)
    3. Sesame Street has a video that can be used to teach sorting by multiple attributes - in this video, two characters are sorting blue squares and stripped circles when they come across a stripped square. This shape doesn't fit in either group so they make a new group for it. 
    4. Sort it out! by Barbara Mariconda and illustrated by Sherry Rogers is a children's story that uses rhyme to explore attributes such as colour, size, texture, shape and material. 


    THE TEXTBOOK SUMMARY
    • Number development is not a finite entity that a students either has or has not, it's development is a life long process.
    • The stages include pre-number and informal number; early number development; number development and counting
    • These stages form the basis of whole-number development and provide the underpinnings for basic facts as well as mental and written computation involving addition, subtraction, multiplication and division of whole numbers
    • Early number content descriptors can be found in the Australian curriculum in the foundation year
    • Young children have vast amounts of early number experience, many experiences of which do not rely on numbers but provide the basis of early number concepts and foundation for later skills. Such experiences are called pre-number experiences.  (Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al., 2012)
    THE REFERENCES
    ACU,. (2016). Learning Environment Online. Leo.acu.edu.au. Retrieved 3 March 2016, from http://leo.acu.edu.au/course/view.php?id=18458

    Australia, E. (2016). Home - Scootle. Scootle.edu.au. Retrieved 3 April 2016, from https://www.scootle.edu.au/ec/p/home

    Australian government,. (2016). Home - The Australian Curriculum v8.1.Australiancurriculum.edu.au. Retrieved 3 April 2016, from http://www.australiancurriculum.edu.au/

    Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al. (2012). Helping children learn mathematics. Milton, QLD: John Wiley & Sons.

    YouTube. (2016). Youtube.com. Retrieved 3 April 2016, from https://www.youtube.com/

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    Friday, 25 March 2016

    WEEK 4 - DIVISION

    THE BIG IDEAS
    The first discussion for week four surrounded a child's prerequisites for division:

    1. An understanding of the concept of division - As discussed in previous weeks, a comprehensive understanding of a concept (in this case division) is crucial to have the ability to preform the task successfully.
    2. An understanding that division is the inverse of multiplication - this is important as the main strategy for division is 'Think multiplication' and therefore the child must understand the relationship between these operations.
    3. An understanding of the symbols - Unlike addition, subtraction and multiplication, division has various symobls. As discussed in the textbook, meaning for symbols is created through exposure (Helping children learn mathematics, 2012 p.196). With this understanding it can be concluded that students must be provided with the time and oppertunity to created meaning for all division symbols.
    4. Have a quick and accurate recall of the multiplication facts - This is important because of the inverse relationship between multiplication and division. 
    5. An understanding of and the ability to write the turn-around facts for multiplication and division - This is important because of the inverse relationship between multiplication and division. 
    The second idea this week is that division questions fall under two catagories:
    1.         Partition is basically sharing a large number into groups to see how many there are in each group. For example, "I have ten lollies and I share them between five friends. How many lollies does each friend get?"
    2.         Quotition is repeated subtraction where a small quantity is repeatedly subtracted from a larger amount in order to find the number of groups needed to divide up the total. For example, "I have ten lollies and give two lollies to each of my friends. How many friends do I have?"
    Resources and strategies for teaching these categories are discussed below. 

    Thirdly, we looked at the properties of zero. This is a concept that must be discussed with 
    students because the previous operations were compatible with zero when division is not. 
    For example, zero cannot be divided into three groups and three cannot be divided into
     zero groups. To explain this to children, it is best to use real world examples such as "I 
    have zero counters and want to share them fairly between my fives friends, how many 
    counters does each friend get?". It is important to note that the answer for this example is 
    not zero, the answer is that it cannot be shared. 

    Video: Multiplication and division - an inverse relationship
    Video: Partition and quotition division
    Link: Why can't I divide by zero?

    THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES
    Helping children learn mathematics (2012) states on page 217 that: 
    "Just as 'think addition' is an important strategy for subtraction, 'think multiplication' is the primary thinking strategy to aid children in understanding and recalling the division facts. Division is the inverse of multiplication; that is, in a division problem you are seeking an unknown factor when the product and some other factor are known. The multiplication table illustrates all the division facts; you simply read it differently." 
    Concept 
    In division, the concept is that we are separating a number into equal parts. The concept of division can be taught using division mats. As there are two types of division, some mats are used specifically for one type while others are multi-functional. Using division mats gives students a visual representation of what is happening when they divide. 





    Skill
    The ability to separate a number into equal parts. This skill can be practiced using resources such as the addition mats discussed above and addition stories. Addition stories are stories that use addition throughout the plot of the book. Teachers can use this resource to allow children to practice the skill of addition in a way that seems new and exciting. 


     Strategies 
    As the main strategy for division is "think multiplication", we use the same strategies as multiplication. These strategies can be practiced using the division mats. 

    Link: Leo - Other examples

    THE LANGUAGE MODEL
    The language model is used to display the relationship between the visual, verbal and symbolic elements of mathematics and forms 'stages' of learning. 

    Student language - During student language, all language used should be familiar to the child and accompanied with familiar objects. This may vary depending on the 'story'. For example, 'There were six birds in two trees, If each tree had the same amount of birds how many birds was in each tree?'. 
    Materials language - At this stage, there is still no introduction of mathematical terms. The language is very similar to student language however the visuals used have become more abstract. For example, 'there are 12 counters and three counters in each group. How many groups are there?'

    Mathematics language - Moving away from 'stories', this stage introduces mathematical terms. For example, 'What does six divided by two equal?'

    Symbolic language - This is the only stage where symbols (including symbolic numbers) are used. An example of a question from this stage would be '6 / 2 =' 





    THE LANGUAGE MODEL FOR ADDITION



    THE MISCONCEPTION

    1. The student sees multiplication and division as discrete and separate operations. His conception of the operations does not include the fact that they are linked as inverse operations. If I came across this situation, I would demonstrate this inverse relationship to the child using family facts. 
    2. The student knows how to divide but does not know when to divide (other than because she was told to do so, or because the computation was written as a division problem). The child may not fully grasp the concept of division, it may be beneficial to go back to the childrens' language stage. 
    Link: Misconceptions and errors in mathematics

    THE ACARA LINK
    Division is first introduced in year two.  
    Strand: Number and Algebra
    Substrand: Number and Place Value
    Code: ACMNA032
    Content descriptions: Recognise and represent division as grouping into equal sets and solve simple problems using these representations
    Elaborations: 
    • dividing the class or a collection of objects into equal-sized groups
    • identifying the difference between dividing a set of objects into three equal groups and dividing the same set of objects into groups of three
    Scootle resource ideas:
    1. Number Line helps students visualize number sequences and illustrate strategies for counting, comparing, adding, subtracting, multiplying, and dividing whole numbers. The number line can be labeled with multiples of any whole number from 1 to 100.
    2. The divider is an online game that helps students link division and multiplication using array models. 


    THE TEXTBOOK SUMMERY
    • Teaching division has traditionally taken a large proportion of time in the primary school curriculum. Now with the increased use of calculators, many educators advocate reducing that attention accorded to it. Nevertheless, children still need an understanding of the division process and division facts. The facts help them to respond quickly to simple division situations and to better understand division and its relationship to multiplication. 
    • Just as 'think addition' is an important strategy for subtraction, 'think multiplication' is the primary thinking strategy to aid children in understanding and recalling the division facts. Division is the inverse of multiplication; that is, in a division problem you are seeking an unknown factor when the product and some other factor are known. The multiplication table illustrates all the division facts; you simply read it differently.
    • Thinking strategies for division are more difficult for children to learn than the strategies for other operations. There is more to remember and regrouping is often necessary. However, 'think multiplication' is an extremely efficient strategy for division which avoids the difficulties that other division strategies involve.  
    THE TEXTBOOK SUMMERY
    ACU,. (2016). Learning Environment Online. Leo.acu.edu.au. Retrieved 3 March 2016, from http://leo.acu.edu.au/course/view.php?id=18458

    Australia, E. (2016). Home - Scootle. Scootle.edu.au. Retrieved 3 April 2016, from https://www.scootle.edu.au/ec/p/home

    Australian government,. (2016). Home - The Australian Curriculum v8.1.Australiancurriculum.edu.au. Retrieved 3 April 2016, from http://www.australiancurriculum.edu.au/

    Reys, Lindquist, Lambdin, Smith, Rogers, & Falle, et al. (2012). Helping children learn mathematics. Milton, QLD: John Wiley & Sons.

    YouTube. (2016). Youtube.com. Retrieved 3 April 2016, from https://www.youtube.com/
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