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/
Posted by at No comments:

Friday, 18 March 2016

WEEK 3 - MULTIPLICATION

THE BIG IDEAS
The first big idea this week surrounded the concept of multiplication (repeated addition) with an emphasis on the importance of understanding and successfully completing addition before beginning multiplication. This is important because the concept of addition is only extended for multiplication. This means that before you begin to teach students multiplication you must first assess their understanding of addition to ensure they have a solid foundation to build upon. 

 Secondly, unlike addition and subtraction their is only one 'type' of multiplication. However there are four different ways to display or model multiplication. These include:

  1. The set model - This is the most common multiplicative structure where you are dealing with a certain number of groups, all the same size. Both the number and the size of the groups are known but the total is unknown. An example of a question using the set model is: "There are three acorns on each of the two plates. How many altogether?"
  2. The array model - Area and arrays are typical examples of multiplicative structure. The area of any rectangle (in square units) can be found  either by covering the rectangle with unit squares and counting the all individually or by multiplying the width by the length. Similarly, in a rectangular array - an arrangement of discrete, countable objects - the total number of objects can be found by multiplying the number of rows by the number of objects in each row.
  3. Measurement model - The measurement model is used when dealing with measurement. An example of a question using this model is “I bought 4 hair ribbons each 2 metres long. How many metres of ribbon did I buy?” 
  4. Combination model - Combination models involve two factors representing the sizes of two different  sets and the product indicates how many different pairs of things that can be formed. An example of this is "I have 3 different coloured shirts and 2 different coloured pairs of trousers. How many different outfits can I make?" (Helping children learn mathematics, 2012)

Video: Multiplication as repeated addition
Video: Set model
Video: Array model
Video: Combinations model
THE MATHEMATICAL PROPERTIES
Null factor
When a number is multiplied by 0, the product will be 0
e.g. 3x0=0 and 0x3=0

Identity property
Any number multiplies by 1 remains the same
e.g.  3x1=3 and 1x3=3 

Commutative property
Also known as 'Turn arounds', this states that two numbers multiplied together have the same total no matter the order of the addends
e.g  3x4=12 and 4x3=12

Associative property
Used when multiplying three or more numbers together, it states that the problem can be solved out of sequence and still arrive at the correct total
e.g.  3x(4x5) is equal to (3x4)x5

Distributive property
Allows us to break up larger multiplication problems into smaller multiplication problems
e.g. 4x7 is equal to (4x5)+(4x2)


THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES

Concept 
In multiplication, the concept is repeated addition. This concept can be taught using a set up as simple as pens in cups, gems in trays or acorns on plates - using a system such as this allows children to see the relationship between addition and multiplication as 3 x 3 is equal to 3 + 3 + 3. 




   

Skill
The ability to successfully complete repeated addition. This skill can be practiced using resources such as the ones suggested above and multiplication stories. Multiplication stories are stories that use multiplication throughout the plot of the book. Teachers can use this resource to allow children to practice the skill of multiplication in a way that seems new and exciting. 














Strategies 
For multiplication, there are four main strategies:

  1. Use counting (for 5x and 10x) - Using our hands, rods or MAB blocks to add groups of  5 or 10
  2. Think real world (for 0x and 1x) - Showing real world examples of '0 groups of' and '1 group of' to demonstrate
  3. Use doubles (for 2x, 4x and 8x) - similar to the addition strategy of doubles, for 2x the number is doubled, for 4x it is double doubled and 8x is double double doubled. 
  4. Build-up (for 3x and 6x) and build-down (for 9x) - For 3x, we build up from 2x (e.g. 3x3= 2x3+3), for 6x we build up from 5x (e.g. 6x3= 5x3+3) and for 9x we build-down for 10x (e.g. 9x3= 10x3-3). 
A resource to assist with the development of these strategies (particularly use doubles and build-up/build-down) with children is to use folding addition cards. These cards allow the teacher to demonstrate each strategy to the children.
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 three birds in three different trees. How many birds were there altogether?' in this story we may use an addition mat, toy birds, puppets, etc.
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 three counters in each of these three trays. How many counters altogether?'

Mathematics language - Moving away from 'stories', this stage introduces mathematical terms. For example, 'What does three multiplied by three 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 '3 x 3 =' 





THE LANGUAGE MODEL FOR MULTIPLICATION
"The same sequence of experiences used for developing understanding of addition and subtraction - moving from concrete to pictorial to symbolic - should also be followed for multiplication and division. 


 THE MISCONCEPTION
  1. The student may have overspecialized his knowledge of multiplication and restricted it to “fact tests” or one particular problem format. If I was to come across this issue I would take the student back to the childrens' or materials language stage and give examples for all methods of multiplication. To avoid students overspecializing, I will ensure I am providing equal amounts of questions using each format.
  2. The student knows how to multiply but does not know when to multiply (other than because he was told to do so, or because the computation was written as a multiplication problem). This student may not understand the relationship between addition and multiplication and may benefit from spending more time and the childrens' or materials language stage with scaffolding and discussion. 

THE ACARA LINK
Addition is first introduced in year two. 
Strand: Number and Algebra
Substrand: Number and Place Value
Code: ACMNA031
Content descriptions: Recognise and represent multiplication as repeated addition, groups and arrays
Elaborations: 
  • representing array problems with available materials and explaining reasoning
  • visualising a group of objects as a unit and using this to calculate the number of objects in several identical groups
Scootle resource ideas:

  1. The array game is an online, interactive game that introduces children to the array model.
  2. The share and group biscuit factory is an online, interactive game that explores the relationship between multiplication and division at the childrens' language stage.
  3. The making arrays game is an online, interactive game that allows children to create arrays using the set number given.
Link: Australian curriculum - Year two
Link: The array game
Link: Share and group the biscuit factory
Link: Making arrays game

 THE RESOURCES AND IDEAS

  1. Pipe cleaner and beads - used for the set model. Each colour has a certain amount of beads (purple has two, green has three, etc). An example of a question where this resource can be used is finding four groups of three. A child would look for the pipe cleaners with three beads (green), pick up four pipe cleaners and count how many there are altogether. 
  2. Chocolate chip cookies - another resource used for the set model, felt circles or the "cookie" represent the group and felt "chocolate chips" represent the number in each group. So, four groups of three would be displayed as four cookies with three chips on each cookie. 
  3. Dice and a grid - used to introduce the array model, one die represents the rows and the other represents the columns which is then coloured in on the grid. The opposite can the be drawn to demonstrate the commutative nature of multiplication (for example, four rows of three and three rows of four).











THE TEXTBOOK SUMMERY
  • There are four distinct types of multiplicative structure:

  1. The set model - This is the most common multiplicative structure where you are dealing with a certain number of groups, all the same size. Both the number and the size of the groups are known but the total is unknown. An example of a question using the set model is: "There are three acorns on each of the two plates. How many altogether?".
  2. The array model - Area and arrays are typical examples of multiplicative structure. The area of any rectangle (in square units) can be found  either by covering the rectangle with unit squares and counting the all individually or by multiplying the width by the length. Similarly, in a rectangular array - an arrangement of discrete, countable objects - the total number of objects can be found by multiplying the number of rows by the number of objects in each row.
  3. Measurement model - The measurement model is used when dealing with measurement. An example of a question using this model is “I bought 4 hair ribbons each 2 metres long. How many metres of ribbon did I buy?” 
  4. Combination model - Combination models involve two factors representing the sizes of two different  sets and the product indicates how many different pairs of things that can be formed. An example of this is "I have 3 different coloured shirts and 2 different coloured pairs of trousers. How many different outfits can I make?" 
  • Before children tackle written strategies for multiplication, they must first have a firm grasp of place value, expanded notation and the distributive property. 
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/
Posted by at No comments:

Friday, 11 March 2016

WEEK 2 - SUBTRACTION

THE BIG IDEA
The big idea this week was the difference between mathematics and numeracy. Understanding this distinction is important as numeracy is a fundamental component of learning, performance, discourse and critique across all areas of the curriculum - not just mathematics. 

Numeracy involves the disposition to use, in context, a combination of: underpinning mathematical concepts and skills from across the discipline (numerical, spatial, graphical, statistical and algebraic); mathematical thinking and strategies; general thinking skills and grounded appreciation of context. On the other hand mathematics is what is taught, for example number, quantity or space. 

Put simply, when mathematics is taught, students develop numeracy as a result of all the mathematical experiences they are afforded. Therefore, numeracy is more than mathematics, it's mathematics in context and involves all aspects of mathematics not just numbers.

With this understanding, the importance of providing students with the opportunity to practice the basic operations in a large variety of forms seems obvious as the more experiences children are exposed will cause numeracy to be developed further. 

Helping children learn mathematics (2012) discusses this concept on page 196: 

"An understanding of addition, subtraction, multiplication and division - and knowledge of the basic number facts for these operations - provides a foundation for all later work with computation. To be effective in this later work, children must develop broad concepts for these operations. This development is more likely to happen if each operation is presented through multiple representations using various physical models. Such experiences help children recognise that an operation can be used in several different types of situations."


Video: What is numeracy?

Video: Math class needs a makeover by Dan Meyer (Bringing 'real life' into math class)

THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES
Helping children learn mathematics (2012) states on page 255 that: 
"Similarly  to addition, children benefit from experiences with problem-solving situations involving subtraction (solved in any way they can) prior to learning standard written methods."
Concept 
In subtraction, the concept is that we know the total and one part of the total with the goal of finding the quantity of the other part. To demonstrate and reinforce this concept with students, an addition mat can be used. For example, "There were six bees in the main bee hive and two flew away. How many were left?"









Skill
The ability to subtract part A from the total in order to find part B. This skill can be practiced using resources such as the addition mats discussed above and subtraction stories. Subtraction stories are stories that use subtraction throughout the plot of the book. Teachers can use this resource to allow children to practice the skill of subtraction in a way that seems new and exciting. 












Strategies 
For addition, there are three main strategies:
    
     1. Counting back for 0-3 (not 4 or more)      e.g. 9 - 3 -> 9, 8, 7, 6 -> 9 - 3 = 6
     2. Halving                                     e.g. 2-1, 4-2, 6-3, 8-4 etc
     3. Use tens                                               e.g. 11-3 -> 10 - 2 = 8 -> 11+3=8

A resource to assist with the development of these strategies with children is to use folding addition cards. These cards allow the teacher to demonstrate the strategies visually. 










 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 seven birds in the tree, if three birds flew away how many birds are left?' in this story we may use an addition mat, toy birds, puppets, etc.
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 seven counters, if I take away three counters how many are left?'

Mathematics language - Moving away from 'stories', this stage introduces mathematical terms. For example, 'What does seven subtract three 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 '7 - 3 =' 





THE LANGUAGE MODEL FOR SUBTRACTION



THE MISCONCEPTION
  1. The student may assume that subtraction is commutative like addition (for example 5-2 = 2-5). To remediate this situation,  I would use an addition mat to demonstrate the difference as well as re-explain the concept of subtraction as we are removing a quantity from the total. 
  2. The student may have overspecialized during the learning process so that she recognises some subtraction situations as subtraction but fails to classify other situations appropriately. For example, the student understands and can correctly perform take away subtraction, however struggles with comparison and missing addend. In a situation such as this, I would re-teach that addends are 'parts' of the total. The student may also not understand the inverse effect between addition and subtraction yet - this may have to be covered.  

THE ACARA LINK
Subtraction is first introduced in year one.  
Strand: Number and Algebra
Substrand: Number and Place Value
Code: ACMNA015
Content descriptions: Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts
Elaborations: 
  • Developing a range of mental strategies for addition and subtraction problems
Scootle resource ideas:
1. The addition and subtraction lesson plan details various activities to assist with the understanding of concepts and practice of skills and includes a detailed list of resources. 
2. Number trains is an online game that uses the count on and count back strategies to order the trains carriges correctly. 


THE TEXTBOOK SUMMERY
  • For each basic addition fact, there is a related subtraction fact. The relationship between them is readily emphasised and learning the basic facts for both operations proceeds using the term "fact families" (1+3=4, 3+1=4, 4-1=3, 4-3=1). 
  • 'Think addition' is the major thinking strategy for learning and recalling the subtraction facts. Encourage children to recognise, think about and use the relationships between addition and subtraction facts.
  • Once children learn 'counting on' for addition, most find the subtraction equivalent of 'counting back' rather easy. 
  • Halving is a strategy that may need to be taught more explicitly as it rests on the assumption that children know their doubles for addition.
  • Counting on is used in subtraction to find the difference between two numbers.
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/


Posted by at No comments:
Subscribe to: Posts (Atom)