THE BIG IDEAS
The first discussion for week four surrounded a child's prerequisites for division:
The first discussion for week four surrounded a child's prerequisites for division:
- 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.
- 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.
- 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.
- Have a quick and accurate recall of the multiplication facts - This is important because of the inverse relationship between multiplication and division.
- 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:
- 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?"
- 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: 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:
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.
"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.
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
- 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.
- 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.
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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