Week 4 - Division

Concept, Skills and Strategies
Division is separating a number into equal groups, however children need to understand the division process can be undertaken in 2 different ways;  partition and quotition3.

• Partition (sharing) - know the total number of groups, we have to find the total. A partition map can be used to demonstrate this and can make inverse links between multiplication and division by turning it around⁵. 
• Quotition (repeated subtraction)- know the total and the number in each group, we have to find the number of groups⁵.

The division algorithm is part of the skills and can be shown using MAB blocks

There is a strong relationship between division and multiplication because division can be stated in terms of multiplication therefore there are several thinking strategies that are similar to multiplication such as: The array model, counting strategy, use of doubles, build up or down5.

Language Model

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Teaching Strategy - Resource

This is an excellent resource that can be used in the classroom to demonstrate either multiplication or division using partition and quotition4. 

Misconceptions

There are many difficulties students might encounter such as there are multiple division symbols where students may get confused. Dividing anything by 0 as it is an undefined operation, children need to understand they cant do the operation. Remainders is also another difficulty students might not understand the concept and become confused, MAB blocks are a good way to show this because we can physically see the remainders2. 

ACARA

Division is introduced into the Australian curriculum as early as year 2 1. 

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References

1. Australian Curriculum and Assessment Reporting Authority. (2014). Foundation to Year 10 Curriculum: Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#cdcode=ACMNA004&level=F 

2. Harris, Andrew. (2001). Mulitplication & Division. Retrieved from http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit5/A&S.pdf

3. Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn Mathematics (1st Australian ed.). Milton: John Wiley & Sons. 

4. Osmond, K (2015, August 28). EDX1280- Learning Resource for Multiplication, Partition Division and Quotition Division. Retrieved from https://www.youtube.com/watch?v=KAOo6wamuso 

5. Week 4 Lecture

Week 3 - Multiplication

Concept, Skills and Strategies

The concept of multiplication is repeated addition where all the parts that are added are equal to each other. Or it could be said; repeated addition of equal groups or sets. There are 4 ways to picture or model multiplication:

1. The set model - good way to display or model to show them the groups. Eg: I had 3 friends which I gave 6 lollies to each friend how many lollies did i give out altogether?
2. The array model (area model) - Eg: the length of a rectangle is 5cms by 3cms. What is the area of the rectangle? This could be represented using materials such as counters you can use them to turn the equation around to be 3x5 instead. 
3. The measurement of length model - Eg: I bought 4 hair ribbons each 2 meters in length. How many meters did i buy?
4. The combinations (cross products) model - Eg: I have 3 different coloured shirts and  different coloured pairs of trousers. How many different outfits can i make? ⁵

The skills are completed through using multiplication algorithm which can be displayed using materials such as MAB blocks which is shown step by step below for 36x4  6

Thinking strategies for multiplication include3: 

• Commutivity
• Counting, doubles,
• Repeated addition
• Build up (x3 & x6) or Build-down (x9),
• Turn- around 7's.
• Patterns
• Multilying by 1 or 0 

Language model

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This is the language model set out in the textbook which demonstrates the language used and symbolic references. Other materials that can be used to assist with demonstrating multiplication include: counters, MAB blocks, learning mats, blocks and almost any other materials that are in quanities and can be manipulated and moved around3. 

Teaching Strategy - Resource 

Teaching multiplication using the array method - resource using a bus to show how many rows and how many people and then uses blocks to represent the equation4.4

Misconceptions 

The misconceptions are similar in solving all mathematical equations where the child may not have fully understood the concept of the operation or they misinterpret the language or symbol2 . 

ACARA

The Australian curriculum introduces multiplication in year 2 1.

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References

1. Australian Curriculum and Assessment Reporting Authority. (2014). Foundation to Year 10 Curriculum: Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#cdcode=ACMNA004&level=F 

2. Harris, Andrew. (2001). Mulitplication & Division. Retrieved from http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit5/A&S.pdf

3. Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn Mathematics (1st Australian ed.). Milton: John Wiley & Sons. 

4. Robinson, BJ (2011, August 10). Year 2 multiplication arrays. Retrieved from https://www.youtube.com/watch?v=37ejU_8lcaY

5. Week 3 Lecture

6. Week 3 Tutorial

Week 2 - Subtraction

Concept, Skills and Strategies

Subtraction is where you start with a large total amount and then some part was removed from it.
The concept is introduced using 'cover-ups' which models 2 parts and then remove one part from the total and what is the part remaining?4 There are 3 types of subtraction that stem from the concept of subtraction these include:

• Take away - know the total and how many are taken away
• Difference or comparison - comparing amounts 
• Missing addened - one of the addends are missing ⁴

We can use MAB blocks to demonstrate the skills just like addition except start with the larger number and then subtract the smaller number from it. 

The same strategies that are used in addition are used in subtraction but in the reverse; count-back, halves, 10's 3. 

Language Model

We need to teach subtraction using the developmental model so that the children understand the concrete material and make their way up using materials and language to get to the symbolic stage4.

Teaching Strategy - Resource 

 Large children books are an excellent resource to introduce concepts of mathematics. It allows the children to use lifelike situations and allows them to count and understand the concept of subtraction and taking away something to show how many are left.



Misconceptions
There are many misconceptions similar to addition, the children need to understand the language used for subtraction therefore before moving onto symbolic language you need to make sure they understand the concept2.


ACARA
The earliest year level where subtraction is introduced in the curriculum is year 1:

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References 

1   Australian Curriculum and Assessment Reporting Authority. (2014). Foundation to Year 10 Curriculum: Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#cdcode=ACMNA004&level=F 

2   Harris, Andrew. (2000) Addition and Subtraction  Retrieved from http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit5/A&S.pdf

3   Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn Mathematics (1st Australian ed.). Milton: John Wiley & Sons. 

4   Week 2 lecture

Week 1 - Addition

ADDITION!

This week the focus was to allow me to understand how children learn mathematics through the 4 language stages (demonstrated below) allowing me to model and build an understanding of how to teach mathematics.

Concept, Skills and Strategies

The concept of addition is modeled through activities and concrete materials that allow student to join and separate collections and quantities 3. I'll aim to model to the children the joining of groups together, providing them with real life situations to give the students experience at their language stage with the concept of addition.

The skill follows on from what we do with the concept, so in the classroom we could model addition by using a place value map and using MAB blocks and joining the sets of blocks together to get the total.

There are 3 main thinking strategies for basic addition facts including count on for 0-3, the use of doubles and the use of 10 for numbers close to 104.


Language Model

We use this triangle to replicate the language model representing the 4 stages of language. Examples of addition that can be used when teaching in the classroom to check where the students are at in the language stage to get them to the symbolic stage include 5:

1. Student language (three birds put with two birds makes five birds altogether)
2. Materials (three counters with two more counters, makes five counters altogether) transfer to them to the materials of mathematics rather than objects such as teddy bears 
3. Mathematics language (three add two equals five) 
4. Symbolic language (3 + 5 = 8) ⁵ 

Teaching strategy - Resource

An excellent teaching resource to be used to model addition in the classroom is using learning mats. We can create one of these relevant to the topic and it gives the children a real life situation to move materials around to understand addition and can be reversed to also teach subtraction5.








Misconceptions
Some misconceptions that might occur include the children not understanding the concept of the operation or failing to understand the place value of the number. The use of place value mats is an excellent way to understand the place value of the numbers and where they should lie in the operation2.


ACARA
The earliest link to addition is found in foundation year of the curriculum1:

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References

1. Australian Curriculum and Assessment Reporting Authority. (2014). Foundation to Year 10 Curriculum: Mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#cdcode=ACMNA004&level=F 

2. Harris, Andrew. (2000) Addition and Subtraction  Retrieved from http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit5/A&S.pdf 

3. Jamieson-Proctor, R., & Larkin, K. (Week 1). "Mathematics as a Language”: A Theoretical Framework for Scaffolding Students’ Mathematical Understanding. Brisbane, Australia: ACU LEO EDMA202/262.

4. Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012). Helping children learn Mathematics (1st Australian ed.). Milton: John Wiley & Sons. 

5. Week 1 lecture