Week 4: Multiplication

The Big Idea: 

Multiplication was the focus this week. Unlike subtraction, there is only one type of Multiplication. However, even though there is only one form of Multiplication there are 4 main strategies for Multiplication that students need to be aware of. Student's will need to be able to demonstrate the algorithm using materials such as place value mats or MAB blocks.

Concepts, Skills and Strategy:

Concept- To understand the Multiplication students need to be familiar with Addition as multiplication is a form of repeated addition. It is important to introduce The Set model when beginning to teach multiplication as it allows the students to gain a visual making it the easiest way for students to understand multiplication. The concept of multiplication is simple, as it is a form of repeated addition all parts are added to equal each other, it is the repeated addition of equal groups or sets. An example would be how :

4+4+4=12

same as

3 x 4 =12

4 Ways to Picture Multiplication

1. The Set: 2 acorns in 3 plates. How many altogether?
2. The Array: Length of a rectangle is 5cm and has a width of 3cm. What is the area? 
3. The Measurement/ Length Model: 4 ribbons each 2cm. How many meters?
4. The Combination Model: 3 coloured shirts, 2 coloured pants. How many outfits can I make? 

Skill- To be able to learn multiplication, a skill needed is to understand the basics of addition. Materials such as counters, MAB blocks and paddle pop sticks can help students learn as the ability to visual problems is a skill that is often required for children to understand. During the Children's Language Stage it may be referred to as plates of, sets of or groups of. However, as they progress and develop high order thinking skills the use of language will also change to a more mathematical way of thinking and use of language. Knowing that they have gained the skills required to move from the Children's Language Stage to the Maths Language stage having the knowledge and ability to know that Multiplied by and By both mean the same thing. As they move onto the Symbolic Language Stage they will also realise that the symbol for multiplication is represented by " x" and that multiplication is the number of groups x how many in the group they will have the skills and knowledge to do more complex multiplications as they progress. 

Strategy- There are 4 strategies for Multiplication 

Counting Strategy- 5 and 10, using hands to count for 5 and both hands for 10. MAB blocks can also be used for 10. Multiplication grid can also be used 

Real World Strategy- Used for 0 and 1. 1 involved group, row, stack, set ( example: 1 flock of 10 sheep). Writing a number sentence and identifying a pattern. 0 used for real world situations ( example: 6 nest no egg. How many eggs did I get?) 

Use Double- Used for numbers 2,4 and 8. Also known as the mental computation strategy knowing the basic double facts for addition. 2 is concept of multiplication as repeated addition. 4 uses the double double strategy and 8 uses the double x 3 strategy ( example: 6x6=12 12x12=24)

Build Up/ Down- Build up only used for 3 and 6 whereas the Build down is used for the number 9. In order to use the Build Down strategy for the number 9 the 10s fact has to be known. 

Misconception: 

There are 3 misconceptions when it comes to Multiplication. These are assuming multiplication will always equal a larger value, they have been taught it is repeated addition so generally they assume the result will be a larger number. Secondly, they have to multiply in the order given which is false for example 5 x13x2, if the student is. to familiar with the 2 digit multiplication they should start with 5 x2 then it will make it easy for them to multiply two of the two digit numbers together. Lastly, the incorrect use of the Order of Operations always assuming multiplication is before division students should perform in the method of which the equation appears in.

Resources:

According to the Australian Curriculum ( ACARA) students in years 3 and upwards should be able to recall multiplication facts of two, three, five and ten and to be able to relate that to division. They should also be able to represent and solve problems that involve multiplication using both mental and written strategies.

Sourced Teaching Strategy:

                    (https://www.homeschoolmath.net/teaching/md/multiplication_algorithm.php)


Textbook Concept, Skills and Strategy: 

The concept of multiplication is developed through concrete, pictorial and symbolic representations. Children need to know the basic of all four operations but mainly addition as well as multiplication facts, using visuals or stories can help children understand what is being asked. The " think addition" strategy proves useful when it comes to multiplication. There are many strategies when it comes to multiplication the Commutativity strategy which is a key strategy in learning multiplication facts, the Skip counting strategy which is effective with the multiples of 2,5 and 10 and the Repeated addition strategy which extends on the Skip counting strategy to factors that are less than 5 ( Reys. R, 2012).

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406568&chapterid=189998

NAP - Minimum standards - numeracy. (2016). Retrieved from https://www.nap.edu.au/naplan/numeracy/minimum-standards

Numeracy Learning Continuum. Retrieved from https://www.australiancurriculum.edu.au/media/1077/general-capabilities-numeracy-learning-continuum.pdf

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 287, 357, 361). Milton, Qld: Wiley.

Week 3: Subtraction

The Big Idea: 

This weeks focus was on Subtraction. Students will need to know the concept of Addition very well in order to understand Subtraction as they are the inverse of each other. It is important not to use the word minus and only refer to take-away and can not be related to other types of Subtraction. There are 3 different types of Subtraction.

Concepts, Skills and Strategy: 

Concept- Understanding the concept of Subtraction the total always needs to be known, it is important to know one part of the total in order to find the missing part. Creating a Subtraction story can also help students with their understanding of the topic, in a subtraction story the total is always on top.

Skill- When trading starting in the ones column this can be done by using materials such as MAB blocks rounding up numbers and counting by moving up the ability to subtract without a calculator.

Strategy- As stated previously a subtraction story demonstrates 3 types of subtraction which students need to be aware of and these are: take-away, difference or comparison and missing addend.

Misconception :

A big misconception is that big numbers are not always at the top in the case of the same number such as: 8-8 or 8-9 which is an impossible calculation. When teaching negative numbers children may get confused an example would be -8 +9=?. Furthermore, children will not be able to describe how they completed the equation or why they did.

Resources:

According to the Australian Curriculum ( ACARA) under NAPLAN under the Calculating section. It states that children in Years 3 and onward should be able to recall basic facts with smaller numbers and use them in both addition and subtraction. Using number facts to up 20, to assist in the students use of adding and subtracting from two digit numbers. Furthermore, students should be able to use both mathematical functions to solve simple problems and link terms to the appropriate operation to be able to recognise problems that only require one operation.("NAP - Minimum standards - numeracy", 2016).


Sourced Teaching Strategy:

                            (Pintrest- https://www.pinterest.com.au/pin/184577284701548376/?lp=true)


Textbook Concepts, Skills and Strategy:

According to the text, subtraction facts that were under 10 were to be taught to children 6 years of age and subtraction facts that were above 10 to be taught to children 7 years old. Borrowing or carrying of numbers were started to be taught to children ages 8 and above. The concept of subtraction are developed through concrete, pictorial and abstract representations, a visual representation using counters, blocks or a number line can be used to demonstrate both addition and subtraction. Using pictures and a number line can help students visualise and demonstrate the idea of subtraction, it is equally as important to note that subtraction problems can be easily thought of as missing addend problems as the same strategy of "think addition" is used. In some instances both operations ( addition and subtraction) can be taught together, as mentioned the " think addition" strategy is vital when learning and understanding subtraction. Children should be encouraged to recognise, think about and use the relationship between both operations in order to find the answer (Reys. R, 2012).

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406565&chapterid=189992

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 286,351). Milton, Qld: Wiley.

Year 3 Mathematics- Work Sample Portfolio. (2011). Retrieved from http://docs.acara.edu.au/curriculum/worksamples/AC_Worksample_Mathematics_3.pdf

Year 3 Work Sample Portfolio. (2014). Retrieved from http://docs.acara.edu.au/curriculum/worksamples/Year_3_Mathematics_Portfolio_Satisfactory.pdf

Week 2: Addition

The Big Idea:

This week the main focus was on Addition  and the importance of informing students that Addition does not necessarily mean that the total is going to be greater than as this is a misconception which will be elaborated on further on in this blog. It is also important to note the use of language at a Materials Language Stage  as the students have yet to gain a clear understanding of the Mathematic Language and Symbols used to express a form of equation. Hence why, avoiding the words " add" and " equals" at a Materials Language Stage is crucial. Simply put, Addition is joining groups to find a total.

Concepts, Skills and Strategy:

The C/S/S when it comes to the process of Addition are as followed:

Concept- The use of objects when explaining the concept of Addition, by joining numbers together. Developing a story-board or games with different themes (eg. counters, mats, MAB blocks) as well as learning through song by using well-known simple songs that children easily identify with to keep them engaged in the topic providing them with the opportunity to discuss the topic.

Skill- Knowledge of the concept and the ability to follow through. A skill is knowing what to do with the concept and how to do it properly. An ability to perform a simple " sum" is considered a skill. Such as 3+8= 11.

Strategy- Method used to complete a task effectively. A strategy which has proven to be ineffective when teaching children is the " finger counting" strategy as it is a distraction for children. The 3 main strategies used for Addition are the : Count-on strategy used for numbers ( 1-3) beyond that this method is proved to be ineffective, Double-up strategy ( used with larger numbers and numbers which are the same) and Use 10 strategy or Build Up strategy ( used for adding numbers that are close to the number 10. An example would be : 5+5= 10 or 10+0=10)

Misconceptions: 

As mentioned briefly, a misconception when it comes to Addition is that the total is always going to be a greater number. However, this is not the case. Children should not be taught that by adding two numbers together the total is going to be greater the reason for doing so is that as they mature and learn more about different uses and different forms of equations they will come in contact with negative numbers and how a negative number + a positive number will equal a negative number and a positive + a positive equals a positive same way as a negative + a negative will equal a positive. That is why installing the idea of Addition will always be equal to a greater number is a false assumption and should not be taught to students that way. 

Resources:

According to the QCAA and Australian Curriculum ( ACARA) under the section titled Number and Algebra students should establish an understanding of language and the process of counting numbers in sequences (Qcaa.qld.edu.au, 2014). Students should be taught how to connect and join numerals and quantities which also includes the number zero to ten. The goal by the end of the Foundation Year in terms of Addition is that the students should be able to analyse and read numbers up to ten and knowing the simple process of addition. ACARA states that in the Foundation level students learning Number and Algebra should be able to:

Understand- Connect names, numerals and quantity by addition
Fluency- Counting numbers in sequence or patterns
Problem Solving- Using materials to come up with a solution for the problem by using familiar counting sequences
Reasoning- Creating patterns and explanation of the process
( Australiancurriculum.edu.au, n.d)

Sourced Teaching Strategy:

( Amsi.org.au, 2011)

Mentioned previously, the use of counters is a very effective method when it comes to teaching Addition is it gives children a visual representation of the process and how it is presented. 

Textbook Concepts, Skills and Strategy: 

This text summarises the Big Idea related to C/S/S in Addition. Before start schooling it is generally assumed that all children will have the basic idea of Pre-number these will help build an understanding of the early number concept and the foundations of basic skills needed later on in their Mathematic journey ( Reys, R. 2012). The ability for a child to know how to count correctly is a skill however, in some cases a child may know how to count and the process involved but are unable to establish the last number is an indication of how many they are. It is encourage for students to organise their objects for a clearer understanding, this will also reduce the amount of errors a child makes such as accidentally double counting. Therefore, they will gradually improve on their skills as they are able to sort out the numbers efficiently and without hesitation. Furthermore, some of the key strategies  mentioned in the textbook that are used in counting are the Count-On strategy which was mentioned previously and the Count-back strategy. Both of theses strategies are useful when it comes to teaching the children Addition and the basic principles of Counting. This strategy is critical when developing Addition. It teaches children to recognise different numbers and how the starting number differs from the next number ( Reys, R. 2012).

References

Jamieson-Proctor, R. (2019). Learning Environment Online: Log in to the site. Retrieved from https://leo.acu.edu.au/mod/book/view.php?id=2406552&chapterid=189986

Mathematics. (2019). Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?

Reys, R. (2017). Helping children learn mathematics (2nd ed., pp. 281,285,343). Milton, Qld: Wiley.

Search - Scootle. (2018). Retrieved from http://www.scootle.edu.au/ec/search?accContentId=ACMNA002

Week 1: The Language Model

The Big Idea:

This weeks tutorial mainly focused on two key ideas how Mathematics is a language and should be taught as such and the effective structure of the Language Model for teaching mathematics in the classroom. There are four main stages of the Mathematical language these include:
1. Children's Language Stage
2. Materials Language Stage
3. Mathematics Language Stage
4. Symbolic Language Stage

It is crucial to follow these stages through in order for the students to develop a clear and concise understanding of the concept to the Mathematical language.

Concepts, Skills and Strategy:

No mathematical C/S/S were discussed this week. However, the four stages of the Mathematical Language Stages were presented and discussed in detail.

Stage 1: Children's Language:  Mathematics is explained to students at a level in which they are able to understand and interpret the basic meaning and structure of the language. An example would be the word " addition" which means to add however, a teacher may use a different more simpler term for the children such as the word " join". This will help the students gage the concept of the word. Placed in a sentence this can be " Martha and Susie joined Jessica for lunch".

Stage 2: Materials Language:  Similar to the Children's Language stage, the Materials Language stage is simply a process of replacing words with objects and being able to visualise produce verbal cues. This can be done by replacing what was said previously with a more visual piece of stimuli such as counters or blocks.

Stage 3: Mathematics Language:  It begins to explain and elaborate on the proper use of the mathematic terminology such as using the word " subtract" instead of " take- away".

Stage 4: Symbolic Language:  This final stage is where symbols are introduced to students as a means to further simplifying the language. Instead of writing the word " add" the symbol " +" should be written to explain what is meant.

Misconceptions:

A misconception I have known in mathematics is that it was taught in a different style and structured differently which until recently I have learnt is not true as it is a universal language and is taught the same or similarly around the world as well as this mathematics should be taught the same as learning a language. At a younger age I have always believed myself to be not good at mathematics and one has to be " really smart" to be able to obtain the information and understanding in mathematics. However, I now realise just like with learning a new skill or sport it takes time to learn, understand and perfect those mathematical skills.

Resources:

The tool I have chosen for the assignment is a Blog Website by the name of " BLOGGER". I have chosen this form media because not only have I used it multiple times and am familiar with the settings but, I believe Blogger is an effective and simple tool to use in the classroom as it is very straightforward and will not confuse the children. As well as this, Blogger has very bright and colour themes this would intrigue the students and make them concentrate on what is being presented. This media platform can be extremely useful in the classroom when it comes to presenting ideas for topics or sharing information gathered. The URL to this blog is ( https://clarissatchia.blogspot.com )
Sourced Teaching Strategy:

( Pintrest: https://www.pinterest.com.au/pin/251146116706710818/)

The picture above shows an excellent teaching resources that we can use in the classroom with kindergarten and primary students. The Number Mat is a combination of being able to spell out the number  " S-E-V-E-N" as well as write and count the number using different materials such as the frogs and bears shown above.

Textbook Concept, Skills and Strategy:

There are multiple Big Ideas introduced in the text. However, one in particular that stood out is What determines the Mathematics being taught: Needs of the child. Catering, adapting and changes in thinking help influence a child's ideas and approach to mathematics. Some of the key concepts that were understood and implemented in the 1920's is that of a child's understanding on subtraction, that facts under 10 were focused on children between the age of 6 and facts over 10 were aimed towards children at the age of 7. The more complex form of subtraction borrowing and carrying a number were aimed towards children 8 and above ( Reys, R. 2012). In mathematics a child needs to be able to  understand simple concepts such as making connections to problems and applying a solution or develop new ways to solve the problem. Fluency is also critical in a child's understanding of mathematics as it helps them develop the appropriate skills required to form a procedure with accuracy, efficiency and flexibility. Problem solving is also critical for children to be able to comprehend and develop as it requires them to be able to make their own choices and decisions formulating their own problems as well as model and investigating the issue, which will lead to communicating their solutions effectively. Lastly, students need to be able to provide a logical reasoning for their solutions such as proof of analysing, evaluating and justifying their answers.