THE BIG IDEAS
The first idea from week one is the concept that mathematics is a language. this is because it contains all the features of a language (e.g. words and symbols that generate meaning in the mind of the user). This concept changed my perspective of mathematics completely. As a numerate being, to explain a mental process that comes so naturally after years of practice becomes difficult as assumptions come into play. Comparing mathematics to another language puts the concept of learning mathematics as someone who has had little exposure back into perspective. As teachers, it it our responsibility to help children connect the language or word (such as equals, greater than, five, add, subtract and so on) to the meaning.
To assist with the concept of teaching mathematics as a language, we were introduced to the language model. This model is used as a visual tool for teachers as a guide to determine which concrete materials, language and symbols would be most appropriate as well as remind teachers that all three elements (concrete/visual, verbal and symbolic) are extremely important at all stages of learning. The language model changed my understanding of teaching mathematics as before this I was unaware of the importance of the slow transition of language used and the large range of objects that could be used for the concrete objects (such as photos). The language model is discussed further later in this post.
Helping children learn mathematics (2012) references the importance of and the interrelationship between language, concrete materials and symbols on page 198 where it states:
Video: Maths is a languageTo assist with the concept of teaching mathematics as a language, we were introduced to the language model. This model is used as a visual tool for teachers as a guide to determine which concrete materials, language and symbols would be most appropriate as well as remind teachers that all three elements (concrete/visual, verbal and symbolic) are extremely important at all stages of learning. The language model changed my understanding of teaching mathematics as before this I was unaware of the importance of the slow transition of language used and the large range of objects that could be used for the concrete objects (such as photos). The language model is discussed further later in this post.
Helping children learn mathematics (2012) references the importance of and the interrelationship between language, concrete materials and symbols on page 198 where it states:
"The move to symbols is often made to quickly and the use of materials dropped too soon. Instead, the use of materials should precede and then parallel the use of symbols with support from pictorial representation. Early on, children should be manipulating materials as they record answers. when children talk and write about what is happening in the given situation, they are helped to see the relationship of the ideas and symbols to their manipulation of materials and the problem itself.
The oral and written language that children learn as they communicate about what they are doing with materials helps them to make pictorial connection between concrete and symbolic understanding of the mathematical operations. By modelling, drawing, talking and writing, the referent for each symbol is strengthened."
Video: Maths isn't hard, it's a language by Randy Palisoc
THE CONCEPT, SKILL, STRATEGIES AND SUPPORTING RESOURCES
Helping children learn mathematics (2012) states on page 251 that:
Concept
In addition, the concept is that we are bringing two or more 'parts' together. The concept of addition can be taught using an addition mat. Addition mats provide a concrete representation of 'parts' being brought together to make a 'whole'. This is assisted with language such as "How many altogether?".
"Fluency with basic addition is a goal for the early years, but gaining fluency depends on many diverse experiences in grouping and counting. As children work with multiple objects, they develop a concrete understanding of addition. They move objects together and use a counting strategy to identify the total number of objects. Initially, they may begin by recounting all the objects; later this strategy often yields to a more efficient strategy of counting on."
Concept
In addition, the concept is that we are bringing two or more 'parts' together. The concept of addition can be taught using an addition mat. Addition mats provide a concrete representation of 'parts' being brought together to make a 'whole'. This is assisted with language such as "How many altogether?".
Skill
The ability to bring the two numbers together. 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
For addition, there are three main strategies:
1. Counting on for 0-3 (not 4 or more) e.g. 9+3 -> 9, 10, 11, 12 -> 9+3=12
2. Adding doubles e.g. 1+1, 2+2, 3+3 etc
3. Use tens e.g. 9+3 -> 9+1=10 + 2 more = 12 -> 9+3=12
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 two 'parts' as well as strategies that can be used to bring them together.
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 two 'parts' as well as strategies that can be used to bring them together.
Link: Leo - Other examples of the addition mat, addition stories and folding addition cards
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 the apple tree and two birds in the pear tree. 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 red counters and two blue counters. How many counters altogether?'
Mathematics language - Moving away from 'stories', this stage introduces mathematical terms. For example, 'What does three add 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 '3 + 2 ='
THE LANGUAGE MODEL FOR ADDITION
THE MISCONCEPTION
- The student may know the commutative property (known as turn-around facts) of addition but fails to apply the knowledge in practice. In this situation, I would use the addition mats to demonstrate that the order of the numbers being added together does not change the sum.
- The student uses the re-count strategy, instead of developing more effective methods. I would talk the child through the count on method. For example, "We already know there are three starfish here, and four more make seven".
THE ACARA LINK
Addition is first introduced in the foundation year.
Strand: Number and Algebra
Substrand: Number and Place Value
Code: ACMNA004
Content descriptions: Represent practical situations to model addition and sharing
Elaborations:
- using a range of practical strategies for adding small groups of numbers, such as visual displays or concrete materials
- using Aboriginal and Torres Strait Islander methods of adding, including spatial patterns and reasoning
Scootle resource ideas:
1. Counting beetles is an addition based computer game that encourages the count on method.
2. The Domino addition app can be downloaded from iTunes and provides children opportunities to practice the count on and adding doubles strategies.
THE RESOURCES AND IDEAS
THE TEXTBOOK SUMMERY
- Addition is the first basic operation introduced to children
- By beginning with physical objects or drawings accompanied by a story, children develop meanings for the operations
- Addition is commutative - changing the order of the addends does not affect the sum
- Counting on is a strategy that can be used when adding up to three, when adding more than three this strategy is inefficient
- Combinations to ten (or rainbow facts) are the nine pairs of numbers that together make ten (for example, 9 +1, 8 + 2, 7 + 3 and so on). Note that the understanding of the commutative property reduces this to five facts
- Adding doubles and near doubles is a strategy where both addends are the same (or almost the same) number. For example, 7 + 7 = 14 and 5 + 6 = 5 + 5 + 1 = 11
- Using ten is a strategy where one addend is broken apart so that one part of it can be used with the other addend to make ten. Then, the remaining part of the first addend is added to the ten ( for example, 8 + 5 = 10 + 3 = 13
- Gaining fluency in addition depends on many diverse experiences in grouping and counting
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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