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/


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