Theory
Haylock (2014) discusses two structures of addition:
Haylock (2014) also discusses four structures of subtraction:
Van de Walle, Karp and Bay-Williams (2014) discuss three structures for addition and subtraction:
It is important that problems are provided in a context that is real to students (Van de Walle et al., 2014). This will help them to accommodate new knowledge and assist them to understand the importance of addition and subtraction in the real world. Additionally, including problems that have zero in the number will provide opportunities to discuss renaming and regrouping.
- Aggregation - the combining of two sets
- Augmentation - when a quantity is increased by another.
Haylock (2014) also discusses four structures of subtraction:
- Partitioning – “situation in which a quantity is partitioned off” (Haylock, 2014, p. 92)
- Reduction – reducing a quantity by another quantity
- Comparison – comparing two quantities, finding the difference
- Inverse-of-addition – determining “what must be added to a given quantity in order to reach some target” (Haylock, 2014, p. 94)
Van de Walle, Karp and Bay-Williams (2014) discuss three structures for addition and subtraction:
- Change – can be 'join' or 'separate' (Join is generally a quantity being added to, while separate is generally “an amount being removed” (Van de Walle et al., 2014, p. 160)
- Part-part-whole – finding the unknown where two amounts are part of a combination
- Comparison - comparing two quantities, finding the difference (can be more or less)
It is important that problems are provided in a context that is real to students (Van de Walle et al., 2014). This will help them to accommodate new knowledge and assist them to understand the importance of addition and subtraction in the real world. Additionally, including problems that have zero in the number will provide opportunities to discuss renaming and regrouping.
Symbols can then be introduced to students.
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20 + 5 = 25 20 - 5 = 15
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A traditional approach to addition and subtraction is standard algorithms or column addition or column subtraction. Students will require knowledge of regrouping in order to solve a problem with a standard algorithm. It is important that students view standard algorithms as one form for solving problems and not the only form. It is necessary to explicitly teach students to start in the ones column as opposed to starting on the left for alternative approaches. (Van de Walle et al., 2014)
Commutative law – a + b = b + a
It is not applicable to subtraction. It is important for students to understand as they can move the quantities so they can count on from the larger quantity. (Haylock, 2014)
Associative law – (a + b) + c = a + (b + c) you can add numbers in any order
Associative law cannot be applied to subtraction (Haylock, 2014)
Commutative law – a + b = b + a
It is not applicable to subtraction. It is important for students to understand as they can move the quantities so they can count on from the larger quantity. (Haylock, 2014)
Associative law – (a + b) + c = a + (b + c) you can add numbers in any order
Associative law cannot be applied to subtraction (Haylock, 2014)
Common Student Misconceptions
- Student may not understand zero as a placeholder and as a result think they cannot complete subtraction
- Student may not understand the concept of rename and regroup
- Student may miscount when counting on or completing addition or subtraction
- Student may confuse the symbol for addition (+) with multiplication (x) or subtraction (-) with division (÷)
- Student may not grasp the concept of place-value and therefore not understand its value
- Student may overgeneralise and use the commutative law or associative law for subtraction
- Some students may not understand that for a comparison problem they need to make separate sets of objects for each quantity
- Language used in problems can cause students to misunderstand. We not only need to teach students what is required but also need to be careful of own language to avoid confusion
Mathematics Language for Addition and Subtraction
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Early years
Teddy Bears Picnic The Early Years Learning Frameworks (EYLF) key numeracy concept (Outcome 5, p. 6) that is met is: student has the ability to count and order numbers; recognise and write numerals; and compare quantities such as 'more than' and 'less than'. Objective: student displays their ability to count two quantities then add them together and count the total quantity. (Adapted from Addition Mat video, n.d.)
Year 1 Solve the chair problem ACMNA015 Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts Objective: students have the ability to solve addition and subtraction problems using counting on and counting back. Year 4 Problem solving ACMNA073 Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems Objective: students will display their knowledge of place-value, including renaming and regrouping to solve a problem. |
Sarah has 5 blue teddies and 3 red teddies, how many teddies does Sarah have altogether?
Sarah has 8 red and blue teddies altogether, 3 teddies are red. How many blue teddies does Sarah have? Poppi has 8 teddies and Lilly has 10 teddies, how many more teddies does Lilly have than Poppi? Poppi has 8 teddies and Lilly has 10 teddies, how many fewer teddies does Poppi have than Lilly? Lilly has 2 more teddies than Poppi. Poppi has 8 teddies, how many teddies does Lilly have? Michael has 15 matchbox cars, he received another 4 for his birthday. How many cars does Michael have altogether? Michael has 19 matchbox cars. He had 15 cars before his birthday. How many cars did Michael receive for his birthday? Louise had 30 pencils. She gave 13 pencils to her friends. How many pencils does Louise have now? Louise has 17 pencils left after giving 13 pencils to her friends. How many pencils did Louise have to start with? (Adapted from Van de Walle et al., 2014) |
Resources
Curriculum Map
The key numeracy concept for addition and subtraction from Outcome 5, page 6 of the Early Years Learning Framework (EYLF) describes students will have the ability to count and order numbers, recognise and write numerals, and compare quantities such as ‘more than’ and ‘less than’.
(Charles Sturt Universtity Early Years Learning Framework Consortium, 2009)
Addition and subtraction are found in the Australian Curriculum from Foundation through to Year 6.
FOUNDATION
YEAR 1
YEAR 2
YEAR 3
YEAR 4
YEAR 5
YEAR 6
(Charles Sturt Universtity Early Years Learning Framework Consortium, 2009)
Addition and subtraction are found in the Australian Curriculum from Foundation through to Year 6.
FOUNDATION
- Represent practical situations to model addition and sharing (ACMNA004)
YEAR 1
- Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015)
YEAR 2
- Explore the connection between addition and subtraction (ACMNA029)
- Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)
YEAR 3
- Recognise and explain the connection between addition and subtraction (ACMNA054)
- Recall addition facts for single-digit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation (ACMNA055)
YEAR 4
- Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems (ACMNA073)
YEAR 5
- Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)
YEAR 6
- Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)
References
Addition Mat [Video file]. (n.d.) Retrieved from http://player.vimeo.com/video/43864756
Charles Sturt Universtity Early Years Learning Framework Consortium. (2009). Document 16 outcome 5: Children are effective
communicators. Retrieved from http://www.earlyyears.sa.edu.au/files/links/16_Outcome_5.pdf
Education Services Australia. (n.d.) Curriculum. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?
layout=3#page=3
Haylock, D. (2014). Mathematics explained for primary teachers (5th ed.). London, England: Sage Publications.
Lawton, F. & Hansen, A. (2011). Number operations and calculation. In S. Hansen (Ed.), Children’s errors in mathematics: Understanding common
misconceptions in primary schools (2nd ed.) (pp. 47-75). Exeter, England: Learning Matters.
Van De Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Charles Sturt Universtity Early Years Learning Framework Consortium. (2009). Document 16 outcome 5: Children are effective
communicators. Retrieved from http://www.earlyyears.sa.edu.au/files/links/16_Outcome_5.pdf
Education Services Australia. (n.d.) Curriculum. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?
layout=3#page=3
Haylock, D. (2014). Mathematics explained for primary teachers (5th ed.). London, England: Sage Publications.
Lawton, F. & Hansen, A. (2011). Number operations and calculation. In S. Hansen (Ed.), Children’s errors in mathematics: Understanding common
misconceptions in primary schools (2nd ed.) (pp. 47-75). Exeter, England: Learning Matters.
Van De Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.