Theory
Alternative theories are number oriented rather than digit oriented, calculated from the left rather than the right, and are flexible options as opposed to the ‘one right way’. They have many benefits which include students have a greater understanding of numbers and as a result there are fewer errors and less reteaching required (Van de Walle, Karp, & Bay-Williams, 2014).
A number of approaches can be used flexibly to solve addition and subtraction problems. They include:
A number of approaches can be used flexibly to solve addition and subtraction problems. They include:
- Near 10 approach – 9 + 5 becomes 10 + 4 or
- Up over 10 strategy – 46 + 8 becomes 46 + 4 + 4
- Double and near double approach – 6 + 6 = 12, 6 +7 = 6 + 6 + 1
- Constant difference method – involves adjusting both numbers by the same amount so it is easier to calculate
- Front end approach – involves partitioning the number into hundreds, tens and ones for example, completing the addition and subtraction
= 500 + 70 + 9
= 579
- Compensation or shortcut strategy– involves temporarily adjusting the numbers in the sum to complete problem
100 - 96 = 4
4 + 4 = 8
104 - 96 = 8
(Haylock, 2014; 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
It is not applicable to subtraction (Haylock, 2014)
Common Student Misconceptions
- Sums that bridge 10, 100 or 1000 can cause issues for mental computation.
- Students may not regroup when subtracting a larger number from the smaller number.
- A double trade for zeros can cause confusion
- 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
Mathematics Language for Alternative Approaches to Addition or Subtraction
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Early years
Counting reels 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 Stay at Home Educator, 2012)
Foundation Building relationships ACMNA004 Represent practical situations to model addition and sharing Objective: Students will develop relationship knowledge of numbers in order to develop reasoning strategies for addition and subtraction. (Adapted from Van de Walle et al., 2014)
Year 6 Household bills ACMNA123 Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers Objective: students will addition and subtraction strategies to solve realistic problems. (Adapted from Victoria Department of Education and Training, 2014)
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Sarah had a personal goal to kick 12 goals at her soccer game. She has kicked 5 goals, how many more goals does Sarah need to kick to reach her goal? Bernie has 34 lollies and Michael has 38 lollies. How many lollies do they have altogether? The school is having a fair and they had 124 people respond to say they were attending the event. On the day of the fair only 98 people attended. How many people did not attend the fair? |
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
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.
Stay at Home Educator. (2012). Introducing your preschooler to addition and subtraction. Retrieved
from http://stayathomeeducator.com/introducing-your-preschooler- to-addition-and-subtraction/
Van De Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Victoria Department of Education and Training. (2014). Real world investigations: Level 6. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/pages/realworld40.aspx#a1
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.
Stay at Home Educator. (2012). Introducing your preschooler to addition and subtraction. Retrieved
from http://stayathomeeducator.com/introducing-your-preschooler- to-addition-and-subtraction/
Van De Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Victoria Department of Education and Training. (2014). Real world investigations: Level 6. Retrieved from
http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/pages/realworld40.aspx#a1