Theory
Standard Algorithms for Multiplication
Short multiplication – formal way to calculate a multiplication with two or more digit number by a single-digit
3
38 38 x 4
x 4
152
Long multiplication – formal way to calculate a multiplication that multiplies two numbers with more than 2 digits in each
26 26 x 34
x 34
780 26 x 30
104 26 x 4
884
Standard Algorithms for Division
Recording written long division algorithms requires explicit teaching as you start with the largest pieces on the left and work to the right, which is the opposite of other standard algorithms.
Short
Short multiplication – formal way to calculate a multiplication with two or more digit number by a single-digit
3
38 38 x 4
x 4
152
Long multiplication – formal way to calculate a multiplication that multiplies two numbers with more than 2 digits in each
26 26 x 34
x 34
780 26 x 30
104 26 x 4
884
Standard Algorithms for Division
Recording written long division algorithms requires explicit teaching as you start with the largest pieces on the left and work to the right, which is the opposite of other standard algorithms.
Short
(Anderson, Briner, Irons, Shield, Sparrow & Steinle, 2008)
Long
Long
(Anderson et al., 2008)
Problem solving
Van de Walle, Karp and Bay-Williams (2014) suggest that a problem is “any task or activity for which the students have no prescribed or memorised rules, nor is there a perception of a specific correct solution method” (p. 56).
Three types of approaches to problem solving:
1. Teaching for problem solving – teaching the skill or abstract concept before solving related problems. Showing the student how to solve problems can cause them to be dependent on instruction for completing further problems. Students learn more when they are active participants in the problem solving process.
2. Teaching about problem solving – teaching students’ processes or strategies for solving problems. This can improve students’ ability to think mathematically.
3. Teaching through problem solving – using contextual problems to develop concept knowledge.
(Van de Walle et al., 2014)
Four step problem-solving process (Polya Framework)
1. Understand the problem
2. Devise a plan – students should have flexibility in how they devise a plan so providing different options will scaffold their development.
a. Use models or pictures, tables or charts
b. Look for patterns
c. Try a simpler form of the problem
d. Guess and check
e. Write an equation
3. Carry out the plan
4. Look back
(Van de Walle et al., 2014, p. 56)
Other strategies
STAR
S – search the word problem for important information
T – translate the words into models, pictures or symbols
A – answer the problem
R – review your solution for reasonableness
(Van de Walle et al., 2014, p. 44)
THINK
The THINK strategy assists students to develop meta-cognitive skills.
T – talk about the problem
H – how can it be solved?
I – identify a strategy to solve the problem
N – notice how your strategy helped you solve the problem
K – keep thinking about the problem. Does it make sense? Is there another way to solve it
(Van de Walle et al., 2014, p. 67)
Textbooks generally provide non-problem-based tasks in which case they should be adapted so they are problem-based.
(Van de Walle et al., 2014)
Problem solving
Van de Walle, Karp and Bay-Williams (2014) suggest that a problem is “any task or activity for which the students have no prescribed or memorised rules, nor is there a perception of a specific correct solution method” (p. 56).
Three types of approaches to problem solving:
1. Teaching for problem solving – teaching the skill or abstract concept before solving related problems. Showing the student how to solve problems can cause them to be dependent on instruction for completing further problems. Students learn more when they are active participants in the problem solving process.
2. Teaching about problem solving – teaching students’ processes or strategies for solving problems. This can improve students’ ability to think mathematically.
3. Teaching through problem solving – using contextual problems to develop concept knowledge.
(Van de Walle et al., 2014)
Four step problem-solving process (Polya Framework)
1. Understand the problem
2. Devise a plan – students should have flexibility in how they devise a plan so providing different options will scaffold their development.
a. Use models or pictures, tables or charts
b. Look for patterns
c. Try a simpler form of the problem
d. Guess and check
e. Write an equation
3. Carry out the plan
4. Look back
(Van de Walle et al., 2014, p. 56)
Other strategies
STAR
S – search the word problem for important information
T – translate the words into models, pictures or symbols
A – answer the problem
R – review your solution for reasonableness
(Van de Walle et al., 2014, p. 44)
THINK
The THINK strategy assists students to develop meta-cognitive skills.
T – talk about the problem
H – how can it be solved?
I – identify a strategy to solve the problem
N – notice how your strategy helped you solve the problem
K – keep thinking about the problem. Does it make sense? Is there another way to solve it
(Van de Walle et al., 2014, p. 67)
Textbooks generally provide non-problem-based tasks in which case they should be adapted so they are problem-based.
(Van de Walle et al., 2014)
Common Student Misconceptions
- Students understanding of place value may not be strong so they will be challenged to know where to place the numbers under the standard algorithm. They may treat a tens value multiplication as though it is a one.
- Students may “lack understanding of the role of zero as a place holder” (p. 66)
- Student may place a zero down on both rows for multiplying by tens or no zeros at all.
x 23
1020
680
1700
(Lawton & Hansen, 2011)
Mathematics Language for Multiplication and Division
|
|
|
|
Activities
|
Word Problems |
Year 4
Bobbing for apples Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (ACMNA076) Objective: Students will develop doubling and halving strategies to solve problem. Year 5 At the fair Solve problems involving division by a one digit number, including those that result in a remainder (ACMNA101) Objective: Students will demonstrate their ability to solve a word problem that involves division. Year 6 Party planning Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123) Objective: Students will use their knowledge of operations with whole numbers to solve a word problem. |
Mrs Smith has a draw with 525 stickers. If she uses 25 stickers per day, how many days will the stickers last? Susie has 48 trading cards that she would like to share between 4 people (herself and her 3 friends). How many trading cards does each person get? |
(Adapted from Australian Mathematical Science Institute, n.d.)
Resources
Curriculum Map
In the Early Years Learning Framework (Document 16, outcome 5, p. 6) children will learn the key learning concept of patterns including the recognition of repeating designs in the environment; and ability to create, copy and extend repeating designs using colours, sounds, shapes, objects, stamps, pictures and actions.
(Charles Sturt University Early Years Learning Framework Consortium, 2009)
Multiplication and division is 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 University Early Years Learning Framework Consortium, 2009)
Multiplication and division is found in the Australian Curriculum from Foundation through to Year 6.
FOUNDATION
- Represent practical situations to model addition and sharing (ACMNA004)
YEAR 1
- Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count by twos, fives and tens starting from zero (ACMNA012)
YEAR 2
- Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and ten from any starting point, then moving to other sequences. (ACMNA026)
- Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031)
- Recognise and represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032)
YEAR 3
- Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)
- Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies (ACMNA057)
YEAR 4
- Investigate number sequences involving multiples of 3, 4, 6, 7, 8, and 9 (ACMNA074)
- Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)
- Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (ACMNA076)
YEAR 5
- Identify and describe factors and multiples of whole numbers and use them to solve problems (ACMNA098)
- Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100)
- Solve problems involving division by a one digit number, including those that result in a remainder (ACMNA101)
- 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)
- Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies (ACMNA129)
- Multiply and divide decimals by powers of 10 (ACMNA130)
References
Anderson, J., Briner, A., Irons, C., Shield, M., Sparrow, L., & Steinle, V. (2008). The origo handbook of mathematics education. Kedron, Australia: Origo
Education.
Australian Mathematical Science Institute. (n.d.). Year 6 Number and algebra: Whole numbers with all four operations. Retrieved from
http://www.amsi.org.au/ESA_middle_years/Year6/Year6_md/Year6_1a.html#intro
Charles Sturt University 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=1
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.
Education.
Australian Mathematical Science Institute. (n.d.). Year 6 Number and algebra: Whole numbers with all four operations. Retrieved from
http://www.amsi.org.au/ESA_middle_years/Year6/Year6_md/Year6_1a.html#intro
Charles Sturt University 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=1
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.