Teaching children to understand mathematical concepts is a difficult task. This research is about how a tutor can help children calculate basic mathematics mentally. It discusses the lesson tools and diagnostics that will help the teacher to identify the learning capabilities of the children. The first part identifies two children who are of the same age. The next part involves conducting diagnostics to help the teacher identify the student’s mental capabilities. The third part of the essay involves reviewing the tasks that the students will complete. The other issue is on development of the lesson and teaching strategy based on the diagnostics results and the children.
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In order to conduct the diagnostics two children Anna and George were identified. George is in upper primary Year 4 while Anna is in lower primary Year 3. According to George’s parent, he is of average academic capability. He has memorised the multiplication table as well as the multiplications of up to number nine. According to Anna’s parents, she is also of average academic capability based on her previous test scores. They however feel that continuous exercises will help her improve.
Planning for the diagnostic tests
The aim of the diagnostic test is to assist the teacher know the students’ capability. One of the students is called Anna in lower primary year three. The aim of the test is to assess whether she has mastered what she learnt in year three according to the Australian curriculum (Dole & McIntosh, 2005). The other diagnostic test is for George in year four. The students are unknown to the teacher who is carrying out the diagnostic test to know the students capability in mental mathematics (Lambdin, Lindquist, Rays & Smith, 2009).
To carry out the diagnostic test the following resources are necessary. The first one is that they must be done in a classroom where there is no noise and the pupil is not distracted from concentrating on the task. A pencil and a piece of paper are necessary for the students to write down the strategies they may use during the mental calculations (Thomson, 2010).
During the diagnostic test, the teacher will not assist or teach the students. The role of the teacher in the test is to provide instructions on how to answer the questions (Thompson, 1999). The students will write the answers and explanations of how they have calculated their answers on the answer sheet provided. The instructions are that the students must show how they got their answers by writing the strategies on the answer sheets (Cathcart, 2001). The aim of these tasks is to identify the mental recall of the students and their mastery of multiplication, division and addition strategies (Sweller, 1994).The other aim is to establish whether the students can model and remodel different numbers (Swan, 2005). The diagnostics therefore will have addition, subtraction, multiplication and division tasks. However, their complexity will differ depending on whether the student is in upper primary or lower primary (Dole & McIntosh, 2005).
The following are the diagnostics tasks that the children should solve. The first diagnostic task is for the lower primary and year two or three pupils can solve it. For the upper primary pupils they will complete the double-digit multiplication and division calculations (Australian Education Council, 1991).
|Lower Primary (Years 1-3)||Upper Primary (Years 4-7)|
|5 + 6||25 + 26|
|9 + 6||38 + 47|
|16 – 8||62 – 48|
|6 x 5||52 x 4|
|24 ÷ 4||124 ÷ 4|
Completion of the lower primary tasks
5 + 6= 11, 5 + 5 +1 = 11, 2+3+3+3 = 11
9+6 = 15, 5+4+6 =15, 5 +5 + 5= 15
16 – 8 =8, 8 +8 – 8 = 8, 10 – 8 + 6 = 8
6 × 5 = 30, 2 × 3 × 5 = 30, 3 × 2 × 5 =30
24 ÷ 4 = 6, 24 ÷ 2÷ 2 = 6, 24 ÷ 2 ÷ 2 = 6
Completion of upper primary tasks
25 + 26 = 51, 25 + 25 +1 = 51, 12 +13 +13 + 13 = 51,
38 + 47 = 85, 30 + 40 +8 + 7 = 85, 35 + 45 + 3 + 2 = 85,
62 – 48 = 14, 50 – 48 + 12 = 14, 60 – 48 + 2 = 14,
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52 × 4 = 208, 50 ×4 + 2 × 8 = 208, 13 ×2 × 2 × 2 × 2 = 208
124 ÷ 4 = 31, 124 ÷ 2 ÷ 2 = 31,
These are computations with a strong sense of numbers sense. It is a way of developing computation shortcuts using the mind (Haylock 2010). It is making calculations in the head without having the assistance of tools such as paper, pencil or a calculator. Mental mathematics assists children to gain understanding of numbers, order and reorder them to create shortcuts (Cathcart, 2001). Mental computations are of essence mainly because they assist the child to develop the mental faculties and have easy time in complex computations on paper. There are different computation strategy depending on the complexity of the numbers involved in the operation and the level of study of the child. These strategies assist the child to have a strong sense of numbers in terms of compatibility and composition of different numbers. The teacher must assist the student to develop a strong sense of number as well as show the students different mental strategies to enable the child to compute mentally.
|Operation||Strategy||Description||Child 1||Child 2||Researcher|
|5 + 6 = 11||To pull numbers apart||This is making addition to the initial number or simplifying the numbers and then adding on the numbers.||5 + 5 +1 = 11, |
2+3+3+3 = 11
|She reordered it to 5 +5 + 1= 11, for the second one she also rearranged it into simpler numbers of addition such as 9 + 6 = 15, 5+4+6 =15.|
|9+6 = 15||Counting upwards||This reordering numbers up to ten and then adding the remaining numbers and then counting the compatible numbers together||9+6 = |
9 + 1 = 10
6 – 1 = 5
10 + 5 = 15
|The child managed to use this mental computation strategy by simplifying the numbers into fives and then adding the additional numbers.|
|16 – 8 = 8||Counting backwards||This is counting up to sixteen, then counting backward eight numbers, and then counting the remaining numbers.||16 – 8 = 8||This strategy is easier to use especially when the child is dealing with two digit numbers and one digit number.|
|6 × 5 = 30,||Looking for the easy numbers||This is simplifying the numbers for easier multiplication.||2 × (3 × 5 =15) = 30||The child has a strong number sense for her to simplify the numbers and then multiply them.|
|24 ÷ 4 = 6||Using number system knowledge||In division, Anna reordered the numbers such as 24 ÷ 2÷ 2 = 6.||24 ÷ 2÷ 2 = 6 |
24 ÷ 2 = 12
12 ÷ 2 = 6
|She felt that it was easier to calculate mentally when using smaller numbers than huge numbers.|
|25 + 26 = 51,||Using the numbers in tens and then adding additional numbers.||This involves putting numbers in tens together, then combining other numbers||20 + 20 +5 + 5 +1 = 51||This strategy applies when dealing with two digit numbers. The child using this strategy must have strong understanding of digits in tens.|
|62 – 48 = 14,||Counting the numbers in tens and then subtracting the number.||This involves counting the sixth ten and then subtracting 48 = 12 and then adding the 2||60 – 48 + 2 = 14, |
50 – 48 + 12 = 14,
|The child understands splitting numbers into tens and calculating separately.|
|52 × 4 = 208,||Multiplying the numbers in tens and then multiplying the additional numbers||It also involves splitting the numbers into two digits, multiplying them and then adding the outcomes||50 ×4 = 200, |
2 × 8 = 8,
200 + 8 = 208
13 ×2 × 2 × 2 × 2 = 208
|The child has a good numbers sense and understanding of numbers in tens.|
|124 ÷ 4 = 31,||Dividing the number in twos||It involves dividing the larger number by a simplified smaller number||124 ÷ 2 = 62 |
62 ÷ 2 = 31,
|This strategy works well with two digit numbers or other higher numbers.|
Mental computation strategies
Different studies indicate that there are different types of mental calculation strategies used by children in addition and subtraction. For instance, 9-3 the child can count nine fingers and then lower three fingers then count the remaining fingers.
The other strategy is that the child can count three numbers from nine and then count the remaining numbers. The child can count for instance three and then count six additional numbers (Sweller, 1994). For example 3 + 9 the child can count from the larger number and add three more numbers. The child can also count from the smaller number adding nine numbers.
The other additional strategy is that of splitting numbers in doubles especially double-digit numbers such as 25 + 26 = 51, 20 + 20 +5 + 5 +1 = 51, this mental strategy is also applicable in the multiplication of numbers.
The other strategy is that of pulling numbers apart for example (5=2+3 so 8+5 = (8+2) +3 = 13). This is applicable for addition operations.
How they solved each question
Splitting numbers into 10s and ones is another mental computation strategy as in the (62 = 6 tens and 2 ones), whereby the number is arranged in six tens and two ones. This applies in operations involving digital numbers either in multiplication or in addition. In division mental computations, the most appropriate strategy is pulling the dividing digit apart and then subdividing those numbers as they are.
5 + 5 +1 = 11, 12 +13 +13 + 13 = 51,
9 + 6 = 15, 5+4+6 =15, 35 + 45 + 3 + 2 = 85,
16 – 8 =8, 8 +8 – 8 = 8 60 – 48 + 2 = 14
6 × 5 = 30, 2 × 3 × 5 = 30 13 ×2 × 2 × 2 × 2 = 208
24 ÷ 4 = 6, 24-÷ 2÷ 2 = 6 124 ÷ 2 ÷ 2 = 31,
For Anna the most basic strategy, which she used, is the reordering of numbers. For instance, in the first question 5 + 6, she reordered it to 5 +5 + 1= 11, for the second one she also rearranged it into simpler numbers of addition such as 9 + 6 = 15, 5+4+6 =15. In subtraction questions she created simpler numbers by rearranging numbers for example 16 – 8 =8, 8 +8 – 8 = 8. In multiplication, she simplified the numbers and then multiplied those numbers in small units for example, 6 × 5 = 30, 2 × 3 × 5 = 30. In division, Anna reordered the numbers such as 24-÷ 2÷ 2 = 6. She felt that it was easier to calculate mentally when using smaller numbers than huge numbers.
George used similar mental computation strategies. His set of questions was complex because he is in upper primary year four. According to Australian curriculum, it is expected that the students will have mastered computation strategies of double digits while in year three. In question one George simplified the numbers to compute the problem as follows: 12 +13 +13 + 13 = 51, 35 + 45 + 3 + 2 = 85. In multiplication question, George reordered the numbers to calculate as follows: 52 × 4 = 208, 13 ×2 × 2 × 2 × 2 = 208. In subtraction the numbers were reordered as follows: 62 – 48 = 14, 60 – 48 + 2 = 14.
Teaching mental mathematics
There are different strategies relating to the mental computations though it is evident that some of the strategies are more dominant compared to others (Lambdin et al., 2009). Most of the students find one of the strategies easy to use in the mental computations (Haylock, 2010). This is shown by the fact that both Anna and George used simplification and reordering of numbers in the mental calculations. This means that they have already formed the multiplication and number patterns in their calculations (Swan, 2005). The teacher needs to use the strategies, which are simple and easy to understand. Such strategies include reordering numbers into simpler numbers and then adding or subtracting.
The teacher must ensure that the strategies contained in the lesson plan are congruent with the strategies that students find easy to use. In this case, counting forward strategy for the single digit additions will be useful in teaching the students mental strategies. The other addition strategy is using tens and then adding additional numbers to the sum total of the tens. These strategies are in the lesson plan. There are other computations strategies such as adding doubles of numbers, Counting up in 10s, and counting down in 10s.
Relate the children’s knowledge to the Australian Curriculum: Mathematics and misconceptions where appropriate
According to the Australian ministry of education, curriculum year three students should have mastered the multiplication of numbers between one and ten. They should also manage to divide numbers between 2 and 100. They should also recall multiplication of numbers from one to ten. In addition, for those in year four they should manage to divide and multiply double-digit numbers. They should be in a position to recognise, rearrange, and regroup number of up to 10, 000. They should recognise the connection between addition and subtraction and know how to continue number patterns of single digit numbers (Australian Education Council, 1991).
For year four children, they should choose appropriate strategies for multiplication and division and recall multiplication of up to 10 × 10. They need to know how to master multiplication of single digits. Students at this level should solve simple purchases problems (Australian Education Council, 1991).
When relating the children’s knowledge to what is required in the Australian curriculum one can confidently say that the curriculum is up to the standard, the teachers have adhered to its provision as the year four, and year three students could quickly complete the set of tasks provided. However, it is worth noting that the diagnostics administered to each of the student is one year lower than the current year of study (Gill, 2005).
The mental computation strategies are important in the development of child learning. Lessons involving mental computations must be taken seriously as this capability determines how well the students will excel in other calculations. This is because multiplication, division, subtraction, and addition are the basics of any mathematical computation (Thomson, 2010).
Australian Education Council (1991). A national statement on mathematics for Australian schools. Carlton, Vic: Curriculum Corporation.
Cathcart, G. (2001). Learning mathematics in elementary and middle schools. Ohio: Merrill Prentice Hall.
Dole, S. & McIntosh, A. (2005). Mental computation: A strategies approach. Hobart: Department of Education Tasmania.
Gill, B. (2005). Teaching mathematics in the primary school. New York: Continuum International Publishing Group.
Haylock, D. (2010). Mathematics explained for primary teachers. London: Sage Publications Ltd.
Lambdin, D., Lindquist, M., Rays, R. & Smith, N. (2009). Helping children learn mathematics (9th edition). New York: John Wiley & Sons.
Swan, M. (2005). Improving learning in mathematics: Challenges and strategies. Sheffield: Department of Education and Skills Standards Unity.
Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, 4, 295-312.
Thompson, I. (1999). Getting your head around mental calculation. Issues in teaching numeracy in primary schools. Buckingham: Open University Press.
Thomson, S. (2010). Challenges for Australian education: Results from PISA 2009. Melbourne: Australian Council for Educational Research.
Mathematics Education 245
Semester 2, 2012
The purpose of this lesson is to assist the pupils learn how to calculate mentally. The aim of the lesson is to assist the pupils understand how to add, subtract, multiply and divide mentally.
The lesson will start with the teacher providing questions to the students and then solving them together. The question will be on the blackboard and the teacher will encourage participation of all the students during the lesson.
What will the children and you do?
The teacher will provide examples of questions and then solve them together with the children. The children will answer prompt questions asked by the teacher.
How will you organise them?
The first lessons will deal with addition, followed by subtraction, multiplication and finally division. The students will do a set of exercises after the examples where they will calculate mentally.
What are some specific questions you will ask?
|25 + 26 =|
|38 + 47 =|
|62 – 48 =|
|52 x 4 =|
|124 ÷ 4 =|
How will you adjust the lesson for diverse learners in your class (specifically, low and high learners)?
To cater for slow learners who may take long time to understand the teacher will have to assess the students frequently to identify them. The next step will involve giving them extra time, involving the parents to assist them after school and having tests to gauge their speed and comprehension.
How will you know what the children learned in terms of the objective/purpose?
To know what the students have learned and their level of understanding there will be formative tests at the end of every week to assess the children’s comprehension. These tests are imperative as they provide feedback to the teacher on the children and their progress in understanding the mental computations. The assessments have to be competitive and providing incentives to those with high scores will motivate them to concentrate and solve the mental computations.
How will you help children to connect and understand the intended learning of the lesson?
To help the students understand the teacher will use visual aids such as number lines to show children the movement and patterns of numbers during addition and subtraction. The number cards will act as visual aids to assist the students understand what they are learning. In addition, using bottles, people or diagrams during addition and subtraction will assist the students relate real life situations with what is learnt in class.