Computation

Computation is the term used to describe any type of information processing (Bobis, Mulligan & Lowrie, 2008).  It involves using a mathematical operation to solve a numerical problem (Origo, 2008).  As teachers, we must put emphasis on the role of children's thinking in their mathematics learning, and build curriculum based on the development of children's concepts (Ell, 2002).  Computation can be separated into 2 categories: computation using tools, and mental computation, the ability to calculate an exact numerical answer without the use of calculating tools (Origo, 2008).  The Queensland Studies Authority (QSA) essential learning's for mathematics state that by the end of year 3, students should be able to:

Use everyday and mathematical language, mental computations, representations and technology to generate solutions and check for reasonableness of the solutions.

Therefore, computation is a skill that must be developed from the early years.  This blog will look at a number of appropriate learning experiences for early childhood students to extend on their knowledge of computation.

Let's take a look at computation using tools, the first phase of computation.  Teachers can use representations to assist in the development of strategies.  In mathematics, we use representations extensively, as they form a concrete image of the problem in the students mind.  In the classroom, teachers can utilise tens frames, hundreds charts, number lines, counters, paddle pop sticks, or any other material for counting, to represent problems to the students. 

Tens Frames: tens frames use 10 as a reference point to compare quantities.  Teachers can begin to use these to develop understanding of simple addition and subtraction problems of numbers under 10.  In the example below, we can see how there are 7 counters placed on the tens frame. 

The frame illustrates 2 different problems to the students: 7 is 3 less than 10, or 10 less 3 is 7.  Most early childhood students have an understanding of the number 10: it's value, and it's concrete amount.  Teachers can expand on this to use tens frames to develop knowledge of addition and subtraction using 2 tens frames, and a strategy called bridge-to-ten.  The bridge-to-ten cluster includes addition or subtraction facts that use the numerals 8 or 9 as a digit, e.g. 8+5=?, or 9-3=?.  Students can use the tens frame to compute the answers for these problems.  In the example below, we use a tens frame to calculate the answer to the addition problem 8+7=?.  Students would use bridge-to-ten to round the 8 up to a 10.  However, in doing so, they understand that the extra 2 must have come from the other number in the equation; therefore we must decrease the 7 by 2.

Students can then calculate the answer by using the addition problem 10+5=?.

Number lines: number lines are diagrams showing numbers arranged in order along a line that is divided into evenly spaced intervals (Origo, 2008).  Similarly to tens frames, students can use number lines to develop understanding of simple addition and subtraction problems.  For this example we will use another bridge-to-ten addition fact, 9+5=?.  The diagram below shows a number line set out from 0-20.  By placing a mark on the number 9, students will set a starting point for the problem.  Moving up one position to 10 will simplify the equation.  By adding 1 to one side of the equation, they must remember to take away 1 from the other side, leaving them with the simplified addition problem 10+4=?.  Students will then move up the remaining 4 positions to the number 14 to arrive at their answer.  This tool can also be used for subtraction.

The second phase of computation is mental computation.   Teachers can build strategies with students to develop an ability to calculate numerical answers without the use of calculating tools.  Through tutorials, we looked at 2 main tools for developing mental computation.  These are old trusted hundreds/99 charts, and empty number lines.  The two tools go hand in hand for building mental computation strategies. 

The following learning experience can be utilised by a teacher in the development of mental computation strategies.  This activity would work well if all students had their own laminated hundreds chart, and a whiteboard marker. 

The teacher can begin the activity by calling out a number e.g. 40.  Students are to place a counter on that number.  Ask the students to add another counter 10 places afterwards.  Students should begin to count 10 places from 40, where they will land on 50.  From here, the teacher can ask if they notice anything about the 2 counters (adding10 is the same as moving the counter down a line.  Teacher should ensure that correct language is used to illustrate this as down will have the same meaning as decrease to most students).  Next, ask the students to begin again on 40.  Ask the students to add another counter 10 places beforehand.  Students should begin to count 10 places backwards from 40, where they will land on 30.  From here they can ask if the students noticed anything about this move (subtracting 10 is the same as moving up a line.  Again, correct language must be used in order not to confuse students, as most will understand moving up to have the same meaning as increase). 

The following is a suggested order of presentation of numbers:

1. Jumping in tens forward and backwards from multiples of ten (e.g. start with 40 - move forwards and backwards in tens.  Ask the students what pattern they can see);
2. Jumping in tens forwards and backwards (e.g. start with 43 - move forwards or backwards in ten.  Ask the students what pattern they can see);
3. Relate this to addition and subtraction (e.g. start with 43 - add 10, add 20, add 30; take away 10. take away 20 etc);

Students will begin to form their own strategies for addition and subtraction using this hundreds chart.  During a tutorial activity, we were asked to start on the number 53.  Then add 10.  The counter moves 1 row down to 63.  Then minus 20.  The counter moves 2 rows up to 43.  From there minus 1.  The counter moves 1 space to the left.  Then add 2.  The counter moves 2 spaces to the right.  A strategy like this will assist early childhood students in adding and subtracting numbers using mental computation.  However, as stated previously, the teacher must be aware of correct language to use with the students in order to not confuse them with direction.

Empty number lines can be used to support what the students have just completed on the hundreds chart.  Similar to conventional number lines, the empty number is not divided by lines at evenly spaced intervals.  The following example illustrates the process a student would use to add 10 to their starting number 43:

It is up to the student to choose the starting point for their number line.  Teachers should make a point of asking students as to why they chose to start where they did.  Students should being to realise that addition problems will be increasing the starting number, and therefore they should choose a point further to the left to allow room to add more quantity.  Subtraction problems will be decreasing the starting number, and therefore they should choose a point further to the right to allow room to subtract more quantitiy.  Similarly, they could use the same process to take away 10 from 53.  Students can now begin to use the empty number line to add and subtract larger 2 digit numbers, using the bridge-to-ten strategy.  The teacher could give the students the equation 48+26=?.  The example below illustrates how the student could calculate this on the empty number line:

The following is a suggested order of presentation of numbers (expanding upon suggestions from hundreds board):

1. Further addition and subtraction, without using bridge-to-ten (e.g. 43+20);
2. Further addition and subtraction, using bridge-to-ten (e.g.47+28, 47+19).

During these activities, the teacher could use the following as focus questions aimed at the student:

• How did you solve that? Show us/tell us
• Did anyone else solve it that way?
• How is it the same?
• Who did it differently?
• How is that different?
• What do you think about those strategies?