Design Research on Addition and Subtraction up to 100
Using Mental Arithmetic Strategies on an Empty Number Line
At the 2nd Grade of SDN Percontohan Komplek IKIP, Jakarta
(by Puspita Sari)
An empty number line is found to be a powerful model to do mental arithmetic strategies flexibly. An empty number line is best to represent children’s informal strategies in counting and it has potential to foster the development of more sophisticated strategies in children (Gravemeijer in Puspita Sari).
Mental arithmetic is suggested to be learned by children before they learn algorithm (column arithmetic) (Beishuizen in Puspita Sari) and it has become more important because it stimulates number sense and the understanding of number relations (McIntosh, Reys, & Reys in Puspita Sari).
1. How do children develop a framework of number relations to construe flexible mental arithmetic strategies?
2. How do the number relations support the flexibility of doing mental arithmetic strategies in solving addition and subtraction problems up to 100?
II. Didactical Domain in Addition and Subtraction Up to 100
There are three levels of students thinking in solving addition and subtraction problems up to 100.
1. The low level, children solve addition and subtraction problems by counting one by one.
2. The next level, children acquire the idea of ‘unitizing’ which means they understand that a number of objects can build another unit.
3. The top level, children should already master the previous concepts and strategies before they are taught algorithm.
III. Mental Arithmetic Strategies on an Empty Number Line
Mental arithmetic is a way of approaching numbers and numerical information in which numbers are dealt with in a handy and flexible way, and characterized by:
1. Working with numbers and not with digits
2. Using elementary calculation properties and number relations
3. Being supported by a well-developed feeling for numbers
4. Possibly using suitable intermediate notes according to the situation, but mainly by calculating mentally.
Three basic form of mental arithmetic:
a. Mental arithmetic by a stringing strategy
b. Mental arithmetic by a splitting strategy
c. Mental arithmetic by a varying strategy
IV. Design Research Methodology
The purpose of the present research is to contribute to an empirically grounded instruction theory for addition and subtraction up to 100.
Phase 1. Preparation and Design
In the first phase of the design research, mathematical learning goals should be elucidated as well as anticipatory thought experiments in which sequences of learning activities are designed and students’ mental activities in engaging the activities are envisioned.
Phase 2. Teaching Experiment
During a teaching experiment researchers and teachers use activities and types of instruction that seem most appropriate at that moment according to the HLT.
Phase 3. Retrospective Analysis
After each lesson, we will analyze the data that we get from the class and use the result of the analysis to develop the next design.
Phase 4. Reliability and Validity
Reliability is about when other person conducts our research, s/he could trace what steps are in it and what decisions have been made, and it would give almost the same results for him/her. In validity, there is internal validity in which triangulation plays important role to test data with HLT.
V. A Conjectured Local Instruction Theory
Conjectured local instruction theory is made up of three components:
1. Mathematical learning goals for children, formulated as below:
a. Children will develop a framework of number relations to construe flexible mental arithmetic strategies.
b. Children will be flexible in solving addition and subtraction problems up to 100 both in context and in a bare number format using mental arithmetic strategies.
2. Planned instructional activities
Unit 1. Celebrating the 63rd Indonesian Independence Day
Unit 2. Measuring
Unit 3. The emergence of an empty number line
Unit 4. Exploring number relations
Unit 5. Exploring addition
Unit 6. Developing addition strategies
Unit 7. Developing addition strategies
Unit 8. Exploring subtraction
Unit 9. Developing subtraction strategies
Unit 10. Developing subtraction strategies
The design research attempts to develop Local Instruction Theory (LIT) an addition and subtraction up to 100 by designing a sequence of activities that help children enhance their mental arithmetic in solving addition and subtraction problems up to 100. An empty number line is proved to be a powerful model to represent children’s thinking in doing mental arithmetic strategies. Developing mental arithmetic using realistic approach is expected to prevent innumeracy.