Ana Isabel Navarro , Estíbaliz Aragón

A NEW METHODOLOGY FOR LEARNING MATH:

OPEN BASED ON NUMBERS ALGORITHM (ABN)

Universidad de Cádiz

ABSTRACT

In the latest PISA report (2012), Spain scored 10 points below the OECD average and 5 points below the European Union. That suggests it is convenient to review Early Mathematics teaching and learning procedures. The Algorithm Based on Numbers (ABN) methodology represents a significant change in the quantity and quality of children’s mathematical achievement. According to Jaime Martínez-Montero (designer of ABN methodology), children can learn faster and in more depth. ABN teaching is based on understanding rather than memorizing concepts. In addition, this new method dramatically improves students´ability, estimating mental calculation and problem solving. This research focused on addition, one of the basic math facts. ABN approach uses a new format to solve addition, and the number facts are always presented as a problem-solving. The addition fact could be represented in three columns. The use of this format is a major change on the ABN calculation method. The students do not work with numerals but use the whole numbers. As specific teaching equipment, ABN uses low cost tools, such us number lines, ten-frames, toothpicks (which represent units, tens and hundreds).

INTRODUCTION

Four decades ago, Ablewhite [2] focused us on the problem of training students with traditional methods based on closed algorithms, emphasizing the complexity of them. With these methods, students learn mathematical procedures without fully understanding the underlying concept. They are able to learn complicated and routine algorithms, but they do not know what practical situations they can resolve with them.

Others authors who are publicated diferents researchs are against the method used in schools to teach Maths. Such as the designer of the Algorithm Based on Numbers (ABN) methodology, Dr. Jaime Martinez-Montero.

There is no much background in the application of the new algorithm models that he proposes. For this reason, we have only their studies, which highlight that the students teaching by ABN methodology improve their learning more than the students who are teaching by the traditional method. Specially in mental calculation than resolving problems.

INTRODUCTION

The Algorithm Based on Numbers (ABN) methodology represents a significant change in the quantity and quality of children’s mathematical achievement. According to Martínez-Montero, children can learn faster and in more depth. In addition, this new method dramatically improves students´ ability, estimating mental calculation and problem solving. Finally, ABN procedure allows them to perform mathematical calculations according to their ability. This new approach develops in students an effective improvement in motivation and positive students attitude to Mathematics.

The theoretical foundation of the ABN method is in the orbit of constructivist models. This constructivist inspiration, changes the center of gravity about teaching and position it in learning. What is important now is what the students can assimilate. It is better students build their own knowledge.

ABN method is based on the perspective of the "Realistic Mathematics Education". The purpose of this perspective is that Maths in schools must be considered as a human activity that has to be nourished on the experiences of students, to be adapted to their personal characteristics and must be related to everyday life and the real needs of students.

ADDITION ALGORITHM

This research focused on addition, one of the basic math facts. ABN approach uses a new format to solve addition, and the number facts are always presented as a problem-solving.

The addition fact of two addends could be represented in three columns: in the left column the student chooses one quantity from any addends; the central column shows the remaining amount it is necessary add from the first addend. And in the right column the result from such partial addition, and the final result. The use of this format is the major change on the ABN calculation method. The students do not work with numerals but use the whole numbers.

638 + 594 =

6 632 600

2 630 602

30 600 632

300 300 932

200 100 1132

100 0 1232

ADDITION ALGORITHM

Below we will explain two examples of the same addition fact, one of them calculate by the traditional method (on the left) and the other by the ABN method (on the right). Calculations are resolved simultaneously in the two methods, step by step, to see the differences between them.

In both calculations we have begun by units, as it is required in the traditional method. In traditional algorithm we wrote the first result 2 (8 + 4) and keep in our mind the numeral that we get in tens (1). In ABN algorithm we do the same: in the first column we wrote 8 because is the units that we have to add. In the middle column we write 630 because it represents the amount remaining of the first addend and in the third column, we write 602 because is the first partial result (8 + 594). (Figure 1)

638 + 594 =

6 3 8 8 630 602

+ 5 9 4

2

Figure 1: Traditional calculation / ABN calculation

ADDITION ALGORITHM

We continuous with the calculation in Figure 2. In traditional method we have added the second column (3 + 9) and we have to add the one we had in mind (1), (3 + 9 + 1), obtained in that way the result 3 and, another time, we keep in our mind (1) that we get to the third column. In ABN method we added the tens (30 + 602) obtaining 632 and knowing that we have 600 for add.

638 + 594 =

6 3 8 8 630 602

+ 5 9 4 30 600 632

3 2

Figure 1: Traditional calculation / ABN calculation

ADDITION ALGORITHM

The calculation ends in Figure 3. In the traditional method we have added the third column (6 + 5) obtaining 12 because we have added the one that we brought from the previous time (6 + 5 + 1) obtaining as a final result 1232. In ABN method we have obtained the same result when we have add the hundreds (600) and calculate (600+ 632).

638 + 594 =

6 3 8 8 630 602

+ 5 9 4 30 600 632

1 2 3 2 600 0 1232

Figure 1: Traditional calculation / ABN calculation

ADDITION FACT

Habitually, the learning progress sequence of the addition facts follows this twelve steps:

1. Basic combinations until 10. The traditional addition fact table. 0+0; 0+1; 0+2; ...; 10+10

2. Three-digit addition facts.3+4+1; 3+8+6; 5+8+9

3. Complete tens plus digits.20+8; 40+5; 60+1

4. Adittion of complete tens.20+30; 50+40; 10+80

5. Addition of complete tens plus incomplete tens.30+25; 80+19; 50+36

6. Incomplete tens plus digits.38+5; 41+7; 84+5

ADDITION FACT

7. Addition of two incomplete tens.43+26; 28+69; 15+64

8. Addition of complete hundreds plus complete tens plus units or complete hundreds plus incomplete tens.

300+40+9; 300+49

9. Incomplete hundreds plus units.357+6; 428+9

10. Incomplete hundreds plus complete tens.357+60; 428+70

11. Incomplete hundreds plus incomplete tens.357+63; 428+74

12. Addition of two incomplete hundreds.357+428; 214+547