How Concrete is Concrete?

How Concrete is Concrete?

By Koeno Gravemeijer

We know that mathematics is abstract, and not easy to accept by students. Because of that many students thinking that mathematics is one of difficult subject. Based on that problems, we have to make mathematics fun and easy for students.   

One ways to solve that problems is to make the mathematics from abstract to real or concrete in their daily life, with visual model or manipulatives model. So there is an interesting contrast between the way we make something concrete in everyday life , and the way we do this in mathematics education.

This is one of example :

Cobb (1989) interview one of the student in first grade, called Auburn. Auburn solve the first task , “ 16 + 9 “ used formal standart additon or column format as in the text book. And she answer 16 + 9 = 15. But when cobb used cookies as context, she answer 25.

This surprisingly, because we now that Auburn still confuse with formal standart addition in the text book, but she can calculate easily when used cookies context.

Based on the Auburn condition, is important to us as mathematics teacher to make visual or manipulative to facilitate students. Manipulatives may be used to help students to express their own thinking. The use of manipulatives will then be cast in terms of scaffolding & communicating. We may take the so called arithmetic rack as an example (Treffers, 1990). And important to help students to construct mathematics knowledge in a bottom-up manner connecting with what the students are familiar with.

To sum up, mathematics should start and stay within common sense', by trying to

foster the growth of what is common sense for the students. In such an approach, tactile and visual models will not be used to make the students “see” the abstract mathematics, instead, material and visual representations may be used by the students as means of scaffolding and communicating their own ideas.

Reference

Treffers, A. (1990). Rekenen tot twintig met het rekenrek (addition and subtraction up to twenty with the arithmetic rack). Willem Bartjens, 10 (1), 35-45.

 

Read More

Solving Problems with The Percentage Bar

Solving Problems with The Percentage Bar

By ; Frans van Galen and Dolly van Eerde

Many students in junior school know about percentage, but yet they often struggle with percentage problems. When they got this problem :

On a bike that normally cost €600 you get a discount of 15%. What do you have to pay?

Surprisingly for 14 tested students, only some students who solve that question correctly. The researcher found that 8 of 14 students written 600 divided by15, the original price divided by the discount percentage. This problem seems to be a standard percentage problem, so how can it be that children in 7 grade still struggle with such a simple problem?

Based on that problem, the researcher tried to introduce percentage bar to solve that problem. When students try to solve percentage problems, drawing a percentage bar has several advantages (van den Heuvel, 2003; van Galen et al,2008; Rianasari et al,2012). First, students can make representation of themselves of the relations between what is given and what is asked. 

Second, the percentage bar offers scrap paper for the intermediate steps in the calculation process. 

 

And the third advantage is that percentage bar offer a natural entry to calculating via 1%.    

From all the figure, showed that percentage bar more meaningful because it gives clear and concrete picture of the relations between the total and its parts.

Reference

Janssen, J., F. Van der Schoot & B. Hemker (2005). Balans [32] van het Reken Wiskundeonderwijs aan het Einde ven de Basisschool 4; Uitkomsten van de Vierde Peiling in 2004. Arnhem: Cito.

Rianasari, V.F., I. K. Budasaya & S. M. Patahuddin (2012). Supporting Students’ Understanding of Precentage, IndoMS. Journal on Mathematics Education, 3(1), 29-40.

Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in Realistic Mathematics Education: An example from a longitudinal trajectory on Precentage. Educational Studies in Mathematics, 54(1), 9-35.

Read More