Summary of the article:

Gravemeijer, K. (2011). How concrete is concrete. *Indonesian Mathematics Society Journal on Mathematics Education, 2*(1), 1-8.

This might be the answer of the previous problem given in “When nine doughnuts price as much as the other ten”. The real problem is that students sometimes think that mathematics that they learn in school has no relation with what they see in everyday life.

Dealing with this problem, teachers try to provide students with manipulative which perhaps help them connect the math to the real-world context. Occasionally, those manipulative seemed very realistic for teachers, unfortunately not for students. Teachers are usually trapped with their own thinking and forget that their students are in not in their level.

People might assume that this should not be a problem since students, later on, will realize how a concept supposed to work as they continue learning on it. Unluckily, such assumption is very difficult to happen. Otherwise, the students will keep guessing whether they have learned the concept correctly. As the result, they would be math anxious.

**What actions should teachers take, then?**

Fruedenthal (1987) answers “*Mathematics should start and stay within common sense*“. More practically by Gravemeijer (in this article), we begin the lesson with what students know and help them construct the math in a bottom-up manner.

**Bottom-up Teaching**

This way of teaching first demand teachers to identify concept-related things which are familiar with students. Smoothly it shifts upper to semi-abstract until it reaches the mathematical representation of the concept (iceberg model would be best example).

In the process from abstract to concrete, scaffolding is often required. This perhaps bridge students to think of the inter-connectivity of math and the real life.

PS.

#this summary is based on my own understanding toward the article.

#several points might not include here (example-case, further explanation on the bottom up manner on a certain topic).

#Therefore, I really suggest the readers to find the real article.

Freudenthal, H. (1987). Mathematics starting and staying in reality. In I. Wirszup & R. Street (Eds.), Proceedings of the USCMP conference on mathematics education on development in school mathematics education around the world. Reston, VA: National Council of Teachers of Mathematics.

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the article by Koeno Gravemeijer is really awesome,, I really suggest you, readers, to find the original one..

there are some practical information on how teacher should teach math.. also examples of how less good math teaching impact to students..