Children and Fractions

In her paper, Warrington discusses some advantages to teaching students in a manner where they learn the material based on their own understanding and not a set of algorithms. The students were willing to solve problems using their prior knowledge and understanding as Warrington illustrates when she first put 1 divided by 1/3 on the board. This kind of teaching also encourages discussion in the classroom. Students debate with one another to discover the error of their reasoning and to come to the correct answer. Warrington's students believed that 1/(2/3) was either 6 or 3/2 and thus discussed the answer until they arrived at the correct answer. One girl solved a problem by doubling the original quantities. Warrington argues that this kind of thinking is not present in students that are told what to do. One student illustrates the autonomy that can be developed when she disagrees with all her peers answers and stands alone.

Although Warrington discusses many advantages to teaching in this manner, she does not discuss the disadvantages. As we saw earlier with Benny, some students who are left on their own to discover the principles do not learn correct principles. This will stunt their ability to learn deeper mathematics later on in life. Also, some students in the classroom will rely on the same students to do the problems and will not be learning at all. When it comes time to take a test they won't know how to proceed. I think it is important in mathematics teaching to teach with both relational and instrumental understanding. That way students know why something works and how it works.

Constructivism

In von Glasersfeld's article about constructivism, he argues how people "construct" knowledge through their experiences. People do not acquire knowledge in a classroom or through lectures. As we live each day, we go through different experiences. From these experiences, we draw conclusions about different subject matters. One does not learn to shoot a basket without having the experience of standing before the basket with a ball and learning the proper technique to shoot the ball. After one has experienced how it feels to stand correctly and shoot, one constructs a knowledge of how to shoot. One also learns how not to shoot, and this also becomes part of the constructed knowledge. Everyone will learn different things from the same or similar situation. Not everyone will draw the same conclusions from their experiences.

In mathematics education, it is important to remember that everyone learns different things from the same situation. When a teacher defines a word or teaches a principle, not every student will understand the word or principle in the same way. Mathematics, as von Glasersfeld states with constructivism, is an individual learning experience. To teach my students, I would instruct them on how to do a certain problem, teaching them why it works and how it works, and then allow them time to do many example problems so they can construct their own understanding of the principle. To ensure they did the problems right, I would review their work thoroughly and give them feedback.