Teaching is Mind Bloggling

When you are trying to solve a problem, it is convenient to have a teacher tell you correct answers, but their are advantages to a teacher not giving his/her students answers at all. For example, in Warrington's paper, she noted that the students never complained that they didn't know how to do the task at hand and they were eager to take risks. That is because the children weren't bogged down with remembering logarithms, they were then free to discover procedures for themselves. Essentially they were able to construct knowledge freely, and on their own. Another advantage is, students are able to build new knowledge from what they already know. For example, the students used their knowledge of dividing whole numbers to learn how to divide fractions.

Though mathematical discourse based on Constructivism beliefs has a lot of advantages, it also has some disadvantages. It is possible that students could make up procedures that are incorrect, like what Benny did. While some students may be able to connect what they are learning to what they have already learned, others may struggle with determining exactly what they are learning. They may just think of the class discourse as a big class discussion where they just talk about theories. It may be hard for some students to know that their method of procuring an answer is sound.

Constructing Knowledge; Not Just Memorization

Constructivism is knowledge "constructed" through our own experiences. We do not "acquire" knowledge, because it is not something that is simply given. Through experience, we form our own representation of the world by using our senses to interpret. As we experience, we continue to modify our view of the world. Knowledge is viable as long as our experiences continue to fit out observation.
As a teacher,I would require my students to do more that retrieve facts, they must be able to apply what they have learned to other situations. Being able to apply "knowledge" (or what they think they know) to other situations will show whether they have really learned something or not.They need to be able to do more than just retrieve a solution to a problem, they need to know the process by which they get that solution. This expectation can be used in mathematics as a result of constructivism, because it has to do with generating something new; like forming new knowledge. Answers are not just simply memorized; they are constructed. More specifically this is called operative knowledge, which is part of constructivism.