With or Without Algorithms

Warrington presents an interesting paper wherein middle-school aged students from all backgrounds find solutions to given fractions without the aid of any 'tricks' or techniques. And they do so with accuracy and without a lack of speed. Indeed, Warrington's class demonstrates the benefits of non-algorithmic learning in that "responses were forthcoming", and "[children could take] a straight computation problem and assign meaning to it by creating a word problem." Fractions seem to make more sense this way, and they have meaning. I also think that by invoking continual classroom discussion like Warrington has done in her classroom that the children will not be so timid to ask questions in the future and are more willing to other people ideas, other points of view for the same problem thus expounding upon their own knowledge and making it stronger.

In contrast however, students without any set of rules they can refer to result in (more often than not) nonefficient work and prone to occasional failure. Warrington admits that it's a lengthy process when she explains that for the understanding of a single problem (4 2/5 divided by 1/3) "went on for some time, and for many it carried over into recess." Also, without a way to check one's answer like you'd be able to with an algorithm or two, answers are less consistent. The obvious example is that given the same problem stated above, the children in Warrington's classroom all got the answer wrong except for one. So although the children were undoubtedly smart, they didn't have a solid way of doing fractions, only ways they came up with themselves.

Constructing Constructivism

Constructivism is a theory of learning by von Glaserfeld where knowledge is not simply received by the senses but is built up on an individual basis by the learning person. It is also the idea that the world we live in may not be reality, but is merely the creation of the subject's experience and observation; when one finds something to be 'true', it enforces the concept that it is real, however there is no way to prove that it is indeed reality. This is why it is theory rather than fact, indeed Glaserfeld points out his own paradox by saying, "to asses the truth of your knowledge you would have to know what you come to know before you come to know it." So you build knowledge upon experience and accept it as truth, whether it is fact or not.

von Glaserfeld then turns to the teacher and states that, "two things are required for the teacher [to build conceptual knowledge in students]: on the one hand, an adequate idea of where the student is and, on the other, an adequate idea of the destination." What Glaserfeld means by this is that teachers must be cognizant of each student and how their progress in understanding the mathematics is going for it is the student who has to do the conceptualizing and operating not the teacher. So the teacher's job is to generate understanding, rather than train specific performance. Because of this if I were to teach using a constructivist method I would encourage and accept student input and participation. I would also encourage students to inquire about what they are learning by asking thoughtful, open-ended questions and encouraging students to ask questions to each other. By promoting individual involvement and interaction each student will learn their own way and will see the perspectives of others further solidifying the understanding of the concepts.