MthEd 117 Blog #5

In Mary Ann Warrington's paper, "How Children Think about Division with Fractions", she gives examples from her class of students at the Atrium School in Watertown, MA, to illustrate the fact that children can develop their own mathematical formulas and ideas when you give them the opportunity to discuss them within the classroom. Warrington believes strongly that "children construct knowledge on the basis of what they already know," and that the best way to teach them is to let them teach themselves through discussion in the classroom. Warrington does not even tell her class the correct answers, but lets them guide themselves to discovering the concepts themselves. The greatest example she gives is when she asked the class what 4 and 2/5 divided by 1/3 was. Many believed that it was 13 and 1/15, but one student disagreed and brought up the idea of 13 and 1/5. As Warrington states: "This response was not only logical and mathematically correct but also a shining example of the autonomy that children develop when encouraged to think for themselves." Giving children the opportunity to think this much and develop their own mathematical ideas and theories can be quite a wonderful way to allow children to explore the world of mathematics.

However, there are flaws with teaching this way. The first is the time constraint. In public schools, this method of teaching would not work in any way. Warrington had actually been teaching fractions to these students for FIVE months, which, in a public school, is the time given to teach MANY different topics. Second, the issue arises, what if that one student hadn't had those thoughts, and nobody had thought that 4 and 2/5 divided by 1/3 is 13 and 1/15 was wrong? Would Warrington give them the correct answer, or would she let them go on thinking the way they were? Third, what about those who just don't have the ability to think about mathematics in this way. For some, math just does not ever click. And last, Warrington said that "children can and do invent ways to do sophisticated mathematics," but sometimes those ways are just plain wrong. Take Benny for an example. He had many ways of doing math, but only some of them were actually logical and correct. Warrington's method of teaching is a wonderful way of letting children discover the world of mathematics, but it would really only work in a private school institution, and never in a public school with the curriculum they have to cover.

Math Ed 117 Blog 4

In von Glasersfeld's paper, he brings up the idea of "constructing knowledge", in opposition to "acquiring" or "gaining" knowledge. The difference between these two ideas may not seem very apparent at first, but von Glasersfeld makes it very clear that the two concepts of how we receive knowledge are quite different. When we talk about "acquiring" or "gaining" knowledge, it is implied that others who have some knowledge, such as in mathematics, can pass this knowledge on to others, and that is how we "acquire" the knowledge. However, von Glasersfeld believes that we "construct" knowledge in life. We cannot know the TRUE reality of the world around us, because everything we "know" comes from experiences that we have had, and how we have filtered them through our own eyes and minds. As we have these experiences, we begin to construct ideas and theories to how the world around us works; and in our case, how mathematics works. We then only change our ideas and theories when we run into a contradiction with another experience. These experiences can include listening to a lecture, doing homework, and experimenting in the world around us. This idea of "constructing knowledge" is like building a brick wall, one brick at a time; each experience we have is another brick to add to the wall. When a brick doesn't fit with the wall we have already constructed, we either have to break the wall down some so it does fit, or we throw the new brick away. Constructivism then becomes a powerful idea in all aspects of learning, and especially applies to mathematical teaching.
In my own teaching, constructivism would make me teach in a variety of ways. Given enough time, I would take each mathematical concept and help the students experience that concept in multiple ways. Taking ideas from the National Council of Teachers of Mathematics, I would give a small lecture to the students, followed by a hands on activity to illustrate the concept. Next, I would do a visual example either on the board or the overhead, including a graphing application if it was applicable. Finally, I would give the students their assignment early so that if they had any more questions, they could ask me in class before they took their homework home with them. This way, hopefully, students would be able to "construct" a correct knowledge of the concept I was teaching; and any contradictions to the ideas they had would be discovered and fixed within that class period. With this method, and with constructivism in mind, I think this will be an extremely powerful way of teaching mathematics to students.