Assignment #5

In her article, Warrington mentions many advantages of allowing children to figure things out on their own, such as children feeling a greater ability to exchange ideas and giving children the capacity to assign greater meaning to numeric problems by creating story problems for them, the biggest advantage put forward is the fact that children, if placed in the correct setting, "can construct knowledge about sophisticated and abstract concepts in mathematics without algorithms." In other words, children in Warrington's class were able to solve quite complex problems, such as 4 2/5 divided by 1/3, in their minds without the use of the 'invert and multiply' rule. Warrington makes the claim that because the students had been actively involved in the discovery process they were able to solve math problems faster and better than students who learn just the algorithms themselves. This ability for children to learn how to solve problems without algorithmic methods is a valuable skill for students to obtain because of the fact that it allows them to delve deeper into the world of mathematics.

The process of constructing knowledge on their own, children truly can learn many things about math. However, there are a few problems with this method of learning. The first is the fact that if a teacher doesn't know the correct way to approach this method of teaching, how can they be expected to "provide learning environments that allow children to be successful"? Another problem is that it would be difficult for a teacher to ensure that every student is learning the necessary skills to allow them to progress on to the next 'step' in the mathematical process. Math is all about utilizing past knowledge to delve deeper into mathematical solutions, so if a child doesn't understand a building block of math, they will find it difficult to advance in math. While I do see the advantages of this style of learning, I do think that it might not be as wonderful as it is a first look.

Assignment #4

In von Glasersfeld's paper on constructivism, rather than making the claim that we acquire or obtain knowledge, he states that we "construct" knowledge through our experiences in life. His use of the word construct helps us see what he is meaning. Just as a construction worker uses a blueprint, or knowledge gained beforehand, to build a house, so do we 'construct' our view of the world through experiences we have had in the past. Continuing the analogy, one does not build a house starting with the roof. A solid foundation is necessary for the walls, doors, windows, support beams, and roof to stand. So it is with our knowledge, we only know the things that we know because of the foundations that we've had. Also, just as a worker has his own 'best' method of construction, so too do we form our own 'best' ideas of what the difference between truth and error are. And, just like the construction worker, we don't change our ideas of truth until we're presented with something that provides a clear counter-example to what we believe. Thus, von Glasersfeld is making the argument that the only way we know things is through experience; hence, it's impossible for us to know the 'truth.'

One important implication that constructivism has on mathematics is how it points out the fact that just because I view a topic one way does not mean that everyone views it the same way. I know that as I have tutored, subjects that I have found to be quite simple to understand, such as rules of exponents, others have really struggled with. After reading this article I've come to learn the importance of trying to see through my student's eyes. Trying to really see where they're coming from and what 'baggage', whether good or bad, they're bringing with them to the proverbial table. As I try to learn more about how people view various aspects of math, I can then modify my teaching to fit into their world views and ideas. By doing this, I can help them create a more accurate view of math and how it works.