Warrington Paper

There are a many advantages to Warrington's take on teaching division of fractions to her students. One is that it develops a relational understanding before it even introduces an instrumental understanding. The students never attempted to "flip and multiply" because that wouldn't have made sense to them yet. Another is that the students estimated before they actually tried to get the right answer so they new an approximate range that their real answer should be in. This also shows that they were developing a relational understanding. Another plus to this technique is that Warrington didn't tell her students if they were right or wrong and let them lead the discussion in class, whether it was in small groups or the whole class. This allows students to feel safe in the classroom and let them share their ideas without the fear of being wrong because no one actually knows if they're right or wrong.

There are disadvantages to this technique as well. One is the fact that it takes a lot of time to go through this kind of discovery process with every kind of mathematics that you come across. In some cases, it would be so much faster to slap an algorithm on the board and call it good. Another disadvantage is that if there's one student who understands right away, they kind of inhibit other students' learning because they are not given the time to figure it out completely on their own. I know if classes that I've been in that sometimes I just wait around for one of the "smart kids" to bust out an answer, sometimes because I have no clue what's going on but sometimes because I'm just lazy.

Constructivism

According to von Glasersfeld, constructivism is the way that we learn. He suggests that all of our learning and our ideas come from past experiences. Our reasoning is not questioned, regardless of what we are taught in lectures, until we have some real life experience that makes us change. Until our conclusions are questioned, they remain correct in our minds. Von Glasersfeld uses the word "construct" rather than "acquire" or "gain" knowledge because knowledge is not something that someone else can hand to us. Knowledge is something that we have to build and modify all on our own.

Similar to what we did in class with the fraction blocks, I think a way that I would bring constructivism into my classroom would be through visual explanations. Concepts and principles don't always make sense in lecture form. If I used hands on experiences, such as the blocks we used in class, the physical aspect of the exercise would shatter any ill-conceived principles or rules that a student might have constructed on their own.