Entry #5

There are many advantages that can result in mathematic classrooms when teachers do not tell their students the right procedures or answers. One advantage is that students are encouraged to think deeply about mathematical concepts which results in a deeper understanding. For example in Warrington's paper she provides us with an instance where a child who realized all parts of 4 2/5 needed to get divided by 1/3 in the problem 4 2/5 divided by 1/3 . The child was able to come up with the correct answer through reasoning and thinking rather than relying on an algorithm. A second advantage of this method of teaching is that students are used to relying on themselves for solutions and thus a typical "I don't know how to do this" does not surface as often. Students learn to develop intellectual autonomy. Warrington showed that even when she moved from 4 divided by 2 to more difficult problems such as 1 divided by 1/3, 1 divided by 2/3, 1/3 divided by 3 etc. that the students did not surrender their thinking, but rather stretched it to find an answer. A third advantage is that students learn how to interact with each other and how to critically think about problems, rather than just listening and memorizing what the teacher tells them. Referring back to 4 2/5 divided by 1/3 in Warrington's paper, we see that the students reasoned with each other and listened to other ideas that would test their own in order to reach a consensus.

While there are many advantages for this style of teaching, disadvantages are also present. One down side is that if students aren't ever told what the right procedure or answer is, they may develop some errors in their thinking as did Benny. Another disadvantage is that some students will understand and develop ideas faster than others, which could mean that not all students will develop intellectual autonomy. The slower students may just rely on others and could even get more confused than if the teacher were to tell them the right answers due to all the different ideas being discussed. In Warrington's paper she says that some students thought 1 divided by 2/3 was 6, but when other students had ideas to disprove this, they had to modify their thinking in deciding 6 was incorrect. However, it could be the case that the students just accepted that 6 was wrong when challenged and went along with the answer of 3/2 without really thinking about why. Some students aren't as apt to speak up, ask questions and share their thinking, which is essential for this style of teaching to be successful. As a result, many students may get lost. Students are often looking for feedback and if the teacher never tells them they are right, it may be hard for them to have the confidence to use their ideas when learning new material. Students may also never find the quick algorithms to help them and could spend lots of thinking time over dividing fractions when that is just one step in obtaining a different end result.

Entry #4

Von Glasersfeld talks about constructing knowledge, which means that everything we experience is interpreted through a lens of things we already know. In other words, knowledge is subjective. There isn't just one form of knowledge that we acquire or gain. In fact, no one can even be sure what knowledge really is, and thus the definition of knowledge becomes a theory in and of itself. The things we learn about and the experiences that we have are stored in our minds a little bit differently than the next person. Even when it seems as though people think about something in the same way, there is no foolproof method to ensure that knowledge is being processed the same way in peoples' brains. Since what we learn is tainted by experiences we have personally had, knowledge is something that is constructed. This causes what is "true" or "correct" to be obscure, which in turn implies that knowledge is viable as long as it does not clash with our experiences.

The principle of constructivism can be implemented in the way we teach. From a teacher's perspective would come a realization that even when it seems as though students understand a concept in the same way that they do, it is really not that obvious. The output students give when assessed may look and sound like the teacher's understanding. However, the students may be interpreting concepts in relation to different experiences, which could lead to an understanding that is not exactly how the teacher wanted them to think about it. Thus, as a teacher, I would try and assess the students in many different ways. I would have them justify and express their thinking by writing in journals, talking with one another, writing on the board, drawing pictures, creating projects, using technology etc. that would help me to grasp how students are thinking. This would also be a means of creating experiences for the students that they can latch onto and associate with the material that is taught. I think it would be useful to ask specific questions that would probe students' thoughts beyond the surface in order to really see how they understand a concept. I would take this approach from the constructivist standpoint in which I realize students are connecting and interpreting concepts based on a variety of experiences, meaning they will probably need different ways to best learn and to best display their diverse ways of thinking.