Blog Entry #5

Two advantages that result from teaching mathematics without revealing specific procedures to students are that students will have a better understanding of why a problem is done a certain way, how to explain it, and more social interaction with/learning from peers. When a student must develop their own procedure for solving a problem, they will understand better why it is done that way. This is illustrated in the article by Warrington as the students explain that they know one divided by one-third equals three. They explain that 3 is the answer not because you invert and multiply the 1/3, but because 1/3 goes into 1 three times. Students taught in this method can explain a procedure and use mathematical jargon quite proficiently as well. This is exemplified by the children in Warringtons classroom as they were asked several times to explain what they had done, and they were able to explain their procedures in a way that the teacher and other students could understand. A third advantage for this method of teaching is that students interact more and share ideas with each other. In Warrington's classroom the students would converse with one another as they tried to form solutions to problems, they would build off of each others ideas and explain to one another how they got their answer. This would provide the opportunity for more students to understand as they talked together.
Some disadvantages for teaching mathematics without revealing specific procedures to students, and possibly not even revealing the right answers, include a possiblitiy for students to formulate incorrect procedures, and possible slower advancement through different mathematical subjects. When students are not given a procedure for solving a problem, but must solve their own, it is quite possible for them to formulate incorrect procedures in the process. An example of this with Warringtons students occured when they approached the problem four and two-fifths divided by one-third. Most all of the students came to the wrong answer because they did not completely understand how to solve the problem and the procedure they created to solve it was slightly incorrect. It also takes students longer to advance through mathematical subects if they must formulate their own mathematical procedures because they have to think more and at a deeper level about how to find a solution. Because of this, it is worrisome that students will not be able to learn all that they need to learn in the amount of time that they need to learn it.

Blog Entry #4

In "Learning as a Constructive Activity" von Glasserfeld describes how we come to know what we know through constructivism. von Glasserfield defines construct knowledge as a process of how we come to know things through experience. The term 'constructing' is used rather than 'acquire' because our knowledge is built through reasoning and experience, not simply gained or acquired through hearing it. As in the view of constructivism, we all gain knowledge through our own experience, von Glasserfeld explains that because of this our own personal knowledge is more of a theory because we do not know if what we know about a subject is the same as what someone else knows. This all can come into play with how teachers teach mathematics today.
As according to constructivism, if we gain knowledge through our experiences, then students must be actively involved in the learning process. If they will not learn simply through hearing, they must have more doing and practicing with mathematics so that they can gain more experience and reasoning with the material. As a teacher I would do this by having more in class assignments where the students can work together while practicing a concept and reasoning together about what is being taught. I would also make it possible for them to ask me questions about their homework assignments so that their reasoning and learning could be guided. Through these ways I would hope they could have more meaningful experiences with mathematics material.