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.
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You did a good job organizing your blog. I might have put enter between the two paragraphs so it was more clear you were starting a new thought. Also, you said "Two advantages" and then listed three. But you did a great job following the prompt.
ReplyDeleteStudents in today's math classes spend a lot of time reviewing. If students understood things well, this time could be dedicated to learning new things. Also, what they've learned in the past would apply now and they have better reasoning that will help them in all their math courses.
I also like to think that students will discover if their procedures are incorrect. I believe the teacher could give them another problem that would make them question their procedure and revise it.
I agree with your statement that Warrington's style of teaching helps students understand why the procedure works. This will also allow the students to understand harder concepts in the future. I didn't really disagree with anything you said. I guess I would have tried more detail about how students could understand the procedures better.
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