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.
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I agree with what you said about students becoming self-reliant in terms of solving problems,this method does force the students into understanding concepts more efficiently.I would think that this method would help solve the problem of students that are afraid to participate getting left behind because it forces them to discuss concepts with their peers. But that is just my opinion. I really agree with the majority of your blog, love your specific examples from the article.
ReplyDeleteI agree with your advantages and disadvantages. I found your disadvantages interesting! i never thought of a lot of them! Like it could be a time constraint for the students in the future. I personally think there needs to be a balance for the students. they need to think constructively and try to figure stuff out on their own but i think they also need to know the algorithms.
ReplyDeleteYou did a great job organizing your blog and giving specific advantages and disadvantages with reasons to back them up.
ReplyDeleteI liked that you brought up that some students are more apt to talk and share their thinking than others. This is true. However, I think that in this classroom, those students are more likely to speak up or ask their neighbor, than they would in a different classroom. Also, having other students explain to them will help all the students involved solidify their understanding.
I agree with the disadvantage of if not being guided and told what they are either doing the problem and methods correctly or incorrectly can greate problems and casue the students to doudt their thinking.
ReplyDeleteI wonder if students reasoning with eachother is an advantage. With no guidance from the instructor their reasonings may be inncorrect.