Blog 7

Maida, P., & Maida, M. (January 2006). How does your doughnut measure up?. Mathematics Teaching in the Middle School, 11 (5), 212-219.

This article was about a middle school teacher who used doughnuts to apply math to real life. The students were asked to estimate the volume of the doughnut by approximating the volume of a cylinder minus the volume of a smaller cylinder from the center. This was a good introduction to calculus and finding exact volumes of different shapes. The children were more interested in the math because it was more hands-on. Each student got their own doughnut and needed to take the measurements themselves. The purpose of the activity to teach the students how to estimate using math and also how to incorporate math into real life situations.

I thought this article was really interesting and I think as a student I would enjoy using math more creatively by doing hands-on activities. It seemed like the students understood what they were doing in relating something they knew how to take the volume of to something more obscurely shaped. The only problem I saw with this activity was the germ factor. I bet it was really unsanitary having students touch all of these doughnuts with different rulers and such.

Blog 6

Switzer, M. J. (2010) Mathematics teaching in the middle school. Bridging the Math Gap, 15 (7), 400-407.

This paper discussed how to be effective in teaching students after they transitioned from elementry school to middle school or middle school to high school. Some ideas the paper presented in order for teachers to be more effective in teaching during the transitory stages of a student's career were knowing different ways to do the same concept and also being able to relate new and old concepts. It is important for teachers to know different ways to do the same concept because students come from previous teachers and schools where many different methods are used to find the same solution, when the different methods are understood, more students can be helped. By relating old and new concepts students get a better understanding of the new concept because they are somewhat familiar with the old. An example the article gave was using tiles to solve multiplication problems and also to solve equations.

I agree with the article, I think it is very important to be aware of the difficulties of transfering to the next level of schooling. This article was effective in allowing teachers to realize what might cause problems for new students and provided ways to avoid these problems. I did wish however, that the article provided more examples of possible problems and solutions. For example if students came from different previous schools, where some had a stronger background on a subject then others, how would a teacher be able to handle the differences in the class?

Blog 5

In this article, Warrington used a teaching method where she had the students rely on their previous knowledge to understand more concepts. Warrington acted simply as the question producer. She would write a question on the board for the students to try to figure out. Then she would allow for discussion and listen to the students converse, asking questions occasionally, but never giving answers. This teaching styles has many advantages, it forces students to think harder about a subject. In order to figure these problems out the students had to have a basic understanding of fractions. I could tell from many of the students' answers that they really did have a good understanding of fractions and I think this learning style helped them think creatively and better understand concepts.
There were some disadvantages too. This learning style was great for students who could figure problems out on their own or who had a solid understanding of the material, but what about those students who were struggling? This approach to learning would not have been beneficial to them. They probably found it difficult to follow what their classmates were discussing and needed extra help from the teacher. Another element of this learning style that I don't understand is, what if the students can't figure the problem out, or what if they figure out the wrong answer and convince the rest of the students to do the same? Will the teacher ever answer questions? I think this method of teaching still has some flaws.

Blog 4

Glasersfeld chose to use the word constructive to describe the learning process because understanding is done through the association of experiences and relating previous knowledge to new knowledge. In this way our learning is constructive because it builds on itself and is different for each person depending on their experiences in life. Under this theory, there is no true knowledge or understanding because there is no way to match what we know with what reality is, everything we learn is filtered through our brain by previous experiences.
Because of these differences in learning, every student has a different understanding of a concept. This could make it difficult to teach students and also to make sure the student understands the concept as intended by the teacher. To make sure a student understands a concept, I think the best way would be to assign a variety of homework problems and ask the student to explain what they think the concept means back to you.

Poor Benny

The main idea of Erlwanger's paper was that the IPI form of mathematics was not effective because it taught math in a way that forced students to work only individually and to react to situations rather than contemplate them. Erlwanger defended this topic by using examples of his interactions with Benny. Benny's explainations showed that he had to determine the rules of mathematics for himself rather than be taught by a trained teacher. Because of this Benny's conception of math was often wrong. Benny's responces also proved that this form of teaching math made students only seek out the answer the key provided rather than think about how to solve the problem rationally.
One part of Erlwanger's paper that I thought was very applicable for today was his opinion of the role of a teacher. According to IPI mathematics, teachers hardly have a role at all, in fact it is some what ironic that they are even called teachers because they don't actually teach the students. IPI mathematics tries to have the students learn individually. Erlwanger emphasized the importance of a teacher in helping children to understand a concept more fully and to assist when needed. I think this idea of teacher participation is very important in a classroom. Students are better able to understand concepts when a teacher is there to explain it to them, or show them what they are doing wrong and how to fix it. I think a lot of people think they are naturally not good at math, while this may be true for some people, I think that the majority of the time this opinion is formed because an individual had a teacher that did not help or encourage them. I think teachers should be more a part of student learning.

Relational vs. Instrumental

Relational understanding and instrumental understanding are both important according to Skemp, but Relational understanding is a better way of understanding because it is more lasting and can be applied to more situations. Both relational and instrumental understand are methods of getting correct answers to questions, the difference is that relational understanding explains not only how to get the right answer, but it also explains why that method works. This type of understanding, according to Skemp, is very important because it is allows students to form a deeper knowledge of a subject, rather than just mechanically solving problems the same way. This form of understanding, relational, is better in enabling students to get correct answers for varying questions, rather than the same problem with different numbers filled in. This is because instrumental understanding only explains how to do a problem. While this has its advantages, they are mostly short-term in that the student is able to get the correct answer quickly. A problem occurs, however, when a different type of question is asked in a given homework assignment, then the student is not able to adapt their knowledge to comprehend the new situation. Skemp believes that both forms of understanding are important and should be taught but instumental understanding can be taught within the more encompassing form of understanding called relational.

MathEd117

Mathematics is the study and interpretation of numbers and how they relate to eachother. Math is very wide encompassing because the study of numbers is so broad. The study of math is often broken down into simpler studies, such as: trigonometry, algebra, and calculus. Everybody learns math in different ways. The best way for me to learn mathematics is to have someone explain the concept to me, then have me try on my own to do problems, asking for help if I need it, then eventually teaching someone else the concept I just learned and applied. My students will learn mathematics best by being able to get individual help after having it explained to the class. I would introduce and explain the new concept to the class, work on some example problems with the class as a whole, then have each student continue to work on problems individually, then allow the students to work in groups as I walked around the room helping those that needed help. I think technology and visuals provided by schools help the students' learning. Often it is visuals that allow people to learn best. A problem within school mathematics, however, is the large size of class rooms. By this I mean the large number of students in each class room. This large size makes it difficult for one teacher to be able to help all the students, making it easy for some students to fall behind. It helps when a teacher has some sort of assistant, but I think individual help is very important. Not all students are confused by the same things. That is why it is so important to talk individually with students and figure out what is confusing to them.