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.