Blog # 7

Switzer, JM. (March 2010). Bridging the math gap. Mathematics Teaching In Middle School. vol 15(7), 400-405.



Switzer gave an argument for the need to know the content and the processes in which elementary schools teach math, in order to better teach students in middle schools and high schools. Switzer wrote about how a teacher called him and asked about these algorithms they were teaching in elementary school that she had not seen before. This lead Switzer to learn about the types of instruction elementary school students received compared to middle school high school students in his district. Switzer wanted to show that it is important for middle school teachers to know what is being taught in elementary schools so that the student's math knowledge can be built upon. Switzer gave the example of the partial product algorithm to multiply integers. This method is an alternative to the usual method of multiplying integers but it allows connection to be made with distribution in multiplication and the placement value of integers. Switzer stated that teacher that know what was previously taught to their students are better able to help them make connection from past material to new material in mathematics and he encouraged teachers and school districts to better communicate their math teaching processes.

Switzer gave a great argument about why it is important for teachers to know what was previously taught to their students, but did not give enough examples. he gave the partial product algorithm and went into great lengths about how it can help students make important connection to deeper mathematical concepts, but this is the only example he gave. I agree with his conclusion that teachers need to learn what elementary schools are teaching, especially since children are learning math faster and at an earlier age now then they did 20 years ago. It would be easy for teachers to get a couple of manuals that give them the idea of what is being taught at lower levels in mathematics. I know that it really helps me when teachers can show me, based on what I have already learned, a new math concept. The only critique I have to the article is that it didn't have enough examples of different algorithms that were being taught in elementary schools.

Blog #5

There are many advantages in Warrington's method of teaching math. One of the advantages is that students learn to associate the problem with real life situation. For example when the students gave examples of the a candy bar being divided or a pie shared among friends. This method of teaching makes students use what they already know to solve problems. Another advantage is that students learn to make logical leaps of understanding on there own. for example, when one of Warrington's students used equivalent ratios to make a problem easier to solve. This is something that will help students for the rest of there life when they are not dependent on a teacher to solve the problem for them.

I also think that there are many disadvantages as well to Warrington's teaching style. One of those disadvantages is that many of the students could be riding on the coat tails of the ones that really understand the topic. Meaning, several students who were struggling may have nodded there heads in agreement with the students who were passionate about their answers, without really understanding it for themselves. Also it is clear from the article that these students already understood fraction and many other mathematical concepts. I don't think this style of teaching would work for all subjects. It would be difficult to introduce new material and not let the students see patterns of how it works.

Blog #4

Glaserfeld had many reason for the theory of constructing knowledge. One of those reasons is that Glaserfeld used is that all the so called knowledge that we gain comes from our senses. Everyone interprets these senses differently and therefor we all construct our own ideas about what we have experienced based on our senses. If we were all acquired knowledge rather than constructing it. Then we would all have the same knowledge because it would be like the knowledge was handed out. If we were all handed a dollar then we would all have a dollar, but if we had to make a dollar then all of our dollars would be different. This is the theory behind Glaserfelds constructivism. That fact that we are all constructing knowledge makes knowledge itself become a paradox. For how can we know what is true knowledge if we are each building it up as we go based on our senses and past experiences. Glaserfeld mentioned that to truly view knowledge unbiasedly, we would need to know what we knew before we knew it. Which is of course impossible.

If I were teaching a math class and I believed in constructivism and I wanted to best help my students learn. I would give a test at the beginning of the year or at the beginning of the term that would help me understand the level that each of my students was at. This test would allow me to see the students work out problems. This way I could see how my student solve problems and get an idea of how they have built their mathematical knowledge. This test would also allow me to see my students past understanding and experiences with math. Since constructivism is always building up on past knowledge. I would be able to help my students by knowing where I should start building.

Blog #3

Erlwanger's main point in the article about Benny was that IPI was failing to teach Benny correct math principles and not only that was impeding Benny's future understanding and enjoyment of mathematics. In the opening paragraphs Erlwanger emphasizes the weakness that are in IPI. He talks about the how IPI weakness stems from how it is taught. Elwanger goes on in the article to mention more then once that in the IPI system the teacher is removed from the his or her role as a guide to students leaning mathematics. Erlwanger feels this is a mistake and almost puts the student and teacher at opposition. The student like Benny gets frustrated at the teacher for only ever following a key of answers. Erlwanger also mentions that Benny learned incorrect math principles from going through the IPI program. Benny made his own reasoning for how math works because there was nobody in the IPI program to tell him differently. As long as his answers were right, he got to move on in the program. Erlwanger felt that the IPI program was important step in education understanding but ultimately fails to teach children mathematics.

One of the most important things Erlwanger taught was importance of the teacher. The IPI program was designed to remove the teacher from the teaching process, and Erlwanger showed that this was a mistake. This is very applicable today because teachers have a huge influence on students attitudes toured subjects, especially math. In my experience, I used to really like chemistry, but I had a horrible teacher a few years ago and it completely ruined the whole subject for me. I still have a bitter taste in my mouth from that class and its subject. On the other hand. I have had great teachers in math that have helped me see the beauty of mathematics.

Blog #2

There are many things that unify and separate Skemp's idea's for relational and instrumental understanding. One of Skemps main ideas about relational understanding is that someone who knows the purpose and the method for the math they are doing. Instrumental is understanding the method, or in other words the process to get the right answer, but without knowing why they are doing it, or why it works. Even though Skemp highly favored relational understanding, he did note that they are not mutually exclusive. To truly understand a principle in mathematics, or to have a relational understanding, one must have the instrumental as well. One must know the method in which to get the answer. Skemp knew that a student must know the method and the purpose to truly have a relational understanding. Some of the main advantages Skemp talked about in regard to relational understanding is that it was easier to remember, it was easier to go from one problem to another, it gave students motivation to learn more. Instrumental understand however, could give more visible positive outcomes. Or in other words, it is easier and faster to see a page full of right answers with instrumental understanding. However, usually if the problems change a little bit, it will be harder for one with instrumental understanding to adapt. Overall there are many reason why a teacher would choose to teach for relational or instrumental understanding. However, Skemp felt that relational understanding would best serve all who learned it.

Blog Entry #1

What is mathematics?



I think mathematics is the study of numbers and relationships between those numbers.



How do I learn mathematics best?



I think I learn mathematics best by examples. I usually need to be shown how to do a math problem before I can understand it. I also learn by repetition, though I probably don't do enough of that. I do not do as well when I have to try and learn math principles out of a book. I do really well when the math problems I am given build upon one another to teach greater principles.



How will my students learn mathematics best?


I think it is important to cover all the learning styles when you teach mathematics. Or at least all the ones that are possible or prudent to the lesson being taught. That way, every student will be able to learn the concepts and not just those students that happen to learn best by a particular style. I also feel it is important to understand how each student learns so that when they ask questions you will be able to tailor your response to that student and help him or her the most.



What are some current practises in school mathematics classrooms that promote students' learning of mathematics?


I feel one of the best current practises is being able to work in groups on math problems. This allows many students to ask questions of each other and learn at a faster rate. I also feel that math games help students learn, because it provides a fun atmosphere and students learn when they are having fun.

What are some current practises in school mathematics classrooms that are detrimental to students' learning mathematics?


I honestly don't know a lot of the current school practices. I do remember that I hated it when a teacher would put the students name on the board who got the highest grades on tests. I felt it led to unhealthy competition and isolationism.