#3

In Erlwanger's paper about Benny's knowledge of fractions, his most important point was that students must have a relational understanding of math concepts. Benny did not know the rules to fraction arithmetic, such as addition, subtraction, multiplication, and division, so he made up his own rules that where not correct. He was not taught the rules, let alone why they work, so he thought that the rules (which he made up) were random and did not make sense. This was confusing for him and led him to think of arithmetic of fractions as a "wild goose chase."

This is, absolutely, still applicable for math teachers today. Students must be taught not only the rules but why they are necessary to come to the correct solution of a problem. Like we read in Skemp's article, relational understanding is easier to remember because it should make sense. If Benny would have had a teacher that taught him the correct rules and why they work, he would have been much happier mathlete!

#2

According to Richard G. Skemp, there are two methods of learning and teaching mathematics. The first way is for instrumental understanding which is an understanding of how to find a solution but not necessarily why it works. Instrumental mathematics, he says, is easier to understand and provides an answer quickly which is why it is appealing to some students and teachers. However, this method is harder to remember since it is based mostly on memorization and students will have a harder time seeing how things relate because they will not have an understanding of how they got the right answer. The second method involves relational understanding. Teachers who teach relational mathematics teach the how and why (how to get to the answer and why the steps taken give the solution). Relational mathematics includes everything in instrumental mathematics and then some. He says that relational mathematics is easier to remember and more adaptable. I know, from my own experience as a student that having teachers go into depth about why a process gives us the answer we are looking for can leave me feeling very overwhelmed, lost, and not interested. However, understanding why the process taken provides the answer helps when learning future concepts. Either way a teacher decides to teach, there must be moderation. Like most things in life, focusing too much on one thing is never good. Both methods will give the process to reach the correct answer so neither way is wrong. Teachers should determine how to teach most effectively considering the students and material that is being taught.

#1

• Math is finding solutions to problems.

• I learn math best by trying to solve problem and then practicing a lot once I know the correct method. I know this because when I try to solve an equation on my own, even if I don't succeed and have to ask someone, I understand the correct method. When the material given to me without the struggle, I have a harder time understand why and how.

• My students will have different learning styles. Some will learn best by doing the problems, listening to them be explained, or by watching them be solved. As a teacher, I need to teach in a manor so that all students will be able to understand.
  

• In some of my math classes the teacher has had the students do problems on the board which is effective because it forces students to practice and to try to solve the problem without the teacher showing how it is done. I have also had teachers make the students try a problem before he or she explains how and why is it solved. This makes the students try to solve it for themselves which will help them understand whether or not they figure it out.
  

• Methods of teaching that are not successful are long lectures where the students do not participate and aren't required to think about how or why the methods and strategies work.