#7

Schultz, K. T. (2009). Soft drinks, mind reading, and number theory. Mathematics Teacher, 103(4), 278-283.

In this article, Schultz describes the experience he had giving his students an opportunity to experience proofs as more than just verification of what they already know. The students in his class were amazed by a mind reading game that used arithmetic of multiple digit numbers to correctly guess a selected number. Having figured out for himself the math behind the magic, Schultz used the curiosity of the students as an opportunity to teach them a different aspect of proofs, using them to find answers and gain understanding. With a little guidance the students were able to figure out the mind reading trick and then proceeded to prove the process to further understand the process. They learned what properly constitutes an accurate mathematical proof and the lesson demonstrated an example of the necessity of a proof naturally arising to gain understanding.

I agree that learning of proofs as a source to gain understanding is very important and useful. The students curiosity seemed to really motivate them to find the answer they were looking for. I am very impressed with Schultz ability to recognize a golden opportunity to get a confusing concept through to his students and to pull through.

#6

Switzer, J. M. (2010). Bridging the math map. Mathematics Teaching in the Middle School, 15(7), 400-405.

Switzer talked about Bridging the Math Gap between elementary, middle, and high school math by building off of what students have already learned. In order to do this, teachers must know what and how their students have learned in previous schooling. This is necessary for students to be able to make connections between concepts from different math classes. Hence, middle school and high school teachers must understand what knowledge their students are starting with and build from there so students are able to see how to use what they already know.

I agree with Switzer's ideas and, as a student, I understand that learning based on information already learned is encouraging and makes new material much easier to understand. I just wonder what a teacher should do if they do not think that the way their students have been taught is the most effective way. I guess, they should determine whether or not it would benefit the students to switch or try to correct what the students have learned to what they believe is the better way.

# 5

In Mary Ann Warrington's paper about teaching division of fractions describes many advantages of teaching constructively. Teaching constructively, requiring the students to figure out mathematical procedures rather than simply teaching them, fosters inventive thinking to find the answer. This was demonstrated by the student who unintentionally used the invert and multiply rule. Another advantage she mentioned was the intellectual autonomy demonstrated by the students. When all of the students had agreed on an answer for a difficult problem, one girl refused to accept that answer because she did not think, or understand how, it could be correct.
Despite the benefits mentioned by Mrs. Warrington, there are also some disadvantages. A class room where everyone understands exactly what is going on is hard to imagine, probably because everyone has been that confused person at some point. For this reason, it seems likely that some of the students in the class room were just listening to other students and going by what they say rather than think about it for themselves. They may not understand and may be given the answers by the other students, which defeats the purpose of teaching constructively.

#4

In von Glasersfeld's theory of constructivism he uses the term constructing knowledge rather than acquiring knowledge. He does this because a student is not given knowledge but rather builds on what he or she already knows. Everything we know is constructed from things we observe and experience in life. For this reason one can not simply acquire information but must be able to relate it to something that they already have concluded to be true.

This is important for teachers today because teachers need to be able to teach in such a way that students can construct knowledge properly. One way to do this would be to relate new math concepts to past concepts that have already been mastered. This will allow students to be able to build on concepts that they have already found to be true. This is a constructivism way of teaching because it promotes constructing knowledge from interpretations that we have already made from experience.

#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.