Blog Entry #3

One of the main points Erlwanger was trying to get across in his paper, "Benny's Conception of Rules and Answers in IPI Mathematics," was the necessity of teacher-student relationships. He wrote about the experience he had interviewing Benny, a sixth grader, who had serious issues in understanding the concepts of decimals and fractions. For several years his misconceptions went unnoticed because of the faulty IPI system which forced kids to learn independently with minimal teacher interaction. By learning everything on these particular computer programs, he was taught to focus on the answers rather than the mathematical processes and concepts. With no teacher to tell him what the meaning and reasons were behind the rules and patterns, he was forced to come up with his own.

I feel that his argument is valid today because teacher-student interaction is crucial to students' education. Although programs and individual work can be very helpful and sometimes even necessary, there must be a balance. If teachers actually work with the students they will be able to notice when students are having difficulties grasping certain concepts. Without this relationship, more and more students will end up like Benny, having false rules and meanings of mathematics coded in their heads.

Blog Entry #2

Richard Skemp's article, "Relational Understanding and Instrumental understanding," discusses the two types of understanding that are widely used throughout mathematics. Relational understanding is knowing the reasons behind the arithmetic. When one is given a problem and understands it relationally, then they know both what to do and why they are applying that specific process. One cannot have relational understanding unless they understand the problem instrumentally as well. Instrumental understanding is knowing the rules, arithmetic and equations used for specific math problems. Although students can get answers right by simply knowing the rules and understanding the problem instrumentally, they will not know the reasons in which those rules are applied. Learning instrumentally can be a much faster way to learn how to solve a problem, but in the long run it is much more difficult to maintain because all it is is memorizing rules. In contrast, relational understanding may take longer for students to grasp and extend the time spent on certain subjects. However, if relational understanding is exercised, students will carry that knowledge with them for much longer because they will have a conceptual foundation that they can build off of.

Blog Entry #1

1. Math. Everyone uses it, but how do you define it? I suppose it would be the attempt of man to form patterns from the numbers, structures and space that we use everyday. There are several forms of mathematics and they can be used for a variety of different purposes. Some may study pure mathematics while others choose to apply it as a science with physics and engineering. Whether we are counting money, building a house, or calculating the force that a spring gives off--we all use math.

2-3. I learn math best when I am given examples of specific problems that I will be dealing with. Once an explanation of the concepts are given, I need to work on problems on my own otherwise I won't understand them fully. I believe that my students would benefit from this method as well. Although working in groups can be helpful, I think it should only be done after each student has time to work out the problem in their own way. Every person works through things differently and I believe it is important to give them that opportunity before providing assistance. Once they have done this, then they can either discuss the problems with classmates or ask me questions that they have. I realize that every student learns in their own way and so questions can be answered specifically for different students' mindsets.

4. It is very helpful when math teachers show different approaches to solve problems and give the students helpful methods to remember the material. The math classes that I learned the most in were taught by teachers who were very interactive with the students and were always willing to work through any obstacle we came across.

5. I think the worst thing for math teachers to do is to give a broad description of the material and concepts and then leave the class alone. Although students should have time to work on the problems themselves, it is useless if the material provided wasn't helpful. My younger sister is in high school right now and constantly needs to call me for help because her teacher never teaches them properly. If an instructor doesn't get to know the students and how they learn things easiest, it can be very detrimental to their education.