Blog Entry #3

Throughout the paper, Erlwanger explored the idea of independent study and showed how it is not a successful way of instruction. He investigated independent study through the system IPI, Independent Program Instruction, which consists of reading, practicing problems, taking quizzes and moving onto new material. Erlwanger’s particular test subject is one of IPI’s top students, Benny, whose mathematical schooling has been taught through this system. It is clear that the IPI system emphasizes Benny’s individualism and self-reliance, but IPI simply does not successfully teach Benny math. As Erlwanger talks with Benny, Erlwanger soon realizes that Benny’s perception of math is skewed. Benny over the course of years has formed many incorrect ideas about mathematical rules and procedures that he truly believes to be correct. Although there is a teacher, there is minimal interaction between Benny and his teacher because the teacher’s primary role is to be a grader that enforces the exact answers according to the key. Therefore, it becomes apparent that as a result of independent study, Benny has concluded many misconceptions and has actually suffered from not having a student-teacher relationship.

The idea of having a teacher-student relationship is applicable today. Without a teacher-student relationship, the student misses out on many opportunities for learning. The teacher plays a key role in the classroom by emphasizing certain ideas, directing discussions, preparing tasks, etc. As a result of independent study, the student simply does not have the chance to learn mathematics in a variety of ways that is possible with a teacher’s direction. While talking with other students about their favorite math classes, many accredited the liking to the teacher. Therefore, I am led to believe that the teacher has a very influential role in the classroom, and without a teacher-student relationship, the students’ success in mathematics will suffer such as in Benny’s case.

Blog Entry #2

According to Skemp, both relational and instrumental understanding can be beneficial or disadvantageous depending on the situation. Relational understanding is defined as knowing both the why and how behind a problem; whereas, instrumental understanding simply addresses the how and relies either on the teacher’s why/reasoning or does do even address the why. As a result, instrumental relies heavily upon memorization or learning of rules and procedures. These rules and procedures are also part of the relational understanding, but the student also learns the why or the reasoning behind using and formulating these rules and procedures. Therefore, relational understanding encompasses instrumental understanding. Consequently, Skemp favors relational understanding; however, he acknowledges there are benefits and downsides of each type of understanding. For example, since instrumental focuses on the rules and procedures it is more simplistic. However, at the same time it becomes complex quite quickly because the students need to memorize numerous rules; often this memorization results in students giving up on and disliking the subject. As a result, relational seems beneficial because less memory work is involved, so students usually have a more pleasurable experience and continue studying voluntarily. Additionally, relational understanding allows for adaptation and flexibility for the student to apply concepts to new problems. Not only may students apply concepts to new problems, but they may also search for and explore new areas. While relational understanding appears superior, there are times when relational understanding is too difficult for a topic or takes too long to achieve. As a result, the more basic instrumental understanding is preferable. Through instrumental understanding, students can more quickly get answers, which produce a greater sense of success and accomplishment. Overall, there are benefits and downsides to both relational and instrumental understanding and each type can be preferable depending on the situation.

Blog Entry #1

Mathematics is the study of numbers and how they relate. These relations are often applied to real life situations and can be expressed in a variety of ways.
I learn the best by being taught the concepts and subject matter first, and then doing practice problems by myself or with minimal assistance. After doing the practice problems, I then like to review the problems with classmates and receive help from the teacher if necessary. I think I learn mathematics best in this way because it is how I have been taught for a majority of my life, and it has worked for me. Also it combines a variety of methods, so I learned the material all the better.
I think my students will learn best in a similar manner. I recognize that students learn in different ways, so I will take that into consideration when I teach. However, I will be most comfortable teaching in a similar manner to how I have been taught and that way includes a variety of learning tactics, so it will probably be most successful in teaching mathematics.
One practice that promotes students’ learning is teaching with a positive attitude and being willing to answer students’ questions. If students feel their opinion and difficulties matter, then they will usually desire to learn. Additionally, I believe that using different tactics such as group work in addition to individual work provides students with the opportunity to learn in different ways, which helps students to better understand the subject matter.
Being rushed and using a negative tone is detrimental to the students’ learning of mathematics. These practices make the students feel the material is more important than the student, and as a result, the teacher does not care about the students and their needs.