Blog Entry #3

I think Erlwanger's most important point in this paper is that the individualism of the IPI program is the biggest downfall of its mathematical teaching system. By taking the reader on a journey with what Benny has learned about mathematics, Erlwanger exposes how individualistic the program is and how mathematics, if learned properly, demands a true teacher to teach students relational mathematics. After spending 4 years learning mathematics through IPI, Benny had learned on his own to distrust the teacher and the answer key by making up his own (false) rules for mathematical basics, especially when dealing the fractions and decimals. Thus, Erlwanger believes that learning individually is not how mathematics should be taught.

I believe that learning should not be an individual experience, which leads me to argue that this point of Erlwanger is applicable in teaching mathematics today because it shows how important a proper math teacher is in today's society. We, as math teachers, need to be trained well enough to teach students relational mathematics, stopping any kind of individual (false) learning as fast as possible in a child's learning lifetime. So not only is it valid to understand that the value of teachers becomes higher as individual learning becomes disintegrated, but we must use stories like Benny's to re-inspire us to continue our mathematical education to help prevent any other child suffering through an individualized program in mathematics.

Blog Entry #2

Skemp revolutionized math educators' studies of mathematics with his discovery of rational understanding versus instrumental understanding, which has led the mathematics educational world to analyze and apply his findings. He found that many students and teachers clash with their views on what mathematics understanding means. Instrumental understanding is learning the how without the why. In mathematics, this is most commonly demonstrated though formulas with no meaning to the students. Rational understanding means that the students are learning how and why to every part of mathematics. According to Skemp, these two distinct types of understanding in mathematics are actually connected. Rational understanding is instrumental learning plus so much more. If students understand rationally, then they understand everything that instrumental understanders know plus the why to mathematics as well as applicable skills used in more advanced mathematics problems. Even though they are connected, both have advantages and disadvantages explored by Skemp. Instrumental understanding happens faster and simpler, while making it harder for students to remember mathematics longer and for them to apply their knowledge to different types of problems they did not see in class. On the other hand, rational understanding, although it is harder to grasp each concept at first, should carry a student on for a longer period of time, as well as teach them the skills they can apply to more advanced problems without learning more rules like instrumental understanding would require. Overall, Skemp deeply defined and explained both types of learning and left the reader able to make their own judgments and decisions about mathematical understanding.

Blog Entry #1

1. What is mathematics?
Mathematics is a subject in school, in which students learn about numbers and problem solving.

2. How do I learn mathematics best? Explain why you believe this.
I learn mathematics best by reading the book, paying attention especially to the sample problems. I study every step, and then I test myself by doing the practice problems. If I get stuck, I go back to the beginning of the chapter and re-read how to do that part before applying it again to the practice homework problems. This is how I learn mathematics best. I know it is the best way because it has produce successful results all through my mathematics past as well as it allows me to not be personally dependent on anyone else to learn mathematics.

3. How will my students learn mathematics best? Explain why you think this is true.
Even though I learn on my own from the book, I believe my students will learn mathematics the best from what I like to call "dramatic learning." Many students in today's society believe they are not good at math, and I do not think reading from a book will help them to improve. They will need someone who can explain the concepts clearly and in a fun, exciting (dramatic!) way. By developing hands-on methods (excluding having them write problems on the whiteboard!), I believe most students will learn mathematics the best because of the visual and more applicable ways of looking at mathematics.

4. What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? Justify your reasoning.
To be honest I do not know a lot of the current practices of mathematics in the classroom, but I know what my math classes did. Nightly homework promotes learning because the students need to try problems on their own in order to really learn mathematics; they cannot simply watch the teacher do a problem during lecture and then be able to reproduce a similar problem on the test without practicing other like problems. So I think daily practice problems are very helping in learning mathematics.

5. What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? Justify your reasoning.
I think one current practice that I noticed in my middle school and high school years in math classes that was detrimental to the learning of mathematics was too much daily homework. There is no need to do 20 problems that are all alike. 3 to 4 problems will suffice. I feel lots of students became overwhelmed at all the homework every night, and therefore decided that they hated math and gave up on trying. I think practice problems are good but within reason. Too much practice will cause students to not try at all and will hinder their learning of mathematics.