Assignment #3

S. H. Erlwanger's paper Benny's Conception of Rules and Answers in IPI Mathematics has many main points that are important to learning mathematics and that are pertinent to the way Benny himself learned math. Yet, the most important idea I feel Erlwanger keeps throughout his paper is the idea of teaching the "how's and why's" to concepts in mathematics. This is what relational understanding is. Through the IPI system, children are learning the processes of different concepts and are just learning how to do or accomplish specific problems given them. This is instrumental learning. This style of learning only teaches children the surface of mathematics. In Benny's experience, because he didn't understand why he was doing these certain processes, he created his own rules and reasons for accomplishing problems the way given him. In examples throughout Erlwanger's paper, he showed how detrimental this can be to a child's life. They begin creating ideas and rules on their own, and don't truly understand the reasons why they are going about mathematics and that specific way.

This main importance throughout Erlwanger's paper of learning relationally instead of instrumentally is still quite valid in today's society. It is still very important for children in the long-run to learn relationally the different concepts in mathematics. The children need to learn why they are using each process and how specifically it relates to the different problems, as well as problems OUTSIDE of what they are doing. This idea is very important in the world today. In learning more than just the process, the child can then connect what they are doing to problems outside the classroom. They will be able to use what they've learned in everyday problems and also be able to connect the different concepts they've learned to past and future education. They will understand how prior knowledge is pertinent to the concepts they are learning in the classroom today, and also remember the processes more in order to apply them to math they learn in the future. Learning relationally, as Erlwanger kept as a main idea throughout his paper, is still as important and valid today as it was when he was researching and studying Benny, a child who didn't have the best mathematics learning experience to help him throughout the rest of his life.

Assignment #2

In the article Relational Understanding and Instrumental Understanding by Richard R. Skemp, two different types of understanding in mathematics were described. Both of these are important and overlap in their description, yet they both have different characteristics that give them advantages and disadvantages. The first style of understanding is instrumental understanding. This is a type where people learn the basic laws of math and possibly the concept of accomplishing a task. They learn the basic steps and memorize things in order to accomplish what is necessary. The other type of understanding in math is called relational understanding. This style overlaps instrumental by people learning laws, concepts and steps on accomplishing a task. Yet, this style is different in the effect that people also learn how to accomplish it, what to do, and why they are using these ideas or this new understanding. Both of these different understandings are very important and have their pros and cons. A disadvantage of instrumental is that people are just learning the basics. They aren't learning why they are using the procedure described, or how someone came up with it in the beginning. This becomes an advantage to someone who learns by relational understanding. As stated earlier, this person can know how and why it is necessary to use a certain procedure for the learned concept. Yet, a disadvantage to relational understanding is that, for many people, it takes time. Learning why and how to apply the idea learned can take quite a while for someone to understand: time they could be using in practicing the task. The advantage to instrumental then takes place of this disadvantage by learning quickly and not taking the time to understand the application. Both of these understandings, though having advantages as well as disadvantages, are important for people and their understanding, yet are just different ways of learning.

Assignment #1

Mathematics. A word that, to many people, means something different. Elementary addition? Middle school algebra? High School calculus? Or how about long story problems? Or real-life measurements and areas? Math is something that is used in everyday life. So therefore each person needs to find their own style of "mathematics". Yet, to me? Math is in and around EVERYTHING.

Mathematics, as I said, can be the simple multiplication used in math classes. Yet, it can also be the dimensions of an object, a room, or a building. Math can also be the weight of a fruit and the mileage to my hometown. Or how about the simple numbers used in math? Counting the money one has, the number of songs someone has in their iTunes, or even the number of friends they have in Facebook. Mathematics is a HUGE concept. It is used everywhere. Whether people realize it or not.

How do I learn math best? This can be quite a difficult question. Yet, I feel the easiest way to learn math is by applying it to my life and the things I do. In explaining a concept, I've always found it easiest to understand when someone describes it by things I enjoy doing: sports, music, etc. This is how I feel my students will learn best. I feel a teacher should take the time to get to know their students and the things they enjoy participating in. They can then describe mathematics in a way their mind works, whether that be through football plays, music or dance beats, or even by using hands-on examples.

One of the things I find math teachers are getting better at is the different ways they are teaching. Many teachers are now understanding that students ALL learn in different ways and therefore use different styles to teach the same concept. My high school calculus teacher was amazing. He knew so many ways to describe the same concept. No one was stumped in the class. If you didn't understand it the first time, in coming to him for help, my teacher could describe it in just the perfect manner for your mind to wrap around the principle.

Some of the small and simple things teachers do or don't do, are actually extremely detrimental to students' learning. One of the things I feel math teachers don't do enough, is like I stated earlier, simply getting to know their students. One of the best teachers I ever had genuinely cared about his students. He wanted us to do well and cared for each of our well-beings. He therefore did all he could to help us use or skills, live up to our potential, and then receive the grade we deserved. If teachers simply got to know the characteristics of their students, the activities they find joy in, and the life they live, I feel students would feel more willing to learn and look to the teacher for help.