Blog #3

"Benny's Conception of Rules and Answers in IPI Mathematics" is a journal written by Elrwanger. In this article Elrwanger spends time with Benny, a student of this IPI program. He notices that Benny has misconceptions of rules with fractions and decimals. For example, Benny thought 2/10 was equal to 1.2. Through questioning Benny, Elrwanger found that Benny had made up rules. This reasoning Benny had made pointed out that students need a relational understanding and a strong one from the very beginning. If they don't have this understanding, then they can become lost and make up their own rules. Benny must not have fully understood a concept, so he made up a rule by himself. Benny had been doing this program since from 2nd-6th grade and he had a lot of misconceptions to unlearn and had a lot of new conceptions to learn.

Relational understanding is really key from the beginning, even now. It is hard to change a student's ideas about a concept once they have made up their own rules, and especially if they have been using this rule for many years, like Benny. I feel it is necessary to really get with the students and make sure they fully understand something. I remember my teachers teaching an idea/concept and then we had a worksheet to enforce the concept. To see if we had understood the concept, they would go over our worksheets herself. A lot of my teachers tried hard to make teach relationally because it is easier to come up with concepts, then having to memorize them all. If you don't give students a good base from the beginning, it is going to be really hard for them to build off of misunderstood information.

Blog Post #2

Thinking back on my years at school, I remember learning math different ways. Somethings I learned without knowing why it worked and others gave me real life application. These different teaching methods are discussed in Richard R. Skemp's article called, "Relational Understanding and Instrumental Understanding". As he goes through the article, he explains what Relational and Instrumental mean, what their advantages are, and how they correlate to each other. Relational Understanding is being able to understand and identify the real life application. This could be as simple as understanding why we use a specific formula to find the circumference of a circle and where we got that formula from. The advantages of this are that is is easier to remember, it can motivate the student to want to learn more, and students can may try to understand new material relationally. The disadvantages are that is takes to long to understand, the teacher has to cut out material, and somethings don't need to be understood. In contrast, Instrumental Understanding can be described as rules without reason. We do something, even though we aren't sure why we do it. For example, we multiply width and height to get the area of a rectangle. Some students don't understand why they are doing it, other than the reason that their teacher told them to. The advantages of this understanding are that it is easier to teach and understand, the rewards are immediate, and the students can get the right answers quickly. The disadvantages are they can be over-loaded with information, the teacher cannot tell the understanding of the concepts, and the students only remember the information for the next test and then lose it. Although Instrumental and Relational are different, they are also very similar. While learning Relational Understanding, you also learn about the Instrumental part of it, whether or not you actually understand it. But is not entirely the other way around. A teacher can almost easily teach Instrumental without teaching any Relational. I think that the best way to teach, is to include both ways of teaching. A good mix of both can help the students and not give them an over-load of material or keep them wondering how it really applies.

Writing Assignment #1

Mathematics is an interesting subject. It is so set, yet so mysterious, due to limited information. I think we can define mathematics, as the numerical system that makes the earth work. It provides us with the information on how fast things move, or how to build a bridge. It lays the foundation for science.

I have found that I best learn mathematics in a structured environment. The more instruction and direction a teacher gives, the better I do. For me this works, because I get less distracted and more focused on the tasks to complete. Plus, I know what the teacher expects of me. It gets me more into the class and I feel the class is more well behaved also, so there are more opportunities for the teacher to explain concepts and not have to control the class behavior.

I really don't know how my students will learn the best. I think it really depends on what type of class you have. I like to have structure, but students can get intimidated with too much and not feel comfortable asking the teacher for help. I do want to have a lot of structure when I teach, but if that doesn't work for my students, then I will probably change to help them learn the best way they can.

Previously I was an intern at a jr. high and I was able to really observe the teacher. I found that the best ways the students learned the concepts and were more apt to learning them, was if we could tie the concept to a real life situation. If the students understood why they would be needing the information later in life, they liked to learn it. Hands on activities were also big factors in helping them get excited. If they knew they were going outside to measure something or just observing objects, they got excited and wanted to learn more. If I was the student, I would want to know why something worked or why I needed the info and so I understand where they are coming from. Plus field trips and so much fun.

I think that one of the worst practices in school is teaching the students that what they learn right now, will not matter later on in life. In all reality it may not, but a lot of times, the students pick up on those things and decided not to learn other things because it "won't matter". I think it is important for the teacher to emphasize that all the concepts the students learn will matter and it is important to learn them.