Writing Assignment #7

Sobel, M.A. (May 2007). Something for everyone. Mathematics Teacher, 100(9), 584-587.

This article is written by the President of NCTM, and he is discussing his favorite article ever written in Mathematics Teacher. He starts by going through some of the his favorite articles he wrote. He hits some of the high points such as what is going right with mathematics through the years and what is decreasing in value. At the end of this article he makes the discovery that each new issue that comes out is his favorite. There is something for everyone in every issue, and he grows to love the new issue every time. He believes that as teachers read these articles they will find something each time that can benefit them as a math teacher.

As math teachers, we need to continue to learn. By reading the new research that is happening in mathematics education, we can learn new ideas to implement in our own classrooms. Our teaching style can never be perfect, but by continually searching for ways to improve we can get closer to perfect. Students will be able to tell if we are trying to find better ways to help them. Reading the Mathematics Teacher is one way to learn new ideas.

Assignment #6

D'Ambrosio B. S., Kastberg S. E., & Viola dos Santos J. R. (Mar 2010). Learning from student approaches to algebraic proofs. Mathematics Teacher, 103(7), 489-495.

Teachers need to help students understand the purpose of proofs and how to format their arguements. The authors spends most of this article examining student's proofs. They believe that by observing student's work we can find out what holes are in their understanding. They took 9 different examples from students and talked about how students simply aren't understanding how to put together a proof.

Students aren't being taught how to approach a proof in the correct way. From examining the nine different examples it is obvious that students aren't understanding the purpose of a proof and what they should be doing. I experienced this in my classes in high school and am still experiencing it in my current proofs class. Often what needs to be proved appears obvious to students, but this isn't the purpose of a proof. By examining how students appoach a proof, it will help teachers better understand how to fill in the holes of a student's knowledge.

Writing Assignment 5

Warrington believes that guiding a classroom without the use of algorithms is the only way to go. This allows students to think on their own and explore math. When students know their teacher isn't going to feed them the right answers it requires them to dig deep into their own math experiences. This also helps students apply what they already know to new kinds of math. Students who come up with their own methods are more likely to retain this knowledge.
As much as I agree with Warrington, I also have many arguements. A whole new math system would have to be developed because although her method is more thorough, it is also much slower. I also feel that the bright students would benefit greatly from this experience, but those who struggle may end up trying to ride on their coattails, which can be difficult. I have had this experience where I have relied on my classmates to figure out how different theories can be explained.

Writing Assignment #4

Von Glasersfeld believes that all of our knowledge is constructed. Knowledge isn't just something that is given to us, but we must process information through our own filter system. We are constructing knowledge everyday in everything we do. Past experiences cause us to view new experiences differently than others who have similar experiences. Knowledge is more of a theory because we never know if what we know is reality. Our reality becomes what we construct through our experiences. Knowledge is viable because it seems to work until we hit a wall and then we must reconstruct it. Therefore, we can never really know if our knowledge is truly correct.
Math teachers need to keep this theory in mind when teaching. When giving examples, they need to be of the variety so that if students have constructed their knowledge wrong they can hit walls and reconstruct it. When teachers use examples that are always the same but different numbers or similar, it makes it more difficult to know if students are making up their own rules that are sure to fail them later on.

Writing Assignment #3

Erlwanger attempts to show that IPI was a system designed to allow students to learn math on their own without the aid of a teacher or discussion with others, and that it failed miserably. The card system allowed each student to work at their own pace. As long as the student was passing the assessment tests each time there was no need to discuss any of the material with anyone else. Benny was doing well and was ahead of most his class. Because of this he never talked to his teacher about his ideas of math. When Erlwanger spoke to him he discovered that because Benny had been working on his own he had invented a million rules about math that had no reasoning behind them except that they sometimes got him right answers on the assessments.
Even today there are classrooms where there is very little teacher student discussion. As long as a student can do well on tests it is assumed they know what they are doing and there is no need for the teacher to interact with them. I think it is important that as I get ready to become a math teacher that I am worried about the individual learner. A teacher needs to know more than what sort of grades their students are achieving. Teachers and students need to build relationships.

Writing Assignment 2

Knowing about rational understanding and instrumental understanding is vital if you want to become the best math teacher you can. Rational understanding is knowing the why behind math. If you have a rational understanding of math then you know where your formulas come from and how to apply them to problems and life. Having an instrumental understanding of math means you know how to plug numbers into formulas in order to get right answers. Instrumental understanding is within rational understanding. You can't truly have a rational understanding of math without knowing the instrumental part of it. Teaching a student an instrumental understanding of math does have its benefits though. Students get more immediate feedback, therefore there is less frustration. Instrumental understanding is also just simply easier to grasp at first. But, without a rational understanding all math is is numbers. Change the situation in a problem and suddenly students with an instrumental understanding can't get right answers any more, whereas those with rational understanding can apply their knowledge to multiple situations. Rational understanding also helps you keep math over time. When a student does finally gain a rational understanding then they have greater satisfaction than when they simply were able to find right answers. Instrumental understanding is important, but it can become useless without gainging a rational understanding.

Writing Assignment 1

Mathematics is a whole new language. It proves how the world works, or at least that is what I have always been told. To me mathematics is organized. There is a right answer to what you are working on. The solution is always there, you just have to find it. Except, sometimes you don't.

I learn mathematics best when there is an unnatural amount of structure. This is obviously not the best method for everyone but it works for me. I need a teacher to make me feel like their class is the most important class I will ever take. Now this may seem silly but if all my teachers make me feel like this then I put in my very best effort in every class.

It has been my experience that when my teachers give me the opportunity to work out problems with them I learn the most. When a teacher makes me feel like they want me to come ask them every one of my questions then I learn more.

In my high school math classes, weekly quizzes were a good thing. They let me know if I needed to work harder and what I was missing. Examples were also crucial. I have a hard time applying theorems without lots of examples guiding me through.

I think that one of the negative aspects of high school math is the negative outlook everyone has towards it. There is this idea in high school that you'll never need the math you are learning. Students lose interest fast and stop trying. You can't stop trying in a math class.