Assignment #3

The goal of Erlwanger's paper is to show the danger of learning math strictly on your own, the problems of not teaching the 'why' behind the 'how', and the faulty conclusions that students obtain from not learning correct principles. According to the IPI system of mathematics, Benny was one of the more advanced students in the program, but upon investigation, Erlwanger discovered many false conceptions that Benny had with regards to math. The fact that Benny in effect 'worked the system', or put down the answer he knew was wanted as opposed to what he felt was correct is a good example of this. By working by himself, Benny had come to many false conclusions with regards to many fundamental parts of math such as fractions and decimals. Because Benny didn't understand why fractions worked the way they do, he made many mistakes. Also, the fact that he didn't have someone checking his work was a large contributor to his coming up with incorrect ideas of math. Had someone been more involved in the learning process, they could have corrected many of his erroneous assumptions.

A main argument that is still valid today is that of the danger of learning math strictly on your own. It's still a valid argument because of the fact that math is a difficult subject to learn and if you have to teach yourself it is easy to create false ideas of how and why math works. I currently am a tutor for some students who are taking some online math classes, and I really don't know how they are expected to learn all the material without someone to explain it to. The language of the text is difficult to understand, the homework is challenging and tests the limits of the concepts, not the basics, also, it's easy to get rules confused and use the wrong rule in a situation. Having witnessed first-hand what not having a teacher does to a student, I know it's SO important to have someone involved in the learning process.

Assignment #2

In Skemp's article, he discusses the pros and cons of two types of mathematical learning; relational understanding, or the understanding of why something works and it's application, and instrumental understanding, or the understanding how to solve a problem. He says that both are valid means of teaching children math, but that the advantages of relational understanding are greater than those of the instrumental approach. The first of the four advantages of relational understanding is the fact that it can be applied to a broader range of problems. The importance of this fact is seen when looking at a disadvantage of instrumental understanding; that of if the problem is slightly different than the norm, the student will get it wrong because they only know how to solve things a certain way. The next advantage mentioned is that it is easier remember things taught relationally as opposed to instrumentally. This may seem contradictory because of the fact that with instrumental learning all you need to do is memorize a formula, but you have to memorize a formula for EVERYTHING instead of just learning how we obtain the formulas in a certain field of math and applying it. Moving on to the third advantage of relational understanding, and that is the fact that this form of learning can be a goal in and of itself; i.e. the reward of figuring out why a certain way of solving works is satisfying to the solvee. This point also leads us to one of the disadvantages of relational understanding, that of the time it takes to achieve the necessary understanding to apply said way of solving. The final advantage of relational understanding talked about is the fact that the satisfaction of discovery will lead others to more discoveries on their own. Now, the advantages of instrumental understanding such as simplicity of understanding and the immediate satisfaction of solving a problem are good, but when held up to those of relational understanding it is clear that relational understanding is the better way of learning.

Assignment #1

• What is mathematics?
  

Math is the study of numbers and how to manipulate them for use in the world. It really is what makes the world we live in work. Without math we'd have no computers, cell phones, cars, amusement parks, movie theaters, plumbing, the list is endless. That's what math is.

• How do I learn mathematics best?

I best learn by example. If I can see how something works it makes doing that thing SO much easier for me. I'm a big fan of the 'plug and chug' method where you're given a set way to solve something and you take that way and 'plug' the various parts in and 'chug' away at solving the problem. I really struggle trying to figure out wordy theorems and definitions. I like knowing what the definitions and theorems are, but then I need to see how they apply to what we're talking about. Learning by example is what works for me.

• How will my students learn mathematics best?

I believe that my students will best learn the same way I learn, by example. It makes sense for this to be true because of the fact that you teach best what you're best equipped to teach, and I'm best equipped to teach by example. As evidenced by these posts, I'm not the most adept person when it comes to words so using examples will best assist me in teaching my students important concepts. I've had quite a bit of tutoring experience and the people that I help have always understood the best when we actually work through a problem instead of them trying to figure it out by me talking about it.

• What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics?

One thing is the obvious fact that it's being taught. You can't learn what you're not taught. :) One practice that I think is very beneficial is group studying. When you are learning a subject with your peers I believe that you can explore more aspects of the subject because you are learning through more than one perspective. I also feel that the tried and true method of daily homework assignments is another great way for students to learn. Math is something that needs to be focused on a little bit each day as opposed to big chunks of learning only once or twice a week.

• What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics?

One of the things that I find most detrimental is the whole 'draw a picture' approach to math. Students in elementary school are being asked to visually represent everything that they do in math, and it's not necessary. I realize that everyone learns differently and that visual representation is one of the best ways to learn; however, I feel that the majority of students would learn math better if they learn it the way it has been taught for years; i.e. 2+2=4. I especially feel that the 'draw a picture' approach is detrimental because of the fact that when students reach higher level classes, such as Algebra 2 and Calculus, there are some things that just can't be represented visually.