Blog #3 Response: Erlwanger's Paper on Benny

Why would someone wish to publish an article on the deficiency of a child's understanding? I believe Erlwanger did this not only to exploit the deficiencies in the Individually Prescribed Instruction (IPI) program itself, but also to stress to the reader the importance of a teacher-student relationship. These two ideas are manifest (1) in the way Erlwanger portrays the unfortunate situation of Benny's tragic misconceptions of decimals and fractions, with the problem partly being due to the fact that the IPI program provided example problems without rules as to why the answers to the examples were such. Benny was forced to concoct rules to solve the problems; and since answers could be represented in different ways, Benny came to believe with confidence that all his answers and rules were correct. (2) Much of Benny's misconceptions could have been avoided if Benny but had one to supervise his progress at a more intimate level. Erlwanger said that "in IPI, teachers are prevented by their role perception from understanding the pupil's conception of what he is doing." The teachers just "go by the key ... what the key says" and don't actually critique the work themselves. If teachers were more involved throughout the course of Benny's past four years of education, they could have had a true perception of Benny's progress.

One main point that is still valid today is the fact that teachers need to be involved with their students' education. For an illustration, I had an algebra class in high school that ran solely on a computer program called Aleks. It allowed students to move at their own pace, be independent, and, similar to Benny's situation, it basically eliminated the role of the teacher. I sympathized with Benny when he said that finding an answer was a chase, a search for patterns. I do not feel I learned as much as I could have if I had had an actual teacher. I appreciated how I could move at my own pace, but when comparing that year to the next (same teacher but in an ordinary class setting), I realize I remember more things from the second year rather than the first. Now, I realize Aleks gave me a more short-term instrumental understanding, while having an actual teacher gave me a more permanent relational understanding.

Blog #2 Response: Instrumental v. Relational Understanding

I found this reading assignment to be rather interesting. It brought up ideas I never have thought about before, yet ideas that actually have been quite applicable in my experience of schooling. While reading through the article, I was able to draw some distinctions between the the topics of discussion: instrumental and relational understanding.

Instrumental understanding is the type where students have been taught the rules and shortcuts to doing mathematics. Although they have these basic tools necessary to solve problems, they do not always know where they are applicable -- that is where relational understanding comes into play.

Relational understanding is more or less the concepts behind mathematical operations. It is the understanding of why, when, and how. It is possible one may get through a mathematical homework assignment with just the instrumental understanding, but they most likely will not know why they are doing the particular steps, or how to apply it to related problems. If one understands the instrumental AND relational, however, they more likely will be able to apply what they know to related problems. Also, they will be able to retain the knowledge for a longer period of time.

In my own experience, teachers who have taught instrumental understanding are the ones who, for example, would teach for you to just memorize the formula a^2+b^2=c^2 for whenever you see a right triangle. Those that have taught relational understanding are the ones who teach us the formula, but then also teach how the formula came about, what a, b, and c represent, why the formula works, and when to use it. When this is done, I have found that even when I have forgotten what the formula was, the relational understanding enables me to contrive it once again.

Blog #1 Response

1. What is mathematics? Mathematics is a language in which to calculate patterns and change, broken down into different categories such as algebra, geometry, and calculus, each represented by their unique symbols. Symbols on a page is no more mathematics than sheet music is music - it takes a learned mathematician to take those symbols off the page and apply them to create real mathematics, as it takes a learned musician to take notes off a page to create music.
   
2. How do I learn mathematics best? I learn mathematics best by a) having a great teacher that can put across the ideas adequately, and b) studying notes and homework to solidify those ideas. If I do not review, much is lost from my learning.
   
3. How will my students learn mathematics best? My students will learn mathematics best by me going through the material carefully and clearly, asking questions and giving quizzes frequently to make sure they understand the material. That way I know what they still need to work on, and when we can move on.
   
4. What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? I think one of the main methods today is testing. The curriculum today is so focused on standardized testing and "getting the grade" that much of the curriculum is aimed towards passing those big tests. In order to pass the big tests, teachers give regular tests in class, which promote students to study and achieve.
   
5. What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? A practice that is detrimental to students' learning today is when teachers ONLY focus on testing. What I mean by this is that some teachers give homework with the aim of giving students practice to prepare for the test, then they do not collect it with the aim to grade it. This generally leads to students slacking off up until right before the test, where there it no time left to adequately learn all the material. Small quizzes and graded homework is important for the learning of the students.