Reading #3: Proof and Progress in Mathematics

While reading the Thurston article, I was struck with the realization of two essential aspects that were missing from the general mathematics high school classroom: socialization and familiar language.
While studying mathematics at the undergraduate level, I can remember being intimidated by the familiar language used by graduate students and professors that caused me more difficulty to understand and relate to the topics being discussed. Thurston comments on the social aspect of math and its important role in the transmission of ideas, however one must already be at a certain level of understanding to continue exploring ideas.
If we apply the same observations to a high school classroom, we might be able to gain some insight into how our students can feel intimidated about approaching the subject. If some students have learned the academic vocabulary and gained the confidence in its application, their peers may feel intimidated and shy from ever trying to understand the topics. The academic language has always presented difficulty for students. Accepting the challenge it presents in social situations, how it can assist learning or be a detriment to learning, and how it can isolate students must now also be considered.


Reading #2: Benny's Conceptions of Rules and Answers

After presenting the case of Benny and his grasp of fractions and decimals using the IPI system, Erlwanger’s observations on what Benny absorbed reminded me of our reading last week. The emphasis on rules and memorizing certain methods for specific problems presents itself as encouraging a strong instrumental understanding of the concepts. The lack of communication between the students and the teacher, the seeming discouragement of it entirely, caused for serious misunderstanding or ability to grasp a concept by the student to be unaccounted for. Thus, this type of instruction prevents any sort of support system to be put in place. Instead of ensuring that a student understands the outlined topics, the method is in fact checking if the student has understood how to master guessing the correct answers. There is certainly no emphasis of understanding the rational behind the correct answers.

It seems to present a valid, although extreme, case as to why teachers should be conscience of the level and extent that instrumental understanding is incorporated in their lessons.  

Reading #1: Instrumental vs. Relational Understanding

Reading this article caused me to associate instrumental learning with memorization where as relational understanding is the ability to grasp a concept and associated and applicable “rules”. As a teacher, it may be less involved to draft a set of rules for students to remember and the required topics can be completed. However, as students explore more difficult and reaching ideas, they are not able to rely on a firm foundation with a relational background of the material to enable them to explore the new information.

In my past experience as a tutor, I often found that my students were either of an instrumental understanding or a rational understanding (but far less so). The difficulty with having “instrumental students” was the increased difficulty in solving new problems that they had yet to memorize the solution. I soon discovered that if I could teach them the basic ideas and concepts that they could apply to any mathematical problem, in essence fill their toolbox, their overall skill and confidence in their abilities grew. They then became able to break down a problem into conceptual ideas they understood to build a solution rather than memorizing a “type” of problem.

The extra time and individual attention needed to take on a relational approach to teach mathematics can be overwhelming, but a student’s overall long-term success could be greatly impacted. Thus, I feel it is important to consider opening that line of discussion with colleagues and exploring the idea of including these practices at all levels of learning.