Assignment #3

In Erlwanger's article, he makes the point to his readers that using Individually Prescribed Instruction (IPI) is not successful when trying to help young students understand mathematics. The sixth grade subject, Benny, is a smart child. As he studied the rules that go along with fractions, he found patterns that were not necessarily correct. For example, since he knew 3 +5 is equivalent to 5+3, then he carried that communitive property over to say that 1/2 is equivalent to 2/1, which is not true. When there is no teacher there to explain why this does not work, Benny has no reason to believe he is doing anytyhing wrong. Right along with this reasoning, when Benny gets the correct answer, he doesn't know why it is correct. He tells Erlwanger that his answers are graded just by a key, and even if the answer is true it will get marked wrong just because of the key. This instills in Benny the idea that mathematics is just applying rules to try to get the same answer as the key, rather than explaining to him that the answers are right, and a teacher can show him why and how. A student-teacher relationship is necessary for students to progress in correct and accurate mathematics as well as to help the students learn to enjoy math.

It is imperative today to keep up this student-teacher relationship to prevent students from "plugging and chugging" with incorrect procedures. This goes right along with the instrumental verses relational understanding, and, in this case, the instrumental understanding does not even consist of the correct rules, let alone the correct understanding of how and why the rules can be applied. If young students today are not given the guidance they need at the early stages of mathematics, then when more concepts are tuaght to them they will not have any foundation to build upon and so their understanding will decrease rather than increase. Our responsibility as educators is to help students learn mathematics, and the only way we will help them is if we are personally teaching them.

Writing Assignment #2

Two types of understanding, as discussed in Richard Skemp's article, include relational understanding and instrumental understanding. Skemp pointed out the importance of recognizing the different meanings of understanding as educators. Relational understanding refers to the "what" and "why" of a concept whereas the instrumental includes memorizing rules and applying them without knowing exactly why or how they can actually be applied. Relational understanding is difficult to teach in the amount of time educators are given, but is definitely the prefered understanding for students to obtain. Intrstrumental understanding is quickly applied by students because they dont need deep explainations. Students often feel more successful with this type of understanding because they get the right answer quickly, without spending time on the "why". The problem, however, comes when the memorized problem comes in a question posed differently from what they memorized. Now instead of just plugging numbers in, they actually need to think about what is going on in the problem, which they never learned. By the definitions, instrumental is actually included in relational understanding, relational just goes deeper into the concept rather than just memorization of rules. Both are important in the learning of mathematics, but the deeper understanding (relational) will prove to be more useful all around application.

Follow up on post #1

Because I never discussed what mathematics is to me, I am posting this new post :) For the most part, mathematics includes numbers, calculations, and variables. These three components are used to understand structures, probablities, speeds, rates, reactions, and many more concepts. In other words, mathematics is a tad bit helpful in each human life.

In my past and current Mathematics classes, I have experienced a number of different teaching styles and techniques. Some have been successful, others have not. Every teaching technique I have experienced, however, has been influencial for my future students because now I know for myself what techniques will help those I teach. When I was first taught algebra, I was given steps and formulas to follow rather than explainations and applications. Because I was young, i needed that direct form of teaching to grasp the algebraic concepts. I think that is a good way to get young students interested and confident in Mathematics. The next level of understanding comes when those steps and formulas transform into meaningful operations needed to compute story problems or information given in an unexpected format. In order to teach this level of understanding, the educator must give examples of many different types of story problems and then allow students to come up with their own so that they can learn to think outside of exactly what they are told. Once a student can analyze a problem on their own and use concepts already taught and grasped, the teacher can become more of a guide and the student will progress more quickly. I have noticed that some educators prefer to only lecture, rather than allow input and participation from students. This environment prevents some learning because without students asking questions and commenting, the second level of understanding cannot be reached. It is still important, however, for a teacher to give students enough information so the ideas that the students are forming on their own are correct and relavant. To summarize, there must be a balance between the information given directly to the student and the information expected for the student come up with on their own. Each student will be at a different level and so it is important for the instructor to become familiar with and understand their students' level of understanding.