Blog Entry #3

In the article about Benny's understanding of mathematics, Erlwanger is suggesting that IPI Mathematics is an ineffective and harmful way to teach math to students. Erlwanger explains the goal of IPI is that students can progress learning objectives in a chronological order. He then describes the process taken by the instructor and student: learning the objective, then being quizzed on it. Erlwanger criticizes the focus on test scores, with an 80% being required for the student to "pass" an objective. In his interviews with Benny, Benny tells him how he feels the key to the test is bad because there are more right answers than are on the key. Benny never thinks he is wrong because of this. As his conclusitory point, Erlwanger gives an interview in which he tried to correct Benny's knowledge and did not succeed because of Benny's previous years of incorrect learning. Erlwanger portrays that IPI was very damaging to Benny and implies it has the potential to be damaging to many other students, he advises against its use.

The overall idea of IPI teaching students individually could be useful today because it allows each student to progress at their own level. This type of individual progress is accomodating to many students because students of all different levels of math can progress individually and therefore better understand the material. Erlwanger strongly discourages the use of IPI, so an alternative style of individual assistance would need to be used to thus accomodate students. The use of many different examples can also be valid in mathematics today, but there needs to be more explanation behind each example. Move slower and allow the student to ask questions. Also, tests are a good examination of students' knowledge, but it is better to do different styles of tests to make sure the student understands. One idea could be having a student create a problem and then explain how he/she would solve it.

Blog Entry #2

In Skemp's article he compares two types of mathematical understanding: relational and instrumental. While these two types have the same end result, the solution to the mathematical problem, they differ in the means of reaching the solution and therefore have different advantages and disadvantages. Skemp describes instrumental understanding as a list of equations, simply plugging numbers into the correct formulas. The advantage of this style of learning is how easily a teacher can teach it and how quickly the student can arrive at the answer. The main disadvantage of instrumental teaching is that the students typically don't understand the reasons behind their solutions, they understand "how" but not "why." Skemp describes relational understanding as having both the "how" and the "why" knowledge. This type of understanding takes a lot longer for the teacher to teach and is thus the reason that it is not used in schools today. This type of teaching has the huge advantage that students understand what they are learning. In this article, Skemp portrays relational as the better of the two understandings.

MthEd Blog Entry #1

Mathematics is the study of numbers and how they can be applied to everyday situations. It is a type of problem solving and even a way of thinking. I learn mathematics best by writing it down, figuring the problem out after I have been taught the correct process to complete the problem. Repetition is also critical for me, the more times i see a certain type of problem, the more i understand how to solve it and why it works. I think my students will learn mathematics best by numerous different methods. All students are different and different techniques work better for each individual student. I believe that teaching by whiteboard encouraging students to follow the steps by writing the problem on their own paper is effective for some. I also think visual aids can help many students. Simple things like hanging posters in the room with equations on them or showing how to do a problem using a picture. Applying the mathematical idea to real life situations helps lots of people too, because they feel it is a productive use of their time to be learning that particular material.

In schools today the beneficial methods of mathematics teaching in place are 10 minute quizzes at the start of class, the encouraging of group work both in and out of the classroom, and having students do certain problems on the board in front of the class. 10 minute quizzes at the start of class were very helpful to all students in my high school math classes in subjects such as the unit circle and derivatives and integrals. The students had to memorize these equations for the quizzes so when the equations were needed other than the quiz it was always fresh on the students mind. Group work in the class can sometimes be a waste of time but when monitored and forced to be productive by an assignment it is very helpful for students to talk with others to better understand the subject. Similarly, it is very helpful when a student does a problem on the board and walks the class through the steps he/she used to arrive at his/her solution. This allows students to hear different explanations of how to solve problems which might be more clear to them.

Some ineffective practices in school classrooms are teaching directly from the textbook and poor distribution of students in classes. Teaching directly from the textbook is boring and loses the attention of students. Use the textbook as an aid to a lesson that does not require students to have the textbook in front of them at all times. A big problem in most high schools is that there are not enough teachers to beneficially teach all the students. The result is that really smart kids end up in classes with those slower to learn. There is nothing wrong with the students that take longer to understand a subject, but their comments can be distracting to students who have already grasped the concept and end up second guessing themselves.