Blog Entry #3

As Erlwanger examined some mathematical conceptions of 6th grader Benny, he concluded that Individually Prescribed Instruction had some serious weaknesses and flaws. IPI was a very ample effort to make mathematics teachable to the individual and was supposed to be beneficial to students in helping them learn math at their own pace. When Benny was asked by Erlwanger to explain some of the rules he had for different mathematical procedures, Erlwanger discovered that Benny had made up many rules of his own to try and make sense of math, and many of these rules were incorrect. Benny had been able to excell through the IPI course with these major misconceptions and this was not the intention of IPI. The teachers were also so uninvolved in IPI that none of Benny's teachers knew he had incorrect ways of doing mathematics, they all believed him to be a star pupil.
Currently in mathematics teaching, teachers can still be very distant from the students in their classes. I feel that in many math courses taken by students today, teachers have interest in the students learning collectively, but not individually. Many teachers ask students collectively if they understand a concept being taught, but they never bother to ask the individual. It seems as if many have an attitude of, "If at least this certain amount of students understand what is going on, then I have done my job, and too bad for the rest." Perhaps that is stated harshly, but it is an attitude that I have percieved. Teachers do not conference with their students about their learning and because of this, students feel less responsible for their learning and fall behind in their classes. Slight teacher interest for each individual student would make success for students more achievable.

Assignment #2

Relational and instrumental understanding are are two different definitions which are used for the word understanding in the field of mathematics. These two types of understanding each have advantages and disadvantages as well as areas where they overlap one another. Richard Skemp described relational understanding as knowing how to solve a problem and why the problem can be solved that way, while instrumental understanding is described by Skemp as only knowing how to solve a problem without understanding why it works. The benefits for relational understanding include being able to adapt what you have learned in one area and apply it to another, the ability to better remember what it is you have learned, becoming an effective goal, and producing a solid base from which to go and gain more mathematical knowledge while exploring the mathematical world. The main disadvantage of relational understanding is that it takes longer to learn the material. There are advantages and disadvantages for instrumental understanding as well. Instrumental understanding is generally easier to learn, faster to learn, and the only way to understand some mathematics because relational understanding would have to include science as well. Instrumental understanding is ill equipped to help a student apply material from one area in mathematics to another, remember what it is they have learned, and be able to solve mathematical problems in more than the one prescribed way. These two types of understanding do overlap one another. Included within relational understanding is instrumental understanding as both types of understanding include the ability to know how to complete the problem, but only relational understanding includes knowing why that works. Relational and Instrumental understanding can also both be successful in helping a student solve a mathematical problem successfully.

Assignment #1

To me, mathematics is a process and tool by which we are able to form relationships between quantities of different forms (such as measurements) and decipher the meaning of numbers including equations and geometry. Mathematics also helps to teach one how to think in processes.
I personally learn mathematics best through step by step explanation and example from someone who is knowledgeable about the subject, such as a teacher. It is really important that I am shown the process of solving a problem, not just given the equation for solving it. I also think that going through several examples is helpful in the learning process, and of course, doing several problems by myself helps me learn as well.
I believe that everyone learns mathematics best through slightly different processes. I think that a variety of examples with explanations and hands on experience is the best way for students to learn. Not all students may understand a concept right away, but when given several types of examples of how to apply a concept, it is likely then that different examples will help different students understand. I think it is then beneficial for students to try different problems that apply that concept with the availability of someone nearby to help if needed. It is also important for teachers to get constant feedback from the class to if they are explaining concepts in a way that they understand and clarifying as needed. It could also be quite helpful to the students learning if a student is occasionally asked to re-explain or discuss a concept to/with other students that was just taught. An environment needs to be present for the students that is comfortable where they feel they can at any time express whether they do or do not understand what is being taught.
In most all of my high school math classes I had teachers who clearly explained concepts and gave ample examples to help students understand what was being taught. They then would give us time in class to begin working homework problems and during this time classmates and the teacher would be available to help with any problem you might have trouble with. These practices in math classrooms are very beneficial to students and will really help them learn the material that is being taught.
I do not believe it to be beneficial at all when a teacher stands at the front of the classroom and simply gives a lecture on the material while writing on the board. In this situation they give a lecture of the material and explain the material in the way that THEY think students will understand, not in a way that the students actually will understand. In this type of scenario they rarely ask if students understand, and even if they do ask, in this type of a situation, students rarely feel comfortable stating that they do not understand.