Mastering Mathematics Needs More Than Just IPI

The main point of Erlwanger's article about IPI mathematics is to get across to the reader some of the inherent weaknesses in the IPI approach to mathematics. IPI mathematics is a scientific approach to teaching which uses assessment, own pace learning, and given rules to teach with very little influence by a teacher. Erlwanger uses the example of a young man names Benny who passed through the IPI program showing good results. Although he was able to make it very far in the IPI program, when assessed personally by Erlwanger, he was found to have a very inaccurate understanding of the material he covered in IPI. The main point that Erlwanger wanted to get across was that since one student was able to do "well" in the program without actually learning the material, a more thorough investigation should be done to determine the usefulness of IPI. Erlwanger's talk with Benny demonstrates that Benny has developed learning habits and views of the math that will slow down his learning progress in the future. He states that teacher should become more involved in the learning process than they are in the IPI approach. Through this investigation, Erlwanger concludes that Benny and other IPI students are taught with too much of an instrumental approach and that they are essentially sent on a "wild goose chase" after mathematical answers.

I think that Erlwanger's argument that teachers are taken out of the picture too much in IPI mathematics is valid and I agree that even today, different approaches to teaching math in a similar mode as IPI, puts teacher involvement to a minimum, which impedes student progress. I talk of teachers who rely too much on workbooks or even textbooks to do the teaching that they are supposed to be doing themselves. I have seen through my personal experience that many students are left behind when this approach is used too much. I really do not have a very good understanding of the concepts of high school geometry due to a teacher who thought that the textbook should do the teaching more than her. Although I received an A grade in the class, I came away with very little real understanding of geometry. Overall, I believe that a balance should be struck between the use of independent type teaching and real teacher involved teaching.

Relational and Instrumental Understanding in Math Education

In Richard R. Skemp’s article, “Relational Understanding and Instrumental Understanding” we find an excellent exposé about the advantages of relational understanding, one of two different types of teaching and learning in math education. Skemp defines relational understanding as learning both the “what” and “why” of mathematics and instrumental understanding as merely learning the rules without learning the reason behind those rules. They are similar in that they both teach the student the rules of math and many times how to apply those rules. Skemp leads his reader to believe that relational understanding includes in itself most or all of instrumental understanding. Both of these teaching types have been used over the years and each has its advantages and disadvantages. Skemp’s opinion is that relational understanding will give to a student a huge advantage in his or her math education by giving them solid bases on which he or she can develop new ideas. On the other hand, Skemp explains that in instrumental understanding, the student does not receive sufficient “why” to continue holding their interest and to help them actually learn and retain usable math, needful throughout their lives. The article explains in great detail this issue which is of most importance to math educators.

My Math Experience

To me, mathematics would be a study using symbols such as numbers to explain what occurs in such things as science and business. It is a way in which we can understand logically the world around us.

Because mathematics is a way to explain our world, I learn math best when I am able to understand where it applies logically in life. Math sinks in better when I am able to wrap my own mind around a concept to the point where I can explain it to others. It is at times detrimental when I am spoonfed a concept.

I think my future students will learn math best as they themselves make discoveries and put forth the effort to wrap their minds are each concept, and as I continually assist them in their endeavors. Then as they practice repeatedly each concept, they will adequately learn each given subject. I believe this will work because this method helps the students use their own mind to discover topics, and with practice, they will be able to remember the given section.

A few practices in school mathematics that really promote students' learning include basic traditions such as giving adequate homework to help the student practice what they have learned, giving reading assignments in which a student can make his or her own discoveries, and the practice of assessing a students learning through tests.

One practice that I believe is detrimental to students' learning in schools today is giving a test over a material and then thinking that the test signifies the end of the learning of the material on the test. Tests should be used to assist teachers in helping students learn each given concept. Another way teachers today are causing students to not learn math is through too much talk, and not enough participation. It is hard for a student to learn a concept if he or she is taking notes too quickly in a class, and is not given the opportunity to try out the concepts themselves in a structured and guided manner.