Entry #3

The main point that Erlwanger is trying to get across is that with the IPI program, Benny just searched to find that rule that would get him to the correct solution that the program was looking for. Benny’s purpose in completing the IPI exercises and assessments was to find the rule, not necessarily to know the reason behind his procedures. Erlwanger argues that this way of reviewing and assessing the students causes the students to come up with their own rules for what they think the procedure is because the students were not taught the concepts behind the procedures. Since this is an individualized program, the students are expected to work mainly on their own. This seems to be causing the problem, especially for Benny. Benny is doing what is necessary to pass the assessments and do well, but he is not discussing with his teachers or peers what he has learned, so his crucial mistakes were not caught early on. Because the students have too much independence, they make up their own rules that fit the examples they see in the IPI programs, and they think this is true for all other problems involving the same mathematical concepts. Erlwanger explains throughout his argument that just because a student can master the skill and content does not mean that they have the correct understanding.


In today’s mathematical teaching, it is still important to remember to make sure that the students understand the concepts that are behind the procedures. When students only understand the procedures, they may be able to master a certain area of mathematics, but they will not be able to make connections using the information they had previously learned. For all mathematical content, it is always important to have the underlying concepts so that one can understand fully the procedures that are taking place. Math does make sense as long as the correct concepts are understood as well as the procedures.

Entry #2

Although instrumental and relational understanding have both advantages and disadvantages, relational understanding is more beneficial to students, especially in the long term. Relational understanding includes knowing not only how to do something, but also knowing the reasoning behind the action. Skemp explains, throughout his argument, that instrumental understanding, the ability to know how to do something, is actually contained within relational, meaning that when a student is learning through relational understanding, that student is still gaining the instrumental understanding as well. Skemp examines why teachers choose to teach instrumental mathematics when clearly relational is the better option because it incorporates both types of understanding and has a longer lasting effect. He found three advantages associated with instrumental understanding. He says that it is easier to understand, the results are more immediate, and the answers could be found more quickly. Along with this, the main disadvantage that Skemp emphasizes in his article is that instrumental understanding only lasts for the short term and is only useful within a limited context. However, he states that relational understanding had four basic advantages. He explains that relational understanding is easier to apply to new tasks, can be an effective goal in itself, and causes the students to reason and understand things for themselves. The last advantage of relational understanding is that it is easier to remember. It takes more time and effort to learn as well as to teach relational understanding, but once it is learned, the result is more lasting. Skemp does not identify any real disadvantages associated with relational understanding, but he explains some of the factors that add to the difficulty of using relational understanding. He says that it is hard to test whether or not a student actually understands relationally just by their test answers, and that sometimes the curriculum is so large that it is hard to be able to make the time to teach everything using relational understanding. Also, he explains that it is difficult to change the ways of both teaching, and the ways for students to learn. The focus of mathematics in schools is often placed on instrumental understanding, so it is difficult to make that switch to relational understanding, even though relational understanding is more beneficial to students. Students will be able to use what they learned from relational understanding because the effects are longer lasting and are easier to connect to other ideas and new tasks.

Entry #1

Mathematics is a way to solve problems. Math is the basic foundation for almost anything you need to do in life. Mathematics is essential knowledge in any sort of education. Many people use numbers, figures, and equations in their everyday life. This is mathematics.

I learn math the best by following examples. If I am taught the basic concept first, and then shown some examples to illustrate the particular concept I am able to apply that concept to many other problems even though those problems might be more intricate. I am a visual learning, so watching someone complete a similar problem helps me learn to solve problems.

I think most students are also visual learners. I think a lot of people learn by examples. When people see how to complete a problem, I think it helps them to see that the problems are just done step by step, and it gives them the confidence to know that they can also complete problems that deal with the same concepts.

One thing that helped me in my high school were the workbooks that we completed while the teacher was lecturing. I don’t know if these workbooks are still out there, but it was really helpful because we had to listen to the teacher while writing down the important concepts that were being taught. Also, there were many examples in these workbooks, so we could practice the concepts with the teacher’s help before attempting our actual homework. These workbooks really helped me to follow along with what the teacher was talking about, and they helped me understand what concepts I had a more difficult time with.

One thing that I had trouble with in high school is when the teacher would only do examples. This sounds contradicting to what I have previously said, but sometimes the teachers just get up do examples without explaining the concept behind it. I had a hard time in high school when my teachers would do this. I think it is important that the teacher explain the underlying concepts before delving into examples and letting the kids loose on the homework problems. It is important that the students understand why they are doing certain operations and steps when working their problems. They need to have that foundation, or else they won’t really learn the material, they will just get enough information to roughly complete the homework.