Entrata tre del blog

In the article "Benny's Conception of Rules and Answers in IPI Mathematics" by S. H. Erlwanger a view of IPI Mathematics is portrayed. The view is a little bit negative as Erlwanger uses the article to show the disadvantages of IPI mathematics as a way of learning and understanding math. To emphasize this point, Erlwanger uses a young student, Benny, and his understanding of mathematics as an example. He show's first how wrong Benny's ideas of the rules of mathematics were. In attempts to understand the assignments and problems he was given, Benny made up his own rules for solving problems with fractions. To drive his main idea home, Erlwanger helped the reader understand why Benny did what he did. All he was learning from was an example and answers to problems. He knew he needed to get from one point to the other, so he created rules to do so, and in this process, he was allowed to teach himself a completely incorrect understanding of mathematics, which Erlwanger proved would be difficult to undo in this young man's mind, along with probably many others.

This study is not just relevant in showing that IPI mathematics is not the best way to teach children this subject. Rather, it shows the important role a teacher plays in a child's education.

From Benny's unfortunate experience, teachers now can see how much of an influence they really have. Not only do they have the responsibility of getting the information to the students, and even more, they must get the information to the students in a way that each student can understand it. Teacher's also have the responsibility and opportunity to make sure children learn concepts and rules correctly. This is because once something is learned, it is difficult to change what ever it may be in the mind of the one who learned it. This is why what Erlwanger emphasized is important to us today.

Second Blog Entry

In Richard R. Skemp's article 'Relational Understand and Instrumental Understanding', he explains both of these ways of learning and points out their advantages and disadvantages. Both of these will be reviewed in this summary. Both of these styles are unique in their own way, yet they are both related. Relational understanding is the big picture, it incorporates everything, including Instrumental learning. In other words, in order to have a completely relational understanding of something, one must need to understand it instrumentally as well. Relational understanding is not only understanding how to do a problem, or applying a formula, but it is understanding why it works the way it does. It is simply knowing why one would do what they were doing. Instrumental understanding is the more basic idea of simply knowing how to do a problem. Skemp reviewed advantages and disadvantages for both. Relational understanding is often times is a great method for helping to understanding something when it is a brand new concept. Also, when trying to remember all of the rules in things (especially math) when the whole picture is understood, it is easy to remember it because a knowledge of why is included. Relational understanding itself can be used as a goal to understand basically the blue prints for different mathematical concepts. On the down side, it is difficult to test if a person actually fully understands the why factor. It can be overwhelming when trying to not that a class will be assessed based on relational learning, it is harder to set and achieve a specific goal, especially when being tested over a basic complete understanding and as a teacher being able to regurgitate what one already knows is quite difficult as well. On the flip side, instrumental understanding has its own positive attributes. One a basic level, it is generally easier to understand something instrumentally. When testing or just trying out a problem, knowing if it is understood instrumentally can be assessed on the spot, and lastly, one can get the correct answer after a short amount of work, because they do not have to think through it. However, this has its downfalls as well. Skemp shows that by simply knowing how to do something, the right answer will not always be obtained. He showed the example through music. It is easy to do parts of music through the basic instrumental way, but without knowing why something is done, it can not be applied in a broader sense. All in all, both styles have their pros and cons, and Skemp does a great job at displaying both.

First Blog Entry

What is mathematics?

Mathematics is the complete study of numbers. Also, it is the application of numbers to any and every situation possible. Math is a way of solving problems through quantifications, and classifying things symbolically and numerically. Math helps to understand everything around us, time, space, and everything in between.

How I learn mathematics best.

I learn math through two ways. First through understanding definitions. By understanding a definition to an equation or even a word, I have a greater idea of how something is to be understood or worked out. The second way is through example. By seeing similar problems, and how they are solved, I am able to understand the underlying concepts of similar types of problems.

How will my students learn math best?

My students will learn math best through example as well. This is true, because in each example, I will be sure that the definitions are understood. Also, by doing multiple examples, I can apply different situations to each problem, so that a great majority of the students that did not understand the first time, will see it in a different light and will be able to pick up on it.

What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics?

I can think of a three main practices in math classrooms that are used to promote students' learning.

First, through independent study. In math classes I have been in, and in others that family members have been in, I have noticed a trend. Often times students are encouraged to read the material before hand or on their own in the classroom to see if sense can be made of concepts through that method.

The next practice I can think of is again, through example. In every class I have seen, teachers will spend a majority of the class doing example problems, to help clarify each of the different concepts.

The last way I can recall is through homework. Basically every math class involves homework. This gives students the opportunity to not only see practice problems, but to apply what they have learned and work on them by themselves.

What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics?

One way that math is taught in classrooms that is detrimental to the students learning, is through pure lecture. If students are not given the opportunity to work out the problems with the teacher, a lot of them will not have the ability to work out a problem on their own.

Another way is by unclarity. An unprepared teacher can stand in front of a classroom and teach something completely off track, or much to advanced for the caliber of students he or she has in their classroom. By doing this, the wrong way, or the overly difficult way is implanted in a students mind, and they will refer back to this often times before thinking of the correct way to solve a problem.