Loss of Connections

IPI (Individually Prescribed Instruction) Mathematics was to imporve pupils mathematics through self discovery, but after a few years of working in the system, it was found to be more detrimental than helpful. S.H. Erlwanger did a study on a students named Benny. Benny was at the top of his class and had been doing IPI for 4 years. As Erlwanger asked Benny about his understanding of factions and decimals, he found Benny had made his own rules for how they work. Benny, for years had been given worksheets with examples, working on them by hisself, and that was his only form or instruction. He worked at his own pace and took a test at the end of each section when he was ready. He had little teacher involvement, only enough to help him when he needed it. Through this, he made up rules that got him correct answers most of the time, but unfortunately he never was told precisly why he got anwswers wrong. So Benny only assumed that his awswers were right because there are many ways to write an anwer, and he simply just didn't put the right one. Since this had gone on for four years, Erlwanger tried to help Benny learn fractions and decimals correctly, but damage had been done. Benny still had problems removing the rules he had already formed. Thus, IPI did more to detriment Benny's learning than to help.
The reason Benny never correctly learned the properties of factions and decimals is because he could never connect them to eachother or to prior knowledge. This could be viewed as Benny recieving a bunch of information with no connection between them, so he tries to make his own connections. This is one thing, for teachers today that must avoided. Teachers need to facilitate learning for their students so they can learn and undestand the connections between information given. For instance, factions and decimals have a particular relation. They represent the same number in different forms. Benny never picked up on this relationship because, as Erlwanger's study suggests, the instruction Benny received emphasized getting the right answer and not the process involved. He made connections he did, so he could get correct answer. If teachers focus on helping students understand the process by which they get correct answers, then students would be able to build on prior knowledge and obtain new knowledge by making proper connections.

Understanding, more than just a word

When does a child truly understand mathematical concepts? Does it have to do with gettign correct answers on homework assignments and tests, or is it more than that? Richard R. Skemp thinks that -- to say a student understands is ambiguous, because "to understand" can have two meanings. According to him, there is relational understanding, which most people are familar with, and there is instumental understanding. Relational understanding means that a person comprehends how to do something and why. Instumental understanding is knowing what to do, but not knowing why you do it. Thus, relational understanding encompasses instrumental understanding. That might be why instrumental understanding is found most common. It is the most easily taught considering the students need only to know how to solve a problem and not why. This means they only need to memorize formulas or rules without understanding why they work. Relational understanding on the other hand, may enable the students to understand why and how, but unfortunately it takes longer to teach. The overall advantage to teaching relational understanding over instrumental is that it enables students to adapt methods to various problems.

When is a student really learning?

1. What is mathematics?
It is a tool used to measure and give meaning to things in this world.
2. How do I learn mathematics best? Explain why you believe this.
I learn best if we go through and solve an example together while the teacher explains the meaning and reason for what we are doing; explaining any of the equations that would be used to solve it. By doing this I am able to put action with reason. After we have gone through a few examples, I need one or two more, so that I can go through in class on my own before we go over it together. Most importantly, when I’m given enough time, I am able to comprehend it and solve it myself. Then I am able to better solidify what I have learned.
3. How will my students learn mathematics best? Explain why you think this is true.
They will learn best when they understand what they are learning. Giving them a formula to use does not necessarily mean they understand why the formula works or exactly how to use it. When they understand the underlying concepts better, then the will be able to learn.
4. What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? Justify your reasoning.
Some teachers choose to lecture mathematics, others use a method of discussion, some use scaffolding to help them, and there are a few who use discovery learning. Lecture is when a teacher simply explains and instructs the class on the concepts to be learned. A discussion consists of an examination of the concepts. Scaffolding is when a teacher helps the students a lot in the beginning and then slowly removes his/her help to where the student does not rely on the teacher. Finally, discovery learning is one of the hardest methods to use. It requires a good amount of time to be given to the students to explore ideas about the concepts being presented, but it allows for the students to discover mathematics for themselves.
5. What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? Justify your reasoning.
Well, I know lecturing is one of them. Lecturing gets the information out there, but it does not allow for student feedback. I does not ensure in any way students are actually understanding what they are supposedly being taught. I also think that when there is no instruction from the teacher at all that some students will lose hope in learning and they will give up on it. Students do need a certain amount of guidance. As a teacher, you need balance. Not everyone learns in the same way, so you have to use different methods of teaching to compensate for this.