One of the most important roles a teacher has is to understand how the student learns and thinks. Teachers must find a way of evaluating not just right answers that students give, but also their overall understanding of mathematic concepts. It actually reminds me of a concept that is taught in Preach My Gospel in the chapter on teaching skills: "Teach for Understanding". It is the teachers responsibility to not just teach the material, but to teach it in a way that the students understand relationally, and not just how to give correct answers. IPI mathematics was not able to do this at all, but rather gave students an idea of math as a guessing game, or a game of chasing right answers. This concept of teaching students so that they truly understand the mathematical concepts will always be important.
Saturday, January 23, 2010
In Erlwanger's article "Benny's Conception of Rules and Answers in IPI Mathematics", he presents evidence that Individually Prescribed Instruction Mathematics, or IPI, is not as effective as the designers of the program would have hoped. Erlwanger uses Benny, a sixth grade student, as an example for how the program has actually hindered the progress of students rather than helped them. As Erlwanger describes the interviews that he had with Benny, he brings up the main flaws of the IPI mathematics program. First of all, IPI mathematics does not do a good job of evaluating the understanding of the students, but rather evaluates correct answers that students give based on rules and examples; it rewards correct answers only, regardless of the way they were obtained. Second, IPI mathematics develops learning habits and views about math that actually hinder progress later. And third, IPI greatly detaches the teacher from their role of for understanding how the student learns and thinks by giving a very rigid and inflexible structure of learning. Benny's example and Erlwanger's study shows that IPI was a good experiment, but it was vastly ineffective.
Friday, January 15, 2010
When we talk about "understanding" in mathematics, it takes us to a very grey area where there are different types of understanding. Richard R. Skemp, in his article "Relational Understanding and Instrumental Understanding", discusses this grey area in the context of two different types of understanding that happen in mathematics, especially in educating students. Skemp states that relational understanding is what can be considered a "true" understanding of the subject matter; it is knowing not only how to solve problems and what to do, but also knowing the why behind it all. Instrumental understanding, however, is knowing a list of rules, and then knowing when to use them to solve problems. At first, it may seem that relational and instrumental understanding are completely disjunct from each other. However, after a closer examination, one realizes that instrumental understanding is actually a smaller part that is completely contained within relational understanding. If someone has a relational understanding of a math concept, then they understand the instrumental side of it as well, knowing rules and shortcuts to use when solving problems; but, they also understand why those rules work and can even derive other rules from their relational understanding. In this way, Skemp shows that teaching mathematics for a relational understanding is a much more empowering way of helping students. There are advantages and disadvantages to both though. Instrumental understanding, in general, is much easier to "understand"; some topics, like multiplying two negative numbers together, are very difficult to understand relationally. Instrumental understanding also allows someone to get the right answers, and get them quickly; which, in turn, gives the students immediate rewards that are quite satisfying to them. However, with that type of understanding, students never understand the "why" in the math they are doing, and will never be able to formulate mathematic concepts on their own. They will always rely on someone else to give them another list of rules and procedures to follow in order to get the right answer. Relational understanding gives students quite an advantage in mathematics because it is much more adaptable to new tasks; when given a new, but related concept in mathematics, students with a relational understanding can reason and manipulate what they know to help them solve a problem that could be considered "outside of the box". Relational understanding is also much easier to remember, and, when gained, motivates students to explore new areas within mathematics on their own steam, greatly reducing the need for the teacher to try and motivate students when teaching. On the other hand, relational understanding is much more difficult for students to grasp at first, and it is also very hard to evaluate whether or not students have gained a relational understanding. Overall, Skemp's article brings up the question, "Which type of understanding should the term 'mathematics' fall under?" To Skemp, only a relational understanding should really be called teaching "mathematics", but the ideas within his article continue to be discussed, and need to be in order for teachers to reach a way to teach the students in the best way possible.
Tuesday, January 5, 2010
What is Mathematics?
To me, mathematics is not just manipulating numbers or using a correct formula to plug a whole bunch of things into. Mathematics includes understanding, exploring, and discovering. It includes looking outside of the box, and creating new ideas that can be tested.
How do I learn Mathematics best?
I learn mathematics best when the teacher has a very structured way of teaching the basic material, then gives examples, and then allows the students to make discoveries about what they have just been taught on their own. Then, as students try to do an assignment, the teacher allows for questions to be asked, and then helps the student when they get stuck.
How will my students learn mathematics best?
Students have many different ways that they learn. I would try very hard to use many different methods to teach students. These would include visual learning, such as writing on the board, hands-on activities that illustrate the principle I am teaching, examples that we would work through in class together, and then time for the students to try it on their own. This would hopefully work by using different styles of teaching to reach the different styles that people learn math the best.
What are some of the current practices in school mathematics classrooms that promote students learning of mathematics?
Many mathematics classrooms these days are doing exactly what I suggested I would do with my students, which is to use a variety of teaching methods to try and reach the different ways that the students would learn. I thoroughly enjoyed my Statistics class where we did many hands on activities that illustrated and demonstrated the principles we were trying to learn. The result was shown at the end of the year when we took the AP Statistics test, and all who took it either got a 4 or a 5 because we understood the concepts so well. As teachers continue to be creative in the way that they teach the material, more students will grasp the concepts that sometimes seem difficult in mathematics.
What are some of the current practices in school mathematics classrooms that do not promote students learning of mathematics?
This, I believe, occurs when teachers fall in to one of two categories. The first are the teachers who simply get up in front of the class, talk for the whole class period, and then give an assignment to the class as the class is leaving. The second are the teachers who do not have a solidified structure. Teachers need to be flexible, but if there is no structure to the class at all, the teacher can lose track of what he or she needs to teach the students, and can also teach things out of order of the best way to teach them. These two practices in school mathematics classrooms make it very difficult for students to learn math, because it does not allow the student to discover or learn for themselves, and doesn't allow them to try the concepts on their own.