Entry #3

In his article about Benny, Erlwanger is trying to convey the weaknesses of an individualized learning system and emphasize the importance for the teacher to initiate interaction with their students. Erlwanger describes the basic precepts of the IPI system to show the individualistic beliefs that underlie its method for students to learn mathematics. Since the IPI focuses on pupil independence, Erlwanger works to show the mishaps it has created within Benny's learning to point out the weaknesses of such an approach. He provides very specific examples of misconceptions Benny has developed such as how 0.3+0.4=0.07. Erlwanger describes how even when a student displays a supposed mastery of the content and skill, this does not necessarily mean that understanding is present. Erlwanger points out that a teacher could encourage Benny to ask questions, discuss, and reflect on his experience with mathematics if a close personal relationship with Benny and his understanding had been developed. Through IPI system, such a relationship does not exist. Benny feels no need to talk with his teacher, and due to the lack of teacher initiated interaction, Benny develops many faulty ideas and conceives math as being different rules for different problems, which creates a chase for the right answer.

One of the points that Erlwanger makes is the need for teachers to initiate interaction with their students. Teachers should provoke deeper thought, reasoning and justification from their students. I believe that this idea is still applicable today. I know that often times I will not even think about certain aspects of the material that I learn unless a teacher points them out to me through questions, examples and discussion. In fact, just this past semester in Abstract Algebra we were learning about elementary and invariant factors. I felt as though I understood the concept and thought I understood the process of how to find these factors. My method produced all of the right answers on my homework, and from an outsiders perspective, it would have looked as though everything was right. Yet even though the work I showed was correct, I did not fully understand. For when I was given a more complicated problem that wasn't quite like all of the rest, my method was not sufficient and did not produce the right answers. When I realized this mishap and learned the proper way of thinking about it, the correct procedure was not very difficult and so it was understandable why someone would assume that I properly understood. However, it was a matter of luck that I happened to spot this because the teacher hadn't asked for any reasoning beyond showing our computations, nor assigned any homework that would have led one to recognize such an error. Because of this experience, as well as others I have had/seen, I feel as though teacher initiated questions, discussion, and reflection can still be improved in classrooms today, even when it seems as though the students have mastered the concepts.

Entry #2

In Richard Skemp's article he explains two meanings of the word understanding and discusses the pros and cons of each meaning. The words relational and instrumental are used to distinguish between the two. Instrumental understanding is knowing what to do. It is understanding how to apply a formula, a procedure, a rule etc. while not fully grasping why the rules work out the way they do. An instrumental understanding is usually easier to come by. Correct answers will be produced and can be provided quickly. The rewards are more immediate and evident. The skills obtained may be all that is necessary in order to understand other areas such as science. Because less knowledge is involved for an instrumental understanding, answers also come in a more reliable fashion. A relational understanding expands beyond instrumental, and the two are not completely unrelated. Relational refers to not only knowing how to do something, but why you do it in such a way. A relational understanding is more adaptable to new problems that don't exactly fit the rule taught. Relational mathematics are easier to remember than instrumental, although harder to learn. It may take more time than is available to understand relationally. With a relational understanding, results are not as easy and quick to obtain. However, unlike instrumental, the need for punishment/rewards are lessened. This is due to the fact that motivation can come from trying to attain a relational understanding. This motivation can also lead people to seek beyond their current level of understanding and explore new material.

Entry #1

What is mathematics?

In general mathematics seems like a group of rules and axioms that are then used and applied in certain ways to solve problems. Mathematics might also be described as a field that studies proportions, measurement, combinations, transformations, properties of numbers and sets using numbers and symbols.

How do I learn mathematics best?

I learn mathematics best when I have already had some exposure to the material beforehand, such as skimming the section prior to class. Whether I give much thought to it or not, at least my mind is somewhat familiar with some concepts so I don't feel completely confused during lecture. I also find it very effective when I am quizzed on material throughout the course in shorter intervals. I feel like this helps me to retain information better and it gets me started on the memorization process. I also feel like I learn best when the teacher goes at a slow enough pace and leaves time for us periodically to think about the material and respond to questions he/she will propose. This helps me to continue thinking during a lecture rather than just mindlessly letting information be copied into my notes.

I also like it when I am given enough time to work out homework problems as well as having many available sources for help such as a TA, office hours, question/answer time in class etc. Often times I think this helps me in more of a mental way. If I know I have enough time and can get help when I am stuck, I am more likely to really give my homework a fair chance and have confidence as I go through and solve problems.

Another technique that I have found very useful is when either the teacher or myself takes the time to connect, relate, classify, integrate etc. the new material to past material. This helps me to find the common ground between certain concepts so that key differences stick out, allowing me to have a better grasp on the material taught.

How will my students learn mathematics best?

I feel like my students will learn mathematics best when they feel like I care about their learning and will help them help themselves succeed. I feel as though often times students are intimidated by the knowledge of their teachers and don't have the confidence to ask questions. So I believe that trying to create a more positive environment would be beneficial.

I also believe that periodic quizzing would help students to more easily retain information. Also, more interactive approaches such as mini group problems or allowing students to write problems up on the board and explain it to their classmates. I think this builds confidence and helps the students to understand the information better. Taking time to review material and connect past ideas is also a strategy that I believe would help students grasp concepts.

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

Allowing students to ask questions on the homework before it is turned in promotes students' learning because they get to have teacher feedback and learn the correct way to do the problems. Also, when teachers allow students to work together, students are learning from others and also solidifying the concepts in their own minds. I believe that periodic quizzing also helps students remember information easier and helps them start preparing for tests. I also think it is beneficial when teachers change things up a bit and throw in worksheets or group projects so that math doesn't become as tedious to the students.

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

Although I said working in groups can promote students' learning, sometimes I believe it can be detrimental because certain students rely on others' knowledge to get by, without understanding the material for themselves. Also, I believe it can be detrimental to a student's learning when they are always meticulously graded on homework. I feel as though homework is still a part of the learning process, and if it is graded so strictly, some students can start to lose confidence in their ability to understand the concepts. Another area that I feel to be detrimental is when too much homework is assigned, as the subject may become more of a burden and the zeal for learning can be lost. I also think it is important for teachers to explain why things are so. Sometimes this step gets lost in teaching students and their understanding becomes more memorization than anything else.