Entry #5

There are many advantages that can result in mathematic classrooms when teachers do not tell their students the right procedures or answers. One advantage is that students are encouraged to think deeply about mathematical concepts which results in a deeper understanding. For example in Warrington's paper she provides us with an instance where a child who realized all parts of 4 2/5 needed to get divided by 1/3 in the problem 4 2/5 divided by 1/3 . The child was able to come up with the correct answer through reasoning and thinking rather than relying on an algorithm. A second advantage of this method of teaching is that students are used to relying on themselves for solutions and thus a typical "I don't know how to do this" does not surface as often. Students learn to develop intellectual autonomy. Warrington showed that even when she moved from 4 divided by 2 to more difficult problems such as 1 divided by 1/3, 1 divided by 2/3, 1/3 divided by 3 etc. that the students did not surrender their thinking, but rather stretched it to find an answer. A third advantage is that students learn how to interact with each other and how to critically think about problems, rather than just listening and memorizing what the teacher tells them. Referring back to 4 2/5 divided by 1/3 in Warrington's paper, we see that the students reasoned with each other and listened to other ideas that would test their own in order to reach a consensus.

While there are many advantages for this style of teaching, disadvantages are also present. One down side is that if students aren't ever told what the right procedure or answer is, they may develop some errors in their thinking as did Benny. Another disadvantage is that some students will understand and develop ideas faster than others, which could mean that not all students will develop intellectual autonomy. The slower students may just rely on others and could even get more confused than if the teacher were to tell them the right answers due to all the different ideas being discussed. In Warrington's paper she says that some students thought 1 divided by 2/3 was 6, but when other students had ideas to disprove this, they had to modify their thinking in deciding 6 was incorrect. However, it could be the case that the students just accepted that 6 was wrong when challenged and went along with the answer of 3/2 without really thinking about why. Some students aren't as apt to speak up, ask questions and share their thinking, which is essential for this style of teaching to be successful. As a result, many students may get lost. Students are often looking for feedback and if the teacher never tells them they are right, it may be hard for them to have the confidence to use their ideas when learning new material. Students may also never find the quick algorithms to help them and could spend lots of thinking time over dividing fractions when that is just one step in obtaining a different end result.

Entry #4

Von Glasersfeld talks about constructing knowledge, which means that everything we experience is interpreted through a lens of things we already know. In other words, knowledge is subjective. There isn't just one form of knowledge that we acquire or gain. In fact, no one can even be sure what knowledge really is, and thus the definition of knowledge becomes a theory in and of itself. The things we learn about and the experiences that we have are stored in our minds a little bit differently than the next person. Even when it seems as though people think about something in the same way, there is no foolproof method to ensure that knowledge is being processed the same way in peoples' brains. Since what we learn is tainted by experiences we have personally had, knowledge is something that is constructed. This causes what is "true" or "correct" to be obscure, which in turn implies that knowledge is viable as long as it does not clash with our experiences.

The principle of constructivism can be implemented in the way we teach. From a teacher's perspective would come a realization that even when it seems as though students understand a concept in the same way that they do, it is really not that obvious. The output students give when assessed may look and sound like the teacher's understanding. However, the students may be interpreting concepts in relation to different experiences, which could lead to an understanding that is not exactly how the teacher wanted them to think about it. Thus, as a teacher, I would try and assess the students in many different ways. I would have them justify and express their thinking by writing in journals, talking with one another, writing on the board, drawing pictures, creating projects, using technology etc. that would help me to grasp how students are thinking. This would also be a means of creating experiences for the students that they can latch onto and associate with the material that is taught. I think it would be useful to ask specific questions that would probe students' thoughts beyond the surface in order to really see how they understand a concept. I would take this approach from the constructivist standpoint in which I realize students are connecting and interpreting concepts based on a variety of experiences, meaning they will probably need different ways to best learn and to best display their diverse ways of thinking.

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.