Wednesday, September 30, 2009

Marked Classroom Instruction

Mr. Dave Hewwitt's approach to classroom instruction is different from what I have seen in my past years- both as a tutor/cram school teacher and as a student. It is an interesting concept, though I am not sure if I'll be able to incorporate it in my own teachings. His teaching approach is quite interactive and engaging- leading to the impossibilty of being unattentive in class.

I appreciated the fact that the classroom environment he created was one that was independent and his teaching methods were decentralized. This approach would take time for the teachers inexperienced and experienced alike to cultivate- as the roles of a typical student and teacher are redefined. As students, they are no longer merely sponges, but rather math scientists. I enjoyed watching this clip as it allowed me insight to how one may approach a class in order to have students learn independently.

One key thing that I noticed and will need to work on if I wanted to use this method is having a LOT of patience. Without the patience, Mr. Hewwitt would not have been able to go from the simple basic building blocks to subtly incorporating higher learning concepts into the lessons. It was very encouraging to note however, that once the basic foundations were set in place (over repetition and drawing out the basic concept), more difficult concepts seemed to come more easily to the students.

One method of technique that I find particularly attractive is his use of incorporating the whole classroom to answer his questions. Instead of picking on ONE student, all the students have the chance to call out the answers in unison- nullifying the chance of a student feeling picked on or being ignored. This led to the classroom atmosphere to be a cooperative learning environment where students worked together as a team to strive towards the same goal instead of a competitive one where they did individual learning. I believe that by doing so, Mr. Hewwitt was successful in forwarding the progress of the students as a whole. Furthermore, this allowed the chance to have the students to self check their concepts and to make sure that they understood what was going on in class.

I believe this is a very ENGAGING and progressive method to incorporate mathematics into the classroom setting.

Sunday, September 27, 2009

Group Summary of Assignment 1

We chose to pose similar questions for the 3 math students and 1 math teacher that we interviewed. Although we asked many more than the ‘5 Burning Questions’ required, the interviews were able to be summarized and contrasted nicely.

The Math teacher we interviewed has been teaching for over 10 years and now mainly focuses on the senior math courses, generally Grades 11 and 12. He is well known among students and faculty and is apparently quite popular within the school environment, many of his former students still keep in touch through email and visit with him surprisingly regularly.

Our student interviewees happened to run the grade gradient and we were able to have one A student (graded at approximately 90 – 95% across all her high school math courses), one B student (regularly scores approximately 75 – 80% on tests) and a C student (who wasn’t terribly concerned with her math scores). It seems presumptuous and slightly disheartening to be classifying our students by their scores, but we thought that it might help to add a sense of significance to their attitudes towards math. Also important is that the A student and the B student are both in Grade 12 Math and the C student is currently in Grade 10 Math. All 3 students are in the Principles of Mathematics stream and have no interests to seriously pursue Math after high school.

The interview with our C-student was shorter compared to the others and was therefore fairly stilted. She informed us that she was only studying math because it was required and would probably have stopped if it had been optional except that her parents are making her. She also indicated that she didn’t like math because ‘[she wasn’t] good at it’ and that she had a peer tutor to get her through the course. However, she there are times where she does enjoy math classes and those are the ‘rare’ times when she feels like she understands a concept. The classes she does enjoy are Music and Science because she ‘gets it’.

Our B-student is a former D-student whose marks were forcibly pulled up after she decided to do math by distance with a private tutor. She indicated that she had been having trouble understanding concepts brought up in class and was too afraid of looking stupid to acknowledge her confusions in class; having a tutor allows her to ask questions immediately and one-on-one which greatly increases her confidence and ultimately her skills. She said that she was constantly unsure of herself and felt lost in the big classroom where many of her peers understood the concepts and it just made her ‘feel dumb’. She anticipates taking math in a higher level institution only if the program she chose required a math prerequisite. She seems to support the idea of some group work within mathematics to add to the lectures to allow her some time to listen to her peers’ ideas, commenting that ‘math just seems so lonely’.

The Grade 12 student who had been pulling an A-grade had a slightly different take on her math class than I had expected. Her attitude towards math was not a question of like or dislike but of competition. This was a class that she felt she could compete in and makes an effort to do so. She likes having other people consider her to be good at it and so she works hard for her grade. She is taking Math because she feels that it is an essential skill for people to have and that it provides her with the option of taking sciences in post-secondary. What surprised me was her attitude towards rectifying confusion during math lectures. She said that she was too intimidated to ask questions during class and would often just relegate her attention to copying down the notes so she could review on her own or ask her teacher after class. According to her, math class would be more accessible if teachers would take small breaks during class to diverge attention elsewhere for a short while. This would allow her time to digest the information and refresh her mind so she could return to the lectures with a renewed concentration. Her ideal math teacher would be kind, understanding and fun.

The interview we had with the teacher was equally informative, if not more so. He likes to teach from a relational standpoint and stresses that a student who aces tests is not the same as the student who really understands the material. It was important for us that he did not think that it was a challenge to get through the curriculum in the proposed time line. In fact, he thought that there was a lot of time and the challenges of teaching were at a far more personal level. He warned us to ensure we are friendly towards our students but not be their friends because teachers are still in a position of authority and must have their students respect them as such. He also advised us to take our practicum seriously and treat it like a ‘real job’, not just as a ‘practice run’.

Our group really enjoyed this project and found that the varied and sometimes surprising answers to our questions helped us to be more aware of how everyone else views math. It is easy, as students who had relatively successful highschool math class experiences, to forget the challenges that others may have. We expect that this will help us in our practicum and future teaching posts to be aware of the difficulties some of our students face and to encourage them and adjust our teaching accordingly.

Saturday, September 26, 2009

Individual Reflection on Assignment 1

From this assignment, I had the opportunity to hear opinions from the perspectives of multiple teachers and students. This was a great opportunity and I had learned many things from this experience.

A few notes that I found really interesting from the students perspectives included:

1. Teachers need to bring his/her own personality to the classroom.
Though I already know that all teachers have their own teaching styles and they are can be equally effective, I forgot how crucial it is to a student's learning. I was surprised at how much emphasis the students had put on this point, and at times even stressing personality and enthusiasm was more important than the material itself! I'm glad many students think this way, as it strengthens my resolve to be a nurturing teacher that sincerely cares about them and is passionate about their future learning.

2. The speed of the classroom instruction should be slower.
I have a slight fear that I might teach too fast for all the students to keep up. In particular, I'm worried that I will speed up in my teaching because of the time constraints to finish teaching the curriculum material. But after talking to these students, I realize that I need to worry less about cramming all the material into their knowledge, but instead, focusing on pacing myself to make sure the students understand what is going on in class. And if necessary, I should making myself available before or after school hours.

Things that I found interesting from the perspective of teachers:

There were many tips that I had picked up from teachers during this assignment. One thing that I found useful was teaching by modeling. In particular, to show how we answer our questions by working through it in class, thinking out loud (in a clear and precise manner), and doing it that way instead of having the students copy things line by line. By doing it this way, there is an emphasis on how we can find solutions in mathematics and not just emphasizing the results. A second point that finally got ingrained into my head was the importance of being FRIENDLY but not being friends with the students. It's OK to be nice, but as teacher candidates, we need to be able to take control of the class and be responsible for them. As well, though I hear it often, I had never realized the value of lesson planning until talking to the teachers. Although a lesson plan doesn't set the lesson into stone and there are always adjustments to be made, it is really important to have guidelines. I had also want to make sure that as a teacher candidate, I treat the practicum not just as a practice run, but as a real job. Because it IS a job and we are responsible for the growth and development of the students. I want to be able to act in the students' best interest.

Although many of these points are things that I had learned or heard before, I had never made a personal connection with it. Though the facts were in my head, sometimes, it was not something that I valued as much because I did not make the concept my own. However, because of the interviews, I had the opportunity to talk and see the opinions of students and teachers firsthand, and as a result, I realized how important it is as teacher candidates to make sure that students and teachers have the same goals. These concepts will be a quite the challenge that I look forward to working on within a month!

On an added note, we had a few more presentations the following day, and I just wanted to remark on a few thoughts that came to mind. With Mike's presentation, they had interviewed teachers and students who had undergone the "Learn on your own Pace" math model. This was a very interesting concept and I believe that one should look into and consider this a great option. It provides flexibility to students who already know what they want to do with their lives and know that math is something they merely need to finish. It also allows students who strive academically to work ahead. It was an interesting proposition that shows other methods of teaching asides from lecturing in class. A second note that I want to reflect on is Jill's group and how they discussed the use of technology. I had found it interesting how many teachers view using technology as a hinder to the lesson. It is something to consider- will the students be able to learn mathematical concepts by using computers in mathematics or will they become distracted and lose the main point and purpose? There is a fine line that we need to be aware of and though technology can be a great aid and tool in working alongside mathematics, it is tricky to teach with technology and not have the students lose the focus and objective of learning a certain concept.

Friday, September 25, 2009

Battle Ground Schools

Battleground Schools written by Gerofsky summarizes three "battles" concerning mathematics in the 20th century that occurred in North America. In particular, there was the Progressive movement, the New Math Reform movement, and the "Math Wars". In general, all three of the debates concern different stances in mathematical education and how we should approach teaching students the subject.

The first period, the Progressivist Movement, emphasized and criticized school mathematics as "meaningless memorized procedures". The crux of the argument was that though the students studied and knew the algorithms to arrive at a solution, there was no flexibility because the students would not know any alternative methods to achieve the answer and did not understand the concept (the WHY) of why one would use the procedure to solve the answer. With this line of thinking, Dewey (one of the leading figures of the Progressivist Movement) revolutionalized the idea of the Montessori classroom. The focus of this classroom was not to memorize and implement, but rather to experiment, inquire, and interpret by having students actively engage in activities. Though many teachers stuck with the tradition of lectures and homework exercises, some classrooms had put Dewey's inspirations into practice.

Over time, the New Math movement began. The trigger for this was the space race between Americans and the USSR. The main fear was that Americans were losing in the space race due to the fact that "school mathematics was not keeping pace with ... research level mathematics". With the goal of creating a generation of "elite rocket scientists", mathematics in education began to change once again. The whole curriculum from grades K-12 was changed and the goal was to "create a unified, logical, highly abstract algebraic structure based on set theory". In order to help the students become familiar with mathematical notations and concepts used in the sciences, many topics from university were taught in from grades K-12. However, there were major problems with the New Math movement. Many teachers had little knowledge of the new material and would have had trouble teaching children. For homework, parents would not be able to aid their children with their homework even in elementary grades. It was difficult to justify why students were taught this material as not every child planned or wanted to become a rocket scientist. In the end, New Math was scrapped as a misguided experiment.

Finally, we come to discuss the Math Wars over the NCTM Standards. This is a "battle" that has started in the 1990's and is still occurring. The focus of the Math Wars once again is, "How do we best educate our young in mathematics?". Should we focus on using a progressive approach or using a traditional method of teaching? With the NCTM Standards in 1980's, there was a focus to have "flexible problem solving skills", using mathematical relationships in different forms, and to use technology as an aid to solve math problems. It was a progressive step to help students appreciate the "beauty of mathematics". However, by 1990's, a "backlash" occured and there was protests against the NCTM Standards. There were people who prefered the traditional and conservative style of teaching.

Hence, over the years, though there has been drastic changes and movements in education and mathematical pedagogy, the question still remains, is it better to have instrumental or relational understanding of mathematics? Is it better to lean on one side or to have a balance of both? I believe that there are both positives and negatives behind conservative and progressive stances in mathematics educaion and a balance should be struck between the two. However, without trying and without experimentation, one can not achieve this goal and "perfect" this way of instruction. When the general public is concerned, change is not always welcomed with open arms, and I believe that is why "battles" over mathematics education is always brought up. One thing that I had not considered much before but am much more aware of now is that the STANCE of which form of teaching is "better" depends much on what goes on with the world around us and the goals of the nation. Do we want to create a generation of rocket scientists? Do we want our children to be "practical" and robots? Would we like them to be philosophical and inquisitive? The culture around us will dictate what the public requires and desires. And what the public desires will inevitably end up effect the way math education is taught.

Monday, September 21, 2009

Two Memorable Math Teachers

Throughout my life, I've had various math teachers- whether it was professional teachers in elementary, high school, and college or the various tutors or friends that had helped me along the way. Today, I'd like to talk about two teachers that had made an impression on me.

My earliest childhood math teacher and I believe the most memorable one is my mother. As a child, I remember her teaching me the basics such as addition, subtraction and multiplication. On a semiweekly basis, we'd go over the material through mandarin textbooks that we brought over from Taiwan cram schools. When I was learning the material, it was frustrating at times for two reasons. One, I had trouble reading the Chinese. And secondly, when I asked her WHY the formulas or algorithms were the way it was, she more often than not could not explain, and told me to memorize it. What was worse was when she told me the reason was "just because that's the way it is". I remember one instance where I was learning about multiplying negatives. In particular, just memorize the fact that two negatives make a positive and THAT was just the way things worked. It was very frustrating not to know the concept and half the time I learned math, I made up my own reasoning for why things worked.

Another memorable teacher I had was in high school. He had a tendency to give an example or question at the beginning of class for us to ponder over, and then went through the classic lecture approach to teaching. But what was different and more engaging to the lesson was the atmosphere he had set up when teaching us using this method. He had set a relaxing and at ease atmosphere that allowed us to ask questions if we needed to and gave us a chance to have class discussion. As well, while he approached the examples he wanted to give to us in class, it was not simply a process of copy and pasting information from his notes to the overhead/whiteboard. Instead, he showed us his way of thinking and worked out the problem with us during the class. By doing so, it helped us stimulate our minds and try to work out the problem in our own way and see if we could obtain the same results as he did. I believe that by giving us the tools we needed to work with first, and then having us work on a problem together in class and seeing if we used the same method of solving a problem- he showed us that there was always multiple ways of deriving an answer.

From the two different experiences, I want to try to reflect on the way I plan to teach. Though I believe in integrating the method of instructional and relational learning, I need to find my balance. I believe that perhaps giving the students the tools to solve the problem is very important, but what is equally important is to give them the concept behind using the formulas and such. As teacher candidates, we should not be having our students blindly using formulas without understanding deeply why the formulas work. As well, giving the opportunity to vary up the lessons such as through visual aids and group work is something that I hope to continuously work on.

Saturday, September 19, 2009

Reflection of Origami Microlesson

Summarizing my peer's evaluation of my lesson:

From my microlesson, the summary and post-tests were too rushed due to the lack of time. For the participatory event, instead of each person taking a turn and then testing it, perhaps each student should have a sheet of paper and participate in folding the crane. As well, it was a bit hard to just let students learn by looking at others doing the steps. Though my explanation of the procedure is clear, I must consider the timing and if a crane may be too complicated to fold in 10 minutes or not. One thing that I didn't notice before was my behaviour. It is good that I have nice eye contact and am friendly, but I use the word "so" too much and need to get rid of that "pause" word.

Write up of my own evaluation of lesson:

I believe that the structure of my lesson and the way I flowed through the BOOPPPS went well. I realize I enjoy telling others about the history of what I'm teaching to others a bit first and it was good that my peers were interested in it. I'm happy with the participation my colleagues had when doing the post test and it was a good way for me to assess how much they learned from the lesson and where I may need to be more clear in instructing my students.

However, if I were to reteach the lesson, I would definitely need to consider how to manage my time better and figure out a way to teach this lesson within the 10 minute time span. As well, even if my time was coming up, next time, if I was short on time- I would not rush my summary and instead, I would tell them the summary and perhaps only have them fold up to half a crane in the post test. Another thing I would have ready would be a diagram with the instructions put up for students that may prefer to learn from paper rather than an instructor or for people that already know how to make certain basic steps of origami. It is beneficial for students to be able to have alternative ways to learning. Finally, to save time, I should keep my introduction less short so I could get into origami folding more quickly!

From my peers, I believe we all agreed that time management is something I need to improve on. Though it is important to go at a pace where students can keep up with, I will then need to be more selective on what I want to teach. As well, though I never realized it before, I have replaced all my "uhm" with "so", and I should get rid of use that word in my oral vocabulary more often.

In summary, though there are many things I need to work on, a few that I will try to improve on first would be my timing and selection of what material to go on. To incorporate more methods of learning during my lesson. And finally, correcting my vocabulary.

Instrumental Understanding vs. Relational Understanding

Thursday, September 17, 2009

BOOPPPS! Project

Min-Chee's ORIGAMI ACTIVITY!

Bridge: The bridge in this activity will be showing my students three different objects, one at a time. First, I will take out an origami rose and ask them what they think it is. Then, a crane, and finally a jar of paper stars. Ask them what in common these objects have with each other. Then talk a bit about the meaning of origami.

Teaching Objectives: From my lesson, I hope that I will be able to incorporate my lesson in such a way that my students will be able to learn in a very hands on fashion and through modeling.

Learning Objectives: The students will be able to learn how to fold an origami object based on their levels and as well, develop an understanding and appreciation for the joy of this folding art.

Pretest: Ask if they have had any folding experience before, and if they did- what have they folded?

Participation: I will demonstrate how to fold the object, and in a circle, each student will take a turn folding.

Post Test: Hand out origami paper to every student, and assess them by seeing if they are able to fold the objects on their own. If they need a helping hand, have the student next to them help them out a bit since teaching one another is a great method of learning.

Summary: By the end of this lesson, the students will hopefully be able to fold an origami creature. If not, they will be able to learn at least a few basic steps of folding.