Reflections

The blog entry that stood out most for me was Education Philosophy Interview with Jason from October 17. While I was quite confused and unsure of what to do when we were first given the task of finding four objects that represented our education philosophy, this activity ended up being one of the more memorable events in the course. Thinking about how my educational attitudes and beliefs could be represented by objects was sorta weird at first but when I found my objects and had the opportunity to explain them to Jason, I was very satisfied and notably happy. I found that associating my beliefs to something concrete really helped me cement my education philosophy in my mind. Seeing Jason’s objects and listening to him explain what each of them meant to him was also interesting. I was nice to hear what he thought was were some of the most important aspects of education and comparing them to my own. I’m not sure if these were the intended outcomes of the interview activity (sort of seemed like it was supposed to be a lesson on interviewing skills) but this is what stuck with me.

Two things that changed:

1. I see myself more as a teacher of students and not just a teacher of math/physics
2. I realize that I have a lot to learn

Two things that have stayed the same:

1. I am very happy with path that I have chosen
2. Getting to classes on time is difficult (This will not be a problem when I start my long practicum!)

Inquiry Article

Title: A Special Kind of Tutor

Author: Linda Brown Anderson

Publication: Teaching Pre K-8

Year of Publication: 2007

Description: Anderson describes how cross-age tutoring can not only increase learning for both tours and tutees but build up a sense of community in the school. She explains how contrary to what many may believe, the tutor benefits the most from tutoring. The most shocking thing about the article is that Anderson teaches students with emotional and behavioural disorders and she implements the peer-tutoring program with these students as tutors. She goes on to say that these students showed improved academic performance following their tutoring. She also describes three successful models in place in the US in both elementary and secondary schools. It may be important to note that this study was done on elementary-aged children.

Questions from Class

On Late Assignments:
While we cannot deduct marks for late assignments, are we able include work habits in our grading criteria where handing in assignments on time would be taken into account?


On Testing Practices:
How effective is having multiple versions of a test within a single classroom at reducing the amount of cheating? Will students find a way to cheat no matter what you do?

Is having a test that takes students the entire block to complete too stressful for students? Or is it better to have a shorter test taking maybe half the block leaving the remainder of the class for other activities?


On Assessment:
Should students who know their material be required to complete all homework assignments? Would it be good to make homework optional and have it considered more as practice questions rather than a form of assessment? Or would that lead to many who need the practice not doing the work because of the lack of extrinsic motivation?

Can students still find value in homework assignments if we don't mark it? Or should we be marking all homework assignments?

Should effort and student ability be considered when marking tests?

Refocusing Our Efforts

Throughout the article, I thought of my experience in high school and what motivated me and from what I remember, from grade 8 to grade 11, how I did on tests was important to me but it was not my primary motivation. For the subjects that I was interested in, it was the intrinsic motivation the pushed me. It think it is important to note that in grade 12, the extrinsic motivation of marks and competition became a big factor in my efforts in school. I wanted the high marks to get into university and I am confident that this I am not the only one that felt this way. I also think this is quite interesting and possibly puzzling that I feel that of all my years of schooling, this was the time that I learned the most. While my grades in that year were the highest I’ve had, I would not say that is the reason why I consider this to be the period of the most learning for me. I think that I say that I learned the most in this period because I believe I have retained the most information. Thinking about the importance of the grade 12 year, I wondered how their study would have gone if they had chosen a Physics 12 class instead of Physics 11. I was glad to see that they mentioned this at the end in their reflections. I think changing the focus to learning will be tough and I think, as pointed out in the last line, that it requires changing the culture of grading.

Using Research to Analyze, Inform, and Assess Changes in Instruction

The first part of the article that stood out for me was when she talked about how she “viewed instructional strategies as hit or miss” but how she had “come to realize that research can inform the selection and implementation of instructional strategies and, as a result, make a huge difference in whether a strategy hits or misses.” This stood out for me because I have heard many times this year that it is inevitable that some our lessons will flop. Taken the wrong way, one could easily just take a flopped lesson to mean that it just wasn't a good day and fail to reflect. I think that it is important to remember that while your lessons may not work out as well as you may have hoped, there is often something that we can adjust that can improve the likelihood of that lesson being effective. With our lessons, it is important to think about why something worked our didn't work so we can change our selection and implementation of instructional strategies.

What was also interesting was the section about higher-level thinking and how he implemented these higher-level questions into his class. In my experience, if you go through any math textbook or test, you will find very few questions requiring deeper thought. We did go through Bloom’s taxonomy in another course but, honestly, I don’t think I would have even considered it when writing a test or assignment just because it is something so uncommon in most math classes. This article really got me to think about how important it is to promote this critical thinking.

Inquiry Project - Student Leadership

What is your question and how did it arise for you?

What are the impacts of increasing student leadership in schools?
How can we improve and implement student leadership programs?

Why is your question significant (to you and/or to others)?

I currently am part of a youth leadership development program at the YMCA and the practicum coordinators at Tupper have recently implemented a student leadership program aimed at involving students in the development of student teachers.

I hope that my findings are able to show how leadership opportunities affects students involvement in schools and if there are any other positive (or negative) outcomes.

What resources will you draw on to explore your question?? (e.g. journal, readings, curriculum/policy documents)

journals, readings, interviews with students and teachers

What do you expect to find out?

I expect to find examples of student leadership programs which have been implemented in secondary schools in North America and how these have impacted the students and the school community. I hope to be able to get these findings from both a researcher, teacher and student perspective.

Reading the Inner Life of the Young Person

I found many parts of this article very relatable. I could easily identify with the students who felt embarrassed for being recognized for doing well. As a student, I did well on math contests which were required for the accelerated math program I was in. When I would be recognized in grade assemblies, I would feel that I was put into the same category as other students who were often considered uncool and often talked about negatively by my peers. I would feel good for doing well and having my name in a plaque but I what was more important to me was how my peers viewed me. When grade 9 ended and I was no longer required to take these exams, I stopped. I didn't want others to know that I would be volunteering spending time doing math. 

I agree with the van Manen’s assertion that “ a good teacher is able to read, as it were, the inner life of the young person.” Looking at my situation above, I believe that our approach as teachers should not be to stop recognizing the efforts of students but to do so in while being sensitive to students’ relationships with their peers. At the same time, I believe we should try our best to remove this uncool stigma attached to math contests and similar activities.

Short Practicum Experience with Math 8

On my short practicum, I taught a class of Math 8. It was a review session coving the multiplication and division of integers. It was interesting because they were required to learn how to use number lines and algebra tiles to show how to solve equations but this was something I had never learned. It was clear to me and their teacher going into the class that they were struggling with using these strategies. I began the class by asking naming certain areas of the chapter (ex. using number lines, using algebra tiles, solving multiplication/division problems using any method, order of operations, word problems, etc.) and how comfortable they were in them and they seemed to have a pretty good idea of what they needed to work on. I found it difficult to fit in everything that I wanted to do. I planned for a 60 minute review session but the quiz that my sponsor teacher planned took longer than expected. I did plan what I needed to cut out but I found that I still didn't have enough time for the shortened plan. In the end, I feel that it would have been better to focus entirely on one area that I had previously determined to be an area that they needed help on.

I also found that I should be more clear with my instructions - especially with grade 8s. It is becoming apparent to me that the more clear I am in explaining in exactly what I would like them to do, the easier it is to keep them on-task.

The Ideal School

I believe a school should be a place where all students can come to learn and grow into responsible citizens. It should be a place where a student feels a sense of belonging. The three most important aspects of a school which promotes unhindered learning and growth would are:

Choice
The ideal school should recognize that students have different goals and not all may want to pursue a career requiring a university degree. Students in the ideal school should have the opportunity to choose. This means that our school must provide the options to them. These choices can include electives (shop, fashion, foods, business, visual arts, etc, etc, etc) but can also include their workload for a year. There should be graduation requirements but these should be adjusted to take student choice into account.

Well-Rounded
In my opinion, school is not just a place for academics. I believe students should have the opportunity to take part in extra-curriculars and electives. For athletics, both intramural and varsity sports should be available and year-round conditioning should be available to all team members. As we discussed in the Choice section, students should have the opportunity to register for electives. The ideal school will not treat electives as lesser subjects and will provide adequate resources for them to run well.

Access to resources
The ideal school recognizes the diverse learning needs of students. They will provide additional support to those with learning disabilities, ESL students and anyone else who may benefit with extra help. Class sizes will be small enough so teachers have the opportunity to connect with each student individually but large enough for group activities and discussions. Homework support is provided both in school and online and counsellors are available for anyone who might like to talk to one. Technology is readily available and its use is promoted if used for educational purposes.

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To my possible inquiry topics list, I would like to add:

the correlation between athletics and academic success
private schools and the availability of electives

Teaching Themes of Care

I believe the teaching themes of care are we really should seriously think about and I think Noddings shows very relevant examples which we can use as a model. In organizing interdisciplinary units on themes of care, I like the variation on his second way where the particular themes which are focused on are up to the teacher and their interests. I believe to effectively implement these strategies into the classroom, the teacher must be wholly invested in the topic. They must show interest and be prepared to handle the big questions that are are sure to come up. I believe this is especially important because if interest is not shown or lessons are unorganized, there is the definite risk of students becoming apathetic towards these important issues.

While I simply don’t see the complete reorganization of the school curriculum that Noddings hopes for, I think there are certainly things we can do which are more achievable and more within our control. The discussion above about organizing interdisciplinary units is surely possible without a complete reorganization of the curriculum. More easily however, and more importantly, in my opinion, is simply modeling the ethic of care. While, as secondary school teachers, we may not see each of our students for as much time as elementary school teachers, I believe we can still have an impact on our students through how we show are care. I think that if we are able to demonstrate a genuine care for all of our students, not only are we being a positive role model but I believe they will naturally be be more open to the topics of caring you choose to share with them.

Mathematics Text

The first part of this text that I could relate with was the difficulty students find with the large number of mathematics symbols and graphics. As a student, I remember being able to develop an understanding of most material quite quickly for my high school years but when I started doing higher level mathematics and I began finding more and more weird symbols being used in the text and in class without explanation, I found myself confused and frustrated. I think that it is important that I remember this and do my best to help students gradually build these symbols into their repertoire. With all mathematical terminology, as this text argues, we should not be afraid to use it, but “allow students to grapple with their ideas and develop their own informal means of expressing them” while “avoid[ing] a premature rush to impose formal mathematical language.” I also liked the section about teachers contributing to students’ confusion of illustrations with definitions. I know I am guilty of always using the same picture for right angle triangles and I think it is a habit I should learn to break.

Something that I often think about is the use of textbooks. In my high school experience, most teachers used them - some a lot some a little - and one teacher was super anti-textbook (he is the Mr. I that I talk about in my memorable math teachers blog entry). I have been leaning towards the no textbook side but I think the text presents a valid argument that with proper instruction of how to use it, the textbook can be a valuable resource.

Microteaching Reflection

One suggestions for improvement that came up an many of the feedback forms was that it would be better to have clearer learning objectives. I feel that one thing that contributed to this unclarity was some confusion in how were going to handle the discussion of the logic games. We had discussed ideas as a group, then typed up a lesson plan, but when it came down to actually implementing it, we found that we lacked the time, we did a bit of improvisation. I found it interesting that a common positive on the feedback forms was that our lesson seemed organized which I’m not sure I would agree with.

I think our group did well on engaging our students. We did not find this too difficult because of our topic but it challenges us to find ways to make math fun for other topics. Another pro was that lesson was mainly student-led. I think that this is related to how many thought our lesson was engaging. I feel that if students feel like they are developing their own understanding and not just sitting listening to someone lecturing them, they are more likely to be engaged and learn. It is important, however, to make sure we provide adequate guidance in these more student-led lessons.

Education Philosophy Interview with Jason

Describe your interview of Jason.

My interview with Jason went well. I see the value in having a pen and paper with you as with some of the questions, he continued talking for a long time. I didn't want to interrupt him so I let him continue but I found it difficult to remember all the ideas. Besides the pen and paper, I think this could have been resolved by having questions that were a bit more focused (My first 'question' was "tell me about yourself" and was followed by a very lengthy response involving all his objects). I also found the importance in listening and that because you cannot predict what your interviewee says, you must always be listening so you are able to build on ideas for which you may want more information.


What is Jason's education philosophy?

In my interview with Jason, he brought out 4 objects.

The first was a book of logic puzzles written in Chinese. He explained that to him, this represented the diversity in the classroom and how we should use different approaches to reach these students. He explained that this diversity can come in a variety of forms (racial, cultural, religious, etc.) and that we should model acceptance for all.

The second was a white out pen which he noted was not seen as often in schools anymore. He said that students are using pencils and erasers or just crossing out their work more often than when he was in school. This, to him, represented the changes in the general teaching methods. He noted that our teaching now is more accommodating of a variety of learning styles. He also said how teaching methods are not universal and that for him, he has moved away from a teaching approach involving mainly a lecture style lesson followed by practice and more towards an approach involving students guided towards developing their own understanding.

The third was a pair of earphones. This represented two ideas: 1) the importance of listening in the classroom and 2) the ability to help students focus by blocking out distractions. He elaborated by saying that as teachers, we must learn to listen to our students; hear where they are coming from so we can see their troubles and then guide them. He also said that students must also listen and we can encourage this by creating an environment that helps them focus. To him, this meant that we block out distractions and this can mean different things to different students. He continued to say that some students like to be alone so we should recognize this and provide them space to learn individually while some are more group learners and social interaction may not be as much as a distraction to them but there is always a balance between group and individual work.

The final object was a battery recharger which represented what it sounds like - recharging. He said that as teachers, we must find the time to relax as our jobs can be tiring. For him, this meant sleeping, taking vacations and making the most of weekends. He also noted that we cannot forget that students need time to recharge as well. We can help them by making class not seem like a routine task by mixing things up: having a variety of activities so students are excited to come to class everyday. This could also mean mixing up homework as well. We can provide them some group projects, individual homework assignments and even some days without homework so they can just take a break.

Being the Interviewer and the Interviewee

As I was reading this article, I kept thinking about why we were given this article to read. Not having a clear perspective to take on as I read this article allowed me to read this article as if I was learning how to prepare to be interviewed as well as learning how to engage in questioning students.

As I was reading this article as an future interviewee, I tried to get an idea of the questions I could expect as well as the thoughts that might be going through the interviewer’s mind. While I may not know the exact questions being asked, knowing that interviewers often like to hear stories from my experience allows me to ready some of my experiences in advance. I also found the idea of interviewers listening to an “inner voice” important to note. Controlling whether I use my public voice or inner voice may not be something that I can easily control but I am sure that just having an awareness of this difference will help in letting my guard down a bit.

I also felt that there was must to be taken from this article as I was reading this as one who would be at the front of the class ‘interviewing’ students. Going back to the idea of listening to an inner voice, I believe this is completely determined by how the interviewee views the environment. If we are able to create a environment which is respectful of the thoughts of others, the responses we receive will be unguarded. This means that we should be careful not to shut down ideas be they wrong or right. The idea of wait time comes up in the section about tolerating silence. What this article brought to my attention that others had not was the idea that our discomfort in silence is projected onto those we are interviewing. If we interrupt their thinking processes with our silence-breaking questions, how are we making those we ask the questions to feel?

For both sides, I believe that while we may prepare questions ahead of time or prepare for questions ahead of time, we must not stick too rigidly to these. The questions we ask and the questions we receive will change according to the responses we give and receive and we must prepare for that.

Flipped Classrooms

A big issue concerning flipped classrooms is the availability of technology. As we cannot assume that all students have sufficient access at home, we first must be able to determine this. Assuming that technology is available, the implementation of flipped classrooms is certainly worth discussing. I believe there are many arguments for and many against and at the moment, I have not taken a firm position on one side or the other.

I believe a very important part of teaching is interaction - both student-student and student-teacher. With flipped classrooms where much of the material is conveyed online, teachers are unable to see where students are having trouble. They are unable to see the looks of excitement when a student grasps a concept nor can they see the looks of confusion when they do not. Can this be addressed in the classroom portion of the flipped classroom? This is a question that I will ask myself continually as I think about the idea of flipped classrooms.

On student-student interaction, flipped classrooms, where the lesson is present online at home, frees up class time which can be used for a greater variety of activities. This can include group work and other activities that allow students to bounce ideas off each other. Concerning this, I can see this being made less effective with students allowed to move at different paces as we will now discuss.

Student are free to move through lectures at their own pace. This means that students who easily understand the material can move ahead while those who need more time are able to rewind and spend more time on topics they have trouble with. An issue with this is related to what was said above - the teacher is not there to determine why the student might be having trouble. Is re-watching a video going to help a student understand better? I believe that sometimes it can but in most cases, it won't. Again, I wonder if this is an issue that can be addressed in the classroom?

The Art of Questioning

Reading this article made me really think about how I tend to ask questions and I feel that I really could use some improvement. I believe that many questions that we ask should involve students thinking about what they know and why that may be important to what they are solving. That means that what we look for as teachers should be more than a correct answer but an understanding of how our students are thinking.

I also think that it is very important that Fairbairn informs us of the prevalent procedure of either answering our own questions or repeating the questions if students do not immediately provide us with an answer. It reminds me of Hoffman and Brahier’s article about the differences between American and Japanese teaching practices and how it is not uncommon in Japanese classrooms for students to work on a single problem for 15 minutes while in North America, students who take even a couple minutes to solve a problem may be seen as slow leading to teachers prematurely butting in an interrupting the student’s though process.

Furthermore, if there are students who know the material very well and are able to answer questions quickly, it does not mean that it is a good time to move on. As teachers, we have to remember that we are teaching to more than just the few “good” students but to a whole class of students who each learns at their own pace. It reminds of facilitating competitive games at camp where teams would race to complete various tasks. When teams had finished their tasks, I would tell them to continue working as if they were not finished so they would not discourage the other teams who were slower. I would often give them an additional challenging problem and let them know that this one was just for fun. In this way, teams who were not yet finished still worked as if they had something to achieve and every team had an opportunity to complete the tasks.

The Geoboard Triangle Quest

There are a few things I found interesting in this article. First, I found it neat how this question was posed by students rather than the teacher. I believe that it is important to foster an environment of curiosity and exploration.

Second, I thought that it was great that not only did the teacher foster an environment of curiosity and exploration but also one of collaboration. It is not the teacher telling them what should be done to find the answer, it is a group working with each other in both their mistakes and their successes. Through this, students learn from each other in a variety of ways and as the teacher noted, many more mathematical terms were used in the discussion which most likely would not have been thought about if students were working individually.

Third. Through this problem, I have seen more of how powerful student exploration in a problem like this can be. This problem allowed students to work with the concepts of congruency, reflection, rotation, right angles triangles, areas, distance and the list goes on. Not only that, but they were also able to learn to work as a group and communicate their ideas. In this case “less is more.”

American vs. Japanese Teaching Tendencies

I enjoyed reading about the differences in the teaching practices between America and Japan. The American structure is very similar to much of what I have experienced as a student. Review old stuff, introduce new stuff with examples followed by assigning homework: the standard format of one of my high school math class. I never have experienced schooling in any other country but I had the impression that mathematics education in east asian countries involved many drills which I am glad to hear is not exactly the case.

The main difference between standard American structure appears to be how teachers handle what Hewitt calls necessary information. The structure in Japan provides the evidence that students tend to learn better when they are given the opportunity to develop their own understanding rather than being simply given the steps required to solve a problem. A statistic that I found a little disconcerting is that U.S teachers gave twice as many definitions and procedures as Japanese teachers. I believe we should try our best to reduce the amount of stating processes which students have the ability to come to understand themselves. Instead, we should provide the activities and environment that enables students to establish their own methods of solving problems. If students struggle, its not a bad thing - it’s an important part of learning that both students and teachers must remember.

Possible Inquiry Topics

A list of topics I may be interested in.

• Computer simulations in science
• BYOD
• Specialized schools such as a math school for the mathematically motivated
• Accelerated and decelerated programs
• Standardized testing - accountability and flexibility
• Group work in mathematics
• Textbook use
• Independent guided studies
• Online courses
• Online learning (ex. Khan Academy)

Conversation

There are many things that we can change and of the features that Eisner describes, I find the idea of increasing individual differences quite interesting. I believe it is important to give students an understanding of a broad range of disciplines however, in doing so, are we holding the “ideal that all students get to the same destinations at the same time.” Are our schools systems designed to give students enough freedom in designing their own path or would we have to develop or alter plans to allow for this?

I felt a little overwhelmed by the number of suggestions Eisner makes in his article however, I feel he concluded it nicely by saying that the features he describes would not be easy to attain. I feel that the most important part of the article was what Eisner presented in the final paragraph where he offers to us an invitation to “begin that conversation in your school, so that out of the collective wisdom of each of our communities can come a vision of education that our children deserve and, through that vision, the creation of the kind of schools that our children need.”

Arbitrary and Necessary

I found this to be a very interesting read. The distinction between the arbitrary and necessary was never really something I thought about before and being made aware of the realms into which these fall, I feel that I am better prepared to teach mathematics.

Being able to determine what is necessary allows me to direct to appropriate topics activities geared towards students using their awareness to understand.

In my experience, I was able to take what Hewitt calls received wisdom and using my awareness, transform this into necessary fact. I fully understand that this is not the case for all or even most students so I must be careful to give activities that open the students’ minds to the concepts and builds on their current awareness.

I must also note that if i choose to take an approach and accompany the received wisdom with an explanation, it is not productive to base my explanation on anything that falls into my awareness but not theirs. As Hewitt states, “A teachers’s explanation is often based upon the teacher’s awareness, and so may use things which students do not find evident - things which are not in the students’ awareness - and so the explanation will not be one which will help those students to educate their own awareness.” As I mentioned in an earlier post, “As I work to become a teacher, I must not forget how a student thinks. Put myself in their shoes and remember what it is like to not know.”

A question I have is whether informing students of necessary mathematics content (received wisdom) can be a preferred method over the alternative in which the teacher works within the realm of awareness.

The Locker Problem

I first solved a problem similar to this a couple years ago. The problem I was given involved 100 lightbulbs and I initially just went lightbulb by lightbulb and seeing whether each was on or off. I remember being too lazy to get out a piece of paper so I just did it in my head. By the time I got to around locker 20, I saw a pattern - all lightbulbs in the positions of perfect squares were on. I realized that this was because the number of times a lightbulb was switched on or off was equal to the number of factors it had and only perfect squares had an odd number of factors (all factors of a number have a corresponding factor which when multiplied by each other equal the number except the square root of a perfect square).

I realize that a student attempting my method may not see the pattern as quickly as I did and may eventually get frustrated and give up. I also imagine that other methods might be more popular. I have seen many listing out the lockers and instead of going locker by locker, they went student by student changing the status of each locker as each student went by it. If a student had this method in mind, the sheer number of lockers might be daunting. If i approached a student who told me that they were having trouble with the number of lockers, I would suggest that they focus on maybe the first 30 lockers and see if they notice anything. Completing this task would hopefully allow them to see a pattern and then hopefully give them the opportunity to think about why that pattern is the answer.

As an extension, you could specify various lockers that you wanted open and then ask for which of the 1000 numbered students should be dispatched to open/close their respective lockers.

Teaching Mission Statement

As a teacher, my mission is to:

1. Motivate students to think and develop understanding
2. Encourage students to strive to reach their full academic potential
3. Provide a safe environment where students are able to express their opinions without fear
4. Be enthusiastic and convey that enthusiasm to my students
5. Encourage involvement in the school and community
6. Promote creativity and the learning from mistakes
7. Work hard and never give up
8. Continually reflect on my teaching and learn from my students

A Perfect Set of Lesson Plans

One thing that has crossed my mind in the past was whether I would be able to develop a perfect set of unit plans for each of my courses and stick to that through the course of my career or at least until the curriculum changed. I see now that if I want to be the best teacher I can possibly be, this can NEVER be my attitude.

When things aren’t working, I must reflect on why, get creative and communicate with my colleagues. When things are going well, I must challenge myself to make things even better.

To be a reflective teacher, I must know my students. If I truly care for my students, as Henderson argues, I must, “[take] the time to help all students discover their individual inclinations and capitalize on them.” How can I relate the subject matter to my students’ past experiences and personal purposes? In asking this question, I can keep myself away from the idea that my students are simply “vessels into which the teacher pours knowledge.” It is important to note that Henderson states that we must help “all students.” Is it possible to practice an ethic of caring but ignore the troublesome students or possibly overlook students who seem to be doing fine without us? This might be difficult but I believe that you cannot practice an ethic of caring if you pick and choose who you will get to know based on how pleasant they are in class.

Difficulties and Challenges Involved in the Chess Board Problem

As a student, I solved the problem first by counting individual squares and then realizing  that there was a pattern. Using the pattern, I was able to solve the problem relatively quickly.

As a teacher, I can see students finding difficulty when trying to determine the number of squares larger than the 1x1 squares. Counting these becomes difficult past this stage as the squares begin to overlap. I would encourage an approach involving finding patterns. To guide them, I could start by helping them with the first step in determining a pattern.

To modify this problem to make it more difficult, you could add another dimension and ask for the number of cubes in a 8x8x8 cube. or ask for the number of triangles in a similarly structured triangle where patterns would be more difficult to find.

Memorable Math Teachers

One of my most memorable math teachers was Mr. P who taught me in grade 11. What I remember most about him were his stories that he brought to math class each day. Some were about his son, some were about his life before coming to Canada and some were about totally random topics. Very few of these stories were related to math but I enjoyed going to math because I knew that I had a teacher who knew how to make us laugh. I must say that his teaching style was not the best. He took a more traditional approach where he would give us a lecture which was followed by class time to work on textbook problems but I came out of it fine because I was able to develop an understanding of the math quite easily.

The following year, I had Mr. I for calculus. He taught us in a way that I had never experienced before. Classes were very interactive and there were times where we were even be using our bodies and voices simultaneously to learn. Lectures, I would say, were non-existent. Even the formal testing I was used to was not present. Our understanding was assessed mainly on our class participation and a few very informal quizzes. I found this teaching style quite interesting and I would say that he succeeded in getting me engaged in the subject matter. I have observed that the lack of formal assessment was a big problem for some students and even some teachers. Students would often question why they got the mark that they got. Furthermore, some students who were very engaged and given high marks were given permission to take courses designed for students showing proficiency in math. There were some cases where students were not preforming well as their math skills were not at the A level which Mr. I had evaluated them to be at. Nevertheless, I believe Mr. I was able to engage his students and encourage us to think in new ways and I will remember him for that.

Experiment and Learn

In my highschool, like the many described by Gerofsky, we had only two teachers for a school of 1500 fully qualified to teach mathematics. I was brought up in more conservative system of mathematics education. Fortunately, this worked well for me. I can easy understand with what was said about those who have “found ways to make sense and understand the mathematics they were presented, and expect that anyone who is good at math should be able to succeed as they did, under a similar system.” Does this mean that how I should teach should model the education practices under which I was taught? As many would argue - no. I must remember there exists a diverse range of learning styles and that a conservative mathematics education will very likely make it difficult or even impossible for many students to thrive. Can I expect students to extract an relational understanding of the material from a class involving mainly drills and memorization of facts and formulas? I don’t think I can.

The alternative to this would then be a progressive approach - a stance involving experimentation which can be “messy, uncertain, and unsettling.” I can also understand the parents’ “worries that their children were being shortchanged by teachers experimenting with their education.” I believe this is a necessary risk if we want to provide students with the best possible education. We must set the example that it is ok to take risks, mess up and learn from our mistakes. It is very important that we learn. We have the responsibility to educate and if each educational experiment we conduct fails, I believe we are failing even if we establish the belief that mistakes are ok.

In short, I believe that, as math educators, we must not be lazy and conform to the traditional ways in which we have been taught. Instead, we must make the effort to challenge students in a variety of ways while at the same time learning and reflecting on our efforts.

Who am I as a teacher?

I believe I am a teacher open to reflection. In doing so, I hope to model similar attitudes in my students. To me, teaching is much more than the transmission of knowledge - it is guiding the personal growth of students by instilling in them values. As a teacher, it is important that I work with all students. It means that when things get tough, I work harder in order to help my students learn. I must put myself in the mind of the student and always remember that I too am a student for life as I have much to learn. It is important to me that my students know that they are able to approach me for support whether it is with a math problem or whether they just need someone to talk to.

What is my goal? Although I have stated that above, I believe it is a question my must continually return to.

On Becoming a Reflective Teacher

Although this article was written in 1984, I believe most if not all of the points made by Grant and Zeichner are relevant today. In particular, the part about teaching all children and not just the ones who best fit your idea of an ideal student.

Putting the time in to being reflective is something into which I have put some thought. What I began to think about was whether a true love of teaching leads to the reflective teaching or if this works the other way around. Also, am I ready to spend the time reflecting to mold my teaching, regardless of how difficult it may be, is a question I am asking myself. At the moment, I really believe that I have the motivation and willingness to do so and based on my commitment to other responsibilities I have held and currently hold, I don’t see this as an unrealistic goal. How I will go about in keeping up these current practices is something I must put more thought into. Continuing to make meaningful connections with students, I believe makes a big difference in keeping myself motivated. I believe these personal connections do not only benefit the individual students but also myself.

Communicating Understanding

What I got most from Thurston’s article is encapsulated in the following line: “The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.” I could easily identify with the image Thurston describes of an audience getting lost in a colloquium talk within the first 5 minutes and sitting silently through the remaining 55 minutes. I can see how effective communication of mathematics with students has does not depend on how big your words are or how vast your vocabulary is but depends solely on how you are able to connect to students in a way that they can understand. We must be able to see the diversity in our schools: diversity in ability and diversity our ways of learning. Considering this, we should see that the language we use should be understandable by students, not only by you. Furthermore, our teaching must accommodate all learners - visual, auditory or kinesthetic. I enjoyed the example of the various ways that the derivative could be understood and how spending time connecting these ideas promoted a fuller understanding of the topic. 

I must keep this in mind: As I work to become a teacher, I must not forget how a student thinks. Put myself in their shoes and remember what it is like to not know.

What does it mean to be educated?

To be educated means many things - in both what we know and what we do. We must always be curious and ready to expand our knowledge. I believe that the more we know, the more we realize we don't know. To be educucated does not mean we know everything; it is having the attitude of a lifelong learner stemming from our realization that our world is vast and new discoveries can be made every day. Furthermore, the educated person is not afraid to make mistakes. They are able to exercise creativity and discover more in that way.

Being educated means being open to others and having the ability to see things from different perspectives. They are well-spoken and are not afraid to share their wealth of knowledge but at the same time, they are able to listen and respect to the opinions of others whether they agree or not.

To illustrate this, our group has come up with this creative representation.

Environmental Education and Creativity in Schools

I have to say that an the topic of this article was not what i was expecting. Honestly, I did not know what to expect but it certainly was not an environmental article which pretty much provides a perfect example of how out of tune I, as well as many others I’m sure, are with our environment. Certainly, we are becoming more environmentally conscious but do we expect all education to be environmental education as Orr urges? Probably not. Nevertheless, I believe environmental education plays an important role in creating eco-conscious citizens and our efforts must be continued not only in the classroom but also throughout the school community.

As for the video, I was quite encouraged by a comment I found from a teacher from my high school posted just this week. He writes, “Personally, i am striving to redefine, facilitate, and reorganize the public education institution's foundations with our students, their parents, my colleagues, and the community.” Although this is quite an ambitious undertaking, it inspires me to challenge myself and my students to think and learn creatively and not just for the sake of information or getting a good grade.

Benny’s Rules: The Problem with IPI Mathematics

From Benny’s Conception of Rules and Answers in IPI Mathematics, I found Benny to have a thorough frustration stemming from his view of what mathematics was. What I found most illustrative of this is when we are told that he said, “I am going to look up fractions, and I am going to find out who did the rules, and how they are kept.” The rules and their inflexibility in his view of mathematics is certainly not the stance I would my students to adopt. Perhaps what is lacking the most in the method of instruction referred to in this article is the discussion with peers and teachers. I believe without this, the study of math can very easily become what I had become for Benny: a fixed set of rules applied to a variety of problems to arrive at the right answer. It seems to me that the effective teacher-student interaction is key to developing an attitude of relational understanding which I believe the joy in math comes from.

Relational Understanding and Instrumental Understanding

What struck me when reading Richard Skemp’s Relational Understanding and Instrumental Understanding was how easily I could relate to the two meanings of understanding. From my experience helping others in math, I found that a big difficulty for many was recalling what they had learned in previous grades or courses. It appears that they do not have the relational understanding explored in the article but only once had some instrumental understanding that has since disappeared. It is clear to me that their background of instrumental mathematics has led them to approach pretty much all mathematics problems with a similar way of thinking however, at higher levels of math, this becomes very difficult as to apply these new rules, they must remember some old rules which have fallen out of their memory. I feel like math to them seems like nothing more than memorizing a bunch of formulas and rules and to break this way of thinking is not an easy task. I’ve also felt at times a need to go back and help them relationally understand the basics but this proves to be a very, very time consuming task and I find it quite understandable why many who “miss a step” have trouble catching up. I think it is a very important task to have students understand relationally as much as possible starting from beginning because from my experience, memorizing is terribly unstimulating while having a true relational understanding can be fun and lead to further exploration.