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.