Hypothetically: 10 Years On...

A letter from a student for whom I was a favourite teacher:

Dear Miss Mackie,

Having just got a new job as an actuary it has made me think back over my mathematical career and contemplate why it has been so successful. Ok, so I put in a lot of hard work, but had it not been for teachers like you who dispelled my initial fear of mathematics I would never have flourished in the subject and gone on to university. I just wanted to acknowledge this and say thank you.

Sincerely,
       Alpha Student

A letter from a student for who did not like me as a teacher:

Dear Miss Mackie,

Now that I am older and wiser I realise how difficult I was as a child. Although I had my reasons for being this way they are no excuse for my behaviour, nor for my treatment of yourself. Having gone back to college and obtained some math qualifications I am aware that you only pushed me because you saw the potential I have. Looking back I now see your teaching methods in a whole new light and as I am aware that my failure in your class pained you I would like to say I am sorry that I did not appreciate this at the time. I hope you are still teaching and that you can be proud of my recent accomplishments.

Best Wishes,
     Beta Student

Considerations:

Our attitude in the classroom will have a great impact on how students feel about the subject not just about us as teachers.

We may have 30 students in a class but a student only has 1 teacher in a class so we will be memorable to them and have more of an influence on their lives than we realise.

Students do not separate their lives from their studies the way adults do and sometimes outside issues may interfere with their learning. We not only need to understand this but have to find a way of dealing with it. We will have to deal with more than just teaching them math and sometimes we may even come across serious issues that our outside our comfort area but these still cannot be ignored.

It is important to make math fun and accessible for all.

As teachers we have the opportunity to inspire the next generation, which is a great priviledge that shouldn't be taken foregranted.

Reflection on Article - "Battleground Schools: Mathematics Education"

Not being from North America myself I was very interested to read this article. I was fascinated by the described battle over mathematics and agree that there is no end in sight. I also believe that there may never be one. As it is not only about the subject but about politics it is a very complex matter and it is natural that there will be differing opinions, we are after all, only human. 

I am not sure that I understand what is meant by "fluency" in this article.  The conservative idea of the goal of mathematics learning is fluency while the progressive idea is understanding. Surely you can't have understanding without fluency, nor fluency without understanding. I believe that fluency for the conservative idea may be the ability to churn out answers but surely if you are "fluent" at this then you have understanding and your knowledge isn't just instrumental. I do not believe that any mathematical proof can be considered a good proof if it is not fluent. Possibly I am not grasping what is meant.

Having gone to school in the UK it was interesting for me to read about Margaret Thatcher. At the age of six I was already very aware of this woman's changes to the school system. We knew her as "Margaret Thatcher the milk snatcher", due to her putting a stop to the government providing us with our free milk at break time.

The final point I would like to make on this article is with regards to the TIMMS study.  I have always known that there is a lot of politics involved in the world of teaching but I've never really given much thought as to why. Upon reading this paragraph a lightbulb went off in my head and I suddenly thought "world domination"! That's what it's all about, having the most money, the biggest secrets and producing the most knowledgeable children so that if needs be you can prove you're the most powerful. Or maybe I've been watching too many movies? 

Feedback and Reflection on 10 Minute Yoga Lesson

As my class size went unexpectedly from 3 to approximately 10 students the lesson plan I had made had to be adjusted last minute. This made me nervous but it was a good learning experience as you never know what will happen in the classroom and you have to always be ready to deal with the unexpected.

The general consensus from my feedback was that I gave good descriptions of the poses and clear instructions. My lack of attention to individuals was brought up but this was to be expected. I had originally planned to go around the class giving feedback and advice as the students were trying the poses, but my class size was too large to manoeuvre around and to get through everyone in the allocated 10 minutes, I therefore decided to eliminate this part of the lesson.

One final point that was brought up was that one person found the lesson difficult. Possibly I should have had a really easy pose that those who were uncomfortable with the lesson could do instead so no-one was left out, this is not something that I foresaw. I have to become more aware of the extreme range of abilities in any classroom.

Results from Student and Teacher Interview

Student Interview

                For our student interview, we talked to a grade 12 student who is tutored by one of our group members. He is an intelligent student, who just does not fully apply himself. With tutoring, he has been able to greatly improve his marks, and he has had teachers in the past from whom he has been able to learn effectively.

                Our first two questions were about what he likes and dislikes about math. He likes to graph and to use the graphing calculator technology, indicating that he is a visual learner. He dislikes problem solving, which we feel is a common issue with many students. He said that this was because there is no predetermined way to change words into equations that was the same every time. To us, this implied that he doesn’t understand math at very deep level.

                Our third question was "What could be done in the classroom to help your learning?". The student said that he wished that the teacher had better and well organized notes. If the teacher is writing on the board, he finds that they often write in a 'non-linear' way. This means that these teachers will write in different locations, go backwards, erase things and write overtop of diagrams. This makes notes difficult to follow after the lectures. The student suggested that teachers will be more effective to use overheads instead of the blackboard. We feel this is significant because teachers need to be aware of these specific learning styles if they are trying to use engaging activities. Some students still need the more traditional teaching strategy of note taking in order to learn effectively.

                The student also mentioned that he likes teachers to give examples that are the same as questions on their tests. We feel that this is an indication that he is used to learning instrumentally, but his teacher is trying to teach relationally.

                Our fourth question was "what would you like to see done in your classroom?". He said that he wanted a more interactive environment. We feel that this means that he wanted a more relaxed and friendly environment, where he felt comfortable talking to his peers. He also mentioned that there are many grade 11's in his class, and that this affects the feel of the classroom. This brought up the importance of the overall mood and feeling that you create within your class. Not only do you have to think about how to organize a lesson and explain concepts, but you also have to consider how students will feel while they are sitting in your class. This is not a simple matter either, for there are many different factors that can affect a classroom environment.

                Our final question was about how he views his overall math education. He found that his view of mathematics changed with every year, and with the individual teacher that he has had. This stresses the importance of our jobs, and the effect (both positive and negative) that we can have on a student. He also said that he felt that elementary school mathematics was a joke, and that nothing was really taught there. We agree with this sentiment as there is no reason why elementary students should not be learning more advanced mathematics because they are perfectly capable. The issue is that teachers cannot assume that students have learned anything in previous years, because they get passed regardless of ability. We also feel that the new curriculum is moving concepts into higher grade levels, while we feel that many of these topics should be taught earlier.

Teacher Interview

                For our teacher interview we talked to the department head at one of the local Vancouver secondary schools. He is fairly young (likely mid to late 30’s) and very involved in the math education community.

                We jumped right in by asking him how he deals with students who are not interested in math. He said that he tries to make the activities and problems as engaging as possible, but admitted that he does not make any special effort to encourage individuals. He implied that choosing to engage in the activities or not is the student’s choice. We found this answer very surprising as it gave us the impression that this teacher almost did not care about such students, and that if they do not want to learn math, then it is not his problem. We feel that reaching out to those students who may have a negative view of mathematics and trying to change their opinions is a very important aspect of teaching math.

                We then asked the teacher if he had any especially effective teaching methods that he would suggest. He said that he likes to teach through investigatory activities, and believes that students learn best through discovering, not being told something. This is something that we have heard a lot about in many of our classes, so it was nice to see that there are teachers putting it into practice. He also said that he likes to give students surprising problems, meaning questions which might have answers that are contrary to what was initially expected. He finds that this really grabs their attention and helps them engage in the material.

                Our third question was about what aspects of the job he finds to be most difficult. He said that his biggest challenge is recognizing his own weaknesses, learning from them, and growing in his teaching. He sees many of his colleagues getting “stuck in a rut” and being afraid of change, therefore he finds it difficult to prevent himself from falling into the same situation. He has tried to integrate new ideas into the classroom, but finds that there is a lot of resistance, both from the students and his fellow teachers. We feel that this is an important thing to recognize, particularly while we are trying to bring in new ideas during our practicum. Everyone gets used to a particular system, and it is very difficult to change.

                Our fourth question dealt with the amount of time that is involved in teaching. He said that he works approximately 55 to 60 hours a week. He stays at school until 5pm everyday, and works in the evenings and on weekends too. There is a lot of marking, although the amount of preparation time required decreases as you gain experience. He is also constantly thinking about teaching, so in that sense, it invades all aspects of his personal life. These are, of course, good things to be aware of as we enter the teaching profession.

                Finally, we asked him, what was most rewarding about his job. He said that he really loves interacting with intelligent, thoughtful and inquisitive students. We agree that this would be a wonderful part of the job, as you would be able to share your love and enthusiasm for the subject matter. However we also feel that helping students who struggle, and seeing them having a major breakthrough with the subject, would be extremely rewarding as well. This may even be more rewarding than interacting with the intelligent students. We got the impression that this teacher focuses a lot of his energy on the bright students, and less on the weaker students. We feel that all students deserve a teacher’s time and energy, and that a teacher can learn a lot from each and every one of their students. 

(This post was created in collaboration with Edward Liao and Shannon Kennedy)

10 Minute Yoga Lesson Plan

Bridge - Yoga benefits the mind and body and is thus applicable to any part of our daily routine

Objectives (Learning) - To successfully string together 3 yoga poses into a fluid movement and to understand the benefits of each pose individually and the practice as a whole

Objectives (Teaching) - To get everyone involved and to learn by doing not just listening, regardless of their fitness level

Pre-test - Explanation of moves and their benefits - 4 minutes

Participatory - Students try moves - 3 minutes

Post-test - Feedback on how they did and what they could improve - 2 minutes

Summary & wrap up - Students put together what they have been taught into a fluid movement while I reiterate the reasoning behind the lesson - 1 minute

(No special materials are required for this lesson)

Reflection on video of David Hewitt's teaching methods

What stood out to me while watching this video was the teachers patience and it's effect on the children's learning.  He allowed the students time when there was a disagreement, time in which they could learn from their peers instead of him preaching at them and time for them to figure out their own mistakes.  His method was also very visual, he moved around pointing at parts of the room, it wasn't all pen and paper, thus the students were more easily engaged and less easily distracted. As the students were more actively involved they were participating more and learning without realising. I believe this form of subconscious learning to be very effective, many times someone has asked me how I know something but I can't recall and more often than not I can't remember things that I know I have been taught. This teacher's methods made sure all the students had the opportunity to participate but without being singled out, which I feel is important not just for their learning but for their confidence also.

Memorable Math Teachers

Throughout high school I was very quiet and kept myself to myself and many of my teachers just left me be, I was to an extent ignored.  Our math classes were split into credit, general and foundation and we were placed depending on our teachers beliefs of our ability. I was placed in the lower class and was there for some time before a teacher actually noticed me asked me how easy I found the work. I was then tested and moved immediately to the highest class. If it had not been for this teacher I may never have made it to university.

My second and final memorable math teacher is such for going out of their way and fighting the system to help me. I was accepted into university on the basis of my fifth year (grade 12) results. Upon being informed of this I dropped out of my sixth year studies at high school in order to obtain full time employment so that I could actually afford to accept my university place. My math teacher thought it a shame that I wouldn't be able to complete my course, especially since that was the subject I would be studying at university. There was however a problem with me sitting the exam as an external student so he took it upon himself to ask the school to keep me on the register and posted worksheets through my mailbox so I could continue my studies at home and sit the exam. I passed and my paper was actually presented to the English exam board as an example of how successful the Scottish system was. I then went on to thoroughly enjoy my first year at university, having already been exposed to many of the concepts throughout my sixth year studies.

Questions I shall pose to a high school student and their math teacher, check back later to read their answers:

Teacher:

1. How do you deal with students who are not interested in math?
2. Do you have any unique or especially effective teaching methods you would suggest to us?
3. What aspect of the job do you find most difficult?
4. How many hours per week do you work and how is this time divided between prep time and class time?
5. What do you find most rewarding about your job?

Student:

1. What do you like most about math, why?
2. What do you like least about math, why?
3. What could be done in the classroom to make it easier for you to learn?
4. What would you like to see or to do in your math class?
5. What do you think of the math teaching you have had throughout school so far? 

Response to ''Relational Understanding and Instrumental Understanding'', Richard R Skemp

I strongly agree that there is instrumental and relational teaching and learning regarding the subject of mathematics. I believe that not all teachers of mathematics who use the method of instrumental teaching are aware they are doing so and this issue must be addressed before any plans can be formulated as to how mathematics is to be taught in the future.

In an ideal world all mathematics would be taught relationally and all students of this method would thrive. However it is a well known fact that we are all individuals with different learning styles, thus does everyone have the ability to learn mathematics if taught relationally from an early enough age? A lot of students who struggle with the subject need it to be taught instrumentally so that they can pass their exams and go on to do what they want to do in life, if this is not mathematically related do we have the right to destroy their hopes and dreams by forcing relational teaching upon them.  This then raises the question of if mathematics is to be taught relationally should it then be mandatory? Possibly both methods of teaching should be used but to different students in different classrooms, then the question arises of when do you decide to separate them; at what age is it appropriate to make the decision that one child cannot succesfully learn relationally, they are after all children, who are all unique and develop differently.

Relational teaching must be implemented at the start of a students education and followed through with. By the time a student reaches their teenage years, a critical time in their life, it would be harmful to their self-esteem to try to switch them from the instrumental teaching methods they are acustomed to.

There is one point in the article to which I disagree, that enlarging a students awareness of possibilities can only be a good thing. For the most part, of course, if we are not aware we cannot make informed choices. However the saying ignorance is bliss is one which I don't think should be forgotten. If opened to too many possibilities some can become over-whelmed and lose focus. We do not wish our students to feel they are drowning in a sea of knowledge and possibilities.

As mathematics teaching stands today, I do not believe the teachers have the time nor the leeway to be able to teach completely relationally, nor am I convinced that this is a good thing for everyone.