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.