Assignment #1 Reflection

Our group presented on the Serpinski Triangles. We focused on two aspects of their design, Cellular Automaton and the use of different cells. Cellular Automaton was taught by showing the class the different rules that generate different triangles, while the concept of creating fractal shapes one level up was taught by having the class play around with physical cells that we cut out. It was interesting to see how one art project could be used to investigate different mathematical topics.

I really enjoyed making the art and getting more experience with the Sierpinski triangle which I have seen many times but never built before. It was interesting to investigate the different aspects of the art project with my group and we realized multiple different angles our project could take. It was a highlight of mine to get to work with Evan and Nandini and collaborate on our ideas. I also enjoyed the process of sharing our art with the class and seeing their interpretation of the extension project which turned out differently than we expected. It reminded me of the benefit of collaborating! My experience as a learner reminded me that Math projects can be fun and engaging, especially when there is creative allowance. 

I think students may enjoy the process of seeing how art and math are connected and getting to be creative. I think our topic could possibly be used in a grade 11 or 12 class when students are learning about patterns! Fractals are a great way for students to see how Math can be seen in nature and design. My experience as a teacher reminded me that being comfortable with lessons going in different directions is important and that students can lead a lesson to interesting discoveries.

Histories and Futures

Student 1

Ms. Christine,

Thank you for being my teacher this year. I had a lot of fun in your class and I could tell that you really cared about all of us. Thanks for being patient and encouraging. I do not hate Math as much as I did last year thanks to you. 

Student 2

Ms. Christine,

I thought the way you taught was confusing and I don't think you could tell when I was struggling. I didn't feel comfortable asking for help and we moved on to new things before I understood what was going on. 

Reflection

My passion is to connect with teenagers and hopefully leave them with a sense that they are important and capable. I hope to be a teacher who is patient, fun and invests in my students. A fear of mine is that kids will "slip through the cracks" and that I will not be aware that they are confused or need help. This is something I want to observe and work on in my practicums - see how teachers meet the needs of students of varying levels. With my background as an EA, I know that certain students have very specific needs. Something I loved about my previous job was that I could adapt my methods in order to help that specific student. Teaching, however, does not allow for that level of adaptation because you have to work with a larger group. I worry that I won't be able to help the kids who need it most. 

Dishes Problem

My strategy was to solve the problem visually and really just count up by 13 until I reached the number I was looking for. I think representation matters and for far too long, certain students have not seen themselves or their culture represented in their education. Although having a problem like this included in a class may not make a huge difference, I think it is important for teachers to use problems from different cultures and perhaps students will feel increasingly comfortable sharing their own culture. In my opinion, using historical problems from different cultures is a much better way of diversifying than coming up with trite problems that just end up appropriating or disrespecting different cultures (this is mostly what I saw in my education). I think different stories and imageries will resonate with different students and either increase or decrease their enjoyment. The main goal is to use a variety of puzzle stories from different cultures in order to communicate that Mathematics was developed in different cultures all over the world - it is a human experience and not limited to one group of people.

Lockhart's Lament response

My interpretation of the central message of this article is that we have stripped Math of its beauty and mystery in our current curriculum and I definitely agree. It struck me to think about all the meaningless nomenclature and symbols we teach students that distract from the point of Mathematics. It would be so much more valuable to engage students in the human experience of problem-solving through Mathematical thinking. There is so much beauty in Math and students are missing out on the process because we are teaching them to follow formulas and a set of steps. 

I think Math does differ from poetry, music or visual arts in that it is not accessible for everyone to appreciate. Anyone could listen to a symphony and experience some awe at what they are hearing however, there needs to be a foundation of knowledge to appreciate the beauty of Math. I don't think we can swing the pendulum to an open curriculum where students just follow their curiosities, but we can definitely teach Math as art and discovery while including fundamental skills. I would also argue that Math sometimes does behave as a science in how it solves certain problems.

I think the idea of relational knowledge is related to what Lockhart is saying but it seems Lockhart's view is more radical. Lockhart thinks we have been distracted from true Mathematics by textbooks and a need for a structured, efficient curriculum.  Lockhart wants to go beyond relational thinking and have students create/discover the relationships!

The locker problem

My strategy with the locker problem was to consider what would happen with the first ten lockers. I noticed that the only lockers that would be closed were all perfect squares. I tested other perfect squares by looking at their factors and saw that the pattern worked. The strategies I used were: a diagram / visual, extrapolation and my knowledge about factors and perfect squares.

Skemp discussion

 

Math teachers

My favourite Math teacher in high school was my pre-calc 11 / calculus teacher. I remember how much she loved Math and how she was well-trained to teach us. She was excited about Math and I think I absorbed that excitement and was able to be interested in the subject. She also gave us lots of opportunities to practice the material and reduced her lecturing time. The last thing that stood out to me as I reflect is that she took time every week to engage in conversations with us about life - she had a question jar and we would talk about all kinds of life questions. I knew she was invested in our development as people as well as math learners.

My least favourite Math teacher was my grade 9 Math teacher. He was not trained in teaching Math and he mostly relied on worksheets. I remember feeling more confused when he lectured and finding mistakes in his work. My takeaway is that students can easily decipher if a teacher is knowledgeable and passionate about the subject they are teaching. I want my students to trust my teaching and also know that I am invested in them.

Christine - Richard Skemp: Instrumental and Relational understandings

    Instrumental ways of understanding Mathematics works for some students and in some situations but it is not the best way to understand a concept or have it be applicable to other fields. Something that excites me about Math education is that it develops critical thinking skills and is connected to many different areas. However, if it is taught purely instrumentally, that application may not be as accessible to students. Reading this reminded me of the times I was in courses at university and started making connections about things I was taught in high school. I was finally able to make relational connections as my understanding grew. I don't think students should have to go into University Math to have a relational understanding of Math. I also reflected on the general dislike of Math I have seen in many students and I wonder if that is because they were taught instrumental knowledge and it seems irrelevant and hard to grasp. Students desire there to be meaning and purpose in what they are doing. I appreciate the map example (page 14) presented by Skemp; students who have an understanding of where a certain skill fits in a bigger picture will be able to connect the skill to different concepts and fields.

    I generally agree with Skemp; relational understanding is deeper knowing and therefore more adaptable and long-lasting. However, sometimes learning instrumentally first makes the relational connections easier to present. Overall, I think having a goal of communicating meaning and connections in Mathematics should be a primary goal in Math education, not just teaching a certain skill.