Micro teaching topic

I will teach my group members how to play the game dutch blitz

Wordy Problem

Brothers and sisters

Have I none - 

But that man’s father

Is my father’s son!

Who is speaking: The only child of his father

Who is ‘that man’: The son of the speaker

What makes this problem difficult/interesting: I think what makes this problem interesting is that it’s really purely a logic problem. There’s no background information other than understanding what a brother, sister, father, son relationship is. It’s different from other math problems in this way, in most math problems you need to be familiar with definitions and theorems to be able to solve them, but with this you only need to be able to think through a question. What I found really great about this problem is that it’s a great introduction to logical thinking. Since you don’t need any background information, it would be a great problem to start the class with to get kids thinking mathematically!

Reflections on the art project

I really enjoyed doing this art project! We chose an art piece by Margaret Kepner, which is based on Ulam's spiral of primes. Pictured to the right is our recreation of her original work. I thought it was really interesting to see numbers represented visually as triangles drawn out in a spiral. The coding was quite a bit of work, and if Kyle hadn't figured out how to do it, we probably would have needed to figure out a different medium to represent this. My job in this was to comment on every line of code what number that triangle represented. It took a long time and was kind of frustrating, but after it was done, it was really cool to notice patterns of where numbers were and I felt like I had a better sense of the artwork.

Choosing how we were going to extend the project was, to me, the most interesting part. We were researching different kinds of numbers and came across Lucky Numbers, which at first I just thought was an interesting type of number, but then we found the connection to the primes and the sieve of Eratosthenes and it was even more interesting and relevant to the piece. We also chose perfect squares as something we could apply to the classroom during a unit on exponents perhaps, this could be a good activity to see numbers represented in a different way. Then we found the similar spirals between the perfect squares and triangular numbers and it was all the more interesting!

This was a really interesting way to see numbers represented in a way you wouldn't often see them. I really enjoyed the process of figuring which triangle represented which number and finding patterns in the artwork.

Locker Problem

When I first looked at this problem, I had a lot of ideas rolling around in my head. I started off thinking about multiples, but that was only a quick thought as I thought there might be a better approach. I realized early that person 501 and onwards would only touch one locker, so I would only need to figure out the first half of the problem and then flip the last half of the lockers.

I decided to work on the first 10 lockers and see if I could spot a pattern.

1:cccccccccc

2:cococococo

3:cooocccooo

4:coocccccoo

5:coococccoc

10:coocooooco

From this, I started to see a pattern that 1, 4 and 9 were all closed. I wanted to try 20 to see if my suspicions were correct and that the perfect squares were all closed.

I got 20: coocoooocoooooocoooo

From there I saw that 16 was also closed, and this was enough for me to say that all the perfect squares are closed, and all others are open.
I thought about this and wondered what makes perfect squares have this property that all of them would be open. I thought back to when I was teaching math last year and how we went over perfect squares and one of the interesting things is that they have an odd number of factors. Every other number will have an even number of factors, so they will all be touched an even number of times, so they will go back to their starting positions. Perfect squares will be touched an odd number of times, so they will change from their starting position.

2 letters from former students 10 years from now

Short note from a student who thought you were the greatest, and a student who struggled in your class. And a summary afterwards of hopes and worries

Ms Ludeman,
I just wanted to let you know how much you impacted my life. Your enthusiasm when teaching math really helped me get engaged. Because I was engaged and wanted to understand, you helped me see that I am good at math! I went on to University in (a math-related field), something I wasn’t sure I would be able to do even though it is something I was always interested in.
Thank you so much for your help!
Future student 1

Ms Ludeman,
Looking back on high school, I realize now where my aversion to math came from. I didn’t understand why you always insisted on going into such detail when I only wanted to learn the formulas, I never cared about the why! I thought you made class much more difficult than it needed to be. I’m good at memorizing formulas and knowing how to apply them. All this other work you had us do was so unnecessary and made me confused. I wanted to let you know, because I think you should simplify your lessons and go back to the proper way of teaching math.
Sincerely,
Future Student 2

Writing these 2 letters, I noticed it was a lot easier for me to write the second one, which leads me to think I have more worries than hopes at this moment. I’m worried that trying to introduce students to logical thinking and getting away from memorizing formulas where possible will be detrimental instead of helpful. What I am most hopeful of is that I can learn from my mistakes and that if I try something and it doesn’t work, I don’t keep trying it. I’m hopeful that introducing students to logical thinking will engage them and get them excited to learn.

Math and me

The first thing I remember about math is being in Grade 1 and being the worst in my class at math. I was the only one who ever got homework and it was really discouraging. I remember the very few lessons where I actually finished the work in class and could play with my friends and being so excited and happy that I understood something. I guess my math progressed throughout elementary school, because the next distinct thing I remember is getting my first set of straight A’s in grade 7. I had a teacher in Grade 7 who had a math degree, and she taught us so well and I think that was the first time I felt good at math and like I understood it.

Moving on to high school, I kept getting better. I think the logic and reasoning needed in high school math really struck a chord with me and allowed me to excel. I got into math honours all through high school and took calculus and got top marks. I had several really amazing teachers who made math approachable and made me feel like I was really good at it. That confidence really helped me when I was picking courses for University.

In university, I originally took general studies which included a math class. I did so well in the math class, my teacher talked to me afterwards and suggested I think about doing a math degree. And then I did. I really enjoyed all the theoretical learning (except for topology) and especially classes I took in group theory. My first math teacher in university offered me a job running tutorial sessions for precalculus and I absolutely loved it. It helped me stay grounded and remember practical applications for what I was doing in my theoretical courses. It was a voluntary class, and I had students who wouldn’t miss a single one. I felt really confident running those tutorials, and I would go after class and do a bunch of homework because I just felt so energized and excited about math.

I then worked for an after school math program in Calgary after I graduated and taught Kindergarten and Grade 9. The Kindergarten class didn’t really want to learn math, and they were understandably tired after a long day at school. My grade 9 class was amazing. The students were smart and excited to learn, and didn’t get discouraged when they didn’t understand something. It was another amazing experience, and I would go home so happy that I could share my excitement about these topics. I know that teaching in schools won’t be the same as teaching at voluntary after school programs and tutorial sessions, but I think that if I can make students as excited as my teachers made me, I’ll be happy.

Mathematical understanding and multiple representations

The thing that convinced me the most in the authors argument was the study comparing students who learned analytically, visually and representationally. The students who learned representationally, or as a mix of both analytic and visual, did by far the best. It makes me think back to my high school days and how I always appreciated it when teachers showed an alternative way to solve problems if the class wasn't working well with the one we had been taught. One of the things that I love about math is that although there is only one answer, there are a million different paths to get there and they are all valid. I think the more methods students are exposed to, the more flexible they will be in their thinking. Math can often involve a lot of shifting ideas. If you've chosen one way to answer a question and get partway through and find that it won't work, you have to be able to switch to another method. This method of teaching will help students develop that mental flexibility.

The main types of mathematical representations in the article compare analytical learning to visual learning or learning with objects such as blocks. One thing my high school math teacher did really well was relate things to humans. When we learned about exponential growth, he showed us in human population and explained why it was so important to understand it. He used diagrams, but his explanation and the way he related it to people made it feel so important to learn, and also so clear. I think this is a strategy that is underutilized in math. Most people don't use exponential growth every day, but I think most people in my class still understand what it is and why it is so important to understand just from a 5 minute discussion in class. Relating things to human beings can really help with understanding, and explaining why it is so crucial to understand really helps with motivation.

Relational Understanding and Instrumental Understanding by Richard Skemp

The first part of this article that made me stop and think is the explanation of instrumental understanding. It made me stop and think back to my own high school days and I felt very grateful for my teachers, because I wanted to learn relationally and that is how they taught. I know a lot of people who get frustrated because their teachers taught them instrumentally, and they didn't understand a lot of things. The next part that made me stop came right after, in the football analogy. I too held the belief that relational is a better method of teaching, and it is interesting to look at instrumental and try to understand it. The last part that really struck a chord with me is in the devil's advocate part when Skemp talks about how one reason it could be argued that instrumental understanding is better is that it is a faster way of teaching. This reminded me of a reading I recently did for another class, which was about learning objectives. According to the article, if students spent half an hour on each learning objective that they are given in school, it would take them 9 extra years of schooling to complete all of them. Skemp talks about an over-burdened syllabi, and this other article talks about making clear and concise learning objectives. I think the 2 could work together.

I think that for the most part, I agree with Skemp. There were so many parts in the article where I stopped because I felt such a connection to what he was saying. So many of my friends dislike math because they had teachers who taught them instrumentally, because they learned instrumentally and didn't understand how to teach relationally. These friends are all people who regularly question things and want to know why they're learning these things and how to apply them and can they be used in these situations. I think math should be taught because it helps develop logical thinking, not because people need to know how to find the area of a triangle. I think teaching relationally can lead to logical thinking much easier than teaching instumentally.

Hello!