28 September - Wallpaper Patterns presentation slides

https://docs.google.com/presentation/d/11ZGjuGOmF7Ho53ACnBbJr88c5K45fpQwiwnYYn5CvaI/edit#slide=id.g99ac6ee6b1_2_54

27 September - Mathematical understanding and multiple representations

What struck me was the following quote: "It is now well accepted that the use of particu-lar modes of representations (e.g., visual or concrete) leads to improvement of students' mathematical abilities and development of their advanced problem
solving and reasoning skills " (p.120). I do believe that citing sources to experiments is a good thing, however, just because someone quotes a source doesn't make the argument valid. While I don't disagree with the author's intention, I will play a bit the devil's advocate and say that, while the use of representation definitely helps, if it's done too much, we can be hurting the child's ability to reason abstractly. Representation could be the equivalent of watching a movie, versus reading the book. When we talk about concepts, for example "a set", sometimes it's best to simply make it up in our own mind, whether we imagine boxes or circles. This very picture-making exercise helps with visualizing concepts. Sometimes the way we visualize things in mathematics makes or breaks whether we'll discover something. Having the right pictures in mind definitely helps. However, if everyone has the same forced picture in their mind, it could become a problem since we are not leaving room for imagination.

The author includes a lot of set representations, especially a grouping of the elements. A representation that is not included is the "tiles" representation of a quadratic equation. I'd use this representation to show that X^2 is an actual square of different size, than let's say 2x.

Sept 23 - Letters to teachers and letters from students

1. Writing about your favorite and least favorite math teachers -- and why.

Favorite Teacher:

I never had a favorite math teacher. That is partly why I decided to become one. I was influenced however by Neil deGrasse Tyson. About 10 years ago, I was watching god vs. religion debates on Youtube and one of his videos popped up. It was about how many people are mathematically illiterate. He mentioned "math is the language of the universe". I was hooked. I was young and I was searching for meaning in life. I spent a great deal reading philosophy which led me unsatisfied, because most philosophical works seem to be someone's opinion on something, and they're not really objective about reality. When I realized that math is what gives us the power of command over nature, as well as understanding the world we live in, I started studying it in my spare time. Eventually I went back to school and 6 years later, I hold a pure math degree. Helping raise mathematical literacy and inspire young kids in the wonders of science, is a way now for me to pay it forward, and to keep the wheel turning.

Least Favorite Teacher:

Mr. X would come to class unprepared. He would then vaguely lecture us, with no enthusiasm and a monotonous voice. Our assignments were never marked on time. Most of them were returned 2 months later, leaving us in the dark about where we were standing in the course. Assignments were also posted late, but the due date remained the same. We had to scramble at the last minute to get a week long assignment in just 2 days. Most of the time I ended up handing in only half of it, due to lack of time. During office hours he was condescending. He would make you feel bad about yourself because you "don't understand", instead of answering your question. You see this a lot in sports coaches too. Some people think that just because you have expertise, now you're absolved from being a decent human being and treating people with respect.

One of the downsides of "tenure".

2. Writing letters from your future students, ten years from now.

Student who liked me:

Dear Mr. Ticu,

I hope you remember me. I was in your Grade 11 class about 10 years ago. I am writing to let you know that I am still on the program, and I am not going to stop. I haven't slept in, in years, and in the past year I've barely missed a handful of workouts, mostly due to being sick. I've been studying on my own time as much as I can to gain knowledge. At 27, I feel stronger and smarter than I have ever been. Who knows how I'll be at 35, or even 40!

My son is 3, and he really enjoys the mathematical puzzles I give him. One day he will be a great mathematician. I myself, I became a firefighter. I always preferred being the silent professional, like you said, and just contribute to my community, and not seek fame or let my ego get in the way.

After my parents passed away, I felt like an orphan in the world. I wanted to feel sorry for myself, but I remembered your words of always looking for the good in things. I figured, that it's good that I'm by myself, that I will be mentally stronger. I focused on building my own family, instead of having a mental breakdown.

I am a completely different person than I was 10 years ago, and I can't thank you enough for it.

By the way, strange how the monsters under my bed never really left. They are under my son's bed, now. It's his time to shine.

Paying it forward,

The Wimpy Kid from Grade 11

Student who didn't like me:

Dear Mr. Ticu,

Not everyone is like you. Not everyone wants to learn math. Not everyone wants to be strong. I don't need to toil to get satisfaction from life. Inheriting money has made me realize that there are many ways to win in life.

Never needed math in my life by the way! I have people working for me who do it. Isn't that being smart and using the resources that you have? I think it is. Right back at ya. It was obvious how you were trying to "inspire us", in all your classes. Maybe being less tryhard would've caught me, but too late now. In life you can't get everyone. Nice try though.  

Oh and about "learning math and discovering the universe"...you know how much a mayfly lives? One day. It has no time to figure out how the world works. If there are aliens who can live even twice as long as us, we'd be the mayflies. So I'm going to enjoy life as much as I can.

I am writing this from my yacht while being surrounded by seven young models in bikinis, by the way. Heck, I can smell your envy from here.

Dan B.

Exit Slip - Sept. 21 - Response to class discussion on Skem's article

I am responding to Alexa's group saying "Instrumental learning first so students understand and are confident using basic facts and methods to have in their math ‘tool box’, to then use to participate in relational learning." I sort of touched on this point in my group's discussion. Instrumental learning can be much like learning grammar. While grammar is important to learn, it's possible to find amazing compositions by very young students, that lack good grammar. Just teaching techniques could possibly be a detriment on brain development. Stimulating the students' brains with questions, and forcing them to reason, can be a great way to stimulate critical thinking at an early age.

EDIT: I ACCIDENTALLY POSTED MY EXIT SLIP ON KARISHMA'S blog.

1000 Locker Problem

 I was a bit confused about the problem. The problem states

• Student #1 closes each locker.
• Student #2 opens each second locker.
• Student #3 changes the state of each third locker (i.e. opens them if they are closed, or closes them if they are open)
• ...and so on, until all 1,000 students have had their turn. 

I think the solution depends on what we mean by "and so on". If we mean that we repeat the pattern above, then the solution looks like this:

All lockers would be closed on the #1000'th iteration.

But what if we mean that #1 and #2 iterations never happen and by "and so on" we mean that #4 to #1000 simply change the states of their numbered locker. Then the solution looks very different. I haven't thought about it yet.
 

Sept 14 - Skemp response

The first thing that made me stop and think, while reading the article, was "field, group, ring, ideal." I have thought of those words myself at some point, especially "field". When I think of "field", something else comes up in my mind (i.e. a grass field), that has nothing to do with the actual mathematical definition. Using the same word can definitely be confusing, especially since "field" is also used in physics (i.e. electromagnetic field.) Another thing that stood out was that the author mentioned there are ways to understand something. I belive it is up to the teacher to identify how his students are learning the material. If it's the case of: "All they want is some kind of rule for getting the answer [and] as soon as this is reached, they latch on to it and ignore the rest." (p.4), then it is obvious that the students were conditioned to this way of thinking in the past. Therefore, it is the job of the teacher to steer hard towards unwinding this faulty way of looking at mathematics.

There is a certain balance that needs to be achieved between instrumental and relational understanding. Why not teach both? In the example of "minus times minus equals plus" it would be certainly possible. While it could be argued that in this case the instrumental way might be efficient, since the students don't have enough mathematical maturity yet. It can also be said that you need to judge which approach is best based on the classroom and even based on the particular student.
I believe that solving the issues raised in the article comes with questioning of why we're teaching what we're teaching in the first place. What is the ultimte goal that we're teaching the technique at hand? To build skill? To build understanding? Somewhere in between these two? If we have a goal in mind, such as "getting students excited about mathematics" then we simply adjust our techniques into reaching this goal. If the goal is "getting students excited about mathematics and learn some technical skills" then we'd require to somehow balance instrumental and relational understanding. Mathematical maturity is real and sometimes, depending on the age of the children, mechanical tasks might be more beneficial.

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