Back in my day as a math student...

The best math teacher I had was actually my math teacher for Principles of Math 11, Principles of Math 12, and Calculus AP (sort of). In fact, I gave him the nickname, and have so-called myself, of Math Man. Math Man was the kind of teacher who recognized he didn't need to hold your hand through work, he believed in the strengths of his students. That came across most prominently in Calculus AP. In fact, I couldn't get into the actual AP class block because I had prior commitments. One day, I asked him for a reference letter. He told me, "I'll give you a reference letter, if you register for Calc AP". After explaining my situation, we came to the agreement that I would come in during my D block spare, which was also his prep block, and he would teach me Calculus for half the block. This was monumental for me. He sacrificed half of his prep time because he believed that, even with the accelerated learning pace (half the time means half the instruction), I could learn the material. It motivated me. He believed in me, and I would NOT let him down. And now look at me, I have a Math degree and am going to become a Math teacher! Math Men for life.

I've never had any horrible math teachers in grade school. So long as they didn't get in my way and fostered the speed at which I learnt math, I was happy. In university however, I've had some doozies. Many professors just aren't equipped to teach: messy printing, don't speak clearly, cannot transmit ideas and concepts well, as well as any number of different problems. Most of my university math degree was me teaching the material to myself, which again, comes back to Math Man, and him facilitating the belief that I CAN do this. I owe a lot to him.

TPI Results

This picture summarizes my TPI results!
There were no real surprises in terms of the five main aspects of teaching and their relative heights to each other. I know that I value the social and emotional development of children a lot more than transmitting knowledge or social reform.
The surprises and interesting results were regarding the relative levels of beliefs, intentions, and actions within transmission. The fact that my beliefs are so much lower than my actions is telling of some sort of outside compulsion to transmit the knowledge, and that I don't value it as much as an aspect of teaching.
This does stimulate some ideas of my teaching. Will this lack of intent and belief come across in my teaching? Will I need to adjust my beliefs? Should I worry more about developing transmission and social reform more? Should I emphasize social reform in a math class? What kind of teacher do I want to be? One that is balanced in these aspects, or is it okay to value and emphasize one aspect more than others? These are questions that I have to answer in the coming year.

Chessboard Extension

Looking at the question "How many squares in a chessboard?" is an interesting and challenging problem. At first glance, I thought to count all the black squares, count all the white squares, and then add up the two totals. Upon further thought, and upon learning that a standard chessboard is 8 x 8 squares, I realized one could just multiply 8 by 8 and get 64 individual squares.
However, it was brought to my intention that there are multiple sizes of squares on a board (1 x 1, 2 x 2, 3 x 3, 4 x 4, 5 x 5, 6 x 6, 7 x 7, 8 x 8). This posed a larger, more complicated problem. Counting is not efficient enough. I had to start look for patterns.
When looking at 8 x 8 squares, there is only one possible square on a chessboard (1^2).
When looking at 7 x 7 squares, there are four possible squares on a chessboard (2^2).
When looking at 6 x 6 squares, there are nine possible squares on a chessboard (3^2).

It is at this point that I noticed the pattern. There would then be 4^2 5 x 5 squares, 5^2 4 x 4 squares and so on.

Thus, through pattern recognition, one can surmise that there would be 1^2 + 2^2 +3^2 + ... + 7^2 + 8^2 total differently sized squares on an 8 x 8 chessboard. This adds up to 204 squares!

My thought process changed when I actually understood what the problem was asking; when I sort of opened my mind to a different understanding of "how many squares"; that there could be different sized squares that aren't as apparent at first glance.

The tools I used were a pencil and paper to draw a mini diagram of a chess board to visualize where squares were oriented. It would've helped to use different coloured pencils to identify different sized squares on the diagram. I also used a calculator to add up all the numbers.

You could extend the problem by cutting out a section of a chess board, and eliminating the occurrence of several squares. Maybe even cut out a rectangular portion of the chessboard to eliminate a number of squares that wouldn't follow a particular pattern (ie. if I just cut out the bottom left quadrant, you can easily figure out how much would be eliminated, but if I cut out a 2 x 4 rectangle, it's a little more difficult).

Sept 21st Reflection on Instrumental vs Relational Discussion

After discussion with my classmates, I have come to realize that incorporating both fluency (instrumental understanding) and meaning-making (relational understanding) is important to a student's learning. Both understandings are valuable in different situations for different individuals.
For example, as discussed in Skemp's article, when considering the area of a field whose dimensions are 20 cms by 15 yards, it is important to be fluent (successfully multiplying 20 by 15) and to understand the meaning (converting the units to be compatible).
It is important to understand that, while having concepts and a relational understanding of a particular topic in math is critical, that information has little use if a student lacks the mechanical and instrumental ability to manipulate those concepts.
It's almost like relational understanding is analogous to nouns in grammatical structure, and instrumental understand is analogous to verbs. A sentence is trivial without the utilization of both.

September 16th Instrumental vs Relational Understanding

The two different notions of understanding confused me. While relational understanding was intuitive enough, instrumental understanding, or the author's articulation of it, was rather confusing. Once he brought up the example of the circle, diameter, and why the formula for the circumference is as it is, I understood the concept of instrumental understanding much better.
I loved his recollection of the teacher who "became suspicious" that his students were not fully understanding relationally what concepts were going into calculating area. The "test" of having different dimensions was a clever and decisive method to check whether his students were understanding instrumentally or relationally. It makes me wonder how I will check to see if my students are understanding relationally, or if I will even notice that they aren't understanding relationally.
The notion of a mismatched teacher with his text is an interesting one. Having an instrumental teacher in general seems like the wrong hire, but the resulting confusion of pairing him with a relationally centered textbook emphasizes the mismatch. It reminds me of many "by the book", needlessly strict teachers that I had in grade school.
I agree with his stance. Relational understanding is better than instrumental understanding. Instrumental understanding is just memorization and calculation. Computers can do that for us. Relational understanding is conceptual intuitiveness. It breeds creativity to branch out into other problems. And further, it strikes motivation and determination to tackle those problems. That, a computer cannot do.

First Post EVER!

Hey! This is my first post ever. I have no idea if I'm doing this right, but give me points for trying. Hi! My name is...What?