My Math Experiences: The Extremes

My best Math teacher:

In grade 6 and 7, I had wonderful teacher, Mr. K. You could tell he really enjoyed teaching, and he would even give us candy on a test day! (He called them "brain vitamins"). During math class, he would make us stand up and learn "math dances," which were silly sayings combined with movements that helped us remember certain mathematical rules. I still remember some of them today! He got us using using tactile tools as well, providing us with blocks and measuring instruments to allow for a more hands-on approach. He also allowed for differentiated learning within the class, grouping different students together and giving them the kinds of questions that would challenge their problem areas.



My worst Math teacher:

I've had quite a few unfortunately bad professors during my undergraduate degree, and they all had a two things in common. One - they would talk very fast and write messily, and I'd always be racing to write down everything (let alone understand any of the lecture). Two - it was obvious that they didn't care about teaching us. The professors would get frustrated at questions, and had no patience for students who asked for things to be repeated or explained again.
That being said, I have had some really engaging math professors, who welcome questions. I just wish there were more of them!

TPI Reflection

Completing the TPI and looking at the results was very interesting and informative for me. I didn't realize my priorities and opinions on quite a few of the topics until I was actually confronted with questions about them. For example, I believe strongly in social reform, but I don't prioritize it in my Math teaching. My main concern is students' social-emotional learning, and their views towards themselves as a self-motivated, confident student. This is also why I believe I scored slightly lower in transmission.
I found it interesting after completing the TPI to think about how I could incorporate more social reform or social justice ideas into my Math lessons. While I still believe focusing on nurture is my main priority, involving social issues could result in projects that involve more relevant, real life problems students can actually care about.
Some ideas I've been thinking about:

• Volume questions related to creating houses with water bottles in Nigeria 
• http://phys.org/news/2011-11-plastic-bottles-nigeria-housing-problem.html
•  Graphing questions related to minimum wage
• Dean, J.  (2007).  "Living algebra, living wage:  8th graders learn from real-world math lessons."  Rethinking Schools, 21(4), 31-35. 
• http://itec-ubc.ca/wordpress/mackowetsky/wp-content/uploads/sites/54/2014/02/Living-Algebra-1.pdf
• Understanding statistics, perhaps related to:
• From http://www.radicalmath.org/main.php?id=SocialJusticeMath#3
• Public Health: AIDS, asthma, health insurance, diabetes, smoking
• Educational access, funding, testing, achievement gaps
• Environment: pollution, hunger, food and water resources
• Welfare
• Immigration 
• Debt, credit cards, minimum payments, interest, loans, etc.
• Lotteries

 

Chessboard Problem

When I first considered this chessboard problem, I immediately jumped to 64 as the number of squares. But once I realized I could also consider squares of bigger sizes, the problem got a bit more interesting! I am a very visual person and I always jump at a chance to draw a picture, so first thing I did was draw a chessboard in my notebook. Next, I wrote down the number of small squares (64) because that was something I already knew. Then, I tackled the next smallest square, 2x2, so I could start systematically working my way through all potential square sizes, smallest to largest, not missing any along the way. I started counting 2x2 squares along the bottom row, and part way along, I realized I was counting the centers of each 2x2 square. So I decided to count how many centers along and how many up there would be in the chessboard, and multiply them together, which happened to be 7(across)x7(up)=49. I did the same method with the 3x3 squares, which I found was 6 middles across and 6 middles up (6x6=36). At this point, I realized the pattern would continue all the way to the 8x8 square, and just wrote down the answers without counting centers.
Finally, it was time to add up all the squares I'd found! To make the mental math a bit easier for me, I added first the numbers that make multiples of five (so 1+9, 4+16, etc.) and then added up the rest. My solution, as you can see from the picture, was 204 squares!

From the above, you can see there were a few changes in my thinking:
- Considering only small, 1x1 squares, to considering many sizes
- Counting individual squares, to counting unique centers
- Counting centers all the centers, to multiplying the number of centers across and up together
- Multiplying centers, to just assuming the continuation of the pattern

Tools I used:
- Drawing a picture
- Use of the center of the squares
- Multiplication and addition

How to extend the puzzle:
- How would the solution change if it was a bigger board? Or if the board was not a square itself (ex. 8x9 board)?
- How would the puzzle change if we were instead looking for rectangles of the form Ax2A? Would there still be a pattern in the answer?
- What if we instead had a board of triangles? (below)

Reflection on Our Instrumental vs. Relational Class Discussion

I was surprised at the evenness in how our class split on the discussion of instrumental vs. relational understanding in a math context. I much prefered relational, though I did see some of the benefits of instrumental, and I found it interesting how some of the points from the opposing side I felt completely different about. For example, when they raised the topic of technology, I thought that was a point in favour for relational understanding. Now, with computers to do most of the grunt work of calculations, I think it is more important for us to know how and why things work instead of the number-crunching, 'rules without reasons.' I also think, for certain concepts at least, learning relationally first, not instrumentally, is much more intuitive, not the other way around.
However, I do agree that, in terms of time-saving, instrumental understanding seems like the clear winner. It is much faster to teach, and when it comes to doing practice problems, it is much quicker. Understanding what multiplication and division is relationally is very important, but eventually you just need to memorize shortcuts to get things done in time! That speed and comfortableness is the "fluency" students need to move on to new topics. And though relational understanding is ideal, with the given constraints of class time, making sure every student can abstractly consider each topic isn't feasible.
I think both understandings are inherently linked to the way with must teach math. If I was teaching logarithmic rules, for example, students would understand how logs relate to exponents, and we would use that relational understanding to explain why log(xy)=log(x)+log(y). This meaning-making is important, but eventually for ease and quickness (i.e. fluency), students will memorize the rule.

Instrumental vs. Relational Understanding

                Skemp’s article solidified some of the ideas that have been vaguely floating around in my brain. While doing my undergrad, some of my favourite math moments came from finally learning where certain formulas came from – for example, when the mystery origins of the ‘area of a circle’ formula were finally proven in calculus class. In fact, proofs were often my favourite parts of my courses, and I didn’t really enjoy ‘rule’ heavy courses (ex. differential equations: “just ‘guess’ this solution!”). When I was in elementary school, I actually hated math because all it was to me was memorizing a bunch of times tables and facts. It wasn’t until I got to high school when I got to do some logical problem solving did I start to enjoy it. Because of this, I related a lot to Skemp’s arguments for relational understanding as a basis for mathematical instruction. I agree that to fully comprehend and appreciate math, having that foundation is crucial. I also believe that, at least in my case, it makes the subject more engaging, making students feel more like a detective and less like a machine. However, I really appreciated his devil’s advocate section, because in some cases, it is still appropriate to use instrumental approaches. And with the time given, it is extremely difficult for a teacher to make sure every student has a relational understanding of all the topics in the jam-packed curriculum.

                Three specific lines stood out to me. One, “well is enemy of better.” If our only focus is for students to perform well on tests, then teaching beyond simple rules seems unnecessary. But if this way of thinking is what is taught, it inhibits further development in the subject, because math builds on itself grade after grade. Without a good foundation, ‘better’ is much more difficult. Two, there can be “a less obvious mismatch … between teacher and text.” I hadn’t thought of this conflict before, but it makes sense. When there is disconnect between a teacher’s style and the text, or other teachers’ styles, this can make it a lot harder for students to understand the material in either relationally or instrumentally! I appreciated this acknowledgment of difficulties of introducing more relational based approaches into instrumental oriented schools. Three, “there are two effectively different subjects being taught under the same name, ‘mathematics.’” This sentence took me a while to muddle through - it is an unusual statement. But it really ties in with the earlier statement about mismatching the two approaches to understanding.

                

First Blog Post (Sept 14th)

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