Pro-D Day

For this year's professional development day, unfortunately I will not be in the province to attend due to my convocation on the same day. However, I am looking to see if there are other PD-related courses as part of BCMT.

My Best and Worst Math Teacher

From my past experience, I have had decent teachers that taught me mathematics. My best teacher was my high school math teacher who taught me math throughout my high school. I have to say that she was the one who motivated me and inspired me to become a math teacher. Even though she was a tough teacher, she really pushed me to work hard in class. In fact, she pushed the entire class to do really well. I guess it's the perk of being in an enriched class than the regular academic classroom. Sometimes, if I answered questions wrong, or take longer to understand certain concepts, she gets disappointed in me or almost felt like ashamed of me for answering the questions wrong, which I took it personally. It is almost as if I'm letting my parents down. This kind of "failure" really impact me in class. Perhaps, she expected a lot from me, and that not getting the correct answers isn't something that I normally do? Sometimes, I get intimidated by her, and the questions she asked in class or on the tests were challenging. However, if it weren't for her tough love and push I wouldn't be able to do well in math as I am now. So, I thank her for that.

In all, I think my worst math teacher is also my best math teacher. If it weren't for her tough love, her lack of patience and her constructive criticisms, I wouldn't be able to push myself to pursue math as my undergraduate study. I think she definitely improved my understanding in math (the technical aspects) for sure. There is another female teacher in high school who taught me Gr.10 math. She has not only taught me the "technical" aspect of math, but also taught me the "soft" side, the personal side, of teaching math. She was so awesome as a teacher that I decided to work with her as a volunteer teaching assistant at her new high school. She was more patient with her students than the first teacher I wrote about in this blog. Both were funny math teachers, even though one is more serious than the other, but they have all pushed me hard to help me reach my goal as a student.

Blog 4: My TPI Reflection

As seen in this TPI Profile Sheet Table, my result seems pretty consistent throughout. On average, I scored 32.6 on the scale, with a standard deviation of 2.65. I scored high on Apprenticeship teaching style and lowest on Social Reform, while Nurturing, Developmental, and Transmission teaching style are fairly consistent. From looking at the results, I would think that I scored high on Transmission because when I was tutoring or teaching students, I often value the subject matter (the knowledge) more highly than the teaching styles. As a student, I would think knowledge is the ultimate key to understanding because without it, there would be nothing to teach the students or us teachers! However, as I read more on other areas of teaching, I start to realize that there are more areas than just simply teaching the contents. As a teacher, we are no longer teaching the subjects, but really teaching the students. The high-scored Apprenticeship area tells me that, as a teacher, I "translate" knowledge into "accessible language" for students. I am here to provide "guidance and direction" to engage learners with ways that they feel more comfortable of learning. I find this result of teaching very true.

I remember when I was tutoring students on math areas that I felt comfortable with sharing, mostly because I learned and understood before as a high school student, I noticed that some students weren't getting the concepts, even if I explained to them in most simplest terms. I realized that how my teacher taught me was way different (and a bit more complicated) than how their teachers had taught them. I guess I didn't expect teachers within the same school board to teach students differently. As a result, they just weren't getting the concepts. To resolve this, I had to adapt my ways of learning to their ways of learning, to "translate" my thinking into a language that is "accessible" to them that they are more familiar with. It is important for me, as a teacher, to read and really understand students and their needs (i.e. strengths and weaknesses) so I would know what best strategies to teach them - to see whether they needed support or additional remedial assistance.

I am quite surprised that I scored lowest on Social Reform. Social Reform, as written on the TPI website, is the "effective teaching [that] seeks to to change society in substantive ways." I had always thought that mathematics is really practical in ways that can build our society. The world we live in is immersed with the concept of math - it can be found in science, engineering, and even arts - they all have math as a fundamental basis. I really encourage students to associate what they are learning to the world around them; to let them apply math in a practical setting is the sole purpose of math studies. As a teacher, it is also important to let students take critical stances on what they are learning (to mature their brain activities). As the guest speaker at one of my other class said to us earlier, "the most difficult child is the one who learns the most." I guess I will have to improve my my social reform teaching style!

Blog 3: How Many Squares Are in A Chessboard?

For this fun activity, we are given an 8x8 chessboard. How many squares are there?

Before we get excited and jumping into conclusions, if you are a visual person like me, you can start by either having the chessboard in front of you, or you can start by drawing an 8x8 grid on a piece of paper.

1. Once you have the visual in front of you, let's start by looking at 1x1 grid.

Think:
How many 1x1 grids are in an 8x8 chessboard?

Visual example.

Answer:
Since there are 8 pieces of 1x1 grid horizontally and 8 pieces of 1x1 grid vertically, we can find the total numbers of 1x1 grids by calculating the area.

There are 64 pieces in total. (You can count them all to be sure!)

2. We now look at 2x2 grid.

Think:
How many 2x2 grids are in an 8x8 chessboard?
Hint: think about how many 2x2 grids are there horizontally and vertically. Include any (overlapping) combinations of 2x2 squares, as seen in red.

Visual example.

Answer:
If you continue to find 2x2 grids, we can see that there are 7 pieces of 2x2 grid horizontally and 7 pieces of 2x2 grid vertically.
There are 49 pieces of 2x2 in total.

3. We now look at 3x3 grid.

Think:
How many 3x3 grids are in an 8x8 chessboard?
Hint: think about how many 3x3 grids are there horizontally and vertically. Again, include any (overlapping) combinations of 3x3 square, as seen in red.

Visual example.

Answer: 
We can see that there are 6 pieces of 3x3 grid horizontally and 6 pieces of 3x3 grid vertically.
There are 36 pieces of 3x3 in total. 

Do you notice any patterns?

See how as the grid size increases by 1, the numbers of pieces of square decreases by 1? As the pattern continues, we can create a handy-dandy table that tracks how many squares are in each case. You do the calculations!

So now that you have counted or calculated how many squares there are from 1x1 grid to 8x8 grid, what's next?

We can add the numbers of grids of all sizes together!

Why?

Let's look at a similar (simple) example: How many squares are in a 2x2 chessboard?

A 2x2 chessboard is consists of 4 pieces of 1x1 grid and 1 piece of 2x2, since the chessboard itself is a square. In total, there are 4+1=5 squares.

Final answer:
Similarly, in an 8x8 chessboard example, we have to add all the numbers of squares together because we are looking for the total number of squares.

In total,
There are 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares in an 8x8 chessboard.

Check: Did you do the calculations right?


This puzzle can be extended if we ask how many rectangles of a certain unit sizes are in the 8x8 chessboard. For instance, we might ask: Find the numbers of 2x3 rectangles are in the chessboard, or find the numbers of 3x2 rectangles in the chessboard. To make this activity even more interesting, we can extend this chessboard to a 3-dimensional objects, like a Rubik's cube! This might pose a challenge to some students, but it is certainly more fun as we get our brains moving!

Blog 2: Personal Reflection on Instrumental vs. Relational Learning

The choice of having an instrumental or relational understanding for students is very subjective among teachers, and I think it really depends on students' learning abilities to see which choice of learning is the best or preferable. In our debate, I think many of us teachers candidates used our learning strategies when we were students and how we think should be a good learning method for our future students. In my opinion, depending on the grades we teach, the overview of math concepts should be introduced first. For example, teachers should introduce the signs of addition (+) or subtraction (-), multiplication, and division. Once it is introduced, or it has been taught "instrumentally", students can then dwell deep into why addition exists, or why division exists, and how these concepts are practical in real life. Provided with these questions, teachers can then explain things such as "When you are at a supermarket, you may want to calculate the cost of each apple, and see which one is the cheapest!", for instance. Teaching students the concept of the order of operations is another example. We first introduce the concepts to students "instrumentally", then we can explain to them why this order of operation, BEDMAS, is important in real life, "relationally".

Furthermore, as we go on to teach at higher grades, complex math concepts such as the Pythagorean Theorem or the vertex form of a quadratic equation, in my opinion, should be taught first "instrumentally" as well. Once students understand fully these concepts and are able to, fluently and correctly, solve questions using the concepts as tools, we can further let them question why and how certain theorems work that way. Of course, the real meaning behind these formulas, proofs and theories is the foundation to a better understanding, but It doesn't necessarily mean that students with instrumental way of learning is wrong or are less knowledgeable than students with relational way of learning. It just means that they both understand the same concepts and that they both know how to apply those concepts as a tool to manipulate them in order to fit their learning framework. In all, I think students have various preferred learning strategies, and it is how they choose to learn that are ultimately important to our math teaching.

Blog 1: Response to Richard Skemp's Article

As I was reading through Richard Skemp’s article, titled Relational Understanding and Instrumental Understanding, some key points raised in his article made me think about my past experiences as a student as I recall my learning techniques during early or secondary classes. Firstly, the fact that Skemp points out the exact issue that I am regretfully of committing – this instrumental way of learning and understanding – in most of my math classes, as a way for me to speed up the learning process. Secondly, I cannot agree more on the points he covered on relational understanding, and how it is a basis for other new concepts and topics. On this point, I realize that the units covered in math classes have all tied to each other in a sequential way. So, it may be harder to learn and understand a new topic without fully understanding the previous topic really well, and on why we study this. For example, we learned the area of a shape by first learning measurement. Thirdly, as Skemp may not have addressed, however important relational understanding is for students, I think that relational understanding of teaching may only be appropriate for certain classes, such as the academic, gifted, IB, or AP classes. Surely it is worthwhile to teach students in applied classes and engage them with relational understanding, as a way to find their potentials, but I think teachers should teach this way one step at a time. By first engaging the students in the materials they are learning, and once they are “hooked” on a particular topic, teachers should then dwell deeper into the reasons and to seek new topics which covers that as a basis.

Overall, I agree with how Skemp stands on this issue because teachers should provide students with a relational understanding of concepts. This is particularly important once students study at post-secondary schools or even doing research for their post-graduate studies. That being said, finding the right class to teach these kind of lessons is something else that I would like to explore.

Welcome to YT's math blog!

Hello everyone! My name is Ying Ting, and this is my first blog.