My Most Influential Math Teacher

          I do not remember exactly when I first enjoyed learning math, but the earliest moment I can recall when I realized my math ability is when I was in 3^rd grade. I joined math/science club where I collaborated with my peers to work on the math and science projects. My 3^rd grade teacher inspired me to join the club because she found my math potential and believed that I would love exploring mathematical problems through a variety of inquiry projects. Indeed, I was a top student in mathematics and always had confidence to solve math problems. In my math/science club, I really enjoyed getting engaged with projects by integrating math into science and vice versa. Not only did I learn how to collaborate with others, but I also realized once again that I had special interests in math, which made me have a dream of becoming a math teacher. At this point in my life, although I am not a math teacher yet, I appreciate my 3^rd grade teacher because she supported me and helped me find my ability/interest in math.

          However, I did not like my middle school math teacher. Although she helped me strengthen myself in math problem solving skills through math contest problems, she was boring when she taught math because all she did was write notes on the board during class. Yet, I liked her notes because they were very organized and clear. Even though I understand the purpose of note-taking for learning, I believe that if I had had a fun activity like some kind of math project, my math class with her would have been much more enjoyable and memorable than it was.

Provincial Pro-D Day

I am attending IB-DP Workshop on Oct. 23rd & 24th.

Teaching Perspectives Inventory

From my TPI result above, I found it interesting to see that my apprenticeship perspective scores fall above the upper line labeled ‘Dominant’, and my transmission and nurturing perspective scores are ‘Back-up’ perspectives. Also, my developmental perspective scores fall below the lower line ‘Recessive’.

Above all, I was quite surprised to see that my developmental perspective are recessive. This is an unexpected result, because I try to think creatively and flexibly when I teach students by using a variety of resources for a better understanding. However, I already knew that my social reform perspective scores would be low since I rarely take risks in my learning. Nevertheless, I should teach students how to take and manage good risks by helping them explore the world around them to create a better society.

While my developmental and social reform perspective scores are low, my transmission, apprenticeship, and nurturing perspective scores are high. Of all, I was pleased to see my transmission perspective with high internal consistency. I think that as an educator, the transmission of knowledge and skills should be based, so the teacher can help students acquire knowledge. In addition to transmission, nurturing perspective is vital since the teacher should encourage and guide students to keep them safe. I expected my nurturing perspective scores would be the highest, because I believe as a future teacher, that it is important for the teacher to find students’ potentials/talents and be aware of their different contexts.

How many squares are there in a 8X8 chessboard?

8X8 Chessboard

This question may be confusing for students because it is actually about the total number of any possible squares in a 8X8 chessboard, not simply the number of 1X1 squares in a 8X8 chessboard. To help students understand better, the teacher can break it into smaller chessboards and find patterns.

Number of squares in 1 X 1 chessboard -> 1 square

Number of squares in 2 X 2 chessboard:

1X1 : 4 squares,   2X2: 1 square   -> total number of squares=4+1=5 squares

Number of squares in 3 X 3 chessboard:

1X1: 9 squares

2X2: 4 squares (look below)

3X3: 1 square

è  Total number of squares = 9+4+1=14 squares.

We can look at the total numbers of squares in each chessboard and see a pattern.

1X1: 1 = 1^2

2X2: 5 = 1^2+ 2^2

3X3: 14 = 1^2+ 2^2+3^2

...

8x8: 1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2=1+4+9+16+25+36+49+64=204

è  There are 204 squares in a 8X8 chessboard.

To extend the puzzle for students:
I will first let students solve the problem by themselves and see the point where they have difficulties in common. Then, I will show them a clear definition of rectangles and squares to help them find possible outcomes in a 8X8 chessboard.
Or, I can draw a different picture of the equilateral triangle, which is divided into smaller equilateral triangles. Then, I will show them how to find possible equilateral triangles within the big equilateral triangle. 

Reflection: Integrating Instrumental and Relational Learning

Based on the class discussion, I realized that as an educator, there is no need to choose between two methods when teaching students. Rather than using one method, integrating both relational and instrumental learning is an effective way of teaching students. However, the order of methods does not matter in teaching as long as students understand logically and construct their own way of thinking to solve problems. For example, when introducing arithmetic series, a teacher can use both methods as below.

Ex) What is the sum of all the odd numbers between 1 and 100?

1+3+5+......+95+97+99 = ?

●Relational method:

è  (1+99)+(3+97)+(5+95)+...  = 100+100+100+...

Since they are odd numbers between 1 and 100, so there are 50 odd numbers.

But two of the odd numbers are added together, so there are 25 of them.

è  100 * 25 = 2500

Therefore, the sum is 2500.

●Instrumental method:

With a formula:

Sn=(n(a1+a2))/2 ,     where Sn=the sum of n terms, a1=first term, an=nth term.

So, when applying this formula to the problem,

S50=(50*(1+99))/2 = (50*100)/2=2500

Without a formula:

1=1=1^2

1+3=4=2^2

1+3+5 =9=3^2

1+3+5+7=16=4^2

We see patterns of n^2. Since there are 50 terms, n=50. -> 50^2=2500.

Not only applying a formula to the problem (instrumental learning), but showing why it works (relational learning) will help students understand the principle, so they will be able to branch out to similar problems. Moreover, even if the formula is not given, a teacher can help them find their own way to solve the problem by making pattern rules. 

My Response to Richard Skemp's Article on Instrumental VS. Relational Ways of Knowing in Mathematics

I find the term “faux amis” very interesting, because this is the word I have never seen it before (1). Along with its definition, he mentions that it is seen in mathematics as well. This makes me relate the idea of “faux amis” to problem solving situations. Just because students have the same problem to solve, it does not mean that they all interpret it in the same way. Students find their own way to solve math problems regardless of what they have been taught. In addition to the meaning of “faux amis” in math context, multiplication of fractions appeal to me as well (3). It is easier for me to see patterns in numbers how or why they work the way they do. The example of multiplication of fractions makes me understand the author’s purpose. Moreover, the “mis-match” between the teacher and student in understanding makes me think for a moment. I am surprised to realize that the way teacher wants students to understand concepts does not always work the way they want.

Based on my personal experience, I understand relationally and instrumentally. However, if I have to choose between these two learning methods, I should say I learn relationally. When I first see formulas, I try to see how the formulas work the way they do and find the relations between variables rather than simply memorizing the formulas. Such a habit of mine makes me remember the formulas for a long time. However, I teach instrumentally most times when I try to explain math problems to students. This is because I am concerned about them getting answers wrong that affect their grades. Indeed, instrumental teaching method is easier to use when helping students understand better; on the other hand, it often causes problems when students can not solve other similar questions since they do not have wide application. Over all, I should take advantages of both relational and instrumental teaching methods to enhance students' understanding. 

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