Chinese plate-guest problem (without using algebra!)

If every two people share a dish of rice, then every person eats 1/2 a dish of rice.
If every three people share a dish of broth, then every person eats 1/3 a dish of broth.
If every four people share a dish of meat, then every person eats 1/4 a dish of meat.
Therefore, a single person eats the equivalent of 1/2 + 1/3 + 1/4 = 13/12 dishes of food.
Since there were 65 dishes in all, we can determine how many guests there were by dividing 65 with 13/12, which is the amount that each guest ate.
65/(13/12) = 60.  Hence, there are 60 guests.
This method only works if all the dishes of food were consumed entirely (that is, there were a perfect number of guests needed to consume all the food).
I think that cultural context certainly makes the question more understandable, and easier to relate to.  This problem presupposes an understanding that all the guests share the food equally, that they eat from common bowls, and that just enough food was put out to feed everyone.  In a western banquet, one would certainly not eat all the food, and just eat selections of the menu.  Underlying assumptions might sometimes be based on cultural context.

Reflection on my best/worst math teacher

The worst math teacher that I have ever had taught me during my grade 9 year in high school.  Objectively, she wasn't a bad teacher.  In fact, most students liked her.  She was our science and biology teacher.  But math 9 didn't require someone whose expertise was in mathematics.  She taught us math, and we were fine with it.  I have always done well in math, being at the top of my class, getting As.  But in grade 9, things were different.  I got a C+, primarily because I didn't try very hard to do well.  Math normally came second nature to me, so getting this grade surprised me a little.  Because my attitude towards school in grade 9 was very lax, I think that the teacher judged that I was not able to do exceptionally well in class.  And she treated me as such.  She never showed faith in my abilities, never gave me a chance to prove myself, and never encouraged me to do better.  Rather, she let me be, and I was fine with that.  However, I think it is a teacher's role to sometimes go beyond a person's comfort zone and to reach out to them.  Even if it's in math class.
The best teacher that I have ever had was my grade 11 math teacher.  He was funny, personable, interested in students' well-being, and had a real passion for math.  His degree was in math and music.  He told us math jokes, encouraged us to get involved in math contests, and was always friendly and good-humored.  He made us want to enjoy math because he enjoyed it himself.  And he gave us the opportunity to do the same.

Reflection on my TPI results

Compared to some of my classmates, whose graphs I have seen, my results are pretty even across the board.  However, this does not mean that there is nothing to say about these results.  First of all, my strongest category is that of transmission (TRANS) of the material.  This does not totally surprise me, for I certainly consider this to be one of the main purposes of teaching.  Do I think it is the primary purpose?  Well, yes and no.  I think that forming a student in a holistic way is done in large part through the process of teaching the material.  So, from that perspective, some of the points given to transmission may perhaps be considered to be being taken from the nurturing category.

I think that the holistic approach that I take to teaching is reflected by my results.  The fact that they are pretty even across the board indicates that I put about equal emphasis on every aspect of a student’s learning, and that when teaching I have multiple goals or purposes in mind, not just a single one.

One more observation that I made about my results is that my beliefs are normally stronger than my action.  I find this interesting.  It is true that I have strong beliefs about teaching and pedagogy, but when it comes to teaching, I do find myself to be quite passive as far as the effect that I may have on my students.  I take a hands-off, let-them-be-free-to-respond approach, which may or may not be for the best.  Experience and time will tell.

October 23rd

On October 23rd, I will be going to the Catalyst BCScTA Conference for Physics Student :)

How many squares are there in a 8x8 square chessboard?

Interesting question!  At first, one might suppose that the answer is obvious - 8x8 is 64, of course!  But the answer is not so simple.  One must ask, how many squares can be made in total - including the squares that can be made out of the other squares.

To answer this question, we definitely want to systematize our approach.  It would be way too complicated to just make random squares here and there, and to try to count them up on our fingers.  In mathematics, we often see patterns develop that can help us to determine complicated questions by first finding the solution to very simple ones.  In some ways, this is like the inductive approach to proving a mathematical statement or equation - first prove n=1, then assume n, and show that n+1 is true.  Let us try to solve this problem by considering the most basic case: a 1x1 square.  Obviously, there is only one way to form a square out of this.

Next, consider a 2x2 square.  In this case, we have the 4 individual squares, plus the one bigger one made up of the smaller ones.  So in total we have 4+1=5.

Then, consider a 3x3 square.  We have the 9 little squares.  Then, if we count the number of 2x2 squares, we find 4.  Finally, we have only 1 3x3 square.  So the total is 9+4+1.

Next, consider the 4x4 square.  We have 16 little squares, 9 2x2 squares, 4 3x3 squares, and 1 4x4 square, for a total of 16+9+4+1=30.

We definitely see a pattern emerging!  The pattern is as follows:

1x1

1x1+2x2

1x1+2x2+3x3

1x1+2x2+3x3+4x4

...

1x1+2x2+3x3+..........nxn, where n is the number of sides of the square.

This can be proven fairly simply by induction.

 

It is interesting - induction can be used to prove a whole series of problems having to do with geometric patterns.  Number theory and geometry (the most abstract and the most concrete topics in mathematics!) combine beautifully in order to determine the solution to these problems.  Wherever there are patterns, there is often a mathematical description to the pattern.

Instrumental vs. relational teaching - A reflection on today's class

Instrumental vs. relational teaching: a reflection of today's class discussion

Sept. 16, 2015

Today we looked at the pros and cons of teaching instrumental mathematics versus relational mathematics. Yet teaching doesn't have to incorporate just one or the other method. Both can be integrated. When teaching a subject, I think that both instrumental and relational should be taught together at all times. In philosophy, we would say that every object has both form and matter -- the form being the essence of the thing, and the matter embodying its accidental (secondary) attributes. I would propose that this analogy of form and matter fits very well with our understanding of relational and instrumental ways of teaching math. The structure (instrumental) is necessary as a scaffold in which to set the understanding of the student (relational). We teach the rules, the boundaries, the restrictions, the equations, while explaining how these equations and rules are derived, and how they reflect and can predict reality. To give but one example, consider the Pythagorean theorem. For a student to understand this theorem, he must first have an understanding of what a right triangle is. The Pythagorean theorem provides us with an equation which relates the relationship between the different sides of the triangle. Once the equation is memorized, and the framework is understood, we can then explore the consequences of this equation by noticing that the hypotenuse is always going to be larger than the other two sides. We can consider one or two different triangles, which we can draw out accurately on the board, and test our equation on them. We can consider extreme cases where one of the sides is almost zero in length. We can show how if we double the two right-angled sides of the triangle, then the hypotenuse is also doubled in length. We can also prove the Pythagorean theorem using the simple visual proof of triangles inscribed in a square. The understanding will follow within the framework of the instrumental equations and rules.

I think too that if the teacher has a very relational understanding of mathematics, then he will communicate his way of understanding to the students. I have noticed this whenever I taught high school math -- I would always investigate the why of what I was teaching. And yet a prudent teacher must know where the middle road is, and how much time ought to be spent simply on imparting meaning vs. intrumental teaching.

Skemp's article reflection

Relational Understanding and Instrumental Understanding

A reflection on Skemp's article

The first thing the author wrote that struck me was his differentiating the two ways in which students learn mathematics.  He maintained that there is an essential difference between the relational and instrumental approaches to mathematics.   I concede that these are definitely two different approaches to the way we study mathematics, but I never considered them separate entities altogether.  The author was successful, I think, in demonstrating that the instrumental approach to mathematics can be very limiting, and that it really should not even be called “mathematics” in the same way as is understood when we talk about relational mathematics.  The second thing that struck me was the author’s summary of the pros and cons of both approaches.  First, he talks about how “relational” mathematics is superior to “instrumental” mathematics.  However, there are some serious limitations in teaching mathematics relationally.  For example, this way of teaching might take a lot longer, and therefore might be impractical.  It made me wonder, where do we draw the line in terms of how much time should be sacrificed just so that we can teach in this way?  Finally, what struck me was the way in which the author radically separated the teaching of instrumental mathematics from the teaching of relational mathematics.  He seems to have a very black and white view.  I think that we have to consider these things more as a spectrum – our teaching will involve both instrumental and relational approaches.  We cannot teach everything intuitively, nor can we teach students only as though they were robots.  I think what the author failed to do was to see the gray area in which we can get the benefits from both ways of teaching math.

This is my new blog