The Dishes Problem & Nursery Rhyme Puzzle

My first reaction to seeing the dishes problem is to scour through methods that are familiar to me, including solve it algebraically:

Suppose the number of guests is denoted by x, then the guests have used:

• x/2 number of rice dishes
• x/3 number of broth dishes
• x/4 number of meat dishes

As the total of the dishes used is 65, these information gives us:

x/2 + x/3 + x/4 = 65

Now, solve for x in the above equation:

6x/12 + 4x/12 + 3x/12 = 65

13x/12 = 65

x = 60

Hence, using algebra gives a solution of 60 guests.

Even though this problem can easily be stated in its equation form without altering the solution, I think presenting it as a word puzzle (and its original form) adds more flavor and relevance to the real world. I also feel that word problems, especially puzzles and riddles, add another layer on top of mathematical logic and concepts. The problem solver now has to dissect the semantics and language to infer what the question is, and then proceed with finding the answer. It is also in our human nature to be motivated by challenges, and reshaping a pure math problem that can be easily solved into something both complicated and playful is a way of changing our attitudes towards what's trivial.

Regarding the nursery rhyme puzzle (at least that's what I think it is), I have two ways of interpreting the cake sharing scenario:

1) If the fiddler, the wife, the piper and the mother are four different individuals, then they could have allocated the three cakes, three half cakes and three quarters any way they had liked because it's not necessary for them to have equal shares.

2) If the fiddler's wife and the piper's mother are one person, then there would be three people, and so each would receive one cake, one half cake and one quarter.

I am leaning towards the latter because if we really want to impose an equality condition, then this allocation is easily attainable. 

Microteaching: Game of soccer (with emphasis on offside rule)

Lesson Topic: The Offside Rule in Soccer

Objectives

• To understand how the game of soccer works (basics)
• To learn the offside rule
• To distinguish between offside/onside situations

Materials

1. Coaching board

2. Black marker and eraser

Beginning (2 minutes)

• Discuss the basic rules of soccer
• Introduce the offside rule

Middle (3 - 5 minutes)

• Explain in detail the offside rule
• Why is it significant?

End (5 minutes)

• Provide students with examples
• Have students distinguish offside positions on activity sheet

Math Puzzle: Father and Son

My interpretation of this word puzzle is that "the man" in reference is the speaker's son.

The way I looked at this is by parsing the sentence into two, with the conjunction used as a separator. The first part says the speaker (let's say A) has no brothers or sisters. The second part says there is a man, and that man's father equals my father's son.

I then parsed this into two phrases ("That man's father" and "My father's son")  with the "is" acting as an equal sign. From the second phrase, I deduced that it means the same as A's father's son, which equates to A. Hence, the first phrase equals. A, and so that man's father is the speaker himself.

What I found interesting in the end is that the first sentence has no effect on the second sentence. Removing either would not change the relationship deduced in the second part.

Math Art Reflection

Before deciding on the Babylonacci artwork by Philippe Leblanc for the first group project, our group had spent a long time browsing the Bridges gallery because many of the choices seemed really interesting to explore. In the end, this was chosen because of two main factors: 1) The concept was relatively simple for the entire class to understand; and 2) We had various ideas initially about how to extend the art piece. In the end, we thought to incorporate the Fibonacci Spiral was a great way to not only create another version of the original work but also spark further discussions.

In the process of replicating the original, we came to a better understanding of the Babylonian number symbols. With the help of the math history class, I was able to grasp the idea very quickly. However, extending the art into our Fibonacci Spiral was more difficult than originally imagined, because we had to plan the limited space we had very carefully. As a result, it became necessary to test out at which point in the sequence the spiral should stop at for it to be aesthetically presentable, so there were some trial and error as well as measurements involved in making our Fibonacci Spiral. This made me appreciate the work done by Mr. Leblanc as planning out this kind of project is more difficult than it seems.

What I was amazed by throughout the entire week of planning and crafting the project was how ubiquitous the Fibonacci Sequence is, and it appears in a wide range of disciplines, which as a group we decided to touch upon in the presentation. Originally I only perceived this as a mathematical concept that coincidentally appeared in music teachings, but the application range is much more than that. Thus, I find that this kind of project can be really helpful to my future students who might not be interested in mathematics to be more engaged with these concepts through interconnections with other subject areas.

The Locker Problem

My approach to the "Locker Problem" is as follows:

1. Start with lockers that will be changed once (i.e. #1, 2, 3 ... ).
2. Then proceed with ones that change twice, thrice, and so on.

Continue with the pattern until we reach 500.

The mechanical approach I would take is to use an Excel sheet.

A Letter to Myself in Ten Years

Letter #1

Dear Mr. Chen,

It has been ten years since I last sat in your classroom, listening to your fascinating stories, your random movie critiques and your funny rants on what went wrong around the world.

From your class, I gained a different perspective on how a real positive classroom environment can be constructed, and I was able to learn a lot from your class. In fact, I became more interested mathematics, and I went on to study mathematics for my undergrad. Now I'm pursuing my PhD and teaching my little ones simple math problems in creative ways.

I want to thank you for inspiring me to love the subject that I have spent years studying and researching.

Next June will be my graduation convocation. I hope your calendar hasn't been taken up by then.

Sincerely Yours,
Your Best Student

Letter #2

Dear Mr. Chen,

First of all, I want to thank you for all these years that you have tried to teach the complexities of mathematics to me. I do realize that while I have certain limitations, I probably had more potential than most realize.

I've always disliked the subject of math, partly because every class I would be just taking notes and drawing graphs that will be left asleep in my notebook for days. I hoped that someone would help me understand what I had just written down, but it seemed like the lecturers would just go on and on, and I was more interested in staring at the clock than the board.

Personally, I respected your teaching and attitude towards your students.

However, in the end, I feel like I have underachieved in mathematics. As I step into the real world and actually work with numbers, mathematics actually seemed easier than I thought. All I needed was some guidance, probably.

Sincerely Yours,
Your Past Student

Mathematics and Me

One of the most memorable moments I've had as a math student growing up happened during my grade 12 year, when I was taught by a teacher who used methods that seemed eccentric at the time. He would often tell us jokes or go over scores from last night's hockey games during the class, or he would just tell us random facts that none of us will remember, but this actually created a positive experience for what is widely considered as a difficult class. First of all, what he did was to create a connection with his students, as well as mental breaks and opportunities for students to relax and absorb from the massive amount of material they have to learn. Secondly, many of the facts or stories he did tell were sometimes related to numerical problems, which made a few students more interested in the subject (including myself) and the class (such as the guys who loved hockey but hated math). Lastly, it was a sharp contrast to what most of us had experienced before, and even if some may have felt uncomfortable with his unconventional teaching approach at first, gradually the class fell in love with a more vibrant math class, especially compared to those of the past. The most important thing? Our academic performance as a class was terrific.

Representation in Mathematics

The one statement in this week's reading that I strongly concur with is the claim that mathematical representations must serve as a cognitive tool rather than the end product of a task. I would also like to add that, in my view, representations are most effective when acting as an accumulation of all the cognitive efforts that helped students to arrive at an answer.

For example, an analogy I have is how we use the GPS to guide us from point A to point B. The representation in this case is not only a series of lines are connected to the two points, but also why each step was taken knowing where point B is on the map, and if one of the "traffic intersections" was blockaded, what alternatives could we take to arrive at the same destination. In most cases, there are multiple ways to get to the final answer (both in terms of geographical location and mathematical problems).

There are two examples mentioned in the reading that I found interesting: 1) the usage of instructional materials like the base 10 blocks to help students with addition; and 2) using a combination of algebraic and pictorial approach to teach trigonometry. The first example, although mostly applicable for students at a early age, requires a high level of instruction (as mentioned in the article, which I agree with) that can successfully map the material representation with the internal math procedure. For the second example, it's conceivable that the referenced study was able to show success when both internal and external representations were used, because trigonometry requires visualizations more than other concepts in mathematics. These examples show that mathematics is both internal and external in its representations, as the author suggests. But in practice, what, when and how we use external representations are context based. They can vary among different teachers and disciplines, and the question whether students prefer external or internal representations should be asked. Teachers also need to be more reflective, as some could conform to using a 'correct' way of representation instead of coming up with ones that are more helpful and creative for students to experiment with. In addition, there could be weaknesses in using too many external representations, as some argue that having more than two different visual representations may be confusing and counterproductive.

One example of external representation that I immediately thought of when I started reading the article was the tree diagram used in teaching probability. This was how I learned probability effectively as a student, so it's something I can relate to. But the tree diagram is only one of thousands of ways to teach probability using external representations. As stated above, as teachers it's important to think about how we use representations to not only help students understand the abstract ideas, but also obtain a blueprint to apply the ideas in a flexible manner.

Instrumental vs Relational Ways of Understanding

The most striking aspect of this week's reading is the example of teaching students music. As someone who studied music and learned to play piano growing up, I went through the same methods that the passage described as "relational" (it's funny that my way of learning a musical instrument was not "instrumental"). Perhaps this also explained why I found the theory part of music tedious and was not very skillful at techniques such as sight-reading and harmony identification. In mathematics, there are similar questions we can ask regarding the two different ways of understanding. There are benefits and disadvantages of treating one as superior over the other. The passage questions whether they're separate subjects or just different ways of looking at the same thing - and I would lean toward the latter. My own experience from my time as a math student and then a tutor dictates that the relational way can definitely be helpful to students who need immediate results and accuracy with future problems that could be perceived to be in unfamiliar territory. The important task is for students to ask why this works? Because relational understanding is especially helpful in developing the critical thinking skills. I feel as teachers, we need to devise a strategy (i.e. visuals, graphics) on how to teach students that creates a mental map of different concepts they have learned and how they can work together, because nothing that is taught within one course is left on a single island. It is also important to consider whether students are receptive to the relational understanding of mathematics, because if they're only thinking about the "instrumental" way, then there's a misalignment between the messenger and the receiver.