Reflection: Pro-D Day

Today many of the math and physics cohorts had the opportunity to attend the e-sports workshop organized by UME Academy. Building upon the idea that "play is learning", the event introduces how computer games can be used as an educational tool, arguing against the stereotypical view that games are harmful to children.

For the first part of the day, we participated in a game-creation activity using the Unity platform, which can sometimes be brought into the classroom to stimulate creativity and improve students' digital literacy. I believe the skills that students can acquire through these kinds of games can be very rewarding when they enter the real world. In the 21st Century, technology skills and creative thinking are two of the most important ingredients in personal success. Encouraging kids to practice their cognitive and creative skills online and experiment with certain software will bring more good than harm in the future.

The second part of the day entails learning about the game of Fortnite, which is incredibly popular among today's children. As teachers, I think it's important to understand the world in which students live in, part of that is being curious about what they are interested in and what they like to do. Using Fortnite as an example, we can get an idea of the type of games they play on a daily basis, and what strategies and artistic expressions they can use by playing these games.

Moreover, teachers were once students as well, and everyone grew up playing games and gaining knowledge through these activities. As modern teachers, we should try to explore what students are able to learn from games rather than denouncing computer games altogether. As mentioned by the presenters today, gaming addiction among children is often a byproduct of various socioeconomic factors, such as the lack of parental guidance and bullying, and many students often use excessive gaming as a means of escaping from reality.

The Scales Problem

The first step I took in approaching this problem is to figure out the mass of the smallest weight, which is one gram because it's required to measure exactly one gram of herb.

Next, I proceed onto the second smallest weight (let's denote its mass as X). This weight can measure X amount of herb, as well as X+1 and X-1 amounts when used in combination with the one-gram weight. I then deduced that the smallest integer X can take is 3, because at X=3, it is possible to measure both two and four grams of herb, whereas if X=2, it can also be used to measure one gram, which is counterproductive.

Using this method, I can also find the second largest weight (let's denote its mass as Y). In a similar fashion as above, this weight can measure Y, Y+3, Y-3, Y+3+1, Y-3-1, Y+3-1, Y+1-3 grams of herb. The smallest amount of these is Y-3-1, which means Y must be 9 (since the small weights can already measure 1,2,3 and 4 grams, Y must be a number that can be used to measure at least 5 grams). Accordingly, the amounts of herb that can be measured using the three weights are: 5, 6, 7, 8, 9, 10, 11, 12 and 13 grams.

Lastly, to find the largest weight, we apply the same rules (let its mass be Z): the four weights used together must be able to measure the remaining amounts of herb (from 14 to 40). Since we know that the smallest amount we need is 14, then Z-9-3-1=14 must hold. Thus, solving the equation gives us Z=27.

So, the four weights are: 1, 3, 9, 27 grams

To extend this problem in the classroom, we can apply the geometric sequence pattern and explore how it may be related to simple addition and subtraction. This can be done by changing the parameters such as stretching the range of the maximum amount of herb, as well as allowing for multiple quantities of the same weight.

Promoting "Flow" in the Classroom

As teachers, I believe it's essential for students to engage in the learning process as much as possible. The ultimate goal is to see students fully immersed in an activity, entering a state of "flow" and deep concentration that allows them to absorb new ideas while working on the task at hand.

One way to create this kind of environment for students is to relate teaching to things or events that are relevant to their lives. An example is that some teachers ask students to participate in interest-driven inquiry projects from time to time, where they will explore various topics that they are curious about and can be related back to the subject area being taught.

Another way to promote "flow" is to arranged extra-curricular activities that allow students to "take a break" from the classroom while still being provided opportunities to learn.

Lastly, as demonstrated in Monday's class, transforming classroom activities into games or even competition can also boost students' motivation to try new things and tackle problems they are challenged to solve.

My Thoughts on the Three Curricula

One thing mentioned in Eisner's article that I found really thought provoking is the notion of rewarding students publicly such as by assigning those with higher grades to honors classes. From my personal experience, I went through a period where my classmates and I often "compared" with one another to demonstrate our superiority, and this kind of competitiveness was only out of pride (for self and family) without any intention to belittle anyone else. But the result is that students do judge each other, and some students who would be placed in honors classes tend to view the rest as inferior. In my opinion, this enabled students to foster attitudes of inequality that could follow them once they enter the real world. As Eisner stated, schools also teach not only what's written in the curriculum, but also the school culture, which itself could count as another curriculum.

Another concern raised by Eisner which made me stop and reflect was the misalignment between schools' promises to children and what students actually obtain through learning. He also inserted a passage from one of Aleksandr Solzhenitsyn's famous works to illustrate that, but takes it further by using the prisoner as an analogy. On this subject, I agree with Eisner's argument that schools have neglected certain intellectual processes that could benefit students in favor of what they have been teaching out of habit. As an economics major, I also believe that the subject area could serve as a useful foundation for students to understand how the social system functions, but it is still seldom taught in classrooms even to this day.

As for the BC curriculum, I can see attempts to modify the conventional approaches to teaching, such as by incorporating visual representations and similar problem-solving processes in mathematics to help students become powerful and creative thinkers. From a personal perspective, a shift away from results-oriented approaches to teaching will do more good for the students down the road.

Reflection: Group Microteaching

For today's group microteaching session, David, Hugo and I decided to focus on introducing the concept of arithmetic sequences to the class. We felt this topic was both simple and interesting enough to talk about within the allotted lesson time.

However, we agreed afterward that our lesson could be improved in several ways. As many of our peers have pointed out in the evaluation forms, the entire lesson could have been cut shorter so that there would be enough time to do the class activity. Also, the transitions between ideas and different stages of the lesson could have been more natural, although this might have been affected by the time constraint. Overall, I feel the entire lesson was well-planned, but the execution of the plan has room to improve.

 

 

 

Lesson Plan: Microteaching

LESSON PLAN – INTRODUCTION TO SEQUENCES

Lesson Overview

This microteaching session is designed to introduce the concept of sequences in mathematics and its possible applications.

Duration

15 minutes

Materials and Equipment Needed for this Lesson

White board, markers & TCs’ cell phone(s)

Lesson Stages

1. Warm-up (2 minutes)

• Background
• In-real life examples: paychecks; as you work more hours, you get paid more.
• Brief Introduction
• Definitions and differences between set, sequence, and series
• Set is a collection of numbers
• Sequence is a set, but with order
• Series is the summation of  all the terms in a sequence
• Today we are focused on arithmetic sequences/series
• Arithmetic sequence is a sequence where the difference between consecutive terms is constant
• Arithmetic series is the summation of a finite arithmetic sequence

2. Presentation (3 minutes)

• General Form
• an=a1+d(n-1)
• an=ak+d(n-k)
• Examples
• 2, 5, 8, 11, …:
  an=2+3(n-1)
• 31, 26, 21, 16, …:
  an=31+(-5)(n-1)
• Applications
• Linear relations
• Fibonacci sequence
• 1,1,2,3,5,8,13,...
• an=an-1+an-2

3. Extension (5 minutes)

• Extension: arithmetic sum
• History of Gauss
• Computing the sum from 1 to 100 for a punishment in primary school.
• General form:
  Sn=n(a1+an)2
  Sn=n(2a1+(n-1)d)2
• Orange equation is used when we know the total number of terms, first term, and the last term.
• Green equation is used when we know the total number of terms, first term and the common difference of successive members.
• Extension: averages: μ=Snn= a1+an2

4. In-class activity (5 minutes)

• Fibonacci sequence “mathic”
• Ask a volunteer to pick two numbers between 100 to 200
• Once he/she list their own Fibonacci sequence up until 10th term, allow him/her to use a cellphone to find the sum.
• While he/she types all the numbers on the phone, find the sum quickly before him/her.
• Compare answers!
• If time allows, ask another volunteer.
• Explain the trick!

5. Evaluation (5 minutes)

Geometry Puzzle

How I can draw 30 equally spaced points on a circle? This was my initial reaction to the puzzle.

I then decided to experiment with other numbers that are easier to plot to find a pattern.

So, realizing that it's much simpler to divide everything into halves and plotting the points that way, I went on to use 2, 4, 8, and then 16 equally spaced points, drew them on a piece of paper using a Starbucks cup and labelled each point clockwise (as shown in photo).

I noticed that for each diagram I have drawn, the pairs of diametrical opposite points look like this:

Number of Points                       Opposite Pairs

Two points                                  point 1 - point 2

Four points                                  point 1 - point 3
                                                    point 2 - point 4

Eight points:                               point 1 - point 5
                                                    point 2 - point 6
                                                    point 3 - point 7
                                                    point 4 - point 8

Sixteen points                             point 1 - point 9
                                                    point 2 - point 10
                                                    point 3 - point 11
                                                    point 4 - point 12
                                                    point 5 - point 13
                                                    point 6 - point 14
                                                    point 7 - point 15
                                                    point 8 - point 16

Consequently, I realized that there is a pattern and formula to find the opposite point for each particular point on the circle. Suppose that there are n equally spaced points, then:

Opposite point = Base point + n/2, when base point < n/2; and
Opposite point = Base point - n/2, when base point > n/2

Using this formula, the opposite point for point #7 on a 30-point circle is 7 + 30/2 = 22, so point #22.

Battleground Schools: My Thoughts

The first thing that resonated with me in this week's reading is the notion that there exists a "math phobic" attitude among North American adults. From my previous work experiences in Canada, I have met several people who shared the same view on math: "If I don't have the talent for it, why bother?", "It's not useful in work, so I shouldn't study it", etc. It can make the work environment somewhat uncomfortable as those who enjoy math have to "tone" down their abilities to work with numbers to avoid offending those who dislike math. Sometimes the workplace can even become disharmonious just because one can come across as "arrogant" and "smart" to others. What's worse is that this attitude can be easily transferred to other people, especially the younger generation.

Secondly, I learned that math only became a focal point of education in the United States during the Cold War years, leading to the New Math movement. Governments saw math as a tool to elevate their nation's economic power and status, and the learning of math was forced upon children to benefit the collective, not the self. To many people who lived through that era, math was treated almost like a religion, but its popularity soon waned as people realized that students cannot be taught these abstract and mainly conservative ideas when they had not even developed an interest in the subject. I think this served as a lesson to policymakers this kind of approach to education would not achieve its desired results (not just in math).

The last section of the reading about the ongoing war over the NCTM Standards also interested me. I also read up on EdReports' recent review and criticism of NCTM's instructional materials, as well as the NCTM's response. In my opinion, arguments over the curricula and methodology can only be conducive to the quality of education obtained by students, and I don't think constructive criticism should end even when there's a widely accepted set of teaching standards.

Microteaching Reflection

Today I had the opportunity to do my first ever microteaching activity, teaching my classmates about the offside rule in soccer. From this, I was able to gain some useful insights and feedback from my peers to improve my teaching techniques.

What Did I Learn?

Before today, I was unsure about how the session would go and what my expectations are, especially due to the 10-minute time constraint. However, once the timer started, I felt more relaxed and quickly became acclimated to the teacher role. Even though the entire activity was brief, I was able to go through all the materials and activities that I had planned. I also had to adjust parts of the lesson spontaneously depending on how others responded to my teaching. I believe this is an important technique a prospective teacher like myself has to master - knowing how to accommodate others while still having your objective in mind.

What Worked Well?

My peers were also extremely engaging throughout the lesson, even though at first I wasn't sure if the topic could spark enough interest. Fortunately, those who watched soccer before were intrigued by the concept of offside. I thought the materials I brought were sufficient to demonstrate what the offside rule is. The visualizations and coaching board were really helpful to the group, and in the end, everyone was able to point out the offside position in soccer.

What Could Have Been Better?

I noticed that many of my peers chose to use media (audio, video, etc.), and those are also helpful in complementing this kind of teaching. I also felt that more flavor could be added to my introduction, because not always will my students be interested in what I'm about to present. This is especially difficult for adolescents as compared to adults, so it's something to work on for the future.

What the group thought about my microteaching: