Micro teaching lesson topic

I will be teaching how we can make shopping paper bags.

Wordy puzzle

Brother's and sisters have I none, but that man's father is my father's son.

 I tried to solve this using the tree diagram and realized that the speaker is the only child and in this puzzle, he is talking about himself. And "That man" is the speaker's son.

This puzzle was not straight forward for me, I had to relate it with myself, draw some tree diagrams to find some clue. I found that such word problems make us think more critically and it is different from just solving the questions based on certain algorithms and formulas.  

Reflection from the math art work

When I was introduced to this assignment by our Professor, I was looking forward to know about the insights that connect math and art together. I browsed the Art Math gallery 2019 and was very amazed to see the various math art forms done by the people. I was looking into each one of them and wondering which one would be best for me to work on. Eventually, we as a group came up with Clayton Shonkwiler artwork, which was about creating a map between the two block letters(polygons) which have the same number of vertices and then we replicated the art by using our initials as block letters.

Initially when I started working on this project, I was not able to relate to the math at high school level with the coding in Mathematica which the scientist used to make this math art. On discussing with my group mates and breaking the art into smaller conceptual segments, I found that this art can be used to introduce a coordinate system, reflecting on the vertices drawn on the graph paper, then extending them to the concepts of reflection and translations. I also found it interesting how beautifully bijective function fits this math art which gives me the opportunity to explain the one-one function and the onto function in depth. This concept of functions, can be explored to discuss the bijectivity of exponential, logarithm and trigonometric functions and gives chance to students to inquire about the possible domain and ranges for which the function is bijective. In addition to this, we can also make the students understand about the Euler’s Equation ((number of vertices)- (number of edges) + (number of faces)) is always equal to 2 using the block letter of their own name.

The overall experience was pretty good. The mapping of initials was a bit time consuming but once it was done, I was really happy to look at it.  My classmates were a very good audience, they actively participated by engaging in the class activity, answering the questions I asked. My teammates were awesome to work with, we helped and shared our ideas with each other. We had good timing and everyone had the chance to participate. But there is always an opportunity to improve and do better. I feel I could have explored the coding behind the art more so that I can make more connectivity and play around for myself. On the other hand, the conclusions of this math art are of great importance for high school students.

The Locker problem

The school has 1000 students and 1000 lockers. On the first day, all the lockers were open. Student 1 closes each locker, Student 2 opens every second locker, Student 3 opens every third locker and so on. After all the 1000 lockers are done. Which lockers are closed or open and why?

Let us start considering 30 at a time

 1.C  C  C  C   C  C  C  C  C  C  C  C  C  C   C  C  C  C  C  C  C  C  C  C   C  C  C  C  C  C

2. C  O  C  O  C  O  C  O  C  O  C  O  C  O  C  O  C  O  C  O  C  O  C  O  C  O  C  O  C  O 

3. C  O  O  O  C  C  C  O  O  O  C  C  C  O  O  O  C  C  C  O  O  O  C  C  C  O  O  O  C  C

4. C  O  O  C  C  C  C  C  O  O  C  O  C  O  O  C  C  C  C  C  O  O  C  O  C  O  O  C  C   C

5. C  O  O  C  O  C  C  C  O  C  C  O  C  O  C  C  C  C  C  O  O  O  C  O  O  O  O  C  C   O

6. C  O  O  C  O  O  C  C  O  C  C  C  C  O  C  C  C  O  C  O  O  O  C  C  O  O  O  C  C   C

7. C  O  O  C  O  O  O  C  O  C  C  C  C  C  C  C  C  O  C  O  C  O  C  C  O  O  O  O  C   C

8. C  O  O  C  O  O  O  O  O  C  C  C  C  C  C  O  C  O  C  O  C  O  C  O  O  O  O  O  C  C

9. C  O  O  C  O  O  O  O  C  C  C  C  C  C  C  O  C  C  C  O  C  O  C  O  O  O  C  O  C   C

10 C  O  O  C  O  O  O  O  C  O  C  C  C  C  C  O  C  C  C  C  C  O  C  O  O O  C O  C   O

11.C  O  O  C  O  O  O  O  C  O  O  C  C  C  C  O  C  C  C  C C  C C  O  O  O  C  O  C   O

12.C  O  O  C  O  O  O  O  C  O  O  O  C  C  C  O  C  C  C  C C  C C  C  O  O  C  O  C   O

13.C  O  O  C  O  O  O  O  C  O  O  O  O  C  C  O  C  C  C  C C  C C  C  O  C  C  O  C   O

14.C  O  O  C  O  O  O  O  C  O  O  O  O  O  C  O  C  C  C  C C  C C  C  O  C  C  C  C   O

15.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  O  C  C  C  C C  C C  C  O  C  C  C  C   C

16.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  C  C  C  C C  C C  C  O  C  C  C  C   C

17.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  C  C  C C  C C  C  O  C  C  C  C   C

18.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  C  C C  C C  C  O  C  C  C  C   C

19.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  O  C C  C C  C  O  C  C  C  C   C

20.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  O  O C  C C  C  O  C  C  C  C   C

21.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  O  O O  C C  C  O  C  C  C  C   C

22.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  O  O O  O C  C  O  C  C  C  C   C

23.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  O  O O  O O  C  O  C  C  C  C   C

24.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  O  O O  O O  O  O  C  C  C  C   C

25.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  O  O O  O O  O  C  C  C  C  C   C

26.C  O  O  C  O  O  O  O  C  O  O  O  O  O  O  C  O  O  O  O O  O O  O  C O  C  C  C   C

and so on,

On careful observation, it is observed that the closed lockers are 1, 4, 9, 16, 25....which are the squares of the numbers starting from 1 to 31. Since square of 32 is 1024 which is bigger than the given number of lockers. So , the closed locker numbers are 1,4,9,16,25,36,49,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625,676,729,784,841,900,961. rest all the lockers are open. It is interesting to see that child at serial number 500 opens or closes the locker just twice because there are only 2 multiples of 500 till 1000, child at 501 opens or closes the locker only once because the next multiple 1002 is beyond 1000. Student number 3 opens or closes the lockers 333 times.Therefore it can be concluded that child opens or closes the locker as many times their serial number goes into 1000.

Letters from two of my future students.

Hii Mrs.Bajwa,

How are you doing?

My name is Jason. I was in your grade 10 class in 2019. I am feeling immense pleasure writing you a note. You were amongst the teachers who really cared about their students and were very approachable and humble. Apart from learning the subject content, I learnt the values and ways of life to become a better person in my life. The qualities of being kind, helping each other, believing in yourself, coping with hard times helped me to lead a peaceful life. When I reflect back, I still remember the math fun activities in our class. You made us all feel so happy. At present, I am not only excelling in my life as a software engineer but also living my life with full enthusiasm and happiness. I still remember your motivation to strengthen our strengths and weakness and realize hidden potential. Thanks for always being there for my support and inspiration.

Kind regards, 

Jason

Hii Mrs.Bajwa,

My name is Kiran. I am writing to you to express my feeling which I was not able to when I  was your student. You was my math teacher in grade 11 in 2019. I remember your math lessons made sense to only those who were already good in math. I wish you had taken care of those who were not excelling in math then it would have served a great help to them emotionally and academically. I still remember the time when I was almost in a state of depression and helplessness. Even though I was trying my best but could not make up with the rest of the class. I really felt like crap at that time. Although I really appreciate you for being very punctual, responsible and organized in class, I wish you had paid more attention, used alternative pedagogies to explain the math concept and created an unbiased environment in the class so that everybody had the opportunity to perform well in math.

Kind regards,
Kiran

As a teacher, my teaching pedagogies, relationship with the students have a huge impact on their life. It affects them both academically and emotionally. My concern is that it might be difficult to build up a relationship with the students who are bit introvert. I might not know what they are going through. Therefore, such situations can be challenging. I sometimes feel worried about dealing with misbehaviors in the classroom. But, I am hopeful that I will use the best possible pedagogies suitable for my students for their excellence.

Mathematics and me

Since childhood, I remember I was good at math. I really liked doing the problem-solving word problems. My mother used to relate math problems real-life to make me ponder on them mathematically such as using money, objects, etc. As I grew up, I wanted to pursue my career in math because I loved solving the questions based on certain algorithms such as solving quadric equations, binomials, etc. I remember the time when I did not like math was when I had to write long theorems in algebra and calculus which really did not make any sense to me. But what made me very passionate about math was my first job at Mathnasium. I found that I approached the questions more analytically and by using representations to explain it to my students, and the happiness on students' faces inspired me every day to become a better teacher.

Entrance Slip: Mathematical understanding and multiple representations

The graphs, charts, visuals represent the external representations which stimulate to understand the underlying idea behind the math problems. I agree with the argument of the author that one can use representation to extend the knowledge to the real-world, for instance, bar charts can be used to see the yearly change in the population trend over time which would otherwise be a challenge to observe the trend. In addition to this, I am really impressed by the author's idea of considering representation as a “social activity”. It means that representation is not a “static end result” which the students have to follow but it is a process that requires a student’s active involvement to reach the representation.

This can be exemplified as the addition of the whole numbers using base ten blocks (each block represents 1, each I by 10 block make 10, each 10 by 10 block makes 100), requires social involvement of teachers and students to think, imagine, internalize, communicate and reproduce to the final answer. The pilot experiment conducted by Tchoshanov reassures me to investigate the student’s prior knowledge on any topic and then engage them to related hands-on or cognitive thinking activities (mapping, plotting, coding) so that they have chance to communicate, explore the underlying mathematical concept.

Examples of mathematical representations that are not included.

Adding ¾+ ¾ can be taught with the help of pies where each pie is divided into 4 parts. The number of parts is the denomination. Therefore, we get 4 quarters (I whole) + 2 quarters (1 half) = 6 quarters (6/4), teaching addition of integers with help of number line ( -12+11= -1 implies starting from -12 on the number line move 11 steps towards positive direction so we reach at -1). Factoring of the quadratic equations can be taught with the algebra tiles, finding the slope of the line by the study of the graph, finding the signs of the trigonometric identities by drawing 4 quadrants in a unit circle. For instance, all the trigonometric identities (sin, cos, tan) are positive in the first quadrant, (sine, cosec) positive in 2^nd qd. rest are negative, (tan, cot) positive in 3^rd qd. Rest are negative (cos, sec) positive in 4^th qd. rest are negative.

                                                                                                          

Relational Understanding and Instrumental Understanding

The article “Relational Understanding and Instrumental understanding” written by Richard R. Skemp made me reflect on my ways of approaching the subject content. I really liked the term “Relational understanding and the instrumental understanding” he used for different levels of understanding. I have been using both approaches depending on the different situations during my school life. Although I firmly believe, it is essential for the students to know the concept behind the rules of their routine actions. If a teacher is accustomed to using the instrumental style of teaching then it would be detrimental for the students who want to have relational understanding. I remember one of my history teachers would deduct marks if we did not answer her exam questions as written in the textbooks. I believe such practices will impoverish the creative minds of students. 

In addition to this, I found it interesting, how a student good in instrumental math understanding can do better in the other subjects where math is required. I believe along with knowing the instrumental ways, one must have deep knowledge in order to solve the problems with much more understanding. The example he uses to explain the importance of having a “cognitive map” of town instead of just knowing the directions to final positions encourages me to take teach math by making connections with interrelated topics so that students have the whole essence of the concept.  A relational understanding of math gives us ample freedom to approach problems in comparison to instrumental understanding which constraints us to solve problems using a certain set of rules.

Going further the contrast discussed between relational mathematics and instrumental mathematics intrigued me to think more about it. Skemp also raises the question about considering relational math and instrumental math as two different subjects or different approaches to the same content. I find the latter more relevant. From my past experience of tutoring math, I have experienced that math made more sense to my students if they were taught the underlined concepts of the problems( for instance -4 -5 = -9 can be taught with the help of number line rather than just telling them the rule of adding and putting a negative sign Infront), hence, they will feel more confident in solving the difficult questions themselves. The substantial difference between the approaches of the types of math can make a significant difference in the student’s core understanding and learning outcomes. Although, I understand, in actual practice, teachers and students are constrained by the time, exam pressures, content completion deadlines which can overlook the relational understanding,. Therefore, we as educators must reflect on our ways of understanding and on our pedagogy in order to induce a relational approach in our students wherever possible.

Introduction

Hello everyone,
I am Jashan. I am looking forward to become a high school Math and physics teacher.