Indigenous aspect of thinking

I was struck by the connection between the geometry and the clothing. I was amazed to realize that that linearity corresponds to the tension in the fiber. Therefore, the math is everywhere, it just requires our awareness and keen observation. It is important to notice how beautifully the grid is used as a metaphor to describe the control and ownership. My second stop was at anything that is real is independent of the grid system. As no grid system can explain the underlying geometry of the shape. Hence I really liked the idea of shifting the focus to understanding the geometry rather than focussing on using the grid system according to our convenience.

The idea of starting off with the simpler knots and then extending to the 3-D knots gives the students an opportunity to go beyond the rectilinear thought and also open their minds towards how the indigenous people get connected with their culture and looking for the mathematics behind the different geometries.

Math fair by MacNair Secondary students

The  Math fair by the MacNair Secondary  School students was amazing. There were such new ideas of representing the things I never thought, such as representing the integers ( their multiplication, division of negative, positive numbers) in the fun and interesting ways like " Did the turtle eat the cookie". I was really impressed by their ideas. I believe that students have a lot of potentials, emerging thoughts, and innovative ideas and they look at things through a different lens which is more meaningful to them. In addition to this, It was pleasing to see how the stories they made led them to solve the linear equations and made more sense to them which was much more effective than simplifying solving for the variable. " Fire and the water metaphors for positive and negative integers were interesting to see. I wish the best for those students and their teacher who is successful to build the student's faith in him. I wish for their success and hope that they would make a difference in the traditional ways of thinking about math and My big thanks to them and to their teacher for imparting new ideas to me and to my fellow mates.

Rat and wine puzzel

I will start with the small number of bootles and rats and then extend.
Let's consider rat 1 and only 2 bottles out of which one of them contains poison.
If the rat 1 drinks bottle 1 and die then bottle 1 contains the poison, but if not them the bottle 2 contains the poison.
Therefore 1 rat is required to test 2 bottles.
If we have 2 rats, then let's figure out how many bottles can be checked.
Rat 1 is given a mixture of 1 and 2
Rat 2 is given a mixture of 2 and 3
If rat 1 and 2 die then the bottle 2 is poisoned.
If rat 1 dies and rat 2 survives then bottle 1 is poisoned.
If rat 1 survives and rat 2 dies then the bottle 2 is poisoned.
If both the rats survive then the bottle 4 is poisoned.
Therefore, with 2 rats I am able to check 4 bottles.

Now let's start with 3 rats.
Rat 1 is given a mixture of 1,2,3,4
Rat 2 is given a mixture of 3,4,5,6
Rat 3 is given a mixture of 1,3,5,7

If rat 1 dies, number 2 contains the poison
If  rat 2 dies, number 6 contains the poison
If rat 3 dies, the number  7 contains the poison
If  all the 3 rats die, then number 3 contains poison
If  only 1 and 2 die the number 4 contains poison
If none dies number 8 contains the poison

Hence till now with 3 rats, I can check 8 bottles.
I can notice the pattern,
If there is 1 rat then the number of bottles check is 2
If there are 2 rats then the number of the bottles checked is 4
If there are 3 rats then the number of the bottles checked is 8

Therefore 2^n( 2 to the power n) where n is the number of rats.
hence 10 rats are enough to check 1000 bottles as 2^10= 1024
but for 9 rats I am not able to figure out as only 512( 2^9) bottles can be checked.

Revised Unit Plan

https://drive.google.com/drive/u/0/folders/1n2fVteD8uOyTbUEGe31iuwwA07Tpkv18

Textbooks and Mathematics

The modality of the linguistics is really important for the textbooks to be useful for the readers. It is interesting to notice that the use of the first-person pronouns( I and We) in the math textbooks helps the readers to get involved in the textbooks and make better connections. The absence of the first-person pronouns makes the textbook really dull and it also distances the reader from what the author wants to explain.  Moreover, the use of second-person pronouns( you) connects the reader directly to mathematics. I remember, during my high school days, I use to connect more to the questions which involved first and second-person pronouns. It helped me to imagine the experience what the authors want to interpret. In addition to this, the use of the graphics and the photographs also make a huge difference in the student's understanding. Although I agree with the author of the article that graphics are more helpful as it generalizes the domain whereas the photograph talks about that particular person. For instance, the example of the hand doing mathematics in the article, lets the readers imagine their own hand doing mathematics and helps them to have a better understanding and experience. I remember my mathematics textbooks in my high school and college was all about numbers and variables which would not make sense to any layman reading the textbook. There were no pictures for the reader to make connections to the real world. Therefore, only the students who loved numbers or were interested in math would excel in math, and the textbooks were not good enough to develop anyone's interest.

According to me, the use of textbooks should not be ruled out completely, some good points can be incorporated by the teachers from textbooks for making their teaching and learning better. Especially, the textbook questions can be used to do more practice. I believe a teacher can play an important role in the interpretation of the information written in the textbook, even if the textbook is not good enough for the student readers. As per my recent experience, while making the unit plan for science, I found BC connection 9 textbook really helpful and I was amazed to see how beautifully the author made connections between the science concepts and the outer world.

Scales Problem

The first weight which I think the vendor should have is 1 g weight. Then I was in a dilemma should I opt for 3 g or 2 g. Initially, I chose 2g and I was able to get a maximum of 3 g but if I chose 3g then 3-1 will give me 2 and 3+ 1 will give 4g. So, till now I thought 1g and 3g should be fine to weigh the herbs till 4g. Now in order to weigh 5 g next highest weigh should be 9 g since, 9-3-1 = 5 g, I tried with 8g but  Using 1, 3 and 8. I was able to get to the maximum at 12 g in comparison to 9,1,3 which leads to 13. For instance, I chose 1,3,8 then the highest weight I figured out was 25g since 25-8-3-1= 13g, But using 1,3,8,25 the maximum I was able to reach was 37g( 1+3+8+25). Therefore, I figured out I should choose 9 instead of 8. Since 1, 3, 9 covered all the weighs from 5g to 13g.
9-3-1=5
9-3=6
9-3+1=7
9-1=8
9+1=10
9+3-1=11
9+3=12
9+3+1=13
The weight should be 27g since 1+3+9+x= 40, so x=27g.
Moreover, using 1,3,9,27 we can have the weighs after 13g.
27-9-3-1=14                        27-9+3=21              27+3-1=29               27+9=36
27-9-3=15                           27-9+3+1=22          27+3=30                   27+9+1=37
27-9-3+1=16                       27-3-1=23               27+3+1=31               27+9+3-1=38
27-9-1=17                           27-3=24                   27+9-3-1=32             27+9+3=39
27-9=18                              27-3+1=25               27+9-3=33                27+9+1+3=40
27-9+1=19                          27-1=26                   27+9-3+1=34
27-9+3-1=20                       27+1=28                  27+9-1=35

Eisner on "Three curricula" taught by all schools.

While reading the article, my first stop was that giving rewards to the students can foster the willingness to perform better in their schools but on the other hand students, those who work for getting the reward don't perform well when they don't see any appreciation.  I have first-hand experience with this ideology. When I came to Canada in 2017, I started working as a math instructor at the Mathnasium of South Surrey( Math learning Centre), where students were getting punches in their rewards card for completing every page. I saw that the rewards acted as the driving force for them to complete more and more math pages and buy some big rewards from the rewards cabinet by redeeming their completed punched cards. On the other hand, there were some students who were not allured by these rewards and worked at their own pace. Therefore, I believe that giving rewards have their own pros and cons. Sometimes rewards can be beneficial to trigger someone towards taking the first initiative. On the other hand, rewards should not be the sole motivation to do something. The students should enjoy the process of accomplishing their goals.

Another thing that really speaks to me is the use of location and time. I remember in school time, subjects like Physical education, drawing, Fine Arts, Dance used to be the last periods of the school time table which subconsciously reinforced that the arts subjects were not as important as science and math. I truly feel that I am a deficit of those talents which I think I could have explored more if enough was devoted to those subjects. Therefore, I believe that enough time should be allotted to all the subjects so that students can explore their interests.

In addition to this, I was happy to realize that how one can even learn from the school time table. The time table teaches the students to cognitively flexible and be able to adapt to the new demands on the schedule. It helps them to understand the importance of punctuality.  Moreover, we as teachers should acknowledge the importance of the implicit curriculum which teaches the student about the social and moral values which will help to become a good human being which I believe is above all the explicit curricula.

Micro teaching reflection.( Factoring trinomials)

Planning: We ( Me and Karmdeep) decided to micro-teach factoring polynomials from Foundation Math 10 and Precalculus. I looked into Foundation Math 10 textbook and found that Factoring polynomial is quite a big unit so decided to teach only factoring trinomials by using the visual method. In addition to this, we tried to think from a student's perspective and reflected on how can the student relate polynomials in real life. Therefore, we used PHET simulations to demonstrate the trajectory of parabola( trinomial). Moreover, we used virtual algebraic tiles to show the factoring of the trinomials on the projector. I should add that Karm deep did a very good job of making the algebraic tiles with colorful card paper which we used for our activity.

What worked well: I feel that our microteaching went okay and the learning objectives of the lesson were met. I am glad that all our students during the microteaching lesson were very engaged. They really liked the idea of using real-life examples of polynomials and exploring the mathematical representation of their path. The PHET simulation further enhanced the understanding of the trinomials. When it came to teaching, I suddenly got passionate, energized and felt as if I am actually teaching a grade 10 students. The activity at the end was very hand-on and we helped our students wherever they needed help and addressed their inquiries.

Areas of improvement: The major area of improvement for me is timing and pacing. In the last microteaching lesson, I finished a bit early but this time we had to drop the example demonstrating the use of zero pairs in making the factors using algebraic tiles. Some of the students were able to identify the use of zero pairs in the activity and took it as a challenging question but some of the students wanted the concept to be discussed before which I think we should have. The reason behind this is that I made the introduction longer, therefore less time was left for the rest of the lesson. The key take away is that I should be mindful of the time and teach accordingly and focus on the learning objectives for the particular lesson. My tendency is to give whatever knowledge I have about that topic to my students to make the explanation more meaningful but I should understand that they will learn the concepts gradually and eventually with time.

Geometric puzzle

It was a little hard for me to draw 30 equally spaced points. So, I thought of visualizing a clock that has 12 equally spaced points and matched the numbers diametrically and found that 1 is opposite to 7, which means 1 matches with ((half of 12) +1)= 7. Similarly, Considering 30 equally spaced point on clock, then matching the numbers which are diametrically opposite to each other, I found that 1 is diametrically opposed to (half of 30) + 1= 16, hence drawing and generalizing , I found that 7 is opposite to (( half of 30) + 7) = 22.
Therefore, the formula becomes, A number diametrically opposite to x in evenly spaced points on circle= (half of evenly spaced points) + x.


This problem can be extended to figure out what if the number of evenly spaced points is odd, how would the formula change.
The value of giving an impossible puzzle is that while solving the puzzle they might get introduced to new insights of math, which did not think before. But on the contrary, it can be sometimes a bit time consuming and frustrating as well for some kids who just want to focus on the curriculum.

I believe if the puzzle has some kind of geometric figures attached, I tend to solve it by actually drawing it out and then inferring the logic behind it and then generalizing it if possible.

Micro teaching factoring trinomials ( Jashan and karmdeep)

https://docs.google.com/document/d/1fJ1WK1Lve0qjCyyguCis01WwdubvuL90VQhQBS4FbwY/edit

Reflection on "Battle ground schools"

The whole contrast between conservative and progressive was interesting to reflect. It was shocking to see that according to conservatives the only reason to study math was just knowing “minimal math survival skills” and the “abstract technical skills” were only for a very small proportion of people whereas the progressive approach of math is to develop the problem-solving skills in all the students without any discrimination. Although I believe that fluency (supported by conservatives) is important in math, during my past school visits, and in my Canadian experience, I have seen the dependency of students on the calculators for simple calculations. Therefore, I think students should be taught in such a way that they understand the logic and then building on the logic they become fluent as well. (For instance, multiplication tables can be understood by thinking in groups, 12× 9 can be explained as 12 groups of 10 take away one group of 12)

I found that it is important for the math teacher to have a good grasp of the math concepts. I can understand how elementary school teachers have to teach everything including math even if they are not comfortable teaching math. But it is important to realize that the young students tend to develop the same attitudes which their teachers have, so it is essential to be always on the progress of learning and overcoming the “math phobia” and the other challenges for the enrichment in both students and teachers learning.

Dewey ‘s thoughtful remarks made me stop and digest them. I found it interesting how the two entirely different approaches (conservative and progressive) can lead to entirely different human beings. The Ones who are “obedient rule followers” and the other ones are reflective” scientific and democratic thinkers”. Students can greatly benefit and develop their mental inquiries by doing things and exploring its corollaries rather than just following the facts. Although I believe, doing practice, following the algorithms and knowing the reasons behind it, plays an important part in understanding math concepts in depth.

The rise and fall of the new math war is another evidence that the conservatives approach focusses on educating certain group of students who can understand abstract mathematics and have the potential to become future scientists, but we know that everyone has different  capabilities, so it is important to recognize those potentials of the students and design the pedagogies to help them explore their talents. In addition to this, I insist that we should also explore modern mathematics topics such as set theory, linear algebra, abstract algebra in addition to calculus to give the wide exposure to the students who want to extend their knowledge in math.

The dishes problem

Let the total number of guests be x

Number of dishes of rice used by the guests= x/2

Number of dishes of broth used by guests= x/3

Number of dishes of meat used by guests=x/4 

Since we know that the total number of dishes are 65
Therefore, there are many ways to find x, out of which two are discussed below :   
   

First method:                                                 Second Method:

x/2+ x/3  +x/4 = 65 (i) Hence, finding the lowest common multiple of 2,3,4 It comes out to be 12 On solving, We get, 6x + 4x + 3x = 780 13x = 780 x= 60, No. of guests = 60

We know that the Right-hand side is a whole number, therefore 2,3,4 must divide x  evenly, which is possible only if x is common multiple of 2,3, and 4. And, the lowest common multiple of 2,3,4  is 12. I found that on substituting 12 instead of x in the equation (i)  does not equal 65 Let’s try 24, 36, 48, 60, 72( multiples of 12) and substitute them to figure out which one satisfies the equation and gives us 65. On solving, I found that. x =60 satisfies x/2 + x/3 + x/4 = 65

Follow up questions:
I believe that it definitely matters if the students are offered examples, puzzles and told the history of mathematics from different cultures. In this way, they can learn cultures of their classmates as well. Nowadays the classrooms are very diverse and children from widely different backgrounds and cultures are at the same platform in the class. Hence, making them aware of each other's cultures along with cognitive math learning is an intelligent step towards creating an inclusive environment. Moreover, teaching math history of different cultures through puzzles or examples is an excellent way of acknowledging the great contributions of the mathematicians from different civilizations and this effort will certainly make the students realize that the math they are studying now is not the works of only one civilization or community, people all over the world have put efforts in to make it more meaningful.

I believe that word problems/ puzzles are really effective to break the monotonicity of solving regular math problems. I have seen some kids getting very excited to solve a puzzle introduced in the classroom. It adds the fun and the spark element in the student's routine. and I can relate that one feels immense happiness and a sense of accomplishment after solving the puzzle. In addition to this, I also believe that too many word problems and puzzles in the classroom can sometimes disinterest some students. The simple reasons for the disinterest can be an unwillingness to put in extra effort to solve the puzzle or think outside the box.

Micro teaching Reflection

Microteaching was really a good exposure towards teaching and then reflecting on what went well and what could have been done better . My topic for microteaching was making of paper bags. I felt that my classmates were well engaged throughout the task of making paper bags. I saw the happiness on their faces after holding their own hand made colorful bags. But what I realized is that although all of them were able to make the bags along with me, we finished the task bit early. I should have made some backup plans.  Although I taught them a way to make handles of their bags, I should have also brought some scissors to actually have them make the handles. I  anticipated that making bags will take a longer time but we finished early. It was due to the fact that somewhere at the back of mind I felt that it will take more than 10 min and I should speed up. Hence, I made the introduction smaller and started making bags. Therefore, in order to cover up the time, I had to extend my closure time more than planned. Overall, I really enjoyed teaching , learned the basic structure of the lesson plan for future lesson plans and also learned to manage the time and the content efficiently. Moreover, feedbacks received from my colleagues are very helpful and appreciable for my future teaching. Moreover, this activity gave me opportunities to learn some very interesting games and strategies that added spark and fun to my beautiful life.

Lesson Plan: Making paper bags

Lesson: Making paper bags

Time Duration: 10 min

Grade: Any

Big Ideas	Ø  Making the best possible use of the things around us.
Objectives	Ø  Students will be able to make shopping paper bags. Ø  Students will be able to realize the variety of different uses of paper bags and its ecofriendly benefits.
Core competencies	Ø  Working collaboratively, communicating with each other and ensuring everyone gets the desired result builds up their relationships and gives them a sense of togetherness. Ø  Thinking critically and making inquiries that how can a plain paper be given a desired shape

Materials Used

Printed colored paper, glue sticks

Lesson Components:	Learning activity	Time allotted
Introduction	Ø  Asking them how useful is the paper bags they think and what if they learn how to make paper bags it would be of great use for them such as storing accessories, gift wraps, etc.	1-2 min
Lesson development	Ø  Everyone is given a paper and glue stick. Ø  Steps are followed by students as led by the teacher to make the paper bag. Ø  It is ensured that everyone is at the same pace such that no one lags behind.	7-8 min
Closure	Ø  What additional can be done to make the paper bags better. Ø  Reflecting on their own art.	1 min

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.