Doolittle: Off the grid

After reading this article, my first stop was that we are so obsessed with grids that we are following the same structures and schedules again and again instead of seeing its failures. But this is not the same with Indigenous cultures, they are not so obsessed with the grid. They have their own natural ways of understanding the situations. It is very important for us to understand that one can tackle some of the failures of the grid but one can not change everything according to himself or herself.

We are only teaching Euclidean geometry in the schools, but there are some other ways as well those can be used to teach concepts in geometry and algorithm. For instance; Knots which is ancient Indigenous tradition of string figures and it is related to three-dimensional geometry of space.
I really like the indigenous people's agricultural tradition of farming without using clocks and calendars and also their definition of territory. Indigenous people like to go with the flow of nature instead of trying to take everything under control. There are a lot many things one can learn from indigenous culture.

We should look beyond the fixed grid and try to see the situation from different perspectives, and should also use alternative geometries, geometries of liberation into our math classrooms.

The wine and rat puzzle

Rat 1 drinks 1-100
Rat 2 drinks 101-200
Rat 3 drinks 201-300
Rat 4 drinks 301-400
Rat 5 drinks 401-500
Rat 6 drinks 501-600
Rat 7 drinks 601-700
Rat 8 drinks 701-800
Rat 9 drinks 801-900
Rat 10 drinks 901-1000
one of them must be die, and the bottles remains to test would be 100 now
Rat 1 drinks 1-10
Rat 2 drinks 11-20
Rat 3 drinks 21-30
Rat 4 drinks 31-40
Rat 5 drinks 41-50
Rat 6 drinks 51-60
Rat 7 drinks 61-70
Rat 8 drinks 71-80
Rat 9 drinks 81-90
Now there are two possibilities, if the one of the rat dies drinking these bottles then there are only 10 bottles to test, otherwise the poison is in remaining 10 bottles
If the rat dies, there are 10 bottles to test again
Rat 1 drinks 1st bottle
Rat 2 drinks 2nd bottle
Rat 3 drinks 3rd bottle
Rat 4 drinks 4th bottle
Rat 5 drinks 5th bottle
Rat 6 drinks 6th bottle
Rat 7 drinks 7th bottle
Rat 8 drinks 8th bottle
we have two remaining bottles, if one of the rat dies than it is clear which bottle is this. otherwise 2 rats can taste remaining bottles and we can check which bottle is poisoned.
This same phenomenon could be used to check the second possibility.

Unit plan and lesson plan: Draft 2

https://drive.google.com/open?id=165DAdZr1mx2MOtG5bQbKKXTZzIh3pySE

Draft unit plan and lesson plans

Below is the link to my unit plan:


https://drive.google.com/file/d/165DAdZr1mx2MOtG5bQbKKXTZzIh3pySE/view?usp=sharing

Thinking about math textbooks

As a former student, I loved to use textbooks. My teachers taught me according to the textbooks. It was very easy for me to follow the structure of the textbooks. I read the definitions given in the starting of the lesson, then I looked at the examples to understand the use of formula to solve the questions. It was very easy for me to solve the worksheet questions by following these two steps. Although I was not able to solve all the question, I needed help of my teachers to understand certain questions. As a teacher, I noticed that it is not easy for each students to understand mathematics by just seeing the definitions and examples given in the textbooks. Moreover, the textbooks usually have 2-3 examples those are not enough for students to understand the whole concept. Linguistic used in textbooks also sometimes confuse students. Moreover, sometimes examples are given as the perspective of others, and students don't feel connected to them.
Previously I thought that textbooks are important for students as they get to know what they are doing, but now my perception has been changed. Now I think that when teachers follow everything given in the textbooks they become so dependent on them. They don't want to try different activities those increase the critical thinking of students. The school, I went for my short practicum doesn't use textbooks. The teachers only get the idea from textbook about the topic, then they plan their own activities, and worksheets suitable for that topic. The activities they choose are connected to the real life situations, and students like to discuss those activities with their peers and teachers. I observed that students learn better in this way. I think this is the reason schools nowadays trying to adapt new techniques of learning  instead of making students text-books dependent.

The Scales Problem

A market vendor sells dried cooking herbs in whole-number amounts from 1 to 40 grams. The vendor has an old fashioned two pan weigh scale, and has exactly four weights of different amounts that allow them to weight out any of these amounts of herbs--without using the herb or any other object as an auxiliary weight.
According to me the values for 4 weights should be 1,2,5,10. When I saw this problem, it reminds me of Indian weight system. Most of the vendors in small cities use two pan weight scale to sell their fruits and vegetables. Although they have more weights, and their weights usually include 1,2,3,5,10,20,50,100. According to me suitable weights those add up to the numbers from 1 to 40 are 1,2,5 and 10. I didn't take 3 because to get 3 we can add 1 and 2 together.
1,2,5 and 10 can add to get any value up to 40. For instance
1gm= 1gm                                                                                                    
2gm=2gm
3gm=1+2
4gm= 1+2=3 and again plus 1gm
5gm=5gm
6gm= 5+1
7gm= 5+2
8gm= 5+2+1
9gm= 5+2+1=8 plus 1gm more
10gm= 10gm
11gm= 10+1
12gm=10+2
13gm= 10+2+1
14gm= 10+2+1 plus 1gm more
15gm= 10+5
it would continue like 10 more than as from 1 to 10
we have to add 10 twice to the calculations from 1 to 10 to get the numbers between 20 and 30, and thrice for numbers from 30 to 40. For instance, 37=10+5+7 once and again add 10 two more times. It is possible to get any number between 1 to 40 by using these four numbers.
Are there several correct solutions?
Yes, there might be several correct solutions to this problem. The other numbers should be 1,3,5 and 10 or 1, 2, 10, and 20. But according to me one number must be 1 and the second number must be either 2 or 3.
Extension to this problem
What should be the 4 weights if the vendor wants to sell dried cooking herbs in whole numbers amount from 1 to 100 grams? Could you use the same weights as you used for 1 to 40 grams? why or why not?

How to promote flow in classrooms

It is very important to maintain flow in our classrooms. It plays a vital role in the development of growth mindset, which is very important for every student to be successful in life. Flow means to completely involve someone in what he or she is doing. To create such an environment for students so that they just fully concentrate on their work instead of keeping track of time, which students usually do when they feel bored in the classrooms. Now the question here is how we can promote flow in classrooms. 

Open ended questions: As I learned in our problem solving class, always give students an open ended questions. Don't tell them that the question has only one particular answer. Let students try different methods to solve the problem. Be ready to extend the problems. Students feel excited in solving these type of problems.

Check prior knowledge: Before giving any question always check the prior knowledge of the students. Challenge them according to their skills, because if we will give hard tasks to the students they will feel anxious and bored. The question should not be too easy or too tough. 

Not tell too much: Feel free to share the relevant information as long as the question remains problematic for the students. Do not share too much information with them.

Clear instructions: Always give clear instructions to the students such as how they will solve the problem; in groups or individually. What the question is. What strategies they need to follow to solve the problem.

Connect problems with their daily lives: Always try to make problems relevant to their daily lives. In this way they feel more connected to the problem and try to solve it with more enjoy and curiosity. 

Personal interest: Check the personal interest of students and allow them to choose activity of their own interest. Always make your students feel comfortable in your classrooms, so that they don't hesitate to share their problem with you.

Moreover, involve them in hand on activities because students learn more by visualizing things. Last but not the least take them to the gardens to teach which would be very helpful to maintain the flow.

Reflection: Elliot Eisner on three curricula all schools teach

I learn something new from every reading. From this reading I got to know the functions of three curriculums those are explicit curriculum, implicit curriculum and null curriculum. These are my three stops.

Explicit curriculum refers to the plan for learning that the school offers to the students. It includes subjects that are offered by school. It has goals and objectives for each subject. These goals include what topics are to be learned, which lesson plan is to be followed, curriculum guides and textbooks, what materials to be used, and what is taught by the teachers Not only this, the schools offer to the community an educational menu of shorts; students can choose subjects of their choice. 

Implicit curriculum is the hidden curriculum which consists of unspoken and unintended decision those are made according to the present situations of the classrooms; for example physical set up of the classrooms, expectations of teachers. After reading the article I came to know that there are both positive and negative results of implicit curriculum. The positive results are that "implicit curriculum of school can teach a host of intellectual and social virtues: punctuality, a willingness to work hard on tasks that are not immediately enjoyable, and the ability to defer immediate gratification in order to work for distant goals can legitimately be viewed as positive attributes of schooling." Although these are not part of formal curriculum, but they are taught in the schools. Some of the negatives of implicit curriculum are when students are given rewards for completing the tasks, they become dependent on rewards. As I studied in the Human development class it is called positive reinforcement, when students are given something as a reward to do something. But as a result of this students start looking for rewards for each activity. If the awards are not given, they are less likely to engage even in an inherently enjoyable activity. Students becomes too compliant. Additionally, Competitiveness through grading system and differentiation of classes into ability groups could have both positive and negative impacts on students. It might be good for some students, but it can also cause others feel less valuable  because students in difficult level classes are more valued by community.

In the end, the null curriculum consists of what is not taught in the schools because it simply has not been traditionally taught.  It has two major dimensions intellectual processes that school emphasis and neglect and the other one is subject areas and contents. The curriculum more emphases on math, English, science, history, and physical education, but other subjects like law, dance, psychology, music are frequently not offered or are not required parts of secondary school program. The null curriculum limits the mind of the students and their ability to achieve intellectual discovery.

Reflection on micro teaching

Through this curriculum micro teaching, I got the chance to learn things which were new for me. I and Jashan decided to do our micro-teaching on factoring trinomials by using algebraic tiles, which is the grade 10 topic in B.C. math curriculum. It was a whole new concept for me. I only knew product sum method for factoring polynomials. I saw few videos to check how we can use algebraic tiles to find out the factors, and I found it very interesting. I realised that students will also find this method very interesting, and it will help them to engage with maths and improve their critical thinking. 

Previously, I thought that our peers would have known the use of algebraic tiles, but it was also new concept for them. I believe we managed to explain the use of algebraic tiles very well as they were able to solve few problems. Jashan connected this topic with physics which was also very interesting and new concept for students. The problem we faced was the time management. We thought that we will manage to explain everything in 15 minutes, but during presentation our introduction went a bit long, so we were not able to explain them the zero pair as we decided to explain. Overall it was very good. I find micro teaching very helpful for me because through micro-teaching I got the chance to teach and reflect on myself. The reflections of others are also valuable for me as those help me to improve my weaknesses.

Group micro teaching lesson plan: Karmdeep & Jashan

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

Geometric puzzle

Extension: 1)If there are 60 equally spaced points on the circumference of the circle, then which number will be diametrically opposite to 23?

    2)  Check if the same problem would work when the points are equally spaced on a triangle instead of circle.

what makes this puzzle truly geometric, rather than simply logical?

This problem is related to circumference and diameter of the circle. All the points are equally spaced on the circumference of the circle, and we need to find out the dramatically opposite points which we can find out for all the numbers. Therefore, this puzzle is truly geometric as it includes the circle and its circumference and diameter.

Battleground Schools

The first thing that stopped me is the Progressivist reform for mathematics through activity and inquiry. Before the development of this movement mathematics was totally conservative, and most of the adults in North America had negative views about mathematics because of their negative experiences of their own schooling. The progressivist reform was based on experimental learning that concentrate on the development of student's talents instead of just focusing on traditional methods of teaching. This type of education would be very helpful for the students, the thing that made me stop is why Dewey's recommendations were never taken up in a wholesale fashion across North American Schooling. The second think that stopped me in the New Math Movement, in this movement the main focus was on precision and correctness . The main motive of New Math was to educate students in a way as they all are going to use mathematics for scientific calculations instead of thinking about those students who might not be interested and capable of solving high abstract mathematical problems. Why?.  Furthermore, the curriculum was reformed according to NCTM standards in which the concepts are presented in a haphazard way. I also read in the article there was the Math War over the NCTM standards, because through these methods students are  able to achieve conceptual understanding without understanding the basic skills.  The question that came to my mind is which method is better? or can we use both methods together for better understanding?

Dishes puzzle

1)Whether it makes a difference to our students to offer examples, puzzles and histories of mathematics from diverse cultures (or from 'their' cultures!)
Yes, Off course the examples, puzzles and histories of  mathematics makes a lot difference in their learning. Our classrooms are diverse, through these puzzles students get to know each others culture, like what kind of problems are there in different cultures. These puzzles also make maths interesting for students as they get something new every time instead of same type of boring mathematical questions.  Also through the histories of mathematics, they get the knowledge how mathematics was developed in different civilizations. They understand the contribution of different nations to the mathematics. It creates the sense of equality in them.

2) Whether the world problem/puzzle story matters or make a difference to our enjoyment of solving it.
Puzzles provide knowledge as well as enjoyment to students. Puzzles offer challenges to our brain in a very enjoyable way. We usually get bored with same type of  problems in mathematics, but puzzles create twist in simple problems and add flavour to simple questions. The questions asked in puzzles create curiosity in listener's mind to solve it. Our mind starts working fast to find the solution, and we feel relaxed after solving the puzzle. 

Reflection on micro teaching

My topic for micro teaching was benefits of stretching in our daily lives. Everybody finds it very interesting and helpful. It was a well planned lesson with everything clear in my mind, but one thing that I lacked is rehearsal which caused nervousness in me. Even though I did rehearsal 2-3 times but I think I should do it more. My reflection on it is that I got nerves in the beginning of the activity because I was expecting it in a small group. Also, the people passing near our circle from other classes distracted me. I really felt very sad about my nervousness after the completion of the activity. But I was not nervous through out the overall activity, I overcame my nervousness in the middle.  Everything went very well in the middle and in the end.
I found it very helpful for me that I got the chance to teach a big group and received their feedback. Most of their feedback was on my nervousness, everything else was good. For the next micro teaching I will try to utilize more time for rehearsal to avoid nervousness.

Micro teaching: lesson plan

                                                     Micro Teaching 

Date:  Oct. 2nd, 2019

Subject:  Benefits of stretch exercises in the office or teaching environments

Duration:  10 minutes

Materials Required:  No material required

                                                                  

Introduction ( 2 -3  minutes ): -   ·         I will discuss the importance of stretching in our daily lives ·         I will ask students if they do any stretching exercises ·         How often we should do these stretches

Middle ( 5 - 6  minutes ):- ·         I will demonstrate some basic stretches ·         The first stretch focuses on neck ·         The second will be for the wrists ·         The third will be on the hands ·         The fourth will stretch the shoulders ·         I will do the activity myself and will ask students to do the stretches with me

Conclusion ( 1 -2 minutes ):- ·         Ask if the students have any questions ·         Ask what students actually learned from this session

Micro teaching topic

I am going to discuss the benefits of exercise in our daily life, and going to demonstrate some basic exercises for healthy living.

Wordy puzzle

From the first sentence we get to know that the person who is describing the puzzle has no brother and sister as he says, "brothers and sisters have I none". From the second sentence we get to know
that the man who is describing the puzzle is talking about himself and "that man" is his son.
The way in which this puzzle is described is very confusing as well as interesting.

Reflection on presentation

We selected the art work by Sharol Nau for our presentation. We found this art work very interesting and engaging. This art work is about "how many triangles do you see" in the given 5 base triangle. This is a very good brainstorming exercise for students. Usually when we see these type of problems, we feel curiosity to get the answer, and our mind starts thinking critically. Therefore, we thought that this art work would engage the classroom, and encourage students to think critically.

After selecting the art work, we first helped each other to completely understand the whole concept. We started thinking about the shape of the triangle, tables and patterns we can use to solve the problem, and the formula that can be used to solve this problem. Once we ourselves were cleared, we all brought different types of ideas to make this project interesting for students. We first selected the 5 base triangle for our activity, then we thought about starting from base 1, then 2 and so on, to help students understand the basics of the whole activity. Basic knowledge is very important to easily understand the whole concept. When we have the total understanding of the concept, we can solve all the other related difficult problems very easily. Then, we decided to explain the concept through table method where students only needed to calculate the upside and downside triangles on different bases, observe the pattern, and then add them together to get the answer. We also did rehearsal before our presentation where we decided who will perform which part.

I decided to make this art work on paper chart. To make it more artistic and easy to understand I choose two different color of papers to separate upside triangles and downside triangles . Each triangle is equilateral with sides 4.5cm and angles 60 degree. We can also make the same project for isosceles triangle. While making this art work I realised that how geometry can be taught to students through this project. They can learn about the properties of isosceles  triangle and equilateral triangle, base of the triangle, patterns followed to solve the problem, triangular numbers through this project.

During our presentation, I realised that we were successful in engaging our class with our presentation. Everybody was able to understand the concept and to find the number of triangles in 5 base triangle.

Locker Problem

Letters from two of your future students

One loved your class!

Hi Karmdeep,

You were my favourite teacher because you were very creative, and always brought something new to learn  in class. I never found your class boring. You increased my interest in maths. You were always there to help me and other students. You were always smiling and very friendly to all the students, I never felt hesitation before asking anything from you.

The other, not so much

Hi Karmdeep,

I didn't like your class because you always wanted us to focus on studies. You didn't  let us do  fun activities in the classroom.

I hope to teach all the students with love and care, so that they enjoy my teaching and learn good things form me. I am little worried about how students will react to my teaching.

Math and Me

When I was in school,
1) I love mathematics. Sometimes I spent my whole day just solving math problems.
2) My favourite teacher was maths teacher.
3) I used to help my friends, my brother and relatives in solving their math problems.
4) I think, I decided to become a math teacher because my friends always told  me that we can easily understand the topic when you teach us.
5) But I was never taught maths in the interesting way like through group discussions, and presentations. But I will try my best to make it more and more interesting for my students.

The role of representations in developing mathematical understanding

According to the author both internal and external representations are interconnected with each other. The examples given by the author to explain this theory convinced me in his argument. In case of internal representations we imagine the numbers and figures in our mind, and when we give them the shape on the paper they become external representations. On the other hand, when we have external representations, and we study and examine them in mind, then these representations become internal representations. Therefore, our mind analyse things internally and then visualize externally. The representational thinking depends on the individual's ability to interrupt, construct and operate effectively with both form of representations.

The article includes the base 10 blocks for multidigit addition, two different ways of presenting the same set of things, and seeing, imagining and analysing patterns of numerical values like Fibonacci series. It is explained in this article through example, one particular mode of representation does not improve student's conceptual understanding and representational thinking.  Students who use both analytical and visual representations are able to solve multiple types of problems.

 Mathematical representation of fractions is not included in this article. Fractions are very easy to understand through pictures. I will ask students what they can see in this picture. I will give them time to observe the picture. There must be different answers. Some will count the whole parts of the circle, some will count the shaded parts, whereas the others will count the unshaded parts of the circle. After that, to calculate the fractions I will ask them to observe how many parts are shaded out of total number of parts. In first case the answer would be 4 out of 5 is shaded, which is the fraction. We can do the same for unshaded parts. I will also explain the concept of numerator and denominator,

Richard Skemp on instrumental vs. relational ways of knowing mathematics

The first thing is that we can connect the concept of Faux Amis with mathematics. In mathematics the word understanding has two different meanings which are relational understanding and instrumental understanding. It depends on a person how he or she takes it. There are all kind of students in the classrooms, some of them want to understand concepts instrumentally because they just don't want to know all the explanation behind the concept. Whereas others want to get the depth-in knowledge of the concept.

Secondly, after reading the examples of instrumental understanding, it made me think that most of the time  I had been taught in the instrumental way. Only the formulas were given to solve the problems. After that I came to realise that why in school I liked some teacher's style of teaching more than other. Some of them teach relationally which was easy to understand, on the other hand, some just gave the formulas to solve the problems. In the end, there is a difference between learning the formula and understanding the formula. If you learn the formula, you may think that you have understand the formula, but in fact you haven't understand it. By learning formula you will only be able to solve similar type of questions.

According to me, instrumental understanding do have some advantages over relational understanding. As it is written in the article, there are some topics those are difficult to understand relationally. Otherwise, teachers should try their best to teach students relationally.

Hello everyone !

This is Karmdeep Bagri