Group Microteaching Lesson: Peers' Feedback and Reflections

 Peers' Feedback

"Never learned before, happy to learn finally,how algebra tiles work," was one of the various responses we got from our peers as feedback for the use of algebra tiles in our micro-teaching lesson, and it was just satisfying for our group to know to have achieved one of the teaching objectives. Almost all of them appreciated the use of manipulative, the algebra tiles, with which we were able to engage all of them actively in the process of teaching and learning. One of the observers pointed out the idea that connecting factorization visually to the area of rectangle can  prove to be a different kind of learning experience for a student.The observers also liked the idea of using the website resource, the virtual manipulative, to make the lesson more interesting.
         Although the lesson was a good one, but it lacked the proper time management, pointed by almost all of them. The introduction, no doubt, was very appropriate, but, was too fast, and I strongly agreed upon their view that it can be hard for a grade 10 student to follow. The pace of the lesson was fast, and students were not be able to get time to think before they could answer any question.

Self Reflection

     
I also found my group's lesson to be an excellent one, but, no doubt, it lacked the time factor. This was the reason that we had to be fast,and the observers couldn't follow the instructions at the beginning of the lesson. A teacher has to be very particular regarding the time factor. Indeed it was  a different kind of experience to teach with manipulative, to make it possible for the students to make sense of the mathematical situations. I also found it very useful to connect factoring with the area of a rectangle, and making it possible for the students to enjoy this connection as a new learning experience. At the end of the lesson we arranged for expressing their views regarding their self evaluation, that is, what they learn today, to be responded together with the question sheet provided to them,and this was indeed, very useful as a teacher as you got to have some insight of your teaching at the end of the lesson.Truly, micro-teaching was a fruitful experience for me as a teacher candidate.

Microteaching Lesson Plan: EDCP 342A

EDCP 342A:  Lesson Plan
Topic:  Factoring Quadratic Trinomials Using Algebra Tiles                                          
Group:  Howard, Maria, Raman                  

Intended Students:    Grade 10  Fundamentals and Pre-calculus

	WHAT	HOW LONG	MATERIALS
BRIDGE	Give everyone a small  sheet of paper.   In 5 seconds, write as many factors of 60.	1 minute
LEARNING OBJECTIVES	Using the algebra tiles, students will be able to: 1. Factor quadratic trinomials, including perfect square trinomials 2. Relate the dimensions of a rectangular area with finding the factors of a trinomial 3. Experience three modes of factoring trinomials:  algebraic method, concrete algebra tiles, and virtual  manipulatives
TEACHING OBJECTIVES	1. Maximum engagement of all students 2. Individual hands-on-learning using math manipulatives  (algebra tiles) 3. Demonstration of using virtual manipulatives in factoring trinomials
PRETEST	Each student will be given a worksheet sheet      1. Factor the trinomial: x^2 + 5x + 6.  Write answer in worksheet. Ask for answer.  Show of hands who got the correct answer.  Ask a student to briefly explain his/her answer.	2 minutes
PARTICIPATORY LEARNING	1. State the learning objectives.  Tie-up bridge and pre-test to objectives. 2. What are the factors of 6? (3 and 2)  How can we illustrate this geometrically? (Draw a 3 by 2 rectangle, divided into 6 squares).  How are factors related to dimensions (of length and width), and product related to area?  (Finding the factors of a number is the same as finding the dimensions of a rectangle whose area is the number).  Will this geometric representation work for finding factors of a trinomial? 3. Distribute/introduce the algebra tiles, as a geometric method of finding factors of trinomials.  Each student will be given a complete set of tiles, with a transparent tile board.  Walk the students through the 3 different tile sizes representing x^2 (green), x (white) and 1 (red).  Explain that x is a variable that can represent any positive number. 4. Assemble 2-green x^2, 5-white x tiles and 2-red 1-tiles.  If all the 9 pieces represent the area of a rectangle, what algebraic expression represents this area?    (2x^2 + 5x + 2)  How can we get the dimensions of this rectangle?  * In your worksheet, complete equation #2:  2x^2 + 5x + 2 = (2x + 1)(x + 2) 5. Empty your tile board.  For our second rectangle, assemble 1- green x^2, 6-white x and 9-red 1-tiles into a rectangle.  What expression represents the area of this rectangle?  (x^2 + 6x + 9).  What are the factors?  (x + 3) and (x + 3).  What do you notice with our rectangle?  (It is a square).  Introduce the perfect square trinomial (PST).  * In your worksheet, complete equation #3:  x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 6. Virtual Manipulatives:    Reiterate that finding the linear factors of a quadratic trinomial is very much related to finding the dimensions of a rectangle that contain the trinomial.  The internet is full of virtual manipulatives that offer fun, creative, and interactive ways of factoring trinomial, which may appeal to today’s technology-savvy students.  Factor x^2 + 7x + 12.  (x + 4)(x+3).	9 minutes	* Algebra tiles * Virtual manipulatives
POST-TEST	Using your algebra tiles, find values of k, where x^2 + kx + 6 factors into 2 binomials. (k = 5, 7).  Write answer in #5 of your worksheet.		* Algebra tiles
SUMMARY & WRAP-UP	Ask students what they have learned today, which should touch the following points: 1. That to the concept of factoring is very much related to finding the dimensions of a rectangle of a given area. 2. That a quadratic trinomial factors only if one can arrange it into a rectangle. 3. That we know that a trinomial is a perfect square if the tiles neatly arranges into a square, with 2 equal dimensions. 4. Ask students to complete # 6 & 7 of their worksheet.  Collect worksheets.	3 minutes

Suggested student worksheet format:
Name:  __________________                                                                                    Topic:  Factoring Quadratic Trinomials
1.    Factor:    x^2  +  5x  +  6
2.    2x^2 + 5x + 2 = (                )(                  )
3.    x^2 + 6x + 9   = (                 )(                  )  =  (                )
4.    x^2 + 7x +  2  = (                  )(                  )
5.    x^2 + kx + 6   = (                   )(                    ) or (                   )(                   )
6.    One thing I learned today is _____________________________________.
7.    Algebra tiles do/do not help in understanding factoring trinomials because_______________________________________________________.

My Response: Formula for Thinking Mathematically

I find these two chapters, "Phases of Work" and "Responses to being Stuck"  to be very-very interesting, and informative, as they have provided the detailed analysis of various stages of dealing with a problem. The three phases of work involved in solving a problem, are really crucial for developing of thinking mathematically, and a teacher must have the knowledge of these stages, as well as enough skill to apply the same during his or her teaching process, in order to achieve the required learning outcome. Although, I also feel that these stages occur naturally while we encounter a problem, but by highlighting them beforehand we can train our students to enjoy the process of discovering beyond the solutions. Consequently, their thinking will be more mathematical.
      I, especially, liked the analysis of the "Attack" phase in 2nd chapter, where the remarkable use of the words "stuck" and "aha" has provided a more practical illustration of a mind's activities while going through this significant stage of exploration. I just want to share my own practical experience of developing mathematical thinking. When I used to teach solving Mathematical problems in India, I always asked my students to write down all the steps involved in solving a problem, together with making down where they got stuck, and how they got the way to come out. These readings have really provided me with concrete information, and I can now use it more effectively to train myself and my students in actually thinking Mathematically.

Division's Boast

Division boasts,
"I am a politician
             ruling over the numbers,
             to break them into pieces,
              using them against one another
Just see  how capable I am 
              to use even the empty headed,
              against the biggest powers,
              And scatter them into,
              in an infinite number of pieces.
You must have come to know who is that empty headed,
               And that one is the Zero."

Divide and Zero (in my views)

Divide

Divide is one of the four mathematical operations. It is used to make divisions. We need this to apply in our daily activities. For example, dividing the things among different people. Sometimes politicians use this in order to create different groups, for example, in the history of India, the Britisher Colonialists used the policy of 'divide and rule' just to scramble the whole unified empire of India in small city-states, and it worked.

Zero

Although placed on the neutral position, zero has a significant part to play.Dividing any number by zero will lead to infinity. It has very significant uses in ordinary language, as one who is out of every thing is regarded as zero. It is the empty set too.we must multiply our stresses with zero just to remove them from our life.

School Mathematics for Citizenship Education

I just enjoyed the article as it has provided an opportunity to view Mathematics as an art as well as science of  society. By highlighting the relationship between school Mathematics and modern society, Elaine Simmt has put forward the need for the strong foundations of Math education in schools, which is crucial for developing the, active, well informed and responsible citizens.  I strongly feel that it is the power of Mathematics that this modern society has been able to acquire its modernity with the strong applications of Mathematics, in its pure as well as applied form. The numbers in the form of quantitative data are used, as a part of statistics, in almost all walks of life. Thus, Math education is the need of the hour in order to make our citizens play their part actively in this highly mathematized society, as well informed and understanding individuals. It is, therefore, required that we must change our way of dealing with the subject, that is the curriculum must be set and instructed in a way that it leads to develop thinking ability in the individuals. Indeed, merely following the rules and getting the right answers aren't going to develop citizenship. I especially like Simmt's views about revising the curriculum ,together with reviewing the ways it is instructed in our classrooms.

       Although the instructional strategies provided by Simmt  are all appropriate for the desired outcome, I find the demand for the explanation to be very significant, as it will help in better understanding of the concept, and also, make the individuals feel confident and learned who will then be able to prove their view point as educated citizens in the society.

Fictional Letters

Following are the imaginary letters that I have imagined to be written by two of my students, one of them liked me a lot, the other who didn't. Take some time to look into my imaginative world, and share my imaginary thoughts.

Student #1

Dear Mrs. R.

I am Grace, one of your students of grade 12, in 2010 batch of ABC Secondary School. Last Sunday, I saw you in the XYZ store, and all my memories of Mathematics journey became fresh. I still remember the very first day in your class when after the introduction, you took a promise of hard work from our (student's) side,and motivated us to do our best in the subject. I used to be an average student with a view of "not so good at Math" before I got you as my Math teacher, but you taught me in a way that I was able to come out of my complexes. With your motivating spirits, extra help and care, I started developing confidence in Math as well as in other subjects too, as you developed my ability to think to get a solution for a problem. You will be happy to know that I have done my Masters in Mathematics, and also did my Bachelor of Education with teachable subject as Mathematics. Presently, I am working as a Mathematics teacher in MM Secondary School. I will always remember you as my role model.

yours student

Grace

Student # 2

Hi Mrs. R.

I am Sam, one of your students of Mathematics from grade 11 in ABC Secondary School. Last weak I happened to come across you, but couldn't manage to talk to you. However, my mind just started remembering those days when I was one of the intelligent students,and if you remember, I always had problems with your teaching style. I used to be very fast at calculations, and getting the solutions, but you always insisted on understanding of the concepts. I also, used to become bored by the different activities that you undertook for the understanding of the slow learners. Now it is been almost 10 years, and I am presently running a small business of my own. I am satisfied in my life.

Your student

Sam

My Reflections to the Fictional Letters

Writing fictional letters is an amazing idea to have a look into our future, and imagining our lives after 10 years is just exciting. Thinking about the brighter, and the darker side of your future in advance,is a kind of valuable experience, that can motivate us to plan ahead our strategies to hope for the better results. I just enjoyed the idea of having a glimpse into my future professional relations.