Mathematical Notation is the use of symbols which stand for specific quantities, according to the problem, to avoid the writing of long words. It is an abbreviation that helps ease the reading of any solution. However, the most impressive thing that attracted my attention is the use of "0 and 1" as a notation to present a solution by one of the fifth graders. It was excellent. The child used a chart where he wrote the first letter of each topping and 0 for the unused topping and 1 for the used. His presentation was successful. It wasn't only that. He, also, was able to use the "four stories towers" to present the "pizza" activity. The combination he made between the two activities was mathematically correct!
When it comes to the second part, where the children were presenting and discussing their work, they were convincing. They were sure about their solutions, defending and insisting. I would say that the children gain these skills from the activities they were involved in. Thus, hands-on-learning strategy does not only inhance students' learning precess, but also their self-confidence, which I myself consider the highest goal.
Saturday, April 10, 2010
Are You Convinced?
To start with, teachers must be able to prove certain formulas or rules because some of these formulas are "undigestable" without proofs. In addition, some other rules need to be worked on with the students step by step, so they will completely get them and gain the ability to use them correctly.
Myself, when our professor gave us the "Building Four Stories Towers with Two Colors without Repeating," I thought of this: 0 1 2 3 4 3 2 1 0
0= no mixture of color, the original color
1= one color interapting the original color
2= two of the same color interapting the original
3= three of the same color interapting the original
4= the interapting color gets over the original color
Then, repeat the same thing from the beginning.
And here appears the idea of "Patterns." Some problems in math need the application of "Backward" thinking to be solved, and this is another strategy shown in mathematic textbooks for Problems Solving.
After watching the video, I became convinced that the best way to deliver a proof to young students is with the use of Hands-On-Learning strategy. It is a method that must be included in teachers' daily lesson plan as much as possible. When a child gets in touch with an activity, he/she will learn and understand at the same time much easier than only reading from the text book. Logically, the brain will process freely when it is not framed with rigid instructions. The children in the video were exchanging ideas in a very relaxed environment. More over, there is a big difference between the outcomes of individual work and team or group work, as part II of video II showed.
In the second part, I saw how the students were able to organize and discuss their solutions between each other in a clear platform. They were comparing their conclusions and justifying them. Each one of them was convinced that his way was the correct path to be followed and applyed as a solution to that activity.
To start with, teachers must be able to prove certain formulas or rules because some of these formulas are "undigestable" without proofs. In addition, some other rules need to be worked on with the students step by step, so they will completely get them and gain the ability to use them correctly.
Myself, when our professor gave us the "Building Four Stories Towers with Two Colors without Repeating," I thought of this: 0 1 2 3 4 3 2 1 0
0= no mixture of color, the original color
1= one color interapting the original color
2= two of the same color interapting the original
3= three of the same color interapting the original
4= the interapting color gets over the original color
Then, repeat the same thing from the beginning.
And here appears the idea of "Patterns." Some problems in math need the application of "Backward" thinking to be solved, and this is another strategy shown in mathematic textbooks for Problems Solving.
After watching the video, I became convinced that the best way to deliver a proof to young students is with the use of Hands-On-Learning strategy. It is a method that must be included in teachers' daily lesson plan as much as possible. When a child gets in touch with an activity, he/she will learn and understand at the same time much easier than only reading from the text book. Logically, the brain will process freely when it is not framed with rigid instructions. The children in the video were exchanging ideas in a very relaxed environment. More over, there is a big difference between the outcomes of individual work and team or group work, as part II of video II showed.
In the second part, I saw how the students were able to organize and discuss their solutions between each other in a clear platform. They were comparing their conclusions and justifying them. Each one of them was convinced that his way was the correct path to be followed and applyed as a solution to that activity.
Following Children's Ideas in Mathematic
How children develop mathematically? This question was the main purpose observed in video I.
The activity which the children paticipated in was absolutely different from direct addition, subtraction, multiplication, or division operations. It pulled them toward unusual experience where their thinking as well as their hands were both involved in finding the solution. This combination let them use two different strategies to solve the given activity: "Drawing a
Picture" or "Making an Organized List." The two startegies are found in mathematic text books in different levels and grades.
During the activity, the students were engaged in an open discussion, revision, comparison, and presentation of their work. Also, they were able to control and extend their solution. Thus, the algebraic thinking was present. It does exist.
How the algebraic thinking exists in students' mind who were never exposed to it in a direct way? The answer is simple. Let's go back in time and try to look at this students at the age of one and half to four years old. During this stage, children devolop their algebraic thinking -observ, analyze, and conclude when they are: playing -either with toy(s) or in the playground, eating, taking baths... on their own or with partner(s), guided or unguided. They are always looking for answers to their why and how questions. How is it my plate, full with food, is now empty? How is it the bucket was filled many times with sand, and then emptied, and yet the castle is getting larger? Brain development not only allows advancement of supporting yourself, but also how to solve problems in different fields, logically.
Back to our video. The question discussed within was, how children develop mathematically? I would ask: How can teachers guide students to applying algebraic thinking (which is already developed in a child, but simply unused) on their own?
How children develop mathematically? This question was the main purpose observed in video I.
The activity which the children paticipated in was absolutely different from direct addition, subtraction, multiplication, or division operations. It pulled them toward unusual experience where their thinking as well as their hands were both involved in finding the solution. This combination let them use two different strategies to solve the given activity: "Drawing a
Picture" or "Making an Organized List." The two startegies are found in mathematic text books in different levels and grades.
During the activity, the students were engaged in an open discussion, revision, comparison, and presentation of their work. Also, they were able to control and extend their solution. Thus, the algebraic thinking was present. It does exist.
How the algebraic thinking exists in students' mind who were never exposed to it in a direct way? The answer is simple. Let's go back in time and try to look at this students at the age of one and half to four years old. During this stage, children devolop their algebraic thinking -observ, analyze, and conclude when they are: playing -either with toy(s) or in the playground, eating, taking baths... on their own or with partner(s), guided or unguided. They are always looking for answers to their why and how questions. How is it my plate, full with food, is now empty? How is it the bucket was filled many times with sand, and then emptied, and yet the castle is getting larger? Brain development not only allows advancement of supporting yourself, but also how to solve problems in different fields, logically.
Back to our video. The question discussed within was, how children develop mathematically? I would ask: How can teachers guide students to applying algebraic thinking (which is already developed in a child, but simply unused) on their own?
Sunday, February 7, 2010
The Lesson Plan / Fifth Grade / Chapter 7
Objective: Adding and Subtracting Fraction with Alike Denominators Correctly
Motivation: Remind the students about the following definitions:
Fraction= division, dividing the numerator by the denominator= decimal
Denominator= the number, which is called denominator, tells us into how many pieces
1 whole has been divided into.
Numerator= the number, which is called numerator, tells us how many parts we will
be using from the pieces of the denominator.
Equivalent= equal value
Material: Pattern Blocks
Procedure: the students will be working in groups of three.
- I will demonstrate to my students how to build three congruent regular hexagon. One is divided into 8 congruent trapezoids, thus the denominator is 8. The second is divided into 12 congruent rhombus, thus the denominator is 12. The third is divided into 24 congruent and equilateral triangles, and here the denominator is 24.
- I will let each student within each group to choose his/her denominator, start writing fraction, ex: 1/8; 3/8; 5/8; 5/12; 7/12; 9/24; 11/24 and so on.
- Add any fraction, either with the same or different denominators. Write the your observations by answering the following question: does adding unlike denominator fractions give you the same part of the same hexagon, knowing that each one was divided into specific number of parts? Was adding fractions of the same denominator gave you the same hexagon? Explain.
- Subtract fractions following the same steps.
Ending Activity: Students can divide a map, for example into congruent squares, so they can draw the same map easily.
Homework: practice book
The Four Story Building
At the beginning, I gave the block to my husband and I explained to him the rules and how he must create as many 4-story building as he can, with the use of 50 orange and 50 black blocks, with out repeating the same order of colors. He tried for about half an hour and all he could accomplish was 12 buildings and he didn't want to go any further. As for me, I built 18 buildings in an hour and half. I thought of a pattern, which is starting by entering the orange color on the black building one, by one, then two by two, then three. When I reached the orange building, I did the opposite. I entered the black blocks one by one, two by two, then three -of course without repeating the same stories. Observe the picture from right to left.
The Changing Colors Rectangle
I took the "brain teaser" with me to school, where I teach and I gave it to two of my students to try it. One is an eighth grader with a B+ level, who did it in less than 3 minutes. The second one was an ADHD fifth grader. He has a good observation, quick answer to tricky questions, and gave up easily. He tried it for about 7 minutes. At first he moved correctly four steps, but then he lost it and he couldn't go back and start from the beginning. He had hard time remembering his own steps. At home, I gave it to my two sons. Anas, who is in third grade he did it in less than 5 minutes and he was able to remember his steps without any mistake. I was impressed. Aassem, who is in 9th grade, in honor classes, he tried for about an hour, giving up sometime and go back to it, then he was able to switch the colors with respect to the rules. But, he didn't want to do it again.
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