Summary
In this article, Tahta talks about the role and value of geometry. He states that unlike algebraic experience that lies in symbols, geometry involves mental images made through sight. He outlines three important aspects of studying geometry.
(1) imagining: seeing what is said
(2) construing: seeing what is drawn and saying what is seen
(3) figuring: drawing what is seen
Everyone has different ways of imagining geometric ideas when listening to verbal descriptions that invoke different types of images - kinesthetic, tactile, aural and visual. Next, Tahta provides two examples to illustrate "construing". The first example shows a 2-dimensional picture of a horse and cart in which some people may construe the wheels on the cart as being elliptical on paper and others may see them as being circular in 3-dimensional depth. To illustrate "saying what is seen", the second example shows a shape made of different triangles and bordered by hexagons. Tahta argues that people verbally describe this same picture differently based on what they see. Sharing your verbal descriptions of a shape or a solid makes other people realize that it is possible to consider the object from multiple acceptable viewpoints. In a geometrical sense, "figuring" is related to projective geometry which involves drawing 3-dimensional objects in 2-dimensions. Tahta states that until the 17th century, the Rennaissance architects and artists had influenced the development of projective geometry where they found new ways of representing projections of an object from different observation points before mathematicians considered cross-sections of solids from different projections. Lastly, the author thinks it is imperative that geometry be included in the math curriculum because children explore math ideas through action, intuition, and an awareness of imagery.
Reflections
A lot of my students are visual learners and enjoy learning algebra in a geometric context. Unfortunately, some important geometry topics have been removed from the math curricula. Some topics, such as conic sections should be brought back to the curriculum to deepen students' understanding of shapes and space essential for learning calculus. I feel that I constantly need to learn more geometry on my own to meet the needs of my students. In particular, I find it difficult to connect geometry to ideas, such as radical expressions and rational expressions.
In Math 8, whenever I teach a lesson on views and nets of prisms, students need to draw nets and do isometric drawings on paper. The students demonstrate their own ways of imagining transformations from 3D to 2D or vice versa. Those students who possess weak perceptual skills have difficulty in visualizing and drawing the sides of a solid from different vantage points. Nonetheless, their images drawn on paper, if fitted together properly, may produce some creative solids that are worthy of discussion. Another exercise that I do is to have one student describe verbally his/her mental image to others. Then, they are asked to sketch, compare, and contrast one another's drawings. This exercise gives the students an opportunity to talk about any common properties and relationships to algebra among their drawings. I find that students are very engaged in debating and construing each other's perspectives in a qualitative way. Whenever possible, I incorporate geometry into certain algebraic concepts through imagining, construing and figuring to improve students' learning.
Saturday, 21 February 2015
Saturday, 14 February 2015
The Experimental Mathematician: The Pleasure of Discovery and the Role of Proof
Summary
In his article, Borwein who is a proponent of computer technology talks about its influence on math proof (deduction). He defines experimental math (induction) as a mathematical method via the use of the computer which facilitates math discoveries, imagination, development of insight and intuition, hypothesis-testing, and lengthy computations. In contrast, proof exhibits the aesthetics of math and de-emphasizes computation. He states that one advantage of computation is to help one see the result of a conjecture in advance to determine whether it merits any proof. As modern computer software generates better graphics and sounds, it makes proof more accessible and computation more accurate. For example, proving the square root of 2 to be irrational pictorially with inscribed triangles shows how easy it is to prove or disprove conjectures on a computer. However, Borwein points out a drawback to the high efficiency of computation which may hinder students and researchers' ways of examining math structure when they can thoughtlessly do enormous computations accurately on a computer. He concludes that computer technology is a good tool for mathematicians but is not a good substitute for proof.
Reflections
I think proof and computation work hand in hand. Proofs are needed to help us make sure that we put theory into practice correctly. They also help us check our understanding and reasoning. At the same time, computation is necessary to bring us back on course when we make mistakes in our proofs. For example, in Pre-calculus 12, when my students and I prove trigonometric identities together, occasionally we make minor mistakes in our proof which prevent us from completing it. Then, we use a calculator to substitute a value for the angle in each step until we find and correct the mistakes. Most of my students are interested in proving identities. However, it comes to formal proofs of math theorems, my students ignore them and go straight to the worked examples. I think because the formal logic which serves as the basis for proofs is not emphasized in high school, students are uninterested in math structure. Time is a major factor for de-emphasizing math proofs in schools. This is also the reason why I cannot emphasize any simple proofs in my learning centre. Besides that, parents expect tutors to teach their children how to do computations for the purposes of school tests.
In his article, Borwein who is a proponent of computer technology talks about its influence on math proof (deduction). He defines experimental math (induction) as a mathematical method via the use of the computer which facilitates math discoveries, imagination, development of insight and intuition, hypothesis-testing, and lengthy computations. In contrast, proof exhibits the aesthetics of math and de-emphasizes computation. He states that one advantage of computation is to help one see the result of a conjecture in advance to determine whether it merits any proof. As modern computer software generates better graphics and sounds, it makes proof more accessible and computation more accurate. For example, proving the square root of 2 to be irrational pictorially with inscribed triangles shows how easy it is to prove or disprove conjectures on a computer. However, Borwein points out a drawback to the high efficiency of computation which may hinder students and researchers' ways of examining math structure when they can thoughtlessly do enormous computations accurately on a computer. He concludes that computer technology is a good tool for mathematicians but is not a good substitute for proof.
Reflections
I think proof and computation work hand in hand. Proofs are needed to help us make sure that we put theory into practice correctly. They also help us check our understanding and reasoning. At the same time, computation is necessary to bring us back on course when we make mistakes in our proofs. For example, in Pre-calculus 12, when my students and I prove trigonometric identities together, occasionally we make minor mistakes in our proof which prevent us from completing it. Then, we use a calculator to substitute a value for the angle in each step until we find and correct the mistakes. Most of my students are interested in proving identities. However, it comes to formal proofs of math theorems, my students ignore them and go straight to the worked examples. I think because the formal logic which serves as the basis for proofs is not emphasized in high school, students are uninterested in math structure. Time is a major factor for de-emphasizing math proofs in schools. This is also the reason why I cannot emphasize any simple proofs in my learning centre. Besides that, parents expect tutors to teach their children how to do computations for the purposes of school tests.
Saturday, 7 February 2015
On Culture, Geometrical Thinking and Math Education
Summary
Gerdes begins with comments on the high illiteracy rate in Africa and the methods of teaching math based on rote memorization which downgraded children's informal practical math skills. Even worse, colonization that took place in the past in Africa had a huge impact on the Mozambicans there and wiped out their unique cultural and mathematical traditions. Later, a "cultural rebirth" recognized and incorporated the Mozambicans' indigeneous mathematics into the new curriculum, emphasizing geometrical forms and patterns of their weaving skills. A new generation of math teachers now investigates and rediscovers the math ideas and the geometrical thinking that their parents and ancestors unconsciously used when weaving various products. One example that the author presents is a square woven button consisting of 4 right triangles whose hypotenuse forms each side of the button and a tilted square in the centre of the button. He explains how to take the button apart in a manner that shows the use of the Pythagorean theorem. Other examples of woven products are funnels, fish traps and soccer balls whose geometric patterns and properties are associated with pyramids, similar figures, polygons, and the sum of all the interior angles of an n-gon.
Reflections
First, the author's statements, "The artisan, who imitates a ... technique, is ... not doing mathematics. But, the artisan(s) who discovered the technique, ... (was) were thinking mathematically. (p140-141)" really resonate with me when I think about the students I tutor. Some of my students tend to imitate how they learn from their math teachers without injecting any creativity into their own problem-solving strategies. The reason is that they fear they will lose marks for not "imitating" their teachers' ways of doing math. Imitating is not learning and telling is not teaching. To ensure "real" learning, I think as an activity, teachers can give their students an opportunity to prepare a lesson plan, deliver it to their peers and demonstrate their creative ways of doing math. Teachers need to be more open-minded about the ways their students do problem-solving.
Second, the weaving skills of the Mozambicans lead me to think of the excellent carving and drawing skills of the Aboriginal Canadians. The First Nations' history in Canada is rich and diverse. Their artworks and sculptures, especially the totem poles, involve a lot of geometric thinking and reasoning, a practical math skill that the indigenous people use unconsciously. Incorporating their artistic works into the math curricula is a way of recognize their contributions to Canada. This makes math more meaningful and interesting to students when they can learn math in the context of Canadian history.
Gerdes begins with comments on the high illiteracy rate in Africa and the methods of teaching math based on rote memorization which downgraded children's informal practical math skills. Even worse, colonization that took place in the past in Africa had a huge impact on the Mozambicans there and wiped out their unique cultural and mathematical traditions. Later, a "cultural rebirth" recognized and incorporated the Mozambicans' indigeneous mathematics into the new curriculum, emphasizing geometrical forms and patterns of their weaving skills. A new generation of math teachers now investigates and rediscovers the math ideas and the geometrical thinking that their parents and ancestors unconsciously used when weaving various products. One example that the author presents is a square woven button consisting of 4 right triangles whose hypotenuse forms each side of the button and a tilted square in the centre of the button. He explains how to take the button apart in a manner that shows the use of the Pythagorean theorem. Other examples of woven products are funnels, fish traps and soccer balls whose geometric patterns and properties are associated with pyramids, similar figures, polygons, and the sum of all the interior angles of an n-gon.
Reflections
First, the author's statements, "The artisan, who imitates a ... technique, is ... not doing mathematics. But, the artisan(s) who discovered the technique, ... (was) were thinking mathematically. (p140-141)" really resonate with me when I think about the students I tutor. Some of my students tend to imitate how they learn from their math teachers without injecting any creativity into their own problem-solving strategies. The reason is that they fear they will lose marks for not "imitating" their teachers' ways of doing math. Imitating is not learning and telling is not teaching. To ensure "real" learning, I think as an activity, teachers can give their students an opportunity to prepare a lesson plan, deliver it to their peers and demonstrate their creative ways of doing math. Teachers need to be more open-minded about the ways their students do problem-solving.
Second, the weaving skills of the Mozambicans lead me to think of the excellent carving and drawing skills of the Aboriginal Canadians. The First Nations' history in Canada is rich and diverse. Their artworks and sculptures, especially the totem poles, involve a lot of geometric thinking and reasoning, a practical math skill that the indigenous people use unconsciously. Incorporating their artistic works into the math curricula is a way of recognize their contributions to Canada. This makes math more meaningful and interesting to students when they can learn math in the context of Canadian history.
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