Summary
The article examines two major arguments - history as a goal and history as a tool. The former is linked with meta-issues which allow one to see math history from a holistic view in terms of its evolution, individual contributions, cultural influences, and social fabric. In contrast, the latter treats math history as an inspirational source to increase students' perseverance and interest in the learning of math related to inner issues which cover concepts, theories, methods and so on.
Jankvist presents three approaches to teaching math history: illumination (integration of historical events), module (lessons dedicated to history) and history-based (sequencing topics in line with chronological order). He also considers two genetic principles for teaching and learning math.
(1) The historical-genetic principle is intended to help students progress in their learning from the lowest to the highest level of complexity as in the historical development of math.
(2) The psychological-genetic principle is based on active learning through discovery.
Reflections
In my opinion, teaching students math history related to meta-issues does not offer students any concrete help in improving their math understanding since historical events give only factual information. When I tutor students, I notice that they pay no attention to the pages about history and go straight to the examples and exercises. These pages seem to serve no purposes for them. On the other hand, in-issues supported by history-as-a-tool arguments are more applicable to the aims of mainstream math curricula which focus on math relationships and structures. How one plans a current lesson, connects it with the prerequisites from past lessons, and guides students from this lesson to the next are generally influenced by the in-issues.
Jankvist suggests that teacher tell struggling students stories about how historical mathematicians stumbled over the course of their learning and remained determined for years to resolve their difficulties. These stories may be inspiring, but I am not sure if they would have any affective effects on struggling students. I believe that struggling students may need direct support from their parents and teachers whom they can trust.
Lastly, the psychological-genetic principle seems to offer math education its current direction. This direction gives students opportunities to explore, discover and invent math concepts using their own strategies. This may also give the teacher opportunities to present historical approaches briefly relevant to the concepts the students investigate. So, the historical approaches may broaden the students' understanding of the math concepts.
Saturday 28 March 2015
Sunday 22 March 2015
Learning Angles Through Movement ... in an Embodied Activity
Summary
This article analyzes whether body-based activities can help learners deepen their understanding of angles and angle measurements. Smith, King, and Hoyte think the body plays an important role in connecting the visual, abstract representations of angles with physical movements. In their empirical study, they worked with 20 grade 3 and 4 students consisting of 9 boys and 11 girls. The researchers used a Kinect for Windows program to design a motion-controlled activity for the students to represent angles with their arms and arm movements which determined one of the 4 pre-defined colors on the screen based on the size of an angle formed. The screen showed both static and dynamic representations of angles formed by the students' arms as they were prompted to position and vary the angles. An on-screen protractor was used to measure angles in degrees for the students to make conjectures about how the screen color changed with the angle size. The results revealed that most of the students achieved higher scores from the pre-test to the post-test. In addition, one of the two interviewed students connected his daily life experiences with body-based representations of angles.
Reflections
Using one's body to learn math concepts certainly has some benefits. Probably, people interested in performing arts and physical education enjoy learning anything such as math through their body movements and senses. For example, once in a while, I have students who have piano-playing skills and connect them with related math concepts. Usually, I notice that especially in curve sketching, they use their fingers to form certain shapes that represent the prominent parts of a graph, allowing them to visualize it more easily. Then, they draw it out on paper.
On the other hand, even if body-based activities have been shown in this study to be helpful in increasing young children's sense of angle concepts, this result may not guarantee their future success in learning geometry. Over the years, I have noticed that in Pre-Calculus 12, many students have a lot of trouble visualizing the size of angles in radian measures even when they can convert between degrees and radians very easily by hand. To deal with this difficulty, they generally need to switch angles from radians to degrees. This makes me wonder if the body-based approach can improve their visualization of angles in radians.
This article analyzes whether body-based activities can help learners deepen their understanding of angles and angle measurements. Smith, King, and Hoyte think the body plays an important role in connecting the visual, abstract representations of angles with physical movements. In their empirical study, they worked with 20 grade 3 and 4 students consisting of 9 boys and 11 girls. The researchers used a Kinect for Windows program to design a motion-controlled activity for the students to represent angles with their arms and arm movements which determined one of the 4 pre-defined colors on the screen based on the size of an angle formed. The screen showed both static and dynamic representations of angles formed by the students' arms as they were prompted to position and vary the angles. An on-screen protractor was used to measure angles in degrees for the students to make conjectures about how the screen color changed with the angle size. The results revealed that most of the students achieved higher scores from the pre-test to the post-test. In addition, one of the two interviewed students connected his daily life experiences with body-based representations of angles.
Reflections
Using one's body to learn math concepts certainly has some benefits. Probably, people interested in performing arts and physical education enjoy learning anything such as math through their body movements and senses. For example, once in a while, I have students who have piano-playing skills and connect them with related math concepts. Usually, I notice that especially in curve sketching, they use their fingers to form certain shapes that represent the prominent parts of a graph, allowing them to visualize it more easily. Then, they draw it out on paper.
On the other hand, even if body-based activities have been shown in this study to be helpful in increasing young children's sense of angle concepts, this result may not guarantee their future success in learning geometry. Over the years, I have noticed that in Pre-Calculus 12, many students have a lot of trouble visualizing the size of angles in radian measures even when they can convert between degrees and radians very easily by hand. To deal with this difficulty, they generally need to switch angles from radians to degrees. This makes me wonder if the body-based approach can improve their visualization of angles in radians.
Sunday 1 March 2015
FLM 1 - 1 (1st issue)
The design of the front cover consists of numerous lines to form different shapes, patterns and other geometric properties, such as symmetry and tessellation. The illustration makes me think that the journal is tailored to elementary children and talks about entirely geometry, or at least certain aspects of geometry in relation to algebra. My prediction is partly correct in that only the first three articles are dedicated to the learning and teaching aspects of geometry with some reference to algebra. All other articles in this volume examine students' arithmetic errors, word problems in arithmetic and other non-algebra topics. Next, I browsed through the pages, read some paragraphs, and noticed some interesting things. The FLM is a Canadian publication involving contributors from Canada, USA and UK (shown behind the front cover), requires no abstracts, accepts writing in both English and French, and prescribes a length of 2500 - 5000 per article as shown on the back cover. There is a total of 9 articles in this volume which, on average, are 5 pages long. In addition to diagrams, tables, graphs, and flow charts, some articles even include excerpts of interviews. In the bibliography sections, the authors quote the works of different researchers from what other authors use.
I think some titles are either too general or not self-descriptive. For example, the title of the first article, "About Geometry", is too broad to reveal the specifics of the article. The unclear title made me read the first four paragraphs to figure out that geometric thinking is the main idea of the article. A lack of subheadings in all the articles except one makes it difficult for the reader to get an idea of content in the next section. I feel that an article is not complete without an abstract which necessarily gives the reader an overview of the article. One positive side of the journal is that especially the 7th and 8th articles provide a collection of detailed diagrams, tables and graphs that clarify the authors' perspectives and major themes in a simple, concise way. I really like the math history, such as Euclid's Elements, incorporated into most of the articles and the articles' informal tone. Finally, students' work samples and analyses of their work, which increase readability and my personal interest, are the best yet.
I think some titles are either too general or not self-descriptive. For example, the title of the first article, "About Geometry", is too broad to reveal the specifics of the article. The unclear title made me read the first four paragraphs to figure out that geometric thinking is the main idea of the article. A lack of subheadings in all the articles except one makes it difficult for the reader to get an idea of content in the next section. I feel that an article is not complete without an abstract which necessarily gives the reader an overview of the article. One positive side of the journal is that especially the 7th and 8th articles provide a collection of detailed diagrams, tables and graphs that clarify the authors' perspectives and major themes in a simple, concise way. I really like the math history, such as Euclid's Elements, incorporated into most of the articles and the articles' informal tone. Finally, students' work samples and analyses of their work, which increase readability and my personal interest, are the best yet.
Saturday 21 February 2015
Is there a geometric imperative?
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.
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 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.
Saturday 31 January 2015
Problem Posing in Mathematics Education
Summary
The authors, Brown and Walter, highlight the importance of problem posing in math education to improve student learning. They begin by defining what appears to be a math problem is not really a problem. Their first example shows that finding Pythagorean triples in x^2 + y^2 = z^2 is not a problem per se because it is not contextualized in any way. In contrast, their next example is a simple triangle which lends itself to problem-posing more easily when there is no other information given. Students may pose problems about this triangle like finding its total area or dividing the triangle into 3 equal parts. Later, they suggest a strategy for posing a problem as in the famous Konigsberg bridge problem where one can keep all the original information of the problem and change only the question. In fact, historically, this original problem when re-posed repeatedly led to the invention of graph theory.
Next, the authors talk about the benefits of posing problems. By engaging with problem-posing, students can see the interrelationship between posing and solving. This activity gives students an opportunity to restructure the situation described in the original problem and allows them to find new connections that they don't expect to see by problem-solving. Finally, other benefits include promoting creativity, incorporating student learning into social settings, and reducing fears and anxieties caused by problem-solving.
Reflections
I do recognize problem-posing as an essential ingredient of both teaching and learning math. It seems to me that problem-solving (like an object) and problem-posing (like a mirror image) are complementary activities that act as "opposites". I associate posing/solving with "opposites" like multiplication/division, squaring/square rooting, differentiation/integration, and so on. I think it is easier to do the former (like walking forward) in each pair than it is to do the latter (like walking backwards). When I tutor students privately in math, they may be struggling with a math concept which prevents them from solving the problem with this concept. I have to admit that in the past, I used to be afraid to take risks when posing problems for students to highlight certain aspects of a math concept. The reason is that I was not sure whether my posed problems would be solvable, yet some of these unsolvable problems offered good teaching moments. In addition, these moments also turned out to be good learning experiences for my students. Now, although I am not afraid to take risks anymore, I intend to give my students the opportunity to pose their own problems by modifying the original problem. Through posing, as they explore ideas, they reflect on every action they take and generate new questions to improve the flexibility of their thinking. On the other hand, most of my
low-achieving students struggle with posing more badly than they do with solving because they think posing a problem is like walking blind-folded.
The authors, Brown and Walter, highlight the importance of problem posing in math education to improve student learning. They begin by defining what appears to be a math problem is not really a problem. Their first example shows that finding Pythagorean triples in x^2 + y^2 = z^2 is not a problem per se because it is not contextualized in any way. In contrast, their next example is a simple triangle which lends itself to problem-posing more easily when there is no other information given. Students may pose problems about this triangle like finding its total area or dividing the triangle into 3 equal parts. Later, they suggest a strategy for posing a problem as in the famous Konigsberg bridge problem where one can keep all the original information of the problem and change only the question. In fact, historically, this original problem when re-posed repeatedly led to the invention of graph theory.
Next, the authors talk about the benefits of posing problems. By engaging with problem-posing, students can see the interrelationship between posing and solving. This activity gives students an opportunity to restructure the situation described in the original problem and allows them to find new connections that they don't expect to see by problem-solving. Finally, other benefits include promoting creativity, incorporating student learning into social settings, and reducing fears and anxieties caused by problem-solving.
Reflections
I do recognize problem-posing as an essential ingredient of both teaching and learning math. It seems to me that problem-solving (like an object) and problem-posing (like a mirror image) are complementary activities that act as "opposites". I associate posing/solving with "opposites" like multiplication/division, squaring/square rooting, differentiation/integration, and so on. I think it is easier to do the former (like walking forward) in each pair than it is to do the latter (like walking backwards). When I tutor students privately in math, they may be struggling with a math concept which prevents them from solving the problem with this concept. I have to admit that in the past, I used to be afraid to take risks when posing problems for students to highlight certain aspects of a math concept. The reason is that I was not sure whether my posed problems would be solvable, yet some of these unsolvable problems offered good teaching moments. In addition, these moments also turned out to be good learning experiences for my students. Now, although I am not afraid to take risks anymore, I intend to give my students the opportunity to pose their own problems by modifying the original problem. Through posing, as they explore ideas, they reflect on every action they take and generate new questions to improve the flexibility of their thinking. On the other hand, most of my
low-achieving students struggle with posing more badly than they do with solving because they think posing a problem is like walking blind-folded.
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