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.
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