Will the Digital Computer Transform Classical Mathematics?
Summary
“Technology is permeated by mathematics.” (Rotman, p 1675) This article describes how mathematics and technology go hand in hand. It discusses the effect of the digital computer on mathematics. The author describes the local and global transformations that have come around since the invention of the digital computer. This article went on to describe how mathematics has changed since computers have been invented, and how it will continue to change as the computer becomes more advanced. The author describes the three theoretical discourses of mathematics: idea, symbol, and procedure. Rotman defines each of these discourses and briefly describes how they affect mathematics. “The digital computer was spawned by modern mathematics and metamathematics.” (p 1676) The author discusses mathematics at a local transformation and a global transformation, and how these transformations started and will continue in the future. In the conclusion portion of the article, the author attempts to sum up his article and clarify the most important parts. “The computer is a machine whose conception, operation, construction and infrastructure are digital: it is antagonistic to the analog and to notions of smoothness and unbroken continuity associated with it.” (p 1688)
Reflection
Again, I found an article that seemed more interesting than it really was. The title of this article made me think that is was going to be about how to use technology (the computer) to change traditional mathematics. Unfortunately, this article did not live up to its title. This article included technical language that made it very difficult to understand. The overall theme of this article was extremely advanced and made one examine our own beliefs about the relationship between computers and mathematics.
The most interesting part of the article was the conclusion. In the conclusion, the author quickly came to the point. “Of course, the computer will transform classical mathematics; surely this is no longer in question. And it is reasonable to predict that it will do so along the same lines–machinic, digital, material–which have determined the changes it has so far wrought.” (p 1688) The author tried to use real world examples to make his point. To me, this was not as helpful as it should have been. Although I struggled reading this article, I still found it interesting and insightful. Once a person got past the technical language, the article made very interesting points about the connection between the digital computer and mathematics.
Reference
Rotman, B. (2003). Will the Digital Computer Transform Classical Mathematics? Philosophical Transactions: Mathematical, Physical and Engineering Sciences. 361(1809), 1675-1690.
Duke Blue Devils #1
Monday, March 8, 2010
Illuminations - Concentration (At Home)
For my second Illumination activity, I played the Concentration game. What I like the most about this game is the fact that you can change what you are looking for - i.e addition, multiplication, fractions, etc. This would be a great review for my students who struggle with visualizing concepts. This game can be played individually or with a partner. The only part I dislike about the partner game is that the game rotates even if you find a correct match. This means that each partner could potentially end up tied so there is little motivation to pay attention to where matches are. Students could just randomly choose until they find a match. Overall, this game will help students visual different math topics. This activity addresses many different TPAK strategies. One is it incorporates knowledge of learner characteristics, orientation, and thinking to foster learning of mathematics with technology. It also uses technology to support learner-centered strategies that address the diverse needs of all learners of mathematics.
Illuminations - Pan Balance: Numbers (In Class)
I explored the illumination on balancing equations. This illumination asks students to recognize that equations with different factors can balance one another. Students can balance addition, multiplication, subtraction, and division equations. They can mix up the equations, balancing addition and subtraction equations, addition and multipllication equations, etc. This activity encorporates one of the TPAK strategies-facilitate mathematics instruction with technology as an integrated tool. Teachers will facilitate technology-enriched, mathematical experiences that foster creativity, develop conceptual understanding, and cultivate higher order thinking skills. Students will be able to see relationships between different mathematical equations and can expand their knowledge. This will help students later in schooling when they reach Algebra and they learn how to balance equations.
Thursday, March 4, 2010
Virtual Manipulatives - Home
Virtual Manipulative - At Home Assignment
I played with the Money under the 3-5 Measurement. The best part of this manipulative is the fact that you can change what you are looking for. First, you can count the total and type in the numbers. Or you can move a certain amount into the box. Finally, you are asked to make $1.00 using different coins and bills. This activity is great to use in the computer lab because each log in is different. No two children will have the same amount at the same time. This means students must know how to count the money. They can't just look at their neighbor. I loved playing around with this manipulative, especially since money is a very difficult concept for 3rd graders to understand. This manipulative will be a great tool to use next year when introducing the concept of money.
Overall, I have looked at many different manipulative on this website and I wish I knew about this many years ago. This website is extremely beneficial to teaching math. I can't wait until testing is over so that my students can spend some more time working with the different concepts.
I played with the Money under the 3-5 Measurement. The best part of this manipulative is the fact that you can change what you are looking for. First, you can count the total and type in the numbers. Or you can move a certain amount into the box. Finally, you are asked to make $1.00 using different coins and bills. This activity is great to use in the computer lab because each log in is different. No two children will have the same amount at the same time. This means students must know how to count the money. They can't just look at their neighbor. I loved playing around with this manipulative, especially since money is a very difficult concept for 3rd graders to understand. This manipulative will be a great tool to use next year when introducing the concept of money.
Overall, I have looked at many different manipulative on this website and I wish I knew about this many years ago. This website is extremely beneficial to teaching math. I can't wait until testing is over so that my students can spend some more time working with the different concepts.
Virtual Manipulative - Class
Virtual Manipulative:
After looking through this site, I saw a clock on I found many things that would have helped with teaching elapsed time. Many of my students struggled with understanding elapsed time, and they have trouble visualizing clocks. The clocks we use, Judy Clocks, only move the hour or the minute hand independently. This virtual manipulative would have been extremely helpful because my students could see the hour and minute moving together. If my students have an opportunity to use this program, then they would gain a better understanding and visual of elapsed time.
After looking through this site, I saw a clock on I found many things that would have helped with teaching elapsed time. Many of my students struggled with understanding elapsed time, and they have trouble visualizing clocks. The clocks we use, Judy Clocks, only move the hour or the minute hand independently. This virtual manipulative would have been extremely helpful because my students could see the hour and minute moving together. If my students have an opportunity to use this program, then they would gain a better understanding and visual of elapsed time.
Abstract #2
Making Mathematics Work for All Children: Issues of Standards, Testing, and Equity
Summary:
This article describes the transition needed from traditional mathematics – i.e. solving problems – to quantitatively sophisticated reasoning. This article discusses the performance gap between white students and students who are categorized as minorities. The author describes how a shift from traditional curriculum to what he calls “reform” curriculum has narrowed the performance gap between whites and minorities. The author gathered data from schools in Pittsburgh, Pennsylvania. “Perhaps most important with regard to equity, not only did average scores on skills, concepts, and problem solving go up, but racial differences in performance diminished substantially” (Schoenfeld, p 17). The author describes how mathematics reform can only occur when curriculum, assessment, and professional development are all aligned. Finally, the author described what it takes to reform mathematics the right way. In order to change mathematics, and to insure the sustained improvement of instruction, the following issues must be addressed: high quality curriculum; stable, knowledgeable, and professional teaching community; high quality assessment that is aligned with curricular goals; stability and mechanisms for evolution.
Reflection:
I found this article extremely interesting in that it describes how traditional curriculum is preparing students, especially minorities, for low paying jobs. Looking back at my educational experience, I realize that throughout my schooling, I was directed towards enriched mathematics courses while many of my minority classmates were directed towards basic courses. Just because of my skin color, it was assumed I would do well in Math. This was not the case. Math has never been my strongest subject and I quickly moved back to basic Math courses. However, many of my minority classmates could easily solve my “enriched” problems quicker than I could.
Another interesting point that made me think was how lacking our society is when it comes to conceptual understanding and problem solving. Unfortunately, this starts at a young age. As a third grade teacher that is responsible for teaching students to pass the CRT’s, I am unable to spend much time on problem solving. The amount of information that must be learned before testing is so vast, that we “stick to the basics” (how to solve problems). There is little time to teach how to figure out a solution, we ask students to give us an answer. By the time students reach the upper grades and are given word problems that involve multiple steps, they are unable to do it properly. Once a habit is formed, it is difficult to change it.
Reference:
Schoenfeld, A. (2002). Making Mathematics Work for All Children: Issues of Standards, Testing, and Equity. Educational Researcher. 31(1), 13-25.
Summary:
This article describes the transition needed from traditional mathematics – i.e. solving problems – to quantitatively sophisticated reasoning. This article discusses the performance gap between white students and students who are categorized as minorities. The author describes how a shift from traditional curriculum to what he calls “reform” curriculum has narrowed the performance gap between whites and minorities. The author gathered data from schools in Pittsburgh, Pennsylvania. “Perhaps most important with regard to equity, not only did average scores on skills, concepts, and problem solving go up, but racial differences in performance diminished substantially” (Schoenfeld, p 17). The author describes how mathematics reform can only occur when curriculum, assessment, and professional development are all aligned. Finally, the author described what it takes to reform mathematics the right way. In order to change mathematics, and to insure the sustained improvement of instruction, the following issues must be addressed: high quality curriculum; stable, knowledgeable, and professional teaching community; high quality assessment that is aligned with curricular goals; stability and mechanisms for evolution.
Reflection:
I found this article extremely interesting in that it describes how traditional curriculum is preparing students, especially minorities, for low paying jobs. Looking back at my educational experience, I realize that throughout my schooling, I was directed towards enriched mathematics courses while many of my minority classmates were directed towards basic courses. Just because of my skin color, it was assumed I would do well in Math. This was not the case. Math has never been my strongest subject and I quickly moved back to basic Math courses. However, many of my minority classmates could easily solve my “enriched” problems quicker than I could.
Another interesting point that made me think was how lacking our society is when it comes to conceptual understanding and problem solving. Unfortunately, this starts at a young age. As a third grade teacher that is responsible for teaching students to pass the CRT’s, I am unable to spend much time on problem solving. The amount of information that must be learned before testing is so vast, that we “stick to the basics” (how to solve problems). There is little time to teach how to figure out a solution, we ask students to give us an answer. By the time students reach the upper grades and are given word problems that involve multiple steps, they are unable to do it properly. Once a habit is formed, it is difficult to change it.
Reference:
Schoenfeld, A. (2002). Making Mathematics Work for All Children: Issues of Standards, Testing, and Equity. Educational Researcher. 31(1), 13-25.
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