Goodman, T. (2010) Shooting free throws, probability, and the golden ratio. Mathematics Teacher, 103(7), 482-487.
Goodman's article focuses on the application of shooting basketball free throws to probability and finding the golden ratio. The article starts with a real-life example of a girl on a high school basketball team and her free-throw shooting percentage. The problem starts with a one-and-one free-throw situation and trying to decide how many points-0,1, or 2-is the most likely outcome. Using a table or a probability tree, students can find which outcome is most likely. The problem can then be expanded to a general case for different shooting percentages or shooting more free throws. Teachers should encourage these expansions so that students can discover patterns and mathematical rules. When analyzing the situation for a general case, students can find which point outcome is most probable based on different shooting percentages. One of the results of this analysis is the discovery of the golden ratio, which exists when for a line segment divided into two parts, the ratio of the shorter part to the longer part is the same as the ratio from the longer part to the entire line segment.
The connection between algebra and shooting free-throws is very innovative and useful in learning many aspects of algebra. When I first read the article, I kept being surprised about how much you could do with a simple example of shooting free-throws. There were so many extensions and applications by changing different aspects of the problem slightly, such as shooting percentage or how many shots were being taken. Expanding the idea to a more general case helped students to know more about probability as well as showing them what to do in future problems if they want to know about the general case. I also really enjoy basketball, which made the article fun to read, but it actually applied to what the students were learning. I also loved when whatever concept we were learning connected to real life. It would also stick in my memory more because I was not working with an abstract concept, but rather something real that I could relate to. This article is an excellent example of applying math to real life.
Thursday, March 25, 2010
Wednesday, March 17, 2010
Otten, S., Herbel-Eisenmann, B.A., & Males, L.M. (2010). Proof in algebra: Reasoning behind examples. Mathematics Teacher, 103(7), 514-518.
Throughout this article, the main focus is on how proving an idea or rule is much clearer and provides for greater understanding than simply providing examples of the rule does. It is mentioned that proofs should not only be used for geometry class, but also for other areas of math such as basic algebra. They do not need to be complex, college-level proofs, but they need to be adequate to prove the rule is true for all cases. The majoritiy of the article is spent on a classroom example of the cross-multiply rule That is, if two fractions are equal, you can multiply the numerator of the first with denominator of the second, and that will be equal to the numerator of the second multiplied with denominator of the first. The teacher first goes about showing that this works through providing a few examples. The article points out that students are easily convinced so they would believe the rule to be true. They might eventually think that examples are a proof or that proofs are unnecessary. When a general case of the rule is presented after examination of several different examples, the students will know that the cross-multiply rule works for all cases and will be most benefited because they will know a certain rule is true.
Including more proofs in lower-level math classes would be beneficial to the students then and in the future. Most of my math experience has been taught through examples, like in the article, which showed that the rule was true for the numbers in the examples. Sometimes I would get confused and assume the rule is true for all numbers when that is really not the case. Doing a proof of a rule would teach students how to find out when a rule works and when it does not. Like the article said, proofs do not really come up until geometry. The idea of a proof is so new to students in geometry that they do not understand how to use them and get frustrated. This leads to a general aversion towards proofs, which I have seen many times. If proofs were introduced sooner, students would be more comfortable with them and be more inclined to use them rather than avoid them. Introducing them sooner would also show that proofs are necessary and can not be replaced by examples. Including proofs would also connect ideas together and make mathematics seem more logical. Often math can seem like a random bunch or rules and procedures, but using proofs can connect ideas and show why a rule works instead of just knowing that the rule works. This would be beneficial to students because they would more likely enjoy math instead of resenting it.
Throughout this article, the main focus is on how proving an idea or rule is much clearer and provides for greater understanding than simply providing examples of the rule does. It is mentioned that proofs should not only be used for geometry class, but also for other areas of math such as basic algebra. They do not need to be complex, college-level proofs, but they need to be adequate to prove the rule is true for all cases. The majoritiy of the article is spent on a classroom example of the cross-multiply rule That is, if two fractions are equal, you can multiply the numerator of the first with denominator of the second, and that will be equal to the numerator of the second multiplied with denominator of the first. The teacher first goes about showing that this works through providing a few examples. The article points out that students are easily convinced so they would believe the rule to be true. They might eventually think that examples are a proof or that proofs are unnecessary. When a general case of the rule is presented after examination of several different examples, the students will know that the cross-multiply rule works for all cases and will be most benefited because they will know a certain rule is true.
Including more proofs in lower-level math classes would be beneficial to the students then and in the future. Most of my math experience has been taught through examples, like in the article, which showed that the rule was true for the numbers in the examples. Sometimes I would get confused and assume the rule is true for all numbers when that is really not the case. Doing a proof of a rule would teach students how to find out when a rule works and when it does not. Like the article said, proofs do not really come up until geometry. The idea of a proof is so new to students in geometry that they do not understand how to use them and get frustrated. This leads to a general aversion towards proofs, which I have seen many times. If proofs were introduced sooner, students would be more comfortable with them and be more inclined to use them rather than avoid them. Introducing them sooner would also show that proofs are necessary and can not be replaced by examples. Including proofs would also connect ideas together and make mathematics seem more logical. Often math can seem like a random bunch or rules and procedures, but using proofs can connect ideas and show why a rule works instead of just knowing that the rule works. This would be beneficial to students because they would more likely enjoy math instead of resenting it.
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