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.
Wednesday, February 17, 2010
Entry #5
Warrington's ideas about students constructing their own knowledge are very innovative and have many advantages. One of these advantages is that the exchange of ideas in the classroom is prevalent. Warrington uses an example about the problem 1 divided by 2/3. When she allowed the class to share how they had gotten different answers, the students could listen to their classmates and either agree that they had gotten the correct answer or find the mistakes they had made. Another remarkable advantage is that students form and discover mathematical relationships on their own without being taught them explicitly. When one girl was doing the problem about chocolate-covered peanuts, instead of using methods similar to her peers, she simply "doubled it and divided by one." She had discovered that the answer is the same even when the equation is doubled without anyone teaching her that relationship. The last main advantage is the "intellectual autonomy" that students develop. They begin to think on their own, like one girl who disagreed with the rest of her class about a certain division with fractions problem. The way that the other students had done the problem did not make sense to her, and she would not let it rest. She explained her reasoning, which happened to be correct, even though she was standing alone from her peers.
There are several disadvantages of Warrington's methods, which inhibits her ideas from becoming more widespread. One of the disadvantages is that generally teachers do not believe it works so they would not be willing to try it. Warrington said that most teachers think that certain algorithims are required for students to learn how to do more complex math. Similarly, if a school was to start teaching in a way that students would have to construct their own knowledge, the students would have to go through a major learning curve because they have never been expected to do that before. Warrington said the response, "I don't know how to do this," was not present in her classroom because the students knew they were expected to find a way to do a problem. For students who have not been learning this way, there would be major resistance to learning on their own. Another more specific disadvantage is that all of the students might not be learning and understanding as well as the top students. In the way Warrington described her classroom, students were allowed to say the answer when they got it and explain their method. Although some of the students were constructing their own knowledge very well, there could be some students who were not and were left confused.
There are several disadvantages of Warrington's methods, which inhibits her ideas from becoming more widespread. One of the disadvantages is that generally teachers do not believe it works so they would not be willing to try it. Warrington said that most teachers think that certain algorithims are required for students to learn how to do more complex math. Similarly, if a school was to start teaching in a way that students would have to construct their own knowledge, the students would have to go through a major learning curve because they have never been expected to do that before. Warrington said the response, "I don't know how to do this," was not present in her classroom because the students knew they were expected to find a way to do a problem. For students who have not been learning this way, there would be major resistance to learning on their own. Another more specific disadvantage is that all of the students might not be learning and understanding as well as the top students. In the way Warrington described her classroom, students were allowed to say the answer when they got it and explain their method. Although some of the students were constructing their own knowledge very well, there could be some students who were not and were left confused.
Wednesday, February 10, 2010
Constructivism
Ernst von Glasersfeld discussed how knowledge is not transferred from one person, book, etc. to another person, but rather that the person interprets and senses the world around them. This is called constructivism. Glasersfeld talks about constructing knowledge rather than acquiring knowledge because he believes there isn't just one specific truth that we take in, but rather that we perceive what is happening around us and convert that into our own truth. Because no one perceives something in exactly the same way as someone else, what every person thinks of as truth is different. We each also go through different experiences so our knowledge of one topic or event could be completely different from another person's knowledge of that same topic or event. This is another way how different people have different ideas of what truth is. Experience plays a large role in not only supplying knowledge and our idea of truth, but also in changing it. We may think that the truth is one way of looking at a specific topic, but then our experience comes along and contradicts our idea of truth. We then have to change our idea of truth to match our new experience. This is how knowledge is constructed and not merely acquired through the transfer of information.
Constructivism can readily be applied in a mathematics classroom through the use of it in making sure students fully understand the idea being taught. Through the idea of constructivism and not being able to simply transfer knowledge from teacher to student, there must be a way to teach mathematics where students discover on their own and construct their own knowledge. Then the teacher needs to be responsible to test a student's knowledge and correct it. This will verify that the student's idea of knowledge and truth is the same as the teacher's knowledge and the rest of the math world's knowledge. I do not want to suggest one way to apply constructivism becauseI think there would be many possibilities of how to do this. Every student learns in a slightly different manner and would discover math knowledge in a different way. Students could make discoveries through doing practice problems in homework and having to explain how and why they did the things they did. This would provide them with a way to make an idea about how a certain algorithim is done. If the teacher sees a flaw in their explanation, he or she could give the student an example problem for the student to solve that will not give the correct answer if the student follows his or her own explanation. When the student solves and the problem and explains it again, he or she will have a new knowledge about that algorithim that matches more closely with that of the teacher. Experience would be changing the student's knowledge.
Constructivism can readily be applied in a mathematics classroom through the use of it in making sure students fully understand the idea being taught. Through the idea of constructivism and not being able to simply transfer knowledge from teacher to student, there must be a way to teach mathematics where students discover on their own and construct their own knowledge. Then the teacher needs to be responsible to test a student's knowledge and correct it. This will verify that the student's idea of knowledge and truth is the same as the teacher's knowledge and the rest of the math world's knowledge. I do not want to suggest one way to apply constructivism becauseI think there would be many possibilities of how to do this. Every student learns in a slightly different manner and would discover math knowledge in a different way. Students could make discoveries through doing practice problems in homework and having to explain how and why they did the things they did. This would provide them with a way to make an idea about how a certain algorithim is done. If the teacher sees a flaw in their explanation, he or she could give the student an example problem for the student to solve that will not give the correct answer if the student follows his or her own explanation. When the student solves and the problem and explains it again, he or she will have a new knowledge about that algorithim that matches more closely with that of the teacher. Experience would be changing the student's knowledge.
Monday, January 25, 2010
Entry #3
Erlwanger believes that understanding the reasoning behind the rules is fundamental in learning mathematics. Benny was a sixth-grade student who had been using the IPI method of math since second grade. The IPI method is basically following example problems and then doing similar problems in order to learn a specific rule. Benny has been successful, even one of the top of his class, but when questioned, Benny states incorrect rules. Repeatedly Erlwanger mentions how Benny was only interested with the rules of math, in this case fractions. Benny had been discovering "magical" rules since he began the program in second grade. Benny thought they had been invented by someone and made to be the ultimate standard for doing mathematics. Erlwanger wants us to know that relational understanding is the most beneficial. We need to know the how and the why of the rules in order to completely understand the rules and be able to use them consistently. Benny said he was on a wild goose chase to find the answers in the key because he thought there were multiple right answers, and he was just trying to find the right one. The IPI program had led him to teach himself incorrect rules about fractions and decimals that he blindly followed. He didn't understand that there should be one correct answer. Benny also thinks that all the rules are set in stone, and that there is only one way to do something. If he had a relational understanding, however, he would be able to find more than one way to do a problem because he would understand how the rule worked.
The whole concept of relational understanding and knowing how a rule works is still valid today. When we know how a learned rule works, we can apply it in new situations and not just in the specific case where it was learned. We also might be able to apply a different rule to a given situation because we fully understand how the situation works. There definitely could be more students like Benny who, for one reason or another, end up teaching themselves incorrect rules but get by with high grades. This should also be applied today. Simply because a student has a high grade does not mean they understand the material and are able to get the correct answer all of the time. Benny was one of the top students in his class, but he had gotten by without someone noticing that he didn't actually understand the material. Today we should be more aware of this and catch it at the beginning. By the time Erlwanger talked to Benny, he had been doing this for four years and could not change his thinking even when Erlwanger went back and tried to teach him. We could prevent cases like Benny if we could assure that the students relationally understand the rules, which could be difficult to do.
The whole concept of relational understanding and knowing how a rule works is still valid today. When we know how a learned rule works, we can apply it in new situations and not just in the specific case where it was learned. We also might be able to apply a different rule to a given situation because we fully understand how the situation works. There definitely could be more students like Benny who, for one reason or another, end up teaching themselves incorrect rules but get by with high grades. This should also be applied today. Simply because a student has a high grade does not mean they understand the material and are able to get the correct answer all of the time. Benny was one of the top students in his class, but he had gotten by without someone noticing that he didn't actually understand the material. Today we should be more aware of this and catch it at the beginning. By the time Erlwanger talked to Benny, he had been doing this for four years and could not change his thinking even when Erlwanger went back and tried to teach him. We could prevent cases like Benny if we could assure that the students relationally understand the rules, which could be difficult to do.
Thursday, January 14, 2010
Relational Understanding and Instrumental Understanding
A major question in mathematics education today is what qualifies truly understanding the material. There are generally two different approaches to understanding: instrumental understanding and relational understanding. Instrumental understanding is having a mathematical rule and being able to use and manipulate it. Relational understanding is having a mathematical rule, knowing how to use it, and knowing why it works.
From the definitions given, relational understanding includes instrumental understanding and more. Instrumental is simply knowing and applying the rule, while relational is knowing and applying the rule while also being able to know why a rule works and connect one rule with another. Both types of understanding give the correct answers, but relational is much more extensive.
Although relational understanding is often thought of to be a better alternative to instrumental understanding, there are advantages and disadvantages of both. Often the advantages of one type of understanding are the disadvantages of the other. Instrumental has three main advantages. The first is that it is easier to understand, often to a great extent. Some topics are difficult to grasp and can much sooner be learned through just using rules and set computations than through knowing why something works the way it does. The second is that the positive results are instantaneous. Once the rule or algorithm is learned, the student can use it to do many problems in that format and get the correct answers. The third advantage of instrumental learning goes along with the second in that the correct answers can be obtained very quickly and consistently. Instead of relational learning, where the thought process is longer in trying to understand the problem, in instrumental learning, once the student is able to follow a rule or algorithm, they can do problems that apply to that rule rapidly and always get the right answers. All three of these are disadvantages of relational understanding. Having to think about why something works the way it does or how it works is much more difficult than using rules. It is also more time consuming and involves a lot more thought. This applies to both teaching relationally and actually doing the problems. For each problem, the student is relying on his or her understanding of the idea to solve the problem, which may be correct or incorrect. When the students simply learn and use rules that are always there for them (instrumental), they will always get the right answer. There are four general advantages to relational understanding that are disadvantages to instrumental understanding. The first is that relational is easily adjusted when a new task is introduced. The students can take what they have already learned and apply and adjust it to the new idea because they understand why it works. This does not work with instrumental because the students can only do problems that fit within the rule they learned. Although a rule may be very similar to one they already learned, they will not discover it on their own because they don't know why the first rule works. The second advantage to relational is that students can more easily remember what they've been taught. If they know general ideas about why computations work the way they do, they can connect them with other ideas and therefore remember them more easily. On the other hand, in instrumental understanding, a student has to remember many separate rules that seem unconnected from each other. The third is that relational learning turns into its own goal. External benefits are not needed as much as they are with instrumental because just through fully understanding the idea the student is rewarded. The last advantage is relational knowledge naturally grows. Once a student understands relationally what has been taught, he or she will want to expand that knowledge and look for new concepts to apply it to. This does not happen in instrumental where the student learns a new rule and is content until given the next rule because he or she does not know how to make the connections to expand his or her knowledge.
From the definitions given, relational understanding includes instrumental understanding and more. Instrumental is simply knowing and applying the rule, while relational is knowing and applying the rule while also being able to know why a rule works and connect one rule with another. Both types of understanding give the correct answers, but relational is much more extensive.
Although relational understanding is often thought of to be a better alternative to instrumental understanding, there are advantages and disadvantages of both. Often the advantages of one type of understanding are the disadvantages of the other. Instrumental has three main advantages. The first is that it is easier to understand, often to a great extent. Some topics are difficult to grasp and can much sooner be learned through just using rules and set computations than through knowing why something works the way it does. The second is that the positive results are instantaneous. Once the rule or algorithm is learned, the student can use it to do many problems in that format and get the correct answers. The third advantage of instrumental learning goes along with the second in that the correct answers can be obtained very quickly and consistently. Instead of relational learning, where the thought process is longer in trying to understand the problem, in instrumental learning, once the student is able to follow a rule or algorithm, they can do problems that apply to that rule rapidly and always get the right answers. All three of these are disadvantages of relational understanding. Having to think about why something works the way it does or how it works is much more difficult than using rules. It is also more time consuming and involves a lot more thought. This applies to both teaching relationally and actually doing the problems. For each problem, the student is relying on his or her understanding of the idea to solve the problem, which may be correct or incorrect. When the students simply learn and use rules that are always there for them (instrumental), they will always get the right answer. There are four general advantages to relational understanding that are disadvantages to instrumental understanding. The first is that relational is easily adjusted when a new task is introduced. The students can take what they have already learned and apply and adjust it to the new idea because they understand why it works. This does not work with instrumental because the students can only do problems that fit within the rule they learned. Although a rule may be very similar to one they already learned, they will not discover it on their own because they don't know why the first rule works. The second advantage to relational is that students can more easily remember what they've been taught. If they know general ideas about why computations work the way they do, they can connect them with other ideas and therefore remember them more easily. On the other hand, in instrumental understanding, a student has to remember many separate rules that seem unconnected from each other. The third is that relational learning turns into its own goal. External benefits are not needed as much as they are with instrumental because just through fully understanding the idea the student is rewarded. The last advantage is relational knowledge naturally grows. Once a student understands relationally what has been taught, he or she will want to expand that knowledge and look for new concepts to apply it to. This does not happen in instrumental where the student learns a new rule and is content until given the next rule because he or she does not know how to make the connections to expand his or her knowledge.
Tuesday, January 5, 2010
Blog Entry #1
Mathematics is everywhere around us. It is a very precise science that relies on formulas and concepts. I'm having a hard time coming up with an exact definition, but it includes measurements, numbers, and symbols. Like other sciences, it is used to explain natural laws.
I learn math the best when a teacher explains the concept and why it is that way. I like to know why we use certain formulas because it helps me to better understand what we're doing. Even though proofs can be long, they are also very helpful when one topic is building on another. I also learn by example. I like to be shown how to do something before I really know how to do it because otherwise I might end up practicing doing it incorrectly. I like to be shown the complete process while someone can explain why they are doing each step. It is not beneficial when someone just does the problem without any explanation.
My students will learn mathematics through a step-by-step explanation given of what the concept is followed by several examples of how to apply the concept. I've found that practicing the concepts is beneficial to remembering and fully understanding the ideas taught. My students will also learn through the opportunity to explain the concepts to each other. That way they will know what part they understand and what part they are still having trouble with.
One of the most important things is having ample time to ask questions. Like we do here, my high school had a math lab where students could go to get help. This provided a way for students to get individualized assistance with questions they had and ideas they were confused with. This can happen if homework time is made available during the class, as well. Students will also learn better if they are interested in the topic. Finding ways, such as news articles or example problems, to connect mathematical concepts to real life promotes students' learning in mathematics.
Some current practices that are detrimental to students' learning of mathematics is when only the teacher is speaking and explaining the concepts with a wall between them and the students. The classroom needs to be an interactive environment and open to questions from the students. Often teachers will just try to get the class through some material until they reach their favorite part of the curriculum without explaining the importance of what had been taught so far. Teachers need to try to keep everything relevant, as much as possible.
I learn math the best when a teacher explains the concept and why it is that way. I like to know why we use certain formulas because it helps me to better understand what we're doing. Even though proofs can be long, they are also very helpful when one topic is building on another. I also learn by example. I like to be shown how to do something before I really know how to do it because otherwise I might end up practicing doing it incorrectly. I like to be shown the complete process while someone can explain why they are doing each step. It is not beneficial when someone just does the problem without any explanation.
My students will learn mathematics through a step-by-step explanation given of what the concept is followed by several examples of how to apply the concept. I've found that practicing the concepts is beneficial to remembering and fully understanding the ideas taught. My students will also learn through the opportunity to explain the concepts to each other. That way they will know what part they understand and what part they are still having trouble with.
One of the most important things is having ample time to ask questions. Like we do here, my high school had a math lab where students could go to get help. This provided a way for students to get individualized assistance with questions they had and ideas they were confused with. This can happen if homework time is made available during the class, as well. Students will also learn better if they are interested in the topic. Finding ways, such as news articles or example problems, to connect mathematical concepts to real life promotes students' learning in mathematics.
Some current practices that are detrimental to students' learning of mathematics is when only the teacher is speaking and explaining the concepts with a wall between them and the students. The classroom needs to be an interactive environment and open to questions from the students. Often teachers will just try to get the class through some material until they reach their favorite part of the curriculum without explaining the importance of what had been taught so far. Teachers need to try to keep everything relevant, as much as possible.
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