Wednesday, March 24, 2010
Correction to the 1st Article's Citation
Guerrero, S. M.. (March 2010). The value of guess and check. Mathematics Teaching in the Middle School, 15(7), 393-398.
Enabling Teachers to Teach
Smith, M. S., Bill, V., & Hughes, E. K.. (October 2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle Schools. 14(3), 132-138.
The authors of the article Thinking through a Lesson: Successfully Implementing High-Level Tasks want teachers to know, though it is hard to think of high-level tasks, it is worthwhile to create these tasks using TTLP (Thinking Through a Lesson Protocol). Teachers may complain that it is overwhelming to plan a high-level task for every day, but the authors suggest that TTLP shouldn't be used every day at first. With the help of other teachers to develop more and more cognitive tasks, TTLP should be used periodically to where eventually a teacher will have a task for every day by accumulating them over time. Though planning a high-level lesson takes a lot of time using TTLP, it is possible to use only part of the TTLP method. The goal of using TTLP is to help teachers change the way they think about a lesson. By using TTLP, teachers will be able to further the use of a high-level task, so that students are able to learn more. Using TTLP will also enable a teacher to know how to respond to complications in the task, thus helping the task to go a lot smoother, and for the goals of the task to be met.
I think TTLP is a great way to create lesson plans that will promote cognitive thinking in students. Though there are a lot of things to think about when planning a lesson using TTLP, I think that the outcome is worthwhile. TTLP helps teachers to think about questions to ask students to further their thinking, and also helps the teacher think more about how students might respond to the task. When a teacher has already looked into where students might have trouble or how they can further a child's thinking if students are having a relatively easy time accomplishing the task, a teacher can then be more prepared for these cases. A teacher will already know how to respond. Knowing how to respond before something happen gives an advantage to the teacher, so that the teacher's task will go smoother, and will help the teacher and students stay focused on the goals of the task.
The authors of the article Thinking through a Lesson: Successfully Implementing High-Level Tasks want teachers to know, though it is hard to think of high-level tasks, it is worthwhile to create these tasks using TTLP (Thinking Through a Lesson Protocol). Teachers may complain that it is overwhelming to plan a high-level task for every day, but the authors suggest that TTLP shouldn't be used every day at first. With the help of other teachers to develop more and more cognitive tasks, TTLP should be used periodically to where eventually a teacher will have a task for every day by accumulating them over time. Though planning a high-level lesson takes a lot of time using TTLP, it is possible to use only part of the TTLP method. The goal of using TTLP is to help teachers change the way they think about a lesson. By using TTLP, teachers will be able to further the use of a high-level task, so that students are able to learn more. Using TTLP will also enable a teacher to know how to respond to complications in the task, thus helping the task to go a lot smoother, and for the goals of the task to be met.
I think TTLP is a great way to create lesson plans that will promote cognitive thinking in students. Though there are a lot of things to think about when planning a lesson using TTLP, I think that the outcome is worthwhile. TTLP helps teachers to think about questions to ask students to further their thinking, and also helps the teacher think more about how students might respond to the task. When a teacher has already looked into where students might have trouble or how they can further a child's thinking if students are having a relatively easy time accomplishing the task, a teacher can then be more prepared for these cases. A teacher will already know how to respond. Knowing how to respond before something happen gives an advantage to the teacher, so that the teacher's task will go smoother, and will help the teacher and students stay focused on the goals of the task.
Thursday, March 18, 2010
There is More to it than Guess and Check
Guerrero, Shannon M. (March 2010). The Value of Guess and Check. Mathematics Teaching in the Middle School, 15, 393-398.
Main Idea/author’s thinking: Guess and check is a powerful problem-solving strategy that helps students make connects from conceptual understanding to algebraic representation. From the method Guerrero has described, students are first required to break down the word problem into it’s components by identifying what they know and what they are trying to find. The next step is to guess values for what they are trying to find and see if it fits the information given. If it doesn’t, then another guess is to be made based on the previous guess. They are to guess until they find an answer that fits all the information given. As they guess, they are to use a table format to organize the information given and the guesses they have been making. The final step is to use the guess and check method to make algebraic representations for the information given. Guerrero claims that it is easier once a student has already figured out the answer to then relate the information given to a symbolic representation. She also believes that through the process of guess and check, students will better learn how to decompose words problem. Thus, better understanding what each word problem is asking.
Though guess and check may be helpful in some situations, I think there are better methods that can be used to help students with solving word problems. Guess and check is helpful when you don’t know how to solve a problem, but it’s not very helpful when the problem is more complex and possibly has to do with fractions. Guessing the right fraction between 0 and 1 can be hard because there are an infinite amount of possibilities. The process of guess and check that Guerrero refers to may be helpful to some, but I think there are more concise ways of breaking down a word problem without have to use guess and check. I think organizing the information similar to the way Guerrero has suggested is a great way to break down a word problem, but I do not think it is necessary to use a guess and check system. That could be skipped so that the next step would be to represent the information given in an algebraic expression, and solve for the unknown(s). Through Guerrero says guess and check can be very effective when first learning how to deal with word problems, she admits that it is not very effective later on when dealing with more complex problems. I think that if you start to teach students through guess and check, they will not want to move on from that. Through that process, they get the answer through guessing first, what more is needed… There would not be much need for algebra after that in a child’s eyes. This also would lead to the problem that some students might start to think math is a game of guess and check. There are obviously better mathematical ways of solving a problem than guess and check.
Main Idea/author’s thinking: Guess and check is a powerful problem-solving strategy that helps students make connects from conceptual understanding to algebraic representation. From the method Guerrero has described, students are first required to break down the word problem into it’s components by identifying what they know and what they are trying to find. The next step is to guess values for what they are trying to find and see if it fits the information given. If it doesn’t, then another guess is to be made based on the previous guess. They are to guess until they find an answer that fits all the information given. As they guess, they are to use a table format to organize the information given and the guesses they have been making. The final step is to use the guess and check method to make algebraic representations for the information given. Guerrero claims that it is easier once a student has already figured out the answer to then relate the information given to a symbolic representation. She also believes that through the process of guess and check, students will better learn how to decompose words problem. Thus, better understanding what each word problem is asking.
Though guess and check may be helpful in some situations, I think there are better methods that can be used to help students with solving word problems. Guess and check is helpful when you don’t know how to solve a problem, but it’s not very helpful when the problem is more complex and possibly has to do with fractions. Guessing the right fraction between 0 and 1 can be hard because there are an infinite amount of possibilities. The process of guess and check that Guerrero refers to may be helpful to some, but I think there are more concise ways of breaking down a word problem without have to use guess and check. I think organizing the information similar to the way Guerrero has suggested is a great way to break down a word problem, but I do not think it is necessary to use a guess and check system. That could be skipped so that the next step would be to represent the information given in an algebraic expression, and solve for the unknown(s). Through Guerrero says guess and check can be very effective when first learning how to deal with word problems, she admits that it is not very effective later on when dealing with more complex problems. I think that if you start to teach students through guess and check, they will not want to move on from that. Through that process, they get the answer through guessing first, what more is needed… There would not be much need for algebra after that in a child’s eyes. This also would lead to the problem that some students might start to think math is a game of guess and check. There are obviously better mathematical ways of solving a problem than guess and check.
Tuesday, February 16, 2010
Teaching is Mind Bloggling
When you are trying to solve a problem, it is convenient to have a teacher tell you correct answers, but their are advantages to a teacher not giving his/her students answers at all. For example, in Warrington's paper, she noted that the students never complained that they didn't know how to do the task at hand and they were eager to take risks. That is because the children weren't bogged down with remembering logarithms, they were then free to discover procedures for themselves. Essentially they were able to construct knowledge freely, and on their own. Another advantage is, students are able to build new knowledge from what they already know. For example, the students used their knowledge of dividing whole numbers to learn how to divide fractions.
Though mathematical discourse based on Constructivism beliefs has a lot of advantages, it also has some disadvantages. It is possible that students could make up procedures that are incorrect, like what Benny did. While some students may be able to connect what they are learning to what they have already learned, others may struggle with determining exactly what they are learning. They may just think of the class discourse as a big class discussion where they just talk about theories. It may be hard for some students to know that their method of procuring an answer is sound.
Though mathematical discourse based on Constructivism beliefs has a lot of advantages, it also has some disadvantages. It is possible that students could make up procedures that are incorrect, like what Benny did. While some students may be able to connect what they are learning to what they have already learned, others may struggle with determining exactly what they are learning. They may just think of the class discourse as a big class discussion where they just talk about theories. It may be hard for some students to know that their method of procuring an answer is sound.
Tuesday, February 9, 2010
Constructing Knowledge; Not Just Memorization
Constructivism is knowledge "constructed" through our own experiences. We do not "acquire" knowledge, because it is not something that is simply given. Through experience, we form our own representation of the world by using our senses to interpret. As we experience, we continue to modify our view of the world. Knowledge is viable as long as our experiences continue to fit out observation.
As a teacher,I would require my students to do more that retrieve facts, they must be able to apply what they have learned to other situations. Being able to apply "knowledge" (or what they think they know) to other situations will show whether they have really learned something or not.They need to be able to do more than just retrieve a solution to a problem, they need to know the process by which they get that solution. This expectation can be used in mathematics as a result of constructivism, because it has to do with generating something new; like forming new knowledge. Answers are not just simply memorized; they are constructed. More specifically this is called operative knowledge, which is part of constructivism.
As a teacher,I would require my students to do more that retrieve facts, they must be able to apply what they have learned to other situations. Being able to apply "knowledge" (or what they think they know) to other situations will show whether they have really learned something or not.They need to be able to do more than just retrieve a solution to a problem, they need to know the process by which they get that solution. This expectation can be used in mathematics as a result of constructivism, because it has to do with generating something new; like forming new knowledge. Answers are not just simply memorized; they are constructed. More specifically this is called operative knowledge, which is part of constructivism.
Saturday, January 23, 2010
Loss of Connections
IPI (Individually Prescribed Instruction) Mathematics was to imporve pupils mathematics through self discovery, but after a few years of working in the system, it was found to be more detrimental than helpful. S.H. Erlwanger did a study on a students named Benny. Benny was at the top of his class and had been doing IPI for 4 years. As Erlwanger asked Benny about his understanding of factions and decimals, he found Benny had made his own rules for how they work. Benny, for years had been given worksheets with examples, working on them by hisself, and that was his only form or instruction. He worked at his own pace and took a test at the end of each section when he was ready. He had little teacher involvement, only enough to help him when he needed it. Through this, he made up rules that got him correct answers most of the time, but unfortunately he never was told precisly why he got anwswers wrong. So Benny only assumed that his awswers were right because there are many ways to write an anwer, and he simply just didn't put the right one. Since this had gone on for four years, Erlwanger tried to help Benny learn fractions and decimals correctly, but damage had been done. Benny still had problems removing the rules he had already formed. Thus, IPI did more to detriment Benny's learning than to help.
The reason Benny never correctly learned the properties of factions and decimals is because he could never connect them to eachother or to prior knowledge. This could be viewed as Benny recieving a bunch of information with no connection between them, so he tries to make his own connections. This is one thing, for teachers today that must avoided. Teachers need to facilitate learning for their students so they can learn and undestand the connections between information given. For instance, factions and decimals have a particular relation. They represent the same number in different forms. Benny never picked up on this relationship because, as Erlwanger's study suggests, the instruction Benny received emphasized getting the right answer and not the process involved. He made connections he did, so he could get correct answer. If teachers focus on helping students understand the process by which they get correct answers, then students would be able to build on prior knowledge and obtain new knowledge by making proper connections.
The reason Benny never correctly learned the properties of factions and decimals is because he could never connect them to eachother or to prior knowledge. This could be viewed as Benny recieving a bunch of information with no connection between them, so he tries to make his own connections. This is one thing, for teachers today that must avoided. Teachers need to facilitate learning for their students so they can learn and undestand the connections between information given. For instance, factions and decimals have a particular relation. They represent the same number in different forms. Benny never picked up on this relationship because, as Erlwanger's study suggests, the instruction Benny received emphasized getting the right answer and not the process involved. He made connections he did, so he could get correct answer. If teachers focus on helping students understand the process by which they get correct answers, then students would be able to build on prior knowledge and obtain new knowledge by making proper connections.
Thursday, January 14, 2010
Understanding, more than just a word
When does a child truly understand mathematical concepts? Does it have to do with gettign correct answers on homework assignments and tests, or is it more than that? Richard R. Skemp thinks that -- to say a student understands is ambiguous, because "to understand" can have two meanings. According to him, there is relational understanding, which most people are familar with, and there is instumental understanding. Relational understanding means that a person comprehends how to do something and why. Instumental understanding is knowing what to do, but not knowing why you do it. Thus, relational understanding encompasses instrumental understanding. That might be why instrumental understanding is found most common. It is the most easily taught considering the students need only to know how to solve a problem and not why. This means they only need to memorize formulas or rules without understanding why they work. Relational understanding on the other hand, may enable the students to understand why and how, but unfortunately it takes longer to teach. The overall advantage to teaching relational understanding over instrumental is that it enables students to adapt methods to various problems.
Tuesday, January 5, 2010
When is a student really learning?
1. What is mathematics?
It is a tool used to measure and give meaning to things in this world.
2. How do I learn mathematics best? Explain why you believe this.
I learn best if we go through and solve an example together while the teacher explains the meaning and reason for what we are doing; explaining any of the equations that would be used to solve it. By doing this I am able to put action with reason. After we have gone through a few examples, I need one or two more, so that I can go through in class on my own before we go over it together. Most importantly, when I’m given enough time, I am able to comprehend it and solve it myself. Then I am able to better solidify what I have learned.
3. How will my students learn mathematics best? Explain why you think this is true.
They will learn best when they understand what they are learning. Giving them a formula to use does not necessarily mean they understand why the formula works or exactly how to use it. When they understand the underlying concepts better, then the will be able to learn.
4. What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? Justify your reasoning.
Some teachers choose to lecture mathematics, others use a method of discussion, some use scaffolding to help them, and there are a few who use discovery learning. Lecture is when a teacher simply explains and instructs the class on the concepts to be learned. A discussion consists of an examination of the concepts. Scaffolding is when a teacher helps the students a lot in the beginning and then slowly removes his/her help to where the student does not rely on the teacher. Finally, discovery learning is one of the hardest methods to use. It requires a good amount of time to be given to the students to explore ideas about the concepts being presented, but it allows for the students to discover mathematics for themselves.
5. What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? Justify your reasoning.
Well, I know lecturing is one of them. Lecturing gets the information out there, but it does not allow for student feedback. I does not ensure in any way students are actually understanding what they are supposedly being taught. I also think that when there is no instruction from the teacher at all that some students will lose hope in learning and they will give up on it. Students do need a certain amount of guidance. As a teacher, you need balance. Not everyone learns in the same way, so you have to use different methods of teaching to compensate for this.
It is a tool used to measure and give meaning to things in this world.
2. How do I learn mathematics best? Explain why you believe this.
I learn best if we go through and solve an example together while the teacher explains the meaning and reason for what we are doing; explaining any of the equations that would be used to solve it. By doing this I am able to put action with reason. After we have gone through a few examples, I need one or two more, so that I can go through in class on my own before we go over it together. Most importantly, when I’m given enough time, I am able to comprehend it and solve it myself. Then I am able to better solidify what I have learned.
3. How will my students learn mathematics best? Explain why you think this is true.
They will learn best when they understand what they are learning. Giving them a formula to use does not necessarily mean they understand why the formula works or exactly how to use it. When they understand the underlying concepts better, then the will be able to learn.
4. What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? Justify your reasoning.
Some teachers choose to lecture mathematics, others use a method of discussion, some use scaffolding to help them, and there are a few who use discovery learning. Lecture is when a teacher simply explains and instructs the class on the concepts to be learned. A discussion consists of an examination of the concepts. Scaffolding is when a teacher helps the students a lot in the beginning and then slowly removes his/her help to where the student does not rely on the teacher. Finally, discovery learning is one of the hardest methods to use. It requires a good amount of time to be given to the students to explore ideas about the concepts being presented, but it allows for the students to discover mathematics for themselves.
5. What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? Justify your reasoning.
Well, I know lecturing is one of them. Lecturing gets the information out there, but it does not allow for student feedback. I does not ensure in any way students are actually understanding what they are supposedly being taught. I also think that when there is no instruction from the teacher at all that some students will lose hope in learning and they will give up on it. Students do need a certain amount of guidance. As a teacher, you need balance. Not everyone learns in the same way, so you have to use different methods of teaching to compensate for this.
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