Sweeney, E. S., & Quinn, R. J. (Jan. 2000). Concentration: Connecting fractions, decimals, and percents. Mathematics Teaching in the Middle School, 5(5), 324-328.
The authors emphasized the importance of connecting mathematical ideas; mainly they wanted students to understand the relationship between fractions, decimals and percents. They noticed that students' definitions of fractions, decimals and percents lacked detail and understanding. While it was clear that all the students had studied these concepts, the students were unable to make connections between them. As a result, the authors proposed the activity, concentration, that would help the students recognize the relationship between the terms. First this activity was done as a whole class with recognizing the shaded area of a circle in terms of a fraction, decimal and percent. The class then composed a table of the different numbers according to various shaded regions. Next, the class continued building the table while expanding their ideas to a 10 by 10 geoboard. At this stage the class broke into groups of 4 and composed equivalent but individual cards with fractions, decimals and percents. At this stage, the smaller groups then played a matching game with the cards. It was through the game that the students became more excitement in the subject matter, but also appeared to be making great progress with the connection of ideas as they quickly made matches.
While the authors nicely planed and carried out the task, the timing of the task did not seem to make sense. It was clear from the beginning that the students had already studied the ideas of fractions, decimals and percents, and they were revisiting the material to make the connection. As a result, it appeared the teacher really should have made the connection earlier when the class first studied the material. It is possible that the students' lack of understanding was impeding their progress in other areas, but if that was the case it was not clear in the article. Despite the timing, it was a well thought out task. The teacher successfully accomplished the goal of having students notice the similarities between the different ideas and deepen their understanding. Furthermore, the students were engaged in the task and began to show confidence in the material, which are good signs in the classroom that the students are learning. Overall, it seemed to be a beneficial task even if the timing did not coincide.
Monday, March 22, 2010
Tuesday, March 16, 2010
Blog Entry #6
McGatha, M. B., & Darcy, P. (Feb. 2010). Rubrics as formative assessment tools. Mathematics Teaching in the Middle School, 15(6), 328-336.
Rubrics help develop students’ understanding and support then as independent thinkers. First, the purpose of rubrics is to benefit the students. Depending on rubric’s requirements, it sends a message to the students about what is important. For example, the authors describe two different rubric approaches: holistic and analytic. Holistic rubrics describe qualities of performance as a whole, which emphasize the thinking processes and overall communication of mathematical ideas. Analytical rubrics focus of essential traits of the task, such as understanding the problem, planning a solution and getting an answer. As a result, depending on the type of rubric, the teacher sends a message to the students about what is important. Additionally, when students create their own rubrics for problems, they come to better understand expectations. The students are able to see what constitutes full credit, so they know how much detail is required and what aspects of a problem solving are important. Lastly, rubrics support students in becoming independent learners. From the rubric, they can notice areas of weakness and see for themselves what aspects of problem solving give them difficultly.
I agree that rubrics can be a great tool to help deepen your students’ understanding. From my experience, I have only seen rubrics in relation to writing, so it at first seemed weird to relate them to mathematics. However, as I thought about it, rubrics are a great way to provide feedback that lets your student know more than if they have the right answer or not. It allows your students to realize areas of strength and weakness, which helps them become a better problem solver. Additionally, I think that it is essential that your students understand what aspects of math are important, and rubrics are a clear way to pass along that message of importance. It will be hard for students to misunderstand the important aspects of a problem when they have a rubric to compare the problem with. Finally, since rubrics are versatile, they can apply to a variety of areas. You can always create a rubric that applies to specific problem or many times a more general rubric can apply. As a result, rubrics are not limited and can help students learn no matter what you are studying.
Rubrics help develop students’ understanding and support then as independent thinkers. First, the purpose of rubrics is to benefit the students. Depending on rubric’s requirements, it sends a message to the students about what is important. For example, the authors describe two different rubric approaches: holistic and analytic. Holistic rubrics describe qualities of performance as a whole, which emphasize the thinking processes and overall communication of mathematical ideas. Analytical rubrics focus of essential traits of the task, such as understanding the problem, planning a solution and getting an answer. As a result, depending on the type of rubric, the teacher sends a message to the students about what is important. Additionally, when students create their own rubrics for problems, they come to better understand expectations. The students are able to see what constitutes full credit, so they know how much detail is required and what aspects of a problem solving are important. Lastly, rubrics support students in becoming independent learners. From the rubric, they can notice areas of weakness and see for themselves what aspects of problem solving give them difficultly.
I agree that rubrics can be a great tool to help deepen your students’ understanding. From my experience, I have only seen rubrics in relation to writing, so it at first seemed weird to relate them to mathematics. However, as I thought about it, rubrics are a great way to provide feedback that lets your student know more than if they have the right answer or not. It allows your students to realize areas of strength and weakness, which helps them become a better problem solver. Additionally, I think that it is essential that your students understand what aspects of math are important, and rubrics are a clear way to pass along that message of importance. It will be hard for students to misunderstand the important aspects of a problem when they have a rubric to compare the problem with. Finally, since rubrics are versatile, they can apply to a variety of areas. You can always create a rubric that applies to specific problem or many times a more general rubric can apply. As a result, rubrics are not limited and can help students learn no matter what you are studying.
Monday, February 15, 2010
Blog Entry #5
The practice of teaching mathematics without telling students the procedures or even the right answers has many advantages. First, students’ interaction and exchange of ideas plays an essential role to the their acquirement of knowledge. The students are able to convince each other of their reasoning or why a certain answer is not feasible; additionally, they build off each others’ ideas in order to come up with the best solution. Secondly, the students are able to adapt and apply their knowledge. The students were able to use what they knew about whole number division and apply it to division of fractions. Furthermore, students modified or adapted their ideas during the class discussion. Next, students develop the skill of problem solving. When first figuring out division of fractions, they started by estimating, and then they used that knowledge in conjunction with their division of whole numbers knowledge in order to solve the problem. Lastly, students are taught to think of themselves. They come up with their procedures and methods for division of fractions, rather than relying on their teacher as the source for invert and multiply. Overall, this teaching method appears to have many advantages for the students.
However, there are also disadvantages associated with this teaching style. It takes longer for the class to go over the material than maybe a more traditional teaching style would. The students spent at least a week working with division of fractions; whereas, the invert and multiple rule could be taught easily in a day or two. As a result, it appears the students would not get through as much material in a year. Additionally, there is a lot of responsibility on the students. In Ms. Warrington’s class the students volunteered answers and participated in “intellectual bantering,” but if students did not participate or did not feel comfortable in the classroom, then this teaching style would be impossible. Therefore, you need willing students and a positive classroom in order for this style to be successful. Lastly, there is great responsibility on the teacher too. The teacher needs to know how the children are thinking, which will take a considerable amount of time and effort. Also, the teacher needs to know when to intervene in a discussion or just let the students struggle. If the teacher is not aware, then the students can easily make incorrect assumptions and get off the path of learning accurate mathematics. This teaching style does appear quite advantageous, but the disadvantages need to be considered too.
However, there are also disadvantages associated with this teaching style. It takes longer for the class to go over the material than maybe a more traditional teaching style would. The students spent at least a week working with division of fractions; whereas, the invert and multiple rule could be taught easily in a day or two. As a result, it appears the students would not get through as much material in a year. Additionally, there is a lot of responsibility on the students. In Ms. Warrington’s class the students volunteered answers and participated in “intellectual bantering,” but if students did not participate or did not feel comfortable in the classroom, then this teaching style would be impossible. Therefore, you need willing students and a positive classroom in order for this style to be successful. Lastly, there is great responsibility on the teacher too. The teacher needs to know how the children are thinking, which will take a considerable amount of time and effort. Also, the teacher needs to know when to intervene in a discussion or just let the students struggle. If the teacher is not aware, then the students can easily make incorrect assumptions and get off the path of learning accurate mathematics. This teaching style does appear quite advantageous, but the disadvantages need to be considered too.
Monday, February 8, 2010
Blog Entry #4
The term constructing knowledge comes from von Glaserfield’s opinion that we gain knowledge through our experiences; our current knowledge is the result of our past experiences. As we have new experiences, we are constantly adding to or adapting our knowledge. Therefore, our knowledge is a “theory” because it can never be complete, since we are constantly having new experiences that change it.
One implication of constructivism for teaching mathematics is that the teacher needs to ask questions in order to assess her students’ understanding. According to constructivism, everyone learns in a different way because we all have had different past experiences, and therefore, we modify or add to our individual experiences. As a result, it is necessary for the teacher to accurately assess the students’ learning. The way for teachers to do this is to ask questions in order to assess their individual understanding. This can be done by asking certain students to justify their answers or by asking similar questions. We learn from constructivism that even if students have the right answer, they might not understand the concept accurately. Therefore, teacher involvement is necessary and one way this can be done is through questioning the students and their understanding of the concept.
One implication of constructivism for teaching mathematics is that the teacher needs to ask questions in order to assess her students’ understanding. According to constructivism, everyone learns in a different way because we all have had different past experiences, and therefore, we modify or add to our individual experiences. As a result, it is necessary for the teacher to accurately assess the students’ learning. The way for teachers to do this is to ask questions in order to assess their individual understanding. This can be done by asking certain students to justify their answers or by asking similar questions. We learn from constructivism that even if students have the right answer, they might not understand the concept accurately. Therefore, teacher involvement is necessary and one way this can be done is through questioning the students and their understanding of the concept.
Friday, January 22, 2010
Blog Entry #3
Throughout the paper, Erlwanger explored the idea of independent study and showed how it is not a successful way of instruction. He investigated independent study through the system IPI, Independent Program Instruction, which consists of reading, practicing problems, taking quizzes and moving onto new material. Erlwanger’s particular test subject is one of IPI’s top students, Benny, whose mathematical schooling has been taught through this system. It is clear that the IPI system emphasizes Benny’s individualism and self-reliance, but IPI simply does not successfully teach Benny math. As Erlwanger talks with Benny, Erlwanger soon realizes that Benny’s perception of math is skewed. Benny over the course of years has formed many incorrect ideas about mathematical rules and procedures that he truly believes to be correct. Although there is a teacher, there is minimal interaction between Benny and his teacher because the teacher’s primary role is to be a grader that enforces the exact answers according to the key. Therefore, it becomes apparent that as a result of independent study, Benny has concluded many misconceptions and has actually suffered from not having a student-teacher relationship.
The idea of having a teacher-student relationship is applicable today. Without a teacher-student relationship, the student misses out on many opportunities for learning. The teacher plays a key role in the classroom by emphasizing certain ideas, directing discussions, preparing tasks, etc. As a result of independent study, the student simply does not have the chance to learn mathematics in a variety of ways that is possible with a teacher’s direction. While talking with other students about their favorite math classes, many accredited the liking to the teacher. Therefore, I am led to believe that the teacher has a very influential role in the classroom, and without a teacher-student relationship, the students’ success in mathematics will suffer such as in Benny’s case.
The idea of having a teacher-student relationship is applicable today. Without a teacher-student relationship, the student misses out on many opportunities for learning. The teacher plays a key role in the classroom by emphasizing certain ideas, directing discussions, preparing tasks, etc. As a result of independent study, the student simply does not have the chance to learn mathematics in a variety of ways that is possible with a teacher’s direction. While talking with other students about their favorite math classes, many accredited the liking to the teacher. Therefore, I am led to believe that the teacher has a very influential role in the classroom, and without a teacher-student relationship, the students’ success in mathematics will suffer such as in Benny’s case.
Wednesday, January 13, 2010
Blog Entry #2
According to Skemp, both relational and instrumental understanding can be beneficial or disadvantageous depending on the situation. Relational understanding is defined as knowing both the why and how behind a problem; whereas, instrumental understanding simply addresses the how and relies either on the teacher’s why/reasoning or does do even address the why. As a result, instrumental relies heavily upon memorization or learning of rules and procedures. These rules and procedures are also part of the relational understanding, but the student also learns the why or the reasoning behind using and formulating these rules and procedures. Therefore, relational understanding encompasses instrumental understanding. Consequently, Skemp favors relational understanding; however, he acknowledges there are benefits and downsides of each type of understanding. For example, since instrumental focuses on the rules and procedures it is more simplistic. However, at the same time it becomes complex quite quickly because the students need to memorize numerous rules; often this memorization results in students giving up on and disliking the subject. As a result, relational seems beneficial because less memory work is involved, so students usually have a more pleasurable experience and continue studying voluntarily. Additionally, relational understanding allows for adaptation and flexibility for the student to apply concepts to new problems. Not only may students apply concepts to new problems, but they may also search for and explore new areas. While relational understanding appears superior, there are times when relational understanding is too difficult for a topic or takes too long to achieve. As a result, the more basic instrumental understanding is preferable. Through instrumental understanding, students can more quickly get answers, which produce a greater sense of success and accomplishment. Overall, there are benefits and downsides to both relational and instrumental understanding and each type can be preferable depending on the situation.
Tuesday, January 5, 2010
Blog Entry #1
Mathematics is the study of numbers and how they relate. These relations are often applied to real life situations and can be expressed in a variety of ways.
I learn the best by being taught the concepts and subject matter first, and then doing practice problems by myself or with minimal assistance. After doing the practice problems, I then like to review the problems with classmates and receive help from the teacher if necessary. I think I learn mathematics best in this way because it is how I have been taught for a majority of my life, and it has worked for me. Also it combines a variety of methods, so I learned the material all the better.
I think my students will learn best in a similar manner. I recognize that students learn in different ways, so I will take that into consideration when I teach. However, I will be most comfortable teaching in a similar manner to how I have been taught and that way includes a variety of learning tactics, so it will probably be most successful in teaching mathematics.
One practice that promotes students’ learning is teaching with a positive attitude and being willing to answer students’ questions. If students feel their opinion and difficulties matter, then they will usually desire to learn. Additionally, I believe that using different tactics such as group work in addition to individual work provides students with the opportunity to learn in different ways, which helps students to better understand the subject matter.
Being rushed and using a negative tone is detrimental to the students’ learning of mathematics. These practices make the students feel the material is more important than the student, and as a result, the teacher does not care about the students and their needs.
I learn the best by being taught the concepts and subject matter first, and then doing practice problems by myself or with minimal assistance. After doing the practice problems, I then like to review the problems with classmates and receive help from the teacher if necessary. I think I learn mathematics best in this way because it is how I have been taught for a majority of my life, and it has worked for me. Also it combines a variety of methods, so I learned the material all the better.
I think my students will learn best in a similar manner. I recognize that students learn in different ways, so I will take that into consideration when I teach. However, I will be most comfortable teaching in a similar manner to how I have been taught and that way includes a variety of learning tactics, so it will probably be most successful in teaching mathematics.
One practice that promotes students’ learning is teaching with a positive attitude and being willing to answer students’ questions. If students feel their opinion and difficulties matter, then they will usually desire to learn. Additionally, I believe that using different tactics such as group work in addition to individual work provides students with the opportunity to learn in different ways, which helps students to better understand the subject matter.
Being rushed and using a negative tone is detrimental to the students’ learning of mathematics. These practices make the students feel the material is more important than the student, and as a result, the teacher does not care about the students and their needs.
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