Tuesday, April 27, 2010
Journal Article Review: Fractions and the funky cookie
Summary:
This article discussed the use of manipulatives, pattern blocks, to help students deepen their understanding of fractions. This was a 5th grade class who received extra help from a math specialist who came in to work with the students according to one agenda; however, their agenda changed as they saw the needs of the students. The specialist realized through the use of guiding questions as they worked with manipulatives that the students needed to understand the concept of fractional parts of a whole item must be of equal size. The teacher used pattern blocks to demonstrate with the use of the "funky cookie" which was a an irregular shape made up of different types of pattern blocks. The teacher asked the students to explain if and how they could share the "funky cookie" between 6 people. They were also asked to construct their own shape that could be shared between 6 people. Through this, the specialist understood that the students needed to understand that in order to share between 6 people, the parts had to be the same size in order to make it equal. The students were able to come to a full understanding of what equal parts of a whole unit meant.
Reflection:
I thought this article was great! Especially after working with pattern blocks in class today in our manipulative activity!! It was helpful to see how the teacher adapted her teaching plans/strategies to fit the students' needs as they arose. It is evident from this article that teachers need to anticipate where the students' thought process are going and how to guide them with the right questions that will be thought provoking and directional to guide them to what they need to know. It is also evident that teachers need to be fluent in their mathematical knowledge and need to have a depth of understanding in math that exceeds that of paper and pencil math so that the teacher can think of more than one strategy to problem solve.
This was a creative approach to help the students come to an understanding and was a great example of running with a teachable moment with the students. I think that this reinforces the benefits of incorporating manipulatives into lessons in order to help students grasp concepts that are difficult to understand. I liked this article a lot and appreciated the way the teacher was flexible in their plans to meet the needs of the students.
Ellington, A. Whitenack, Joy (2010). Fractions and the funky cookie.
Teaching Children Mathematics 16(9), 534-539.
This article discussed the use of manipulatives, pattern blocks, to help students deepen their understanding of fractions. This was a 5th grade class who received extra help from a math specialist who came in to work with the students according to one agenda; however, their agenda changed as they saw the needs of the students. The specialist realized through the use of guiding questions as they worked with manipulatives that the students needed to understand the concept of fractional parts of a whole item must be of equal size. The teacher used pattern blocks to demonstrate with the use of the "funky cookie" which was a an irregular shape made up of different types of pattern blocks. The teacher asked the students to explain if and how they could share the "funky cookie" between 6 people. They were also asked to construct their own shape that could be shared between 6 people. Through this, the specialist understood that the students needed to understand that in order to share between 6 people, the parts had to be the same size in order to make it equal. The students were able to come to a full understanding of what equal parts of a whole unit meant.
Reflection:
I thought this article was great! Especially after working with pattern blocks in class today in our manipulative activity!! It was helpful to see how the teacher adapted her teaching plans/strategies to fit the students' needs as they arose. It is evident from this article that teachers need to anticipate where the students' thought process are going and how to guide them with the right questions that will be thought provoking and directional to guide them to what they need to know. It is also evident that teachers need to be fluent in their mathematical knowledge and need to have a depth of understanding in math that exceeds that of paper and pencil math so that the teacher can think of more than one strategy to problem solve.
This was a creative approach to help the students come to an understanding and was a great example of running with a teachable moment with the students. I think that this reinforces the benefits of incorporating manipulatives into lessons in order to help students grasp concepts that are difficult to understand. I liked this article a lot and appreciated the way the teacher was flexible in their plans to meet the needs of the students.
Ellington, A. Whitenack, Joy (2010). Fractions and the funky cookie.
Teaching Children Mathematics 16(9), 534-539.
Manipulatives Blog
I have come to understand manipulatives more through this class. I cannot remember a time that I used manipulatives in math growing up in school. I am very enthusiastic about using manipulatives in my classroom because I feel that they assist the student in deepening their understanding of mathematical concepts and gives them a visual representation that will further their knowledge. I believe that what you hear, you'll forget; what you see, you'll remember; what you do, you learn. I think that all three of these steps are vital to a students' learning experience, but the latter is often forgotten to be made part of their curriculum.
1. How do you know students deepen their understanding while using manipulatives?
You know the students deepen their understanding while using manipulatives by their interactions and conversations while they are participating in the activity. As the students are working through manipulative activities, they should be talking out their processes and you can usually tell when a student is confused by the look on their faces. The students are more likely to have the "aha!" moment when using manipulatives because they can connect the mathematical ideas that they know in their mind with the experience and visual representation. Students understand more by doing what they are learning.
2. How do you know if the students can transfer their understanding from manipulatives to other situations?
You will be able to know if the students can transfer their understanding from manipulatives to other situations by asking them guiding questions where they will have to think through practical, real world application questions. They will then problem solve and communicate through discussion ways that they can take what they are learning by using the manipulatives and connect it with the real world.
3. How can you assess that understanding or growth?
You can assess their growth by requiring reflection to be a part of their activities. This way, the students will be incorporating the communication process standard and you can easily tell what their thought processes are about math. You can then assess their ability to utilize their mathematical language in an appropriate manner. Their understanding and growth can be assessed by setting up activities that are meaningful to their learning. A teacher can tell if the student is growing and learning based on their interactions with other students in working together using critical thinking skills to problem solve.
4. When students work in groups, how do you hold each youngster accountable for learning?
When students work in groups, I think that it is important that students have individual work to show that they were working and putting forth effort that is acceptable and contributes to their learning. I would have the students document what they are learning as they go through the different activities. I would also have the student not only write about what they are doing (methods), but what they are learning, and also what observations they make along the way. I would then have the students write a synthesis of their information in the form of a reflection that includes conclusions that they reached from their observations and also their thoughts about it.
5. When students work in groups, how do you assess each youngster's depth of understanding?
When students work in groups, their depth of understanding is critical as manipulatives should deepen their understanding, not provide a way to do simply hands on activities without any academic gain. Teachers should pay close attention as they are engaging in the activities and also take time to go around the room to ask the students questions that are thought provoking for the student which will cause them to give more than a surface/product answer. The student should be able to explain the process and make conjectures of how what they are doing relates not only within mathematics, but also interdisciplinary.
6. How are you improving students' problem solving skills with the manipulatives?
Students will need to know how to think critically in problem solving not just in math or in school, but as it relates to their daily lives and futures as professionals. Students learn to problem solve when working with manipulatives because they are activating their mind to think beyond what seems obvious. Students have to use manipulatives to solve problems and this causes the students to learn how to use many approaches and a variety of strategies to solve problems. There is usually more than one way to do things and this is true for problem solving. There is usually more than one way to solve a problem and some ways work better than others. Using manipulatives also helps students to refine their skills of determining the best strategy to problem solve and sometimes through guess and check/trial and error.
7. Why do they say not "hands - on" but "hands - on; minds - on"?
If activities are hands on only, students aren't learning to the depth at which their potential is. Activities need to be thought provoking and engaging of the mind as they work with their hands so that they can correlate knowledge/information to experience. A teacher should ensure that they are promoting equity within the education they are giving to the student in that they need to give the students solid instruction that give the student a firm foundation from which to build upon. If the students don't have good instruction and are given manipulatives, they will end up confused and won't get the max out of what they are supposed to be learning. They won't be able to make the connections and conclusions that are intended for them. Manipulatives must be coupled with strong teaching.
8. How do the process standards relate to the use of manipulatives?
Problem Solving:
Manipulatives incorporates the use of every process standard. Students exercise criticial thinking skills when using manipulatives in that they have to think beyond what is "expected or obvious" to come up with creative solutions. Students use a variety of problem solving strategies to come to a solution. Manipulatives also focuses more on the process rather than the product which is important when it comes to deepening mathematical understanding.
Communication
Students naturally use the process standard of communication when working with manipulatives/activities, especially when working with groups. As the students are encouraged to talk through their mathematical thought processes to reach a solution they are furthering their ability to communicate using the language of math. Their communication skills are sharpened as they engage in discussion and written reflection.
Reason and Proof
Students are asked to give rationale for their thinking when working with these activities. They should be talking through their processes, making conjectures and testing their predictions to see if they have understanding. Students should be working through problems that are meaningful and lend to explanation and justification of their work.
Representation
manipulatives are a great way to incorporate representation. Students can use manipulatives to represent data, mathematical equations, shapes, patterns, etc.
Connections
Students can make connections especially with the real world as they problem solve and work with manipulative activities. They are able to connect mathematical ideas together to see how mathematics is a coherent whole.
In class today, Monique and I did three different manipulatives which I thought were good. We did the snap blocks, unifix cubes and pattern blocks. I would have liked to experiment with the other manipulatives but we ran out of time. I really liked the snap blocks because they were so versatile and had many colors. It was easy to find activities that could be incorporated at every grade level with these. I also really liked the pattern blocks as there were a lot of things you could do with them. I thought with the pattern blocks; however, that it could be helpful to have many colors of the same shape (e.g. triangles would have yellow, green, orange, red, blue) so that if a student wanted to do representation of a pie graph for data analysis, it would be less confusing. I do understand that it is also important for them to be the same color. I also appreciated our discussion in class about what to do when students had deficiency in fine motor skills and thought that the suggestions were very good and insightful.
I really have enjoyed learning more about manipulatives in this class and I'm excited to purchase my own and implement them in the classroom.
1. How do you know students deepen their understanding while using manipulatives?
You know the students deepen their understanding while using manipulatives by their interactions and conversations while they are participating in the activity. As the students are working through manipulative activities, they should be talking out their processes and you can usually tell when a student is confused by the look on their faces. The students are more likely to have the "aha!" moment when using manipulatives because they can connect the mathematical ideas that they know in their mind with the experience and visual representation. Students understand more by doing what they are learning.
2. How do you know if the students can transfer their understanding from manipulatives to other situations?
You will be able to know if the students can transfer their understanding from manipulatives to other situations by asking them guiding questions where they will have to think through practical, real world application questions. They will then problem solve and communicate through discussion ways that they can take what they are learning by using the manipulatives and connect it with the real world.
3. How can you assess that understanding or growth?
You can assess their growth by requiring reflection to be a part of their activities. This way, the students will be incorporating the communication process standard and you can easily tell what their thought processes are about math. You can then assess their ability to utilize their mathematical language in an appropriate manner. Their understanding and growth can be assessed by setting up activities that are meaningful to their learning. A teacher can tell if the student is growing and learning based on their interactions with other students in working together using critical thinking skills to problem solve.
4. When students work in groups, how do you hold each youngster accountable for learning?
When students work in groups, I think that it is important that students have individual work to show that they were working and putting forth effort that is acceptable and contributes to their learning. I would have the students document what they are learning as they go through the different activities. I would also have the student not only write about what they are doing (methods), but what they are learning, and also what observations they make along the way. I would then have the students write a synthesis of their information in the form of a reflection that includes conclusions that they reached from their observations and also their thoughts about it.
5. When students work in groups, how do you assess each youngster's depth of understanding?
When students work in groups, their depth of understanding is critical as manipulatives should deepen their understanding, not provide a way to do simply hands on activities without any academic gain. Teachers should pay close attention as they are engaging in the activities and also take time to go around the room to ask the students questions that are thought provoking for the student which will cause them to give more than a surface/product answer. The student should be able to explain the process and make conjectures of how what they are doing relates not only within mathematics, but also interdisciplinary.
6. How are you improving students' problem solving skills with the manipulatives?
Students will need to know how to think critically in problem solving not just in math or in school, but as it relates to their daily lives and futures as professionals. Students learn to problem solve when working with manipulatives because they are activating their mind to think beyond what seems obvious. Students have to use manipulatives to solve problems and this causes the students to learn how to use many approaches and a variety of strategies to solve problems. There is usually more than one way to do things and this is true for problem solving. There is usually more than one way to solve a problem and some ways work better than others. Using manipulatives also helps students to refine their skills of determining the best strategy to problem solve and sometimes through guess and check/trial and error.
7. Why do they say not "hands - on" but "hands - on; minds - on"?
If activities are hands on only, students aren't learning to the depth at which their potential is. Activities need to be thought provoking and engaging of the mind as they work with their hands so that they can correlate knowledge/information to experience. A teacher should ensure that they are promoting equity within the education they are giving to the student in that they need to give the students solid instruction that give the student a firm foundation from which to build upon. If the students don't have good instruction and are given manipulatives, they will end up confused and won't get the max out of what they are supposed to be learning. They won't be able to make the connections and conclusions that are intended for them. Manipulatives must be coupled with strong teaching.
8. How do the process standards relate to the use of manipulatives?
Problem Solving:
Manipulatives incorporates the use of every process standard. Students exercise criticial thinking skills when using manipulatives in that they have to think beyond what is "expected or obvious" to come up with creative solutions. Students use a variety of problem solving strategies to come to a solution. Manipulatives also focuses more on the process rather than the product which is important when it comes to deepening mathematical understanding.
Communication
Students naturally use the process standard of communication when working with manipulatives/activities, especially when working with groups. As the students are encouraged to talk through their mathematical thought processes to reach a solution they are furthering their ability to communicate using the language of math. Their communication skills are sharpened as they engage in discussion and written reflection.
Reason and Proof
Students are asked to give rationale for their thinking when working with these activities. They should be talking through their processes, making conjectures and testing their predictions to see if they have understanding. Students should be working through problems that are meaningful and lend to explanation and justification of their work.
Representation
manipulatives are a great way to incorporate representation. Students can use manipulatives to represent data, mathematical equations, shapes, patterns, etc.
Connections
Students can make connections especially with the real world as they problem solve and work with manipulative activities. They are able to connect mathematical ideas together to see how mathematics is a coherent whole.
In class today, Monique and I did three different manipulatives which I thought were good. We did the snap blocks, unifix cubes and pattern blocks. I would have liked to experiment with the other manipulatives but we ran out of time. I really liked the snap blocks because they were so versatile and had many colors. It was easy to find activities that could be incorporated at every grade level with these. I also really liked the pattern blocks as there were a lot of things you could do with them. I thought with the pattern blocks; however, that it could be helpful to have many colors of the same shape (e.g. triangles would have yellow, green, orange, red, blue) so that if a student wanted to do representation of a pie graph for data analysis, it would be less confusing. I do understand that it is also important for them to be the same color. I also appreciated our discussion in class about what to do when students had deficiency in fine motor skills and thought that the suggestions were very good and insightful.
I really have enjoyed learning more about manipulatives in this class and I'm excited to purchase my own and implement them in the classroom.
Saturday, April 17, 2010
Thursday, April 15, 2010
Errors
I was glad to have done this activity in class this semester. It was so interesting to see students' work and their thought processes. I realized through this activity that I learned just like these students -- a set of rules and regulations without knowing why the rules worked. It is important to be able to see where students are messing up in their math work as a result of a wrong understanding about their mathematical processes.
I felt that this was a very helpful activity to prepare me to be an excellent educator because I have more of an understanding of how and why students think the way they do. It's amazing how learning math in the way that "it has always been done" gives a faulty foundation in math.
I really appreciated when we would work through a problem and then use manipulatives to show how the problem worked because it helped me to visualize and comprehend why rules that I have been taught worked.
I learned not to teach students shortcuts, use manipulatives as much as possible, and to show the processes, explain why they work, and the product and not just the product.
TECHNOLOGY BLOG:
I think I've used technology more in this class than in all of my other classes. I am so thankful to have had the opportunity to implement technology into my learning experience as a future educator. It is overwhelming when I think about how technologically advanced this next generation is whom I will be teaching. I am still nervous about using the smart board in the classroom, but before this class, I had never come in contact with one, so I feel like I am at a point where I am comfortable enough to play around with it and figure it out whereas before I wasn't at all. I feel like I've grown a lot in my understanding in how it works and also in understanding the importance of it in the classroom as it gets the students to actively engage in learning.
I've noticed that I have learned a lot about formatting Word documents because of the different requirements in our assignments in ETE 339 as well. I've never used a Google document before and the thought of it scared me a little, but I create them all of the time. I also use them outside of school in my organization for things such as sign up sheets and brainstorming ideas sheets. This tool is so handy to have in collaborating with others to accomplish a common task/goal.
I remember the calculators that I had in high school and I thought that they were technologically advanced and I graduated in 2000. As we looked at the calculators in class and went through the various activities yesterday, I was amazed that one could do so many things on a calculator and that they had become so versatile. I never would have dreamed that a calculator could set up a data table, find the lcd, gcd, etc.
These things goes to show that technology has come such a long way and should be implemented in the classroom. If you would have asked me in January if I was excited to use technology in the classroom, unfortunately, I would have responded the same as most teachers today that they didn't know enough about it to use it effectively. I think that at least now, though I'm not an expert in technology, have gotten enough exposure to it and the possibilities within the classroom that technology can be used that I am excited to try it out in my own class. I know that I will be learning as I go, but teachers are life long learners anyway. I am glad that I had the opportunity to experiment and practice with technology in this class. I feel that I've grown a lot in this area.
I felt that this was a very helpful activity to prepare me to be an excellent educator because I have more of an understanding of how and why students think the way they do. It's amazing how learning math in the way that "it has always been done" gives a faulty foundation in math.
I really appreciated when we would work through a problem and then use manipulatives to show how the problem worked because it helped me to visualize and comprehend why rules that I have been taught worked.
I learned not to teach students shortcuts, use manipulatives as much as possible, and to show the processes, explain why they work, and the product and not just the product.
TECHNOLOGY BLOG:
I think I've used technology more in this class than in all of my other classes. I am so thankful to have had the opportunity to implement technology into my learning experience as a future educator. It is overwhelming when I think about how technologically advanced this next generation is whom I will be teaching. I am still nervous about using the smart board in the classroom, but before this class, I had never come in contact with one, so I feel like I am at a point where I am comfortable enough to play around with it and figure it out whereas before I wasn't at all. I feel like I've grown a lot in my understanding in how it works and also in understanding the importance of it in the classroom as it gets the students to actively engage in learning.
I've noticed that I have learned a lot about formatting Word documents because of the different requirements in our assignments in ETE 339 as well. I've never used a Google document before and the thought of it scared me a little, but I create them all of the time. I also use them outside of school in my organization for things such as sign up sheets and brainstorming ideas sheets. This tool is so handy to have in collaborating with others to accomplish a common task/goal.
I remember the calculators that I had in high school and I thought that they were technologically advanced and I graduated in 2000. As we looked at the calculators in class and went through the various activities yesterday, I was amazed that one could do so many things on a calculator and that they had become so versatile. I never would have dreamed that a calculator could set up a data table, find the lcd, gcd, etc.
These things goes to show that technology has come such a long way and should be implemented in the classroom. If you would have asked me in January if I was excited to use technology in the classroom, unfortunately, I would have responded the same as most teachers today that they didn't know enough about it to use it effectively. I think that at least now, though I'm not an expert in technology, have gotten enough exposure to it and the possibilities within the classroom that technology can be used that I am excited to try it out in my own class. I know that I will be learning as I go, but teachers are life long learners anyway. I am glad that I had the opportunity to experiment and practice with technology in this class. I feel that I've grown a lot in this area.
Monday, April 12, 2010
Journal Article Review:
It is important that teachers are familiar with technology applications that are available to learners so that they can implement them within the classroom to enhance the learning of their students. The use of technology allows students to have feedback immediately so that they know what they understand and what they need to work on. It is also an interactive approach to learning and allows students to have a visual representation. These programs have the ability to be customized and can even have bilingual options which helps English Language Learners.
The article gives multiple examples of different uses of technology in a lesson. The outcomes of using technology in teaching are very positive. I appreciated this article because I find myself to be one of those people who doesn't like technology all that well. It is good for me to be able to see the benefits and different applications that are available for learning. I can see myself using this in my classroom.
The article gives multiple examples of different uses of technology in a lesson. The outcomes of using technology in teaching are very positive. I appreciated this article because I find myself to be one of those people who doesn't like technology all that well. It is good for me to be able to see the benefits and different applications that are available for learning. I can see myself using this in my classroom.
Journal Article Review: Supporting Language Learners
In this article, several techniques were discussed to become more effective in teaching English Language Learners. ELL's are becoming increasingly prevalent in the United States and teachers need to teach with awareness in order to provide an education that is equitable. The techniques that were discussed were: using advanced organizers, developing student's vocabulary, using visual cues, adjusting teacher talk, promoting low-anxiety classrooms, and using predictable routines. The authors of this journal also expressed the need for teachers to keep high expectations on these students because they will rise to the challenge and meet expectations.
I thought that this article was very helpful because it gave practical examples of how to help these students to get a quality education. All students have the right to learn, so it is our responsibility as teachers to ensure that students are able to engage within the classroom just like the other students. I will definitely take the information given in this article and keep it for future reference as it had excellent suggestions.
I thought that this article was very helpful because it gave practical examples of how to help these students to get a quality education. All students have the right to learn, so it is our responsibility as teachers to ensure that students are able to engage within the classroom just like the other students. I will definitely take the information given in this article and keep it for future reference as it had excellent suggestions.
Wednesday, March 24, 2010
Assessment Article Review ~ Using Learning Logs in Mathematics: Writing to Learn
This article was written to inform educators of the benefits of using learning logs/journals in the classroom. The author of the article stated that Learning logs function to assist students to reflect on what they are learning and learn while they are reflecting on what they are learning. As students use learning logs, they are actually keeping a running account of their understanding and thought processes in regards to math. This shouldn't be focused on grammar, but rather the content to ensure that the student is becoming proficient in their learning. This is also a great way for educators to track what the student is learning, what areas in which they need extra help, and to see how the lesson went from that day.
Some teachers mentioned their reasons for not using learning logs were class time and teacher time. They expressed the issue of it taking so much time out of the day that they would rather use an assessment method that is more direct and that takes less time. The article mentions that the students could be given a time limit of 10-15 minutes depending on how long the students need, and it would take about 5 minutes per journal to read through and make comments. It is very important that teachers make comments that are full of positive reinforcement because it helps the students to attain a positive disposition towards math. This positive feedback from the educator also empowers the student and makes them want to strive for further excellence.
Reflection:
I think that this could be a great way to track how the students are processing the content being learned. It is also a great way to keep them engaged in mathematical reasoning and communication which is key to understanding math. I was thinking that it would be neat to incorporate technology by having the students develop an online blog for their journals. I don't think that this should be the only assessment technique used as it is not a complete and thorough type of assessment. It should rather be used in conjunction with another type to ensure both quality and quantity in assessment.
McIntosh, M., Draper, R. (2001). Using Learning Logs in Mathematics: Writing to Learn. Mathematics Teacher 94(7), 554-555
Some teachers mentioned their reasons for not using learning logs were class time and teacher time. They expressed the issue of it taking so much time out of the day that they would rather use an assessment method that is more direct and that takes less time. The article mentions that the students could be given a time limit of 10-15 minutes depending on how long the students need, and it would take about 5 minutes per journal to read through and make comments. It is very important that teachers make comments that are full of positive reinforcement because it helps the students to attain a positive disposition towards math. This positive feedback from the educator also empowers the student and makes them want to strive for further excellence.
Reflection:
I think that this could be a great way to track how the students are processing the content being learned. It is also a great way to keep them engaged in mathematical reasoning and communication which is key to understanding math. I was thinking that it would be neat to incorporate technology by having the students develop an online blog for their journals. I don't think that this should be the only assessment technique used as it is not a complete and thorough type of assessment. It should rather be used in conjunction with another type to ensure both quality and quantity in assessment.
McIntosh, M., Draper, R. (2001). Using Learning Logs in Mathematics: Writing to Learn. Mathematics Teacher 94(7), 554-555
Sunday, March 21, 2010
Journal Article Review: Identifying Logical Necessity
This article explains the increasing importance of teachers being able to make sound judgments in regards to the logic that is often used in mathematical arguments. Logical Necessity is defined as the condition for which conclusions follow necessarily from premises. Throughout the article, examples of student work is given. These examples include problems from which individual's have to determine the correct solution based on the correct logic. They are to give their own logic and then they also look at students' logic to determine the correct response.
It is shown that through feedback from the instructor concerning specific ways the student can improve, the individual will improve and correct logic will start to become more natural. It is vital that the instructor not only show the student where they are wrong, but must give them direction in how to improve through proof and reasoning. Logical Necessity should not only be taught in the elementary school, but in other subjects as well in order to more thoroughly equip themas they progress through other grades.
According to Stylianides's conception of proof, it is a mathematical argument, a connected sequence of assertions for or against a mathematical claim that follows: statements, argumentation, and representation.
It is proven that teachers who exercise logical necessity in their reasoning and proof will also be able to instill this skill in their students. It is important to teach learners how to reason answers through logic and proof; not just what the answers are.
It is shown that through feedback from the instructor concerning specific ways the student can improve, the individual will improve and correct logic will start to become more natural. It is vital that the instructor not only show the student where they are wrong, but must give them direction in how to improve through proof and reasoning. Logical Necessity should not only be taught in the elementary school, but in other subjects as well in order to more thoroughly equip themas they progress through other grades.
According to Stylianides's conception of proof, it is a mathematical argument, a connected sequence of assertions for or against a mathematical claim that follows: statements, argumentation, and representation.
It is proven that teachers who exercise logical necessity in their reasoning and proof will also be able to instill this skill in their students. It is important to teach learners how to reason answers through logic and proof; not just what the answers are.
Journal Article Review: The Value of Guess and Check
This was an interesting article that talked about the Guess and Check method. This method has proved to be advantageous for students in the middle grades; especially as they work through word problems. Many times, students give up and get frustrated when it comes to word problems because they don't actually understand what is being said because they don't have a working knowledge of mathematical language. These students end up following a set of guidelines and formulas instead of comprehending mathematical application and critical thinking.
This article suggests that students use guess and check which will enable the student to develop a conceptual strategy to make sense of word problems and to find the appropriate solution, pattern, or equation. Students start out with a series of questions that guide them through the process. Through this process, students develop a strategy and skill that is useful to them in all facets of life. This allows learners to go from an objective view of mathematics to more of a subjective approach as they delve into symbolic representation. Learners must not only know how to use this approach, but know when it is the most useful tool and when it is not.
Guess and Check is an incredible tool for students to grasp because it brings a direct correlation between understanding concepts and symbolic representation of their knowledge. It is also a way of not only finding solutions to problems, but rather getting into the habit of exercising critical thinking within math to create mathematical equations.
This article suggests that students use guess and check which will enable the student to develop a conceptual strategy to make sense of word problems and to find the appropriate solution, pattern, or equation. Students start out with a series of questions that guide them through the process. Through this process, students develop a strategy and skill that is useful to them in all facets of life. This allows learners to go from an objective view of mathematics to more of a subjective approach as they delve into symbolic representation. Learners must not only know how to use this approach, but know when it is the most useful tool and when it is not.
Guess and Check is an incredible tool for students to grasp because it brings a direct correlation between understanding concepts and symbolic representation of their knowledge. It is also a way of not only finding solutions to problems, but rather getting into the habit of exercising critical thinking within math to create mathematical equations.
Wednesday, March 3, 2010
Video Blog #2
Lesson Analysis 1: Identify several strategies the teacher used to orchestrate and promote student discourse in this lesson.
The teacher would get down on the students’ level and ask them guiding questions that made them think critically through their own problems. This got them thinking together to come up with their answer. It also helped them talk through how they were getting their answers through justification. The teacher also gave other examples while talking with the students that helped them to make deeper mathematical connections within their groups to fully understand their own math problems.
Lesson Analysis 1: Provide 3 examples of evidence that students have learned the mathematics being taught.
The teacher would get down on the students’ level and ask them guiding questions that made them think critically through their own problems. This got them thinking together to come up with their answer. It also helped them talk through how they were getting their answers through justification. The teacher also gave other examples while talking with the students that helped them to make deeper mathematical connections within their groups to fully understand their own math problems.
1. One of the students was able to present her group’s work with excellent articulation of not just the answer, but how the answer correlates to the graph. She student was able to make accurate connections and patterns from the graph.
2. As the teacher asked the students how they got their answer, they were able to tell the sequence they generated and also how they got the equation. They were able to justify their answers.
3. The students were actively engaging in conversation with the teacher about their mathematical reasoning. They were using critical thinking to process the mathematical equations.
Reflective Task 1: Describe the student-teacher interactions during the task debriefing discussions and assess the effectiveness of these interactions.
The teacher asks the students questions that prompt the students to clarify what they have said. The instructor allows the students to “teach” her their thought processes and when she gives them verbal clues that she understands where they are coming from, she gives them different angles to think about. I think this is a very effective technique because it helps the students to think through their thought process and fine tune their strategies. It also challenges the students to think deeper and gives them more ways they could approach the problem.
The teacher would get down on the students’ level and ask them guiding questions that made them think critically through their own problems. This got them thinking together to come up with their answer. It also helped them talk through how they were getting their answers through justification. The teacher also gave other examples while talking with the students that helped them to make deeper mathematical connections within their groups to fully understand their own math problems.
Lesson Analysis 1: Provide 3 examples of evidence that students have learned the mathematics being taught.
The teacher would get down on the students’ level and ask them guiding questions that made them think critically through their own problems. This got them thinking together to come up with their answer. It also helped them talk through how they were getting their answers through justification. The teacher also gave other examples while talking with the students that helped them to make deeper mathematical connections within their groups to fully understand their own math problems.
1. One of the students was able to present her group’s work with excellent articulation of not just the answer, but how the answer correlates to the graph. She student was able to make accurate connections and patterns from the graph.
2. As the teacher asked the students how they got their answer, they were able to tell the sequence they generated and also how they got the equation. They were able to justify their answers.
3. The students were actively engaging in conversation with the teacher about their mathematical reasoning. They were using critical thinking to process the mathematical equations.
Reflective Task 1: Describe the student-teacher interactions during the task debriefing discussions and assess the effectiveness of these interactions.
The teacher asks the students questions that prompt the students to clarify what they have said. The instructor allows the students to “teach” her their thought processes and when she gives them verbal clues that she understands where they are coming from, she gives them different angles to think about. I think this is a very effective technique because it helps the students to think through their thought process and fine tune their strategies. It also challenges the students to think deeper and gives them more ways they could approach the problem.
Sunday, February 14, 2010
Turtle Pond - Applet Review
Get the Turtle to the Pond
K-2
http://illuminations.nctm.org/LessonDetail.aspx?id=L396
Summary: This is an application geared for grades K-2. This is a simple game developed to help students apply measurement and basic geometry knowledge. The students are given directional buttons which include moving forwards, backwards, rotating the turtle 90 degrees left or right. The students get to choose how many units to move their turtle. They have to get the turtle to the pond, and then the turtle splashes in the pond. The students could also document their paths on paper by drawing the turtle and his journey to the pond and writing our the steps including the measurement units. The students will be able to write directional words including right, left, forwards, backwards. They will also be able to use the applet to create an appropriate path to the pond.
Critique:
I think this is a really great applet for K-2. It is very age appropriate with a good amount of challenge. This has the strength of letting learners have interaction with what they are learning. They can also determine the level of difficulty by pushing different buttons that put barriers in the way and also that take the grid off of the page. This could be difficult for students who are directionally challenged. This would be a good use for students as an extension to what they are already learning in class.
K-2
http://illuminations.nctm.org/LessonDetail.aspx?id=L396
Summary: This is an application geared for grades K-2. This is a simple game developed to help students apply measurement and basic geometry knowledge. The students are given directional buttons which include moving forwards, backwards, rotating the turtle 90 degrees left or right. The students get to choose how many units to move their turtle. They have to get the turtle to the pond, and then the turtle splashes in the pond. The students could also document their paths on paper by drawing the turtle and his journey to the pond and writing our the steps including the measurement units. The students will be able to write directional words including right, left, forwards, backwards. They will also be able to use the applet to create an appropriate path to the pond.
Critique:
I think this is a really great applet for K-2. It is very age appropriate with a good amount of challenge. This has the strength of letting learners have interaction with what they are learning. They can also determine the level of difficulty by pushing different buttons that put barriers in the way and also that take the grid off of the page. This could be difficult for students who are directionally challenged. This would be a good use for students as an extension to what they are already learning in class.
Pan Balance - Applet Review
Pan Balance - Numbers
3-5
http://illuminations.nctm.org/ActivityDetail.aspx?ID=26
Summary:
This is an application that is geared towards grades 3-5 dealing with number operations. In the applet, the students are given a balance with one pan on each side of the balance. A simple problem is entered into the red pan such as 7+3 and it will show the sum of 10. Then in the blue pan the student should enter something that will be it's equivalent such as 5x2 and the answer 10 will appear. The problems with solutions will then appear on the right hand column called the balanced equations area because it was equivalent. If a student does not enter problems that yield the same answer, the equation with the higher number will appear heavier causing the balance to tilt one side; then it will not go into the balanced equations area. Students will be able to type mathematical equations that are equivalent. Students will also be able to recognize the difference between equivalent and non equivalent equations.
Critique:
I think that this application would be an effective use in the classroom to reinforce lessons done in class about equivalent equations. This application could be used for one person or as a two person game. This is good for those who are fluent in numbers because they don't give you any numbers to start of. The student enters their own equations without any prompting from the applet. The weakness of this is that students who need help with numbers and operations may not know where to begin, so extra help would be needed. This could be a good tool for those who are gifted/talented because it would give an extra challenge.
Critique:
3-5
http://illuminations.nctm.org/ActivityDetail.aspx?ID=26
Summary:
This is an application that is geared towards grades 3-5 dealing with number operations. In the applet, the students are given a balance with one pan on each side of the balance. A simple problem is entered into the red pan such as 7+3 and it will show the sum of 10. Then in the blue pan the student should enter something that will be it's equivalent such as 5x2 and the answer 10 will appear. The problems with solutions will then appear on the right hand column called the balanced equations area because it was equivalent. If a student does not enter problems that yield the same answer, the equation with the higher number will appear heavier causing the balance to tilt one side; then it will not go into the balanced equations area. Students will be able to type mathematical equations that are equivalent. Students will also be able to recognize the difference between equivalent and non equivalent equations.
Critique:
I think that this application would be an effective use in the classroom to reinforce lessons done in class about equivalent equations. This application could be used for one person or as a two person game. This is good for those who are fluent in numbers because they don't give you any numbers to start of. The student enters their own equations without any prompting from the applet. The weakness of this is that students who need help with numbers and operations may not know where to begin, so extra help would be needed. This could be a good tool for those who are gifted/talented because it would give an extra challenge.
Critique:
Wednesday, February 10, 2010
Article Review: Techniques for small-group DISCOURSE
I chose the article, Techniques for small-group DISCOURSE, taken from the Teaching Children Mathematics Journal. This article featured the importance of reasoning and critical thinking in mathematics through group discourse or discussion. Too many of students develop a negative disposition because of the way instructors teach mathematics. In order for mathematical instruction to be effective, the teacher must apply various methods which include: critical thinking, reasoning, open communication, question asking, making conjectures, constructing and assessing mathematical arguments.
According to this article, there are many things a teacher can do to keep the students actively engaged in learning math. The teacher's effectiveness in group discourse can either make or break a student's disposition towards math. One of the techniques teachers can use is asking questions. Teachers need to master the art of question asking. They need to ask thought-provoking, guided questions that will yield critical thinking, and the sharing of learners' own ideas as well as listening to others.
The authors of this article took the reader through some examples of students engaging in small group discourse. The small group discussions were then evaluated to come up with techniques that work and to assess things that the teacher did that didn't work so well.
It was found that when teachers require the students to give explanations for their work, students become more fluent in mathematical ideas and are more open to other ways of justifying the answer. Students should be redirected from just giving a final answer and encouraged to think through the process. It would be advantageous for the learner to consider multiple ways of arriving at a solution instead of having one particular way of doing things in mind and shutting out other creative approaches.
I think that this approach to instruction within mathematics is a very useful tool. I believe that students would learn better by internalizing through critical thinking and justification.
Kilic, H., Cross, D., Ersoz, F., Mewborn, D., Swanagan, D. and Kim, J. (2010). Techniques for small-group Discourse. Teaching Children Mathematics 16 (6), 350-357
According to this article, there are many things a teacher can do to keep the students actively engaged in learning math. The teacher's effectiveness in group discourse can either make or break a student's disposition towards math. One of the techniques teachers can use is asking questions. Teachers need to master the art of question asking. They need to ask thought-provoking, guided questions that will yield critical thinking, and the sharing of learners' own ideas as well as listening to others.
The authors of this article took the reader through some examples of students engaging in small group discourse. The small group discussions were then evaluated to come up with techniques that work and to assess things that the teacher did that didn't work so well.
It was found that when teachers require the students to give explanations for their work, students become more fluent in mathematical ideas and are more open to other ways of justifying the answer. Students should be redirected from just giving a final answer and encouraged to think through the process. It would be advantageous for the learner to consider multiple ways of arriving at a solution instead of having one particular way of doing things in mind and shutting out other creative approaches.
I think that this approach to instruction within mathematics is a very useful tool. I believe that students would learn better by internalizing through critical thinking and justification.
Kilic, H., Cross, D., Ersoz, F., Mewborn, D., Swanagan, D. and Kim, J. (2010). Techniques for small-group Discourse. Teaching Children Mathematics 16 (6), 350-357
Article Review: Calculus in the Middle School?
I read the article entitled Calculus in the Middle School? found in the Mathematics Teaching In the Middle School journal. The authors of this article were presenting the idea of incorporating calculus earlier in a learner's education. Just as Algebra is being taught at earlier academic levels, so should calculus. It has been proven successful to teach algebra as a way of thinking to elementary students as a way of preparing them for Middle School. Just as Algebra is viewed as an introductory course to ready young learner's minds for higher level math, Calculus is viewed in the same way. Calculus is used in High School to prepare individuals for College level math. It is also used to weed out those who would not be successful for particular fields of study based on their performance in these types of math courses.
The article presents two main ideas in Calculus that should be taught in Middle School math called differentiation and derivative. They suggest that calculus needs one strong basis of competency in order to be strong in these classes. That baseline content area is the study of mathematical change. If students understood mathematical change, they would be set up for success upon entering these calculus classes.
In the article, the authors pose a hypothetical situation in a classroom where the teacher introduces calculus to the students without their knowing. The students work the problem step by step together while activating mathematical discourse and reasoning. Through their discourse, they come to a working knowledge of each step until they complete the last step of the calculus problem that is found in a calculus textbook. The teacher then tells the class what they have done.
REFLECTION:
I think that this is a good idea; however, since the article uses a pretend scenario, it's hard for me to imagine how the classroom discussion would actually occur. I could see this being very strategic and effective, but I would be interested to see how the students respond in a real life setting. I think that the article poses an excellent point and idea which should be entertained.
Barger, R., McCoy, A. (2010). Calculus in the Middle School? Mathematics Teaching in the Middle School 15(6), 348-353
The article presents two main ideas in Calculus that should be taught in Middle School math called differentiation and derivative. They suggest that calculus needs one strong basis of competency in order to be strong in these classes. That baseline content area is the study of mathematical change. If students understood mathematical change, they would be set up for success upon entering these calculus classes.
In the article, the authors pose a hypothetical situation in a classroom where the teacher introduces calculus to the students without their knowing. The students work the problem step by step together while activating mathematical discourse and reasoning. Through their discourse, they come to a working knowledge of each step until they complete the last step of the calculus problem that is found in a calculus textbook. The teacher then tells the class what they have done.
REFLECTION:
I think that this is a good idea; however, since the article uses a pretend scenario, it's hard for me to imagine how the classroom discussion would actually occur. I could see this being very strategic and effective, but I would be interested to see how the students respond in a real life setting. I think that the article poses an excellent point and idea which should be entertained.
Barger, R., McCoy, A. (2010). Calculus in the Middle School? Mathematics Teaching in the Middle School 15(6), 348-353
Sunday, February 7, 2010
Wednesday, February 3, 2010
PBL Article Review
I chose the article, How to buy a car 101, in which a 7th grade class does a PBL on buying cars. They are given a set of guidelinesfor the project and parameters for the type of car needed. The students are given 2 weeks to do research by looking at dealerships and cars online, contacting banks and dealerships, etc. The students had an opportunity to take mathematics and get their hands dirty in a real life situation in which they could apply what they had learned and learn new ideas and concepts.
I thought that the teacher who put this PBL together did an excellent job of adhering to the content standards that were set forth. He/she also did a great job of making the problem both interesting and applicable to the students as they are at the age of dreaming about driving and owning their very own car. I liked how the teacher set the boundaries and guidelines for the learners, had an example of the final product, and let the students take control of the rest. The article emphasized the importance of letting the learners take over in the classroom in taking initiative.
In learning through PBL, students get a better understanding of the material they are learning because they are applying it to their everyday lives or real life situations.
Flores, C. (2006). How to buy a car 101. The National Council of Teachers of Mathematics. 12(3), 161-164.
I thought that the teacher who put this PBL together did an excellent job of adhering to the content standards that were set forth. He/she also did a great job of making the problem both interesting and applicable to the students as they are at the age of dreaming about driving and owning their very own car. I liked how the teacher set the boundaries and guidelines for the learners, had an example of the final product, and let the students take control of the rest. The article emphasized the importance of letting the learners take over in the classroom in taking initiative.
In learning through PBL, students get a better understanding of the material they are learning because they are applying it to their everyday lives or real life situations.
Flores, C. (2006). How to buy a car 101. The National Council of Teachers of Mathematics. 12(3), 161-164.
Problem-Based Learning Review
After much study and review, I have found that problem-based learning is an instructional method that helps teachers to adhere to the standards. PBL is also heavily concentrated on real life situations and problems that are interesting and relevant to the student. Problem based learning is student centered and allows students to take a problem and work with it making necessary changes or additions as they see fit. There is no particular outcome or one right answer. The goal of this type of instruction is to have many different possible outcomes so that students can grapple with mathematics methods.
Problem Based Learning has a lot of benefits. This technique of instruction includes integrating knowledge not just in mathematics but in other situations which allows the learner to think critically through reasoning. Through using PBL, students feel equipped to take on tasks that may be a challenge. This also allows students to think through the small practical steps leading up to the solution.
Problem Based Learning has a lot of benefits. This technique of instruction includes integrating knowledge not just in mathematics but in other situations which allows the learner to think critically through reasoning. Through using PBL, students feel equipped to take on tasks that may be a challenge. This also allows students to think through the small practical steps leading up to the solution.
Friday, January 29, 2010
Article Review: The answer is 20 Cookies. What is the question?
In the article, The answer is 20 cookies. What is the question?, a new approach to problem solving was presented to teachers from different classrooms. They were then given the assignment to try it in their classrooms to see what would happen. This approach uses the working backwards strategy where the answer is given and you have to determine ways of getting the answer by creating the problem. The teachers used this technique with their students and stood amazed. For instance, one student was given the answer 245. That problem that the student constructed was, “There are 245 red cars are in a parking lot. 200 blow up how didn’t blow up? (Barlow and Cates, 2007)
The teachers had a carefully selected math problem that would yield critical mathematical reasoning and the use of prior knowledge. Because the teachers had selected a problem that would have a wide variety of solutions, it acted as a prompt to get the students to think creatively and to think of new ways to incorporate information they already knew. They were enabled as learners to take existing knowledge and come up with a problem that was innovative on their own. (Barlow and Cates, 2007) This is supported by the standard in that teachers are encouraged to promote problem solving by compiling innovative information and prior knowledge. They do this by selecting math problems carefully that will help their students think outside the box.
Some of the students when given the answer 20 cookies wrote down “How many cookies?” Others would give the answer 10+10. (Barlow and Cates, 2007) Although they were giving an answer, they needed more explanation in order to determine the appropriate solution to the problem. In the article, teachers found that sometimes there is a need for further instruction and guidance and the students can alter problem solving strategies and methods. They could then relate to the answer given to come up with a creative and well thought out problem. The problem solving standard supports this principle of being flexible to changing strategies because different problems will need different strategies and the students need to be able to know how to differentiate between the strategies in that way. The students also need to be open to reconfiguration in their problems because there could be a better way of solving.
In the article, the students had to come up with the problem, as the answer was already given. They were encouraged to be creative in their thinking in order to come up with a problem that would be effective. Their problems were about real life situations which drew correlations between math and real life which caused them to think of how to problem solve in other situations. (Barlow and Cates, 2007) The problem solving standard reinforces this idea in that students should be able to come up with valuable solutions to problems that they will face not only in math, but in real life situations.
Barlow, A. T. and Cates, J. M. (2007). The answer is 20 cookies. What is the question? Teaching Children Mathematics 13(5), 252-255.
The teachers had a carefully selected math problem that would yield critical mathematical reasoning and the use of prior knowledge. Because the teachers had selected a problem that would have a wide variety of solutions, it acted as a prompt to get the students to think creatively and to think of new ways to incorporate information they already knew. They were enabled as learners to take existing knowledge and come up with a problem that was innovative on their own. (Barlow and Cates, 2007) This is supported by the standard in that teachers are encouraged to promote problem solving by compiling innovative information and prior knowledge. They do this by selecting math problems carefully that will help their students think outside the box.
Some of the students when given the answer 20 cookies wrote down “How many cookies?” Others would give the answer 10+10. (Barlow and Cates, 2007) Although they were giving an answer, they needed more explanation in order to determine the appropriate solution to the problem. In the article, teachers found that sometimes there is a need for further instruction and guidance and the students can alter problem solving strategies and methods. They could then relate to the answer given to come up with a creative and well thought out problem. The problem solving standard supports this principle of being flexible to changing strategies because different problems will need different strategies and the students need to be able to know how to differentiate between the strategies in that way. The students also need to be open to reconfiguration in their problems because there could be a better way of solving.
In the article, the students had to come up with the problem, as the answer was already given. They were encouraged to be creative in their thinking in order to come up with a problem that would be effective. Their problems were about real life situations which drew correlations between math and real life which caused them to think of how to problem solve in other situations. (Barlow and Cates, 2007) The problem solving standard reinforces this idea in that students should be able to come up with valuable solutions to problems that they will face not only in math, but in real life situations.
Barlow, A. T. and Cates, J. M. (2007). The answer is 20 cookies. What is the question? Teaching Children Mathematics 13(5), 252-255.
Problem Solving
Problem Solving is used anytime the means for attaining an appropriate solution is unknown. Problem Solving enables students to glean from prior knowledge, and wrestle with problems to find the correct conclusions. The great thing about problem solving is that it helps students understand more mathematical ideas and concepts, but it also helps students grasp ways of processing to problem solve in real life situations. Problem solving is a huge part in math and it should penetrate every area.
Compile Innovative Information through Problem Solving
It is imperative that teachers choose wisely the problems they present their learners. These problems should be worthwhile and be relevant to their lives in order to introduce new mathematical concepts. Students can engage in math that is close to home in that what they are learning is useful to them. Students will grasp new concepts with less difficulty if they can draw correlations to their own experience.
Create effective solutions to problems that come up in math and other situations
Those who excell at problem solving tend to bring situations under scrutiny and there pose problems based on their observations. Assumptions can be made by looking at a problem with little investigation, but the further one delves into the problem the more accurate the outcome will be. Educators have the responsibility of cultivating an environment within the classroom in which students are free and encouraged to create problems that align with their own lives to solve. They should be exhorted to investigate, discover new insights, try new things, and take risks in the classroom.
Relate and adjust many effective problem solving strategies and methods
There are many different problem solving strategies which include: working backwards, guess and check, the use of diagrams and patterns, etc. Instructors should be teaching these strategies to their students along with other problem solving methods. They should assign tasks in class that use a variety of strategies so that students can compare different ways to solve a problem and thereby the students become fluent in using them by the time they reach the higher grades. Different types of problems are going to require different types of problem solving methods. Classroom discussion about problem solving is going to make this a norm in the classroom and help students to be able to internalize what they have practiced and observed.
Observe the process of problem solving and be able to draw personal insight and application
Effective problem solvers are continually engaged in observing their actions and processes, making changes that they see that are necessary, read and reread, and ask questions. They also keep track of their progress and are open to change because there may be a better way of doing things. When the teacher asks probing questions to make sure his or her students understand concepts, and that the learners have gone into much depth in their thinking, they create an atmosphere where students feel free to assess their own work and thinking. It is very important that learners lay hold of reflective practice. By doing this, they will become more efficient and effective at problem solving not just in math, but in their own lives.
Compile Innovative Information through Problem Solving
It is imperative that teachers choose wisely the problems they present their learners. These problems should be worthwhile and be relevant to their lives in order to introduce new mathematical concepts. Students can engage in math that is close to home in that what they are learning is useful to them. Students will grasp new concepts with less difficulty if they can draw correlations to their own experience.
Create effective solutions to problems that come up in math and other situations
Those who excell at problem solving tend to bring situations under scrutiny and there pose problems based on their observations. Assumptions can be made by looking at a problem with little investigation, but the further one delves into the problem the more accurate the outcome will be. Educators have the responsibility of cultivating an environment within the classroom in which students are free and encouraged to create problems that align with their own lives to solve. They should be exhorted to investigate, discover new insights, try new things, and take risks in the classroom.
Relate and adjust many effective problem solving strategies and methods
There are many different problem solving strategies which include: working backwards, guess and check, the use of diagrams and patterns, etc. Instructors should be teaching these strategies to their students along with other problem solving methods. They should assign tasks in class that use a variety of strategies so that students can compare different ways to solve a problem and thereby the students become fluent in using them by the time they reach the higher grades. Different types of problems are going to require different types of problem solving methods. Classroom discussion about problem solving is going to make this a norm in the classroom and help students to be able to internalize what they have practiced and observed.
Observe the process of problem solving and be able to draw personal insight and application
Effective problem solvers are continually engaged in observing their actions and processes, making changes that they see that are necessary, read and reread, and ask questions. They also keep track of their progress and are open to change because there may be a better way of doing things. When the teacher asks probing questions to make sure his or her students understand concepts, and that the learners have gone into much depth in their thinking, they create an atmosphere where students feel free to assess their own work and thinking. It is very important that learners lay hold of reflective practice. By doing this, they will become more efficient and effective at problem solving not just in math, but in their own lives.
Wednesday, January 27, 2010
The Variable Machine
The activities in these video clips serve the purpose of showing that the values of the variables have an effect. In the activity, the students can change the variables associated with the given values in order to increase or decrease. The activity is also a hands on learning tool which causes the students to work with problem solving both individually as well as collaboratively. This also enables the students to grapple with mathematical problems, ideas, and solutions; giving them a foundational understanding of math as it connects as a whole.
Discourse #1
How effectively does the teacher use questioning to help students develop mathematical understanding?
The teacher does an excellent job of asking students questions that do not yield "yes" and "no" answers. She asks questions that guide the learners without giving answers. When the students are stuck, but on the right track to understanding, she prompts her students with guiding questions that helps them figure things out on their own. I love how the teacher answers the students' questions by asking them questions that help them find the answer they were trying to get.
Discourse #2
What techniques and strategies are used to orchestrate and promote student discourse, and how effective are these strategies?
The teacher gives brief and concise instruction for the students on what they will be doing and then lets the students get started working in groups. As the students are working she walks around the room which keeps the students on task and asks questions to groups as well as to the classroom as a whole. The teacher also listens to the students' solutions and explanations to ensure their understanding. I think that these methods are highly effective in student discourse. The teacher asked strategic questions that caused the students to expand their horizons and think creatively.
Student Learning #1
What strategies were used to assess student understanding?
The teacher used an informal approach to assessment because she didn't feel that it would be effective to have a formal assessment. The educator could walk around to each group and listen to different discussions to see if the students understood. As the students would work in their groups, different questions and ideas would arise which would give the teacher insight and understanding to where the student's knowledge was. The teacher would also have students share what they got for their solutions and how they arrived at their conclusions which would also act as a means of assessment.
I thoroughly enjoyed watching the video and thought that the activity was very creative. I felt like the activity was productive and worthwhile. As I was watching the videos I kept thinking that I never learned math in this manner. We never reasoned with other people or with the whole class how we reached our conclusions. I think if I had learned this way, I would have enjoyed math a lot more. I was encouraged to help students go deeper in their understanding of math and broaden their thinking. I appreciated being able to think critically about the videos in correlation to the questions that were being asked so that I knew what I was looking for. This was a very helpful task.
Discourse #1
How effectively does the teacher use questioning to help students develop mathematical understanding?
The teacher does an excellent job of asking students questions that do not yield "yes" and "no" answers. She asks questions that guide the learners without giving answers. When the students are stuck, but on the right track to understanding, she prompts her students with guiding questions that helps them figure things out on their own. I love how the teacher answers the students' questions by asking them questions that help them find the answer they were trying to get.
Discourse #2
What techniques and strategies are used to orchestrate and promote student discourse, and how effective are these strategies?
The teacher gives brief and concise instruction for the students on what they will be doing and then lets the students get started working in groups. As the students are working she walks around the room which keeps the students on task and asks questions to groups as well as to the classroom as a whole. The teacher also listens to the students' solutions and explanations to ensure their understanding. I think that these methods are highly effective in student discourse. The teacher asked strategic questions that caused the students to expand their horizons and think creatively.
Student Learning #1
What strategies were used to assess student understanding?
The teacher used an informal approach to assessment because she didn't feel that it would be effective to have a formal assessment. The educator could walk around to each group and listen to different discussions to see if the students understood. As the students would work in their groups, different questions and ideas would arise which would give the teacher insight and understanding to where the student's knowledge was. The teacher would also have students share what they got for their solutions and how they arrived at their conclusions which would also act as a means of assessment.
I thoroughly enjoyed watching the video and thought that the activity was very creative. I felt like the activity was productive and worthwhile. As I was watching the videos I kept thinking that I never learned math in this manner. We never reasoned with other people or with the whole class how we reached our conclusions. I think if I had learned this way, I would have enjoyed math a lot more. I was encouraged to help students go deeper in their understanding of math and broaden their thinking. I appreciated being able to think critically about the videos in correlation to the questions that were being asked so that I knew what I was looking for. This was a very helpful task.
Saturday, January 23, 2010
Article Review
In the article, Support preservice teachers' reasoning and justification, they give examples of classroom settings, and students working through problems. Throughout these examples students are instructed that they will be using prior knowledge and build off of that in order to complete the problems. The teachers are reinforcing within their classroom content that has been presented and expounding on it in order for students to make connections. In the teaching principle, one of the goals of an effective teacher is to help the students be able to correlate previous knowledge to current content that they are learning so that they can see the overall picture of math integrated.
The students are also collaborating in their work to draw from each learner's knowledge and to contribute. This idea was presented in the teaching principle because students should not just be doing busy work, but working together to do meaningful and worthwhile activities. When students work together they learn more and communicate more effectively a solution to a problem.(p218)
In the classroom segment C, the teacher acted as a facilitator. She waited for repsonses, asked questions that triggered more thoughtful explanations, etc. This is an idea presented in the principle because an effective teacher asks thought provoking questions that will spur students on to think more critically.
According to the article, students are to take action to direct their learning in that they ask questions, explain mathematical thinking, understand sources of mathematical ideas, and are responsibile for their own learning.
The article says that teachers should have a deep understanding of the functions of math and the relationship between numbers, operations, functions and how they relate to another. The principle states the same, that the teacher should have a fluent understanding of math as a whole and how it is intertwined.
The article is aligned well with the teaching principle. It was interesting to read this article and see exactly how the teaching principle was applied in these classroom situations with the given problems.
Rathouz, M. M. (2009). Support preservice teachers’ reasoning and justification. Teaching
children mathematics 16(4), 214-221.
The students are also collaborating in their work to draw from each learner's knowledge and to contribute. This idea was presented in the teaching principle because students should not just be doing busy work, but working together to do meaningful and worthwhile activities. When students work together they learn more and communicate more effectively a solution to a problem.(p218)
In the classroom segment C, the teacher acted as a facilitator. She waited for repsonses, asked questions that triggered more thoughtful explanations, etc. This is an idea presented in the principle because an effective teacher asks thought provoking questions that will spur students on to think more critically.
According to the article, students are to take action to direct their learning in that they ask questions, explain mathematical thinking, understand sources of mathematical ideas, and are responsibile for their own learning.
The article says that teachers should have a deep understanding of the functions of math and the relationship between numbers, operations, functions and how they relate to another. The principle states the same, that the teacher should have a fluent understanding of math as a whole and how it is intertwined.
The article is aligned well with the teaching principle. It was interesting to read this article and see exactly how the teaching principle was applied in these classroom situations with the given problems.
Rathouz, M. M. (2009). Support preservice teachers’ reasoning and justification. Teaching
children mathematics 16(4), 214-221.
The Teaching Principle
Upon reading the Teaching Principle, I learned that in order to be an effective teacher in the area of mathematics, one must possess a strong knowledge base and glean from it to approach teaching while being versatile. The teacher must view every student as a person and a learner; therefore, since not everyone learns the same, a variety of techniques need to be implemented when teaching to ensure quality education.
The teacher must have two specific types of mathematical knowledge. The first type is knowledge about the whole of mathematics. Instructors should understand overall themes, ideas that are conducive to appropriate grade levels, and goals and standards. This type of knowledge lends itself to aid in curricular decisions, helping students understand better, and connect math concepts together. The second type of mathematical knowledge is Pedagogical knowledge. This helps instructors grasp how the student learns math. Teachers should be fluent in a variety of teaching strategies. This also helps in knowing how to approach classroom management and also organizational skills within the classroom.
I also learned that teachers must have a firm understanding on how mathematical ideas relate to one another and to be familiar with the common roadblocks children have with their understanding of math in order to bring insight that will cause students to draw the accurate conclusions.
Teachers hold the responsibility for ensuring their students an environment in which to learn math. They are responsible through their own actions, conversations, behavior, etc. how their students will then think about math. Teachers should be facilitating conversations that will lead the students to think for themselves in drawing conclusions about math, problem solving, and thinking critically about math. Educators should also be choosing activities that are productive for the student to learn and put their knowledge in to practice. The activities should also include real life situations so that the students can deepen their understanding of the content.
Being an effective teacher goes beyond the practical implications to understanding students, listening to them, and observation. When teachers engage in reflective practice they are evaluating themselves as teachers and the classroom to know how to better their teaching strategies and bring more creativity that will be more conducive to the learner. Reflection is vital in the classroom to create the best learning environment possible.
The teacher must have two specific types of mathematical knowledge. The first type is knowledge about the whole of mathematics. Instructors should understand overall themes, ideas that are conducive to appropriate grade levels, and goals and standards. This type of knowledge lends itself to aid in curricular decisions, helping students understand better, and connect math concepts together. The second type of mathematical knowledge is Pedagogical knowledge. This helps instructors grasp how the student learns math. Teachers should be fluent in a variety of teaching strategies. This also helps in knowing how to approach classroom management and also organizational skills within the classroom.
I also learned that teachers must have a firm understanding on how mathematical ideas relate to one another and to be familiar with the common roadblocks children have with their understanding of math in order to bring insight that will cause students to draw the accurate conclusions.
Teachers hold the responsibility for ensuring their students an environment in which to learn math. They are responsible through their own actions, conversations, behavior, etc. how their students will then think about math. Teachers should be facilitating conversations that will lead the students to think for themselves in drawing conclusions about math, problem solving, and thinking critically about math. Educators should also be choosing activities that are productive for the student to learn and put their knowledge in to practice. The activities should also include real life situations so that the students can deepen their understanding of the content.
Being an effective teacher goes beyond the practical implications to understanding students, listening to them, and observation. When teachers engage in reflective practice they are evaluating themselves as teachers and the classroom to know how to better their teaching strategies and bring more creativity that will be more conducive to the learner. Reflection is vital in the classroom to create the best learning environment possible.
Friday, January 22, 2010
The Assessment Priniciple/Article: On-the-Spot Assessments
The Assessment Priniciple highlights important points concerning the major part of giving assessments to students of all ages, such as improving students learning. It's important for teachers to understand that assessments do not dictate how smart students are, but it judges their performance on that particular subject matter. The Assessment Principle briefly describes how teachers should be careful when deciding which assessment to administer to students. Since every student learns in different ways, teachers should make assessments a normal routine as part of the classroom instruction, instead of a distraction to the classroom environment. As expressed in the reading, assessments should develop students learning and is an important tool for teachers to use when making informative decisions.
I think this article describes how students use their estimation, prediction, identification, observation and number thinking skills. Although this article strongly informs readers about the five main points listed above, it strongly supports how students observe and predict. In the activity explained in the article, the teacher began by recognizing that her students had observation problems which became another problem for them to estimate quantity. Four canisters labeled A-D were filled with differently sized candies. Students were told to predict how many candies were filled in each canister. At the end of the assessment, the teacher suggested when they are told to predict how many candies are in a jar, they should ask the question, "How big are the candies?" Asking a question that is relevant to the study is a change in how math can be taught and assessed.
I think this article describes how students use their estimation, prediction, identification, observation and number thinking skills. Although this article strongly informs readers about the five main points listed above, it strongly supports how students observe and predict. In the activity explained in the article, the teacher began by recognizing that her students had observation problems which became another problem for them to estimate quantity. Four canisters labeled A-D were filled with differently sized candies. Students were told to predict how many candies were filled in each canister. At the end of the assessment, the teacher suggested when they are told to predict how many candies are in a jar, they should ask the question, "How big are the candies?" Asking a question that is relevant to the study is a change in how math can be taught and assessed.
Subscribe to:
Posts (Atom)