This article is based on an activity within a study in which 50 kindergarten, first and second grade teachers asked each of their students at the beginning of the year to create a sign for the door of the classroom that would tell visitors how many students were in the class. Some students made tallies or wrote numerals while others created graphs, used pictures or even made representations using money. Perhaps most intriguing and surprising to the teachers were the number of students who payed special attention to the race or gender of themselves and their classmates. One drawing included each class member with accurate skin and hair color. Another student created a bar graph including three bars, one for students with white skin, one for students with tan skin, and one for students with brown skin. This activity and others like it help teachers to understand the students ethnicity, culture and home life, which does in fact have an impact on how the student learns mathematical representations.
This is how I found the article to relate to the main points of the process standard of representation:
1) Students need to use traditional methods of representation to solve problems as well as create unique representations that are meaningful to them.
This article shows many examples of student work in which children used their out of school experiences and cultural background in order to create their own unique representation for the number of students in the class. The article also provides some examples of students' explanations of their work, which helps to further understand why the students choose to represent the class in the way that they did (McCulloch, 2009).
2) Students should use representations as a way of organizing and understanding mathematical concepts.
The example activity in this article helped many students to further understand one to one correspondence through the use of inventive representations. Some students also focused on other concepts through their representations such as grouping, charting, graphing and estimating (McCulloch, 2009).
3) Students will be able to apply their understanding of representations not only to mathematics but also to the world around them.
The article proved to teachers of mathematics that this is indeed true through the unique and surprising results of the study performed. Emphasis was placed on the fact that students' experiences out of school have a large impact on their thinking in school (McCulloch, 2009).
4) New technological tools provide additional methods of representation and allow students to better understand more challenging concepts.
The activity done in the article did not include any use of technology. However, teachers could have used technology by having students to create their representation using any number of computer programs that allow students to "draw" or "paint" using the mouse of the computer. This could be done instead of a pencil and paper drawing or in addition to the original representation.
McCulloch, A. W., Marshall, P. L. and DeCuir-Gunby, J. T. (2009). Cultural capital in children’s
number representations: Reflect and discuss. Teaching children mathematics 16(3), 184-
189.
Thursday, January 28, 2010
Process Standard: Representation
I found the following key points within the process standard of representation:
1) Students need to use traditional methods of representation to solve problems as well as create unique representations that are meaningful to them.
2) Students should use representations as a way of organizing and understanding mathematical concepts.
3) Students will be able to apply their understanding of representations not only to mathematics but also to the world around them.
4) New technological tools provide additional methods of representation and allow students to better understand more challenging concepts.
1) Students need to use traditional methods of representation to solve problems as well as create unique representations that are meaningful to them.
2) Students should use representations as a way of organizing and understanding mathematical concepts.
3) Students will be able to apply their understanding of representations not only to mathematics but also to the world around them.
4) New technological tools provide additional methods of representation and allow students to better understand more challenging concepts.
Wednesday, January 27, 2010
4th Grade Lesson: Variables (Video)
In this lesson fourth grade students, who have not had any experience working with variables, are instructed on how to make and use “variable machines”. These easy to make machines are made out of strips of paper and help students to understand the effect that changing the value of one variable has on another variable. The students work in groups of four, each with their own machines, to determine the sum of the value of all of the letters in their names. As the lesson progresses, students manipulate the variables to spell a variety of words and to control the sum. The main purpose of this lesson was to expose the students to variables for the first time and to teach them the effect of changing a variable.
1) Describe how the teacher’s questioning, and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment.
The teacher’s questioning is definitely a positive contribution to the classroom learning environment and a vital part of the lesson. She has taught her students over the course of the school year to think about and discuss how they should go about solving mathematical problems and why. She challenges her students to explain their thought process in a clear and organized fashion in front of the class. In fact, her class has grown so accustomed to this practice that they do not seem to view it as a challenge, but rather something that is expected. The teacher speaks in a kind tone of voice and expresses genuine interest in what each group is doing. This creates a caring and safe classroom atmosphere where students are more inclined to speak up in class. The teacher offers praise for their verbal responses and probes them for more detail when necessary.
2) What techniques does the teacher use to determine whether students have learned the material you are teaching?
In this lesson the teacher uses a very informal method of assessment. She simply walks around the classroom making sure to visit each table and talks with her students about what they are doing and why. In one video she even mentions to the other teachers that she was in particular looking for children to be able to tell her, “well if I assign this letter with that number, then this other letter will be worth ___ points”. The core idea is that if you change the value of a variable in an equation, then the value of all of the variables in that equation will also change accordingly.
3) Describe the primary task in this lesson and identify the mathematical skills and concepts that this task is designed to develop.
The primary task in this lesson is to create and use a “variable machine” in order to assign numbers to variable letters in a word so that they add up to either the highest or the lowest amount possible. One major mathematical skill needed in this exercise is the understanding of cause and effect. Another more basic but necessary skill is to be able to add and subtract large numbers with a calculator, as it is allowed in this particular lesson. This task is also designed, in large part, to help students understand the concept of assigning a number value which can change to a letter.
I feel that viewing this video was a positive contribution to my learning as a future teacher of math. This helped me to visualize and contextualize the concepts of active learning, cooperative learning, inquiry based learning and mathematical discussion that we have focused on in class. Seeing this teacher in action has given me a better idea of how to be a math teacher who facilitates active learning and discussion, rather than reads from the textbook and hands out worksheets. I thought this was an excellent lesson and I would definitely use it in my own classroom.
1) Describe how the teacher’s questioning, and the manner in which student responses are handled, contribute or do not contribute to a positive classroom learning environment.
The teacher’s questioning is definitely a positive contribution to the classroom learning environment and a vital part of the lesson. She has taught her students over the course of the school year to think about and discuss how they should go about solving mathematical problems and why. She challenges her students to explain their thought process in a clear and organized fashion in front of the class. In fact, her class has grown so accustomed to this practice that they do not seem to view it as a challenge, but rather something that is expected. The teacher speaks in a kind tone of voice and expresses genuine interest in what each group is doing. This creates a caring and safe classroom atmosphere where students are more inclined to speak up in class. The teacher offers praise for their verbal responses and probes them for more detail when necessary.
2) What techniques does the teacher use to determine whether students have learned the material you are teaching?
In this lesson the teacher uses a very informal method of assessment. She simply walks around the classroom making sure to visit each table and talks with her students about what they are doing and why. In one video she even mentions to the other teachers that she was in particular looking for children to be able to tell her, “well if I assign this letter with that number, then this other letter will be worth ___ points”. The core idea is that if you change the value of a variable in an equation, then the value of all of the variables in that equation will also change accordingly.
3) Describe the primary task in this lesson and identify the mathematical skills and concepts that this task is designed to develop.
The primary task in this lesson is to create and use a “variable machine” in order to assign numbers to variable letters in a word so that they add up to either the highest or the lowest amount possible. One major mathematical skill needed in this exercise is the understanding of cause and effect. Another more basic but necessary skill is to be able to add and subtract large numbers with a calculator, as it is allowed in this particular lesson. This task is also designed, in large part, to help students understand the concept of assigning a number value which can change to a letter.
I feel that viewing this video was a positive contribution to my learning as a future teacher of math. This helped me to visualize and contextualize the concepts of active learning, cooperative learning, inquiry based learning and mathematical discussion that we have focused on in class. Seeing this teacher in action has given me a better idea of how to be a math teacher who facilitates active learning and discussion, rather than reads from the textbook and hands out worksheets. I thought this was an excellent lesson and I would definitely use it in my own classroom.
Tuesday, January 26, 2010
Friday, January 22, 2010
Journal Article: "Teacher as Musician versus Teacher as Composer"
The article "Teacher as Musician versus Teacher as Composer" examines an analogy that compares textbook authors to musical composers and teachers to musicians. The author of the article, Margaret R. Meyer, examines the issue of teachers taking on the role of composers in addition to musicians in that they "adapt" portions of the lesson for a number of reasons.
The principle of teaching as described by NCTM stresses the importance of teachers putting a great amount of thought and intention into the learning experiences they create for students. This article discusses this by explaining the need to "adapt" the materials and lessons provided (Meyer, 2009).
Similarly, the principle of teaching as described by NCTM states that teachers must be flexible in their lesson plans. This aspect of teaching is applied in the article in Meyer's explanation that nearly all lessons will need to be at least slightly altered to meet the specific needs of students (Meyer, 2009).
Also included in the NCTM description of the principle of teaching is the importance of reflective practice and continual improvement. This concept is discussed in the article in Meyer's mention of veteran teachers who teach the same inventory of lessons in nearly the exact same way year after year, believing in their tried and true methods (Meyer, 2009). This is an example of teachers who do not use reflective practice.
Meyer, M. R. (2009). Teacher as musician versus teacher as composer. Mathematics teaching in
the middle school 15(2), 70-73.
The principle of teaching as described by NCTM stresses the importance of teachers putting a great amount of thought and intention into the learning experiences they create for students. This article discusses this by explaining the need to "adapt" the materials and lessons provided (Meyer, 2009).
Similarly, the principle of teaching as described by NCTM states that teachers must be flexible in their lesson plans. This aspect of teaching is applied in the article in Meyer's explanation that nearly all lessons will need to be at least slightly altered to meet the specific needs of students (Meyer, 2009).
Also included in the NCTM description of the principle of teaching is the importance of reflective practice and continual improvement. This concept is discussed in the article in Meyer's mention of veteran teachers who teach the same inventory of lessons in nearly the exact same way year after year, believing in their tried and true methods (Meyer, 2009). This is an example of teachers who do not use reflective practice.
Meyer, M. R. (2009). Teacher as musician versus teacher as composer. Mathematics teaching in
the middle school 15(2), 70-73.
Teaching Principle
Important points from the teaching principle:
1. A student's attitude towards and knowledge of math is largely dependent upon the experiences involving math that their teachers provide them with. Teachers must use their knowledge of mathematics and of pedagogy to carefully and intentionally shape these experiences in order to ensure that students learn math in a positive and meaningful way.
2. Teachers must be flexible in the teaching of mathematics, which includes adapting lessons to the students' needs and taking advantage of teachable moments. In addition, effective teachers of mathematics engage in reflective practice and work to improve their teaching techniques.
3. It is important that teachers create an environment in which it is believed that each and every student can and will learn math. Similarly, it is important that this environment be one in which inquisitive and critical thinking in math is encouraged and discussed.
1. A student's attitude towards and knowledge of math is largely dependent upon the experiences involving math that their teachers provide them with. Teachers must use their knowledge of mathematics and of pedagogy to carefully and intentionally shape these experiences in order to ensure that students learn math in a positive and meaningful way.
2. Teachers must be flexible in the teaching of mathematics, which includes adapting lessons to the students' needs and taking advantage of teachable moments. In addition, effective teachers of mathematics engage in reflective practice and work to improve their teaching techniques.
3. It is important that teachers create an environment in which it is believed that each and every student can and will learn math. Similarly, it is important that this environment be one in which inquisitive and critical thinking in math is encouraged and discussed.
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