This article is well worth the time for any math teacher to read. In one article, a wealth of knowledge and research is shifted to the readers, examples supporting the text.
The Rational Number Project (RNP), a group focused on children's learning tendencies of fractions, decimals, proportions, and ratios, developed a curriculum using the Lesh Translation model, which depicts five areas where representations should be used: real world situations, manipulatives, pictures, spoken symbols, and written symbols. Students should be able to represent any rational number (e.g. 0.26) in each of these categories. They should also be able to translate a representation from any one category into another representation from a different category. Theoretically, the students will develop a deeper understanding of rational numbers as opposed to rote memorization of how to manipulate them (adding, subtracting, etc.)
The crux of the RNP curriculum was using the traditional 10 x 10 grid. Students connected their spoken language of rational numbers (e.g. twenty-sixth hundredths, not point twenty-six!) with the pictorial representation. They also wrote the numbers with written symbols and translated the grid to a number line - another picture representation. The group discovered through their research that the 10 x 10grid was the most effective representation for students to begin studying rational numbers with. They found that students' initial ideas about decimals or other forms of rational numbers was a crucial element in how deeply they came to understand them.
The more I read about representation, the more I can see the how America's traditional math curriculum and teaching approach hindered mathematical reasoning. I just hope that when I become a teacher I will not become so caught up in preparing students for high-stakes testing that I cannot truly teach math from the "bottom up" and the "inside out."
Cramer, K. A., Monson, D. S., Wyberg, T., Leavitte, S. and Whitney, S. B. (2009). Models for initial decimal ideas. Teaching children mathematics 16(2), 106-116.
Wednesday, January 27, 2010
Representations
Representations are used daily in math instruction, yet students and teachers sometimes hardly realize that those representations are indeed...representations. That is, they are symbols used in place of a complex process as well as a function. For example, the + sign, we are told, tells us that 2 + 2 equals 4. In reality, that symbol represents a complex process - a mathematical concept.
Representations are not in and of themselves the process, but they are fundamentally important in learning and understanding mathematics. If an addition sign would not be the established representation for adding, we would inevitably create our own symbol at some point in life to describe a process in short hand. When we as teachers hand the children those representations, they quickly embrace the symbol and its function before grasping the process. Teachers should encourage students to create their own representations - perhaps even before the established mathematical symbol is presented them. This would help students think about the process and approach a meaningful symbol to represent that process, and would probably help the students to remember math better.
In addition to letting the students devise their own representations, teachers should encourage the students to share their representations with other students to increase flexibility in different types of representations for the same concepts. The article suggests that students tend to miss the connection between a circle divided into three equal parts, a division bar, and a fraction bar. If they are presented with a rigid, one-symbol approach to math, they will not be able to make appropriate connections between mathematical concepts.
Finally, once students are able to create representations and translate them to a variety of contexts, they should be able to use them as models for real-life situations. A model, as pointed out in the article, is an ideal picture of an object or a phenomenon. They give people something to work towards - a desired end. Many occupations use models, many of them containing mathematical representations, to engineer their product or goal. It is clear, then, that representations should not be used as an object, but as building blocks to deeper, more creative thinkers.
NCTM (2000). Representations. Principles and Standards for School Mathematics (pp. 76-70). Reston, VA: NCTM.
Representations are not in and of themselves the process, but they are fundamentally important in learning and understanding mathematics. If an addition sign would not be the established representation for adding, we would inevitably create our own symbol at some point in life to describe a process in short hand. When we as teachers hand the children those representations, they quickly embrace the symbol and its function before grasping the process. Teachers should encourage students to create their own representations - perhaps even before the established mathematical symbol is presented them. This would help students think about the process and approach a meaningful symbol to represent that process, and would probably help the students to remember math better.
In addition to letting the students devise their own representations, teachers should encourage the students to share their representations with other students to increase flexibility in different types of representations for the same concepts. The article suggests that students tend to miss the connection between a circle divided into three equal parts, a division bar, and a fraction bar. If they are presented with a rigid, one-symbol approach to math, they will not be able to make appropriate connections between mathematical concepts.
Finally, once students are able to create representations and translate them to a variety of contexts, they should be able to use them as models for real-life situations. A model, as pointed out in the article, is an ideal picture of an object or a phenomenon. They give people something to work towards - a desired end. Many occupations use models, many of them containing mathematical representations, to engineer their product or goal. It is clear, then, that representations should not be used as an object, but as building blocks to deeper, more creative thinkers.
NCTM (2000). Representations. Principles and Standards for School Mathematics (pp. 76-70). Reston, VA: NCTM.
Fourth Grade Video
The fourth grade video addresses variables - a difficult topic for young children to understand. They have not worked extensively with variables up to this point, so the newness of the topic might intimidate them and inhibit learning. To help the students build understanding of variables - a mathematical concept - the teacher incorporated a lesson devoted to creating a variable machine. The variable machine was simply two strips of paper - one with the alphabet written vertically on the lined paper, the other with the numbers 0 - 25. Placed side-by-side and attached to eachother, each number was paired with a letter, or variable.
Reflective Task 1 says, describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught. I thought this primary task was valuable because it focused on the concept of variables before delving into the function of variables. One math curriculum that I have had to use is Saxon (in tutoring). Saxon math spends far too little time on developing concepts, while drilling problem after problem of various algorithms. The students become confused as to when to subtract from what side, where and when to place the decimals, and so often, I have found myself helping the students survive the class rather than helping them understand. I certainly can see the value of spending ample time on the concept of variables prior to using variables on paper.
Reflective Task 1 under discourse says, 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. First of all, I noticed that the teacher encouraged thinking by telling the class to "Talk to me about what you were thinking." Apparently, her students were comfortable with this manner of academic speaking and explaining, because immediately one of the students responded, explaining her group's thought process much more adeptly than many adults could do. I noticed that some of the teacher's questions were lower level thinking questions, but many of them required evaluation, synthesis, or analysis. For example, "How could you make your table's names have a higher value?" challenged the students to analyze what they had already done, and synthesize that information to create a new pattern.
Lesson Analysis 2 says, Give two examples of evidence that students are having difficulty understanding the mathematics being taught. One of the tables included a little girl who could not explain why her group changed the numbers so that the low numbers were at the beginning of the alphabet. This indicates that she does not understand the concept of variables - that the letters are simply representations of the numbers, and by assigning numbers to certain letters, the value of the letters will change. Another group decided to increase their scores by changing the length of their names. They thought that if they would include their middle name, the values would increase. This indicates that they were not comprehending the main objective of the lesson. Their goal was to get a higher score in any way possible. They did not realize that the only way the scores could change without lengthening their names would be to assign different numbers to different letters (variables).
Overall, I benefited from watching an experienced teacher teach mathematics in a constructivist environment. I have wondered many times how the principles of mathematics could really be applied in the classroom, but seeing it on the video gave me a glimmer of hope that it can be done. I learned that if I am to be an effective teacher, I will have to branch off of the textbook and worksheets. I will need to facilitate mathematical discussion and explanations, and I will need to provide a student centered environment in which understanding is discovered - not spoonfed.
Reflective Task 1 says, describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught. I thought this primary task was valuable because it focused on the concept of variables before delving into the function of variables. One math curriculum that I have had to use is Saxon (in tutoring). Saxon math spends far too little time on developing concepts, while drilling problem after problem of various algorithms. The students become confused as to when to subtract from what side, where and when to place the decimals, and so often, I have found myself helping the students survive the class rather than helping them understand. I certainly can see the value of spending ample time on the concept of variables prior to using variables on paper.
Reflective Task 1 under discourse says, 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. First of all, I noticed that the teacher encouraged thinking by telling the class to "Talk to me about what you were thinking." Apparently, her students were comfortable with this manner of academic speaking and explaining, because immediately one of the students responded, explaining her group's thought process much more adeptly than many adults could do. I noticed that some of the teacher's questions were lower level thinking questions, but many of them required evaluation, synthesis, or analysis. For example, "How could you make your table's names have a higher value?" challenged the students to analyze what they had already done, and synthesize that information to create a new pattern.
Lesson Analysis 2 says, Give two examples of evidence that students are having difficulty understanding the mathematics being taught. One of the tables included a little girl who could not explain why her group changed the numbers so that the low numbers were at the beginning of the alphabet. This indicates that she does not understand the concept of variables - that the letters are simply representations of the numbers, and by assigning numbers to certain letters, the value of the letters will change. Another group decided to increase their scores by changing the length of their names. They thought that if they would include their middle name, the values would increase. This indicates that they were not comprehending the main objective of the lesson. Their goal was to get a higher score in any way possible. They did not realize that the only way the scores could change without lengthening their names would be to assign different numbers to different letters (variables).
Overall, I benefited from watching an experienced teacher teach mathematics in a constructivist environment. I have wondered many times how the principles of mathematics could really be applied in the classroom, but seeing it on the video gave me a glimmer of hope that it can be done. I learned that if I am to be an effective teacher, I will have to branch off of the textbook and worksheets. I will need to facilitate mathematical discussion and explanations, and I will need to provide a student centered environment in which understanding is discovered - not spoonfed.
Thursday, January 21, 2010
Gifted students and equity
In "The Gifted Student," three primary concepts are addressed in light of two gifted students, Caroline and Craig. First, teachers who differentiate instruction so that all students - including the gifted - are respected, engaged, challenged, and allowed to be creative, have happier classrooms with greater participation and enthusiasm. Second, the article discussed gifted students' perceptions of their math classes, noting that differentiated instruction reaching even gifted students' needs gave them positive perceptions of math education. Finally, the article provides ideas for math teachers working with students of a variety of learning levels - virtually all teachers - and gives suggestions for how they might handle the high demands of teaching such a diverse population.
This article is outstanding - clear, practical advise and examples for teachers striving to guide learners to the highest possible level according to their individual needs. I loved it. I especially appreciated the four principles for differentiating instruction - respect, engagement, challenge, and opportunity for creativity and flexibility - because, as the article suggests, gifted students are so often deprived of all four of these educational "rights." I well remember feeling disdained simply "because everything was so easy for you," they would say(not as a truly gifted student, but as a student who excelled). The truth was not so - the truth was that the information presented me was not engaging, challenging, and did not allow for creativity. Simply put, it was boring. So in a much smaller way than truly gifted students experience, I experienced being deprived of these four principles. I believe that if teachers would keep these principles posted in their minds and hearts, America's leading mathematicians would become the world's leading mathematicians. But until instruction is differentiated, we will remain a subservient academic society.
Chval, K. B. and Davis, J. A. (2009). The gifted student. Mathematics Teaching in the Middle School 14 (5), 267-274.
This article is outstanding - clear, practical advise and examples for teachers striving to guide learners to the highest possible level according to their individual needs. I loved it. I especially appreciated the four principles for differentiating instruction - respect, engagement, challenge, and opportunity for creativity and flexibility - because, as the article suggests, gifted students are so often deprived of all four of these educational "rights." I well remember feeling disdained simply "because everything was so easy for you," they would say(not as a truly gifted student, but as a student who excelled). The truth was not so - the truth was that the information presented me was not engaging, challenging, and did not allow for creativity. Simply put, it was boring. So in a much smaller way than truly gifted students experience, I experienced being deprived of these four principles. I believe that if teachers would keep these principles posted in their minds and hearts, America's leading mathematicians would become the world's leading mathematicians. But until instruction is differentiated, we will remain a subservient academic society.
Chval, K. B. and Davis, J. A. (2009). The gifted student. Mathematics Teaching in the Middle School 14 (5), 267-274.
Equity by Lana Stoller
"The Equity Principle" discusses the core ideas which make equity possible in middle schools. Among the reasons for promoting equity in schools, students will rise to the expectations laid out for them. It then becomes the challenge for educators to maintain high expectations for all students - not just the sects which are stereotypically proficient in math. In addition, all students need opportunities and support to learn. This does not imply that equal types and levels of assistance are given to all students, but rather, that appropriate assistance is given for every student to learn and attain to the high expectations established. A third principle for equity is that all classrooms are provided with the necessary resources to meet a vast array of students' needs. English Language Learners will be assesed according to their mathematical understanding with adequate support to make up for their language barriers, while manipulatives and supplemental programs will aid students with disabilities. Finally, teachers should participate in professional development programs and should continually learn, seeking to understand the diverse population of students they encounter and how to best meet their educational needs.
I thought this article was a good reminder that equity is truly a primary concern in math education today. After comparing the standardized math test scores of students from two different socioeconomic statuses, I was disturbed, knowing that the potential in the low-income schools is so great, though the performance was so low. If these principles could be applied to a greater degree, I believe that a wider population of Americans would reach their goals. A classroom teacher has power in her hand if she could apply the principles of high expectations for all, providing opportunities for all to learn, providing appropriate instruction in light of differing needs, and participating in her own professional development. It may not be as easy for her to provide necessary resources in a low income setting, but she could do her part by using manipulatives and available instructional tools.
NCTM (2000). Equity. Principles and Standards for School Mathematics (pp. 11-13.). Reston, VA: NCTM.
I thought this article was a good reminder that equity is truly a primary concern in math education today. After comparing the standardized math test scores of students from two different socioeconomic statuses, I was disturbed, knowing that the potential in the low-income schools is so great, though the performance was so low. If these principles could be applied to a greater degree, I believe that a wider population of Americans would reach their goals. A classroom teacher has power in her hand if she could apply the principles of high expectations for all, providing opportunities for all to learn, providing appropriate instruction in light of differing needs, and participating in her own professional development. It may not be as easy for her to provide necessary resources in a low income setting, but she could do her part by using manipulatives and available instructional tools.
NCTM (2000). Equity. Principles and Standards for School Mathematics (pp. 11-13.). Reston, VA: NCTM.
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