Sally K. Roberts and Carla Tayeh
March 2010, Volume 15, Issue 7, Page 406
http://www.nctm.org/eresources/view_media.asp?article_id=9178
In this article, teacher candidates are challenged to make a booklet of reasoning and proof which show that they know how to communicate mathematical concepts and ideas to a larger audience. Their instructors do not allow them to say, "You know what I mean" even if it is obvious that they do know what the student means. It is important for teacher candidates to be able to understand a problem at a deeper level than by simply giving the correct answer. Later in the article, middle school students were challenged to use reasoning and proof to solve a problem without a clear-cut answer. Some of the students found an incorrect answer using faulty reasoning. Some of the students came up with the correct answer using good reasoning, but without providing adequate defense or proof to show how they found the answer. Finally, some of the students used good reasoning and proof to find and convey their answers to a wide audience.
I think that reasoning and proof can be the most challenging, yet important, component in mathematics. Because communicating our reasoning takes time and practice, teachers should begin to help students communicate their thoughts at a young age. I hope that I can resist the temptation to nod at a student and say, "Yep. You have the right idea. I know what you are trying to say. Good job." Students need to be given time to sort through their reasoning - to justify their thinking. This is perhaps as important as coming up with the correct answer.
Monday, March 22, 2010
Sand and Water Table Play
Teaching Children Mathematics
March 2010, Volume 16, Issue 7, Page 394
In this article, the teacher emphasizes the value of play in mathematical learning of young children. Children left alone to play may not develop mathematically, but children playing with a teacher in close proximity prompting them to think while playing offers great possibilities. This teacher kept a table in her kindergarten classroom filled with sand. She placed geometric containers, beakers, and bottles in the sandbox. At first, the students were allowed to play and the teacher simply observed the students' discourses and mathematical play, and considered possible math interventions that would naturally flow from the students' play. Then, the teacher played with the students. She asked only a few simple questions to enrich their play - to get them thinking about their play. Finally, when she noticed a frustration with some mathematical concept within the students' playing, she developed lessons from those frustrations to meet the state standards for kindergarten mathematics.
I am always in favor of hands-on learning, as long as the play truly causes learning and is not simply play. This article provided a good example of how to use play to interest the children and to make their learning relevent to them. I think that I will attempt to use such student-centered learning in my future classroom of (hopefully) intermediate or middle school students. I may not provide as much time for free-play with older grades, but I hope to allow a measure of freedom so that I can learn the subjects that matter most to my students, thereby creating lesson that they can relate to.
March 2010, Volume 16, Issue 7, Page 394
In this article, the teacher emphasizes the value of play in mathematical learning of young children. Children left alone to play may not develop mathematically, but children playing with a teacher in close proximity prompting them to think while playing offers great possibilities. This teacher kept a table in her kindergarten classroom filled with sand. She placed geometric containers, beakers, and bottles in the sandbox. At first, the students were allowed to play and the teacher simply observed the students' discourses and mathematical play, and considered possible math interventions that would naturally flow from the students' play. Then, the teacher played with the students. She asked only a few simple questions to enrich their play - to get them thinking about their play. Finally, when she noticed a frustration with some mathematical concept within the students' playing, she developed lessons from those frustrations to meet the state standards for kindergarten mathematics.
I am always in favor of hands-on learning, as long as the play truly causes learning and is not simply play. This article provided a good example of how to use play to interest the children and to make their learning relevent to them. I think that I will attempt to use such student-centered learning in my future classroom of (hopefully) intermediate or middle school students. I may not provide as much time for free-play with older grades, but I hope to allow a measure of freedom so that I can learn the subjects that matter most to my students, thereby creating lesson that they can relate to.
Portfolio Assessment
Damiani, V.B. (2004). Portfolio assessment in the classroom. In Helping Children at Home and School II: Handouts for Families and Educators (S3, 129-131). Retrieved from http://www.nasponline.org/communications/spawareness/portfolioassess.pdf
Using portfolios is an authentic method for assessing student work, from primary grades through college levels. A well-done portfolio contains a collection of student work from a specific topic or theme. Portfolios are valuable in that they show what students can do - not just what they know.
The portfolio process involves several steps:
1. Decide on a purpose or theme for the portfolio
2. Consider what samples of student work might best illustrate the goal being assessed.
3. Determine how samples will be selected.
4. Decide whether to assess both the process and the product, or just the product.
5. Develop an appropriate scoring system.
6. Share the scoring system with the students.
7. Engage the student in discussion about the product.
Portfolio assessment has several advantages and challenges.
Advantages -
1. Assesses what students can do - not just what they know.
2. Engages students actively
3. Fosters student-teacher communication
4. Fosters depth of exploration and learning
5. Helps parents understand the educational process
6. Provides goals for student learning
7. Offers an alternative to traditional testing
Challenges -
1. Establishing scoring systems that are reliable over time and with different raters
2. The process is time consuming.
3. Portfolios portray depth of student learning, but not breadth - assessment is over a narrow theme.
4. Keeping the process fair, when some students may have advantages in their homes, such as parental support and access to technology.
5. It is difficult to compare the students' learning to students across the nation.
6. Students may not accurately self-assess their work if they know it is being assessed by the teacher.
Using portfolios is an authentic method for assessing student work, from primary grades through college levels. A well-done portfolio contains a collection of student work from a specific topic or theme. Portfolios are valuable in that they show what students can do - not just what they know.
The portfolio process involves several steps:
1. Decide on a purpose or theme for the portfolio
2. Consider what samples of student work might best illustrate the goal being assessed.
3. Determine how samples will be selected.
4. Decide whether to assess both the process and the product, or just the product.
5. Develop an appropriate scoring system.
6. Share the scoring system with the students.
7. Engage the student in discussion about the product.
Portfolio assessment has several advantages and challenges.
Advantages -
1. Assesses what students can do - not just what they know.
2. Engages students actively
3. Fosters student-teacher communication
4. Fosters depth of exploration and learning
5. Helps parents understand the educational process
6. Provides goals for student learning
7. Offers an alternative to traditional testing
Challenges -
1. Establishing scoring systems that are reliable over time and with different raters
2. The process is time consuming.
3. Portfolios portray depth of student learning, but not breadth - assessment is over a narrow theme.
4. Keeping the process fair, when some students may have advantages in their homes, such as parental support and access to technology.
5. It is difficult to compare the students' learning to students across the nation.
6. Students may not accurately self-assess their work if they know it is being assessed by the teacher.
Wednesday, March 3, 2010
Video analysis - graphing
The purpose of the activities in this video is for students to represent graphs in a variety of ways and to extract meaning from the graphs. The students first create a story about the graph, which helps them learn that the graphs depict a relationship between two quantities. Next they create tables as another representation of the graph, and finally they develop an equation to show a fourth representation. The content of the graph is therefore shown as a graph, a table, a story, and an equation – four representations!
Lesson Analysis 1: 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 for students to create a sequence for their classmates and teachers to figure out missing numbers. In order to meet that goal, students will develop the skill of representing a concept in several different ways and problem solving to find a generalized equation for that sequence.
Reflective Task 1: Describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught.
I think the primary task in this lesson was very appropriate because the students had to use various representations to display the meaning of the graph, communicate mathematically in groups, and problem solve to figure out a sequence that would be a challenge even for their teacher. Not only were the process skills used, but the students developed a better understanding of the relationship between two variables in a graph because they attached personal meaning to their graphs through stories.
Lesson Analysis 2: Describe what the teacher does to support learning while students are working in groups.
The teacher supports learning by asking guiding questions to assist the students’ thought processes. She did not simply tell them the answers or tell them how to think, she guided their learning through questions. She also encouraged the students to communicate mathematically with each other, telling them to “talk about it in your groups.” As she rotated between groups, she listened to their thinking patterns to understand where they were misinterpreting or forgetting some important aspect of their problem.
Once again, I benefited from watching the videos because I could actually see the teacher leading the children to understanding. It is helpful to witness someone carrying out the strategies we learn in school. Observing her decisions motivated me to one day guide students to learning through the process standards rather than teach them the way I was taught.
Lesson Analysis 1: 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 for students to create a sequence for their classmates and teachers to figure out missing numbers. In order to meet that goal, students will develop the skill of representing a concept in several different ways and problem solving to find a generalized equation for that sequence.
Reflective Task 1: Describe how appropriate you think the primary task in this lesson is for developing an understanding of the mathematics being taught.
I think the primary task in this lesson was very appropriate because the students had to use various representations to display the meaning of the graph, communicate mathematically in groups, and problem solve to figure out a sequence that would be a challenge even for their teacher. Not only were the process skills used, but the students developed a better understanding of the relationship between two variables in a graph because they attached personal meaning to their graphs through stories.
Lesson Analysis 2: Describe what the teacher does to support learning while students are working in groups.
The teacher supports learning by asking guiding questions to assist the students’ thought processes. She did not simply tell them the answers or tell them how to think, she guided their learning through questions. She also encouraged the students to communicate mathematically with each other, telling them to “talk about it in your groups.” As she rotated between groups, she listened to their thinking patterns to understand where they were misinterpreting or forgetting some important aspect of their problem.
Once again, I benefited from watching the videos because I could actually see the teacher leading the children to understanding. It is helpful to witness someone carrying out the strategies we learn in school. Observing her decisions motivated me to one day guide students to learning through the process standards rather than teach them the way I was taught.
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