The article "My Bar Graph Tells a Story" detailed a five day lesson in which a class of diverse second graders explored the relationship between qualitative and quantitative bar graphs. For the first three days of the lesson the teacher guides the students through various activities introducing the relationship between qualitative and quantitative bar graphs. Students fill in pre-made blank graphs and measure them using unifix cubes in corresponding colors. During the final two days the culminating activity was for students to match common nursery rhymes and stories to qualitative bar graphs.
I thought that this article was extremely detailed and specific in the manner in which they presented the idea of the lesson. The authors seemed to ramble on with the specifics, which I find not to be very useful to teachers. I think that most teachers would adopt the basic idea of this lesson, but then alter it to fit the needs of their students. The general idea, objectives, and methods of the lesson are of a high quality. This lesson would definitely help students to strengthen their ability to communicate using mathematical language, and to interpret graphs with and without labels.
Monday, March 22, 2010
Poematics: Exploring Math Through Poetry. Mathematics Teaching in the Middle School
The article "Poematics: Exploring Math through Poetry" details a lesson in which fifth and seventh graders write their own poems about math topics. First, teachers explained two types of poems, haikus and limericks, and showed examples of these two types of poems. Then, students created their own poetry, choosing any mathematical concept they found interesting to write about in the format of either a haiku or a limerick. Some students had difficulty thinking of topics to write on or getting started writing; however, in the end all students were engaged in writing. The authors suggested having students peer edit each others' poetry to improve the level of accuracy in the future. Using writing, especially poetry, in math class allows students to use their creativity to express their ideas more freely.
I thought that this article provided a very new and innovative idea that could be relatively easily implemented in mathematics classes of a variety of levels. This activity does not require any special materials or supplies, or any extensive preparation, which makes it easy to implement. Also, the activity could be done at a range of grade levels, because it does not focus on any one particular mathematics topic. The activity does not take up a large amount of time, and can be incorporated into any mathematics unit. I think that students could benefit from this lesson as soon as they have a basic understanding of poetry and can write poetry on their own. Finally, this lesson is beneficial because it forces students to reflect on what they have learned and think creatively and conceptually.
I thought that this article provided a very new and innovative idea that could be relatively easily implemented in mathematics classes of a variety of levels. This activity does not require any special materials or supplies, or any extensive preparation, which makes it easy to implement. Also, the activity could be done at a range of grade levels, because it does not focus on any one particular mathematics topic. The activity does not take up a large amount of time, and can be incorporated into any mathematics unit. I think that students could benefit from this lesson as soon as they have a basic understanding of poetry and can write poetry on their own. Finally, this lesson is beneficial because it forces students to reflect on what they have learned and think creatively and conceptually.
Wednesday, March 3, 2010
Video Analysis 2: 7th grade graphing
The main purpose of the activities in this lesson was for students to identify and demonstrate the relationship between two variables in an equation. Also, students learned how equations, ordered pairs, tables, and graphs are related and how they are used. Lastly, students developed their abilities to identify patterns and formulate equations or "rules" from patterns based on real life scenarios.
1. How do you determine whether group work is appropriate and effective?
I believe that group work is effective if students are actively learning and working together to do so. I think it is important that all students within the group are learning and benefiting from the group work. Group work can be very effective in lessons that involve investigation of a new concept. In a small group, students are able to bounce ideas off of one another so that they hit fewer dead ends. Also, in instances where it is beneficial for students to explain the process in which they found their answer, or why their answer is what it is, group work can be very effective.
2. What criteria do you use to determine whether or not to use a particular task with your class?
First and foremost I consider what the task is teaching the students. In other words, what will my students come away with after doing this task? Then I ask myself, does this align with the state and NCTM standards for this grade level? I also consider the level at which my students are at and the concepts that they have mastered, are still learning, and have not yet been exposed to. Where would this task fit into those categories? Is it logical to do this task now, or at another time during the year? Also, it is important to me that my students do tasks that are directly related to the real world. I will consider this when evaluating a task.
3. Describe how you generally deal with student mistakes and misconceptions that arise during a lesson?
This topic is discussed during one of the videos in which Ms. Allen was being interviewed after the lesson. One of the interviewers offers a compliment to Ms. Allen on how she deals with "errors" by bringing them to the attention of the class and having students talk about the error. I think that this is an extremely effective strategy; however, it must be used with caution as not to embarrass students. Ms. Allen explains that this is common practice in her classroom and students are accustomed to it. It is clear through watching her video that she has a good rapport with the children and an encouraging classroom environment, two things that are vital for this strategy to be effective. Allowing students to really understand why the mistake they made was incorrect will prevent them from making similar errors in the future. Bringing this to the attention of the entire class will help all of the students to also avoid the same error.
It is clear to me why NCTM has chosen this lesson as an exemplary one to place on their website. What first strikes me as most different from how I was taught, yet most like how I am currently being taught to teach is the emphasis on students talking about math. Since I have not been taught in this way, it is helpful for me to view videos of this style of teaching to better understand how it is actually done in the classroom. I also liked the fact that all of the problems the students did were connected to real life scenarios that the children could relate to. For example, starting with ten dollars and earning three dollars each week is something most seventh graders could easily do by doing household chores or helping a neighbor. Real world connections to math were also evident during the first part of the lesson in which students worked in groups to develop stories to premade graphs. I thought this was an excellent start to the lesson and helped make graphs meaningful to students.
1. How do you determine whether group work is appropriate and effective?
I believe that group work is effective if students are actively learning and working together to do so. I think it is important that all students within the group are learning and benefiting from the group work. Group work can be very effective in lessons that involve investigation of a new concept. In a small group, students are able to bounce ideas off of one another so that they hit fewer dead ends. Also, in instances where it is beneficial for students to explain the process in which they found their answer, or why their answer is what it is, group work can be very effective.
2. What criteria do you use to determine whether or not to use a particular task with your class?
First and foremost I consider what the task is teaching the students. In other words, what will my students come away with after doing this task? Then I ask myself, does this align with the state and NCTM standards for this grade level? I also consider the level at which my students are at and the concepts that they have mastered, are still learning, and have not yet been exposed to. Where would this task fit into those categories? Is it logical to do this task now, or at another time during the year? Also, it is important to me that my students do tasks that are directly related to the real world. I will consider this when evaluating a task.
3. Describe how you generally deal with student mistakes and misconceptions that arise during a lesson?
This topic is discussed during one of the videos in which Ms. Allen was being interviewed after the lesson. One of the interviewers offers a compliment to Ms. Allen on how she deals with "errors" by bringing them to the attention of the class and having students talk about the error. I think that this is an extremely effective strategy; however, it must be used with caution as not to embarrass students. Ms. Allen explains that this is common practice in her classroom and students are accustomed to it. It is clear through watching her video that she has a good rapport with the children and an encouraging classroom environment, two things that are vital for this strategy to be effective. Allowing students to really understand why the mistake they made was incorrect will prevent them from making similar errors in the future. Bringing this to the attention of the entire class will help all of the students to also avoid the same error.
It is clear to me why NCTM has chosen this lesson as an exemplary one to place on their website. What first strikes me as most different from how I was taught, yet most like how I am currently being taught to teach is the emphasis on students talking about math. Since I have not been taught in this way, it is helpful for me to view videos of this style of teaching to better understand how it is actually done in the classroom. I also liked the fact that all of the problems the students did were connected to real life scenarios that the children could relate to. For example, starting with ten dollars and earning three dollars each week is something most seventh graders could easily do by doing household chores or helping a neighbor. Real world connections to math were also evident during the first part of the lesson in which students worked in groups to develop stories to premade graphs. I thought this was an excellent start to the lesson and helped make graphs meaningful to students.
Monday, February 15, 2010
Applet Review: Deep Sea Duel
Deep Sea Duel. NCTM Illuminations. http://illuminations.nctm.org/ActivityDetail.aspx?ID=207
The objective of this applet is for students to win the game by selecting a specified amount of numbered flash cards to equal a sum, before "Okta" the octopus opponent does so. Students must use addition skills, problem solving skills, planning ahead, and defensive playing strategies in order to be successful in this game. This game has varying levels, which can accommodate students in grades 3-8. Students or teachers can choose to play with either 16 cards or 9 cards and can play on easy or hard levels and with "Okta" set on "nice" or "nasty" playing. This game can be quite challenging because of the many higher order thinking skills required and the unique nature of the game. This applet is presented in a fun and kid-friendly manner, and it makes learning fun and intriguing for young students.
On another note, the game can be quite confusing for students (0f any age). The rules of the game allow a player to select a variety of cards, while only a designated number (3 or 4, depending on if the game is played with 9 or 16 cards total) of the cards selected will count towards the final sum. For example, a player choosing to play with 9 cards could have selected "10, 7, 1, 11, 6" in order to make the sum of 14. Only the numbers 7, 1, and 6 would count towards the sum of 14. I think that this is a very confusing concept for children that has very little practical application. To teach students that only some numbers within a group count towards the sum seems to contradict other, more practical concepts within the math curriculum. Also, the program did not allow the player move on to another problem once the problem had been solved. The only navigation button reset the same problem for another try. The only way I found to begin a new problem was to go back to the main menu settings.
The objective of this applet is for students to win the game by selecting a specified amount of numbered flash cards to equal a sum, before "Okta" the octopus opponent does so. Students must use addition skills, problem solving skills, planning ahead, and defensive playing strategies in order to be successful in this game. This game has varying levels, which can accommodate students in grades 3-8. Students or teachers can choose to play with either 16 cards or 9 cards and can play on easy or hard levels and with "Okta" set on "nice" or "nasty" playing. This game can be quite challenging because of the many higher order thinking skills required and the unique nature of the game. This applet is presented in a fun and kid-friendly manner, and it makes learning fun and intriguing for young students.
On another note, the game can be quite confusing for students (0f any age). The rules of the game allow a player to select a variety of cards, while only a designated number (3 or 4, depending on if the game is played with 9 or 16 cards total) of the cards selected will count towards the final sum. For example, a player choosing to play with 9 cards could have selected "10, 7, 1, 11, 6" in order to make the sum of 14. Only the numbers 7, 1, and 6 would count towards the sum of 14. I think that this is a very confusing concept for children that has very little practical application. To teach students that only some numbers within a group count towards the sum seems to contradict other, more practical concepts within the math curriculum. Also, the program did not allow the player move on to another problem once the problem had been solved. The only navigation button reset the same problem for another try. The only way I found to begin a new problem was to go back to the main menu settings.
Applet Review: Angle Sums
Angle Sums. NCTM Illuminations. http://illuminations.nctm.org/ActivityDetail.aspx?ID=9
The objective of this applet is for students to be able to manipulate shapes. Also students will be able to identify the relationship between the number of sides/angles in a shape and the sum of the angles formed by the shape. Students will also be able to identify the relationship between angles within a shape and the concept that the sum of the angles within a shape is constant. The applet allows students to choose a shape (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon) and then manipulate the lines and angles by clicking and dragging any point of the shape. Angles are numbered and color coded with a key on the right side of the shape. The key contains the exact measurement of each angle and the sum of all angles. The applet is simple and easy to use, as well as colorful and atheistically appealing. There are no complications, this applet is very straightforward.
I think that this applet could be useful in student learning, although it would need to be used in a very structured, supervised manner. With little guidance, or thought provoking questions, many students may just "play" with the application, gaining few mathematical understandings. A teacher could use this tool along with a mini lesson or a "record" sheet for students to record angle measurements. A teacher-led conclusion or discussion of learning after using the applet would be crucial to students' learning. Viewing with a critical eye, I feel that this applet is a bit too simplistic and boring. It does not seem to engage students in learning, and in depth thinking seems to be optional when using this applet, as it does not require any computations or problem solving to use the tool.
The objective of this applet is for students to be able to manipulate shapes. Also students will be able to identify the relationship between the number of sides/angles in a shape and the sum of the angles formed by the shape. Students will also be able to identify the relationship between angles within a shape and the concept that the sum of the angles within a shape is constant. The applet allows students to choose a shape (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon) and then manipulate the lines and angles by clicking and dragging any point of the shape. Angles are numbered and color coded with a key on the right side of the shape. The key contains the exact measurement of each angle and the sum of all angles. The applet is simple and easy to use, as well as colorful and atheistically appealing. There are no complications, this applet is very straightforward.
I think that this applet could be useful in student learning, although it would need to be used in a very structured, supervised manner. With little guidance, or thought provoking questions, many students may just "play" with the application, gaining few mathematical understandings. A teacher could use this tool along with a mini lesson or a "record" sheet for students to record angle measurements. A teacher-led conclusion or discussion of learning after using the applet would be crucial to students' learning. Viewing with a critical eye, I feel that this applet is a bit too simplistic and boring. It does not seem to engage students in learning, and in depth thinking seems to be optional when using this applet, as it does not require any computations or problem solving to use the tool.
Wednesday, February 10, 2010
Journal Summary: Transitions from Elementary School to Middle School Math
There are many changes that occur during the jump from elementary school to middle school that it can be hard for students to adjust and often results in a dip in academic achievement. These changes occur not only in the actual math content that the students are expected to learn, but also in the way the content is presented. Teachers often have very different teaching styles and procedures in the classroom in the elementary school versus the middle school. There are even noticeable differences in textbooks manufactured for a middle school versus an elementary school. Compounded with a new physical environment and a new social environment this can be quite a challenge for many students. However, there are specific things that teachers can do to help ease this transition. Perhaps the single best thing that teachers in the grades surrounding the transition is to visit each other’s classrooms and observe their teaching. When teachers at either level notice drastic differences they can then work to either prepare students for this change or ease students more slowly into this change, depending on which setting they are teaching in. If an in person visit is not possible, viewing a videotapes of a teacher in a classroom one grade level up or down can be a good alternative.
I found this article to be very interesting and relevant to my future teaching. Although I was aware that the transition to middle school can be difficult, I was not aware of all of the specific changes that occur. For example, I thought it was particularly interesting that textbooks are so noticeably different between fifth grade and sixth grade. The article even points out that some companies manufacture different textbooks for sixth grade depending on if sixth grade is situated in an elementary or middle school setting. As a future elementary teacher, I will keep these important aspects of transition in mind. I think that it is an excellent idea to visit a classroom in the middle school where your students may be the following year, and I sincerely hope to do so if I am teaching the uppermost grade in the elementary school setting. I believe that this would be most useful to do near the beginning of the school year, so that the elementary teacher gains a better idea of what specifically her students should be able to do in exactly one school year. Similarly, the middle school teacher will be dealing with the transition issues at the beginning of the school year, and this would be a good time for her to solicit advice from the elementary teacher.
Schielack, J. and Seeley, C. (2010). Transitions from Elementary School to Middle School Math. Teaching Children Mathematics. 16(6), 358-362.
I found this article to be very interesting and relevant to my future teaching. Although I was aware that the transition to middle school can be difficult, I was not aware of all of the specific changes that occur. For example, I thought it was particularly interesting that textbooks are so noticeably different between fifth grade and sixth grade. The article even points out that some companies manufacture different textbooks for sixth grade depending on if sixth grade is situated in an elementary or middle school setting. As a future elementary teacher, I will keep these important aspects of transition in mind. I think that it is an excellent idea to visit a classroom in the middle school where your students may be the following year, and I sincerely hope to do so if I am teaching the uppermost grade in the elementary school setting. I believe that this would be most useful to do near the beginning of the school year, so that the elementary teacher gains a better idea of what specifically her students should be able to do in exactly one school year. Similarly, the middle school teacher will be dealing with the transition issues at the beginning of the school year, and this would be a good time for her to solicit advice from the elementary teacher.
Schielack, J. and Seeley, C. (2010). Transitions from Elementary School to Middle School Math. Teaching Children Mathematics. 16(6), 358-362.
Journal Summary: Rubrics at Play
Rubrics are useful for a number of purposes: to assess students, to provide feedback to students, and to plan instruction. Similarly, there are many different varieties of rubrics and a multitude of methods to use rubrics effectively with students. Formative assessments are based on more specific criteria, and therefore give students more beneficial feedback, whereas summative assessments serve the purpose to assign a letter grade or number to a student's overall quality of work. Rubrics are also categorized as either holistic, analytic, specific, or general. Holistic rubrics, a method of summative assessment, give one overall score of the student's work. Analytic rubrics, on the other hand, include more specific areas in which students receive a score for. Specific rubrics are created solely for one task or assignment, as opposed to general rubrics which can be used for many similar or related tasks. General rubrics can be given to students before beginning the assignment, because the answer is not included on these rubrics. Also, some teachers find it helpful to allow students to assist in the process of developing a rubric. This holds students more accountable for their work and keeps them motivated to improve their work to the next level as described on the rubric.
I found this article to be helpful in explaining the many different ways that a rubric can be used. I was not previously aware that there were so many types of rubrics, probably due to the fact that many of my past teachers and professors have used similar types of rubrics. Also, I found the section that described how a teacher included her students in the process of developing a rubric. Although the teacher did mention that this took an entire day of class time for math, I feel it was a worthwhile activity. This is an idea that I will hold on to and will seriously consider adopting for my own classroom. I believe that it empowers students and helps them to understand how grades are derived. Similarly, I found the idea of general rubrics to be of particular interest to me as a special educator. At first, I was skeptical that a general rubric could be effective; however, it is beneficial in that it is more practical for reasons of efficiency. In a special education classroom, I may have students who are all doing work on different levels or in a different format. A more general rubric will allow me to more easily adapt it to each individual student's needs.
McGatha, M. B. and Darcy, P. (2010). Rubrics at Play. Mathematics Teaching in the Middle School. 15(6), 328-336.
I found this article to be helpful in explaining the many different ways that a rubric can be used. I was not previously aware that there were so many types of rubrics, probably due to the fact that many of my past teachers and professors have used similar types of rubrics. Also, I found the section that described how a teacher included her students in the process of developing a rubric. Although the teacher did mention that this took an entire day of class time for math, I feel it was a worthwhile activity. This is an idea that I will hold on to and will seriously consider adopting for my own classroom. I believe that it empowers students and helps them to understand how grades are derived. Similarly, I found the idea of general rubrics to be of particular interest to me as a special educator. At first, I was skeptical that a general rubric could be effective; however, it is beneficial in that it is more practical for reasons of efficiency. In a special education classroom, I may have students who are all doing work on different levels or in a different format. A more general rubric will allow me to more easily adapt it to each individual student's needs.
McGatha, M. B. and Darcy, P. (2010). Rubrics at Play. Mathematics Teaching in the Middle School. 15(6), 328-336.
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