This article, written by 2 university professors, focused on increasing mathematical reasoning, and justification in minority students. The authors implemented an informal, after-school mathematics program. During the program they focused on a variety of fraction based concepts and had the choice to work individually, in pairs, or in groups. The goal for students was to problem solve then convince their peers of their answer. Two specific instances were explained in which this was successfully accomplished. The article concludes with 5 suggestions for teachers to out these ideas into action: Give students options for grouping, differentiate instruction through giving adequate time and extension options, let students share their ideas with the class, choose tools and tasks carefully, and hold high standards for all students.
I think that this article makes very excellent points. I am a huge proponent of the belief of the belief that every student can learn and these teachers should not only hold high expectations for all students, but also make these expectations known to students. No wonder many minority students are not achieving at the same level of their minority counterparts, since they often are being held to lower standards in the classroom and are not being given adequate opportunities to share in "thoughtful mathematics opportunities". I particularly liked the 5 suggestions to teachers at the end of the article. So often, I read educational journal articles and am left wondering, "How exactly can I implement this in a way that will work for me?" This article spells it out for teachers.
Thursday, April 29, 2010
Tuesday, April 27, 2010
Examining Math Manipulatives
1. How do you hold every student accountable for learning while using manipultives?
Teachers must put purposeful effort into holding each student accountable for their learning with manipulatives. There are a number of effective ways to do so. First, you could have students record some or all of the work that they do with the manipulatives onto a sheet of looseleaf paper or onto a teacher made "record sheet". This is different from a work sheet in that students are recording what they do with the manipulatives, not simply using the manipulatives to solve teacher made problems. Another way to hold students accountable is to have students journal after using manipulatives. Journal topics may vary and may include things such as "what did you learn at this center?", "What was your favorite and least favorite thing you did at this center?", or "did this manipulative help you with a math concept? Explain how it did or did not, and which concept". Finally, teachers could have students share what they have learned through using the manipulatives with the class. This is a great way to get students talking about math.
2. Why is the new emphasis on "Hands on, Minds on", instead of simply "Hands on".
All manipulatives are hands-on by nature. However, this does not mean that because students are using the manipulative, they will learn math. Manipulatives should be used as a tool for instruction, the key word being instruction. Students should be instructed or guided in some fashion in order that they are using the manipultive to help them meet the math objectives of the lesson. Only then is the activity both hands on and minds on. Handing first graders a set of base-ten blocks does not guarantee that they will use the manipulative in any way involving math. In fact, I would predict that given this situation (with no prior instruction using base 10 blocks) many first graders would build intricate buildings and towers. Building towers has educational value, though it is not mathematics and likely does not meet the learning standards of the lesson. In order to achieve a true hands on, minds on experience for students, teachers must couple good instructional strategies with the use of manipulatives. Similarly, as stated above teachers must hold all students accountable for their learning while using manipulatives.
3. How are the process standards used with the use of manipulatives?
Representation immediately sticks out to me when considering the use of the process standards in learning with math manipulatives. The nature of a manipulative is such that the manipulative itself is a way to represent math concepts. Showing students multiple representations, including 3 dimensional objects, is a key to a solid understanding for many students. Communication can very easily be incorporated into the use of manipulatives by having students work together in small groups to explore math concepts with manipulatives. Also, sharing ideas and findings with the class after an investigation activity using manipulatives is a way to improve students' communication of math concepts. Manipulatives can also incorporate the process standard of connections. This can be done especially with real world manipulatives, such as money. Problem solving is included in the use of manipultives when the teacher allows students to investigate how to use the manipulative to solve a problem, rather than providing direct step-by-step instructions on what to do and how to do it. Lastly, students should use manipulatives as tools to explain their reasoning of and to show proof that they have the correct answer.
Teachers must put purposeful effort into holding each student accountable for their learning with manipulatives. There are a number of effective ways to do so. First, you could have students record some or all of the work that they do with the manipulatives onto a sheet of looseleaf paper or onto a teacher made "record sheet". This is different from a work sheet in that students are recording what they do with the manipulatives, not simply using the manipulatives to solve teacher made problems. Another way to hold students accountable is to have students journal after using manipulatives. Journal topics may vary and may include things such as "what did you learn at this center?", "What was your favorite and least favorite thing you did at this center?", or "did this manipulative help you with a math concept? Explain how it did or did not, and which concept". Finally, teachers could have students share what they have learned through using the manipulatives with the class. This is a great way to get students talking about math.
2. Why is the new emphasis on "Hands on, Minds on", instead of simply "Hands on".
All manipulatives are hands-on by nature. However, this does not mean that because students are using the manipulative, they will learn math. Manipulatives should be used as a tool for instruction, the key word being instruction. Students should be instructed or guided in some fashion in order that they are using the manipultive to help them meet the math objectives of the lesson. Only then is the activity both hands on and minds on. Handing first graders a set of base-ten blocks does not guarantee that they will use the manipulative in any way involving math. In fact, I would predict that given this situation (with no prior instruction using base 10 blocks) many first graders would build intricate buildings and towers. Building towers has educational value, though it is not mathematics and likely does not meet the learning standards of the lesson. In order to achieve a true hands on, minds on experience for students, teachers must couple good instructional strategies with the use of manipulatives. Similarly, as stated above teachers must hold all students accountable for their learning while using manipulatives.
3. How are the process standards used with the use of manipulatives?
Representation immediately sticks out to me when considering the use of the process standards in learning with math manipulatives. The nature of a manipulative is such that the manipulative itself is a way to represent math concepts. Showing students multiple representations, including 3 dimensional objects, is a key to a solid understanding for many students. Communication can very easily be incorporated into the use of manipulatives by having students work together in small groups to explore math concepts with manipulatives. Also, sharing ideas and findings with the class after an investigation activity using manipulatives is a way to improve students' communication of math concepts. Manipulatives can also incorporate the process standard of connections. This can be done especially with real world manipulatives, such as money. Problem solving is included in the use of manipultives when the teacher allows students to investigate how to use the manipulative to solve a problem, rather than providing direct step-by-step instructions on what to do and how to do it. Lastly, students should use manipulatives as tools to explain their reasoning of and to show proof that they have the correct answer.
Monday, April 26, 2010
Errors Reflection
I think that analyzing common math errors that students make is an important component of a high quality teacher education program. Too many math teachers either were never taught how to carefully analyze student work to find the origin of their student's errors, or they do not take the time to do so. Too many teachers use the very ineffective teaching methods of "Louder and Slower" and "More of the Same", without spending the time and energy to understand where the student's misconception is. Though collecting adequate samples of a each student's work and carefully searching for common errors is very time consuming, I believe it will save in time spent teaching the student to understand the concept and achieve the correct answer. Once a teacher has identified a mistake that a student continually makes, he or she can hone in on correcting that particular piece of the puzzle.
Studying common student errors in math methods class helped make me a better future math teacher in two ways. First, it helped make me aware of particular mistakes that students commonly make. Second and more importantly, it taught me the skill of analyzing students work in order to identify a common error. I now know the types of errors students often make. Similarly, I know ways to prevent and correct these errors, such as teaching estimation and using manipulatives.
Studying common student errors in math methods class helped make me a better future math teacher in two ways. First, it helped make me aware of particular mistakes that students commonly make. Second and more importantly, it taught me the skill of analyzing students work in order to identify a common error. I now know the types of errors students often make. Similarly, I know ways to prevent and correct these errors, such as teaching estimation and using manipulatives.
Sunday, April 25, 2010
Overall Use of Technology in the Math Classroom
Using a wide variety of technologies throughout this course has helped open my eyes to the many ways that technology can assist in teaching and learning math. That said, it has also made me more critical of the use of technology in the classroom. It is my personal belief that everything should be done in moderation, especially when trying something new. New technologies should be used in conjunction with old methods that have proven to be effective teaching tools. Students should be learning using math applets, calculators, SMART boards, and other new technologies, as well as more conventional learning tools such as paper and pencil, base ten blocks (physical blocks to pick up in your hands, not on the computer), unifix cubes, and pattern blocks. Using many different learning tools that are both high and low tech is one way in which a teacher can teach to a variety of learning styles and preferences. For many students, the use of technology is motivating, while for others it is frustrating.
More specifically, I think that the SMART board can be a positive teaching tool, though I can think of very few applications of the SMART board that can not be done with other technologies that are more commonly found in classrooms (i.e. projector, overhead, dry erase board, computer). One application that I do see the SMART board as uniquely useful for is the ability to project a graph, make notes on it, and then save the page. One of my main frustrations in using the SMART board is the inaccuracy of writing with the pen. I think it is important to model good handwriting to students, and I can not do this with the SMART board pen.
More specifically, I think that the SMART board can be a positive teaching tool, though I can think of very few applications of the SMART board that can not be done with other technologies that are more commonly found in classrooms (i.e. projector, overhead, dry erase board, computer). One application that I do see the SMART board as uniquely useful for is the ability to project a graph, make notes on it, and then save the page. One of my main frustrations in using the SMART board is the inaccuracy of writing with the pen. I think it is important to model good handwriting to students, and I can not do this with the SMART board pen.
Monday, April 12, 2010
Making Techonology Work
In this article, 3 authors from Croatia explain why technology is necessary to incorporate into math education and how it can enhance the math curriculum. In summary, technology is beneficial because it enables students to work on more complex, often real-life, tasks, motivates students, and helps students visualize math concepts. However, technology is only useful in teaching when it's use has been thoughtfully evaluated and planned out. The authors described several specific programs and tools and their suggested uses in the classroom. The Geometer's Sketchpad is particularly liked by students and teachers because it allows students to focus more on overall concepts, rather than spending too much time on measuring and computations. Also, the article described a game in which students learn to differentiate when they should and should not use a calculator. Finally, the authors describe various real-life problems that students are able to analyze using various computer programs. Without these programs assistance, these problems would often be too advanced or too time consuming to implement.
I really enjoyed reading this article, as it has brought me one step closer to embracing more forms of technology for use in the math classroom. The article was extremely well written and concise, yet included excellent points and provided useful examples. I particularly liked the game in which students race to see if a computation can be solved faster by using a calculator or by doing mental math. I think that this is an excellent way to show students, rather than tell them, when they should choose to use a calculator as opposed to mental math. However, the idea of racing bothers me, especially the idea of having two students race against each other. What if one student is simply faster than another? This would skew results not to mention likely be the cause of hurt feelings. To alleviate this I may consider having students time themselves with a stop watch doing the problem first mentally, then using the calculator. This would not eliminate being timed, but at least it would minimize competition.
Soucie T., Radovic N. and Svedrec R. (2010). Making Technology Work. Mathematics Teaching in the Middle School, 15 (8), 467-471.
I really enjoyed reading this article, as it has brought me one step closer to embracing more forms of technology for use in the math classroom. The article was extremely well written and concise, yet included excellent points and provided useful examples. I particularly liked the game in which students race to see if a computation can be solved faster by using a calculator or by doing mental math. I think that this is an excellent way to show students, rather than tell them, when they should choose to use a calculator as opposed to mental math. However, the idea of racing bothers me, especially the idea of having two students race against each other. What if one student is simply faster than another? This would skew results not to mention likely be the cause of hurt feelings. To alleviate this I may consider having students time themselves with a stop watch doing the problem first mentally, then using the calculator. This would not eliminate being timed, but at least it would minimize competition.
Soucie T., Radovic N. and Svedrec R. (2010). Making Technology Work. Mathematics Teaching in the Middle School, 15 (8), 467-471.
Smorgasbord of Assessment Options
In this article a fifth grade teacher from Virginia explains the importance of student centered assessments which match the "target achievement" or goal. She uses a Virginia Standard of Geometry to focus her example assessments on. Assessment should produce useful information for both the students and the teachers and should influence future teaching. Student centered assessment means that the assessment centers around the concepts which have been learned, instead of what has been taught. Similarly, learning is a process which is scaffolded, and assessment should reflect this by assessing what has already been learned and pointing towards the "next step". Lastly, teachers must carefully choose assessment that provide the most valuable information in the least amount of time invested in assessing.
I most definitely agree with all of the main points of the article. I particularly identified with the point the author made on page 467 that thought the process and products may vary from classroom to classroom or student to student, the content should still remain the same in accordance with the standards. Likewise on page 468 the author outlines how student centered assessment is extremely beneficial in meeting the needs of all students, including those with special needs, those who are bilingual, and those who are gifted. As a special education major, I can see how creating student centered assessment is vital to gaining information which is useful to both the teacher and the student.
Bacon, K. A. (2010). Smorgasbord of Assessment Options. Teaching Children Mathematics.
458-469.
Wednesday, March 24, 2010
Portfolio Assessment
This article details how two middle school teachers began using portfolios as a way to enhance communication between parents, students, and teachers. The two teachers were looking for a way to emphasize student learning and growth, rather than letter grades. To do so, they designed two lessons on triangles using the "backward" design approach, and approach in which the teacher begins the lesson planning process by determining what they would like their students to gain from the lesson and which standards they will be meeting. The teacher then goes on to determine how they will assess the student and finally creates a procedure in which to teach students the material. After the two lessons on triangles were completed, students created portfolios, showcasing their work. The portfolios included a letter to their parents explaining their learning, examples of their work and assessments with commentary from students, a letter from the teacher, and a student/parent reflection sheet. Overall, both teachers felt that the portfolios and "backward" design was very beneficial. The portfolios increased parent understanding of what their children were learning as well as increased communication between parents and students. Another benefit is that students feel pride and ownership of their work when using portfolios. Also, the authors emphasized the fact that portfolios demonstrate whether or not students have met state and national standards more completely than some other methods assessment.
Britton, K. L. and Johannes, J. L. (2003). Portfolios and a Backward Approach to Assessment. Mathematics Teaching in the Middle School 9(2), 70-76.
Britton, K. L. and Johannes, J. L. (2003). Portfolios and a Backward Approach to Assessment. Mathematics Teaching in the Middle School 9(2), 70-76.
Monday, March 22, 2010
My Bar Graph Tells a Story: Teaching Children Mathematics
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.
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.
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.
Wednesday, February 3, 2010
PBL, Part 3: Comparison of Example PBLs
The first PBL I read was created for 7th-8th graders and was entitled "Lounging Around". In this PBL, students are given the assignment to create a new study/lounge area for all of the students in their school to use during free time. Students were required to budget and tell exactly what furniture and supplies they would include in the area. They were also required to use their prior knowledge of area and perimeter to be sure that all of the furniture fit properly. The final product was a scale model of their lounge area and a presentation. Students were assessed based on a rubric including their presentation, final project, and reflection of the project.
The second PBL I reviewed was created for 5th-6th graders and was entitled "Redo the Zoo". In this project each group member assumed a role, choosing from: Zoologist, Architect, Accountant, Horticulturalist, and Builder. In this PBL, students went on a field trip to the local zoo, and had a guest speaker from the zoo come in to introduce the project to the class. Students were required to propose a plan to redesign and construct the layout of the zoo buildings and exhibitions within a designated budget and time constraint. Students were also required to consider the needs of the various animals in their zoo when planning. The students were assessed informally during class work time and through the teacher reviewing journal entries. Also, the students created a portfolio and scale model as well as gave a presentation, which was all assessed with a rubric. Finally, students wrote a reflection on the PBL process.
I feel that the "Lounging Around" PBL was a bit too basic for 7th and 8th graders to spend 16 days on. I think that this was a very strong idea, because it is relevant to the students' lives and interesting to them; however, I feel that they should have included more requirements or components of the project. On the other hand, I found the "Redo the Zoo" PBL to be a bit out of reach for 5th and 6th graders. As a college student, I would feel lost given the assignment to budget $32 million dollars over five years for the construction of a zoo. I feel that something on a smaller scale, perhaps redoing one exhibit of the zoo, would be more manageable for this grade level. The zoo is a good idea, considering many 5th and 6th graders have been to a zoo and are interested in animals. I also thought that given the amount of mini lessons this group included in their PBL, 15 days was too short of a time frame to complete the project. Students need to be given adequate time in school to work together.
Reviewing the two PBLs solidified my understanding of what a PBL is and should look like. However, I was surprised by the length and depth of the projects. I was also surprised at the amount of structure, planning, and direct teaching that seemed to be involved. I was under the impression that this was kept more to a minimum because PBLs are student lead (in theory).
Both of the PBLs I reviewed were similar in that they required students to plan the design of a specific area. This has me wondering if all PBLs for the math methods class are required to follow this format. However, there were distinct differences between the two PBLs. "Lounging Around" included more open ended guiding questions than "Redo the Zoo" did. Also, the questions for "Lounging Around" seemed to be listed in an order that would foster chronological thinking, while "Redo the Zoo" contained guiding questions that seemed to be in a very illogical order.
As mentioned earlier, I felt that "Lounging Around" was not quite complex and challenging enough for 7th and 8th graders. I would add more mini lessons and more requirements, such as finding a building/contracting company to paint or make renovations necessary. This would have to be budgeted in also. In addition, students could be asked to add recreational games or computers. They would need to consider what would get the most use and analyze the cost of these items to decide which items are best.
I felt the "Redo the Zoo" was an excellent idea, but too large scale for 5th and 6th graders to complete in 15 days. If I were to change this PBL, I would have each group design a different exhibition of their choice within the zoo. Also, I thought that this group had an extremely messy idea web that needs some color coding and organization. Arrows should not dart all the way across the page.
I believe that in both of these PBLs math is the main focus of the project. However, the "Redo the Zoo" project also contains a significant amount of science in the research of the animals and thier needs. Similarly, "Lounging Around" includes quite a bit of art in the consideration of the decorations for the area. Both projects require intensive budgeting, area mapping, plotting on graphs, and other mathematical concepts.
Both PBLs included writing in a journal daily, which I feel is an appropriate method of assessing and evaluating math processes on an informal level. However, I did not find any mention in the "Lounging Around" PBL that the teacher was going to use informal assessment to review the journals, as was indicated in "Redo the Zoo". I feel that both final rubrics are more general and address the final product and presentations, rather than specific math concepts learned within the PBL unit. Students needed to use specific math concepts in order to complete their projects and meet the standards described in the rubrics. So, the rubrics are indirectly assessing those math concepts. However, teachers must keep in mind that since this is a group project and many students are collaborating, evaluating the final product is not an accurate method of measuring if a specific student understands a particular math concept.
The second PBL I reviewed was created for 5th-6th graders and was entitled "Redo the Zoo". In this project each group member assumed a role, choosing from: Zoologist, Architect, Accountant, Horticulturalist, and Builder. In this PBL, students went on a field trip to the local zoo, and had a guest speaker from the zoo come in to introduce the project to the class. Students were required to propose a plan to redesign and construct the layout of the zoo buildings and exhibitions within a designated budget and time constraint. Students were also required to consider the needs of the various animals in their zoo when planning. The students were assessed informally during class work time and through the teacher reviewing journal entries. Also, the students created a portfolio and scale model as well as gave a presentation, which was all assessed with a rubric. Finally, students wrote a reflection on the PBL process.
I feel that the "Lounging Around" PBL was a bit too basic for 7th and 8th graders to spend 16 days on. I think that this was a very strong idea, because it is relevant to the students' lives and interesting to them; however, I feel that they should have included more requirements or components of the project. On the other hand, I found the "Redo the Zoo" PBL to be a bit out of reach for 5th and 6th graders. As a college student, I would feel lost given the assignment to budget $32 million dollars over five years for the construction of a zoo. I feel that something on a smaller scale, perhaps redoing one exhibit of the zoo, would be more manageable for this grade level. The zoo is a good idea, considering many 5th and 6th graders have been to a zoo and are interested in animals. I also thought that given the amount of mini lessons this group included in their PBL, 15 days was too short of a time frame to complete the project. Students need to be given adequate time in school to work together.
Reviewing the two PBLs solidified my understanding of what a PBL is and should look like. However, I was surprised by the length and depth of the projects. I was also surprised at the amount of structure, planning, and direct teaching that seemed to be involved. I was under the impression that this was kept more to a minimum because PBLs are student lead (in theory).
Both of the PBLs I reviewed were similar in that they required students to plan the design of a specific area. This has me wondering if all PBLs for the math methods class are required to follow this format. However, there were distinct differences between the two PBLs. "Lounging Around" included more open ended guiding questions than "Redo the Zoo" did. Also, the questions for "Lounging Around" seemed to be listed in an order that would foster chronological thinking, while "Redo the Zoo" contained guiding questions that seemed to be in a very illogical order.
As mentioned earlier, I felt that "Lounging Around" was not quite complex and challenging enough for 7th and 8th graders. I would add more mini lessons and more requirements, such as finding a building/contracting company to paint or make renovations necessary. This would have to be budgeted in also. In addition, students could be asked to add recreational games or computers. They would need to consider what would get the most use and analyze the cost of these items to decide which items are best.
I felt the "Redo the Zoo" was an excellent idea, but too large scale for 5th and 6th graders to complete in 15 days. If I were to change this PBL, I would have each group design a different exhibition of their choice within the zoo. Also, I thought that this group had an extremely messy idea web that needs some color coding and organization. Arrows should not dart all the way across the page.
I believe that in both of these PBLs math is the main focus of the project. However, the "Redo the Zoo" project also contains a significant amount of science in the research of the animals and thier needs. Similarly, "Lounging Around" includes quite a bit of art in the consideration of the decorations for the area. Both projects require intensive budgeting, area mapping, plotting on graphs, and other mathematical concepts.
Both PBLs included writing in a journal daily, which I feel is an appropriate method of assessing and evaluating math processes on an informal level. However, I did not find any mention in the "Lounging Around" PBL that the teacher was going to use informal assessment to review the journals, as was indicated in "Redo the Zoo". I feel that both final rubrics are more general and address the final product and presentations, rather than specific math concepts learned within the PBL unit. Students needed to use specific math concepts in order to complete their projects and meet the standards described in the rubrics. So, the rubrics are indirectly assessing those math concepts. However, teachers must keep in mind that since this is a group project and many students are collaborating, evaluating the final product is not an accurate method of measuring if a specific student understands a particular math concept.
PBL, Part 2: Review of a Website
Dr. De Gallo at the University of California, Irvine explains the method of PBL teaching and learning in his web page entitled "What is Problem Based Learning?". PBL is student centered, meaning that the problem is in some way relevant to the student's life and that students have some input in designing the goals of the project. Relevant, student centered learning increases students' motivation to learn. Also, it is important to realize that students' prior knowledge will affect how they go about acquiring information and solving a problem in a PBL scenario. Teachers must try to understand their students prior knowledge and learning styles in order to provide the best guidance possible to their students. Also in an effort to coach their students, teachers ask questions of students that require them to reflect throughout the PBL process. A typical PBL problem is usually in the form of a specific "case" or real world scenario. PBL situations are very contextual and in order to reach a solution students must learn the content and apply it to the given situation.
This was a very informational web page that discussed many of the controversies involved with Problem Based Learning, such as the students' abilities and inabilities to identify what they need to learn. The article also discussed in detail the benefits of PBL, emphasizing students' increased motivation to learn. I appreciated the concise manner in which the material was presented. However, I thought that this article was lacking in that it did not outline any method of solving a PBL problem. It would have been beneficial for the readers if this had been included. Also, I felt that a brief example of a PBL scenario and students' solution would have helped readers contextualize the ideas presented. It seemed to me that pieces of this article were geared towards those with little to no knowledge of PBL, while other sections were geared for those who had some previous knowledge and were looking for a in depth analysis of PBL.
De Gallo & Grant, H. What is Problem Based Learning? Retrieved from http://www.pbl.uci.edu/whatispbl.html
This was a very informational web page that discussed many of the controversies involved with Problem Based Learning, such as the students' abilities and inabilities to identify what they need to learn. The article also discussed in detail the benefits of PBL, emphasizing students' increased motivation to learn. I appreciated the concise manner in which the material was presented. However, I thought that this article was lacking in that it did not outline any method of solving a PBL problem. It would have been beneficial for the readers if this had been included. Also, I felt that a brief example of a PBL scenario and students' solution would have helped readers contextualize the ideas presented. It seemed to me that pieces of this article were geared towards those with little to no knowledge of PBL, while other sections were geared for those who had some previous knowledge and were looking for a in depth analysis of PBL.
De Gallo & Grant, H. What is Problem Based Learning? Retrieved from http://www.pbl.uci.edu/whatispbl.html
PBL, Part 1: What is it and where is it used?
Problem Based Learning, or PBL, is a method of learning that centers around an "ill structured" and messy problem, which students must work in small groups to solve. In using this method, the teacher takes on the role of facilitator and coach, rather than dispenser of information. PBLs empower students to take control of their learning and to investigate to find the information and resources necessary to help them develop a solution. A typical procedure for students to follow when approaching a PBL usually begins with examining and identifying the problem, identifying what they already know, and identifying what they need to find out. Students then investigate and research, with guidance from the teacher or mentor, to gain to knowledge necessary. Next, students develop many possible solutions and work together to determine the best one. Finally, students present their solutions and reflect upon the process. PBLs are being used in classrooms across the content areas and across many grade levels, including higher education. Teachers report that while creating PBLs and giving up some control of the classroom to students is challenging at first, they feel it is worth the sacrifice. Students learn more and remember more when they research to gain information and develop solutions, rather than being "spoon fed" the information in the format of a lecture, worksheet, or textbook.
Thursday, January 28, 2010
Representation Journal: Cultural Capital in Children's Number Representations
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.
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.
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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