Core Proposition #2
Teachers know the subjects they teach and how to teach those subjects to students.
Element A: Teachers appreciate how knowledge in their subjects is created, organized and linked to other disciplines.
Artifact: Algebra II Astronomy Project - Student Example and Project Introduction
Event: Linking Algebra II to Astronomy.
Description: As a high school mathematics teacher, it is critical that I understand the math the students see prior to arriving to my class and how the math I am teaching them will prepare them for the math they will see after leaving my class. That is why I apply the Curriculum Principle listed in the National Council of Teachers of Mathematics (NCTM) book Principles and Standards for School Mathematics. The Curriculum Principle states that “a curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades” (NCTM, 2000, p. 14). When a curriculum is coherent, it effectively organizes and integrates important mathematical ideas so that students can see how ideas build on, or connect with, other ideas, and in result, enables them to develop new understandings and skills (NCTM, 2000). In addition to having organized curriculum, I strive to link my content to other disciplines. The following scenario describes how I link Logarithms in Algebra II to a lesson students are learning in Astronomy through a project.
Students in Algebra II are involved in a chapter that teaches them about Logarithms (logs). Students complete sections on rewriting logs and exponential functions, evaluating logs, graphing logs, applying properties of logs, and solving log equations. Once the students have learned the skills necessary, they are introduced to a project that shows how logs are used in astronomy. (See handout for project introduction).
First, students are introduced to the definitions of the terms used in astronomy. Next, the students and I work a couple of problems that they would solve if they were in an astronomy class. Two equations include apparent magnitude, absolute magnitude, and distance or diameter. It is often used to calculate a star’s distance. If the students know two of the three components, they can solve for the missing value. Lastly, in groups, the students are to create an instruction manual to give to their peers that explain how to manipulate an equation that they have been using in their astronomy class using logarithms. The manual should take the students’ peers through the process of changing the equation from logarithm form to exponential form and why it is beneficial to change from one form to the other. The project’s instructions and guidelines for grading are discussed before the students are released to work on their projects.
Analysis: A strategy that promotes achievement according to Cole (1995) is integrating mathematics with other content areas. Students who have experiences that link mathematics to other subject areas are able to apply previously acquired knowledge to new situations (Cole, 1995). Projects that focus on real-life issues make mathematics more relevant to students. I believe this project did just that. The students achieved more through application and they saw a connection with logarithms and real-life. This project was successful because students accurately applied what they learned in class and were able to explain it to their peers. I always tell my students that when you can explain and/or teach a concept to others, you truly understand it. In addition to showing students where logarithms are used in real life, this assignment proved to be authentic and effective because it gave the students an audience to present to and gave them the opportunity to interact with others while developing the manual. When students know that their work will be put on display for an audience, they take more pride in their work. This student example shows how students made their manual appealing for their audience (their peers), gave reasoning for manipulating the equation using logarithms, and accurately explained the steps in doing so.
Reflection: In the future, I will incorporate this project into my Algebra II Logarithm Unit again. The students developed some very creative and interesting manuals. It wasn’t until the project that I was asked some of the deeper questions about logarithms from my students. They seemed more engaged and more excited. Whenever I can avoid the question “When is this ever used in real-life”, I will. This was an instance where the project answered that question for me.
A change that I will make in the future for this project, however, is developing a more detailed rubric for the students to work from. The rubric will have the categories of “Appeal”, “Reasoning”, and “Accuracy”. Each category will have a description of what exactly will earn them a specific score. This will leave no questions about what I expect in the final product. Each year I do this, I hope to improve the implementation of this project.
Description: As a high school mathematics teacher, it is critical that I understand the math the students see prior to arriving to my class and how the math I am teaching them will prepare them for the math they will see after leaving my class. That is why I apply the Curriculum Principle listed in the National Council of Teachers of Mathematics (NCTM) book Principles and Standards for School Mathematics. The Curriculum Principle states that “a curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades” (NCTM, 2000, p. 14). When a curriculum is coherent, it effectively organizes and integrates important mathematical ideas so that students can see how ideas build on, or connect with, other ideas, and in result, enables them to develop new understandings and skills (NCTM, 2000). In addition to having organized curriculum, I strive to link my content to other disciplines. The following scenario describes how I link Logarithms in Algebra II to a lesson students are learning in Astronomy through a project.
Students in Algebra II are involved in a chapter that teaches them about Logarithms (logs). Students complete sections on rewriting logs and exponential functions, evaluating logs, graphing logs, applying properties of logs, and solving log equations. Once the students have learned the skills necessary, they are introduced to a project that shows how logs are used in astronomy. (See handout for project introduction).
First, students are introduced to the definitions of the terms used in astronomy. Next, the students and I work a couple of problems that they would solve if they were in an astronomy class. Two equations include apparent magnitude, absolute magnitude, and distance or diameter. It is often used to calculate a star’s distance. If the students know two of the three components, they can solve for the missing value. Lastly, in groups, the students are to create an instruction manual to give to their peers that explain how to manipulate an equation that they have been using in their astronomy class using logarithms. The manual should take the students’ peers through the process of changing the equation from logarithm form to exponential form and why it is beneficial to change from one form to the other. The project’s instructions and guidelines for grading are discussed before the students are released to work on their projects.
Analysis: A strategy that promotes achievement according to Cole (1995) is integrating mathematics with other content areas. Students who have experiences that link mathematics to other subject areas are able to apply previously acquired knowledge to new situations (Cole, 1995). Projects that focus on real-life issues make mathematics more relevant to students. I believe this project did just that. The students achieved more through application and they saw a connection with logarithms and real-life. This project was successful because students accurately applied what they learned in class and were able to explain it to their peers. I always tell my students that when you can explain and/or teach a concept to others, you truly understand it. In addition to showing students where logarithms are used in real life, this assignment proved to be authentic and effective because it gave the students an audience to present to and gave them the opportunity to interact with others while developing the manual. When students know that their work will be put on display for an audience, they take more pride in their work. This student example shows how students made their manual appealing for their audience (their peers), gave reasoning for manipulating the equation using logarithms, and accurately explained the steps in doing so.
Reflection: In the future, I will incorporate this project into my Algebra II Logarithm Unit again. The students developed some very creative and interesting manuals. It wasn’t until the project that I was asked some of the deeper questions about logarithms from my students. They seemed more engaged and more excited. Whenever I can avoid the question “When is this ever used in real-life”, I will. This was an instance where the project answered that question for me.
A change that I will make in the future for this project, however, is developing a more detailed rubric for the students to work from. The rubric will have the categories of “Appeal”, “Reasoning”, and “Accuracy”. Each category will have a description of what exactly will earn them a specific score. This will leave no questions about what I expect in the final product. Each year I do this, I hope to improve the implementation of this project.
Element B: Teachers command specialized knowledge of how to convey a subject to students.
Artifacts: Authentic Assignments Paper, Project Picture Gallery
Event: Algebra II Tennis Ball Parabola Project
Description: With my advanced Algebra II students this year, I decided to have the students mix two of my favorite things together in the form of a project: tennis and math! I coach high school tennis along with teaching math at my school. My vision was for the students to create an illusion of a tennis ball bouncing in my classroom. When a tennis ball bounces, its path is a parabola.
This was my first year doing this project so I decided to not try to create all of the instructions, guidelines, steps, etc. myself. I decided to let the students have some choice and creativity with the project. Basically, I bought the supplies (fishing wire, paper clips, tape, sewing needles and provided used tennis balls from the season) and told the students they were to make a tennis ball parabola that was mathematically correct in 5 days. The rest was up to them. As a class of 10 students, they decided where it should be located in my classroom, if it should just be one parabola or if it should have the effect of bouncing, and how to go about creating the product.
The first couple of days moved very slowly. Students were brainstorming about how it should look and what the best plan of action was. I was simply there monitoring and putting in my two cents every now and then. Students started to assign tasks to themselves and others. I had some students taping off the wall as they decided it should act as a big coordinate plane. I had other students taking initiative with sewing the fishing wire through the tennis balls (thank goodness because I wasn’t quite sure how that was going to work). The other students started playing around with numbers, values, and equations on the classroom whiteboard. A student or two acted as the messengers from each group of students to the other groups. Some students stepped up as leaders and others acted as followers. After some trial and error, the whole task just started clicking.
As the first 3 days were mostly brainstorming and trial and error, the last two days of this project were mostly construction. The students had figured out what they wanted to do, and how they wanted to do it. The “numbers” students were telling the “fishing line” students how long to make the strings, and then the “fishing line” students were telling the “coordinate plane” students were to hang the balls. By the end, I was extremely happy with the result. Here are two pictures of the result.
Analysis: To me, specialized knowledge means taking a subject area and doing more with it than just teaching the facts. It means making the subject real, applicable, and going above and beyond for the students. Creating this project did just that. It was an authentic assignment. Authentic assignments are extremely beneficial to all students and even more for English Second Language learners. Students can see, hear, feel, perform, create, and participate in order to make connections and construct personal, relevant meanings (Echevarria & Short, 2013). This was especially important for my student from Korea, Brian. He was one of the 10 students in this Algebra II class who made the parabola. At the time this project was constructed, Brian had only been in the U.S. for three months. Even though he was very bright in math, I feel that he struggled with the language barrier. This project was very helpful to him to learn the unit’s vocabulary and build his English language skills.
I feel that this project was successful and authentic for several other reasons as well. For starters, this project showed the students where math was used in real life. If someone was wanted to create architecture in the form of a parabola, they would need to know the characteristics of a parabola and its function. Therefore, the students found meaning and a purpose with this assignment. Also, the students were engaged throughout the whole process. They were extremely excited to face this challenge of being my first group of students to create this project. They wanted to prove that they could do it, and they had the choice to do it however they felt was best. Another successful attribute of this project is that it required my students to take risks. I did not give them much guidance at all and the students had to problem solve just like if they were given a task from their future employer. They had to try things that may have not worked out and learn from failure. Learning from trial and error is a very important experience for students to have. Lastly, I knew this project was successful because the students claimed ownership of their work. They knew that their peers, other teachers, and their parents would see their final product since it remained in my classroom the entire year. They were proud of their hard work.
Included here are pictures of students working throughout the process. As you can see, students were all actively engaged, taking risks, learning from trial and error, and taking pride in their work. What is not pictured here is how intrigued my Algebra I students were with the project, the discussion we had about Algebra I concepts and Algebra II concepts, and all the requests to do a project like that for them.
Reflection: Not only did this project require the students to take risks, but it was a risk for me as the teacher. I like to be very organized, know where my students are heading, and have clear instructions and expectations for my students. With this project, not all of this happened. In fact, I had no idea how this was going to turn out and I certainly didn’t give my students an organized list of instructions. This was a huge risk for me, and I am glad I took it. This project was a great learning experience for my students and it was inspirational for my Algebra I students. In the future, I will allow my students to create the parabola they want and offer input from previous years only if the students want to hear it. I am guessing that each year students will do something new and different. Students are more creative than we give them credit!
Furthermore, I want to try to incorporate more projects like this in the future. I want to do this for more units and for all of my classes, not just Algebra II. With the reaction from my Algebra I students I think I will allow them to create a linear function on another wall. The possibilities are endless.
Description: With my advanced Algebra II students this year, I decided to have the students mix two of my favorite things together in the form of a project: tennis and math! I coach high school tennis along with teaching math at my school. My vision was for the students to create an illusion of a tennis ball bouncing in my classroom. When a tennis ball bounces, its path is a parabola.
This was my first year doing this project so I decided to not try to create all of the instructions, guidelines, steps, etc. myself. I decided to let the students have some choice and creativity with the project. Basically, I bought the supplies (fishing wire, paper clips, tape, sewing needles and provided used tennis balls from the season) and told the students they were to make a tennis ball parabola that was mathematically correct in 5 days. The rest was up to them. As a class of 10 students, they decided where it should be located in my classroom, if it should just be one parabola or if it should have the effect of bouncing, and how to go about creating the product.
The first couple of days moved very slowly. Students were brainstorming about how it should look and what the best plan of action was. I was simply there monitoring and putting in my two cents every now and then. Students started to assign tasks to themselves and others. I had some students taping off the wall as they decided it should act as a big coordinate plane. I had other students taking initiative with sewing the fishing wire through the tennis balls (thank goodness because I wasn’t quite sure how that was going to work). The other students started playing around with numbers, values, and equations on the classroom whiteboard. A student or two acted as the messengers from each group of students to the other groups. Some students stepped up as leaders and others acted as followers. After some trial and error, the whole task just started clicking.
As the first 3 days were mostly brainstorming and trial and error, the last two days of this project were mostly construction. The students had figured out what they wanted to do, and how they wanted to do it. The “numbers” students were telling the “fishing line” students how long to make the strings, and then the “fishing line” students were telling the “coordinate plane” students were to hang the balls. By the end, I was extremely happy with the result. Here are two pictures of the result.
Analysis: To me, specialized knowledge means taking a subject area and doing more with it than just teaching the facts. It means making the subject real, applicable, and going above and beyond for the students. Creating this project did just that. It was an authentic assignment. Authentic assignments are extremely beneficial to all students and even more for English Second Language learners. Students can see, hear, feel, perform, create, and participate in order to make connections and construct personal, relevant meanings (Echevarria & Short, 2013). This was especially important for my student from Korea, Brian. He was one of the 10 students in this Algebra II class who made the parabola. At the time this project was constructed, Brian had only been in the U.S. for three months. Even though he was very bright in math, I feel that he struggled with the language barrier. This project was very helpful to him to learn the unit’s vocabulary and build his English language skills.
I feel that this project was successful and authentic for several other reasons as well. For starters, this project showed the students where math was used in real life. If someone was wanted to create architecture in the form of a parabola, they would need to know the characteristics of a parabola and its function. Therefore, the students found meaning and a purpose with this assignment. Also, the students were engaged throughout the whole process. They were extremely excited to face this challenge of being my first group of students to create this project. They wanted to prove that they could do it, and they had the choice to do it however they felt was best. Another successful attribute of this project is that it required my students to take risks. I did not give them much guidance at all and the students had to problem solve just like if they were given a task from their future employer. They had to try things that may have not worked out and learn from failure. Learning from trial and error is a very important experience for students to have. Lastly, I knew this project was successful because the students claimed ownership of their work. They knew that their peers, other teachers, and their parents would see their final product since it remained in my classroom the entire year. They were proud of their hard work.
Included here are pictures of students working throughout the process. As you can see, students were all actively engaged, taking risks, learning from trial and error, and taking pride in their work. What is not pictured here is how intrigued my Algebra I students were with the project, the discussion we had about Algebra I concepts and Algebra II concepts, and all the requests to do a project like that for them.
Reflection: Not only did this project require the students to take risks, but it was a risk for me as the teacher. I like to be very organized, know where my students are heading, and have clear instructions and expectations for my students. With this project, not all of this happened. In fact, I had no idea how this was going to turn out and I certainly didn’t give my students an organized list of instructions. This was a huge risk for me, and I am glad I took it. This project was a great learning experience for my students and it was inspirational for my Algebra I students. In the future, I will allow my students to create the parabola they want and offer input from previous years only if the students want to hear it. I am guessing that each year students will do something new and different. Students are more creative than we give them credit!
Furthermore, I want to try to incorporate more projects like this in the future. I want to do this for more units and for all of my classes, not just Algebra II. With the reaction from my Algebra I students I think I will allow them to create a linear function on another wall. The possibilities are endless.
Element C: Teachers generate multiple paths to knowledge.
Artifacts: Absolute Value Functions and Their Graphs Lesson Plan
Event: A 3 day lesson over absolute value functions and their graphs.
Description: This Absolute Value Functions and Their Graphs Lesson Plan maps out a 3 day unit over absolute value functions and their graphs for 9th graders in Algebra I. This is just an example of what all of my units are like. In each unit I strive to allow all my students of all levels to experience a variety of teaching strategies to help them reach their learning goals. Within this specific unit, students were involved in activities that included cooperative learning, technology, and instruction that fit several learning styles and multiple intelligences of my students.
Throughout the unit students were assessed using formative and summative assessments. Formative assessments include 3 homework assignments over the 3-day unit, body aerobics activity, response boards, and a matching card activity. The summative assessment for this unit was the chapter test that included absolute value questions. This summative assessment came at the end of the unit, not at the end of the 3-day unit.
Analysis: This unit was successful because students experienced a variety of learning strategies and multiple intelligences that catered to their strengths or improved their weaknesses. As mentioned in Element B of Core Proposition #1, Howard Gardner’s multiple intelligences (linguistic, musical, spatial, kinesthetic, mathematical, interpersonal, intrapersonal, and naturalistic) need to be included in lessons as much as possible. For example, the Verbal-Linguistic intelligence was represented through students’ openings and closings when students wrote reflections. The Body-Kinesthetic intelligence was incorporated with the body aerobics that took place each day. The Interpersonal intelligence was implemented with a cooperative learning memory matching card activity. And of course, the Logical-Mathematical intelligence is put in to practice every day, all day, since we are working on math!
This unit also catered to my students’ learning styles. A learning style is a way a person goes about learning. The four dimensions that make up the learning style model are sensing, thinking, feeling, and intuition (Silver, Strong, & Perini, 2000). From those dimensions, people fall somewhere in a quadrant that tells them their learning style: Master Style, Interpersonal Style, Understanding Style, and Self-Expressive Style. Mastery Style learners (or Sensing-Thinking learners) did well with the steps I gave them in the notes portion of the unit. They don’t ask “Why?” They just want to know what to do and how to do it. Interpersonal Style learners (or Sensing-Feeling learners) are concerned with completing a task when it makes them or others feel good in addition to having an effect on people’s lives. They like to work in groups and feel comfortable working with others. The Matching Card activity played to their strength. Self-Expressive Style learners (or Intuitive-Feeling learners like to explore and answer the question “What would happen if…?” If the student is not interested in the task, they may not complete it. The incorporation of the graphing calculator “Investigations” activity was good for these students. They were free to work at their own pace to discover what makes an absolute value function’s graph move up, down, left, right, upside-down, etc. They were able to play around with the problems and discover things for themselves.
Furthermore, the variety of activities allowed for formative assessment. With several formative assessments, I was able to adjust my lesson plans if I felt it was needed. Sometimes lesson plans need to be adjusted and sometimes they do not. No plans needed to be adjusted during this unit.
Reflection: Choosing to include so many different paths to the understanding of absolute values equations and their graphs was successful and I plan on doing the same in the future. The students really enjoy the variety of instruction, activities, and formative assessments. It keeps the class lively and avoids monotony. This is beneficial to me as the teacher as well. I don’t get bored with the same thing over and over, I avoid the frustration that comes along with students not putting in the effort because they are bored, and I am a much better/positive teacher because of it.
Description: This Absolute Value Functions and Their Graphs Lesson Plan maps out a 3 day unit over absolute value functions and their graphs for 9th graders in Algebra I. This is just an example of what all of my units are like. In each unit I strive to allow all my students of all levels to experience a variety of teaching strategies to help them reach their learning goals. Within this specific unit, students were involved in activities that included cooperative learning, technology, and instruction that fit several learning styles and multiple intelligences of my students.
Throughout the unit students were assessed using formative and summative assessments. Formative assessments include 3 homework assignments over the 3-day unit, body aerobics activity, response boards, and a matching card activity. The summative assessment for this unit was the chapter test that included absolute value questions. This summative assessment came at the end of the unit, not at the end of the 3-day unit.
Analysis: This unit was successful because students experienced a variety of learning strategies and multiple intelligences that catered to their strengths or improved their weaknesses. As mentioned in Element B of Core Proposition #1, Howard Gardner’s multiple intelligences (linguistic, musical, spatial, kinesthetic, mathematical, interpersonal, intrapersonal, and naturalistic) need to be included in lessons as much as possible. For example, the Verbal-Linguistic intelligence was represented through students’ openings and closings when students wrote reflections. The Body-Kinesthetic intelligence was incorporated with the body aerobics that took place each day. The Interpersonal intelligence was implemented with a cooperative learning memory matching card activity. And of course, the Logical-Mathematical intelligence is put in to practice every day, all day, since we are working on math!
This unit also catered to my students’ learning styles. A learning style is a way a person goes about learning. The four dimensions that make up the learning style model are sensing, thinking, feeling, and intuition (Silver, Strong, & Perini, 2000). From those dimensions, people fall somewhere in a quadrant that tells them their learning style: Master Style, Interpersonal Style, Understanding Style, and Self-Expressive Style. Mastery Style learners (or Sensing-Thinking learners) did well with the steps I gave them in the notes portion of the unit. They don’t ask “Why?” They just want to know what to do and how to do it. Interpersonal Style learners (or Sensing-Feeling learners) are concerned with completing a task when it makes them or others feel good in addition to having an effect on people’s lives. They like to work in groups and feel comfortable working with others. The Matching Card activity played to their strength. Self-Expressive Style learners (or Intuitive-Feeling learners like to explore and answer the question “What would happen if…?” If the student is not interested in the task, they may not complete it. The incorporation of the graphing calculator “Investigations” activity was good for these students. They were free to work at their own pace to discover what makes an absolute value function’s graph move up, down, left, right, upside-down, etc. They were able to play around with the problems and discover things for themselves.
Furthermore, the variety of activities allowed for formative assessment. With several formative assessments, I was able to adjust my lesson plans if I felt it was needed. Sometimes lesson plans need to be adjusted and sometimes they do not. No plans needed to be adjusted during this unit.
Reflection: Choosing to include so many different paths to the understanding of absolute values equations and their graphs was successful and I plan on doing the same in the future. The students really enjoy the variety of instruction, activities, and formative assessments. It keeps the class lively and avoids monotony. This is beneficial to me as the teacher as well. I don’t get bored with the same thing over and over, I avoid the frustration that comes along with students not putting in the effort because they are bored, and I am a much better/positive teacher because of it.
My Goals:
1) To improve projects that link mathematics to other disciplines that I already implement in my curriculum.
2) To implement more projects, in which some link mathematics to other disciplines, for more units in each of my classes each school year.
3) To continue to use a variety of teaching methods to cater to the variety of my students’ learning styles.
1) To improve projects that link mathematics to other disciplines that I already implement in my curriculum.
2) To implement more projects, in which some link mathematics to other disciplines, for more units in each of my classes each school year.
3) To continue to use a variety of teaching methods to cater to the variety of my students’ learning styles.
References
Cole, R. (1995). Educating everybody’s children: Diverse learning strategies for diverse learners. Alexandria, VA: ASCD.
Echevarria, J., Vogt, M., & Short, D. (2013). Making content comprehensible for English learners: The SIOP model. 4th ed. Boston: Pearson Allyn & Bacon.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Silver, H. F., Strong, R. W., & Perini, M. J. (2000). So each may learn: Integrating learning styles and multiple intelligences. Alexandria, VA: Association for Supervision and Curriculum Development.
Cole, R. (1995). Educating everybody’s children: Diverse learning strategies for diverse learners. Alexandria, VA: ASCD.
Echevarria, J., Vogt, M., & Short, D. (2013). Making content comprehensible for English learners: The SIOP model. 4th ed. Boston: Pearson Allyn & Bacon.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Silver, H. F., Strong, R. W., & Perini, M. J. (2000). So each may learn: Integrating learning styles and multiple intelligences. Alexandria, VA: Association for Supervision and Curriculum Development.