Phase III: Action Plan
I. Identify desired results
The heart of my ImagineIT project will beat with four valves: why the real number line (RNL) matters, what comprises it, how we can see it through new lenses and how it has utility in mathematics and in the world.
The RNL is one of the most basic structures of mathematics, something so foundational that we often take it for granted and are content with a surface understanding of it. But like good poetry, it has many layers and offers opportunities for deep levels of understanding.
Nearly every topic I teach in algebra can be connected back to the RNL: operations with real numbers, solving equations, graphing equations, slope and slope-intercept form, systems of equations, exponential functions & quadratic equations. When solving equations, we can plot our answer(s) on the RNL – except for the cases when we cannot, as when there is no solution. When comparing slopes of lines, the farther we move in either direction of zero, the greater the steepness of a given line. If I was limited to teaching just two concepts each year, they would be these: understanding equations and slope. They’re robust topics that have connections to other mathematical disciplines, such as geometry, statistics, calculus and physics, and the world teems with applications of equations and slope. And, both concepts connect in a natural and meaningful way to the RNL.
Part of the essential knowledge base deals with rational numbers. Becoming comfortable with fraction arithmetic is necessary to grasp each of the topics in the previous paragraph. Knowing how to place fractions on a number line is important for a variety of reasons: it provides context or a reference point, it allows us to compare numbers, it offers a way to make sense of results, like , etcetera. Without this essential knowledge base, one simply cannot move beyond a 3rd grade understanding of mathematics.
Developing a solid knowledge base and deep understanding of the RNL means creating a systematic approach to connecting each topic of study to this structure. Allowing students time to explore the line, something I haven’t tried before, seems worthwhile. For instance, what observations about the line can be made? What different types of numbers lie on the line, and why do these numbers matter? Keeping this activity open-ended and exploratory given a few parameters can pique student curiosity and build buy-in. Students can walk and talk the number line in multiple ways. Here’s one: If we zoom in to 0 and 1, place our feet at 1, and move halfway to 0, landing us at one-half, and then cut that distance in half, and each subsequent distance in half, will we ever reach 0? What does this show us and mean? This can motivate operations with fractions, which can inform our understanding of equations, slope, systems of equations, and so on.
In the end, I envision each unit’s connection to the RNL will lead to a deeper understanding of and appreciation for it. Students can represent this knowledge in various ways – through story telling or song or debate or pictures – that allow them to interact with the line in creative, meaningful ways. Placing the burden on students to connect this structure to their own lives, present or future, will allow the line to percolate in their minds outside of the 51 minutes I see them each day.
II. Determine acceptable evidence (performances of understanding)
I like the idea of starting with a traditional pre-test followed by multiple ‘post-tests’ throughout the year. Tracking student performance on a line graph will show valuable trends. Are students able to retain understanding of the RNL over time? At what point did they understand it the best? What supports were put in place that might explain this?
I also like the idea of surveys, perhaps given via Survey Monkey, to allow students to self-assess. This would draw out Wiggins’ & McTighe’s idea of having self-knowledge. This process of metacognition would allow students and me to identify and constructively criticize some of their thought processes. This information could naturally lead to small group discussions followed by whole class talks. (Of course, the surveys would be anonymous, which should allow students to feel safe in diving deep into their thinking.) I envision the whole class discussions to be centered on clearing up amnesia and fantasia and offering ways to stave off inertia. Misconceptions would be brought to the front and would allow the class to critique ‘facts’ and gaps in understanding.
Having students conduct interviews* with each other to tease out misconceptions will be a productive alternative to surveys. I can use the idea of surveys with equations and interviews with slope. I can also incorporate an interview with several students in front of the whole class as demonstrated through the improvisation workshop. This can be a nice way to review for a summative assessment featuring the RNL. Creating videos of interviews and showing them to the class might allow for more whole class discussion centered on clearing up misconceptions about each topic as it relates to the line.
I’d like to build on a debate format I’ve used with equations and the difference between a solution of and no solution. Tying this to the RNL is an obvious connection. Creating this format for exponents, especially zero and negative exponents (the bane of many former students’ existence), is something I will do to try to (a) bring out misconceptions in an engaging way and (b) build buy-in to what is often perceived as a dry topic.
Building empathy into each activity is something that I can do with the surveys and debate most easily. I plan to give students the example of multiple infinities, which happens to also connect to the RNL. I’ll show them the video by Dennis Wildfogel, ask them their thoughts and then walk them through how difficult it was for me to wrestle with these ideas. Empathy, I’ll explain, comes from remembering that what’s now easy to understand wasn’t always so. I’ll coach students to be mindful of this throughout the year, especially when they’re exposed to the (mis)conceptions of their classmates.
The common thread among pre-test/post-test analysis, surveys, interviews and debates is that they make public the common misconceptions, and each of these platforms connects to the curriculum in an authentic way. I would like to use these formats once per unit and evaluate performances in mostly a formative way. Students would receive feedback verbally in most cases, though I could summarize the class’ findings – or better yet, have the students do this – on posterboard.
*I know, I’m stealing the ideas listed in the directions – but they’re good ideas that gave me food for thought!
III. Plan learning experience and instruction
A. Context
I work as a high school math teacher. I am afforded a Mobi Interwrite device that connects to an LCD projector. This allows me to project what I write on the Mobi as I circulate through the classroom, which helps with classroom management, engagement, and one-on-one assistance. Constraints include not being a 1:1 school and not having my own classroom; that is, I share multiple rooms with colleagues. The focus of the project will be my 9th grade Regular Algebra I classes. Generally, I teach kind students who are reluctant learners of math. Many have trouble with integer or fraction arithmetic and number sense, so for them, algebra is a big leap. This is evidenced by last year’s average math ACT score, 18.6, which is about 2.5 points lower than the statewide or nationwide average. On a given day, more than 50% of my students ‘do’ the homework, though I’m skeptical how much is actually learned based on summative assessments. That said, nearly every student works in class, and I’ve found that the more experiential I can make the math, the more willing the students are to put forth a good-faith effort. My technology support is adequate, though burned out LCD projector bulbs can mean the only working technology in the room is my laptop and students’ cell phones.
B. Content
I want my students to become more comfortable with, and have a deeper understanding of, numbers. We often call this number sense. Some students shut down the moment they see a fraction or a negative number. These are meaningless to them, and this is something I want to change. By building comfort and understanding with manipulating numbers, some of the bigger ideas of algebra (exponents, equations, slope) are more likely to be understood because students aren’t bogged down, or lost, in the arithmetic.
For the sake of brevity I’ll only focus on exponents. Each year, students struggle mightily with really understanding the rules despite my effort to make the topic concrete (‘when in doubt, write it out’ is our refrain) and logical (we walk through the reasons why positive integer bases raised to negative or zero exponents must equal the results they do). Making exponents more experiential is a new tack. Working in the structure of the RNL will tie something students know, the coordinate plane, with something they don’t, exponents.
C. Pedagogy
I think that a combination of pedagogical approaches will allow students multiple access points to the RNL and how each topic connects to it. Storytelling – creating a picture book – can allow students a creative way to cement their understanding of the dry laws of exponents. Affording students the time and space to bat around ideas of equations or slope through structured debate will bring to life topics that usually fall into the realm of pencil and paper work. Interviewing students – or perhaps having students interview a number, with research about significance, applications, relationship to other numbers, etc. – can also bring to life an idea that we jot in our notebook or that just sits on a number line. Capturing and holding students’ attention with respect to infinity, undefined and zero, and how each relates to exponents, equations and slope, is more likely to happen when students interact with the material through storytelling, interviews, surveys and debates. Gallery walks will allow students to showcase their thought processes with each topic and how it relates to the RNL. Each of these ideas incorporates aspects of Heath’s & Heath’s “Teaching that Sticks” traits: simple, unexpected, concrete, credible, emotional and story.
D. Technology
Accessing Prosser’s computer lab or cell phones to conduct research is one way in which we can use technology to build toward the activities mentioned above. Having students record one another during interviews, and then posting these interviews on YouTube, will allow me to share ideas with the entire class in a different way. Allowing students time to explore with Desmos in order to build their understanding of slope and slope-intercept form is something I’m excited to try. I need to play around with it, but PowToon looks like a platform that could allow students to showcase the concepts they’re wrestling with and making sense of. Perhaps I won’t be able to try all of these ideas this year, but the idea that I can implement some of them excites me.
E. Summary & Demonstration
The Total PACKage: I feel that my plan incorporates TPACK organically and dynamically. Nothing feels forced, although much of this is new to me. The technology component is the weakest, in part because of its newness and my constraints. What do you think?
The teaching demonstration will focus on the RNL and how I plan to relate each topic this year to it in a meaningful way. I plan to use PowToons for the demonstration. I’ll give an overview of the four valves that comprise the heart of the RNL: why the RNL matters, what it’s made of, how we can see it through new lenses and how it has utility in mathematics and in the world. I’ll share examples of performances of understanding, including the class’ first project. I may even share my i-Video to set the stage. Part of my motivation for sharing this will be to show a creative idea while reassuring students (during the demonstration with students) that difficulty with execution of the idea, which was my issue, is part of the process.
The first project will focus on the real number line and the relationship among the numbers on it. Students will tell a creative story and draw pictures representing their story. This activity, while seemingly simple, will require students to include integers, rational numbers and irrational numbers in their story. These are definitions they likely do not know, which is part of the point of the project. Marks will be based on creativity, variety of numbers and the relationship among those numbers.
The heart of my ImagineIT project will beat with four valves: why the real number line (RNL) matters, what comprises it, how we can see it through new lenses and how it has utility in mathematics and in the world.
The RNL is one of the most basic structures of mathematics, something so foundational that we often take it for granted and are content with a surface understanding of it. But like good poetry, it has many layers and offers opportunities for deep levels of understanding.
Nearly every topic I teach in algebra can be connected back to the RNL: operations with real numbers, solving equations, graphing equations, slope and slope-intercept form, systems of equations, exponential functions & quadratic equations. When solving equations, we can plot our answer(s) on the RNL – except for the cases when we cannot, as when there is no solution. When comparing slopes of lines, the farther we move in either direction of zero, the greater the steepness of a given line. If I was limited to teaching just two concepts each year, they would be these: understanding equations and slope. They’re robust topics that have connections to other mathematical disciplines, such as geometry, statistics, calculus and physics, and the world teems with applications of equations and slope. And, both concepts connect in a natural and meaningful way to the RNL.
Part of the essential knowledge base deals with rational numbers. Becoming comfortable with fraction arithmetic is necessary to grasp each of the topics in the previous paragraph. Knowing how to place fractions on a number line is important for a variety of reasons: it provides context or a reference point, it allows us to compare numbers, it offers a way to make sense of results, like , etcetera. Without this essential knowledge base, one simply cannot move beyond a 3rd grade understanding of mathematics.
Developing a solid knowledge base and deep understanding of the RNL means creating a systematic approach to connecting each topic of study to this structure. Allowing students time to explore the line, something I haven’t tried before, seems worthwhile. For instance, what observations about the line can be made? What different types of numbers lie on the line, and why do these numbers matter? Keeping this activity open-ended and exploratory given a few parameters can pique student curiosity and build buy-in. Students can walk and talk the number line in multiple ways. Here’s one: If we zoom in to 0 and 1, place our feet at 1, and move halfway to 0, landing us at one-half, and then cut that distance in half, and each subsequent distance in half, will we ever reach 0? What does this show us and mean? This can motivate operations with fractions, which can inform our understanding of equations, slope, systems of equations, and so on.
In the end, I envision each unit’s connection to the RNL will lead to a deeper understanding of and appreciation for it. Students can represent this knowledge in various ways – through story telling or song or debate or pictures – that allow them to interact with the line in creative, meaningful ways. Placing the burden on students to connect this structure to their own lives, present or future, will allow the line to percolate in their minds outside of the 51 minutes I see them each day.
II. Determine acceptable evidence (performances of understanding)
I like the idea of starting with a traditional pre-test followed by multiple ‘post-tests’ throughout the year. Tracking student performance on a line graph will show valuable trends. Are students able to retain understanding of the RNL over time? At what point did they understand it the best? What supports were put in place that might explain this?
I also like the idea of surveys, perhaps given via Survey Monkey, to allow students to self-assess. This would draw out Wiggins’ & McTighe’s idea of having self-knowledge. This process of metacognition would allow students and me to identify and constructively criticize some of their thought processes. This information could naturally lead to small group discussions followed by whole class talks. (Of course, the surveys would be anonymous, which should allow students to feel safe in diving deep into their thinking.) I envision the whole class discussions to be centered on clearing up amnesia and fantasia and offering ways to stave off inertia. Misconceptions would be brought to the front and would allow the class to critique ‘facts’ and gaps in understanding.
Having students conduct interviews* with each other to tease out misconceptions will be a productive alternative to surveys. I can use the idea of surveys with equations and interviews with slope. I can also incorporate an interview with several students in front of the whole class as demonstrated through the improvisation workshop. This can be a nice way to review for a summative assessment featuring the RNL. Creating videos of interviews and showing them to the class might allow for more whole class discussion centered on clearing up misconceptions about each topic as it relates to the line.
I’d like to build on a debate format I’ve used with equations and the difference between a solution of and no solution. Tying this to the RNL is an obvious connection. Creating this format for exponents, especially zero and negative exponents (the bane of many former students’ existence), is something I will do to try to (a) bring out misconceptions in an engaging way and (b) build buy-in to what is often perceived as a dry topic.
Building empathy into each activity is something that I can do with the surveys and debate most easily. I plan to give students the example of multiple infinities, which happens to also connect to the RNL. I’ll show them the video by Dennis Wildfogel, ask them their thoughts and then walk them through how difficult it was for me to wrestle with these ideas. Empathy, I’ll explain, comes from remembering that what’s now easy to understand wasn’t always so. I’ll coach students to be mindful of this throughout the year, especially when they’re exposed to the (mis)conceptions of their classmates.
The common thread among pre-test/post-test analysis, surveys, interviews and debates is that they make public the common misconceptions, and each of these platforms connects to the curriculum in an authentic way. I would like to use these formats once per unit and evaluate performances in mostly a formative way. Students would receive feedback verbally in most cases, though I could summarize the class’ findings – or better yet, have the students do this – on posterboard.
*I know, I’m stealing the ideas listed in the directions – but they’re good ideas that gave me food for thought!
III. Plan learning experience and instruction
A. Context
I work as a high school math teacher. I am afforded a Mobi Interwrite device that connects to an LCD projector. This allows me to project what I write on the Mobi as I circulate through the classroom, which helps with classroom management, engagement, and one-on-one assistance. Constraints include not being a 1:1 school and not having my own classroom; that is, I share multiple rooms with colleagues. The focus of the project will be my 9th grade Regular Algebra I classes. Generally, I teach kind students who are reluctant learners of math. Many have trouble with integer or fraction arithmetic and number sense, so for them, algebra is a big leap. This is evidenced by last year’s average math ACT score, 18.6, which is about 2.5 points lower than the statewide or nationwide average. On a given day, more than 50% of my students ‘do’ the homework, though I’m skeptical how much is actually learned based on summative assessments. That said, nearly every student works in class, and I’ve found that the more experiential I can make the math, the more willing the students are to put forth a good-faith effort. My technology support is adequate, though burned out LCD projector bulbs can mean the only working technology in the room is my laptop and students’ cell phones.
B. Content
I want my students to become more comfortable with, and have a deeper understanding of, numbers. We often call this number sense. Some students shut down the moment they see a fraction or a negative number. These are meaningless to them, and this is something I want to change. By building comfort and understanding with manipulating numbers, some of the bigger ideas of algebra (exponents, equations, slope) are more likely to be understood because students aren’t bogged down, or lost, in the arithmetic.
For the sake of brevity I’ll only focus on exponents. Each year, students struggle mightily with really understanding the rules despite my effort to make the topic concrete (‘when in doubt, write it out’ is our refrain) and logical (we walk through the reasons why positive integer bases raised to negative or zero exponents must equal the results they do). Making exponents more experiential is a new tack. Working in the structure of the RNL will tie something students know, the coordinate plane, with something they don’t, exponents.
C. Pedagogy
I think that a combination of pedagogical approaches will allow students multiple access points to the RNL and how each topic connects to it. Storytelling – creating a picture book – can allow students a creative way to cement their understanding of the dry laws of exponents. Affording students the time and space to bat around ideas of equations or slope through structured debate will bring to life topics that usually fall into the realm of pencil and paper work. Interviewing students – or perhaps having students interview a number, with research about significance, applications, relationship to other numbers, etc. – can also bring to life an idea that we jot in our notebook or that just sits on a number line. Capturing and holding students’ attention with respect to infinity, undefined and zero, and how each relates to exponents, equations and slope, is more likely to happen when students interact with the material through storytelling, interviews, surveys and debates. Gallery walks will allow students to showcase their thought processes with each topic and how it relates to the RNL. Each of these ideas incorporates aspects of Heath’s & Heath’s “Teaching that Sticks” traits: simple, unexpected, concrete, credible, emotional and story.
D. Technology
Accessing Prosser’s computer lab or cell phones to conduct research is one way in which we can use technology to build toward the activities mentioned above. Having students record one another during interviews, and then posting these interviews on YouTube, will allow me to share ideas with the entire class in a different way. Allowing students time to explore with Desmos in order to build their understanding of slope and slope-intercept form is something I’m excited to try. I need to play around with it, but PowToon looks like a platform that could allow students to showcase the concepts they’re wrestling with and making sense of. Perhaps I won’t be able to try all of these ideas this year, but the idea that I can implement some of them excites me.
E. Summary & Demonstration
The Total PACKage: I feel that my plan incorporates TPACK organically and dynamically. Nothing feels forced, although much of this is new to me. The technology component is the weakest, in part because of its newness and my constraints. What do you think?
The teaching demonstration will focus on the RNL and how I plan to relate each topic this year to it in a meaningful way. I plan to use PowToons for the demonstration. I’ll give an overview of the four valves that comprise the heart of the RNL: why the RNL matters, what it’s made of, how we can see it through new lenses and how it has utility in mathematics and in the world. I’ll share examples of performances of understanding, including the class’ first project. I may even share my i-Video to set the stage. Part of my motivation for sharing this will be to show a creative idea while reassuring students (during the demonstration with students) that difficulty with execution of the idea, which was my issue, is part of the process.
The first project will focus on the real number line and the relationship among the numbers on it. Students will tell a creative story and draw pictures representing their story. This activity, while seemingly simple, will require students to include integers, rational numbers and irrational numbers in their story. These are definitions they likely do not know, which is part of the point of the project. Marks will be based on creativity, variety of numbers and the relationship among those numbers.
