Phase 2
Implementation
Before I began this research project I noticed a few unsettling things in my classroom. First, the summative assessments used in math did not effectively measure my students’ mathematical understanding. Second, there are gaps in my students’ mathematical understanding. Third, my students are not cognizant of their learning (or in the gaps that exist). It was with this research that I hoped to improve these conditions within my classroom.
During Phase 1, I responded to these problems by incorporating mathematical content that emphasizes the three strands of mathematical proficiency (conceptual understanding, procedural fluency, and problem-solving), by creating the opportunity for my students to record and reflect on their learning each day (Daily Reflections), and by implementing an assessment tool (portfolios) that allow students to construct evidence of their learning. From Phase 1, I learned the following in regard to my research questions:
There were a few findings from Phase 1 that did not respond to my research questions. However, I still felt that they must be addressed in Phase 2. These findings include the following:
During Phase 1, I responded to these problems by incorporating mathematical content that emphasizes the three strands of mathematical proficiency (conceptual understanding, procedural fluency, and problem-solving), by creating the opportunity for my students to record and reflect on their learning each day (Daily Reflections), and by implementing an assessment tool (portfolios) that allow students to construct evidence of their learning. From Phase 1, I learned the following in regard to my research questions:
- my students require additional support in developing their conceptual understanding and problem-solving skills
- my students have gained a more positive disposition toward math
- the changes from Phase 1 positively impacted my students ideas about what it means to be good at math
- my students require additional support in developing their conceptual understanding and problem-solving skills
There were a few findings from Phase 1 that did not respond to my research questions. However, I still felt that they must be addressed in Phase 2. These findings include the following:
- my students are inclined to learning math in the traditional way.
- my students may not be able to express their learning in words.
- my students may not understand the Portfolio Task prompts.
Moving from Phase 1 to Phase 2
Phase 2 implementation felt a lot different than Phase 1 implementation. The way I did things and my students’ attitudes were different. I was more confident in the interventions because of the changes I had made. Unlike in Phase 1, I didn’t feel as though I had to convince my students the changes would benefit them. This is due in part to the fact that the changes were made based on my observations and their feedback. The changes I made had to do with the Daily Reflections, the ways I interacted with students and the condition of the Portfolios. I also think that the interventions I made in Phase 1 (which were continued in Phase 2) were no longer “new.” Since my students had spent close to a month interacting with math through the three strands and they had already completed a portfolio, it as as though this was the new norm of our classroom.
Daily Reflection Changes
During Phase 1, I noticed that the majority of my students were not responding to the questions in the Daily Reflections. Instead, they would share their experience working through the Problem of the Day, but their response did not answer the questions they were asked. For example, rather than documenting how they solved the problem, one student wrote,
“For the first three problems, I figured out my problems fast but for the other two I had a hard time.”
Responses similar to this were prevalent. I did not know why my students did not answer the questions. I was not sure whether it was because they did not know how to answer them or because they felt they did not have to. I thought that one way to get to the heart of the issue would be to make the task more accessible. To support my students to answer the questions, and effectively record their learning each day I made two changes. First, I changed the format of the Daily Reflections. Instead of being structured like this:
“Explain the Problem. How did you solve the Problem? What struggles did you have while solving the Problem? What exactly did you learn?”
“Explain the Problem. How did you solve the Problem? What struggles did you have while solving the Problem? What exactly did you learn?”
For Phase 2, I structured them like this:
1. Explain the Problem.
2. How did you solve the Problem?
3. What struggles did you have while solving the Problem?
4. What exactly did you learn?
I felt that this structure makes it clear that students were expected to answer each question and fill the space under each question. According to Lattimer, “providing [students] with space to write and respond to text improves comprehension and facilitates thoughtful response” (Lattimer, 10). In addition to an ample amount of space, the questions are clearly broken up into sections, which, I believe, make it more clear that each one must be addressed.
On the first day of Phase 2, I informed my students of this structural change when it was time for them to complete their Daily Reflection. Several students remarked that the new structure was “better” and that they liked “this one more because there is more space to answer each question.” There were still a few students who would not answer one or two of the questions, but 90% of my students responded to each question every day. Additionally, they were more engaged while completing the Daily Reflections in Phase 2 than in Phase 1. On Day 4 of Phase 2, I wrote in my observation notes that there were “very few students off task. Majority is focused. Many flip back to work.”
The second change I made to the Daily Reflections involved incorporating the questions into our class discussions. Once Phase 1 was implemented, we would begin the class by reading through the Problem for the day, discuss all important information and make sense of the tasks involved in the Problem. During Phase 2, the class was structured the same way, but I explicitly incorporated the Daily Reflection questions into the conversation. For example, instead of simply reading through the Problem, I would ask for a volunteer to “Explain the Problem.” After students completed the tasks and we were having our summarize discussion, rather than asking “who would like to share their work?” I asked, “Who can tell us how they solved the Problem?” I also asked students “what struggles they had while solving the Problem?” and “what exactly did you learn?” My intention was that if students were verbally/orally exposed to these questions and possible responses to the questions, they would be more prepared to answer them on their own.
At first, I was unable to determine if this change helped my students. I was not sure whether the conversation or the structural changes described above were supporting my students to answer the questions. Halfway through Phase 2, I turned to my students for guidance. In an informal way, I separately asked two students “What do you think of our conversations at the end of class? Do they help at all?” One student replied,
1. Explain the Problem.
2. How did you solve the Problem?
3. What struggles did you have while solving the Problem?
4. What exactly did you learn?
I felt that this structure makes it clear that students were expected to answer each question and fill the space under each question. According to Lattimer, “providing [students] with space to write and respond to text improves comprehension and facilitates thoughtful response” (Lattimer, 10). In addition to an ample amount of space, the questions are clearly broken up into sections, which, I believe, make it more clear that each one must be addressed.
On the first day of Phase 2, I informed my students of this structural change when it was time for them to complete their Daily Reflection. Several students remarked that the new structure was “better” and that they liked “this one more because there is more space to answer each question.” There were still a few students who would not answer one or two of the questions, but 90% of my students responded to each question every day. Additionally, they were more engaged while completing the Daily Reflections in Phase 2 than in Phase 1. On Day 4 of Phase 2, I wrote in my observation notes that there were “very few students off task. Majority is focused. Many flip back to work.”
The second change I made to the Daily Reflections involved incorporating the questions into our class discussions. Once Phase 1 was implemented, we would begin the class by reading through the Problem for the day, discuss all important information and make sense of the tasks involved in the Problem. During Phase 2, the class was structured the same way, but I explicitly incorporated the Daily Reflection questions into the conversation. For example, instead of simply reading through the Problem, I would ask for a volunteer to “Explain the Problem.” After students completed the tasks and we were having our summarize discussion, rather than asking “who would like to share their work?” I asked, “Who can tell us how they solved the Problem?” I also asked students “what struggles they had while solving the Problem?” and “what exactly did you learn?” My intention was that if students were verbally/orally exposed to these questions and possible responses to the questions, they would be more prepared to answer them on their own.
At first, I was unable to determine if this change helped my students. I was not sure whether the conversation or the structural changes described above were supporting my students to answer the questions. Halfway through Phase 2, I turned to my students for guidance. In an informal way, I separately asked two students “What do you think of our conversations at the end of class? Do they help at all?” One student replied,
“Yeah, I like it. Usually, I don’t get the problem enough to write about it in the reflections. When we go over it together then I know what to write.”
The other student told me that,
“Talking about the problem, like our struggles and what we learn, and then writing about it is a waste of time. It would be better if we only did one.”
From these I gathered that this change benefitted some students, by supporting them to make sense of their learning, and that some students might not need to do it because they are able to complete the Daily Reflections on their own. I decided to continue implementing this change.
Changes to My Interactions
As I mentioned in Phase 1, my students harbored negative feelings toward the changes I made. They did not think that they would benefit from the changes. Consequently, there was a lot of tension between my students and me. I felt that these feelings detracted from the experiences my students had. I wanted to change this in Phase 2. I wanted to adapt the changes I had made so that my students would still benefit from them, but also know that I value and hear their opinions. To set this in motion, I decided to learn about my students’ experience. In a student feedback form I asked the following question, “The last few weeks of math have been different than the rest of the year. What changes worked for you? Why? Which changes did not work well for you? Why?” From my students’ responses I noticed four themes.
- Students did not learn the new content because there was not enough support from the teacher.
- Recording their learning through the Daily Reflections helps students retain the knowledge they constructed.
- Less focus on the test and more focus on the content allows students to prioritize learning over getting a good grade.
- All students working at the same pace on the same content increase morale.
“I think what helped me was the daily reflections so I can just go back and remember what I worked on because before I did not have anything to remind me.”
“I like the fact that we didn’t need to take tests every single friday which was a relief and I feel like I could focus more on my math.”
“I liked how everyone was working at the same pace so I didn’t feel rushed and everyone could have helped me.”
Themes #2-4 reinforced my initial intentions with this research project. I was glad to see that, although my students resisted the changes in Phase 1, they were beginning to realize the benefits. Themes #2-4 made me feel confident about the changes that I had made. I had lost this confidence throughout Phase 1, but regained it going into Phase 2. This confidence helped me be comfortable interacting with my students again. I was no longer defensive nor did I feel as though I had to battle them everyday. I believe that the positive outlook I had during Phase 2 transferred to my students. They did not challenge me. Sure, there were a few students who would express their discontent now and again, but the complete opposition from Phase 1 had diminished.
Theme #1 was an eye opener for me. As I explained in Phase 1, I learned that my students were struggling to develop their conceptual understanding and problem-solving skills, but I had not considered that it was due to the fact that I was not supporting them enough. In the student feedback form, one student wrote that, “Honestly, this has been by far the worst weeks this year for math. I don't understand what to do, there isn't really any help and the questions and problems are confusing and I don't understand. None of the changes have worked for me at all and practically everything we have done just went over head. Nothing is working for me at all.” Another wrote, “The changes haven’t worked for me. I haven’t learned anything because it isn’t explained well and it wasn’t really helpful. I think that this stuff should be explained more instead of not explaining it at all.” It became clear to me that my students need more support when interacting with the mathematical content through the three strands in order to learn.
I did some reflecting of my own to determine how I could do this. I realized that in order to support my students to become proficient in mathematics, I cannot simply present them with content that emphasizes conceptualizations, procedures and problem-solving. Effective math instruction depends on the interactions of the teacher, the students and the mathematical content; it is not one piece, rather all the different ways the three pieces connect (Kilpatrick et al, 2001). In Phase 1, I only had one of these pieces: the content. I wanted to change this going in to Phase 2.
I thought that if I could interact with my students in ways that supported them to access and engage with the content, they would be more successful in developing their mathematical proficiency. To do this, I implemented a “Focusing” approach to my interactions with students. “A focusing-interaction pattern requires the teacher to listen to students’ responses and guide them based on what students are thinking rather than how the teacher would solve the problem” (Herbel-Eisenmann & Breyfogle, 2005). I thought this approach would be the most effective because it would allow me to meet students at their level and follow them through their construction of their understanding.
While interacting with my students in this way, I noticed that many of them were frustrated. After working with one student on a problem that involved creating a linear equation from given information, I wrote in my teacher journal that “[the student] was put off by my approach. She was unwilling to respond to my questions, which were meant to help her make sense of the problem without telling her how to do it. It was obvious that she was frustrated because she did not know what to do. After I prodded her almost five times about one question, she replied ‘I don’t know! Can’t you just tell me what to do? Then I can do it.’’ I feel safe to say that almost half of my students experienced similar feelings to this student. I did my best not to abandon my efforts, because from the research I did, I believed that it would benefit my students in the long run.
Theme #1 was an eye opener for me. As I explained in Phase 1, I learned that my students were struggling to develop their conceptual understanding and problem-solving skills, but I had not considered that it was due to the fact that I was not supporting them enough. In the student feedback form, one student wrote that, “Honestly, this has been by far the worst weeks this year for math. I don't understand what to do, there isn't really any help and the questions and problems are confusing and I don't understand. None of the changes have worked for me at all and practically everything we have done just went over head. Nothing is working for me at all.” Another wrote, “The changes haven’t worked for me. I haven’t learned anything because it isn’t explained well and it wasn’t really helpful. I think that this stuff should be explained more instead of not explaining it at all.” It became clear to me that my students need more support when interacting with the mathematical content through the three strands in order to learn.
I did some reflecting of my own to determine how I could do this. I realized that in order to support my students to become proficient in mathematics, I cannot simply present them with content that emphasizes conceptualizations, procedures and problem-solving. Effective math instruction depends on the interactions of the teacher, the students and the mathematical content; it is not one piece, rather all the different ways the three pieces connect (Kilpatrick et al, 2001). In Phase 1, I only had one of these pieces: the content. I wanted to change this going in to Phase 2.
I thought that if I could interact with my students in ways that supported them to access and engage with the content, they would be more successful in developing their mathematical proficiency. To do this, I implemented a “Focusing” approach to my interactions with students. “A focusing-interaction pattern requires the teacher to listen to students’ responses and guide them based on what students are thinking rather than how the teacher would solve the problem” (Herbel-Eisenmann & Breyfogle, 2005). I thought this approach would be the most effective because it would allow me to meet students at their level and follow them through their construction of their understanding.
While interacting with my students in this way, I noticed that many of them were frustrated. After working with one student on a problem that involved creating a linear equation from given information, I wrote in my teacher journal that “[the student] was put off by my approach. She was unwilling to respond to my questions, which were meant to help her make sense of the problem without telling her how to do it. It was obvious that she was frustrated because she did not know what to do. After I prodded her almost five times about one question, she replied ‘I don’t know! Can’t you just tell me what to do? Then I can do it.’’ I feel safe to say that almost half of my students experienced similar feelings to this student. I did my best not to abandon my efforts, because from the research I did, I believed that it would benefit my students in the long run.
Seeing Portfolios in a New Light
From the experience of Phase 1, it was apparent that my students did not think portfolio assessments could benefit their mathematical learning. Due to a few obstacles, I was unable to report that my students had benefitted from compiling Portfolios during Phase 1. These obstacles include misunderstandings and inaccessibility to the portfolio tasks. Note that, although I did not see students benefit from portfolio assessments, they were not disserviced by them.
During Phase 2, I sought to change my students’ perceptions of portfolio assessments and alter the tasks so they would be more accessible to all students. I believed these changes would allow my students to benefit from portfolio assessments. To do this, I started with gaining insight into my students’ thoughts about the Portfolio Project. As I explained in Phase 1, my students were not on board with the project. They did not want to analyze or reflect on their learning. Because of this, I felt as though they were not putting forth an honest effort, which only resulted in them not benefitting from the experience.
During the first week of Phase 2, I asked my students to answer the question “What did you think of the Portfolio Project?” on a note card. I encouraged them to be honest and constructive. I asked students to not write their names on their responses so that they could be anonymous.
Before I read the responses, I braced myself. I knew that since Phase 1 was such a confrontational time, the responses could be harsh. It was difficult for me to read them. At times, I doubted the effectiveness of my efforts and intentions during Phase 1, and questioned their continuation. After taking a step back, I was able to separate my practice from my students’ responses. I noticed several themes to the responses. In general, my students felt that
A. The Portfolio Project helped reinforce learning.
B. Reflecting on reflections is pointless.
C. More time to revise the responses and better execution would improve the project.
During Phase 2, I sought to change my students’ perceptions of portfolio assessments and alter the tasks so they would be more accessible to all students. I believed these changes would allow my students to benefit from portfolio assessments. To do this, I started with gaining insight into my students’ thoughts about the Portfolio Project. As I explained in Phase 1, my students were not on board with the project. They did not want to analyze or reflect on their learning. Because of this, I felt as though they were not putting forth an honest effort, which only resulted in them not benefitting from the experience.
During the first week of Phase 2, I asked my students to answer the question “What did you think of the Portfolio Project?” on a note card. I encouraged them to be honest and constructive. I asked students to not write their names on their responses so that they could be anonymous.
Before I read the responses, I braced myself. I knew that since Phase 1 was such a confrontational time, the responses could be harsh. It was difficult for me to read them. At times, I doubted the effectiveness of my efforts and intentions during Phase 1, and questioned their continuation. After taking a step back, I was able to separate my practice from my students’ responses. I noticed several themes to the responses. In general, my students felt that
A. The Portfolio Project helped reinforce learning.
B. Reflecting on reflections is pointless.
C. More time to revise the responses and better execution would improve the project.
“I liked how we were able to go back and reflect on our work, it was a way of being able to understand your learning and explaining it because you know it. It made me find out things I did that I didn’t even notice that helped me and everything we did connected.”
“To be honest, I didn’t like it very much we were reflecting on reflections and it seemed like it wasn’t that helpful.”
“Honestly, I was not a huge fan. It was extremely short notice, and it could have been more organized because I was confused. I had no idea what to write. It’s a good idea, it just wasn’t executed well.”
Theme A reinforced my initial intentions with the Portfolio Project. My goal was that students would deepen their learning by providing evidence for what they believe they learned. The notecard responses that addressed Theme B, encouraged me to continue the Portfolio assessment aspect of my research.
From Theme B, it became apparent that my students had not realized the benefits of reflecting and analyzing what they learn. Rather than learning more about math, this project was meant to provide students with the opportunity to take a step back and see what they learn through the experience of Phase 1. Additionally, they were not charged with the task of “reflecting on their reflections,” as so many of my students expressed. Instead, they were responsible for explaining how their work demonstrated what they learned. On student wrote, “The portfolio project was not helpful or useful to me at all. It was reflecting on my reflections, which was just a rewording of the previous reflection and a waste of my time. How can one reflect on a reflection deeply? I could have spent the time learning math.”
It was obvious to me that I needed to clarify this misunderstanding. To do so there were a few actions I took. First, I reworded some of the questions of the Portfolio Task. During the first Portfolio Project (which will now be referred to as Portfolio #1) the Daily Reflection task said, “Choose one set of Daily Reflections, either from Investigation 1 or Investigation 2. Explain how these reflections capture your learning for the Investigation.” For the second Portfolio Project (which will now be referred to as Portfolio #2), the Daily Reflection Task said,
“1. Choose one set of Daily Reflections, either from Investigation 3 or Investigation 4.
2. How does what you wrote for all three reflections show what you learned throughout the entire investigation?”
I made similar revisions for each Task in Portfolio #2. I believe that the revised Tasks were more clear about what students were charged with.
The second action I took was to engage students about the misconception they had developed through Phase 1. Before we began Portfolio #2, I explained to my students that what they were supposed to be doing was not “reflecting on reflections,” but rather, providing evidence of what they had learned. I used the Daily Reflection Task as an example. I explained that from the way it was written in Portfolio #1, it could be assumed that they were supposed to reflect on their Daily Reflections. However, they way it was worded in Portfolio 2, made it clear that they were supposed to state what they learned and explain how their Daily Reflections shows what they learned. About this time in my classroom, I wrote in my teacher journal that “mostly everyone seemed to get it. I saw many nods, and a few students verbally expressed how much clearer the tasks were.”
Theme C began to get at one large scale change I knew I had to make for Portfolio 2. In addition to more time to complete the project, my students needed more structure. Toward the end of Phase 1, when my students were creating Portfolio 1, it was a free-for-all. I presented the assignment to my students and gave them four days to complete it at their leisure. For Phase 2, I wanted there to be more structures in place to support my students to be successful in Portfolio 2. As a result, I implemented the following changes. First, I mandated a Portfolio Protocol, which consisted of the following:
1. Write response
2. Self-critique, then make revisions.
3. Compare response to Portfolio 1, then make revisions.
4. Peer critique, then make revisions.
5. Submit Portfolio.
From Theme B, it became apparent that my students had not realized the benefits of reflecting and analyzing what they learn. Rather than learning more about math, this project was meant to provide students with the opportunity to take a step back and see what they learn through the experience of Phase 1. Additionally, they were not charged with the task of “reflecting on their reflections,” as so many of my students expressed. Instead, they were responsible for explaining how their work demonstrated what they learned. On student wrote, “The portfolio project was not helpful or useful to me at all. It was reflecting on my reflections, which was just a rewording of the previous reflection and a waste of my time. How can one reflect on a reflection deeply? I could have spent the time learning math.”
It was obvious to me that I needed to clarify this misunderstanding. To do so there were a few actions I took. First, I reworded some of the questions of the Portfolio Task. During the first Portfolio Project (which will now be referred to as Portfolio #1) the Daily Reflection task said, “Choose one set of Daily Reflections, either from Investigation 1 or Investigation 2. Explain how these reflections capture your learning for the Investigation.” For the second Portfolio Project (which will now be referred to as Portfolio #2), the Daily Reflection Task said,
“1. Choose one set of Daily Reflections, either from Investigation 3 or Investigation 4.
2. How does what you wrote for all three reflections show what you learned throughout the entire investigation?”
I made similar revisions for each Task in Portfolio #2. I believe that the revised Tasks were more clear about what students were charged with.
The second action I took was to engage students about the misconception they had developed through Phase 1. Before we began Portfolio #2, I explained to my students that what they were supposed to be doing was not “reflecting on reflections,” but rather, providing evidence of what they had learned. I used the Daily Reflection Task as an example. I explained that from the way it was written in Portfolio #1, it could be assumed that they were supposed to reflect on their Daily Reflections. However, they way it was worded in Portfolio 2, made it clear that they were supposed to state what they learned and explain how their Daily Reflections shows what they learned. About this time in my classroom, I wrote in my teacher journal that “mostly everyone seemed to get it. I saw many nods, and a few students verbally expressed how much clearer the tasks were.”
Theme C began to get at one large scale change I knew I had to make for Portfolio 2. In addition to more time to complete the project, my students needed more structure. Toward the end of Phase 1, when my students were creating Portfolio 1, it was a free-for-all. I presented the assignment to my students and gave them four days to complete it at their leisure. For Phase 2, I wanted there to be more structures in place to support my students to be successful in Portfolio 2. As a result, I implemented the following changes. First, I mandated a Portfolio Protocol, which consisted of the following:
1. Write response
2. Self-critique, then make revisions.
3. Compare response to Portfolio 1, then make revisions.
4. Peer critique, then make revisions.
5. Submit Portfolio.
During Phase 1, there were students who “finished” their Portfolios within the first day of the assignment. I encouraged them to receive feedback from their peers and make revisions before they submitted their final versions, but very few students actually did. I was less concerned with how much time they spent on the project, and more concerned with the thoughtfulness of their reflection and the depth of their analysis. The Portfolio Protocol ensured that my students would complete at least two revisions on their work. I wrote it on the board before we began working on Portfolio 2 and referred students to it as they completed each task. It became apparent that they knew they had to complete the checklist before they could submit their work. After the second day of working on this project, students went right to the Self-Critique without me having to prompt them to do so.
In addition to the Portfolio Protocol, students were to compete a specific task each day. On Day 1, we worked on Task #1. On Day 2, we worked on Task #2, and so on until the end of the week. In establishing a class focus each day, I hoped that the solidarity would encourage my students to share their work. It definitely seemed to do so. On Day 3 of the Portfolio Project, I wrote in my observation notes that a “student asked table partner for feedback on wording.” Interactions like these showed me that my students were interested in producing quality responses.
At the beginning of each day, I presented the students with an example response for the task. I pulled strong responses from Portfolio 1 and was given permission by those students to anonymously show their work. A student would read the response out loud and we would discuss “what was strong about the response and what we could do to strengthen it.” I did not participate in this discussion, but rather facilitated and made sure each student shared an idea they had. There were two things I hoped to accomplish through this exercise.
1. I wanted my students to determine what makes a response strong.
2. I wanted my students to realize that the examples I showed them were not perfect.
From these accomplishments, my students would have ownership over the criteria, and have the ability to respond to the tasks in their own voice.
While students shared their ideas, I took notes. From their opinions and suggestions, I create the Self-Critique question list students were to complete during Step 2 of the Portfolio Protocol. My students had brilliant ideas about the quality of the examples and how to strengthen them! I was blown away. They were able to make claims about the examples and pull evidence right from the example to justify their claims. One student explained that
In addition to the Portfolio Protocol, students were to compete a specific task each day. On Day 1, we worked on Task #1. On Day 2, we worked on Task #2, and so on until the end of the week. In establishing a class focus each day, I hoped that the solidarity would encourage my students to share their work. It definitely seemed to do so. On Day 3 of the Portfolio Project, I wrote in my observation notes that a “student asked table partner for feedback on wording.” Interactions like these showed me that my students were interested in producing quality responses.
At the beginning of each day, I presented the students with an example response for the task. I pulled strong responses from Portfolio 1 and was given permission by those students to anonymously show their work. A student would read the response out loud and we would discuss “what was strong about the response and what we could do to strengthen it.” I did not participate in this discussion, but rather facilitated and made sure each student shared an idea they had. There were two things I hoped to accomplish through this exercise.
1. I wanted my students to determine what makes a response strong.
2. I wanted my students to realize that the examples I showed them were not perfect.
From these accomplishments, my students would have ownership over the criteria, and have the ability to respond to the tasks in their own voice.
While students shared their ideas, I took notes. From their opinions and suggestions, I create the Self-Critique question list students were to complete during Step 2 of the Portfolio Protocol. My students had brilliant ideas about the quality of the examples and how to strengthen them! I was blown away. They were able to make claims about the examples and pull evidence right from the example to justify their claims. One student explained that
“This response is strong because they provided details. Instead of just saying that ‘I thouroughy explained all of the questions I was asked’ they said that ‘I thoroughly explained all of the questions that I was asked while also using examples to help show my work and understanding for it.’ They even add an example after that!”
Another explained that
“Even though they say what they learned, they don’t prove they learned it. They said ‘We learned about the three ways to solve a linear equation which were graphing, substitution and combining, or elimination.’ But if they had said what those are or maybe the steps, then we would know they really know it.”
We ran through this exercise at the beginning of each day, immediately before students began working on the task for the day. After Day 2, the suggestions were of the same form (e.g., give more details, go more in depth, etc.), but we continued with this exercise.
I was confident in the changes I made to Portfolios in Phase 2. I believed that from addressing the misconceptions my students had, from listening to and incorporating their feedback, from making the tasks more accessible, and from clarifying my expectations, my students saw the Portfolio Project in a new light. I was confident that they were on board with completing the assignment and better suited to do so during Phase 2.
I was confident in the changes I made to Portfolios in Phase 2. I believed that from addressing the misconceptions my students had, from listening to and incorporating their feedback, from making the tasks more accessible, and from clarifying my expectations, my students saw the Portfolio Project in a new light. I was confident that they were on board with completing the assignment and better suited to do so during Phase 2.
Findings
Recall that my overall goal for this research was to determine the effects portfolio assessment that emphasizes conceptual understanding, procedural fluency and problem-solving skills, on students’ dispositions toward math, their overall mathematical understanding, and their perception of what it means to be good at math. In the following section, I will address my findings in regard to these areas.
Dispositions Toward Math
During the fourth day of working on Portfolio 2, I noticed something I have never experienced in my classroom. It was silent. Every single one of my students was engaged, on-task, and focused. The only sound to be heard was the click of the keys on the keyboards. I reveled in the beauty of a completely focused class for a few minutes before I realized the harsh contrast between the current state of my classroom and the state of it when I tried to implement Portfolio 1. On that day, my class was on the brink of chaos. I don’t think there was a day all year when I had less control. Every single one of my students was off-task and distracted. They did not want to do the Portfolio Project, and when my students don’t want to do something, they don’t do it. It was extremely difficult for me to get them on-task. I struggled with it all week. Yet during Portfolio 2, everyone was engaged, everyone was focused, and everyone was on-task. Right away my inquisitive mind asked, “Why? What changed? What’s different now?”
I posed these questions to my mentor teacher and to several of my students. There were two general themes I noticed from their responses.
1. The end of the year is fast approaching. Portfolio 2 is the last math assignment and it is worth a significant amount of points. My students were so focused because they have to be, they have to complete the assignment.
2. The changes I made to the Portfolio Tasks (rewording/eliminating questions and providing examples) made the assignment more doable.
Needless to say, I really hope Theme #2 is the real reason! It would be great to know that the changes I made between Phase 1 and Phase 2 supported my students to be successful. Regardless of the cause, never in the entire school year had my students been as interested, focused, or on task than they were when completing that project. My students’ interest in completing Portfolio 2, as well as the stark contrast between their interest from Phase 1 and Phase 2, demonstrates a more positive disposition toward mathematics. Rather than resisting the changes made in Phase 1 (as was the case in Phase 1) my students embraced the changes in Phase 2. They exhibited an inclination toward analyzing their work and reflecting on their mathematical learning. I believe the more positive disposition is a result of the changes I made between Phase 1 and Phase 2. While Phase 2 began, I sought to improve the conditions of the changes to establish a middle ground with my students. I made this clear to them. During Phase 2, they knew that we had to continue with the interventions I had put into place, but that I was willing to make changes based on their feedback. This showed them that I heard them and that I cared for them.
Finding #1 Students are more inclined to engage with content when their voices are heard and they know they are cared for.
Mathematical Understanding
To uncover my students’ mathematical understanding from Phase 2, I analyzed my students' Daily Reflections, their work throughout Phase 2, and Portfolio 2.
From reading through their Daily Reflections from Phase 2 it became apparent that my students were making sense of the content each day and that they were able to put their learning into words. However, I noticed that the language they used was more informal and less academic. For example, in response to the question “How did you solve the problem?” one student wrote,
Dispositions Toward Math
During the fourth day of working on Portfolio 2, I noticed something I have never experienced in my classroom. It was silent. Every single one of my students was engaged, on-task, and focused. The only sound to be heard was the click of the keys on the keyboards. I reveled in the beauty of a completely focused class for a few minutes before I realized the harsh contrast between the current state of my classroom and the state of it when I tried to implement Portfolio 1. On that day, my class was on the brink of chaos. I don’t think there was a day all year when I had less control. Every single one of my students was off-task and distracted. They did not want to do the Portfolio Project, and when my students don’t want to do something, they don’t do it. It was extremely difficult for me to get them on-task. I struggled with it all week. Yet during Portfolio 2, everyone was engaged, everyone was focused, and everyone was on-task. Right away my inquisitive mind asked, “Why? What changed? What’s different now?”
I posed these questions to my mentor teacher and to several of my students. There were two general themes I noticed from their responses.
1. The end of the year is fast approaching. Portfolio 2 is the last math assignment and it is worth a significant amount of points. My students were so focused because they have to be, they have to complete the assignment.
2. The changes I made to the Portfolio Tasks (rewording/eliminating questions and providing examples) made the assignment more doable.
Needless to say, I really hope Theme #2 is the real reason! It would be great to know that the changes I made between Phase 1 and Phase 2 supported my students to be successful. Regardless of the cause, never in the entire school year had my students been as interested, focused, or on task than they were when completing that project. My students’ interest in completing Portfolio 2, as well as the stark contrast between their interest from Phase 1 and Phase 2, demonstrates a more positive disposition toward mathematics. Rather than resisting the changes made in Phase 1 (as was the case in Phase 1) my students embraced the changes in Phase 2. They exhibited an inclination toward analyzing their work and reflecting on their mathematical learning. I believe the more positive disposition is a result of the changes I made between Phase 1 and Phase 2. While Phase 2 began, I sought to improve the conditions of the changes to establish a middle ground with my students. I made this clear to them. During Phase 2, they knew that we had to continue with the interventions I had put into place, but that I was willing to make changes based on their feedback. This showed them that I heard them and that I cared for them.
Finding #1 Students are more inclined to engage with content when their voices are heard and they know they are cared for.
Mathematical Understanding
To uncover my students’ mathematical understanding from Phase 2, I analyzed my students' Daily Reflections, their work throughout Phase 2, and Portfolio 2.
- Daily Reflections
From reading through their Daily Reflections from Phase 2 it became apparent that my students were making sense of the content each day and that they were able to put their learning into words. However, I noticed that the language they used was more informal and less academic. For example, in response to the question “How did you solve the problem?” one student wrote,
“I would cross out a drink and a $ I would do the same with all of them and then the only letters left were for tacos and for money.”
This student is explaining the Elimination Method of solving a system of Linear Equations. It is obvious that although this explanation is accurate, it is not expressed using the appropriate mathematical language.
Finding #2 Although the changes I made to the Daily Reflections supported my students to express their learning, they need additional support in expressing their learning in academic English.
For assignments that contained problems that were strictly procedural and/or application problems (which calls for problem-solving skills), 79% of my students attempted the problems. In other words, 21% of students did not attempt these problems. In regard to these students, I wrote the following in my teacher journal: “Could not write equations or inequalities. Failed to make sense of info. Could not graph equations. Could not write a system of equations or inequalities. Could not graph an inequality…Struggled to write inequalities and equations in some cases…Majority of problems not attempted.”
Additionally, 43% of my students were to able to complete all of the procedural and problem-solving exercises. In regard to these students, I generally wrote that these students were “successful in problems that only have one equation/inequality...tried the substitution method but did not execute it properly...Wrote the correct equations. Did not try to write inequalities...Wrote the wrong equation and graphed it incorrectly... Could not write and then graph the inequality. It was not attempted. Could not write a system of linear inequalities or graph them or find the solutions. It was not attempted.”
I also found that although the unit ended in Phase 2 with graphing systems of Linear Inequalities, 100% of my students could not graph a system of linear inequalities. In general, my students did not attempt these problems on the homework or on the quiz. In short, there were various skills more than half of my students were not proficient in at the end of the unit. It was clear that my students had not developed a proficiency in Systems of Linear Equations/Inequalities or Linear Inequalities.
Finding #3, my students need more exposure to and interactions with Systems of Linear Equations/Inequalities and Linear Inequalities in order to improve their understanding.
Finding #2 Although the changes I made to the Daily Reflections supported my students to express their learning, they need additional support in expressing their learning in academic English.
- Student Work
For assignments that contained problems that were strictly procedural and/or application problems (which calls for problem-solving skills), 79% of my students attempted the problems. In other words, 21% of students did not attempt these problems. In regard to these students, I wrote the following in my teacher journal: “Could not write equations or inequalities. Failed to make sense of info. Could not graph equations. Could not write a system of equations or inequalities. Could not graph an inequality…Struggled to write inequalities and equations in some cases…Majority of problems not attempted.”
Additionally, 43% of my students were to able to complete all of the procedural and problem-solving exercises. In regard to these students, I generally wrote that these students were “successful in problems that only have one equation/inequality...tried the substitution method but did not execute it properly...Wrote the correct equations. Did not try to write inequalities...Wrote the wrong equation and graphed it incorrectly... Could not write and then graph the inequality. It was not attempted. Could not write a system of linear inequalities or graph them or find the solutions. It was not attempted.”
I also found that although the unit ended in Phase 2 with graphing systems of Linear Inequalities, 100% of my students could not graph a system of linear inequalities. In general, my students did not attempt these problems on the homework or on the quiz. In short, there were various skills more than half of my students were not proficient in at the end of the unit. It was clear that my students had not developed a proficiency in Systems of Linear Equations/Inequalities or Linear Inequalities.
Finding #3, my students need more exposure to and interactions with Systems of Linear Equations/Inequalities and Linear Inequalities in order to improve their understanding.
- Portfolio 2
In total, 16 students demonstrated their proficiency in the strand they had chosen. They did this by constructing evidence of their understanding.
“In Math Journal 11, it asked me two questions. The first problem describes a graph with one linear inequality connected by only two variables. It asks how to find more solutions using the graph. In my response, I explained that the way to find more solutions was to graph the inequality like an equation and then shade above or below the line. If it is a > you shade above the line, if it is a < you shade below the line. The shaded part shows all solutions to the inequality. The second question asks how you may graph/find solutions to a given system of linear inequalities. I explained in my response that you could graph each line separately and then see where they intersect…”
“The problems that I had to solve for investigation 3.2 were very fun to me. These homework problems were fun because I had to solve them with inverse operations. The homework asked you to solve each inequality. To solve an inequality, you use inverse operations like you were solving an equation. So let’s say you have the problem that looked like this 12 < 7x - 2 to solve this problem you have to isolate x. To start you add 2 to each side. Now your inequality should look like 14 < 7x. Now we have to get rid of the 7 so you divide on each side by 7. Now your inequality is 2 < x. This is your final answer to the inequality…”
“This problem asked me to find some possibilities for the number of front-pade ads and the number of indeside-page ads a club could place with their $30 if front-page ads cost $2 each an inside-page ads cost $1 each. It also asked me to write a system of linear equations to model the situation and then graph the system of equations. How I personally recognized the important information was by looking for the numbers so I ooked at their $30 budget, $2 front-page ads, then the $1 inside-page ads, and that they wanted to advertise 20 times at least. From multiplying and adding I found that if they published five front-page ads then they would pay $10. If they published ten inside-page ads they would spend $10 so all together they would spend $20. They would have $10 left over…”
Of the 12 students (43%) who did not demonstrate their proficiency, only 3 expicitly stated that they did not understand the work they were analyzing. Below are examples of different students' response to the conceptual understanding task, the procedural fluency task, and the problem-solving task.
“For math journal 11 we were asked how you could use a graph to find solutions of the inequality? and how you could use a graph to find the solutions of the systems. I felt pretty confused on how you could use a graph in order to find solutions of the systems. I didn’t really get what it was asking but I remembered from one of the lessons we discussed that the solution to a graph is always on the line.”
In regard to the other 9, I cannot definitively state whether or not they constructed an understanding of the content. This is due to the fact that in their responses, these students did not talk specifically about their work, nor did they provide the details necessarry for me to see their mathematical understanding.
“For the homework problem 4.2, we were asked to use inequalities to solve a problem about a math club who wants to make a profit of at least $200 by selling games, for $10, and puzzles for $8. We first were told to think of some possibilities for what it could have been, and then we were told to write an inequality to show how many of each item must be sold. Finally, we were asked to graph our inequality. I completed it with clear answers, as simple as the problem itself. Although I do have an understanding and familiarity with linear inequalities, I did most of the work in my head. The problems did not require a lot of work. I did it in my head, and graphed and shaded correctly.”
Because this student does not specifically talk about the work he did to solve this problem I cannot determine whether or not he is actually proficient. This was the case with 25% of my students. Finding #3 It was difficult to absolutely determine my students’ mathematical proficiency from their responses to Portfolio 2.
At this point, I am unable to confidently explain the impact portfolio assessments had on my students’ mathematical understanding. However, I can speak to the positive impact they had on my students in terms of their abilities to explain mathematics in writing and analyze their learning experience.
There were several changes that I made to the Portfolio Project during Phase 2. I made these changes in the hopes that my students would benefit from the experience more than they did in Phase 1. In general, all of my students had written stronger responses for Portfolio 2. I noticed that 83% of the submissions for Portfolio 2 included responses that answered the questions present in the Tasks. Additionally, 26% of my students analyzed their learning experience in Portfolio 2 rather than just stating it. Both of these statistics increased dramatically from Portfolio 1 to Portfolio 2. Finding #4 The changes made for Portfolio 2 supported students to reflect on and analyze their learning.
Perceptions of what it means to be good at math
My students completed Portfolio 2 at the beginning of the last week of school year. I had created a final student feedback form for them to complete once they submitted Portfolio 2. The form focused on self-assessment questions. I asked students about the improvements they made between Portfolio 1 and Portfolio 2, how completing the portfolios affected their understanding of linear equations and linear inequalities, and what they learned about themselves as math learners by completing the portfolios. Additionally, as I did with the other two student feedback forms that took place before and after Phase 1, I asked “What does it mean to be good at math?” and “How are you good at math.” I have included these questions in all three forms because I have been interested in how this perception changes as they grew through the interventions, and because it is important to me that all students realize they are good at math.
During the last week of the school year at my school, students take part in Presentations of Learning (POLs). Students prepare for POLs for at least a month in advance, and they take up the entire week. That being said, instruction ends before the last week of school. I was unaware of this. I was under the impression, that I would see my students as a class during the last week of school. I had planned to have them fill out the student feedback form at that time. Since this was not the case, my students were unable to do so. I emailed the form to my students and asked them to complete it, but I think because I was unable to hold them accountable for completing it, they were not compelled to do so. As a result, I was unable to learn about how their perceptions of what it means to be good at math has changed from Phase 1 to Phase 2.
At this point, I am unable to confidently explain the impact portfolio assessments had on my students’ mathematical understanding. However, I can speak to the positive impact they had on my students in terms of their abilities to explain mathematics in writing and analyze their learning experience.
There were several changes that I made to the Portfolio Project during Phase 2. I made these changes in the hopes that my students would benefit from the experience more than they did in Phase 1. In general, all of my students had written stronger responses for Portfolio 2. I noticed that 83% of the submissions for Portfolio 2 included responses that answered the questions present in the Tasks. Additionally, 26% of my students analyzed their learning experience in Portfolio 2 rather than just stating it. Both of these statistics increased dramatically from Portfolio 1 to Portfolio 2. Finding #4 The changes made for Portfolio 2 supported students to reflect on and analyze their learning.
Perceptions of what it means to be good at math
My students completed Portfolio 2 at the beginning of the last week of school year. I had created a final student feedback form for them to complete once they submitted Portfolio 2. The form focused on self-assessment questions. I asked students about the improvements they made between Portfolio 1 and Portfolio 2, how completing the portfolios affected their understanding of linear equations and linear inequalities, and what they learned about themselves as math learners by completing the portfolios. Additionally, as I did with the other two student feedback forms that took place before and after Phase 1, I asked “What does it mean to be good at math?” and “How are you good at math.” I have included these questions in all three forms because I have been interested in how this perception changes as they grew through the interventions, and because it is important to me that all students realize they are good at math.
During the last week of the school year at my school, students take part in Presentations of Learning (POLs). Students prepare for POLs for at least a month in advance, and they take up the entire week. That being said, instruction ends before the last week of school. I was unaware of this. I was under the impression, that I would see my students as a class during the last week of school. I had planned to have them fill out the student feedback form at that time. Since this was not the case, my students were unable to do so. I emailed the form to my students and asked them to complete it, but I think because I was unable to hold them accountable for completing it, they were not compelled to do so. As a result, I was unable to learn about how their perceptions of what it means to be good at math has changed from Phase 1 to Phase 2.
"Next Steps" Lingering Questions
As I mentioned above, Phase 2 ended just before my school year did. Thus, there is not time for me to conduct a Phase 3, although there are a few questions that I am still curious about. If I had the time, the ways in which I would address these Lingering Questions would be my Next Steps.
Lingering Question #1: What are the long term effects portfolio assessment has on students’ overall mathematical understanding, on students’ dispositions toward math, adn on students’ perceptions of what it means to be good at math?
Between the experiences of Phase 1 and Phase 2, I noticed positive effects in these areas. This research took place for approximately 6 weeks. I am curious as to how these three areas would be affected if protfolio assessment was used as the summative assessment for an entire school year. If I extrapolate the positive results I noticed through my research, it is safe to say that students would be more positively impacted. However, until I do this I cannot say for sure.
To answer this question, I would continue the interventions I made in Phase 1 and Phase 2. To determine the effects portfolio assessments have on all three areas, I would put the same data sources in place and collect data at regular intervals. During the year I would track individual student’s progress in all three areas. At the end of the year I would look at the data collectively and determine the overall effects.
Lingering Question #2: What are the long term effects of continuous Daily Reflections?
As was the case with Lingering Question #1, I noticed strong benefits in my students’ metacognition. They and I attribute this to the fact that each day they would record their learning experience through the Daily Reflections. Since these reflections had such a strong impact on my students, I am interested in learning the effects they have when implemented continuously for an extended period of time.
To explore this question, I would continue to implement the Daily Reflections as I did in Phase 2. Throughout the year I would collect data on my students’ abilities to record and reflect on their learning as well as their ability to demonstrate their mathematical proficiency. I would track individual student’s progress and collectively analyze the data to learn the effects Daily Reflections have on learning.
Lingering Question #3: What are the best ways to support students to develop their mathematical understanding in terms of their conceptual understanding, procedural fluency, and problem-solving skills?
After Phase 1 and Phase 2, I felt as though I had not effectively supported my students to develop their mathematical understanding in regard to the three strands. Because students developing a proficiency is very important to me, I feel driven to determine the best ways to support students to do so. I am aware that it depends on the learning experiences I create for my students as well as the ways I encourage them to interact with each other, the content, and me.
To answer this question, I would reach out to my Professional Learning Community about effective strategies they use to support their students to develop their mathematical understanding in terms of the three strands. I would also research effective strategies and try to implement them in my classroom. I would be careful not to continue a strategy that was not working for my students.
In the next and final section I will discuss the implications, limitations and reflection of my research.
Lingering Question #1: What are the long term effects portfolio assessment has on students’ overall mathematical understanding, on students’ dispositions toward math, adn on students’ perceptions of what it means to be good at math?
Between the experiences of Phase 1 and Phase 2, I noticed positive effects in these areas. This research took place for approximately 6 weeks. I am curious as to how these three areas would be affected if protfolio assessment was used as the summative assessment for an entire school year. If I extrapolate the positive results I noticed through my research, it is safe to say that students would be more positively impacted. However, until I do this I cannot say for sure.
To answer this question, I would continue the interventions I made in Phase 1 and Phase 2. To determine the effects portfolio assessments have on all three areas, I would put the same data sources in place and collect data at regular intervals. During the year I would track individual student’s progress in all three areas. At the end of the year I would look at the data collectively and determine the overall effects.
Lingering Question #2: What are the long term effects of continuous Daily Reflections?
As was the case with Lingering Question #1, I noticed strong benefits in my students’ metacognition. They and I attribute this to the fact that each day they would record their learning experience through the Daily Reflections. Since these reflections had such a strong impact on my students, I am interested in learning the effects they have when implemented continuously for an extended period of time.
To explore this question, I would continue to implement the Daily Reflections as I did in Phase 2. Throughout the year I would collect data on my students’ abilities to record and reflect on their learning as well as their ability to demonstrate their mathematical proficiency. I would track individual student’s progress and collectively analyze the data to learn the effects Daily Reflections have on learning.
Lingering Question #3: What are the best ways to support students to develop their mathematical understanding in terms of their conceptual understanding, procedural fluency, and problem-solving skills?
After Phase 1 and Phase 2, I felt as though I had not effectively supported my students to develop their mathematical understanding in regard to the three strands. Because students developing a proficiency is very important to me, I feel driven to determine the best ways to support students to do so. I am aware that it depends on the learning experiences I create for my students as well as the ways I encourage them to interact with each other, the content, and me.
To answer this question, I would reach out to my Professional Learning Community about effective strategies they use to support their students to develop their mathematical understanding in terms of the three strands. I would also research effective strategies and try to implement them in my classroom. I would be careful not to continue a strategy that was not working for my students.
In the next and final section I will discuss the implications, limitations and reflection of my research.