Phase 1
Implementation
Phase 0/Pre-Phase 1
During Phase 0 or Pre-Phase 1, I set out to introduce my students to the different strands of mathematics we would be exploring in Phase 1, because, as I stated in my Action Plan, the majority of my students do not understand the strands well enough to explain them. I learned this through the initial needs assessment. In addition to an introduction to the strands, I planned to give my students the opportunity to practice reflecting on their work. In other words, they were going to create a miniature portfolio.
Phase 0 or Pre-Phase 1 did not go according to plan. From Day 1, I experienced an incredible amount of push back from my students. I introduced the three strands in the form of a simple example. My students explored the procedure of addition, the concept of addition, and then applied addition to solve a problem. Because the example was quite simple (2+2=4), they had no trouble grasping the ideas of the three strands. However, when it got to the part where I explained that we were going to begin emphasizing all three strands rather than just the procedural and the reasons why, they lost it. The majority of my students voiced their discontent about the shift in content focus. Many could not keep from questioning the changes and expressing their strong opposition to them. Some opposed emphasizing the three strands. Others opposed reflecting on their work and/or on their learning. One student said,
Phase 0 or Pre-Phase 1 did not go according to plan. From Day 1, I experienced an incredible amount of push back from my students. I introduced the three strands in the form of a simple example. My students explored the procedure of addition, the concept of addition, and then applied addition to solve a problem. Because the example was quite simple (2+2=4), they had no trouble grasping the ideas of the three strands. However, when it got to the part where I explained that we were going to begin emphasizing all three strands rather than just the procedural and the reasons why, they lost it. The majority of my students voiced their discontent about the shift in content focus. Many could not keep from questioning the changes and expressing their strong opposition to them. Some opposed emphasizing the three strands. Others opposed reflecting on their work and/or on their learning. One student said,
“I see why it’s a good idea to learn the concepts behind the math, like to be more well-rounded or whatever, but how will it help me get a good score on the SAT? The SAT doesn’t have questions like ‘why is two plus two four?’ All I have to know is that two and two is four. I don’t see why we have to change. It’s just going to be more work for nothing.”
Another said, in regard to the reflective component of the changes,
“If I can solve a quadratic equation, I obviously know how to solve a quadratic equation. I don’t need to reflect on it. I already know it. If I didn’t, I wouldn’t have been able to solve it in the first place. This seems like more busywork. And just like the Math Journals it’s going to be a huge waste of time.”
Several others went on to challenge the changes. I spent the entire class period trying to answer the confrontational questions and to persuade my students to the see the benefits these changes could have on them as math learners. It was important to me that they know the reasons why math was going to change in our classroom, and that they were on board. And so, I had no problem taking so much time to respond to their concerns. In spite of my efforts and my intentions, I did not convince them and they were not on board. It is very possible that I alienated them more than they would have been if I had not taken the time to inform them.
On that day, I feel that the majority of my students closed their minds to the benefits of the changes I planned to set forth. They wrote off all the changes before they were even implemented. I should state here that there were a handful of students who embraced the changes, put forth an honest effort, and truly excelled in their learning as a result. The others opposed the changes and challenged them (and me) everyday. Soon into Phase 0, I had become exhausted by the constant battle. I began questioning my motives. I was no longer confident in the changes I had made.
On that day, I feel that the majority of my students closed their minds to the benefits of the changes I planned to set forth. They wrote off all the changes before they were even implemented. I should state here that there were a handful of students who embraced the changes, put forth an honest effort, and truly excelled in their learning as a result. The others opposed the changes and challenged them (and me) everyday. Soon into Phase 0, I had become exhausted by the constant battle. I began questioning my motives. I was no longer confident in the changes I had made.
Phase 1
I always look forward to spending time with my students. As awful as it is to say, there was a brief period when I dreaded the upcoming hour of math because I knew what was in store for me. As Phase 1 began, I decreased the explicit mention of the three strands. Although the content incorporated the strands, we did not discuss them as a class. The only change I had to continue calling attention to was the Daily Reflection. I knew that a significant portion of my students did not see the value in the Daily Reflections. This was evident from their responses and from their attitudes/levels of engagement while completing the Daily Reflections. During the first week of Phase 1, I wrote in my teacher journal that, “most students seem bored, looking around, not engaged, few words written.” There were, however, four or five students who were engaged in the Daily Reflections. They were focused on the task at hand, and dedicated to write honest and thoughtful reflections. One of these students wrote the following in response to the question “What exactly did you learn?”
“One thing that I learned was that it is helpful to draw out the problem and use the elimination method because for myself it helps to be able to visualize the problem.”
During Phase 1, before we began the Portfolios, I felt as though I tried to fly under the radar to keep the peace with my students. As I explained above, I did everything I could to avoid referring to the changes that had taken place in my classroom. This was not possible once we began the Portfolio Project. The intent behind this project was for students to reflect on and analyze their learning and their work for the two previous weeks. For more details, please see my Action Plan. I introduced the Portfolios by going through the Portfolio Project Overview. I explained that reflecting on learning means that a person spends time thinking deeply about what they have learned. For this project, I expected them to look at themselves critically and to use this as an opportunity to become aware of their learning. I stated this explicitly before I went on to explain the Portfolio Tasks.
After the first day of working on the Portfolios, and seeing the little progress my students had made, I decided to reduce the Portfolio Project slightly. Rather than mandating that they complete six tasks, I decided that they would complete Task #1, Task #6, and one of Task #2, Task #3, and Task #4. I completely eliminated Task #5. Task #1 focused on the Daily Reflections; Task #6 was a reflection on students’ overall learning; Tasks #2-4 addressed one of each of the strands; Task #5 focused on a previous test. I was very interested in learning about my students overall learning experience, and their interpretation of their Daily Reflections. This would be achieved by reading my students’ responses to Tasks #1 and #6. I also wanted my students to have the opportunity to explore one of the three strands, and thought they would be successful if they were able to choose the strand they were interested in. This would be achieved by reading my students’ responses to Task #2, #3, or #4.
As it turned out, the week I had dedicated to completing the Portfolio Project in class also happened to be the week my school piloted the Smarter Balanced Assessment. Consequently, I only had two hours, rather than five, when I would see my students for math that entire week. I made it clear that they would have to work on the Portfolios outside of class. I encouraged them to receive peer feedback on their responses and to see me with any questions they had. Since the end of the year was quickly approaching, we did not have the time to extend this project. My students worked on their Portfolios up until the end of Friday that week. The only questions I received were when my students would quickly show me the length of text they had written as and ask “Is this enough?” I heard this question at least ten times during this project. Each time, I made my response loud enough for the entire class to hear. “It has to be long enough for you to answer each question thoroughly and in detail.” Thoroughly and in detail became my mantra that week.
The resistance my students displayed in Pre-Phase 1/Phase 0 was still evident during the Portfolio Project. Rather than explicit opposition, it took the form of complete disinterest. I could not get my students to focus on their assignment and be productive. There was one day when my students were the worst behaved they had been all year. Literally every one of my students was off task. Pandemonium was taking place in my classroom. This was especially frustrating because of the lack of time we had to dedicate to math that week. I tried to bargain with them. I offered them five minutes of free time for them being productive for the remainder of the class period. Looking back, it is clear to me that they were not engaged because they were not interested.
After the first day of working on the Portfolios, and seeing the little progress my students had made, I decided to reduce the Portfolio Project slightly. Rather than mandating that they complete six tasks, I decided that they would complete Task #1, Task #6, and one of Task #2, Task #3, and Task #4. I completely eliminated Task #5. Task #1 focused on the Daily Reflections; Task #6 was a reflection on students’ overall learning; Tasks #2-4 addressed one of each of the strands; Task #5 focused on a previous test. I was very interested in learning about my students overall learning experience, and their interpretation of their Daily Reflections. This would be achieved by reading my students’ responses to Tasks #1 and #6. I also wanted my students to have the opportunity to explore one of the three strands, and thought they would be successful if they were able to choose the strand they were interested in. This would be achieved by reading my students’ responses to Task #2, #3, or #4.
As it turned out, the week I had dedicated to completing the Portfolio Project in class also happened to be the week my school piloted the Smarter Balanced Assessment. Consequently, I only had two hours, rather than five, when I would see my students for math that entire week. I made it clear that they would have to work on the Portfolios outside of class. I encouraged them to receive peer feedback on their responses and to see me with any questions they had. Since the end of the year was quickly approaching, we did not have the time to extend this project. My students worked on their Portfolios up until the end of Friday that week. The only questions I received were when my students would quickly show me the length of text they had written as and ask “Is this enough?” I heard this question at least ten times during this project. Each time, I made my response loud enough for the entire class to hear. “It has to be long enough for you to answer each question thoroughly and in detail.” Thoroughly and in detail became my mantra that week.
The resistance my students displayed in Pre-Phase 1/Phase 0 was still evident during the Portfolio Project. Rather than explicit opposition, it took the form of complete disinterest. I could not get my students to focus on their assignment and be productive. There was one day when my students were the worst behaved they had been all year. Literally every one of my students was off task. Pandemonium was taking place in my classroom. This was especially frustrating because of the lack of time we had to dedicate to math that week. I tried to bargain with them. I offered them five minutes of free time for them being productive for the remainder of the class period. Looking back, it is clear to me that they were not engaged because they were not interested.
Findings
Recall from my research question and subquestions, that I am interested in exploring the effects Portfolio Assessments have on students’ overall mathematical understanding, dispositions toward math, and ideas about what it means to be good at math. In this section I will address my findings in regard to these three areas, as well as other patterns I noticed throughout Phase 1.
Mathematical Understanding
As a math teacher, I am very interested in my students’ mathematical understanding. I want to see their thinking. Specifically in regard to Phase 1 of my research, I wanted to see the ways my students made sense of math that emphasizes conceptual understanding, procedural understanding and problem-solving. For the entire school year up until Phase 1 began, which was in the beginning of May, my students were accustomed to a more traditional way of learning math, one that consisted of my “teaching” the students through example procedural problems. The learning activities for Phase 1, which I obtained from “It’s in the System” a curriculum unit created by Connected Math, were student-centered and inquiry-driven. They emphasized students constructing their learning for themselves and doing so in regard to the three strands instead of only procedural fluency.
In the hopes of seeing their thinking, I explored my students’ work. At the end of the first week of Phase 1 I had in-class activities, homework assignments, a Math Journal entry, and a quiz to analyze. (Note that the quiz was used formatively and not summatively.) The homework assignments included procedural, conceptual and problem-solving type elements. I noticed that for an assignment that addressed procedural and problem-solving practices, sometimes the procedural portion would be completed while the problem-solving would not be. For example, consider the following problem:
For a fundraiser, students sell calendars for $3 and posters for $2. Suppose the goal is to raise $600. One equation relating the calendar and poster sales to the goal of earning $600 is 3c + 2p = $600. Suppose the company donating the calendars and posters decided they can donate a total of 250 items.
a. What equation relates the number of calendars and posters to the 250 items donated?
b. Graph both equations on the same grid. Find the coordinates of the intersection point. Explain what this point tells you about the fundraising situation.
In this problem, students have to write an equation from the given information. Then they have to graph the given equation and the one they created to find the intersection point of the lines and explain what this point means in regard to the problem. In general, my students graphed the given equation, but did not write the second equation. As a result, they were not able to graph both lines or determine the intersection point.
I observed similar actions from at least once for 48% of my students on their homework from the first week of Phase 1. This gave me the impression that although my students understood how to execute the procedures they learned, they were not comfortable applying it to a problem.
The quiz my students took at the end of the each week included conceptual, procedural and problem-solving pieces. In analyzing my students’ work from the first week, I found that 13% of them were proficient in all three strands. They were able to make sense of the underlying concepts, execute procedures and solve the problems on the quiz. I also found that 55% of my students were only successful with procedural fluency. These students correctly answered the procedural problems. However, only one student out of these twelve students did not attempt the conceptual, procedural, and problem-solving questions on the quiz. All the others attempted to make sense of the content. I also found that 9% of my students were successful with the problem-solving and the procedural questions and were unsuccessful with the conceptual question. These students responded to the question, but were not able to make sense of the solution in regard to its implications. Lastly, I found that 23% of my students were unsuccessful with all three strands.
In analyzing my students’ quiz from week 2, I found that only one student was successful in all three strands. This student was apply to accurately execute procedures, make sense of the present concepts and solve the problems. I also found that 50% of my students were unsuccessful with all three strands. From the work these students submitted, I could see that they tried to make sense of the math-- every student attempted all the problems. However, it seemed that they became confused and did not know how to progress. This was apparent from the work my students did. Many of them began the problems, but did not complete them. I also found that 32% of my students were successful only with the procedural problems. This showed me that these students have a strong procedural understanding, but they need more support in the conceptual and problem-solving areas. The table below illustrates the major themes from the quizzes in Phase 1.
Mathematical Understanding
- Student Work
As a math teacher, I am very interested in my students’ mathematical understanding. I want to see their thinking. Specifically in regard to Phase 1 of my research, I wanted to see the ways my students made sense of math that emphasizes conceptual understanding, procedural understanding and problem-solving. For the entire school year up until Phase 1 began, which was in the beginning of May, my students were accustomed to a more traditional way of learning math, one that consisted of my “teaching” the students through example procedural problems. The learning activities for Phase 1, which I obtained from “It’s in the System” a curriculum unit created by Connected Math, were student-centered and inquiry-driven. They emphasized students constructing their learning for themselves and doing so in regard to the three strands instead of only procedural fluency.
In the hopes of seeing their thinking, I explored my students’ work. At the end of the first week of Phase 1 I had in-class activities, homework assignments, a Math Journal entry, and a quiz to analyze. (Note that the quiz was used formatively and not summatively.) The homework assignments included procedural, conceptual and problem-solving type elements. I noticed that for an assignment that addressed procedural and problem-solving practices, sometimes the procedural portion would be completed while the problem-solving would not be. For example, consider the following problem:
For a fundraiser, students sell calendars for $3 and posters for $2. Suppose the goal is to raise $600. One equation relating the calendar and poster sales to the goal of earning $600 is 3c + 2p = $600. Suppose the company donating the calendars and posters decided they can donate a total of 250 items.
a. What equation relates the number of calendars and posters to the 250 items donated?
b. Graph both equations on the same grid. Find the coordinates of the intersection point. Explain what this point tells you about the fundraising situation.
In this problem, students have to write an equation from the given information. Then they have to graph the given equation and the one they created to find the intersection point of the lines and explain what this point means in regard to the problem. In general, my students graphed the given equation, but did not write the second equation. As a result, they were not able to graph both lines or determine the intersection point.
I observed similar actions from at least once for 48% of my students on their homework from the first week of Phase 1. This gave me the impression that although my students understood how to execute the procedures they learned, they were not comfortable applying it to a problem.
The quiz my students took at the end of the each week included conceptual, procedural and problem-solving pieces. In analyzing my students’ work from the first week, I found that 13% of them were proficient in all three strands. They were able to make sense of the underlying concepts, execute procedures and solve the problems on the quiz. I also found that 55% of my students were only successful with procedural fluency. These students correctly answered the procedural problems. However, only one student out of these twelve students did not attempt the conceptual, procedural, and problem-solving questions on the quiz. All the others attempted to make sense of the content. I also found that 9% of my students were successful with the problem-solving and the procedural questions and were unsuccessful with the conceptual question. These students responded to the question, but were not able to make sense of the solution in regard to its implications. Lastly, I found that 23% of my students were unsuccessful with all three strands.
In analyzing my students’ quiz from week 2, I found that only one student was successful in all three strands. This student was apply to accurately execute procedures, make sense of the present concepts and solve the problems. I also found that 50% of my students were unsuccessful with all three strands. From the work these students submitted, I could see that they tried to make sense of the math-- every student attempted all the problems. However, it seemed that they became confused and did not know how to progress. This was apparent from the work my students did. Many of them began the problems, but did not complete them. I also found that 32% of my students were successful only with the procedural problems. This showed me that these students have a strong procedural understanding, but they need more support in the conceptual and problem-solving areas. The table below illustrates the major themes from the quizzes in Phase 1.
Finding #1 This data shows me that my students require additional support developing their conceptual understanding and problem-solving skills.
- Portfolio
I was very interested in learning about my students’ experiences during Phase 1 through their Portfolios. Their responses were an opportunity for them to really analyze their learning experience. I I was surprised to find that 42% of my students’ responses did not respond to the questions within each task. Instead, they reflected on their experience. For example, instead of responding to the conceptual understanding questions that pertain to a specific Math Journal in Task #2, a student wrote about her experience writing all the math journals for the entire year. The questions for Task #2 are:- How does your response shows your conceptual understanding of the Investigation? How did you make sense of the mathematical ideas? How do you know they are true? If there are pieces to the Math Journal that you misinterpreted, explain why. How did writing the journal entry help you understand Linear Equations?
“[The math journals] are usually easy because I know the answers. If I don’t then it’s more difficult.”
Almost half of my students committed similar oversights. Rather than reflecting on their work and analyzing their learning, they shared their experience with me. It was beneficial to gain insight into their experience during Phase 1, but when my students do not respond to the task at hand, I cannot assess their learning.
Finding #2 My students misinterpreting the prompts tells me they do not understand what they are asking.
For the 58% of students who respond to the task at hand, none of them provided enough specific detail to demonstrate their mathematical understanding. It was common for them to reference what they did in their work, but they did not pull specific examples from it. Nor did they discuss the mathematics behind what they did.
Finding #2 My students misinterpreting the prompts tells me they do not understand what they are asking.
For the 58% of students who respond to the task at hand, none of them provided enough specific detail to demonstrate their mathematical understanding. It was common for them to reference what they did in their work, but they did not pull specific examples from it. Nor did they discuss the mathematics behind what they did.
“I had to answer two questions for the math journal, the first being about solving simple equations that were mainly about the second one being about solving a system of linear equations. I made sense of the mathematical ideas mainly by using the past knowledge that I had acquired during the past unit and using what I learned on the systems of equations, the way I knew the ideas were by going back and looking at my packet so I knew what the basic prompt was and also just reviewing my past knowledge on similar problems…”
In this response, the student was supposed to be demonstrating his conceptual understanding of the unit in terms of the big ideas, as well as explain the process through which he came to understand the big ideas. This student begins to brush at the surface of his conceptual understanding. However, because he does not refer specifically to the big ideas or how he understands them, it is not possible for me to assess his conceptual understanding. This was the case for 58% of my students.
Since my students did not showcase their mathematical understanding in the Portfolios, it is impossible for me to assess it. As was the case with the Daily Reflections, I cannot definitively state whether my students did not construct an understanding, or if they simply failed to express their understanding. Thus, I have not learned the effects portfolio assessments have on students' mathematical understanding.
Going into week 2 of Phase 1 I was explicit about what I expected for the Daily Reflections. I instructed students to answer each question included in the prompt and to provide enough detail so that someone could understand their daily experience just from reading the reflection. For the most part, my students responded to the Daily Reflections prompt during week 2. I noticed something new from the second set of Daily Reflections: my students struggled to explain their work/learning for the day. Rather than going into detail recalling what they did, they would claim that they “solved it” or “did it.” Responses like these limited the insight I gained. Since they did not communicate what they did, I was unable to discern if they knew what they did or why they did it and if they could not express it in words, or if they did not know. I was unsure of whether it was because they could not answer them or simply that they did not. When a student is able to explain the mathematics behind their thinking coherently it demonstrates a solid understanding. Hence, I needed to ascertain whether or not my students were able to record their learning.
Finding #3 These responses showed me that my students may not be able to express their learning in words.
Since my students did not showcase their mathematical understanding in the Portfolios, it is impossible for me to assess it. As was the case with the Daily Reflections, I cannot definitively state whether my students did not construct an understanding, or if they simply failed to express their understanding. Thus, I have not learned the effects portfolio assessments have on students' mathematical understanding.
- Daily Reflections
Going into week 2 of Phase 1 I was explicit about what I expected for the Daily Reflections. I instructed students to answer each question included in the prompt and to provide enough detail so that someone could understand their daily experience just from reading the reflection. For the most part, my students responded to the Daily Reflections prompt during week 2. I noticed something new from the second set of Daily Reflections: my students struggled to explain their work/learning for the day. Rather than going into detail recalling what they did, they would claim that they “solved it” or “did it.” Responses like these limited the insight I gained. Since they did not communicate what they did, I was unable to discern if they knew what they did or why they did it and if they could not express it in words, or if they did not know. I was unsure of whether it was because they could not answer them or simply that they did not. When a student is able to explain the mathematics behind their thinking coherently it demonstrates a solid understanding. Hence, I needed to ascertain whether or not my students were able to record their learning.
Finding #3 These responses showed me that my students may not be able to express their learning in words.
Dispositions Toward Math
I knew that the changes made in Phase 1, specifically, the change in content, would be difficult for my students to get used to. From grades kindergarten through eighth they were learning math in the traditional way--with the teacher showing them procedures and steps to follow. The math in Phase 1 emphasized conceptual understanding and problem-solving skills in addition to procedural fluency. I wanted to track their dispositions toward math through Phase 1. To do this, I issued a student feedback form at the end of Phase 1. The form included two questions that access my students’ dispositions toward math. The first is “What is the best way for you to learn math?” 38% of my students explained that they learn math best when it is taught the traditional way. They prefer when the teacher demonstrates how to solve a problem, explicitly lists out the steps, and then does several examples. The following excerpts illustrate this point.
“The best way for me to learn math is when the teacher goes through every step.”
“If it was explained in more detailed and we got to do a example after that.”
“I like having a brief description as to how to solve the problem, 2 examples, and then 4 problems to work on. That's how I've pretty much learned everything so far.”
Only 15% of my students think they learn math best when they interact with and try to make sense of it. They also mentioned how developing a conceptual understanding helps them learn. These ideas align more closely with the way math was presented during Phase 1.
“The best way for myself to learn math is to be interactive with problems. It helps me to be able to see pictures because they help me visualize the concept and what I need to find out.”
Finding #4 The trends in these responses show me that my students are still inclined to learning math in the traditional way.
The second question from the student feedback form that accesses my students' dispositions toward math is "How are you good at math?" This question was also included in the initial needs assessment. On that form, only 29% of my students thought that they are good at math. Additionally, 41% of my students felt that they possessed average skills/abilities in math. After Phase 1, 46% of my students expressed that they are good at math or that they are getting better at math, while only 19% felt that they possessed average skills/abilities in math. This data shows a positive impact on students' dispositions toward math.
Finding #5 My students are developing more positive dispositions toward math.
Perceptions about what it means to be good at math
Recall that an overwhelming portion of my class (59%) believe that to be good at math a person must pass or correctly answer all the questions on the test. I learned this during the initial needs assessment before Phase 1. I had hoped that by emphasizing the thinking process through the new learning activities and by reflecting on and recording their learning, my students’ ideas about what it means to be good at math would have changed. To ascertain their ideas in this regard, I issued a student feedback form at the end of Phase 1. The form included questions about the experience during Phase 1, as wTell as “What does it mean to be good at math?” and “How are you good at math?” To my delight only one student of twenty-six responded that to be good at math means “you have to get ten out of ten on the math test.” I was surprised to see this student respond this way because when asked what it means to be good at math the first time this student said “to be be good at math is learning different math problems and challenge yourself and practice.” The other sixteen of seventeen people had various responses as to what it means to be good at math. They included,
The second question from the student feedback form that accesses my students' dispositions toward math is "How are you good at math?" This question was also included in the initial needs assessment. On that form, only 29% of my students thought that they are good at math. Additionally, 41% of my students felt that they possessed average skills/abilities in math. After Phase 1, 46% of my students expressed that they are good at math or that they are getting better at math, while only 19% felt that they possessed average skills/abilities in math. This data shows a positive impact on students' dispositions toward math.
Finding #5 My students are developing more positive dispositions toward math.
Perceptions about what it means to be good at math
Recall that an overwhelming portion of my class (59%) believe that to be good at math a person must pass or correctly answer all the questions on the test. I learned this during the initial needs assessment before Phase 1. I had hoped that by emphasizing the thinking process through the new learning activities and by reflecting on and recording their learning, my students’ ideas about what it means to be good at math would have changed. To ascertain their ideas in this regard, I issued a student feedback form at the end of Phase 1. The form included questions about the experience during Phase 1, as wTell as “What does it mean to be good at math?” and “How are you good at math?” To my delight only one student of twenty-six responded that to be good at math means “you have to get ten out of ten on the math test.” I was surprised to see this student respond this way because when asked what it means to be good at math the first time this student said “to be be good at math is learning different math problems and challenge yourself and practice.” The other sixteen of seventeen people had various responses as to what it means to be good at math. They included,
“I think for someone to be good at math it means that they understand the problems and are able to show their work and explain it without needing much help.”
“To be able to adapt to new ways of doing math”
“That you understand the whole process without any struggles.”
I noticed a pattern within the responses for these two questions. 35% of my students equated being good at math with understanding how to solve problems. For example, one wrote that, “I think you need to be able to problem solve well and actually know how to do the problem. Also understand it and know the procedure to do the problem.” Another wrote that, “I think that it means that you understand how to do the problems. I think it means that you have done the work and you know how to do it. It doesn't necessarily mean that you get every questions right but it means that you understand the content.” In the initial needs assessment only half of these students made similar connections. The other half connected being good at math with getting the answer quickly or learning the content easily.
Finding #6 This change in opinion shows me that the learning my students experienced during Phase 1 has positively impacted their ideas about what it means to be good at math.
Finding #6 This change in opinion shows me that the learning my students experienced during Phase 1 has positively impacted their ideas about what it means to be good at math.
Next Steps
Now that Phase 1 is complete, I can begin to consider what I hope to learn in Phase 2. At this point, I feel as though my research question and subquestions have not been answered. Consequently, I will use Phase 2 as an opportunity to investigate the effects portfolio assessments have on students’ overall mathematical understanding, dispositions toward math, and perceptions of what it means to be good at math. In other words, for Phase 2, my research questions will remain the same.
What happens when a portfolio is used as a summative assessment in mathematics?
1. What impact does emphasizing conceptual understanding, procedural fluency, problem-solving, and mathematical reasoning through portfolio assessment have on students’ overall mathematical understanding?
2. What impact does portfolio assessment that emphasizes metacognition have on students’ dispositions toward mathematics?
3. What impact does this portfolio assessment have on students’ perception of what it means to be good at math?
Additionally, the structure of my class and the interventions I established in Phase 1 will remain the same. Specifically, my students will continue to record their learning through the Daily Reflections, the math we work on will include conceptual understanding, procedural fluency and problem-solving, and after two weeks they will compile another portfolio. The differences for Phase 2 will respond to the findings I explained above. I believe that Findings #1 and #2 can be addressed by implementing the same change. The same goes for Findings #4 and #5.
Finding #1 This data shows me that my students require additional support developing their conceptual understanding and problem-solving skills.
Finding #4 The trends in these responses show me that my students are still inclined to learning math in the traditional way.
Since the work my students’ completed in Phase 1 shows they are struggling to develop their conceptual understanding and problem-solving skills in regard to Linear Equations, I believe they require extra attention in these areas. I need to support them more in making sense of the conceptual and problem-solving areas. Additionally, I recognize that since my students are accustomed to learning math the traditional way, they prefer it. However, I am still confident that the new way we are doing math will benefit them in the long run. I want to continue this new way of math into Phase 2. That being said, I need to do a better job of meeting my students at their level. To achieve both of these, I will consistently show and remind my students of the “big picture” of the current unit, assist them in making connections between the different topics, encourage them to identify similarities and differences between the different topics, and ask them to explain their thinking. According to Marzano, Pickering, and Pollock (2001), these strategies significantly impact student understanding. Since my students are not used to this new way of learning math, I feel as though they require more scaffolding. Rather than expecting them to make sense of the content on their own, I need to be mindful of the fact that they are not used to this and do not know how to do it. I need to be more proactive about meeting them at their level and providing them with the supports they require to develop their conceptual understanding and problem-solving skills.
Finding #2 These responses showed me that my students may not be able to express their learning in words.
Finding #3 My students misinterpreting the prompts tells me they do not understand what they are asking.
I am unsure of the causes of these findings. 1. They do not understand what the prompts are asking. 2. They cannot answer the prompts. 3. They are able, but simply do not respond to the prompts. The cause of the trends will inform the way I respond to the trends. That being said, it is necessary that I uncover the cause during Phase 2. To do this, I will restructure the Daily Reflection and Portfolio Tasks. Rather than displaying the prompts in paragraph form, I will number each question and provide space for each question to be answered. I believe that this change in structure will support students to recognize that each question within the prompt must be addressed. From here, I will ascertain whether or not my students are able to respond to the prompts.
Finding #5 My students are developing more positive dispositions toward math.
Finding #6 This change in opinion shows me that the learning my students experienced during Phase 1 has positively impacted their ideas about what it means to be good at math.
I do not need to address these findings in Phase 2 in terms of making a change. However, I will continue to track my students’ dispositions and perceptions through Phase 2.
Please read on to learn about Phase 2!
What happens when a portfolio is used as a summative assessment in mathematics?
1. What impact does emphasizing conceptual understanding, procedural fluency, problem-solving, and mathematical reasoning through portfolio assessment have on students’ overall mathematical understanding?
2. What impact does portfolio assessment that emphasizes metacognition have on students’ dispositions toward mathematics?
3. What impact does this portfolio assessment have on students’ perception of what it means to be good at math?
Additionally, the structure of my class and the interventions I established in Phase 1 will remain the same. Specifically, my students will continue to record their learning through the Daily Reflections, the math we work on will include conceptual understanding, procedural fluency and problem-solving, and after two weeks they will compile another portfolio. The differences for Phase 2 will respond to the findings I explained above. I believe that Findings #1 and #2 can be addressed by implementing the same change. The same goes for Findings #4 and #5.
Finding #1 This data shows me that my students require additional support developing their conceptual understanding and problem-solving skills.
Finding #4 The trends in these responses show me that my students are still inclined to learning math in the traditional way.
Since the work my students’ completed in Phase 1 shows they are struggling to develop their conceptual understanding and problem-solving skills in regard to Linear Equations, I believe they require extra attention in these areas. I need to support them more in making sense of the conceptual and problem-solving areas. Additionally, I recognize that since my students are accustomed to learning math the traditional way, they prefer it. However, I am still confident that the new way we are doing math will benefit them in the long run. I want to continue this new way of math into Phase 2. That being said, I need to do a better job of meeting my students at their level. To achieve both of these, I will consistently show and remind my students of the “big picture” of the current unit, assist them in making connections between the different topics, encourage them to identify similarities and differences between the different topics, and ask them to explain their thinking. According to Marzano, Pickering, and Pollock (2001), these strategies significantly impact student understanding. Since my students are not used to this new way of learning math, I feel as though they require more scaffolding. Rather than expecting them to make sense of the content on their own, I need to be mindful of the fact that they are not used to this and do not know how to do it. I need to be more proactive about meeting them at their level and providing them with the supports they require to develop their conceptual understanding and problem-solving skills.
Finding #2 These responses showed me that my students may not be able to express their learning in words.
Finding #3 My students misinterpreting the prompts tells me they do not understand what they are asking.
I am unsure of the causes of these findings. 1. They do not understand what the prompts are asking. 2. They cannot answer the prompts. 3. They are able, but simply do not respond to the prompts. The cause of the trends will inform the way I respond to the trends. That being said, it is necessary that I uncover the cause during Phase 2. To do this, I will restructure the Daily Reflection and Portfolio Tasks. Rather than displaying the prompts in paragraph form, I will number each question and provide space for each question to be answered. I believe that this change in structure will support students to recognize that each question within the prompt must be addressed. From here, I will ascertain whether or not my students are able to respond to the prompts.
Finding #5 My students are developing more positive dispositions toward math.
Finding #6 This change in opinion shows me that the learning my students experienced during Phase 1 has positively impacted their ideas about what it means to be good at math.
I do not need to address these findings in Phase 2 in terms of making a change. However, I will continue to track my students’ dispositions and perceptions through Phase 2.
Please read on to learn about Phase 2!