Portfolio Assessments in an 8th grade Mathematics classroom
One Tuesday during this school year one of my students (I’ll call him Chauncy) asked me if he could have a practice test. I was surprised by his request because we usually take practice tests on Thursdays in preparation for the tests on Fridays. I asked Chauncey if he had gotten through all the work for the week. He said that he didn’t need to “waste [his] time doing the packet” because he only needed “to know how to do those 10 [problems on the test].” Needless to say, I was shocked; shocked because Chauncey was right.
From the way my cooperating teacher and I structured our 8th grade Math class, students only had to correctly answer 10 computational problems to demonstrate their “mastery” of the content. In other words, our class focus is on the test; when students pass the test, we take that to mean they mastered the mathematics. However, correctly answering ten computation problems cannot demonstrate a student’s mastery of a mathematical topic. It can only prove that the student can perform the particular procedures on the test.
On that Tuesday, I gave Chauncey the practice test, as he had requested. He correctly answered one out of the ten problems. It was evident that he did not understand the content. What was surprising was the disconnect between what he understood and what he thought he understood. He was not prepared for the test, and he didn't even know it. I experienced something similar as an undergraduate in Mathematics. The most advanced class I took was an introduction to Lie Theory (which can be summarized as continuous groups). Going into the first exam, I was less than confident in my understanding of Lie Theory, but I thought I knew enough to pass the test. Turns out, I, like Chauncey, was greatly mistaken. It wasn’t until I met with the professor and we began to discuss what I knew and understood that I realized I didn’t understand any of it. It took me thinking about what I knew, and trying to explain it to someone else for me to begin to understand my lack of understanding.
From the way my cooperating teacher and I structured our 8th grade Math class, students only had to correctly answer 10 computational problems to demonstrate their “mastery” of the content. In other words, our class focus is on the test; when students pass the test, we take that to mean they mastered the mathematics. However, correctly answering ten computation problems cannot demonstrate a student’s mastery of a mathematical topic. It can only prove that the student can perform the particular procedures on the test.
On that Tuesday, I gave Chauncey the practice test, as he had requested. He correctly answered one out of the ten problems. It was evident that he did not understand the content. What was surprising was the disconnect between what he understood and what he thought he understood. He was not prepared for the test, and he didn't even know it. I experienced something similar as an undergraduate in Mathematics. The most advanced class I took was an introduction to Lie Theory (which can be summarized as continuous groups). Going into the first exam, I was less than confident in my understanding of Lie Theory, but I thought I knew enough to pass the test. Turns out, I, like Chauncey, was greatly mistaken. It wasn’t until I met with the professor and we began to discuss what I knew and understood that I realized I didn’t understand any of it. It took me thinking about what I knew, and trying to explain it to someone else for me to begin to understand my lack of understanding.
Needs Assessment
The interaction with Chauncey made me realize that
1. the assessments in my classroom do not effectively measure my students’ mathematical understanding,
2. there are gaps in my students’ mathematical understanding, and
3. my students are not cognizant of their learning.
It became quite clear that I must improve the conditions of the mathematical learning taking place in my classroom. Before I made any changes to the class structure, I wanted to connect with my students to learn about their perceptions of math. I was curious what they think it means to be mathematically proficient and to demonstrate their proficiency. I decided to digitally issue a needs assessment (as a student feedback form) as a way to gain insight. The needs assessment included the following questions:
1. What is math?
2. What does it mean to be good at math?
3. How are you good at math?
4. How well can you explain the idea of “conceptual understanding” in math?
5. How well can you explain the idea of “procedural fluency” in math?
6. How well can you explain the idea of “problem solving” in math?
Question #1 was meant to uncover my students’ foundational understanding of math. I am interested in this because this foundation influences how students interact with math. For example, if I believe that math is performing calculations, then I will see my interactions with math as only performing calculations. On the other hand, if I believe math is a problem-solving tool, I will see my interactions with math as collecting different strategies and learning different ways to solve problems.
Questions #2 and #3 were meant to uncover my students’ perceptions of what it means to be good at math. Their idea of what it means to be good at math will frame how they interpret their mathematical understanding. For example, if my idea of what it means to be good at math means knowing several different ways to solve one problem, then I will know I understand math when I am able to do so.
Questions #4, #5, and #6 were meant to provide me with insight into the different areas of mathematics. Before I move forward with this research project, I will need to know how my students understand the different pieces of math and fit them together. This will be explained more below.
My students responded to the survey individually on computers. I gave them plenty of space during this time so that they would feel comfortable answering honestly. The questions I included begin to flush out students’ perceptions of what math is, what it means to be good at math (proficient in mathematics), and their basic understanding of the different strands of mathematics.
At this point, I was particularly curious about what my students think it means to be good at math. Of the 29 students who responded to the survey, 17 (59%) equated being good at math with passing the math test. Below is a list of a few responses to the questions “What does it mean to be good at math?” and “How are you good at math?”
1. the assessments in my classroom do not effectively measure my students’ mathematical understanding,
2. there are gaps in my students’ mathematical understanding, and
3. my students are not cognizant of their learning.
It became quite clear that I must improve the conditions of the mathematical learning taking place in my classroom. Before I made any changes to the class structure, I wanted to connect with my students to learn about their perceptions of math. I was curious what they think it means to be mathematically proficient and to demonstrate their proficiency. I decided to digitally issue a needs assessment (as a student feedback form) as a way to gain insight. The needs assessment included the following questions:
1. What is math?
2. What does it mean to be good at math?
3. How are you good at math?
4. How well can you explain the idea of “conceptual understanding” in math?
5. How well can you explain the idea of “procedural fluency” in math?
6. How well can you explain the idea of “problem solving” in math?
Question #1 was meant to uncover my students’ foundational understanding of math. I am interested in this because this foundation influences how students interact with math. For example, if I believe that math is performing calculations, then I will see my interactions with math as only performing calculations. On the other hand, if I believe math is a problem-solving tool, I will see my interactions with math as collecting different strategies and learning different ways to solve problems.
Questions #2 and #3 were meant to uncover my students’ perceptions of what it means to be good at math. Their idea of what it means to be good at math will frame how they interpret their mathematical understanding. For example, if my idea of what it means to be good at math means knowing several different ways to solve one problem, then I will know I understand math when I am able to do so.
Questions #4, #5, and #6 were meant to provide me with insight into the different areas of mathematics. Before I move forward with this research project, I will need to know how my students understand the different pieces of math and fit them together. This will be explained more below.
My students responded to the survey individually on computers. I gave them plenty of space during this time so that they would feel comfortable answering honestly. The questions I included begin to flush out students’ perceptions of what math is, what it means to be good at math (proficient in mathematics), and their basic understanding of the different strands of mathematics.
At this point, I was particularly curious about what my students think it means to be good at math. Of the 29 students who responded to the survey, 17 (59%) equated being good at math with passing the math test. Below is a list of a few responses to the questions “What does it mean to be good at math?” and “How are you good at math?”
- “To be good at math I think that it means that you understand the problems and you can [pass] the test on a regular [basis].”
“I feel that I am pretty good [at math] as I get good test scores and I get a lot of work done.”
“I don't consider myself good at math. I never have. I've never gotten 100% or more on a math test, and I believe you are not good at math if you can't ace one test out of hundreds.”
- “...Doing well in tests I guess can show if you're good at the math you went over...”
“In my opinion ‘to be good’ at math means that you pass every single test… with no confusion, and that [you’re] confident about...passing the next upcoming test.”
It is apparent that the majority of my students view math and being good at math in a way similar to Chauncey. I believe the structure of my classroom has enabled this. I teach math in an 8th grade Math/Science classroom in a charter school in San Diego, CA. Within the school, math is taught traditionally. What I mean by this is that the chosen curriculum focuses on procedural fluency. Additionally, teachers present the content through teacher-led examples that students copy down. Then, students perform the procedures on their own by following the steps dictated by the teachers. This teaching practice does not allow for making connections between topics, critical thinking, or applying different strategies to problems. Students complete daily homework assignments, but the majority of their math grade is determined by their test scores.
Research Goals
As a mathematician and math educator, I recognize that math is more than performing computations or procedures. There are different strands of mathematics and in learning mathematics, it is crucial for students to understand the interdependence of the strands. According to Jeremy Kilpatrick, Jane Swafford, and Bradford Findell (the authors of Adding it Up: Helping Children Learn Mathematics), for students to truly succeed in their mathematics learning, it is necessary that they become proficient in the strands of mathematics. For use in my research I will address the following strands (which I adapted from Adding It Up…):
1. Conceptual Understanding: comprehension of mathematical concepts
2. Procedural Fluency: the ability to carry out procedures effectively and accurately
3. Problem-Solving: the ability to solve and represent mathematical problems
(From this point on, being proficient in mathematics will refer to being proficient in the three strands listed above.)
After reflecting and connecting with my students, it is evident that the focus of our class is on procedural fluency. I want to find ways to encourage my students to develop and demonstrate their mathematical proficiency. Since I plan to expand the focus of our math learning through this research project, I realize I must restructure the assessments in my class in order for my students to truly demonstrate their mathematical understanding.
From the research I have done, I understand that assessments are used to gather evidence of students’ understanding. The most effective assessment tools allow students and teachers to better understand the learning that has taken place. (Powell, 2013). Additionally, they must support students to regulate their learning by making sense of their successes and struggles. In order for students to regulate their learning, they must become aware of their learning and know themselves as learners (Tanner & Jones, 2003). Metacognition in assessments of math learning will require students to reflect on and analyze their learning.
Needless to say, the assessment tools that can satisfy these goals are alternative rather than the traditional test. One type of alternative assessment is a portfolio. Powell (2013) frames portfolio assessments as purposeful collections of work that students reflect on, evaluate and analyze to demonstrate their learning. I believe that portfolio assessments will be the most effective tool in assessing my students’ mathematical proficiency throughout this research project because it will provide them with the opportunity to interpret their learning.
I will explain the ideas of mathematical proficiency and portfolio assessments in greater detail in my Literature Review. Also, I will expand on how I plan to implement portfolios into my class in my Action Plan.
1. Conceptual Understanding: comprehension of mathematical concepts
2. Procedural Fluency: the ability to carry out procedures effectively and accurately
3. Problem-Solving: the ability to solve and represent mathematical problems
(From this point on, being proficient in mathematics will refer to being proficient in the three strands listed above.)
After reflecting and connecting with my students, it is evident that the focus of our class is on procedural fluency. I want to find ways to encourage my students to develop and demonstrate their mathematical proficiency. Since I plan to expand the focus of our math learning through this research project, I realize I must restructure the assessments in my class in order for my students to truly demonstrate their mathematical understanding.
From the research I have done, I understand that assessments are used to gather evidence of students’ understanding. The most effective assessment tools allow students and teachers to better understand the learning that has taken place. (Powell, 2013). Additionally, they must support students to regulate their learning by making sense of their successes and struggles. In order for students to regulate their learning, they must become aware of their learning and know themselves as learners (Tanner & Jones, 2003). Metacognition in assessments of math learning will require students to reflect on and analyze their learning.
Needless to say, the assessment tools that can satisfy these goals are alternative rather than the traditional test. One type of alternative assessment is a portfolio. Powell (2013) frames portfolio assessments as purposeful collections of work that students reflect on, evaluate and analyze to demonstrate their learning. I believe that portfolio assessments will be the most effective tool in assessing my students’ mathematical proficiency throughout this research project because it will provide them with the opportunity to interpret their learning.
I will explain the ideas of mathematical proficiency and portfolio assessments in greater detail in my Literature Review. Also, I will expand on how I plan to implement portfolios into my class in my Action Plan.
Research Questions
These questions will drive my research:
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?
In the following section I will review the Literature on mathematical proficiency and assessments in math before I move forward with my research.
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?
In the following section I will review the Literature on mathematical proficiency and assessments in math before I move forward with my research.