Literature Review
After I conducted my first needs assessment, I sought out various studies and articles, that were aligned with my research focus, to help shape the actions I will take in my classroom. As I have mentioned, my aim is to support my students to develop their mathematical proficiency and provide a well-rounded summative assessment that permits them to demonstrate this proficiency. My hope was that the existing literature would help me get a sense of the status of assessments in general, in math in particular, and how assessments that align with mathematical proficiency can assess students’ mathematical proficiency.
Why I’m concerned about mathematical proficiency
K-12 students are not developing a proficiency in mathematics from the education they receive. In general, “too few students in our elementary and middle schools are successfully acquiring the mathematical knowledge, the skill, and the confidence they need to use the mathematics they have learned” (Kilpatrick et al., 2001). Furthermore, the focus of mathematical learning is narrow, and tends to emphasize procedural understanding. As a result, many students view mathematics as “a collection of disconnected, standard procedures...rather than [as] a problem-solving tool” (Boaler, 2002). This problem is significant because it negatively affects student achievement. Students memorize a set of procedures and haphazardly apply them in the hopes of producing the correct answer.
“When considering what it means to know mathematics, most people think of one’s knowledge of...procedures. Of course, these things are central to knowing mathematics. But mathematics is a domain in which what one does to frame and solve problems also matters a great deal. Simply ‘knowing’ concepts does not equip one to use mathematics effectively because using mathematics involves performing a series of skillful activities, depending on the problem being addresses” (Ball et al., 2003).
This series of skills has hence been referred to as mathematical practices. Ball et al. (2003) claim that continuous exposure to and opportunities to interact with the mathematical practices “are essential for anyone learning and doing mathematics proficiently.” Some examples of mathematical practices include making sense of problems, reasoning in different ways, and constructing arguments National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010).
The idea of emphasizing mathematical practices has been given more attention since the adoption of the Common Core State Standards (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). Within the mathematics standards, the CCSS include the Standards for Mathematical Practice, which “describe varities of expertise that mathematics educators at all levels should seek to develop in their students…[and] rest on important ‘processes and proficiencies’ with longstnading importance in mathematics education” (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). The authors of the CCSS connect the mathematical practices with mathematical proficiency.
The idea of what it means to be mathematically proficient was conceptualized by a group led by Jeremy Kilpatrick in 2001. Their outlook on effective mathematics education was published in Adding it up: Helping children learn mathematics in 2001. They propose that for students to be proficient in math, they must demonstrate their competency in conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (Kilpatrick et al., 2001). Together these are referred to as “the strands of mathematical proficiency,” and are defined as follows:
Since 2001, prominent figures in mathematics education reform, including Jo Boaler, Deborah Loewenberg Ball and the authors of the CCSS, have adopted and advocated the strands of mathematical proficiency. It is understood that the five strands of mathematical proficiency are interrelated and interconnected. Rather than focusing on each one individually, a comprehensive and effective mathematics education will support the overlapping and interaction of all five strands. “The core issue is one of balance and completeness, which suggests that school mathematics requires approaches that address all strands” (Ball et al., 2003).
Implications for my research For my students to develop their mathematical understanding, I must engage them in learning experiences that incorporate the five strands of mathematics as well as expose them to the mathematical practices. It is not enough to focus on one of the five strands. I must support my students to develop in each area. Additionally, I must encourage them to engage with the content in terms of the mathematical practices.
The idea of emphasizing mathematical practices has been given more attention since the adoption of the Common Core State Standards (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). Within the mathematics standards, the CCSS include the Standards for Mathematical Practice, which “describe varities of expertise that mathematics educators at all levels should seek to develop in their students…[and] rest on important ‘processes and proficiencies’ with longstnading importance in mathematics education” (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). The authors of the CCSS connect the mathematical practices with mathematical proficiency.
The idea of what it means to be mathematically proficient was conceptualized by a group led by Jeremy Kilpatrick in 2001. Their outlook on effective mathematics education was published in Adding it up: Helping children learn mathematics in 2001. They propose that for students to be proficient in math, they must demonstrate their competency in conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (Kilpatrick et al., 2001). Together these are referred to as “the strands of mathematical proficiency,” and are defined as follows:
- conceptual understanding—comprehension of mathematical concepts, operations, and relations
- procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
- strategic competence—ability to formulate, represent, and solve mathematical problems
- adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
- productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
Since 2001, prominent figures in mathematics education reform, including Jo Boaler, Deborah Loewenberg Ball and the authors of the CCSS, have adopted and advocated the strands of mathematical proficiency. It is understood that the five strands of mathematical proficiency are interrelated and interconnected. Rather than focusing on each one individually, a comprehensive and effective mathematics education will support the overlapping and interaction of all five strands. “The core issue is one of balance and completeness, which suggests that school mathematics requires approaches that address all strands” (Ball et al., 2003).
Implications for my research For my students to develop their mathematical understanding, I must engage them in learning experiences that incorporate the five strands of mathematics as well as expose them to the mathematical practices. It is not enough to focus on one of the five strands. I must support my students to develop in each area. Additionally, I must encourage them to engage with the content in terms of the mathematical practices.
How mathematical proficiency and assessments align
Jo Boaler claims that the view of what it means to be proficient in math is narrow. In general, the focus is on passing the test. Educators who have adopted the five strands of mathematical proficiency argue that for students to be proficient in math, they must demonstrate their competency in conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (Kilpatrick et al., 2001, Boaler, 2002).
If we subscribe to this view of what it means to be proficient in math, it is necessary to consider how evidence of this proficiency is obtained. Boaler poses the following question:
If we subscribe to this view of what it means to be proficient in math, it is necessary to consider how evidence of this proficiency is obtained. Boaler poses the following question:
“Is success on a short, procedural test the measure we want to adopt to assess the effectiveness of our student’s learning? In other words, do these tests assess the sort of knowledge use, critical thought, and reasoning that is needed by learners…?” (Boaler, 1999).
According to Gerald Klum, the purpose of assessments has narrowed in the last ten years. Rather than being used to gather evidence of what an individual student knows and/or is capable of, they are being used to make large-scale decisions (Klum, 2013). He goes on to argue that tests can not demonstrate a student’s true mathematical proficiency because they cover surface level information without diving deep into the content and calling on students’ multiple abilities. The traditionally held view of what it means to understand math--getting the correct answers on the test--supported procedural tests as an appropriate assessment tool. However, if we move forward and expand our view of mathematical proficiency, we must reconsider how students’ mathematical understanding is assessed.
Howard Tanner and Sonia Jones argue that “target setting based on summative assessment often results in students (and teachers) emphasizing...a focus on test performance rather than understanding” (Tanner & Jones, 2003). They go on to argue that rather than target setting as the framework for assessments, students, not teachers, institutions, or government agencies, should have the opportunity to use the information gained from assessments to analyze what they know, what they need to know, and how they can fill in the gaps. In other words, the ultimate goal of assessments should be for students to regulate their learning. In regulating their learning, students need to be metacognitive; they need to be aware of what they know and form beliefs about their strengths and weaknesses.
Implications for my research Since I plan to emphasize the five strands of mathematical proficiency during my students’ experiences with mathematics, I must implement an assessment tool that allows students to demonstrate and realize their understanding in terms of the five strands.
Howard Tanner and Sonia Jones argue that “target setting based on summative assessment often results in students (and teachers) emphasizing...a focus on test performance rather than understanding” (Tanner & Jones, 2003). They go on to argue that rather than target setting as the framework for assessments, students, not teachers, institutions, or government agencies, should have the opportunity to use the information gained from assessments to analyze what they know, what they need to know, and how they can fill in the gaps. In other words, the ultimate goal of assessments should be for students to regulate their learning. In regulating their learning, students need to be metacognitive; they need to be aware of what they know and form beliefs about their strengths and weaknesses.
Implications for my research Since I plan to emphasize the five strands of mathematical proficiency during my students’ experiences with mathematics, I must implement an assessment tool that allows students to demonstrate and realize their understanding in terms of the five strands.
Responses to this problem
Richard Stiggins et al. argue that there are two types of assessment, assessment of learning and assessment for learning, and they both have their place in education (Stiggins et al., 2007). They define assessment of learning as “[measuring] achievement status at a point in time for purposes of reporting…”, and define assessment for learning as “[supporting] ongoing student growth.” Assessment for learning is similar to formative assessments, but includes descriptive (rather than evaluative) feedback that supports students to fill in the gaps of their knowledge and meet the established goals in the assessments of learning. From this argument, it is apparent that to truly assess student learning at the end of a unit, teachers must provide students with descriptive feedback throughout the unit to support them to reach the goals set in place. When there is a lack of focus on assessments for learning, assessments of learning are not meaningful to students.
Gerald Klum argues in favor of alternative assessments over traditional assessments. “Performance assessments [a type of alternative assessment,] are broader in scope than the traditional pencil and paper assessments, since they require and incorporate the entire context of a task.” (Klum, 2013). Furthermore, performance assessments provide multiple ways for students to demonstrate their newly acquired skills and understanding which results in a focus on the learning process rather than on speed and accuracy.
Portfolios are considered a type of performance assessment. Terry Powell defines a portfolio as “a specific collection of material and documents with the purpose of documenting a specific range of performance over a period of time” that include “a component of self-evaluation” (Powell, 2013). It is important that portfolios have a purpose and do not become meaningless collections of student work. Paulson et al. have a similar vision for portfolios. They claim that “a portfolio is a purposeful collection of student work that exhibits the student’s efforts, progress, and achievement,” and that students must choose the work they will include and reflect on the work (Paulson et al., 1991). When constructed this way, portfolios do two things: allow students to learn about their learning, and allow teachers to understand the learning process for individual students (Paulson et al., 1991).
Powell argues that teachers and students are better able to understand a student’s learning in analyzing a student’s portfolio than in analyzing a traditional assessment. Portfolios allow students to construct evidence of their understanding that is meaningful to them. This enhances the students’ quality of learning as well as allows teachers to truly see the achievements their students have made. Portfolios, as opposed to traditional assessments (tests), are effective in gauging the understanding students with ADHD achieve because of their inherent visual nature.
Implications for my research The opportunity for students to create a portfolio from carefully chosen pieces of work and self-reflections/analyses of their work will allow students to realize their learning and demonstrate their mathematical understanding. The portfolios can be used as an assessment tool.
Read on to learn about my Action and Assessment Plan for Phase 1!
Gerald Klum argues in favor of alternative assessments over traditional assessments. “Performance assessments [a type of alternative assessment,] are broader in scope than the traditional pencil and paper assessments, since they require and incorporate the entire context of a task.” (Klum, 2013). Furthermore, performance assessments provide multiple ways for students to demonstrate their newly acquired skills and understanding which results in a focus on the learning process rather than on speed and accuracy.
Portfolios are considered a type of performance assessment. Terry Powell defines a portfolio as “a specific collection of material and documents with the purpose of documenting a specific range of performance over a period of time” that include “a component of self-evaluation” (Powell, 2013). It is important that portfolios have a purpose and do not become meaningless collections of student work. Paulson et al. have a similar vision for portfolios. They claim that “a portfolio is a purposeful collection of student work that exhibits the student’s efforts, progress, and achievement,” and that students must choose the work they will include and reflect on the work (Paulson et al., 1991). When constructed this way, portfolios do two things: allow students to learn about their learning, and allow teachers to understand the learning process for individual students (Paulson et al., 1991).
Powell argues that teachers and students are better able to understand a student’s learning in analyzing a student’s portfolio than in analyzing a traditional assessment. Portfolios allow students to construct evidence of their understanding that is meaningful to them. This enhances the students’ quality of learning as well as allows teachers to truly see the achievements their students have made. Portfolios, as opposed to traditional assessments (tests), are effective in gauging the understanding students with ADHD achieve because of their inherent visual nature.
Implications for my research The opportunity for students to create a portfolio from carefully chosen pieces of work and self-reflections/analyses of their work will allow students to realize their learning and demonstrate their mathematical understanding. The portfolios can be used as an assessment tool.
Read on to learn about my Action and Assessment Plan for Phase 1!