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Education leaders, politicians, researchers, and teachers have been attempting to define proficiency in mathematics for years. Within the current educational system, the most popular measure of a student\u2019s ability in the discipline of mathematics boils down to a single score on a single test administered once a year. That is not enough.<\/p>\n
Though standardized testing has been a part of education for many years<\/a>, No Child Left Behind really made the push to reduce student achievement to a single cut score and label students as \u201cbelow,\u201d \u201capproaching,\u201d \u201con-level,\u201d or \u201cabove.\u201d This has had a ripple effect on nearly every aspect of education<\/a>, from the perceptions of the student and their opportunities to teacher evaluations and district-level funding.<\/p>\n But the impacts go beyond that to define not only what math should be taught but how it should be delivered. High-stakes assessments place an overemphasis on product over process and memory over cognition<\/a>. To meet the demands of the standardized tests, narrowed learning progressions are designed<\/a> to teach content as quickly as possible so students have a fair run at the test each year. This reduces mathematics to a rigid timeline of skills and shortcuts, rather than a rich landscape of concepts that develop and connect to each other organically. Because most classrooms are forced to rush down the prescribed track, computational speed and accuracy become the primary measure of a student\u2019s success\u2014and are often mistakenly equated with proficiency. All this rushing to perform also keeps teachers from opportunities to make genuine connections to the big ideas, transition naturally between concepts, or teach flexibly without regard to the traditional sequence.<\/p>\n Standardized testing isn\u2019t going away, so the best place to start the search for a better approach is with redefining what it means to be proficient. What if the true measure of a student\u2019s proficiency in mathematics was more natural, authentic, and comprehensive?<\/p>\n In Adding It Up: Helping Children Learn Mathematics<\/em><\/a>, Jeremy Kilpatrick, Jane Swafford, and Bradford Findell propose a five-strand measure of successful learning in mathematics. The five interdependent strands of proficiency are:<\/p>\n How much more powerfully could a student\u2019s achievement in mathematics be described using these five strands? Imagine what a teacher could do with such nuanced understandings about a student, as opposed to what they can do with a single percentage of multiple-choice questions answered correctly. Breaking a student\u2019s math journey into five categories allows for richer feedback on strengths and areas of growth.<\/p>\n Consider a student who has been labeled \u201cabove grade level\u201d in math on the past three years of standardized assessments. Perhaps this student is truly accurate and efficient in their calculations with all four operations (that is, procedural fluency) but sometimes has difficulty with transferring a concept to another unfamiliar context (adaptive reasoning) or understanding why the procedure they have memorized and can quickly reproduce actually works (conceptual understanding). This is, in fact, the story of many \u201cadvanced\u201d students in elementary school. Though computationally fluent, many have difficulty explaining their reasoning, connecting ideas, or representing thinking in multiple ways, all skills not measured by standardized tests. Without multiple measures of proficiency, this oversimplified view of \u201cabove grade level\u201d stunts the student\u2019s opportunities to grow in other areas and gives parents and even peers a false sense of what success in mathematics looks like.<\/p>\n Now consider a student who has been labeled \u201cbelow grade level\u201d for the past three years. Since intervention opportunities have been provided based on this test result, the student has spent significantly more time than the \u201cabove grade level\u201d student on building concrete and pictorial models to develop their understanding of how the math works. This has resulted in a stronger ability to choose the right strategy and tool for a situation (strategic competence). But if we rely solely on a test focused on procedural fluency, this student\u2019s growth in other areas\u2014and eventual gains in fluency\u2014may go unnoticed.<\/p>\n We are doing a disservice to both of these students when we expect a single measure of their ability to dictate their entire capacity for finding success in the discipline. Therefore, our mindset about how to assess students and report on these aspects of proficiency in mathematics needs to change.<\/p>\n High quality assessment consists of providing a balance<\/a> of formal and informal opportunities to show understanding. Rather than retrofitting one of the strands into an already existing assessment, I encourage you to design new assessment tasks that target a specific strand.<\/p>\n Below are a few sample strong tasks that can measure proficiency or growth, depending on the assessment\u2019s purpose and timing. In presenting and categorizing the tasks in the ways I have, I also hope to illuminate some defining traits of strong assessment practices:<\/p>\n For more great assessment task ideas, try out Open Middle problems<\/a>, our free Formative Conversation Starters<\/a>, or even student discourse as formative assessment<\/a>.<\/p>\n Task: Given the number 3\/8 on the open number line as shown, locate the number 4\/3.<\/p>\n Task: Place the numbers 0\u20139 exactly once into the spaces in the expressions below such that each expression has a value of 32.<\/p>\n Task: A class is building a rectangular garden. They only have 72 feet of fencing to go around the garden. What dimensions of the garden would guarantee the most area for planting?<\/p>\n This task assesses a student\u2019s ability to formulate, represent, and solve a mathematical problem. No matter the solving strategy, the student should be encouraged to model the relationship in some way and rely on geometric and algebraic reasoning to come to an authentic conclusion.<\/p>\n Task: A woman runs two races of equal length. In the second race, she finishes in 80% of the time it took her in the first. Does this mean she ran 20% faster? Explain.<\/p>\n This task assesses a student\u2019s ability to analyze a proportional relationship and deeply consider the actual difference between a 20% reduction in time and a 20% increase in speed, though these are sometimes incorrectly interpreted as the same thing.<\/p>\n Task: Design a math game for younger students that helps them learn about multiplication. What kind of game would you create? What is most important in the experience younger students have with your game and why?<\/p>\n This task assesses a student\u2019s ability to think creatively about what aspects of mathematics would be important to convey to younger students. They will, in turn, project their own values in learning mathematics and show confidence in their own ability to learn and use math.<\/p>\n Though I\u2019ve spent a considerable number of words criticizing the improper application of standardized test results in this article, I believe they are still a viable metric as long as they are a) applied to a holistic understanding of the student in the right ways and b) used as one of many data points that tell a more comprehensive story of a student\u2019s learning trajectory. In fact, this is how we recommend schools testing with MAP\u00ae Growth\u2122<\/a> use the assessment\u2019s data.<\/p>\n Collecting data year-round from multiple perspectives strengthens teaching and aligns understanding among teachers, parents and other caregivers, and students. Here are some great times to apply the data.<\/p>\n Knowing specifically what to reinforce and build from with each student is a powerful tool. Labeling students as \u201cbelow\u201d or \u201cabove\u201d or \u201cgood at fifth-grade math\u201d doesn\u2019t inform how to push them to develop multiple dimensions of proficiency.<\/p>\n For more ideas on how to use MAP Growth and other assessment data when planning instruction, visit our archive of posts on using assessment data<\/a> here on Teach.<\/em> Learn. Grow.<\/em><\/p>\nAn alternate model for assessing proficiency<\/h2>\n
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Student examples<\/h2>\n
Sample tasks for assessing the five strands<\/h2>\n
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Conceptual understanding<\/em><\/h2>\n
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This task assesses a student\u2019s ability to use spatial reasoning to accurately place a number on the number line relative to another. Students must understand the relationship between 3\/8 and 1 whole (both conceptually and visually) before they can use the relationship between 1 and 4\/3 to plot that point. Student performance on this task will reveal misunderstandings or validate student capability in grappling with the concepts of magnitude and division of a whole into equal parts.<\/p>\nProcedural fluency<\/em><\/h2>\n
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This task assesses a student\u2019s ability to efficiently and accurately evaluate expressions step-by-step using the order of operations in a creative way. While many traditional assessment items can measure a student\u2019s ability to follow a procedure and compute an answer accurately, creating a more rigorous task such as this reveals more about how the student applies known processes or approaches to an unfamiliar situation. How the student begins this task can be just as telling, if not more, than their ability to create three accurate expressions.<\/p>\nStrategic competence<\/em><\/h2>\n
Adaptive reasoning<\/em><\/h2>\n
Productive disposition<\/em><\/h2>\n
When to apply the data from five-strand assessments<\/h2>\n
When planning instruction<\/em><\/h2>\n
When making decisions about student grouping<\/em><\/h2>\n