Mathematics

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Cognitively-Based Assessment of, for, and as Learning (CBAL) MATHEMATICS SUMMATIVE ASSESSMENTS SAMPLE ITEMS

Overview The CBAL Mathematics project has developed prototype tasks and supporting materials to assess middle school students’ abilities to think and communicate mathematically; that is: •

to generate, interpret, apply, and revise mathematical models;

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to create, interpret, and integrate mathematical representations; and

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to construct mathematical justifications, provide supporting examples, pose conjectures, evaluate mathematical arguments, develop informal methods of proof, and use inference.

In addition to the mathematical processes above, each prototype task focuses on one or more major ideas with the content domains of numbers and operations; algebra; geometry; measurement; and data, statistics, and probability. Ratio and proportional reasoning, the concept of variable, the concept of equality, functions, and variability within data sets are examples of major ideas that are represented within the tasks. Each task, consisting of 5 to 15 questions, is based on a scenario that is engaging to students and presents relevant mathematics to assess the content domain and processes described above. All questions in a task are scaffolded to facilitate the collection of information on students’ understanding at various levels of key concepts. The tasks are accompanied by supporting materials that assist teachers in pinpointing students’ misunderstandings and suggest ways that instruction might be adjusted to address those misunderstandings. Many of the tasks include online tools, such as simulations that help students to better visualize the scenarios and gain additional important insights into the mathematical concepts. The following sample contains selected questions from one of the eighth-grade tasks in the data, statistics, and probability content domain.

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SAMPLE TASK In this sample task, titled “On the Road,” students work with a variety of data displays including tables, dotplots, boxplots, histograms, bar graphs, and scatterplots. They are asked to describe distributions of data (in terms of minimum and maximum values, the range of the data, the median, and the spread). In the final part of the task, students evaluate related data from a variety of data displays to create a mathematical justification in paragraph form to support a recommendation that they have identified. The opening screen below presents the scenario for this task, along with the big idea or question that students have to address at the end of the task.


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This is one of the earlier questions in the task.

What is Being Assessed: This question assesses the Use and Interpret Data Displays objective. Several of the early questions in the task address reading and interpreting basic information from different data displays. There are two ways to respond to this question. Students may either use the boxplot to determine the approximate values of the median and quartiles and then identify the values in the table that are closest to those, or they may reorder the values in the table and locate the quartiles and median in that reordered data set. Scoring: When tasks are used as summative assessments, questions or parts of questions within each task are scored using rubrics that are appropriate to the level of complexity of the question. In this question, the rubric includes a level of partial correctness for part (b), since some students may understand the concept of median but might not be able to fully express it. Each of the other parts is scored as correct or incorrect. When tasks are used as formative assessments, each question in the task is accompanied by supporting material for teachers that includes information on what mathematical content is being assessed by the task, guidelines for evaluating student responses across questions to ascertain overall understanding, and suggestions for adjusting instruction to address issues that are identified as a result of teachers’ analyses. Comment: If a student responds correctly only to parts (a), (c), and (d) – but not (b) – this may indicate that the student is at a lower level of understanding than a student who responds correctly to all four parts, since (b) requires a slightly deeper understanding of median. Several online tools are available in this task including calculators, tools to help students construct graphs, and a sort feature for data tables.


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In addition to looking at the data on areas of the regions in earlier questions, this question presents information on the population of each region to be considered in making a recommendation to the delivery service.

What is being assessed: This question also assesses the Use and Interpret Data Displays objective. It is an example of a question in which students must not only be able to interpret a graph and read data from it but also be able to apply knowledge of the scenario of the task (i.e., know that more people usually means more potential business). Scoring: After selecting two regions in the first part of the question, the justification offers an opportunity for the students to connect their interpretation of the graph to the scenario. A response that establishes a strong connection earns complete credit. A student may earn partial credit by demonstrating recognition that more people in a region may translate into more business. Comment: There are several questions like this one in this task that tap students’ abilities to understand and interpret one or more data displays in the context of the scenario.


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In this follow up to the preceding question, students are asked to consider two data displays.

What is being assessed: This question assesses the objectives of Integrate Representations, Use and Interpret Data Displays, and Make Connections. Students must be able to correctly relate the portions of the bar graph that are related to the ‘provide the most amount of business’ goal to the corresponding relevant features of the boxplot. Scoring: The first part of this question is scored correct or incorrect, while the latter part assesses students’ ability to relate the relevant portion of the boxplot to the scenario. Comments: In this question, it is important that students understand the need to look at each data display carefully, in terms of the information being presented, and realize which parts are relevant to the problem under consideration.


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The following four views show the final question in this task, along with the data displays that the student encountered while working through the entire task. While working on this question, students can access each data display by clicking on one of the four tabs shown on the left side of the screen above the display.


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What is being assessed: This question assesses the objectives Construct a Mathematical Argument, Provide Supporting Examples, and Make Connections. It builds on the experience that students have acquired in earlier questions analyzing a wide range of data displays in a way that contributes to informing their response to this culminating question. Knowing which data and graphs can be used to effectively support a conclusion is also an important part of the goal for this question. Scoring: A fully-justified response includes reasonable recommendations and will integrate data from multiple sources to make valid comparisons and comments on trends in the data. Less complete responses present some correct comparisons, but will not fully utilize all of the information to support the proposed recommendations in an appropriate way. Comment: In this question, students should understand that a well-justified recommendation will not be based on a single source (i.e., graph, statistic, etc.), since what may be the obvious or best choice when using one source may not be the best choice when using a different source. The available sources collectively suggest that Region 9 is a good choice for opening a business relative to the stated goal, but the second choice is less clear, since there is more than one plausible option. To arrive at a reasonable second choice, students will have to balance several considerations, making context appropriate compromises as part of the decision-making process.


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