**Are the Smarter Balanced Assessments better assessments?** (continued)

**Mathematics**

The overall emphasis of the mathematics assessment is on demonstrating college and career readiness in mathematics (grade 11) or demonstrating progress toward college and career readiness (grades 3-8).

The suggested uses of the overall mathematics score include calculating AYP, “declaring that a high school student may enter credit-bearing Math courses in college or university”, evaluating the effectiveness of programs, or determining “whether or not a teacher or principal is in need of improvement.” [MCS p. 23] In a bit of an understatement, the authors note that “[t]he examples listed above, in many cases, can be characterized as having relatively high stakes for those affected by the outcome.”

Students will receive a total mathematics composite score (made of subscores from each of the claims as indicated below). Subscores from each claim will also be reported. SBAC is taking an integrated approach to mathematics and will not be reporting subscores such as algebra, geometry, computations with fractions, or place value. [MCS pp.21-22]

The assessment is based on the content of the Common Core State Standards and on the following mathematical practices summary [see MCS p. 79 and see MIS3-5, MIS6-8, or MISHS for further explanations of each of the practices]:

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

Claim #1 [40% of composite score]: CONCEPTS AND PROCEDURES: *Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.*

Grade 3: Multiplication and division within the 10 x 10 times table. Addition and subtraction within 1000 (Singapore Primary Mathematics covers addition and subtraction within 10,000 in grade 3). Solve one-step word problems using time, liquid volume (liters), and mass/weight (grams and kilograms) (Singapore Primary Mathematics covers liquid volume in liters, milliliters, gallons, quarts, pints, and cups; mass/weight in kilograms, grams, pounds, and ounces; and, length in kilometers, meters, centimeters, miles, yards, feet, and inches in grade 3). Find areas by counting unit squares, not using multiplication (Singapore Primary Mathematics uses multiplication to calculate area in grade 3). Geometry tasks may involve shapes not explicitly mentioned in the standards. [MCS pp. 30-33]

Grade 4: Multiplication (up to four digit by one digit or two digit by two digit) and division (up to four digit dividends and one digit divisors) focusing on student strategies (Singapore Primary Mathematics also includes three digit by two digit multiplication in grade 4). One step word problems involving addition and subtraction of fractions with like denominators and multiplication of a fraction by a whole number (Singapore Primary Mathematics includes addition and subtraction of fractions with unlike denominators in grade 4). Express fractions with denominator of 10 or 100 as decimals and compare decimals to hundredth place (greater than/less than/equal to) (Singapore Primary Mathematics covers decimals to the thousandth place and asks students to express numbers such as 3 and 3/8 or 4/25 as decimals in grade 4). [MCS pp. 34-36]

Grade 5: Add and subtract fractions with unlike denominators. Multiply fractions and divide a fraction and a whole number. Student may be expected to build visual models and provide explanations of multiplication of fractions. Finding the volume of right rectangular prisms with whole number edge lengths (Singapore Primary Mathematics covers this in grade 4). [MCS pp. 37-39]

Grade 6: Ratios and percents (Singapore Primary Mathematics begins covering these topics in grade 5). Divide fractions by fractions. Find common factors and multiples. Solve problems involving area (triangles, special quadrilaterals, polygons), surface area, and volume (right rectangular prisms)(Singapore Primary Mathematics also covers perimeter and area of circles and parts of circles in grade 6). [MCS pp. 40-42]

The complete target assessments for this claim for each grade level can be found here. [MCS pp. 30-51] (The grade level content for all four claims can be found in Appendix A. [MCS pp. 79-86])

Claim #1 will be assessed with “SR items, TE items/tasks, and short CR items/tasks that focus on a particular skill or concept.” [MIS3-5 p. 16]

Claim #2 [20% of composite score]: PROBLEM SOLVING: *Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. *

This claim applies the content of claim 1 (from current or prior grade levels) to Mathematical Practices 1, 5, 7, and 8. [MIS3-5 pp. 22-24]

There are four problem-solving assessment targets [MCS pp. 59-60]:

- “Apply mathematics to solve well-posed problems arising in everyday life, society, and the workplace” (students find their own solution path).
- “Select and use appropriate tools strategically” (students have to make a choice about which tools to use).
- “Interpret results in the context of a situation” (reporting an answer that makes sense—like no fractional vehicles for problems of determining how many vehicles are needed to carry a given number of passengers).
- “Identify important quantities in a practical situation and map their relationships (e.g., using diagrams, two-way tables, graphs, flowcharts, or formulas).”

Claim #2 “will be assessed using a combination of SR items, TE items, CR items/tasks, and ER items/tasks that focus on making sense of problems and using perseverance in solving them.” [MIS3-5 p. 20] Tasks could include problems in pure mathematics, design problems, or planning problems (design problem with a time dimension). [MIS3-5 p. 24] “Items/tasks must be real-world or scenario-based (e.g., fantasy) and should take 5-15 minutes to solve.” [MIS3-5 p.24]

Sample short CR task: the student is given information about how much time it will take to create small and large toys, and how much the sale of each toy will earn for charity then asks the student to determine how many of each toy should be made to maximize the amount earned for the charity and calculate how much will be earned. [MCS p. 60]

Sample ER task: the student is directed to describe how the volume of a water tank made from a 6 ft by 6 ft sheet of metal will depend upon the size of squares cut from each corner of the sheet prior to folding the sheet into the tank form. The student is asked to sketch and explain a graph and then write an algebraic formula for the volume. The student is then asked how to cut the sheet to create the largest volume tank possible. [MCS p. 61]

Claim #3 [20% of composite score]: COMMUNICATING REASONING: *Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.*

This claim applies the content of claim 1 (from current or prior grade levels) to Mathematical Practices 3 and 6. [MIS3-5 pp. 29-30]

There are seven communicating reasoning assessment targets [MCS pp. 65-66]:

- “Test propositions or conjectures with specific examples.”
- “Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures.”
- “State logical assumptions being used” (students may be asked to supply missing information through research or estimation).
- “Use the technique of breaking an argument into cases” (under what conditions would a propositions be true or not true).
- “Distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in the argument—explain what it is.”
- “Base arguments on concrete referents such as objects, drawings, diagrams, and actions.”
- “At later grades, determine conditions under which an argument does and does not apply. (For example, area increases with perimeter for squares, but not for all plane figures.)”

Claim #3 “will be assessed using a combination of SR, CR, TE, PT, and ER items/tasks that focus on mathematical reasoning. Some tasks will require students to construct chains of reasoning without specific guidance being provided throughout the task.” [MIS3-5 p. 27] Tasks could include proof and justification tasks (argue why a given proposition is or is not true), critiquing tasks (correct and improve given flawed reasoning), or mathematical investigations (given a phenomenon, students are asked to formulate conjectures and then prove one). [MIS3-5 pp. 30-31] “Tasks should be designed to take 10-20 minutes to solve.” [MIS3-5 p. 31]

Sample ER task—proof and justification: Given a diagram, prove that when the figure is unfolded, the shape is a rhombus. [MCS pp. 66-67]

Sample ER task—critiquing: “Max bought 2 items in a sale. One item was 10% off. One item was 20% off. Max says he saved 15% altogether. Is he right? Explain.” [MCS p. 67]

Sample ER task—mathematical investigation: Students are presented with the phenomenon that some numbers can be written as the sum of two or more consecutive whole numbers (for example, 6=1+2+3) and are directed to find out all they can about writing numbers as sums of other consecutive whole numbers and then to write an account of their investigation. [MCS p. 68]

Claim #4 [20% of composite score]: MODELING AND DATA ANALYSIS: *Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. *

This claim applies the content of claim 1 (from current or prior grade levels) to Mathematical Practices 2, 4, and 5. [MIS3-5 pp. 36-38]

There are seven modeling and data analysis assessment targets [MCS pp. 72-73] :

- “Apply mathematics to solve problems arising in everyday life, society, and the workplace.”
- “Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem.”
- “State logical assumptions being used.”
- “Interpret results in the context of a situation.”
- “Analyze the adequacy of and make improvements to an existing model or develop a mathematical model of a real phenomenon.”
- “Identify important quantities in a practical situation and map their relationships (e.g., using diagrams, two-way tables, graphs, flowcharts, or formulas).”
- “Identify, analyze and synthesize relevant external resources to pose or solve problems.”

Claim #4 “will be assessed both by performance tasks (each lasting up to 100 [or 120] minutes) and by a collection of 5 to 8 extended-response items/tasks which focus on modeling and data analysis. ER tasks should be designed so that a successful student will complete them in 10-20 minutes.” [MIS3-5 p. 34] Tasks could include make decisions from data (select data, analyze, and draw conclusions), make reasoned estimates, plan and design (similar to claim #2 tasks but with information missing that will require student research or estimation to fill in), evaluate and recommend, or interpret and critique. [MIS3-5 p. 38]

Items/tasks in Claim #4 assess student expertise in choosing appropriate content and using it effectively in formulating models of the situations present and making appropriate inferences from them. . . . . Items and tasks of this sort require students to apply mathematical concepts at a significantly deeper level of understanding of mathematical content than is expected by Claim #1. Because of the high strategic demand that substantial non-routine tasks present, the technical demand will be lower–normally met by content first taught in earlier grades, consistent with the emphases described under Claim #1. [MCS p. 71]

Sample ER task—making reasoned estimates: determine whether a roll of toilet paper 100 feet long and 4 inches wide would be enough to completely cover a person playing a party game “wrap the mummy.” Student must estimate the average size of an adult person. [MCS p. 76]

Sample ER task—plan and design: Plan and justify a class trip (on a fixed budget) that includes the cost of buses, admission, and lunch using brochures from various parks. [MCS p. 76]

If you happen to be interested in grade 8 mathematics, Appendix D contains an assortment of sample items and tasks. [MCS pp. 94-145]