DDI and Assessments in Mathematics: Designing Assessments that
Provide Meaningful Data At the 6-12 Level February 4, 2014
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Session Objectives Know the elements of a high-quality Common
Core-aligned mathematics assessment Know how to create and choose
assessment tasks that reflect a balance of rigor and give
comprehensive information about students at all levels
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Agenda Introduction Sorting Task: Why Do We Assess? Balance of
Rigor: 7.RP.1 Evaluate and Modify Your Assessment, Part I Students
At All Levels: 7.RP.1 Evaluate and Modify Your Assessment, Part
II
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Introduction
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Why Assess At All? To learn information about students
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Why Assess At All? 1. Compare the assessments (Documents A, B,
and C). What kinds of information would we learn about students
from each one? 2. What are the strengths and weaknesses of
each?
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So What kinds of information do we want to learn about our
students?
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Two Big Ideas Well Talk About Today 1. The assessment should
reflect a balance of rigor, helping us understand students
procedural abilities, conceptual understanding, and abilities to
apply. 2. The assessment should give us information about students
at all levels.
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Other Considerations (Not Our Focus Today) Focus: Strong
majority of questions assess the Major Work of the grade/course
Coherence: Supporting Content is assessed in ways that engage
students in the Major Work Practices: Items signal the Standards
for Mathematical Practice (not necessarily all items)
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What Not To Do:
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1. Balance of Rigor Procedural Fluency Conceptual Understanding
Application
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From the Publishers Criteria:
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Procedural Fluency $64,000 Question: Are there items that ask
or imply that students do math procedures? Some good verbs: solve,
add, subtract, divide, multiply, graph, compute, find Practice
Connections MP.7 Look for and make use of structure MP.8 Look for
and express regularity in repeated reasoning *Note fluency
expectations named at each grade level. While these are not named
for HS level courses there should still be procedural
assessment.
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But what does mathematical understanding look like? One
hallmark of mathematical understanding is the ability to justify,
in a way appropriate to the students mathematical maturity, why a
particular mathematical statement is true or where a mathematical
rule comes from. There is a world of difference between a student
who can summon a mnemonic device to expand a product such as (a +
b)(x + y) and a student who can explain where the mnemonic comes
from. The student who can explain the rule understands the
mathematics, and may have a better chance to succeed at a less
familiar task such as expanding (a + b + c)(x + y). Mathematical
understanding and procedural skill are equally important, and both
are assessable using mathematical tasks of sufficient richness.
From the New York State Common Core Learning Standards
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Conceptual Understanding $64,000 Question: Are there items that
ask students to explain their thinking? Some good verbs: Explain,
determine, prove, show, compare, justify Practice Connections MP.3
Construct viable arguments and critique the reasoning of others.
MP.6 Attend to precision. *More important in standards that begin
with understand.
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Application $64,000 Question: Does the item have a real-world
context? Some good examples: Single-step word problems, multi- step
word problems, model-building Practice Connections MP.1 Make sense
of problems and persevere in solving them. MP.2: Reason abstractly
and quantitatively. MP.4: Model with mathematics. *Application
Paradox: The more directed the task, the more we can be sure it
aligns to the standard, but the less authentic it becomes.
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7.RP.1 Lets look at a standard together. What would assessment
items based on procedural, conceptual, and application aspects look
like?
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Three Free, Public On-Line Resources To Support A Balance of
Rigor 1. www.illustrativemathematics.org 2. EngageNY: Curriculum
Modules 3. EngageNY: Annotated Items
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Illustrative Math Task
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Annotated Item
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Module Assessment
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Evaluate and Modify 1.Does your assessment reflect a balance of
rigor? 2.What needs to be added or modified in your assessment?
3.Can you generate an item or two that would improve the rigor of
your assessment?
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2. Students at All Levels
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From the K-8 Publishers Criteria: The natural distribution of
prior knowledge in classrooms should not prompt abandoning
instruction in grade level content, but should prompt explicit
attention to connecting grade level content to content from prior
learning. To do this, instruction should reflect the progressions
on which the CCSSM are built. For example, the development of
fluency with division using the standard algorithm in grade 6 is
the occasion to surface and deal with unfinished learning with
respect to place value. Much unfinished learning from earlier
grades can be managed best inside grade level work when the
progressions are used to understand student thinking.
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Three Free, Public On-Line Resources To Support Learning About
Students at All Levels 1.EngageNY: Performance Level Descriptions
2.Common Core Learning Standards 3.Progressions
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Performance Level Descriptions Describe the trajectory of
knowledge and skills at each grade level. Provide rich information
about what students in each performance category know and can do.
Serve as basis of discussion among educators about the specific
knowledge and skills that distinguish the just barely Level 3
proficient student, or the just barely Level 2 student.
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Performance Level Descriptions
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Important Questions What performance levels does the item tell
us about? How could the item be modified to tell us information
about different performance levels?
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Annotated Item
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Use Cluster and Domain Headings
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Illustrative Math Task
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Evaluate and Modify 1.Does your assessment give information
about students at a variety of levels? 2.What needs to be added or
modified in your assessment? 3.Can you generate or modify an item
or two that would better differentiate your assessment?
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To Consider: Cluster Level Alignment Solve the paradox / More
authentic tasks Standards are not written at uniform grain size
Avoid micro-sizing (using twigs to build a fire that burns down our
tree) Easier data management PLDs are written at the cluster
level
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AET
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Two Big Ideas Revisited 1. The assessment should reflect a
balance of rigor, helping us understand students procedural
abilities, conceptual understanding, and abilities to apply. 2. The
assessment should give us information about students at all
levels.
