Domains of Study/Conceptual Categories

Learning Progressions/Trajectories

2010 Alabama Course of Study: Mathematics

College- and Career-Ready Standards

Aligned with college and work expectations Written in a clear, understandable, and consistent

format Designed to include rigorous content and

application of knowledge through high-order skills Formulated upon strengths and lessons of current

state standards Informed by high-performing mathematics

curricula in other countries to ensure all students are prepared to succeed in our global economy and society

Grounded on sound evidence-based research

CHARACTERISTICS

2010 Alabama Course of Study: Mathematics

Coherent Rigorous Well-Articulated Enables Students to Make Connections

Articulated progressions of topics and performances that are developmental and connected to other progressions.

Conceptual understanding and procedural skills stressed equally.

Real-world/Situational application expected.

Coherence

Key ideas, understandings, and skills are identified.

Deep learning stressed.

FOCUS

Mathematical Content Format

K-8 9-12

Grade

Domain

Cluster

Standard

Course

Conceptual Category

Domain

Cluster

Standard

What is the difference?

Domain Cluster Standards

Domain: Overarching “big ideas” that connect content across the grade levels.

Cluster: Group of related standards below a domain.

Standards: Define what a student should know (understand) and do at the conclusion of a course or grade.

What is the difference?

Illustrate progression of increasing complexity from grade to grade.

Organize standards within each grade.

Note: Domains typically span a few grades.

K-8 Domains

MathematicsDomains of Study by Grade

K 1 2 3 4 5 6 7 8 Counting and Cardinality

Operations and Algebraic Thinking

Number and Operations in Base Ten

Measurement and Data

Geometry

Number and Operations: Fractions

Ratios and Proportional Reasoning

The Number System

Expressions and Equations

Statistics and Probability

Functions

May appear in multiple grades.

Illustrate progression of increasing complexity from grade to grade.

K-8 Clusters

Grade 1 Grade 2

Add and subtract within 20.5. Relate counting to addition and subtraction

(e.g., by counting on 2 to add 2). [1-OA5]

6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). [1-OA6]

Add and subtract within 20.2. Fluently add and subtract within 20 using

mental strategies. (See standard 6, Grade 1, for a list of mental strategies.) By end of Grade 2, know from memory all sums of two one-digit numbers. [2-OA2]

K – 8 Cluster

Content standards in this document contain minimum required content.

Each content standard completes the phrase “Students will.”

Reflect both mathematical understandings and skills, which are equally important.

Content Standards

How Many Skills?7. Read and write multi-digit whole numbers using

base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. [4-NBT2]

4.7.1. Read multi-digit whole numbers using base-ten numerals.

4.7.2. Read multi-digit whole numbers using number names.4.7.3. Read multi-digit whole numbers using expanded form. 4.7.4. Write multi-digit whole numbers using base-ten

numerals.4.7.5. Write multi-digit whole numbers using number names.4.7.6. Write multi-digit whole numbers using expanded form4.7.7. Compare two multi-digit numbers based on meanings

of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Grade Level Overview

Critical Area

Grade Level Narrative

Grade

Level Focus

CRITICAL AREAS BRING FOCUS

TO THE NEW STANDARDS

Identify Two to Four Areas of Concentrated Study.

Bring Focus to the Standards. Provide the Big Ideas for Building Curriculum and Guiding Instruction.

CRITICAL AREAS

INVESTIGATING CRITICAL AREAS

In grade-level groups, analyze the critical areas for your grade.

Underline the key words and phrases that help summarize this area.

Design a poster that describes the focus of your grade level.

The Big Picture

K-2Number and number sense.

3-5 Operations and Properties(Number and Geometry)Fractions

6-8Algebraic and Geometric ThinkingData Analysis and using Properties

High School

Functions, Statistics, Modeling and Proof

MathematicsDomains of Study by Grade

K 1 2 3 4 5 6 7 8 Counting and Cardinality

Operations and Algebraic Thinking

Number and Operations in Base Ten

Measurement and Data

Geometry

Number and Operations: Fractions

Ratios and Proportional Reasoning

The Number System

Expressions and Equations

Statistics and Probability

Functions

What Are Learning Trajectories? And What Are They Good For?—Save the Last Word for Me!!!Read the excerpt from Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction

Identify 3 ideas that you are willing to talk about with colleagues.

Highlight the location in the text where these ideas appear.

Learning Progressions/Trajectories

Confrey (2007)“Developing sequenced obstacles and challenges for students…absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise.”

CCSS, p. 4 “… the development of these Standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.”

Directions for Save the Last Word for Me Designate a facilitator and timekeeper. A volunteer begins by reading the sentence(s) from the

text that embody one of his/her selected ideas. The speaker does not comment on the text at this point.

The individual to right of first speaker takes up to one minute to comment on the selected text.

The next two individuals also take up to one minute to comment on the initial speaker’s idea.

The individual selecting the idea has up to 1 minutes to react to colleagues’ ideas and to talk about why she or he thought this was important.

Another group member introduces one idea, and the group follows the same protocol. Continue until all members have shared or until time is called.

Learning Trajectories – sometimes called learning progressions – are sequences of learning experiences hypothesized and designed to build a deep and increasingly sophisticated understanding of core concepts and practices within various disciplines. The trajectories are based on empirical evidence of how students’ understanding actually develops in response to instruction and where it might break down.

Daro, Mosher, & Corcoran, 2011

Starting PointEnding

Point

Startin

g Point

Ending Point

K 1 2 3 4 5 6 7 8 HS

Counting and Cardinality

Number and Operations in Base Ten Ratios and Proportional Relationships

Number and Quantity

Number and Operations – Fractions

The Number System

Operations and Algebraic Thinking

Expressions and Equations Algebra

Functions Functions

Geometry Geometry

Measurement and Data Statistics and Probability Statistics and Probability

Learning Progression Framework

Investigating the Domains/Conceptual Categories

Domains provide common learning progressions.

Curriculum and teaching methods are not dictated.

Standards are not presented in a specific instructional order.

Standards should be presented in a manner that is consistent with local collaboration.

What Goes On The Chart Paper? Summary and/or representation of how

the concept of the use of numbers grows throughout your grade band.

Easy for others to interpret or understand. Visual large enough for all to see. More than just the letters and numbers of

the standards – include key words or phrases.

Let’s Talk About It!

As a table group, consider your journey through the 2010 ACOS as you studied the concept of the use of numbers K-12. What did you learn? What surprised you? What questions do you still have?

Learning Progressions

K 1 2 3 4 5 6 7 8 HS

Counting and Cardinality

Number and Operations in Base Ten Ratios and Proportional Relationships

Number and Quantity

Number and Operations – Fractions

The Number System

Value of Learning Progressions/Trajectories to Teachers

Know what to expect about students’ preparation.

More readily manage the range of preparation of students in your class.

Know what teachers in the next grade expect of your students.

Identify clusters of related concepts at grade level.

Clarity about the student thinking and discourse to focus on conceptual development.

Engage in rich uses of classroom assessment.

Review the Format2003 ACOS 2010 ACOS

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Contains a glossary

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ALSDE Office of Student LearningCurriculum and Instruction Section

Cindy Freeman, Mathematics SpecialistPhone: 334.353.5321

E-mail: cfreeman@alsde.edu

Contact Information