Mathematics Framework Instructional Strategies

Questions and Answers

What are the three key instructional shifts in the CA CCSSM?

Focus, coherence, and rigor

Which of the following is NOT one of the aspects of rigor required by the CA CCSSM?

Application

Procedural skill and fluency

Conceptual understanding

Memorization of formulas (correct)

The CA CCSSM require the application of mathematical concepts and procedures throughout all grade levels. Students are expected to use mathematics and choose the appropriate concepts for __________ even when they are not prompted to do so.

application

The teaching of mathematics should proceed in a strictly linear order according to the CA CCSSM.

False Signup and view all the answers

How do teachers ensure coherence in mathematics instruction?

By linking major topics across grades and ensuring vertical connections. Signup and view all the answers

Which of the following practices is included in the Standards for Mathematical Practice?

Make sense of problems and persevere in solving them Signup and view all the answers

What is the primary focus for instruction in grade five according to the framework?

Developing fluency with addition and subtraction of fractions. Signup and view all the answers

What is the first principle of instructional design for students to achieve learning with understanding?

Instruction is organized around the solution of meaningful problems. Signup and view all the answers

Which instructional model emphasizes self-discovery?

Implicit Instruction Signup and view all the answers

What are the five cognitive stages in the 5E model?

Engage, Explore, Explain, Elaborate, Evaluate Signup and view all the answers

The ______ phase in the three-phase model allows students to work independently.

closure Signup and view all the answers

Singapore Math focuses on developing students' problem-solving abilities.

True Signup and view all the answers

What is a key component of cooperative learning in mathematics instruction?

Students working as partners or in mixed-ability groups Signup and view all the answers

What does cognitively guided instruction emphasize?

Student-generated strategies and reasoning Signup and view all the answers

How does problem-based learning engage students?

Through guided and unguided inquiry Signup and view all the answers

The five practices for orchestrating productive mathematics discussions include anticipating, monitoring, selecting, ________, and connecting.

sequencing Signup and view all the answers

Match the following instructional strategies with their descriptions:

Appointment Clock = Students partner to make appointments for discussions. Carousel-Museum Walk = Groups post sample work on the wall for others to review. Signup and view all the answers

How much money does Marissa have?

Students need to calculate the total amount she saved. Signup and view all the answers

What story problem can be written about how Marissa spends her money?

A story problem that describes a scenario where Marissa spends her savings. Signup and view all the answers

What tools can students use to understand perimeter?

Rubber bands, geoboards, graph paper. Signup and view all the answers

What is the area that students should find rectangles for?

12 square units. Signup and view all the answers

What should students compile based on their findings?

An organized list or a table of possibilities. Signup and view all the answers

What two payment options does Leonard suggest to his mother?

$1.00 per day for two weeks or doubling his pay starting from 1¢. Signup and view all the answers

What should students calculate for Ms. Olsen's sidewalk?

The area of the isosceles trapezoid and the total cost of the sidewalk. Signup and view all the answers

What must students avoid when calculating the area of Ms. Olsen's sidewalk?

Misuse of the length marked 7.2 feet. Signup and view all the answers

How can students determine how many jerseys to order for Bill's baseball team?

By comparing the total costs from both companies using equations. Signup and view all the answers

The cost for Print It per jersey is ____.

21.50 Signup and view all the answers

What does the task of estimating areas of countries involve?

Using centimeter grids Signup and view all the answers

What estimation methods were used based on the world map?

Various methods including visual estimation and geometric reasoning. Signup and view all the answers

What is the role of students during the Charades activity?

Students act out word problems while others try to solve the problems. Signup and view all the answers

What is the purpose of the Clues (Barrier Games) strategy?

One partner communicates a problem to the other partner who cannot see the picture. Signup and view all the answers

In the Coming to Consensus strategy, what do group members do?

They share their individual ideas and come to a consensus before revealing it to the class. Signup and view all the answers

What do explorers and settlers do in this instructional strategy?

Explorers seek settlers to discuss math terms or problems. Signup and view all the answers

What activity do students engage in during the Find My Rule strategy?

Students find the person who matches their card with a problem or rule. Signup and view all the answers

What do students do in the Four Corners activity?

Students form groups in corners based on categories related to a topic. Signup and view all the answers

In the Give One, Get One strategy, what do students exchange?

Students exchange ideas about steps to solve a problem. Signup and view all the answers

During the Inside/Outside Circle activity, what do students do?

Students share information to solve problems by rotating partners. Signup and view all the answers

What is the goal of the Jigsaw strategy?

Students become experts on different concepts and teach those concepts to other students. Signup and view all the answers

What is the purpose of using KWL as a diagnostic tool?

To determine what students already Know, what they Want to know, and what they need to learn. Signup and view all the answers

What do students do during the Line Up activity?

Students line up in a particular order as given by the teacher. Signup and view all the answers

How does the Numbered Heads Together strategy ensure accountability?

Each group member knows the answer to the problems or questions asked by the teacher. Signup and view all the answers

In the Quiz, Quiz, Trade strategy, what do students do?

Students quiz each other and trade cards to reinforce their understanding. Signup and view all the answers

What is the main outcome of a Socratic Seminar?

Students engage in rigorous dialogue to seek deeper understanding of complex ideas. Signup and view all the answers

What is the Think-Pair-Share strategy used for?

Students think about a question, pair up, and share their responses with each other. Signup and view all the answers

How does the Think-Write-Pair-Share strategy differ from traditional Think-Pair-Share?

Students write down their response before sharing it with a partner. Signup and view all the answers

During the Whiparound activity, what do students do?

Students write down responses and share them one at a time without repeating previous answers. Signup and view all the answers

What do students do in the Wraparound activity?

Students share one idea in a sequential manner after writing about a topic. Signup and view all the answers

What kind of representations can be beneficial in math instruction?

Visual representations such as diagrams, concept maps, graphic organizers, and flowcharts. Signup and view all the answers

What are concrete models used for in math instruction?

To help students make sense of problems and understand mathematical concepts. Signup and view all the answers

What role does interactive technology play in math instruction?

It can assist in teaching and enhancing student engagement, but must align with educational standards. Signup and view all the answers

Study Notes

Overview of Instructional Strategies

Aimed at enhancing teachers' repertoire of instructional strategies, not prescribing specific methods.
Effective teaching aligns strategy choice with material and student needs.
Lessons must engage all students and enhance mastery of the California Common Core State Standards for Mathematics (CA CCSSM).

Lesson Structure

Lessons should typically last 50-60 minutes, excluding homework.
Regular exposure to new information is essential for building conceptual understanding and practice of skills.
Teaching requires careful sequencing to ensure all standards are addressed without strict linearity.

Key Instructional Shifts

Focus: Concentration on grade-level standards for deeper understanding.
Coherence: Attention to learning progression across grades and linking major topics within grades.
Rigor: Development of conceptual understanding, procedural skill, and application.

Rigor in Mathematics Education

Rigor consists of three aspects:
- Conceptual Understanding: Teaching mathematical concepts in various contexts to promote meaningful learning.
- Procedural Skill and Fluency: Structuring practice for procedural mathematics, ensuring fluency is achieved through well-planned instruction.
- Application: Encouraging students to use mathematics in real-world contexts, reinforcing relevance and motivation.

Standards for Mathematical Practices

Eight practices that promote expertise in mathematics include:
- Problem-solving perseverance.
- Reasoning abstractly and quantitatively.
- Constructing viable arguments.
- Modeling with mathematics.
- Using appropriate tools strategically.
- Attending to precision.
- Looking for and making use of structure.
- Expressing regularity in repeated reasoning.
Instruction must integrate these practices effectively, creating a safe space for discourse.

Critical Areas of Instruction

Critical Areas guide instruction at each grade level, with a focus on:
- Developing fluency and understanding of fractions.
- Integrating decimal operations into the place-value system.
- Understanding volume measurements.
Each grade-level chapter includes a reference for planning instruction, illustrating focus, coherence, and rigor.

General Instructional Models

Various instructional models help align teaching with CA CCSSM and the critical instructional features.
Effective instruction should be organized around problematic tasks, provide meaningful learning scaffolds, offer ongoing assessment, and promote collaboration.
A combination of instructional models is necessary to cater to diverse student populations in California classrooms.### Instructional Models
Instructional models vary from explicit to implicit instruction, affecting teacher roles and student learning.
Explicit Instruction: Teacher-centered approach, focuses on direct assistance and skills mastery.
Interactive Instruction: Combines explicit and implicit methods, balancing teacher and student regulation of learning.
Implicit Instruction: Student-centered, emphasizes self-discovery and personal meaning construction.

5E Model

Comprises five stages: engage, explore, explain, elaborate, and evaluate.
Teachers serve as facilitators, guides, and models to encourage inquiry and creative thinking.
Focuses on interaction, problem-solving, and effective communication among students.

Three-Phase Model

A structured strategy for direct instruction involving three phases:
- Phase 1: Introduction and explanation of new concepts.
- Phase 2: Guided practice leading to independent application.
- Phase 3: Independent work and assessment of understanding.

Singapore Math

Emphasizes number sense, mental math skills, and conceptual understanding through a concrete-to-pictorial-to-abstract progression.
Prioritizes mastery before revisiting concepts, focusing on model drawing for problem-solving.

Concept Attainment Model

An inductive approach where students categorize ideas/objects based on given examples and non-examples.
Encourages hypothesis testing, critical thinking, and active participation in discussions.

Cooperative Learning Model

Promotes teamwork in problem-solving, allowing students to share reasoning and critique others' arguments.
Supports diverse learners by providing structured group tasks while teachers monitor and assist.

Cognitively Guided Instruction

Students explore various problem-solving strategies, justify their reasoning, and explain their thought processes.
Encourages ownership of learning, fostering a positive disposition toward mathematics.

Problem-Based Learning

Engages students in real-world problem-solving using guided or unguided inquiry.
Promotes perseverance, reasoning, and intellectual independence through collaborative learning.

Student Engagement Strategies

Vital for maintaining focus and enhancing learning environments.
Examples include Number Talks (mental math discussions) and Five Practices for Mathematics Discussions (anticipate, monitor, select, sequence, connect).

Student Engagement Techniques

Appointment Clock: Students schedule discussions using a clock format.
Carousel-Museum Walk: Groups present work on poster boards while others evaluate.
Charades: Students act out math problems for peers to solve.
Clues (Barrier Games): Pairs communicate problem-solving information using barriers to enhance understanding.
Coming to Consensus: Groups collaboratively find solutions and present them to the class.
Explorers and Settlers: Engages students in exchanging definitions and solving problems in designated roles.

Importance of Communication

Mathematical discourse enriches understanding and connects everyday language with mathematical concepts.
Writing enhances thinking and serves as an assessment tool, revealing students’ thought processes.### Classroom Engagement Strategies
Find Your Partner: Students receive cards with rules and examples, matching them through circulation. Promotes inductive/deductive reasoning and problem-solving skills.
Four Corners: Students gather in designated corners based on topic interest, then respond to problems related to the categories, providing an interactive grouping experience.
Give One, Get One: Students brainstorm solutions to a problem, sharing and receiving steps with partners, encouraging collaboration and collective knowledge-building.
Inside/Outside Circle: Concentric circles facilitate peer dialogue, where students ask and answer questions, enhancing communication and mathematical reasoning.
Jigsaw: Students become experts on specific concepts and teach others, promoting collaborative learning and peer instruction.
KWL Chart: Used for assessing what students know, want to know, and need to learn, helping tailor instruction to student readiness.
Line Up: Students line up by criteria (e.g., smallest to largest), explaining their reasoning, fostering quick thinking and communication.
Making a List: Pairs of students create lists of steps needed to solve complex problems, aiding in logical reasoning skills.

Cooperative Learning Strategies

Numbered Heads Together: Each student contributes to group problem-solving, ensuring accountability and collective understanding.
Quiz, Quiz, Trade: Students quiz each other with prepared cards, reviewing material while fostering cooperative learning.
Socratic Seminar: In-depth discussions on complex ideas or scenarios, encouraging deeper understanding through critical thinking and defense of viewpoints.
Team Share: Teams collaborate on problems and present their solutions, encouraging group participation and articulation of mathematical reasoning.
Think–Pair–Share: Students think, then pair-share their responses, enhancing their ideas through discussion and refining understanding through collaboration.
Think–Write–Pair–Share: Variations allowing students to write and then share their ideas, promoting clarity and structured responses.

Tools for Mathematics Instruction

Visual Representations: Diagrams, graphic organizers, and flowcharts help illustrate relationships in mathematical concepts, enhancing student comprehension.
Concrete Models: Manipulatives provide hands-on learning opportunities, aiding students in understanding both basic and complex concepts like the Pythagorean Theorem.
Interactive Technology: Engaging software and applications can enhance learning if they align with educational standards and goals.

Example Mathematical Tasks

Marissa's Savings: A multi-step problem involving monetary amounts that teaches practical application of arithmetic and problem-solving.
Understanding Perimeter: Hands-on activities with rubber bands and geoboards help students grasp the concept of perimeter through exploration and pattern recognition.
After-School Job: A problem encouraging students to analyze two payment options and use of numerical patterns to solve real-world scenarios, integrating concepts of exponential growth and decision-making.

Overall Objective

Strategies are designed to promote engagement, accountability, and collaboration among students while providing effective tools and tasks that resonate with real-world applications, ensuring comprehensive understanding of mathematical concepts.

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Description

Explore the instructional strategies outlined in the Mathematics Framework for California Public Schools. This chapter aims to enhance teachers' approaches to teaching mathematics from Kindergarten through Grade Twelve, providing a variety of strategies to support diverse learning needs.