Instructional techniques for teachers

• This chapter aims to broaden teachers' instructional techniques

• It does not enforce specific methods

• Teachers should match strategies to the material, suiting individual student requirements

• Effective lessons in a standards-based curriculum engage all students in activities designed to master specific standards

• Lessons should last 50-60 minutes daily, not including homework

• The aim is for all students to be prepared for college and careers by mastering standards

• This aligns with the CA CCSSM and the mathematics framework

• Lessons should regularly introduce new information

• They should build understanding, practice skills, and reinforce previously taught content

• Math teaching must be sequenced and organized

• All the standards must be taught, while prerequisite skills lay the foundation for advanced learning

• Learning should not be strictly linear

• Students don't need to master each standard before moving on

• Practice leading to mastery can be incorporated into new problems that are both challenging and strengthen understanding and fluency in math

• Key instructional shifts, standards for mathematical practice and critical areas of instruction will be addressed

Key Instructional Shifts

• Understanding the differences between CA CCSSM and previous standards is essential for implementing California's math standards
• Focus, coherence, and rigor are the three key principles the standards are based on
• Focus should be on grade-level standards
• Coherence should attentively link learning across grades and major topics
• Rigor should develop understanding and fluency and should follow grade level standards
• Teachers should concentrate on these three principles to develop a common understanding and implement the CA CCSSM
• Every grade-level chapter of the framework starts with a summary of these principles

Standards for Mathematical Content

• Emphasize key content, skills, and practices at each grade level
• Support three major principles: focus, coherence, and rigor

Focus

• Instruction should be focused on grade-level standards
• The content scope in each grade (K-12) should be narrowed
• Students should have a deeper experience with the remaining content
• Postsecondary instructors value mastery of prerequisites, not shallow exposure to irrelevant topics

Coherence

• Instruction should attentively link learning across grades and major topics within grades
• Vertical connections are crucial for student progress in math education
• Major content should develop over time
• It is important to think across grades

Rigor

• Instruction should develop conceptual understanding, procedural skill/fluency, and application
• Educators need to pursue all three aspects with equal intensity in each grade's major work
• "Understand" sets expectations for conceptual understanding
• "Fluently" sets expectations for fluency
• "Real-world problems" and the star [★] symbol indicate opportunities for applications and modeling
• Three aspects of rigor are critical for daily and long-term instructional goals

Conceptual Understanding

• Teachers must teach concepts from multiple angles
• Mathematics is more than mnemonics or procedures
• Students demonstrate understanding by applying concepts, writing, and speaking about their understanding
• A benefit is that students understand procedures, algorithms, and why math is meaningful

Procedural Skill and Fluency

• Teachers need to structure class time and homework time for students to practice procedural skills
• Students develop fluency in core areas (addition, subtraction, mulitplication and division)
• They are able to understand and work with more complex concepts
• Fluency requires planning and practice, and is the result of learning progressions

Application

• CA CCSSM requires application of math concepts throughout all grades

• Students have expectations to use math and choose appropriate concepts, even if not prompted

• Teachers should provide opportunities to apply math to real world situations to motivate students

• Students can transfer math to their daily lives and future careers

• Other subjects should ensure students use grade appropriate math

• These aspects should be taught in balance

• The CA CCSSM offer balance

• Application makes math meaningful, and it maintains engagement

Standards for Mathematical Practice (MP)

• Describe expertise for mathematics educators to develop in students at all levels

• Rest on problem solving, reasoning and proof, communication, representation, and connections are important "processes and proficiencies" in math education

• Strands of mathematical proficiency specified in the National Research Council's report Adding It Up include:

• Adaptive reasoning

• Strategic competence

• Conceptual understanding (comprehension of mathematical concepts, operations, and relations)

• Procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and appropriately)

• Productive disposition (habitual inclination to see math as sensible, useful, and worthwhile, with diligence and one's own efficacy)

• Instruction must incorporate these standards

• Teachers should analyze their curriculum

• They should identify where content and practice standards intersects

• Chapters of the framework contain connection examples

• It isn't possible to address every MP standard in every lesson

• MP standards are interconnected, making it difficult to address a single MP standard in a given lesson

• MP standards establish "habits of mind" that should be explicitly taught

• For example, students are not expected to know what a viable argument would look like from the outset

• The teacher and other students set the expectation level by critiquing reasoning

• Teachers are responsible for creating a safe atmosphere in which students can engage in mathematical discourse

• Higher mathematics courses have increased levels of mathematical argument, the increased use of language, logical connections, explanation of arguments, and communication of reasoning

• The teacher guides these skills,

Critical Areas of Instruction

• At the beginning of each grade level chapter in the framework is a brief summary of critical areas

• In Grade 5, instructional time focuses on developing fluency with addition/subtraction of fractions, and understanding multiplication/division of fractions

• Instruction also focuses on extending division to two-digit divisors through use decimal fractions in the place-value system

• Instruction should help develop understanding of operations with decimals to hundredths and fluency with whole-number and decimal operations

• Instruction focuses on developing understanding of volume

• Students fluently multiply multi-digit whole numbers using the standard algorithm

• Critical Areas of Instruction are examples of focus, coherence, and rigor for each grade level

• Critical Area (1) in grade 5 involves using the understanding of equivalent fractions and fraction models to develop fluency with fraction addition and subtraction

• Critical Area (1) is connected to Critical Area (2) in grade 5, with students relating their understanding of decimals as fractions to the rules for multiplying and dividing decimals

• An example is that students further their understanding to add and subtract all types of fractions in grade give

• In Critical Area (1) of grade 5 standards, students apply their understanding of fractions and fraction models

• In Critical Area (1), students develop fluency calculating the sums and differences of fractions

• In third critical area, students solve real-world problems related to volume

• Critical Areas of Instruction should be a reference for planning instruction

• They are listed at the beginning of each grade-level chapter

• Additional examples appear throughout the grade-level chapters

• Each grade-level chapter includes a table highlighting content emphases at the cluster level

• Instructional time should be given to “Major” clusters and listed standards

General Instructional Models

• Teachers are presented with delivering instruction aligned to the CA CCSSM

• Teachers should pay attention to Key Instructional Shifts, the Standards for Mathematical Practice, and critical areas of instruction at each grade level

• This section describes several general instructional models, with particular strengths related to the aforementioned instructional features

• Classroom teachers deliver instruction, how students learn to provide useful information to both teachers and developers of instructional resources

• A combination of models and strategies should be considered to optimize student learning

• Cooper (2006, 190) lists four overarching principles of instructional design for students

• Instruction is organized around the solution of meaningful problems

• Instruction provides scaffolds for achieving meaningful learning

• Instruction provides opportunities for ongoing assessment with feedback, revision, and reflection

• The social arrangements of instruction promote collaboration, distributed expertise, and independent learning

• Mercer and Mercer (2005) suggest that instructional models may range from explicit to implicit instruction

Explicit Instruction

• The teacher serves as the provider of knowledge
• Includes much direct teacher assistance
• Follows the teacher regulation of learning
• Utilizes directed discovery and direct instruction
• Includes task analysis and is behavioral

Interactive Instruction

• Instruction includes both explicit and implicit methods
• Has balance between direct and non-direct teacher assistance
• Implies shared regulation of learning and guided discovery
• Is strategically instructed and has balance between part-to-whole and whole-to-part
• Cognitive/metacognitive

Implicit Instruction

• Facilitates learning by creating situations in which students discover knowledge and construct their meanings

• Has non-direct teacher assistance

• Follows student regulation of learning and self-discovery

• Is self-regulated and a unit approach

• Holistic

• The type of instructional models used will depend on the learning needs of students and the math being presented

• Explicit instruction supports practice to mastery, and the teaching and development of skills/procedural knowledge

• Implicit models link information to student's background knowledge, which develops understanding and problem-solving abilities

5E (Interactive) Model

• It can be linked to introduce, investigate, and summarize
• It has a recursive cycle of five cognitive stages in inquiry-based learning: engage, explore, explain, elaborate, and evaluate
• Teachers facilitate creative thinking, problem solving, interaction, communication, and discovery
• Teachers initiate thinking processes, inspire positive attitudes toward learning, motivate, and demonstrate skill-building techniques
• Teachers help bridge language gaps between children, and to foster individuality, collaboration, and personal growth

Three-Phase Model

• Represents a structured and sequential strategy used in direct instruction
• Effective for teaching information and basic skills
• Phase 1: teacher introduces, demonstrates, or explains new concept
• Phase 1: teachers ask question and check understanding
• Phase 2: intermediate step to result in the independent application of the new concept
• Phase 3: students work independently and receive opportunities for closure; is an assessment to understand what they are learning

Singapore Math

• Emphasizes the development of strong number sense, mental-math skills, and a deep understanding of place value
• Based on Bruner's (1956) principles which has progression from concrete experience using manipulatives to a pictorial stage, and finally to the abstract level or algorithm
• Gives students a solid understanding of basic mathematical concepts
• Concepts are taught to mastery, then revisited
• Focuses on the development of students' problem-solving abilities
• Has a strong emphasis on model drawing

Concept Attainment Model

• Interactive, inductive model, asks students to categorize ideas or objects according to critical attributes
• Teachers provide examples and non examples
• Students develop and test hypotheses about the exemplars and analyze utilized teaching processes
• Students may be asked to categorize polygons and non polygons using a definition
• The teacher is in control of the lesson by selecting, defining, and analyzing the concept
• The teacher encourages student participation through discussion and interaction
• It may be used to introduce, strengthen, or review concepts, and as formative assessment

Cooperative Learning Model

• An important component of the mathematical practice standards says students work together to solve problems
• Students assess their efforts
• Construct viable arguments, communicate reasoning, and critique the reasoning of others (MP.3)
• Teachers guide students toward learning outcomes
• Implicit instruction; students working as partners or in mixed-ability groups to complete tasks
• It assists teachers by addressing the needs of diverse student populations
• Teacher presents the group with a task/problem and sets up student activities
• While students work together, the teacher monitors progress and assists student groups when necessary

Cognitively Guided Instruction

• Calls for teachers to present students consider different ways to solve a problem
• The model uses student-generated strategies (plastic cues, counting on fingers, using knowledge of number facts)
• Asks students to explain their reasoning process publicly
• Students are expected to explain and justify their strategies
• Students and the teacher decide if a strategy is viable
• Students are challenged to use reasoning to solve the problem and find their own solutions
• Teachers are required to open their instruction to students’ ideas and guide them

Problem Based Learning

• Emphasizes making sense of problems and persevering in solving them (MP.1)
• Emphasizes reasoning abstractly and quantitatively (MP.2) and solving problems everyday life, society, and the workplace(MP.4)
• Problem-based (interactive) learning/project/inquiry-based learning provide students engage in mathematical inquiry by collecting information and testing hypotheses
• The basis of the instructional model is the attribution of John Dewey's model of reflective thinking
• Identify and clarify a problem
• Form hypotheses
• Collect data
• Analyze and interpret the data to test the hypotheses
• Draw conclusions
• Two approaches: guided and unguided inquiry
• Guided inquiry: teacher provides data and questions students to arrive at a solution
• Unguided inquiry: students take responsibility for analyzing data and coming to conclusions
• Problem-based learning works to have students solve real-world application problems
• Teacher poses the problem, assists, and monitors progress
• Students learn to think/show resourcefulness and creativity through problem-based activities
• Students must be allowed to make mistakes when problem solving
• Teachers create a classroom atmosphere that recognizes errors as inevitable
• Student errors become the basis of furthering understanding through discussion and feedback

Strategies for Mathematics Instruction

• Teachers develop a repertoire of instructional strategies
• Mathematics teachers, professional learning from county offices of education, California Mathematics Project, other math education professionals, and internet are good resources to build repertoire

Discourse in Mathematics Instruction

• MP standards call for students to make sense of problems (MP.1), construct arguments (MP.3), and model with mathematics (MP.4)
• Students are expected to communicate their understanding, receive feedback, and progress
• Students relate language to math language when students communicate their learning through discussions/writing
• Writing organizes ideas
• Writing provides a unique window to students' thoughts and the way a student is thinking about an idea

Number/Math Talks (Mental Math)

• Refers to classroom conversations around purposefully crafted computation problems that are solved mentally

• Problems are designed to elicit specific strategies that focus on number relationships and number theory

• Students are given problems and are expected to mentally solve them, accurately, efficiently, and flexibly

• Students share and defend solutions and strategies

• They collectively reason/build connections through these solutions and defenses

• Number talk can be conducted in five to fifteen minutes

• During a number talk:

• The teacher writes a problem on the board

• The teacher gives students time to solve

• Students continue finding new strategies

• Students indicate strategies by raising their fingers

• All answers are recorded on the board

Five Practices for Orchestrating Productive Mathematics Discussions

• Smith and Stein (2011) identified five practices for teachers to advance mathematical understanding:
• Anticipating
• Monitoring
• Selecting
• Sequencing
• Connecting
• Facilitating productive math discussions requires planning
• The teacher should anticipate likely responses
• Poses the problem
• Monitors responses as students work individually, in pairs, or in small groups
• Pays attention to strategies used
• The teacher conducts the "share/summarize" and selects a student to present work
• Sequences the sharing
• Highlights the mathematical goal
• Deliberately asks questions to connect responses to ideas

Student Engagement Strategies

• Building a list of student engagement strategies is useful
• Student engagement helps students remain focused
• Good classroom management and effective teaching/learning come from student engagement

Appointment Clock Example

• Partners set up appointments for discussions or work
• Students are given a page with a clock printed on it
• Students set appointments to discuss math problems

Carousel-Museum Walk Example

• Groups post work on wall
• Leader stands to the work group stands near work while the rest circulate
• Each group given a poster board and a math word problem
• After word problems completed, each leader explains the work when other walk around
• The other students walk around and evaluate

Charades Example

• Acting out a scenario with groups
• Students act out word problems while others try to solve them

Clues (Barrier Games) Example

• Partners communicate information
• Each student gives another problem to try to solve with information provided
• Students sit at barrier

Coming to Consensus Example

• Share ideas and come to a consensus
• Members provide answers and steps
• When the group come to a consensus, they reveal it

Explorers and Settlers Example

• Half explore and half are settlers
• Students are designated as explorers with terms/problems
• The settlers students have the definitions/answers
• Explorers seek correct answers/discuss information

Find My Rule Example

• Match related cards
• One partner has a card with rule
• Practicing reasoning/grouping/developing skills

Find Your Partner Example

• Matching cards in some way
• Example cards: rectangles, prime number of 37

Four Corners

• Assign each corner a category
• Students write which category they are most interested in and form groups
• If teacher calls out 6, students in corners 2 and 3 will move
• The activity ends with a prime-number answer, and each return to their seat

Give One, Get One

• Circulate, give one idea, and receive one
• Students name the steps
• Partner A and B exchange, repeat, and raises hands

Inside/Outside Circle

• Concentric circles create partners
• Teacher poses a question
• They discuss and rotate and repeats, and rotate
• Inner-circle students ask a question

Jigsaw

• A group is assigned to teach group, a portion of test/teach remainder
• In factoring jigsaw students teach
• Students become experts on a concept in factoring, and teach

KWL (Know, Want, Learn)

• A sets the stage
• The teacher asks students know, want, and learn
• Determine student readiness and KWL chart

Line up (Class Building)

• In a particular order
• Students align by numbers
• Square root, fraction, decimal

Making a List

• In a multi-step or word problem
• The goal is to name steps

Numbered Heads Together

• Cooperative Strategy
• Each student in a team works together and is assigned a random number
• Each team is given a problem and student explains their method

Quiz, Quiz, Trade

• Paired to quiz each other
• Review Vocabulary/Discuss/Improve skills

Socratic Seminar

• Rigorous dialogue that deepens understanding
• Discuss/Explain/Defend

Team Share

• Share Product with Teams
• Students explain their class' assigned part

Think-Pair-Share

• Think Alone, then with a partner
• Students practice and and refine their response

Think-Write-Pair-Share

• Students are asked to think, write, pair, and share is used when response is needed

Whiparound

• The teacher poses a question that calls for multiple answers. The class writes as many answers
• Teacher and each classmate shares one their response. When called on, students must supplement

Wraparound

• Each student write ideas about a topic and repeat's another student's statement

Tools for Mathematics Instruction

• Tools help to make concrete
• The section highlights some tools

Visual Representations

• Teachers utilize structures, argue, model, and strategically use tools
• Diagrams, concept maps, graphic organizers, math drawings, and flowcharts show diagrams

Concrete Models

• Utilize manipulates so that students make sense of problems

Interactive Technology

• Teachers view teaching applications and software to be sure any technology used adheres to the CCSSM
• Teachers should look for multiple sources of information

Marissa’s Savings

Understanding Perimeter

• The following table - Area (square inches) Length (inches) Width (inches) Perimeter (inches) 12 1 12 26 12 2 6 16 12 3 4 14 12 4 3 14 12 6 2 16 12 12 1 26

After-School Job: Leonard needs to earn money

Ms. Olsen’s Sidewalk

Baseball Jerseys: Bill is going to order new jerseys

Real-World Problems

• Teachers don't use real-world to serve math; they use math to serve and address real-world
• CA CCSSM do not want students mimicking real-world; that the goal is that students solve problems
• A task is still a task to complete In “Exploring World Maps,” based from the California Mathematics Project (2012), students show the mastery by solving problems in a real world area: a. Which area did you estimate to be larger—Mexico or Alaska? b. Approximately how many times can Greenland fit into Africa? c. Do you feel confident in your estimations? d. What estimation methods did you use? e. Now that you know the actual areas (students are provided with the actual areas prior to an- swering this question), what surprised you the most? f. How does the location of the equator affect how this map is viewed?