Concepts Verse Procedures- Algebra 2 Analyzing Mathematics through Multiple Lenses

This resource has been created to support Concepts and Procedures in Mathematical Instruction as stated in Claim 1 of the Common Core

State Standards: It is important to see how concepts link together and why mathematical procedures work the way they do.

Building Concept Building Procedure Explicit or implicit understanding of the principles that govern a domain

and of the interrelations between pieces of knowledge in a domain.

Ideas, relationships, connections, or having a “sense” of something.

Learning that involves understanding and interpreting concepts and the relations between concepts.

To Know why something happens in a particular way.

Action sequences for solving problems.

Like a toolbox, it includes facts, skills, procedures, algorithms, or methods.

Learning that involves only memorizing operations with no understanding of underlying meanings.

To know how something happens in a particular way.

(Literacy Strategies for Improving Mathematics Instruction: Joan M. Kenney)

Conceptual knowledge

Procedural knowledge

Reasoning, transforming,

applying, etc.

Observing, problem solving,

explaining, experimenting,

etc.

Worked examples,

comparison prompts, self-

explanation prompts,

ordering of lessons, etc.

Long-Term Memory

In Working Memory

Student Behavior

Learning Environment

Concepts Verse Procedures- Algebra 2 Analyzing Mathematics through Multiple Lenses

Building Concept Building Procedure

5 Instructional Shifts

Students provide strategies rather than learning them from the teacher.

Teacher provides strategies “as if” from students.

Students create the context.

Students do the sense making.

Students talk to students

The Eight Mathematical 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. Classroom Environment

Grouping and collaboration conversations

Teacher teaches through a task, rather than to the task

Teacher uses “just in time” scaffold

Exploration and discovery process to allow students to interact with the mathematics

Classroom where student errors are valued to help capture misconceptions

The Concept of Mathematics should be taught first, and then the procedures become the tools that allow students to apply the math to real-world problem solving efficiently. As seen in the graphic on the front page of this document, they are equally important. Procedural Fluency

Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving

Research suggests that once students have memorized and practiced procedures that they do not understand, they have less motivation to understand their meaning or the reasoning behind them

The development of students' conceptual understanding of procedures should precede and coincide with instruction on procedures.

All students need to have a deep and flexible knowledge of a variety of procedures, along with an ability to make critical judgments about which procedures or strategies are appropriate for use in particular situations

In computation, procedural fluency supports students' analysis of their own and others' calculation methods, such as written procedures and mental methods for the four arithmetic operations, as well as their own and others' use of tools like calculators, computers, and manipulative materials

Practice should be brief, engaging, purposeful, and distributed.

Analyzing students' procedures often reveals insights and misunderstandings that help teachers in planning next steps in instruction.

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Concepts Verse Procedures- Algebra 2 Analyzing Mathematics through Multiple Lenses

Building Concept Building Procedure

Tasks, Questions, and Evidence

Select appropriate Tasks to support identified learning goals.

Facilitate productive Questioning during instruction to engage students in the Mathematical Practices.

Collect and use student Evidence in the formative assessment process during instruction.

Tasks – connect to learning goal and help identify misconceptions.

Questions – highlight mathematical practices and uncover misconceptions.

Evidence – describes misconceptions and guides necessity of providing scaffolding and offering extensions.

Planning with the Curriculum

When lesson planning, make connections throughout the unit to build coherence across lessons and across concepts.

Plan with the correct focus when selecting problems from the lesson within the module. There are problems within each lesson that address concepts. Allow students to use inquiry type processes.

Concept problems are found in Big Ideas: o Engage and Exploration o Inquiry based o Error Analysis o Predict, Reason, and Justify Problems o Critique the reasoning of others o Structures in Mathematics

Teaching Procedures to the Tasks

Use procedures to help build the capacity to solve problems with efficiency while also applying the procedure to a concept.

Always align and make connections to why and when certain procedures work within context.

Direct instruction and guided instruction problems for a specific purpose.

Specific feedback for specific focus and use of rubrics for evidence.

Planning with the Curriculum

When lesson planning, make connections to the relevance and purpose of the procedural problems, so students can make transfer of deep learning.

Procedure problems are found in Big Ideas o Maintaining Mathematical Proficiency o Monitoring Progress o Independent Practice Problems

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Concepts Verse Procedures- Algebra 2 Analyzing Mathematics through Multiple Lenses

F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of

the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include:

intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums;

symmetries; end behavior; and periodicity.

Building the Concept Building Procedural Skill

Decide whether the function is a polynomial function. If so,

write it in standard form and state its degree, type, and leading

coefficient.

Section 4.1 Page 162 and 163 Big Ideas

Describe the degree and leading coefficient of the polynomial

function using the graph.

Section 4.1 Page 162 Big Ideas

Concepts Verse Procedures- Algebra 2 Analyzing Mathematics through Multiple Lenses

N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Building the Concept Building Procedural Skill

Simplify the expression.

Section 5.2 Pages 245 and 246 Big Ideas

Section 5.2 Pages 248 and 250 Big Ideas

Concepts Verse Procedures- Algebra 2 Analyzing Mathematics through Multiple Lenses

A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may

arise.

Building the Concept Building Procedural Skill

Solve the equation. Check your solution.

Section 5.4 Page 262 Big Ideas

Section 5.4 Page 267 Big Ideas