Concepts Verse Procedures- 8th Grade Analyzing Mathematics

Concepts Verse Procedures- 8th Grade 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


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

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

Effective teaching practices provide experiences that help students to connect procedures with the underlying concepts and provide students with opportunities to rehearse or practice strategies and to justify their procedures.

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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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 Go Math o Engage and Exploration o H.O.T. (Higher Order Thinking) problems o Inquiry based Problems o Error Analysis o Predict, Reason, and Justify Problems o Multiple Representations o Critique the Reasoning o Critical Thinking Problems

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 Go Math o Your Turn o Monitoring Progress o Independent Practice Problems

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8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion, which repeats eventually into a rational number.

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33

= 1/27.

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33

Building the Concept Building Procedural Skill

Section 1.1 Page 14 Go Math Section 1.1 Page 9


8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 =

3–3 = 1/33 = 1/27.


Section 2.1 and Mixed Review Pages 40 and 60 Go Math Section 2.1 Page 36 Go Math


8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the

coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the

vertical axis at b.


Section 3.1 Page 78 Go Math Sections 3.1 and 4.1 Pages 75 and 97 Go Math


8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the

function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate

of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.


Find the slope and y-intercept of the line represented by table and

graph.

Section 4.2 Page 108 Go Math Section 4.2 and 4.3 Pages 106 and 110 Go Math

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Concepts Verse Procedures- 8th Grade Analyzing Mathematics