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The Six Principles & what each means
Equity  Teaching
 Assessment
 Curriculum
 Learning

The Five Content Standards
 Number & Operations
 Algebra
 Geometry
 Measurement
 Data Analysis & Probability

The Five Process Standards & what each means
 Problem Solving –
 how students learn mathematics
 Reasoning & Proof –
 logical thinking should determine if and why answers are correct; providing a
 rationale should be a part of every answer
 Communication –
 talk about, write about, explain and describe mathematical ideas
 Connections –
 2 ways: within and among mathematics ideas; to the real world and other
 disciplines
 Representation –
 symbols, charts, graphs, manipulatives, and diagrams

Four Features of a Productive Classroom Environment

The Mississippi Mathematics Framework structure

the verbs of doing math
 explore represent explain
 investigate formulate predict
 conjecture discover develop
 solve construct describe
 justify verify use

What is basic in mathematics
Every day, students must experience mathematics that makes sense

What is Constructivism
Students must be active participants in the development of their own understanding; i.e. in the “construction” of their understanding.
 To construct and understand a new idea, students make connections between old ideas and the new one.
 Students are not “blank slates”; they do not absorb ideas.
 Constructing knowledge requires reflective thought – actively thinking about an idea, sifting through existing ideas

Five different representations of mathematical ideas
 pictures
 written symbols
 Manipulative models
 Real world situations
 oral language

What understanding means and how it exists
Is a measure of the quality and quantity of the connections that an idea has with existing ideas.
 Think of “understanding” as existing on a continuum:
 Ideas are highly connected > Ideas are isolated completely
 Relational Understanding Instrumental Understanding

Relational Understanding
Ideas are highly connected

Instrumental Understanding
Ideas are isolated completly

Benefits of relational understanding
 It is intrinsically rewarding
 It enhances memory
 There is less to remember
 It helps with learning new concepts & procedures
 It improves problemsolving
 It is selfgenerative
It improves attitudes and beliefs.

Conceptual Understanding
knowledge that results from relationships constructed internally and connected to already existing ideas.

Procedural knowledge
knowledge of the rules and the procedures that one uses in carrying out routine mathematical tasks and of the symbolism that is used to represent mathematics.

What is problem solving
The process involved to solve a problem or situation for which the individual who confronts it has no procedure that will guarantee a solution

Benefits of problem solving
Use of higherorder thinking questions

DOK levels
 1. Recall
 2. Skill/Concept
 3. Strategic Thinking
 4. Extended thinking

Multiple Entry Point Problems
problems that can be approached in several different ways depending on the ability and learning style of the student

Drill
Drill refers to repetitive, nonproblembased exercises designed to improve skills or procedures already acquired.

Practice
Practice refers to different problembased tasks or experiences, spread over numerous class periods, each addressing the same basic ideas

Homework
 Homework communicates the importance of conceptual
 understanding to both students and parents
Homework is a parent’s window to your classroom.
 When homework is a task or problem, discussing the homework
 should be just the same as the discussion or discourse portion of a lesson

Role of the textbook
 Teach to the big ideas, or concepts, not the pages
 Consider the conceptual portions of lessons as ideas or inspirations for planning more problembased activities. The students do not actually have to “do the page”.
 Let the pace of your lessons through a unit be determined by student performance and understanding (rather than the artificial norm of 2 pages a day).
 Use the ideas in the teacher’s edition.
Remember that there is no law saying every page must be done or every exercise completed.

What is Assessment?
The process of gathering evidence about a student’s knowledge of, ability to use, and disposition toward mathematics and of making inferences from that evidence for a variety of purposes

Purposes of assessment
 To Monitor Student Progress
 To Make Instructional Decisions
 To Evaluate Student Achievement
 To Evaluate Programs

Assessment Standards
The Mathematics Standard
 The Learning Standard
 The Equity Standard
 The Openness Standard
 The Inferences Standard
The Coherence Standard

