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The Six Principles & what each means
The Five Content Standards
- -Number & Operations
- -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
- 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
- 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
Ideas are highly connected
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 problem-solving
- -It is self-generative
-It improves attitudes and beliefs.
knowledge that results from relationships constructed internally and connected to already existing ideas.
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
-Introduction of topics with problem solving
- -Inclusion of non-routine and application problems
-Use of higher-order thinking questions
- 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 refers to repetitive, non-problem-based exercises designed to improve skills or procedures already acquired.
Practice refers to different problem-based tasks or experiences, spread over numerous class periods, each addressing the same basic ideas
- -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 problem-based 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
-The Mathematics Standard
- -The Learning Standard
- -The Equity Standard
- -The Openness Standard
- -The Inferences Standard
-The Coherence Standard