NCTM Principles & Standards Flashcards
Six Principles:
Must be integrated into lessons in their overarching themes:
Equity Curriculum Teaching Learning Assessment Technology
Equity:
High expectations and strong support for all students.
Curriculum:
Must be coherent, focused on important mathematics and well articulated across the grades.
Teaching:
Understanding what students know and need to learn and then challenging and supporting them to learn it well.
Learning:
Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.
Assessment:
Should support the learning of important mathematics and furnish useful information to both teachers and students.
Technology:
`Essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning.
The Five Content Standards:
Each encompass specific expectations organized by grade bands:
Numbers and Operations Algebra Geometry Measurement Data Analysis and Probability
Even though this is the NCTM standards you must know your state specific math standards. (State has authority to select their own standards).
Process standards from the Principles and Standards:
Problem Solving Reasoning and Proof Communication Connections Representation
Problem Solving:
Build new mathematical knowledge through problem solving.
Solve problems that arise in mathematics and other context.
Apply and adapt a variety of appropriate strategies to solve problems.
Monitor and reflect on the process of mathematical problem solving.
Reasoning and Proof:
Recognize reasoning and proof as fundamental aspects of mathematics.
Make and investigate mathematical conjectures. (informed guess or hypothesis)
Develop and evaluate mathematical arguments and proofs.
Select and use various types of reasoning and methods of proof.
Communication:
Organize and consolidate their mathematical thinking through communication.
Communicate their mathematical thinking coherently and clearly to peers, teachers and others.
Analyze and evaluate the mathematical thinking and strategies of others.
Connections:
Recognize and use connections among mathematical ideas.
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
Recognize and apply mathematics in contexts outside of mathematics.
Representations:
Create and use representations to organize, record and communicate mathematical ideas.
Select, apply and translate among mathematical representations to solve problems.
Use representations to model and interpret physical, social and mathematical phenomena.
Standards:
Whether your state or district uses the Common Core… or other state standards, it is important to understand how to read and apply math standards in order to plan good math instruction.
Objective:
An objective is written from a standard and the lesson must align with the objective.
Differentiation Considerations:
1) . The degree of assisstance
2) . How structured the task is
3) . The complexity of the task given
3) . The complexity of process
Helps for Gifted Students:
Move quickly is subject and content
Student led and student directed problem solving
Populations to Plan for:
Reluctant Learners
Students with Learning Disabilities
Gifted Learners
ELL Students
CSA:
Concrete, Semi-Concrete, Abstract Sequence.
Peer Assisting:
On an “As-needed” instead of a predetermined sequence.
Special needs students shouls switch roles so they are able to “Teach” to an older peer or a peer who has a better conceptual understanding of the subject.
Think Alouds:
Teacher talking through steps on how to work out a problem. This is a higher level of learning.
Can’t form mental representations of mathematical concepts:
Can’t interpret a number line
Has difficulty going from a story about a garden plot (to set up a problem on finding area) to a graph or dot paper
How should I teach?
• Explicitly teach the representation—for example, exactly how to draw a diagram
• Using larger versions of the representation (e.g., number line) so that students can move to or interact with the model
Difficulty accessing numerical meanings from symbols (issues with number sense):
Has difficulty with basic facts; for example, doesn’t recognize that 3 + 5 is the same as 5 + 3, or that 5 + 1 is the same as the next counting number after 5
How should I teach?
• Explicitly teach multiple ways of representing a number showing the variations at the exact same time
• Use multiple representations for a single problem to show it in a variety of ways (blocks, illustrations, and numbers) rather than using multiple problems
Challenged to keep numbers and information in working memory:
- Loses counts of objects
- Gets too confused when multiple strategies are shared by other students during the “after” portion of the lesson
- Forgets how to start the problem-solving process
How should I teach?
• Use ten-frames or organizational mats to help them organize counts
• Explicitly model how to use skip counting to count
• Jot down the ideas of other students during discussions • Incorporate a chart that lists the main steps in problem solving as an independent guide or make bookmarks with questions the students can ask themselves as self-prompts
Lacks organizational skills and the ability to self-regulate:
- Loses steps in a process
- Writes computations in a way that is random and hard to follow
How should I teach?
• Use routines as often as possible or provide self-monitoring checklists to prompt steps along the way
• Use graph paper to record problems or numbers
• Create math walls they can use as a resource
Misapplies rules or overgeneralizes:
- Applies rules such as “Always subtract the smaller from the larger” too literally, resulting in errors such as 35 − 9 = 34
- Mechanically applies algorithms—for example, adds 7/8 and 12/13 and gives the answer 19/20
How do I teach?
Always give examples as well as counterexamples to show how and when “rules” should be used and when they should not
• Tie all rules into conceptual understanding; don’t emphasize memorizing rote procedures or practices.
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Activities for Moderate and Severe Disabilities:
Number and operations:
• Count out a variety of items for general classroom activities.
• Create a list of supplies that need to be ordered for the classroom or a particular event and calculate cost
• Calculate the number of calories in a given meal.
• Compare the cost of two meals on menus from local restaurants.
Activities for Moderate and Severe Disabilities:
Algebra:
• Show an allowance or wage on a chart to demonstrate growth over time.
• Write an equation to show how much the student will earn in a month or year.
• Calculate the slope of a wheelchair ramp or driveway
Activities for Moderate and Severe Disabilities:
Geometry:
• Use spatial relationships to identify a short path between two locations on a map.
• Tessellate several figures to show how a variety of shapes fit together. Using tangrams to fill a space will also develop important workplace skills like packing boxes or organizing supplies on shelves.
Measurement
Activities for Moderate and Severe Disabilities:
Data analysis and probability:
• Survey students on favorite games (either electronic or other) using the top five as choices for the class. Make a graph to represent and compare the results.
• Tally the number of students ordering school lunch.
• Examine the outside temperatures for the past week and discuss the probability of the temperatures for the next days being within a particular range.
Drill and Practice:
Used as review, is best if limited to 5 to 10 minutes
Accommodations for Special Education Students :
Honor Use of Native Language.
Write and State Content and Language Objectives.
Build Background.
Use Comprehensible Input.
Comprehensible Input:
The message you are communicating is understandable to students.
Build Background:
Similar to building on prior knowledge, but it takes into consideration native language and culture as well as content.
Honor Native Language:
Students can communicate in their native language while continuing their English language development.
Write and State Content and Language Objectives:
Tell what they will be learning. Don’t give away what they will be discovering, but state the larger purpose and provide a road map.
Accommodations for Gifted Students:
Acceleration
Enrichment
Sophistication
Novelty
Novelty:
Introduces completely different material from the regular curriculum and frequently occurs in after-school clubs, out-of-class projects, or collaborative school experiences
Sophistication:
Raising the level of complexity or pursuing greater depth.
Enrichment:
Go beyond the topic of study to content that is not specifically a part of the grade-level curriculum but is an extension of the original mathematical tasks
Acceleration:
Can either reduce the amount of time these students spend on aspects of the topic or move altogether to more advanced and complex content
