Instruction Flashcards
Integrated teaching
multiple concepts are used in problem-solving at once (current best practices)
Non-proportional manipulatives
objects that are not proportional to each other with respect to shape and size. Often all of the items are the same size.
ex: counting tokens
Informal Deduction (Geometric Learning Progressions)
Level 2. Students recognize attributes and properties that objects share and are able to follow logical arguments about the relationship among these properties.
Example: Problem solving that involves reasoning
teacher wait time
the silence that often comes after a question has been asked but before students have finished considering their answer and/or find the courage to speak up
line graph
a visual representation of data which shows change over time or in response to a manipulated variable
verbal representation
word problems and verbal descriptions of how to solve a problem or what the solution means
Example: “We know that Sam gets $10 each week for allowance, so let’s make that a constant. And we know that Sam wants to save $150 to buy a new bicycle, so that’s a constant, too. But, what we don’t know is how long does Sam need to save - let’s make that X. So the equation is 10x = 150.”
Rigor (geometric learning progressions)
Level 4. Students are able to work in different axiomatic systems.
Example: Writing mathematical proofs using formal language. A college level geometry course is taught at this level.
Decimal grid
a model used to visualize multiplication between a whole number and a decimal; involves a 10 x 10 grid in which each square represents 0.01 and the entire grid is equivalent to 1
concrete operational stage
the third stage of Piaget’s Theory of Cognitive development, occurring from 7 years old to adolescence, in which children begin to think logically and use inductive reasoning
Visualization (Geometric Learning Progressions)
Level 0. Shapes are recognized by their appearances. Students can’t identify specific attributes or properties of the shapes. They might identify some characteristics but are not able to use them to categorize or sort shapes.
Example: Working with different sizes of the same shape
Tape diagram
diagrams useful for visualizing operations with fractions; each rectangle is equivalent to 1 and subdivided into increments representing the numerators and denominators of fractions
Symbolic representation
A model using symbols or variables to display a mathematical concept.
Example: Formula
Bar graph
a visual representation of data which compares values in different categories
Example: the number of people who prefer each genre of movie
symbolic/representational stage
Drawing pictures or symbols to represent numbers in an equation
Example: Squares
Constructivism
Learning new behaviors by adjusting our current view of the world
Example: Research projects
word wall
An on-going bulletin board with terms used frequently in the classroom; words are often added as they are introduced
number line
a straight line where each number is equal distance from the next one
Concrete representations
Using physical pieces to represent mathematical problems
Example: Manipulatives
homogeneous group
group comprised of individuals working on the same level
Abstract thinking
Using numbers or letter variables in an equation
Example: 13x=y
Deduction (Geometric Learning Progressions)
Level 3. Students are able to construct and understand formal proofs.
heterogeneous group
group comprised of individuals working on various levels
numeric representation
A model using numbers to display a mathematical concept.
pie chart
a graph in which a circle is divided into sectors that each represent a proportion of the whole. Pie charts are helpful when displaying the relative distribution of categories.
set model
a model in which a set of objects represents the whole, and a subset of those objects (typically shown with a separate color) represents a fraction of the whole
double number line diagram
Consists of two parallel number lines, each representing a different unit
Used to visualize and compare ratios
Think-Pair-Share
Active learning activity in which the teacher provides a prompt, the students consider it individually (THINK), then pair up and brainstorm responses or solutions (PAIR), and then the students then share their results with the class (SHARE).
Proportional manipulatives
objects that are proportional to each other with respect to shape and size
Ex: tangrams
Compartmentalized Teaching
concepts taught one at a time in isolation of other concepts (no longer recommended)
flexible grouping
grouping students based on their learning needs or interests
Analysis (geometric learning progressions)
Level 1. Students begin to identify characteristics and attributes of shapes. They use appropriate vocabulary to describe attributes and are less concerned with characteristics (such as the orientation and size of shapes).
manipulatives
Objects used by students to illustrate and explore mathematical concepts, such as to represent numbers in an equation
Example: Blocks, Coins
area model (fractions)
a diagram used to visualize mathematical operations, especially multiplication and division
behaviorism
learning theory rooted in the notion that all behaviors are learned through interaction with the environment
Graphic / Pictorial Representation
graph or picture that serves as a visual model of a mathematical equation
Example: Number line
Cognitivism
learning new behaviors by connecting current knowledge with new knowledge
Example: Teaching fractions by talking about pizza slices
