A Transition to Abstract Mathematics (2nd
Edition, 2014) – Solutions Manual – Maddox
Objectives - answers-The teacher understands how students learn mathematical skills and uses
that knowledge to plan, organize, and implement instruction and assess learning.
MOVE FROM CONCRETE TO ABSTRACT - answers-Mathematics should be taught conceptually.
Children should learn the "big ideas" behind what they do before they learn to do it. Concrete
instruction is connected to students' real experiences, and uses activities that students can see,
hear, taste and touch. Abstract ideas require children to use their imaginations or their brains
only, without help from pictures or real objects. With each new idea taught, good teachers start
with easier, more reality-based thinking and move to more imaginary ideas.
Manipulatives - answers-Manipulatives are any
objects that can be
touched and moved
(manipulated) to assist
understanding.
Manipulatives can take
the form of counters such as small toys or buttons used to help a kindergarten student add, 100
noodles glued onto construction paper to show place value, or a circle folded and cut into
eighths to assist with understanding fractions.
Research Basis: - answers-Ausabel, Bloom, Brownell, Gagne, Piaget, Vygotsky, etc. Keep in mind
that it is not so much the name, but what their research tells us. The Competency does not
specify the researcher.
1
, Learning Modalities: - answers-Visual, Auditory, Kinesthetic
Types of learning:
Association— - answers-words or symbols Example: Even very young children associate the
word "triangle" with the ∆, without knowing the attributes and properties of a triangle.
Types of learning:
Concept— - answers-relational or concrete attributes.
Example: Similar figures have relational attributes. The corresponding angles are equal and the
ratios of corresponding sides are equal.
Types of learning:
Principle - - answers-generalizations, developed rules Example: The area of a trapezoid is
developed from the concept of a trapezoid and the area of triangles, rectangles, and/or
parallelograms.
Types of learning:
Problem Solving— - answers-putting together concepts and principles to solve a problem new
to the learner
Example: Given a composite figure the student determines the area using the areas of triangles
and rectangles.
Development of Learning: - answers-Instructional Moves: Linked between and among the
following using modeling, describing and recording by both teacher and student.
• Concrete—manipulatives, models, hands-on
• Pictorial—pictures, diagrams, graphs, technology
• Abstract—symbols, words
2
Edition, 2014) – Solutions Manual – Maddox
Objectives - answers-The teacher understands how students learn mathematical skills and uses
that knowledge to plan, organize, and implement instruction and assess learning.
MOVE FROM CONCRETE TO ABSTRACT - answers-Mathematics should be taught conceptually.
Children should learn the "big ideas" behind what they do before they learn to do it. Concrete
instruction is connected to students' real experiences, and uses activities that students can see,
hear, taste and touch. Abstract ideas require children to use their imaginations or their brains
only, without help from pictures or real objects. With each new idea taught, good teachers start
with easier, more reality-based thinking and move to more imaginary ideas.
Manipulatives - answers-Manipulatives are any
objects that can be
touched and moved
(manipulated) to assist
understanding.
Manipulatives can take
the form of counters such as small toys or buttons used to help a kindergarten student add, 100
noodles glued onto construction paper to show place value, or a circle folded and cut into
eighths to assist with understanding fractions.
Research Basis: - answers-Ausabel, Bloom, Brownell, Gagne, Piaget, Vygotsky, etc. Keep in mind
that it is not so much the name, but what their research tells us. The Competency does not
specify the researcher.
1
, Learning Modalities: - answers-Visual, Auditory, Kinesthetic
Types of learning:
Association— - answers-words or symbols Example: Even very young children associate the
word "triangle" with the ∆, without knowing the attributes and properties of a triangle.
Types of learning:
Concept— - answers-relational or concrete attributes.
Example: Similar figures have relational attributes. The corresponding angles are equal and the
ratios of corresponding sides are equal.
Types of learning:
Principle - - answers-generalizations, developed rules Example: The area of a trapezoid is
developed from the concept of a trapezoid and the area of triangles, rectangles, and/or
parallelograms.
Types of learning:
Problem Solving— - answers-putting together concepts and principles to solve a problem new
to the learner
Example: Given a composite figure the student determines the area using the areas of triangles
and rectangles.
Development of Learning: - answers-Instructional Moves: Linked between and among the
following using modeling, describing and recording by both teacher and student.
• Concrete—manipulatives, models, hands-on
• Pictorial—pictures, diagrams, graphs, technology
• Abstract—symbols, words
2
