Classroom Applications of Cognitive Psychology

The recently-developed science of human learning reveals and confirms many principles concerning how people acquire learning initially, how we recall learning, how learners apply knowledge to new situations, and the role of attention in every aspect of learning.  This site is dedicated to helping teachers become even more effective.  Many of us started teaching before cognitive psychology emerged (early 1980's).  Many more teachers endured relatively useless "educational psychology" courses as part of pre-service training and in-service workshops.  Those who have learned how to survive the hard way will wonder why the information in this web site wasn't available when they started teaching.  Never mind -- onward.

This web site is also dedicated to what can be accomplished with the power to share.
As knowledge in general is power, mathematics knowledge is more power than other knowledge, in our day. Mathematics education has more effect on a person’s lifetime income than knowledge in any other discipline, per the U.S. Bureau of Labor Statistics. In mathematics education lies ability and confidence to create in any field, from engineering to visual and performing arts.

·         Sharing this power has gone the way of sharing any other real power. True, we’ve learned to share money, and call it welfare. We have tried to compensate for errors of history (for example, Jim Crow laws) by equal-employment opportunities. But opportunities to earn one’s own money via efficient, effective education have not been shared, and have been deliberately suppressed by some.

·         The strength to share genuine opportunity -- not monetary or employment handouts -- is a spiritual strength. The spiritual power to share genuine economic and political power is behind this web site.

Site Outline

             I.    Psychology of Learning A  Introduction to Cognitive Psychology   


                        a.  Principles of Learning  (brief)


b.  Attention


                        c.  Learning Principles from Cognitive Psychology Textbooks

           II.   Psychology of Learning B: Verbal and Visuospatial Sense Development


                        a.    Origami

                        1.  Presentation Outline


                        2.  Origami, Spatial Relations, and Gender


                        3.  Paper Balloon


                        4.  Cootie Catcher


                        5.  Space Shuttle


                        6.  Crane

                        b.   Ambiguous Drawings


            c.  Abstract of Research: Effects of Origami and Ambiguous Drawing on Visuospatial Sense Development

         III.   Psychology of Learning C: Initial Learning

            a.  Varied Alphabet Contexts


            b.  Varied Counting Contexts


            c.  Contexts for Multiplication of Algebra Variables


            d.  Legend Beautiful Example


            e.  Information Processing Summary

         IV.    Psychology of Learning D:  Recall

                        a.  Association 


1.  Associative Networks and Saxon’s Practice Structure

                                    2.  How students recall -- and what is NOT involved


                        3.  Drama and Story

                        b.  Review


                        1.  Johnston-Uhl Experiment


                        2.  Thorndike Support for Correct Practice Structure


                        3.  Writing Your Own DARES

                        4.  A Great Teacher’s Further Development

                        c.  Stress (benevolent)

           V.    Psychology of Learning E:  Applications

                        a.    Problem-Solving


                        1.  AA Preface and Contents


                        2.  Chapter 1 Beginnings


                        3.  Chapter 2 Domain Knowledge


                                    a)  MATHCOUNTS prose


                                    b)  MATHCOUNTS outline


                                    c)  Strategy: The Harmonic Mean in Problem-Solving


                                    d)  Strategy: The Cantor Heuristic


                                    e)  Strategy: Assume an Ideal Case


                                    f)  Strategy: Look for a Hidden Assumption


                        4.  Chapter 3 Main Problems


                        5,  Chapter 4 Miscellaneous Problem Lists


                                    a)  Aha Insight and Catch Questions


                                    b)  More Quotes


                                    c)  GAMES Genres and Captain Pi


                                    d)  The Seven or Eight Bridges of Konigsberg


                                    e)  Catch Questions from Mathematical Nuts


                                    f)  Examples from Puzzles-dot-com


                                    g)  Tavern Puzzles


                                    h)  Support and Verification


                        6.  Chapter 5 Verification and Linkages


                                    a)  House Doors: Change Form


                                    b)  Horseman


                                    c)  Cow Goat Goose Grass


                                    d)  Kinds of Proof


                                    e)  Differences Strategy


                                    f)  Recursion Aid


                                    g)  Induction Logic  Details for Classic Problems (A)


                                    h)  Induction Logic  Details for Classic Problems (B)


                                    i)  Induction Logic  Details for Classic Problems (C) Subsets


                                    j)  Polygons Regions Analysis


                                    k)  Hats Logic Problem


                                    l)  Proof by Exhaustion: Knights Moves


                                    m)  Chain Links Payments - - Minimal Weights - - Old Timer


                                    n)  Constraints and Balancing Act


                                    o)  Nine Circles Analysis


                                    p)  Nine Circles Solution


                                    q)  Recursion Inventory of Problems and Issues


                        7.  Z Appendices


                                    a)  References



                        b.   Adventures in Mathematics

         VI.    Psychology of Learning F:  Associative Networks

                        a.    Connections


1.      Awesome Initiatives Elementary

a)     Numeration: Writing Numbers in Columns of 25

b)      Numeration: Reading & Writing Whole Numbers & Decimal Fractions

c)      Numeration: Roman Numerals

     CC.  K2 Roman Numerals chart organization

d)      Numeration: Significant Digits

e)      Fraction: Common Fraction Reducing Stages

ee.  Fraction Reducing table

f)       Fraction: Making the Fraction Concept  Digestible

g)      Operations:  Addition concept and implications

h)      Operations: Four Operations Development

i)        Geometry: Classification/Relationship (Geom Only)

j)                     ii.  Classification/relationship geometry definitions

k)      Geometry: Distance/Area/Volume combined best

l)        Geometry: Dist area vol gearshift plus sectorArea arcLength (JrHi)

m)    Geometry: Metric Units

n)      Geometry: Clarifying Area and Volume concepts (teachers only?)

       mm.  Geometry: Generalization/Discrimination  Basic Shapes

o)      Applications: Arithmetic Word Problems Resolved by TAPS

p)      Applications: Algebraic Thinking From Arithmetic

q)      Applications: Sample Spaces

r)       Applications:  Tree Diagrams:  When to + or × probability?

2.         Awesome Initiatives Secondary

a)   Algebra: Clarifying the Varied Roles of Variables

b)   Algebra: “Where Did That ONE Come From?”

c)   Algebra:  Addends -- Products -- Powers

d)   Algebra: Add Like Kinds

e)   Algebra: Arithmetic Catalysts for Algebra Word Problems

f)     Algebra: Customary Function Language

g)   Algebra: Quadratic Formula Development and Graph Connection

h)   Algebra: Parent Functions I II III IV

i)     Algebra: Transformations on Parent Functions with examples

j)     Algebra: Odd and Even Functions Extravaganza

k)   Algebra: Function Inverse Extravaganza Activity  

l)     Algebra: Solving Extravaganza with Logarithm Equations

m) Algebra: Rational Functions Extravaganza

n)   Geometry: Per area vol gearshifting  enhanced: sector area & arc length

o)   Geometry: Pythagorean Triples per Sierpinski

p)   Geometry: Angle Measures in Circles

q)   Geometry: Ratios in Similar Solids

r)    Geometry: Dilbert Head -- Prism Surface Area & Volume

rr.   Dilbert Grandson  hexagon base

s)    Geometry: Deductive Structure

t)     Geometry: Form and Number Definitions Chart

u)   Geometry: Proof Strategies “from the board”

v)   Geometry: Cevians and CLRS best Jan 2010

w) Geometry: Statement Classification and Truth Tables

x)   Geometry: Representation of Means

y)   Trig: Reciprocal Function vs. Inverse Function

z)    Analytic Geometry: Conics Constructed by Manipulation

3.         Essential Defenses Elementary

a)   Numeration:  Whole Number Operations;  Terminology;  Meanings

b)   Numeration: GCF and LCM Organized

c)   Operations:  Division Practice  Basic to Complexity   Parts 1 and 2

d)   Fractions: Ruler for Mandy -- Inch Ruler and Equivalent Fractions

e)   Fractions: Equivalent proportions  dial

f)    Fractions: Four Operations Chart

g)   Geometry: Circle Attributes Schema

h)   Geometry: Polygon Congruence Distinction/Discrimination (JrHi)

i)     General: Permutation and Combination Concept Development

j)     General: Set Relationships  DICES   and Core Concepts

jj.  Set Relationships only

k)   General: Core Concepts in Mathematics

4.         Essential Defenses Secondary

a)   Numeration:  Numbers Outlined

b)   NumerationAlgebra  But It Looks Easy   operations with ONE and ZERO

c)   Algebra: Distributivity and NON Distributivity (and Mutations)

d)   Algebra: Exchanging Greater-Than with Less-Than

e)   Algebra: Rational Expressions vs. Rational Equations

f)     Algebra: When Do Solutions Disappear in Algebra?

g)   Algebra: Three Key Ideas for Understanding Radicals

h)   Algebra: Six Connections for Systems of Equations

i)     Algebra: Two Takes on Completing Square

j)     Algebra: GCF LCM  Organized  

k)   Algebra: GCF  LCM  all methods

l)     Algebra: GCF  LCM  poster with blanks  secondary

m) Algebra: Parabola Extravaganza

n)   Algebra: Domain and Domain Issues

o)   Algebra: Consecutive Integers Reminders

p)   Geometry: Regular Polygons and Tessellations

q)   Geometry: Surface Area and Volume

r)    Geometry: Principles of Negation

s)    Analytic Geometry: Slope/Equation/Chart/Graph/Definition Extravaganza

t)     Analytic Geometry: Conics Connections

u)   Trig: Function Values for Nice Angles

v)   Trig: Development of Identity Cluster

w) Calculus: Derivatives and Integrals List

x)   General: Distinctions Between Probability Situations

y)   General: Properties of Operations and Relations

5.        Inspiring Rainbows Starscapes Elementary

a)   Classification/Relationship: Numbers, Statements, Sets, Operations

b)   Early Similarity, Proportion, Slope, Trig

c)   Concept of Denominator/Denomination

d)   Fractal Forms From Platonic Solids

e)   Reasoning --  Induction and Deduction

6.        Inspiring Rainbows Starscapes Secondary

a)   The Concept of Order in Mathematics

b)   Conics Illustrated

c)   Surface Area Progression

d)   Types of Variation

e)   Concept Nodes in Arithmetic and Algebra  

f)     Definition Chain from Abstract Algebra

7.        Connections to Other Disciplines

a)   Art:  Alternation of Attention   Impossible E   Triangle    Cage

b)   Soccer:  Shootout and Probability

c)   Science:  What science students need to know about fractions

d)   Citizenship:  Simpsons Paradox

e)   Theology:  Amazing Grace and Cantor’s Definition of Infinity

f)     Psychology:  Wallas & Poincare, Problem-solving per Cognition Principles

g)   Application Ratio-Prop-PerCent Problems w Connecting Strategy

h)   Elementary experience of math: sine curve

                        b.   A Tapestry of Geometry Objectives

       VII.   Psychology of Learning G  Generalization and Discrimination

     VIII.   Psychology of Learning H  Affective Outcomes

        IX.    DARES

                        a.  Grade 1

                        b.  Grade 2

  Grade 3

                        d.  Grade 4

  Grade 5

  Grade 6

                        g.  Grade 7

                        h.  Grade 8


                        1.  Days 1-25


                        2.  Days 26-50


                        3.  Days 51-100


                        4.  Days 111-154

            i.  Carencro DARES instructions


            j.  Writing Your Own DARES


            k.  Science DARES Examples (early grades)


            l.  Science DARES Fifth Grade - - one week’s examples

          X.    Incremental Geometry

         XI.    Teacher Resources

   Reading Questions: Blue Pelican Java

  Slot Outlines

            c.  Remedial College Algebra

        XII.    The War for Learning: Social and Political Implications


            a.  A Storming of Our Pythagorean Aristocracy


            b.  Declaration of Non-Dependence


            c.  Social and Political Implications of Mathematics Education


            d.  Changes Recommended: Elementary Grades


                        e.  Changes Recommended: Secondary Grades

                        f.  Professor Hardknocks Speaks: What My Methods Courses Never Told Me


                        g.  Educator Quiz  


                        h.  Educator Quiz Key


                        i.  A Christian Calling in Public Education


                        j.  On Numerical vs Criteria Regulation of Math Literacy


                        k.  Chapter 24 on John Saxon


                        l.  What Most of Us Need re: Learning

      XIII.    Humor


                        a.  Invitation: Return to Pig Run Island


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