Implications of cognitive science research for mathematics education
Robert S. Siegler
Summarizes conclusions we can draw from cognitive science research in eight areas:
Mathematical understanding before children enter school
 Some understanding universal; others depend on environment
 Cardinality: number of objects in a set
o First month of life: cardinality appears universally; can discriminate one
object from two, and two from three (probably by subitizing)
o Five months: understand consequences of adding and subtracting one
o Three or four years: can discriminate four objects from five or six; setcounting (can assign numbers to larer sets than can be subitized)
o Three – five years: five principles of counting mastered
 One-one principle: assign one and only one number to each
 Stable order principle: always count in same order
 Cardinal principle: the last number indicates number in set
 Order irrelevance principle: can count objects in any order
 Abstraction principle: these principles apply to any set of objects
 Ordinality: relative sizes of sets (concepts of more or less)
o Also begins in infancy, but slightly later (12 – 18 months) than cardinality
o 16- to 18-month olds: rudimentary understanding of concepts of more or
less
o Four – five years: children from middle class background can solve,
“Which is more: 6 oranges or 4 oranges?”
 Children from low-income backgrounds have little or no sense of
single-digit magnitudes when they enter first grade
Pitfalls in mathematics learning
 Three areas with prominent misconceptions
o Long subtraction – particularly when top number includes zero
o Comparison of fractions – due to not thinking of the magnitude
represented by each fraction, as well as confusion over relation of symbols
(such as decimal point) to magnitude
o Algebraic equations – often due to treating equation as symbol
manipulation rather than representation of real-world numbers
 Importance of cognitive task analysis (analyze particular procedures and concepts
to be learned) to promote effective learning
Cognitive variability and strategy choice
 Children use a variety of strategies from early in learning, and continue to use
both less and more advanced ones for many years
o Efforts to change this are often unsuccessful
o They learn better when they are allowed to choose their strategy
 From second grade onward, they shift their strategies to meet the demands of the
task
Individual differences

As early as first grade, kids can be divided three groups based on their strategy
choices: good students, not-so-good students, and perfectionists (Siegler, 1988)
o About half the not-so-good students were eventually classified as having
math disabilities by fourth grade in this study
 Math disabilities are combination of limited background knowledge, limited
processing capacity and limited conceptual understanding
Discovery and insight
 Children discover new strategies and concepts when existing approaches are
succeeding as well as when they are failing
 So not all discoveries require “discovery learning” situations (they can arise when
things are going well)
Relations between conceptual and procedural knowledge
 They are correlated
 A substantial percentage of children first gain conceptual understanding and then
procedural competence… and another substantial percentage does the opposite
 So which should we teach first? One study (in press) says conceptual knowledge
Cooperative learning
 Sometimes helpful, sometimes not
 Shown helpful in one study only when experimenter gave feedback concerning
which answer was right; then single best predictor of learning was enthusiasm of
partner’s reactions!
Promoting analytic thinking (set of processes for identifying causes of events) and
transfer
 Analytic thinking is the cause and consequence of purposeful engagement
 It is a constructive process that promotes transfer
 Can be promoted by asking children to explain both why correct answers are
correct and why incorrect answers are incorrect
“Neither controlled scientific experimentation nor theoretical analyses automatically
translate into prescriptions for classroom instruction.” (p. 230)
Japanese emphasis on continuous improvement would do well in U.S. classrooms
 Groups of teachers work together to perfect the way they teach certain concepts
and procedures