Herbert Spencer Lecture a rational approach to education: integrating behavioral, cognitive, and brain science

Herbert Spencer Lecture

• A rational approach to education: integrating behavioral, cognitive, and brain science

Metabolic Brain Images

Oddity with Trial Unique Objects

Learning A Non-Verbal Oddity Task

Open Field Navigation Task

Learning an Open Field Navigation Task

Development of Expert/Novice Knowledge (Means & Voss 1985)

Imaging Number Processing: An early study

Triple Code Model of Number Processing

What kinds of evidence support the model?

• Evidence derives from four kinds of studies:

• Numerical competence of normal and gifted adults

• Development of numerical competence in children

• Animal studies of sensitivity to numerical parameters

• Neuropsychological studies of brain-lesioned patients

Examples of Supporting Evidence

• Adult performance on single-digit operations (2 + 3, 4 x 7)

• Response time to solve such problem shows the problem size effect and tie effect
• Calculation time correlates with the product of the operands or square of their sum except for ties ( 2 + 2, 4 x 4) which show constant RT
• These patterns are explained by duration and difficulty of memory retrieval from a stored lexicon.

• Children’s performance on single-digit addition

• RT for younger children is proportional to the sum
• RT for older children is proportional to the smaller addend
• Younger children use the count-all strategy, while older children use the count-on from larger addend strategy.

• Pigeons and rats can be taught to discriminate two “numerosities”

• Discrimination is easier when the distance between the two numerosities is larger
• Animals, like humans, manifest a “distance effect” when making numerical comparisons.
• Thus, animals, like humans, use an analogue representation in making numerical comparisons.

Neuropsychological Inference

• Task

• Reading number words aloud

• Writing number words to dictation

• Responding to verbally to questions of numerical knowledge

• Comparing orally presented and spelled out number words

• Comparing Arabic numerals

• Making proximity judgments of Arabic numerals

• Reading a thermometer

• Solving subtraction problems

• Solving multiplication problems

Experimental Design for Brain Mapping Study of Number Processing

Number Tasks: Activated Brain Areas

Shaywitz et al. 1998, 2002

Evidence of Training Studies

• Numeracy

• Numeracy requires integrating three representations of number
• Learning problems arise from inadequate integration of these representations
• Training studies show learning problems remediable when representations and their integration are taught explicitly (Resnick, Case & Griffin)

Linking Number Words to Magnitudes

• Learning first formal arithmetic

Kindergartner’s Performance on Number Knowledge Test (% Correct)

Mean Scores (s.d) on Number Knowledge Test Pre- and Post Number Worlds Instruction

Linking Number Words with Visual Arabic Numerals

• Learning Arabic algorithms for multi-digit computation

Linking Calculation with Counting

• Arithmetic Bugs

• Smaller from larger:

• 930

• - 653

• 433

• Borrow from zero:

• 602

• - 437

• 265

• Borrow across zero:

• 602

• - 327

• 225

The Problem of Pre-existing Representations

• Learning fractions

Understanding Fractions

The Promise of Pre-existing Representations

• Teachers’ misrepresentations and teaching algebra

From Arithmetic to Algebra