1. Teaching Arithmetic – Two Traditions New Approach – Strategy Based
Drill & Practice
2 + 3 = ? 4 – 2 = ? 5 – 3 = ?
4 + 7 = ? ? = 7
Meaningful Approach
New Approach – Strategy Based

2. PRINCIPLES OF DRILL AND PRACTICE
SIMPLE TO COMPLEX (order of problem presentation)
DISTRIBUTED PRACTICE BETTER (e.g., if you are going to study for one hour, break it down into two 1/2 hour or 3 twenty minutes segments)
MIXED PRACTICE BETTER (at some point, when the student had some initial practice with individual problem types)

3. Associative Strengths for Simple Addition Problems 4-5 yr olds
Problem Answer
other

4. PRINCIPLES OF THE MEANINGFUL APPROACH
USE CONCRETE EXAMPLES, IMAGERY
(to support understanding)
WHEN A STUDENT HAS DIFFICULTY, DIAGNOSE CONCEPUTAL CONFUSION
BUILD UP UNDERSTANDING BY RELYING ON VISUAL AND ENACTIVE REPRESENTATION (see examples in the next few slides)

5. TEACHING POSITIONAL NOTATION
Array of Beads - Concrete Representation
Color Coded Labels - Intermediate Step
1 0 0
2 0 7
Superimposed Labels 1 2 7

6. ADDITION WITH CARRYING
Color Coded Squares - Intermediate Step
Short cut
Algorithm 1
2 4 6
3 7 3
100 10 10 1 1 1
100 10 10 1 1 1
100 10 10 1 1 1 + 1 1 1 1
100
100 10 10 10 1 1 1 10 10 10
100 10

7. PROMPTING MEANINGINGFUL TRANSFER: THE PARALLELOGRAM PROBLEM
Ask the child to find the area
Give the child scissors and a paper cutout of a parallelogram h b
Insight:
All parallelograms can be changed into rectangles

8. PROMPTING MEANINGFUL TRANSFER: THE PARALLELOGRAM PROBLEM
Transfer to novel problems

9. Single Digit Arithmetic – Strategy Based Options
2 + 3 = ? = ?
6 + 5 = ? = ?
Counting On Your Fingers
Counting – Min Strategy
Infer Answer From Knowledge of Related Problems
Retrieving Answer From Memory
Decomposing Into Easier Problems

10. Single-Digit Arithmetic - Developmental Changes
Children begin with the simplest counting strategy
Move to the “min” counting strategy and more sophisticated strategies like decomposing problem into easier ones
Increasing use of retrieval from memory
Increasing speed and accuracy

11. Why Are Some Single-Digit Arithmetic Problems Easier/More Difficult?
Biased presentation – Adults like to use 1 + 1, 2 + 2, etc
Associative interference – Problems like prompt the next number “5”
Some problems are easier to count to correct answer

12. Single Digit Arithmetic - Key Developmental Question
“How Do Children Move Effectively to Quick and Accurate Responses With Age?
Answer: They adaptively choose between direct retrieval from memory and their host of back-up strategies
“How Do Children Know When to Go With Direct Retrieval, When With a Back-up Strategy?”

13. Model of Strategy Choice – Key Features
Associative Strength
the more often you produce the correct
answer, the stronger the link between the
problem (e.g., 2 + 3) and the answer (5)
Confidence Criterion
the point at which the child is confident to
state a retrieved answer

14. Model of Strategy Choice .7 .7 .5 .5
Associative Strength .3 .3 .1 .1

15. Arithmetic – Individual Differences
differences could be in associative strengths
differences could be in confidence criterion
three kinds of students
“high speed and accuracy” students
“perfectionist” students
“low accuracy” students

16. Mathematical Disabilities
6% of students labeled as having mathematical disabilities
less exposure to numbers before entering school
less conceptual knowledge of counting, arithmetic operations, and place value
less working memory capacity

17. Arithmetic – General Difficulties
understanding of principles
e.g., the inversion principle
a + b – b = ?
4-5 year olds can understand this if working with objects
But, with arithmetic problems, understanding doesn’t appear until 11 yrs of age

18. Arithmetic – General Difficulties continued
context effects
e.g., word problems
“Joe has three marbles. Tom has five more marbles than Joe. How many marbles does Tom have?”
Children often read this as “Joe has 3 marbles and Tom has 5” dropping the relational term
e.g., study of Brazilian children

19. + + + SAMPLE MATH “BUGS” I. Addition - What’s wrong??? 23 18 45 + + +
II. Subtraction What’s wrong???

20. MORE SAMPLE MATH “BUGS”
III. Rounding off What’s Wrong???
(to nearest tenth)
(to nearest tenth)
(to nearest hundredth)
(to nearest hundredth)
IV. Understanding Fractions (represent the shaded bars as a mixed number & a fraction) What’s Wrong???
1 1/ / / /12

21. The Need for Genuine Understanding!
Complex Arithmetic
The Need for Genuine Understanding!
Problem: Estimate the answer to 12/13 and 7/8. You will not have time to solve the problem with paper and pencil.
Answer Age Age 17
% %
% 37%
% %
% 15%
Don’t Know % 16%

22. The Need for Genuine Understanding! And Representational Fluency!
Math Problem Solving
The Need for Genuine Understanding!
And Representational Fluency!
Problem: “There are six times as many students as professors at this university.”
6s = p vs p = s
37% of freshmen engineering students at a major state university could not write the correct equation representing the situation.