Key Concepts in Math Education

Levels of Number Understanding

Level 1

At this level, number figures are symbols linked to real-world objects.

Level 2

Children try to find correspondence between written figures and quantitative aspects of objects.

Level 3

Numbers, especially single-digit numbers, represent counts of objects. Two-digit numbers are not yet seen as composed of constituent figures.

Level 4

Two-digit numbers systematically represent the total quantity, but each digit is treated as a separate number without understanding place value. For example, in 16, '6' might represent six objects and '1' might represent one object. Alternatively, '6' might represent sets of 6, and '1' might relate to a single unit or group.

Level 5

Each digit in a two-digit number represents an amount determined by its place or position (place value).

Traditional Calculation Technique

(Framing and Subsequent Subtraction)

This technique, using the rule of zeros, leads to the final algorithm.

Example: Dividing 15,580 by 63. This involves subtracting multiples of 63 (e.g., 200 times 63, 40 times 63, 7 times 63), with a remainder of 19.

Indirect Length Measurement

Using an intermediary (Didactic Variable) when comparing objects, focusing on the ratio between measuring units (bands).

Measurement Strategies

Transporting the small band pattern, marking the big band using the small one, and repeatedly applying the small band pattern.

Distributive Property Context

Building with squares (e.g., area model).

Didactic Variables

Constancy of numbers in a grid, size of numbers, etc.

8x7, 8x5, 6x5, 7x6 ....................
7(8+6) + 5(8+6)
8(7+5) + 6(7+5)
.....................
(7+5) x (8+6) and backwards

Objectives: Grouping and Place Value

Using the number of tens and hundreds that constitute a number for a collection, using grouping by 1s and 100s for counting.

Didactic Variable (Chalks/Visuals)

Lack of order in intermediate units.

Types of Problems

• Compositions of measures
• Transformations of measures
• Static relationships between measures
• Two-measure problems
• Composition of transformations
• Transformations and composition of relative states