Adding it up- Chapter 5 and 6, Chinese base ten and language, make a 10, base-10 blocks have quanity, but chips not.

Chapter 5 THE MATHEMATICAL KNOWLEDGE CHILDREN BRING TO SCHOOL

Early research on children’s understanding of the mathematical basis for

counting focused on five principles their thinking must follow if their counting is to be mathematically useful

1. One-to-one: there must be a one-to-one relation between counting words and objects;

2. Stable order (of the counting words): these counting words must be recited in a consistent, reproducible order;

3. Cardinal: the last counting word spoken indicates how many objects are in the set as a whole (rather than being a property of a particular object in the set);

4. Abstraction: any kinds of objects can be collected together for purposes of a count; and

5. Order irrelevance (for the objects counted): objects can be counted in any sequence without altering the outcome. 

Chapter 6 DEVELOPING PROFICIENCY WITH WHOLE NUMBERS

Adding it up:- 4 Chapter 4 THE STRANDS OF MATHEMATICAL PROFICIENCY

 

Adding it up-3, difficulty of zero

Adding It Up -2: Chapter 3 NUMBER: WHAT IS THERE TO KNOW? Page 97, xy, 2 3/2, f(x)

 

The handshake problem can be approached by bringing in ideas from other parts of mathematics. If the people are thought of as standing at the vertices of an eight-sided figure (octagon), then the question again becomes geometric but in a different way: How many segments (sides and diagonals) may be drawn between vertices of an octagon? The answer again is 28, as can be verified in the picture below.

As often happens in mathematics, connections to geometry provide a new way of approaching the problem: Each vertex is an endpoint for exactly 7 segments, and there are 8 vertices, which sounds like there ought to be 7 × 8 = 56 segments. But that multiplication counts each segment twice (once for each endpoint), so there are really half as many, or 28, segments.

In still another mathematical domain, combinatorics—the study of counting, grouping, and arranging a finite number of elements in a collection—the problem becomes how to count the number of ways to choose two items (people shaking hands) from a collection of eight elements. For example, in how many ways can a committee of two be chosen from a group of eight people? This is the same as the handshake problem because each committee

of two corresponds to a handshake. It is also the same as the octagon problem because each committee corresponds to a segment (which is identified by its two endpoints).

A critically important mathematical idea in the above discussion lies in noticing that these are all the same problem in different clothing. It also involves solving the problem and finding a representation that captures its key features. For students to develop the mathematical skill and ability they need to understand that seemingly different problems 

are just variations on the same theme, to solve the problem once and for all, and to develop and use representations that will allow them to move easily from one variation to another, the study of number provides an indispensable launching pad.

Adding It Up -1: Helping Children Learn Mathematics

Adding It Up: Helping Children Learn Mathematics

National Research Council, Division of Behavioral and Social Sciences and Education, Center for Education, Mathematics Learning Study Committee

National Academies Press, Nov 13, 2001 - Education - 460 pages

Adding It Up explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years.

The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. With examples and illustrations, the book presents a portrait of mathematics learning:

• Research findings on what children know about numbers by the time they arrive in pre-K and the implications for mathematics instruction.
• Details on the processes by which students acquire mathematical proficiency with whole numbers, rational numbers, and integers, as well as beginning algebra, geometry, measurement, and probability and statistics.

The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics.

See more discussion about Number line start from 87

NOTE: The finite decimals, also called decimal fractions, were first discussed by Stevin, 1585/1959