What characterizes good learning in mathematics and how can adults facilitate such learning?

What characterizes good learning in mathematics and how can adults facilitate such learning?

Counting and extending the number system

A numeral system (or system of numeration) is a writing system for expressing numbers? that is a mathematical notation for representing numbers of a given set? using graphemes or symbols in a consistent manner. It can be seen as the context that allows the numerals "11" to be interpreted as the binary symbol for three? the decimal symbol for eleven? or as other numbers in different bases.

Ideally? a numeral system will:

Represent a useful set of numbers (e.g. all whole numbers? integers? or rational numbers)

Give every number represented a unique representation (or at least a standard representation)

Reflect the algebraic and arithmetic structure of the numbers.

For example? the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits. However? when decimal representation is used for the rational or real numbers? the representation may not be unique: many rational numbers have two numerals? a standard one that terminates? such as 2.31? and another that recurs? such as 2.309999999... . Numerals which terminate have no non-zero digits after a given position. For example? numerals like 2.31 and 2.310 are taken to be the same? except for some scientific contexts where greater precision is implied by the trailing zero. Numeral systems are sometimes called number systems? but that name is misleading? as it could refer to different systems of numbers? such as the system of real numbers? the system of complex numbers? the system of p-adic numbers? etc. Such systems are not the topic of this article.

Extending

In your study of numbers? you have focused on operations (+? -? 3? and 4) with whole numbers? fractions? and decimals. In this unit? you will learn about some important new numbers in the number system. Suppose you start with a number line showing 0? 1? 2? 3? 4? and 5.

language issue in mathematics

Algebra is written in a symbolic language that is designed to express mathematical thoughts. This website describes a text for a course that emphasizes language skills. Most math courses concentrate on what is said? this one concentrates on how is it is said so you can understand what it says. This course emphasizes all the ways that mathematics is used to express thoughts in algebra and higher-level classes. It will make you far better at math regardless of your current level.

Mathematical language skills include the abilities to read with comprehension? to express mathematical thoughts clearly? to reason logically? and to recognize and employ common patterns of mathematical thought. Even as an experienced mathematics professor? teaching the "Language of Mathematics" allowed me to learn new ways to explain to students the nature of mathematics? its interpretation as a language? and its notation. Esty's text provides a wonderful exploration of the deep issues involved in understanding and teaching even the most elementary mathematical concepts. Working with the text has and will ...