The first number systems began when the one-to-one matching that precedes counting was no longer efficient, creating the need to represent the concept of quantity with symbols. They were inelegant and difficult to use since they lacked the essentials needed for a number system, such as a limited number of symbols, repetitive patterns, and open-endedness (the ability to represent any quantity). Some had simple rules, but an inordinate amount of symbols which taxed the human memory. Some had a manageable number of symbols, but many rules. Over many years, the current base-ten positional numeration system developed and it meets the necessary criteria for an efficient number system (Schwartz, 2008).
However, it is complex. For example, the mere act of counting involves the following elements:
1. Locate a collection of objects on which to act
2. Understand that there is such a thing as quantity
3. Utter a correct sequence of words
4. Identify a matching object from the collection for each word uttered
5. Omit no objects in the collection
6. Identify no object more than once
7. Stop uttering words when each item in the collection has been identified
8. Declare that the last word uttered is the “count” of the characteristics of the objects themselves
9. Understand at the end of the process that the last word uttered establishes a characteristic of the collection rather than a characteristic of any specific object (Schwartz, 2008).
Our numeration system uses the ideas of basic digits (symbols), base, and location. The genius of using a base amount as a grouping constant is that the groups themselves become the countable units (Schwartz, 2008). Units are counted, then groups are counted (of the base amount), and then groups of groups, etc. The basic rule is to form a countable group whenever you have the base amount. A repetitious pattern is formed that allows numbers of any size to be represented, while the number of symbols to memorize remains manageable (Schwartz, 2008). The idea of location allows for the reuse of the symbols in a new location within the numeral. The location of the symbol yields information regarding the size of the group being counted. These three ideas (basic digits, base, and location) are the powerful ideas behind our current numeration system and permit the creation of representations of any size quantity. The ideas of quantity and base are abstract concepts, while the ideas of symbols and location (positional notation) are representations of concepts (Schwartz, 2008).
Teachers must possess the mathematical knowledge to understand how students learn mathematics as well as an understanding of the trajectory of mathematics content. Understanding the trajectory enables us to effectively plan and make instructional decisions to impact important mathematical learning. This continuum
contains K-3 Number and Operations in Base Ten standards, Number and Operations – Fractions standards for grade 3, Counting and Cardinality standards for K, and Number and Operations and Algebraic Thinking standards K-3.