“Number sense, like common sense is difficult to define or express simply.  It refers to an intuitive feel for numbers and their various uses and interpretations.  Number sense also includes the ability to compute accurately and efficiently, to detect errors, and to recognize results as reasonable.  People with number sense are able to understand numbers and use them effectively in everyday living.  Good number sense is also recognizing the relative magnitudes of numbers and establishing referents, or benchmarks, for measures of common objects and situations in their environments.” -Helping Children Learn Mathematics (page 138).

The Number and Operation Standard of the Principles and Standards for School Mathematics (NCTM) includes several concepts and skills related to beginning number sense.

http://standards.nctm.org/docment/appendix/numb.htm

Even though it is difficult to define, what we do know is good number sense is necessary for future success in math.  Some of the skills young students must develop are:

Prenumber Concepts:

1.    Classifying- The ability to sort objects based on attributes.

2.    Patterns- The ability to create, construct, and describe patterns.

3.    Comparisons- The ability to compare the amount of objects.

4.    Conservation- The ability to conserve the value of a number.

           http://coe.sdsu.edu/eet/Articles/piaget/index.htm

5.    Group Recognition- The ability to recognize how many objects is in a small group of objects (5 or 6) and name them properly. (This skill to “instantly see how many” in a group is called subitizing).

Number Concepts:

Number sense refers to a person's general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems.

1.    The ability to count objects.  Counting processes reflect various levels of sophistication, beginning with rote counting and eventually leading to rapid skip counting forward and backward.  Counting skills are often refined throughout our lives.

2.    Competence with and understanding of the numbers 0 through 10 are essential for meaningful later development of larger numbers.

3.    Understanding relationships between sets of objects (such as more, less, equal).

4.    The ability to say and write the digits correctly.

5.    Understanding of other math notation (+, -, =).

6.    The ability to use cardinal, ordinal, and nominal numbers.

Place Value:

1.    Understand how our numeration system works.

a.    Place value: the position of a digit represents its value.

b.    Our system is organized on a Base 10 system.  Once you reach ten in a place you need to trade and make a new collection.

c.    Zero (0) is a symbol in our system and allows us to represent the lack of something a collection in a number.  For example, 405 means there is nothing in the tens collection or column.

d.    Additive Property- Numbers can be summed with respect to place value.  For example, 123 names the number that is the sum of 100 + 20 + 3.

2.    The ability to count larger numbers.  (Note- when learning to name larger numbers – the teens often prove to be the most difficult because they do not match the rest of our number names.  In fact in Japan the name for 13 translates to something like one ten and 3 and 23 translates to something like two tens and 3.  Therefore, it may be better to skip the teens and work on higher numbers at first and then come back to the teens later.)

a.    See patterns like adding 10 doesn’t change the value of the ones column.

b.    Trading and renaming- moving from one collection to another is one of the most important skills for young students to acquire.  The pattern of what happens after 9 (or 19, 29, etc) can be very difficult for students to get, but they are critical for future computation.

3.    The ability to read and write numbers.

4.    The ability to manipulate numbers to make mental math and estimation possible. 

5.    To recognize that sometimes numbers are exact and others are approximations.   (A few examples of an approximation would be attendance at a football game, distance to a planet).  Measurement is always an approximation because it can always be made more precise.  If a pencil is 5 inches long, on a closer measurement it might be seen as 4 ¾ inches long, on an even closer look it might be seen as 4 5/8 inches long.  This is an important idea that leads into the need for fractions and decimals in our number system.

I took most of the ideas for this Question and Answer from Chapters 7 and 8 of Helping Children Learn Mathematics (7^th Edition) by Reys, Lindquist, Lambdin, Smith, and Suydam.  ISBN 0-471-15163-7