Mathematics Education

EMTH 215

Chapter 5: Developing Understanding of Numeration

(Summary Notes)


Themes--Major Ideas


Learning Objectives


After studying Chapter 5, the student will be able to:

Chapter Overview

The main thrust of Chapter 5 is to present ideas for helping students develop number sense and an understanding of the Hindu-Arabic numeration system. Numerous discovery-oriented activities are described throughout the chapter to help students construct a sound understanding of our numeration system.

In the chapter, number systems and numeration systems are characterized with detailed consideration given to five characteristics of the Hindu-Arabic system of numeration. These are: base ten, positional or place value, multiplicative principle, additive principle, and zero as a place holder.

A number of activities in the chapter are about developing place value understanding. Particular attention is given to grouping activities, grouping by tens, developing two-digit numbers, introducing base-ten blocks, and using place value mats. A distinction is made between proportional and non-proportional materials used to model numbers. Tasks to assess place value knowledge are described. The research of Kamii and Joseph (1988) and Ross (1986, 1989) on place value is considered.

Teaching considerations about the development of relationships among large numbers are presented. These focus on constructing, comparing, reading, and writing large numbers, number periods, magnitude of numbers, counting to the thousands period and beyond, writing consecutive numbers, writing numbers in expanded notation, rounding numbers, and estimating numbers. For each of the above ideas, appropriate activities are described.

A final section of the chapter focuses on ancient numeration systems, namely, the Babylonian, Egyptian, Mayan, and Roman systems. A study of these by upper elementary and middle school students can help develop a better understanding and appreciation of our own system.



Some extra notes about Numeration--stimulated by what is in Chapter 5:


"A numeration system is a system that enables one to record and thereby communicate one's ideas about number." We need to understand the structure of our numeration system.


Understanding Place Value

  1. grouping (e.g., in fours, how many fours and how many left over--use beans and cups etc. or unifix cubes). Count objects into groups to determine how many groups and count from the groups to determine how many objects. Whatever groupings the children are doing they should be talking about it in their social groups ("tell what you have done").

  2. grouping in tens, children can understand the system of numerals (e.g., 19 is followed by 20 and not 110) without knowledge of number names. Some materials that are useful for grouping by tens are:
  3. place value--proportional materials (e.g., pre-bundled sticks, base ten materials) beginning with concrete--beans or buttons--one-to-one correspondence exists between the material and the number being represented--moving to popsickle sticks, then to base ten rods etc. with the scoring on them for verification, then to unscored rods. Place value non-proportional materials--e.g., centimetre cubes on a place value mat--position of cube on the mat is important; colored chips on a chip trading board; abacus, money.

  4. developing two-digit numbers--representative experiences (e.g., unifix or multilink cubes; popsickle sticks or toothpicks and bands; interlocking centimetre cubes)--children should physically make two-digit numbers up to 100 and especially the 100 square.

  5. introducing Base-Ten Blocks--children can be told that these blocks are ones that don't come apart--easier to count with--demonstrate. Have the children represent numbers with the Base Ten material--e.g., 57 is 5 ten rods and 7 unit cubes--encourage diversity; 4 ten rods and 17 units--look for different modeling representations of the same number. Ask children to represent in the simplest way possible a certain number or show children pictures of base-ten blocks and have them write the numeral.

  6. place value (or organizational) mats should not be headed tens and ones as a ten rod in the tens column actually means 100. Initially they should have no column headings so that children can use the rods and ones, then they can have the headings but use only centimetre cubes in each column as the heading now says what the cubes are units of. "The introduction of a place value mat signifies a move from the concrete type of modeling to the semi-abstract type."

  7. introducing non-proportional materials--trading game in whatever base desired, using centimetre cubes; also the chip trading game, with colours representing numbers. Both games can be adding or subtracting games. An abacus and money can also be used to develop an understanding of place value.


Stages in place value development
(1) association of 2-digit numerals with the quantity they represent (e.g., 28 means the whole amount)
(2) identification of positional names but not know what each digit represents (e.g., 28 is 8 ones and 2 tens)
(3) identification of face value of digits in a numeral (e.g., in 28 the 2 is 2 tens and the 8 is 8 ones, but maybe not know that the 2 tens is twenty)
(4) a transitional stage when true understanding of place value is constructed
(5) level of understanding the structure of the numeration system

Three-Digit Numbers


Developing Number Relationships


Understanding Large Numbers


Expanded notation

In two or more parts in primary grades (e.g., 34 is 30 + 4 or 3 tens + 4). In upper elementary grades expanded form (e.g., 674 = (6 x 100) + (7 x 10) + (4 x 1). Gr. 6 & 7 exponential notation [e.g., 4692 = (4 x 10 x 10 x 10) + (6 x 10 x 10) + (9 x 10) + 2 which in turn equals (4 x103) + (6 x 102) + (9 x 10) + 2]

Rounding numbers

Number approximations--read rules for rounding and look at activities in chapter.

Estimating

Give students opportunities to estimate (e.g., beans in a jar; parents at the school concert)

Consolidating Number Skills


Other Numeration Systems

Students could study another numeration system and then invent one of their own.


To Think and Talk About

  1. Distinguish between number systems and numeration systems.
  2. Write three statements about what the symbol "10" means to you. What should young students learn about the symbol "10?"
  3. 85 means eighty and five and 85 means 8 tens and 5 ones. Which statement shows greater understanding of 2-digit numbers? Do the statements show understanding of place value?

Relevant NCTM Curriculum Standards

Grades K-4--Standard 6--Number Sense and Numeration; pages 38-40.

Grades 5-8--Standard 5--Number and Number Relationships; pages 87-90.

What we will try to do in class today:

  1. We will begin with a counting activity in a base that is NOT Base 10--in order to extrapolate the salient characteristics of a numeration system--rules that will apply to all numeration systems.
  2. We will play the game "Race to 100" and the reverse "Race to Zero."
  3. We will play the Skemp activity "Number Targets."
  4. We will do some basic activities with Base 10 materials (e.g., representing two, three and four digit numbers in different ways--see activity 5-1 and figure 5-7).
  5. You can look at the section on Thinking and Writing about Numbers (p. 97).
  6. We will look at how to demonstrate expanded notation and how to write large numbers.
  7. Everyone should explore activities 5-7; 5-8; 5-9; 5-10; 5-11.
  8. Everyone should read through the section on Consolidating Number Skills (pp. 102-104) as some excellent activities are listed.