Developing an Understanding of the Number System

Dr Jenny Young-Loveridge

Education Studies Department, University of Waikato, Private Bag 3105, Hamilton, New Zealand

(e-mail: educ2233@waikato.ac.nz)

Paper presented at the Australian Association for Research in Education – New Zealand Association for Research in Education Conference on Research in Education, 29 November

– 2 December 1999, Melbourne, Australia

(Paper ID: YOU99550)

Abstract

This paper presents the findings of two research projects which explore children’s understanding of the number system. The first study looked at nine-year-old (Year 4) children’s understanding of place value and other related concepts. Some of the notable findings include the following: only about half of the nine-year-olds were able to demonstrate a satisfactory understanding of place-value; approximately one third of the nine-year-olds counted on by ones to increment a number by ten; relatively few (less than one third) children could do 2-and 3-digit written subtraction problems with vertical presentation but more were successful when the problem was presented orally using a money context; some children were successful with 2- and 3-digit written addition problems with vertical presentation, despite having little, if any understanding of place value. The second study looked at seven-year-olds’ understanding of the number system with a view to developing diagnostic assessment tools. The findings of both studies are considered in relation to a developmental framework for the acquisition of numeracy which shows how children’s understanding about the number system becomes increasingly sophisticated as their thinking develops.

Earlier this decade, the Third International Mathematics and Science Study (TIMSS) showed disappointing results for New Zealand children at the middle primary school level (Years 4 and 5). Particular weaknesses were identified in the area of Whole Numbers, Measurement, and Fractions and Proportionality. In 1997 the Mathematics and Science Taskforce was set up to advise the Government and the Ministry of Education on how to improve the teaching of Mathematics and Science in New Zealand schools. The Taskforce identified Place-value Understanding as an area of particular concern. A paper on the development of place-value understanding was prepared for a Mathematics Education Research seminar held in June 1998, to address topics identified by the Taskforce as having high priority (see Young-Loveridge, 1999a). The first of the two studies reported here commenced in the final term of that year to explore children’s understanding of place value at the Year 4 level.

Late in 1998, a new literacy and numeracy goal for young New Zealanders was announced by the Minister of Education. The goal stated that "by 2005, every child turning 9 will be able to read, write and do maths for success". It was not clear just what "doing maths for success" meant or how it would be measured. However, a crucial aspect of this must surely be a good understanding of the number system and the ability to use that knowledge to solve problems within a variety of contexts (including measurement, statistics, algebra, and

geometry). The announcement of the goal led the following year to the research being extended downwards to younger children, with the aim of developing assessment tasks which could help teachers to identify children who might need extra help in those early years of school, in order to meet the goal by their ninth birthday. Gilmore (1999), reporting on the Submissions to the Green Paper on Assessment for Success in Primary Schools, found strong support for the idea that additional diagnostic tools need to be developed for use in primary schools.

As part of the paper on the development of children’s place-value understanding, a developmental framework was constructed to show how children’s understanding about the number system becomes increasingly sophisticated as their thinking develops (see Young-Loveridge, 1999a, 1999b). The framework was developed from the work of several other researchers (eg, Karen Fuson, Constance Kamii, Lauren Resnick, Leslie Steffe), and integrates many important features of their models. The framework provides teachers with a means of understanding the developmental progression which occurs for children in moving from beginning to competent mathematical thinking, and could assist them in helping children "do maths for success". Having a framework for how children’s thinking develops could help teachers plan appropriate learning experiences for the classroom which could assist their pupils in reaching the numeracy goal. Research in the United States and Australia has shown that teachers who were given this kind of framework were better able to help their pupils build on their mathematical thinking (Carpenter, Fennema & Franke, 1996; Wright, 1997). The framework consists of four stages, each characterised by a major shift in ways of thinking about numbers (see Figure 1). There is room within the framework for expansion and elaboration to include other components of mathematics, such as decimal fractions.

This paper reports on the data gathered from the two groups of children which looks at their understanding of the number system. Although many of the tasks given to the two groups were different because of their age difference, the framework provides a common link between them.

Children

The participants in this project were two groups of children from a decile 5 school in a large regional city in the North Island of New Zealand. The first group consisted of a complete cohort of 97 children nearing the end of Year 4 (age range: 8.5 to 9.9 years; average age: 9.0 years), while the second group consisted of 81 children in the middle of either Year 2 or Year 3, aged 7.0 to 8.1 years (average age = 7.5 years). The ethnic composition of the two groups was similar, with slightly more New Maori children in the younger group than the older one.

Composition of the two groups

Year 4 Years 2/3

N % N %

Gender

Boys 38 39 34 42

Girls 59 61 47 58

Total 97 100 81 100

Ethnicity

NZ European/Pakeha 60 62 47 58

NZ Maaori 16 16 21 26

Pacific Islands 6 6 5 6

Asian 8 8 5 6

African 6 6 3 4

Other 1 1 0 0

Decile 5 school with 600+ pupils

Findings

The numbers and percentages of children who could do each of the tasks is shown in the Appendices.

Determining Framework Levels

The Year 4 children were categorised according to which of the four levels of the framework they seemed to fit best. Below is a description of each level and the percentage of Year 4 children who were at that level.

Level

1 Unitary Concept

Building knowledge of number word sequences, counting processes, part-whole relationships, numerals using a unitary (by ones) strategy (16%).

2 Ten-structured concept

Partitioning of multi-digit number into a whole decade and extra ones, and groupings of ten ones used to solve problems more efficiently (24%)

3 Multi-unit Concept

Recognition that units of tens and ones can be counted separately (ie, place-value understanding), and can be traded and exchanged (eg, 10 ones for one 10, or one 10 for 10 ones) (20%)

4 Extended Multi-unit Concept

Consolidation of place-value understanding and generalisation to units larger than ten. Recognition that units can be any power of ten (40%).

The seven- to eight-year-olds seemed to fit into one of the first three levels identified for the Year 4 children. In addition to these three main levels, three transitional levels intermediate between each of these main levels were added to the framework to enable finer discriminations to be made among the younger children.

Level

0 Needs to learn a lot more about the basics such as counting, reading and writing numerals, and working with part-whole relationships with small quantities (3%)

1 Unitary Concept

Has a good grasp of the basics for small numbers but still works by ones for problems with multi-digit numbers (27%)

1.5 Is just beginning to use composite units (units larger than one) such as tens for some problems with multi-digit numbers (10%)

2 Ten-structured Concept

Makes good use of composite units to solve problems with multi-digit numbers (ie, has good ten-structured understanding) (11%)

2.5 Is just beginning to understand the idea of place value and the significance of position for a digit within a multi-digit number (11%)

3 Multi-unit Concept

Has a reasonably good understanding of place value and the meaning of an individual digit within a multi-digit number according to its position (38%)

Nine-year-olds

The following section outlines the main findings from the study with Year 4 children.

• No more than about half of the nine-year-olds were able to demonstrate or explain place-value understanding satisfactorily

• Approximately one third (32%) of the children counted on by ones to increment a number by ten

• Relatively few children could do 2- and 3-digit subtraction with renaming (29% & 12%, respectively)

• Children were better at forming sets using units of different denominations when the context was money ($1 coins, $10 & $100 notes) than when working with beans (loose beans, and beans in bags of 10 & boxes of 100)

• Many children (82%) knew immediately that 10 and 6 more was 16, but almost a third (29%) of them counted on by ones to work out the number that is ten more than 7 (a problem of going over the decade break)

• Some children could do multi-digit addition with renaming using the written algorithm, yet had little, if any, place-value understanding

• Some children used effective strategies for addition and subtraction with money (verbal problems), but made errors on corresponding problems presented as written computation (vertical presentation)

Seven- to eight-year-olds

These children were an average of 18 months younger than the Year 4 children, and this was reflected in their performance. For example, two-thirds (68%) of the nine-year-olds could rote count by tens to two hundred, compared to just under half (44%) of the younger children. Virtually all (89%) of the nine-year-olds could produce $125 using $1 coins, and $10 and $100 notes, compared to only 45% of the younger children. However, there were many very capable seven- and eight-year olds whose understanding of the number system was considerably more advanced than that of the least capable nine-year-olds, consistent with other studies which have shown enormous diversity in levels of understanding among any particular age cohort of children.

The following section discusses some of the key assessment tasks which have emerged from this study. Several tasks proved useful in identifying children’s level of understanding about the number system.

Adding a single-digit number to a whole decade

This task was presented as a hypothetical situation involving addition of two groups of lollies. For example:

[Let’s pretend that you’ve got some lollies.] What if you had 10 lollies, and then I gave you 7 more lollies. How many lollies would you have altogether?

In the second task, the children were told they had 20 lollies, then were given 8 more lollies. Children’s responses to this task fell into two main categories. Those who understood the ten-structured nature of the number system immediately responded 17 or 28, respectively. Those children who did not respond immediately seemed to use some form of counting by ones strategy (see Appendices for individual profiles). The most efficient of these counting strategies was counting on mentally from the whole decade. Other children counted on, using physical materials, most commonly their fingers. A small group of children began counting from one, and counted the whole decade as well as the single-digit quantity being

added to it. It was interesting to note that adding on to 20 was easier than adding onto 10 (68% cf. 54% used ten-structured understanding), supporting the idea that numbers in the "teens" are in fact more difficult to conceptualise than the decades from 20 onwards, because of their irregularity. Virtually all of the children with solid place-value understanding gave an immediate response, indicating that they had used ten-structured understanding, whereas virtually all of the children working at the unitary level used some kind of counting strategy to find the answer. The advantage of this task over those deliberately designed to check for use of a particular strategy (eg, tasks used in the SENA assessment for Count me in Too; see Curriculum Support Directorate, 1999) is that the children seems to use the most advanced strategy at their disposal. Children who can’t recall number facts immediately call upon counting strategies. Children who can’t count mentally produce their fingers as physical materials to help them with counting. This task is very simple and quick to give, and yields valuable information about children’s understanding. It is effectively the reverse of "unique partitioning" by which a multi-digit number is partitioned into a whole decade and left-over ones (see Young-Loveridge, 1999b).

Constructing Quantities with different Denominations of Money

Virtually all of the children with ten-structured understanding and with a good understanding of place value were able to use the most efficient strategy to create a particular quantity (ie, used the largest denominations in order to construct the quantity with the fewest items of money; (see Appendices for individual profiles). Between a quarter and a third of children at the unitary level could use larger denominations of money to produce a two-digit quantity ($16 & $31) more quickly and accurately. However, very few were able to produce a 3-digit quantity ($125) using larger denominations. A common response on the latter task was to produce a $100 note and count out 25 $1 coins (6 children). For those children who could not produce the 2-digit quantities correctly, a follow-up counting task was given to them. They were asked to say "How much money altogether?" in response to a $10 note and 4 $1 coins, then 2 $10 notes and 3$ coins. Only children at or below the unitary level needed to be given this additional task. Three children simply counting the number of different items ignoring their different values, and came to the answer "5" in both cases. Two other children counted by ones to 10 for the $10 note, before counting on for the $1 coins.

Linking individual digits in a multi-digit numeral with quantities

This task was an adaptation of Sharon Ross’s task with the buttons where the child first counts a quantity, then circles the items which correspond to each of the digits in a multi-digit numeral. Instead of wasting valuable time drawing buttons before being asked to circle the buttons corresponding to each digit, an array of small boxes was printed on paper and presented to the child. The boxes were arranged in rows of ten, with left over "ones" boxes in the bottom row. The children were first asked to "count the boxes to determine how many there were and to write that number on the line. The interviewer circled the left-most digit and asked the child to show how many boxes does this part of [14]] mean? Can you circle them with your pencil". Children who did not understand what was expected of them were then shown a model which began with:

And ended with:

Children’s strategies were observed, and for those who were using at least a ten-structured strategy, a 3-digit problem was also presented. A large space was left below ten rows of ten boxes to distinguish 100 boxes from "leftover" tens and ones. The advantages of this technique were that it enabled the presentation of materials to be standardized for each child, it gave children with ten-structured understanding an opportunity to use that knowledge to complete the task more quickly, and it allowed place-value understanding for 3-digit numbers to be explored in addition to the 2-digit numbers.

Positional Value

As well as using money for constructing quantities using different denominations ($1, $10, $100) money was also used to explore aspects of place value. Children were given a series of 3-digit numerals which consisted of the digits 1, 2, and 3 in different combinations. They asked to show the value of a particular digit in a particular position using the different denominations of money. A common response by children at the unitary level of understanding was to use only the $1 coins. This meant that they were correct for the two problems involving ones (33% & 42%). Quite a substantial proportion of children at the unitary level (38%) could also show the value of 1 in the "tens" position. This task

differentiated children with good place-value understanding (level 3) from those with ten-structured understanding (level 2). All of the children in the highest group responded correctly on all problems involving $10 and $100 notes, whereas children with ten-structured understanding were only slightly better than those with unitary understanding (see Appendices for individual profiles).

Skip Counting

The pattern of results for the three groups suggest that skip counting must be mastered before children can progress towards ten-structured understanding. Virtually all of the children with ten-structured understanding and good place-value understanding could count by tens to at least 100, whereas only about half of the children at the unitary level could do this. Counting by fives was slightly harder than counting by tens but the majority of children in the two higher groups could do this.

Recalling Number Facts

The children with good place-value understanding also had better number fact knowledge than the other two groups. Two main groups of number facts were explored: knowledge of the combinations which make ten, and doubles for digits between 4 and 9. Again, children were asked to imagine the quantities. For example:

Let’s pretend you’ve got some lollies. Say you have 1 lolly, how many more lollies would you need to have 10 lollies altogether?

Performance varied according to the size of the two quantities which together made ten. The easiest problems were "5 and what make 10?" and "1 and what make ten?" Few of the children at the unitary level could recall these particular number facts, apart from the problem involving "5", which half of the low group and virtually all of the other two groups could get. Knowledge of doubles was also explored, showing that doubles for four and five were known by virtually all children at or above level 2, and by approximately half of the children at the unitary level.

Mental Computation with Ten

These tasks involved operations with ten or five. For example:

How much did I spend altogether, if I spent 15c on one lolly and 10c on another lolly?

How much change would I get, if I used a 50c piece to pay for a 10c lolly?

If 4 children each spend 5c, how much money is that altogether?

If there were 10 biscuits to share among 5 children, how many biscuits would each child get?

Virtually all of the children with good place-value understanding were able to solve problems involving addition or subtraction of 10 by using composite units (units larger than one), whereas few if any of the children at the unitary understanding could do this. Between a third and half of the children with ten-structured understanding could add or subtract ten, without counting by ones (see Appendices for individual profiles).

Future Research

The data presented here give a good picture of the "typical" or "average" child in the two age groups studied. The next step is to use these tasks with children from disadvantaged backgrounds who may need a lot of help to meet the Government’s goal for nine-year-olds. Currently interviews are under way with seven- and eight-year-old children in three schools in low income areas (ie, deciles 1 and 2). These children are the neediest and most vulnerable of our children and the extent to which their performance differs from that of children in a decile 5 school will show areas which need much more attention. After more than 130 interviews, the impression given is that few of these children have ten-structured understanding, let alone place-value understanding. Just how "doing maths for success" will be defined is not clear, but in order for every nine-year-old to meet the Government’s goal, the criteria for "success" will need to be set extremely low, or massive efforts will need to be put into intervention over the next five years. Otherwise it seems likely that substantial numbers of our children will fail to meet the numeracy goal by 2005.

References

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Appendices

Numbers and Percentages of Year 4 Children who could do each of the tasks using particular strategies

% Lev 1

% Lev 2

% Lev 3+

N % (n=16) (n=23) (n=58)

1. Writing Numerals

16 95 98 100 96 98

31 95 98 94 96 100

214 80 83 50 78 93

308 88 91 63 91 98

2 005 73 75 31 65 91

7 000 87 90 69 91 97

9 867 48 50 13 26 67

48 315 43 44 13 26 60

60 001 38 39 38 30 43

90 000 59 61 31 48 74

2. Linking Multi-digit Numerals to Objects

16 66 68 13 52 90

31 56 58 0 30 85

125

2=20 49 51 6 22 76

1=100 47 49 0 26 71

267

6=60 32 33 0 13 50

2=200 31 32 0 13 48

3. Reading Numerals

16 95 98 88 100 100

31 96 99 94 100 100

125 91 94 69 96 100

267 92 95 81 91 100

308 88 91 56 91 100

2 005 71 73 38 52 91

3 481 68 70 25 52 90

9 867 69 71 25 52 91

7 000 90 93 69 91 100

10 476 63 65 19 48 85

60 001 58 60 25 39 78

48 315 65 67 19 48 88

90 000 70 72 31 65 86

4 628 531 9 9 0 0 15

8 000 000 40 41 6 44 52

4. Verbal Explanation for Place Value

1 in 14 62 64 0 17 71

2 in 267 49 51 13 13 74

6 in 267 45 46 6 4 74

3 in 3 481 43 44 6 9 69

4 in 3 481 45 46 13 4 72

8 in 3 481 42 43 6 4 71

% Lev 1

% Lev 2

% Lev 3+

N % (n=16) (n=23) (n=58)

5. Forming Sets using Grouped Objects

(Beans in bags of 10 & boxes of 100)

31 by tens & ones 74 76 19 87 90

by ones 11 11 44 0 5

125 by hundreds, tens & ones 34 35 6 13 52

by tens & ones 34 35 0 57 36

267 by hundreds, tens & ones 38 39 6 17 57

by tens & ones 24 25 0 35 28

6. Mental Computation Up & Over 10, 100, 1000

6 + ? = 10 number fact 53 55 25 39 69

by counting 38 39 56 57 28

2 + ? = 10 number fact 51 53 19 22 74

by counting 38 39 69 61 22

7 + ? = 10 number fact 57 59 38 39 72

by counting 36 37 50 57 26

1 + ? = 10 number fact 81 84 38 78 98

by counting 11 11 38 22 0

10 + 6 = number fact 82 85 38 87 97

by counting 13 13 56 9 3

40 + ? = 100 number fact 52 54 19 26 74

by counting 21 22 0 39 21

30 + ? = 100 number fact 49 51 13 22 72

by counting 20 21 6 39 17

200 + ? = 1000 number fact 42 43 13 4 67

by counting 9 9 0 9 12

% Lev 1

% Lev 2

% Lev 3+

N % (n=16) (n=23) (n=58)

7. Using Place Value with a Calculator

198 - remove 9 42 43 0 4 71

214 - remove 2 49 51 0 9 83

3 481 - remove 4 39 40 0 4 66

9 867 - remove 9 38 39 0 4 67

48 315 - remove 8 23 24 0 0 40

8. Money

$31 90 93 69 96 98

$125 86 89 63 83 98

$267 80 83 38 74 98

9. Mental Computation with Money

75c + 15c 60 62 31 39 79

$3.25 + $2.85 47 449 19 17 69

50c - 35c 36 37 13 22 50

$5.00 - $3.95 17 18 6 4 26

10. Computation using TO & HTO Boxes

31 + 16 52 54 6 39 72

75 + 15 47 49 0 30 69

52 - 31 54 56 0 44 76

50 - 35 21 22 0 4 35

236 +142 29 30 0 4 48

325 +285 23 24 0 9 36

357 - 123 33 34 0 9 53

500 - 395 13 13 0 4 21

11. Single-digit Sums & Differences

9 + 8 = number fact 51 53 0 30 76

by counting 31 32 63 52 16

8 + 5 = number fact 33 34 6 13 50

by counting 47 49 75 61 40

5 + 9 = number fact 55 57 13 26 79

by counting 28 29 69 39 17

16 - 9 = number fact 17 18 0 4 28

by counting 28 29 25 35 28

14 - 8 = number fact 25 26 0 13 38

by counting 31 32 13 44 33

15 - 9 = number fact 27 28 6 17 38

by counting 23 24 19 30 22

% Lev 1

% Lev 2

% Lev 3+

N % (n=16) (n=23) (n=58)

12. Written Computation (vertical)

75 + 15 75 77 44 83 85

325 + 285 54 56 31 65 59

50 - 35 28 29 6 13 41

500 - 395 12 12 0 0 21

13. Rote Counting by:

Tens to 200 66 68 31 65 79

to 100 28 29 56 35 19

Hundreds to 2000 23 24 0 4 38

to 1000 37 38 31 30 43

14. Place Value Blocks

75 + 15 60 62 19 39 83

325 + 28 50 52 6 22 76

50 - 35 31 32 6 13 47

500 - 395 22 23 0 13 33

% Lev % Lev % Lev

1 2 3+

N % (n=16) (n=23) (n=58)

15. Incrementing by:

one

7 93 93 81 93 100

16 92 95 75 93 100

31 91 94 75 87 100

214 89 92 63 87 100

1 321 83 86 44 74 100

10 476 82 84 44 70 100

ten

7 76 78 31 65 97

16 74 76 31 57 97

31 70 72 19 48 97

214 57 59 13 26 85

1 321 52 54 6 22 79

10 476 50 52 13 26 74

Increment by ten using counting on by ones

31 32 31 65 19

one hundred

7 42 43 0 9 67

16 41 42 0 4 69

31 42 43 0 4 71

214 46 47 0 4 78

1 321 39 40 0 0 67

10 476 38 39 0 0 66

one thousand

7 37 38 0 0 64

16 36 37 0 0 62

31 36 37 0 0 62

214 35 36 0 0 60

1 321 38 39 0 0 66

10 476 30 31 0 0 53

ten thousand

7 27 28 0 0 47

16 29 30 0 0 50

31 24 25 0 0 41

214 27 28 0 0 47

1 321 26 27 0 0 45

10 476 25 26 0 0 43

one hundred thousand

7 18 19 0 0 31

16 18 19 0 0 31

31 18 19 0 0 31

214 19 20 0 0 33

1 321 12 12 0 0 21

10 476 12 12 0 0 21

Multi-digit Addition & Subtraction with Renaming

Kind of Problem 75+15 325+285 50-35 500-395

Verbal Money Problems 62 48 37 18

TO & HTO Boxes 48 24 22 13

Written Computation

(Vertical Presentation) 77 56 30 12

Place Value Blocks

(Horizontal Presentation) 60 53 33 23

Composition of the Sample

N %

Gender

Boys 38 39

Girls 59 61

Total 97 100

Ethnicity

NZ European/Pakeha 60 62

NZ Maori 16 16

Pacific Islands 6 6

Asian 8 8

African 6 6

Other 1 1

Language

Native Speakers of English 76 78

Non English Speaking Background 7 7

English Speakers of Other Languages 14 14

School's Decile Ranking = 5