Content uploaded by Stefan L. Meyer-Baumgartner

Author content

All content in this area was uploaded by Stefan L. Meyer-Baumgartner on Jun 29, 2018

Content may be subject to copyright.

Number line and simple fractions

Stefan Meyer

University of Applied Sciences of Special Needs Education, Zurich, Switzerland

Introduction

Research aims and hypotheses

EXPERIMENT Levels of Insight or Correspondence

Discussion

Participants

90 children ranging in age from 4;11 years to 15;5 years (M= 9;5) were clinically

interviewed. Master students of the University of Applied Sciences of Special Needs

Education were introduced in this method of clinical interviewing.

Development of insight in rational numbers

refers to changes in the correspondences between meanings and signs.In

the educational field the choice of representations is mostly centered on

enactive or iconic representation.Number line as ahighly abstract (iconic-

symbolic) «Mitteilungszeichen »(sign of message, Nietzsche;sign of

relation, Peirce) is less explored.Number line seams to be an indicator of the

higher levels of the «abstraction réfléchissante »of fractions, akind of red

thread of understanding numbers and mathematical education.

Procedure

The pretest.Every child passed apretest drawing and explaining anumber line: ”Draw

anumber line and tell me what it is.”

When children had few or no ideas about the number line, the interviewer could offer 3

differently detailed information's.

The experimental Task.After the pretest, all participants completed one experimental

tasks:the fraction task:

“Look, here on the number line is 1, there is 2. I ask you now:which of these cards

belongs between 1and 2?”

(move a finger between 1 and 2 on the number line)

Since 1980 it has been well documented that students even at the age of 12

find difficulty in using number lines to work with fractions (Watanabe, 2002;

Padberg, 2002). Watanabe concludes «…that number lines do not help

students develop a sense of fractions as numbers but that number-line

representations make sense only to those students who already understand

fractions as numbers » (p. 462).

Sinclair et al. (1988) reported how preschool children create and read

notations of natural numbers. In Brizuela’s (2006) interesting study

kindergarten-and first grade children had to explain their notations for

fractions and they had to show the different numbers on the number line.

Brizuela found three groups of meanings of fractions: « half is a little bit »;

different understandings across different contexts (partitioning cookies or

pizzas); similar understanding across different contexts.

Young children generates meanings for fractional numbers, number lines and

contexts. There must be bridges of arguments in the « abstraction

réfléchissante » between the natural and the rational numbers.

Our study explored the development of correspondences between simple

fractions and the number line. What do children know about number line and

what kind of conceptual arguments will be produced for ordering a mixed

number ( 1 ½ )?

We also explored if the development of insight correlates with grades of

schooling, types of classes (including special education) and sex.

Differing from Brizuela we excluded contextual manipulatives and

concentrated on the number line. Differing from Watanabe we postulate that

every correspondence with the number line makes sense, not only the right

understanding of some numbers. Differing from Moss & Case (1999) number

line is an open tool rather than an object of ordered training.

Brizuela, B. M. (2006). Young Children's Notations For Fractions. Educational Studies in

Mathematics, 62(3), 281-305.

Moss, J., Case, R. (1999). Developing Children's Understanding of the Rational Numbers: A

New Model and an Experimental Curriculum. Journal for Research in Mathematics Education,

30(2), 122-147.

Padberg, F. (2002). Didaktik d er Bruchrechnung (3. Aufl.). Heidelberg: Spektrum

Akademischer Verlag.

Parrat-Dayan, S. (1980). Etude génétique de l'acquisition de la notion de moitié. Thèse à la

Faculté de Psychologie et des Sciences de l'Éducation pour obtenir le grade de docteur en

psychologie, Université de Genève, Genève.

Piaget, J., Henriques, G., Ascher, E. (Hrsg.). (1990). Morphismes et Catégories. Comparer et

Transformer. Lausanne: Delachaux et Niestlés.

Saxe, G. B., Taylor, E.V., McIntosh, C., Gearhart, M. (2005). Representing Fractions with

Standard Notation: A Developmental Analysis. Journal for Research in Mathematics

Education, 36(2), 137-157.

Sinclair, A., Mello, D., Siegrist, F. (1988). La notation numérique chez l'enfant. In H. Sinclair

(Hrsg.), la production de notations chez le jeune enfant(S. 71-97). Paris: Presses

universitaires de France.

Watanabe, T. (2002). Representations in Teaching and Learning Fractions. Teaching Children

Mathematics, 8(8), 457-563.

Which is it... ? (point with the finger at the cards 0, ½, 1 ½ , 3) -or does nothing go

between? (point at the “none-card”)

Or: What belongs between the 1 and the 2? (Child moves a card)

Explain (tell) me: why did you take that card?

References

RESULTS

36 % of all subjects knew spontaneously what the number line is.

In the randomly selected sample of 44 subjects (see Fig. 2) we found a large correlation

between the performance in the pretest (concept of number line) and the knowledge of

simple fractions (Kendalls τbis (tau -biserial) = -.55, p < .05).

The big majority of the children in the “minus” -number line–group (70% of n=44) was

not able to understand and to place 1 ½ on the number line.

The first offer of information (drawing just a line) effected that 13 children could order the

mixed fraction correctly. The second offer (drawing a line and number 1 to 3) helped 5

children. The last offer (drawing a line and number 1 to 10) was helpful for 2 children.

Differing from Watanabe (2002) talking about number line helps to understand a mixed

fraction (Moss & Case (1999) .

The age of the children and the grades correlated also with the insight in simple fractions.

No correlations has been observed between the types of schooling, sex and the

understanding of simple fractions.

5 levels of hypothetic constructs of correspondences of perceptions and

logico-arithmetical reasoning were found:

Level 1 represents answers about perceptions of the material, there is no

insight in the number line and the given set of numbers.

Level 2 is defined by the counting-and comparison-scheme of natural

numbers. The symbols of the fractions are not integrated. There are also

arguments about addition of natural numbers.

Level 2a integrates experiences with scales (meter) or with the partitioning

of cookies in combination with the symbols of fraction. The cardinality of the

fractions is not developed.

Level 3 integrates the correct seriation (counting and cardinality) of the

natural and the rational numbers on the number line. 1 ½ is explained as the

half between 1 and 2.

Level 3a contains the perfect understanding of the presented fractions in

combination with logico-arithmetical operations (part-whole-relation,

addition, multiplication, or division). 1 ½ can be correctly explained as a

decimal.

Our results support Parrat-Dayan’s (1980) and Brizuela’s view that

understanding of fraction is a gradual process. In the setting of a clinical

(flexible) interview children constructed logico-mathematical

correspondences.

The differences or the correctness of understanding conventional notations

could be classified in different levels. The levels represent a growing

complexity of insight in natural and rational numbers and operations.

The use of the number line provokes operations (handling, reasoning) and

insight in the correspondence between mental and iconic-symbolic

representation.

Children constructed their own aspects of fractions on the topic of the

number line. They used natural numbers to explain the mixed fraction

(counting). They interpreted parts of the conventional notations. They also

used seriation, scales, arithmetic operations, part-whole-thinking (Saxe et

al., 2005) and decimals.

Developing and exploring this clinical interview offered a psychological view

on the development of constructing correspondences (Piaget et al., 1990).

Pragmatic consequences for the research and the education of mathematics

should be:

-Use number line as an open tool not as a manipulative

-Enhance research of logico-arithmetical reasoning about fractions in the

classrooms

-Root mathematical education (dialogue, cooperation, games) in

children’s constructions of correspondences rather than

rooting in manipulatives.

http://www.interview.hfh.ch Stefan.Meyer@hfh.ch / Geneva 2008

Fig. 1

Number line and number

cards in the experimental

task

Fig. 2 Number line and levels of understanding 1 ½ (n=44)

Zahlenstrahl und einfache Bruchzahlen

Stefan Meyer

Interkantonale Hochschule für Heilpädagogik, Zürich

Einleitung

Research aims and hypotheses

EXPERIMENT Levels of Insight or Correspondence

Discussion

Participants

90 children ranging in age from 4;11 years to 15;5 years (M= 9;5) were clinically

interviewed. Master students of the University of Applied Sciences of Special Needs

Education were introduced in this method of clinical interviewing.

Die Entwicklung der Einsicht in rationale Zahlen

refers to changes in the correspondences between meanings and signs.In

the educational field the choice of representations is mostly centered on

enactive or iconic representation.Number line as ahighly abstract (iconic-

symbolic) «Mitteilungszeichen »(sign of message, Nietzsche;sign of

relation, Peirce) is less explored.Number line seams to be an indicator of the

higher levels of the «abstraction réfléchissante »of fractions, akind of red

thread of understanding numbers and mathematical education.

Procedure

The pretest.Every child passed apretest drawing and explaining anumber line: ”Draw

anumber line and tell me what it is.”

When children had few or no ideas about the number line, the interviewer could offer 3

differently detailed information's.

The experimental Task.After the pretest, all participants completed one experimental

tasks:the fraction task:

“Look, here on the number line is 1, there is 2. I ask you now:which of these cards

belongs between 1and 2?”

(move a finger between 1 and 2 on the number line)

Since 1980 it has been well documented that students even at the age of 12

find difficulty in using number lines to work with fractions (Watanabe, 2002;

Padberg, 2002). Watanabe concludes «…that number lines do not help

students develop a sense of fractions as numbers but that number-line

representations make sense only to those students who already understand

fractions as numbers » (p. 462).

Sinclair et al. (1988) reported how preschool children create and read

notations of natural numbers. In Brizuela’s (2006) interesting study

kindergarten-and first grade children had to explain their notations for

fractions and they had to show the different numbers on the number line.

Brizuela found three groups of meanings of fractions: « half is a little bit »;

different understandings across different contexts (partitioning cookies or

pizzas); similar understanding across different contexts.

Young children generates meanings for fractional numbers, number lines and

contexts. There must be bridges of arguments in the « abstraction

réfléchissante » between the natural and the rational numbers.

Our study explored the development of correspondences between simple

fractions and the number line. What do children know about number line and

what kind of conceptual arguments will be produced for ordering a mixed

number ( 1 ½ )?

We also explored if the development of insight correlates with grades of

schooling, types of classes (including special education) and sex.

Differing from Brizuela we excluded contextual manipulatives and

concentrated on the number line. Differing from Watanabe we postulate that

every correspondence with the number line makes sense, not only the right

understanding of some numbers. Differing from Moss & Case (1999) number

line is an open tool rather than an object of ordered training.

Brizuela, B. M. (2006). Young Children's Notations For Fractions. Educational Studies in

Mathematics, 62(3), 281-305.

Moss, J., Case, R. (1999). Developing Children's Understanding of the Rational Numbers: A

New Model and an Experimental Curriculum. Journal for Research in Mathematics Education,

30(2), 122-147.

Padberg, F. (2002). Didaktik d er Bruchrechnung (3. Aufl.). Heidelberg: Spektrum

Akademischer Verlag.

Parrat-Dayan, S. (1980). Etude génétique de l'acquisition de la notion de moitié. Thèse à la

Faculté de Psychologie et des Sciences de l'Éducation pour obtenir le grade de docteur en

psychologie, Université de Genève, Genève.

Piaget, J., Henriques, G., Ascher, E. (Hrsg.). (1990). Morphismes et Catégories. Comparer et

Transformer. Lausanne: Delachaux et Niestlés.

Saxe, G. B., Taylor, E.V., McIntosh, C., Gearhart, M. (2005). Representing Fractions with

Standard Notation: A Developmental Analysis. Journal for Research in Mathematics

Education, 36(2), 137-157.

Sinclair, A., Mello, D., Siegrist, F. (1988). La notation numérique chez l'enfant. In H. Sinclair

(Hrsg.), la production de notations chez le jeune enfant(S. 71-97). Paris: Presses

universitaires de France.

Watanabe, T. (2002). Representations in Teaching and Learning Fractions. Teaching Children

Mathematics, 8(8), 457-563.

Which is it... ? (point with the finger at the cards 0, ½, 1 ½ , 3) -or does nothing go

between? (point at the “none-card”)

Or: What belongs between the 1 and the 2? (Child moves a card)

Explain (tell) me: why did you take that card?

Literatur

RESULTS

36 % of all subjects knew spontaneously what the number line is.

In the randomly selected sample of 44 subjects (see Fig. 2) we found a large correlation

between the performance in the pretest (concept of number line) and the knowledge of

simple fractions (Kendalls τbis (tau -biserial) = -.55, p < .05).

The big majority of the children in the “minus” -number line–group (70% of n=44) was

not able to understand and to place 1 ½ on the number line.

The first offer of information (drawing just a line) effected that 13 children could order the

mixed fraction correctly. The second offer (drawing a line and number 1 to 3) helped 5

children. The last offer (drawing a line and number 1 to 10) was helpful for 2 children.

Differing from Watanabe (2002) talking about number line helps to understand a mixed

fraction (Moss & Case (1999) .

The age of the children and the grades correlated also with the insight in simple fractions.

No correlations has been observed between the types of schooling, sex and the

understanding of simple fractions.

5 levels of hypothetic constructs of correspondences of perceptions and

logico-arithmetical reasoning were found:

Level 1 represents answers about perceptions of the material, there is no

insight in the number line and the given set of numbers.

Level 2 is defined by the counting-and comparison-scheme of natural

numbers. The symbols of the fractions are not integrated. There are also

arguments about addition of natural numbers.

Level 2a integrates experiences with scales (meter) or with the partitioning

of cookies in combination with the symbols of fraction. The cardinality of the

fractions is not developed.

Level 3 integrates the correct seriation (counting and cardinality) of the

natural and the rational numbers on the number line. 1 ½ is explained as the

half between 1 and 2.

Level 3a contains the perfect understanding of the presented fractions in

combination with logico-arithmetical operations (part-whole-relation,

addition, multiplication, or division). 1 ½ can be correctly explained as a

decimal.

Our results support Parrat-Dayan’s (1980) and Brizuela’s view that

understanding of fraction is a gradual process. In the setting of a clinical

(flexible) interview children constructed logico-mathematical

correspondences.

The differences or the correctness of understanding conventional notations

could be classified in different levels. The levels represent a growing

complexity of insight in natural and rational numbers and operations.

The use of the number line provokes operations (handling, reasoning) and

insight in the correspondence between mental and iconic-symbolic

representation.

Children constructed their own aspects of fractions on the topic of the

number line. They used natural numbers to explain the mixed fraction

(counting). They interpreted parts of the conventional notations. They also

used seriation, scales, arithmetic operations, part-whole-thinking (Saxe et

al., 2005) and decimals.

Developing and exploring this clinical interview offered a psychological view

on the development of constructing correspondences (Piaget et al., 1990).

Pragmatic consequences for the research and the education of mathematics

should be:

-Use number line as an open tool not as a manipulative

-Enhance research of logico-arithmetical reasoning about fractions in the

classrooms

-Root mathematical education (dialogue, cooperation, games) in

children’s constructions of correspondences rather than

rooting in manipulatives.

www.hfh.ch unter Links “Das flexible Interview” / Stefan.Meyer@hfh.ch / Genf 2008, Zürich 2009

Fig. 1

Number line and number

cards in the experimental

task

Fig. 2 Number line and levels of understanding 1 ½ (n=44)