PosterPDF Available

Number line and simple fractions

Abstract

This exploratory research should give insight in the ways of how children read, interpret and place the mixed number of 1 ½ on the number line. We also explored if the development of insight correlates with grades of schooling, types of classes (including special education) and gender. 90 children from preschool to grade 9 (5;11 to 15;5 years) were clinically interviewed.
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 fléchissante »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 linegroup (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 fléchissante »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 linegroup (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)
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