OPPORTUNITIES TO LEARN MATHEMATICS WHILE
PLAYING TRADITIONAL DICE GAMES
Hedwig Gasteiger
Ludwig-Maximilians-Universität Munich
Early mathematics education should ensure that, on the one hand, children acquire
the essential prerequisites for further mathematical learning. On the other hand, it
should allow children to learn in a way which is appropriate to their specific age.
Play activities meet these requirements of subject- and child-orientated mathematical
learning in the early years. They can foster the development of mathematical learning
in kindergarten and in school sustainably. Results of an intervention study about
learning mathematics while playing traditional board games in kindergarten confirm
this assumption. The findings can be illustrated by the results of video analyses of the
play situations. Due to these analyses the mathematical learning processes which
occurred during the play activities can be investigated in more detail.
Key-words: early mathematics education, play, natural learning situations
INTRODUCTION
During the last century many educators, psychologists or social theorists have been
deeply involved in the discussion whether introducing children to mathematics in
early childhood education is “inappropriate, unnecessary, or even harmful” (Balfanz,
1999, p. 3). However, there is now a broad consensus that promoting mathematics
education is a very important task for the early childhood teachers. Therefore, many
concepts for early mathematics education have been developed in recent years.
Taking a closer look at these concepts, it can be seen that they differ considerably in
pedagogical background and in quality. So it is still an open question how early
mathematics education should be designed and organized.
GUIDING PRINCIPLES FOR THE ORGANIZATION OF EARLY
MATHEMATICS EDUCATION
Below, some guiding principles for the organization of early mathematics education
will be described normatively. They are based on scientific findings of various
disciplines like mathematics, pedagogy or psychology.
First of all, it is necessary to choose carefully the mathematical content taught in the
early years. It is recommended to align all efforts of early mathematics education
with the “big ideas” of mathematics (NAEYC, 2002, p. 6). These are “overarching
clusters and concepts and skills that are mathematically central and coherent,
consistent with children’s thinking, and generative of future learning” (Sarama, &
Clements, 2009, p. 16). To guarantee continuity of learning, the mathematical content
should be taught “effectively in some intellectually honest form” (Bruner, 1999, p.
33). This means that the underlying “fundamental structure of a field of knowledge”
(Bruner, 1999, p. 31) should be clear – only then children have a chance to
understand what they learn and to relate early mathematical learning to mathematical
learning in school and life contexts. Teaching mathematical contents in a simplified
manner, as it is inherent in some early childhood mathematics programs (e. g.
Friedrich, & de Galgóczy, 2004), can be counterproductive if the fundamental
mathematical structure gets lost.
Moreover, early mathematics education should respect children’s learning processes.
It is known that children differ considerably in their mathematical achievement in the
early years due to social background, family factors and the quality of home learning
environment (Anders, Grosse, Rossbach, Ebert, & Weinert, 2012, p. 207). If school
starters show low mathematical achievement, e. g. low patterning competencies, they
rather have difficulties while learning mathematics at school (Lüken, 2011). We even
know that children who focus on numerosity in their very early years have better
skills to recognize and produce small amounts at the age of four (Hannula, &
Lehtinen, 2001). Therefore, children’s mathematical learning processes should be
observed conscientiously and all efforts of early mathematics education should attend
to children’s individual stages of mathematical development.
Today it seems self-evident that early mathematics education should be based on a
constructivist perspective and that learning or teaching processes should be oriented
to children’s specific age. Results of the EPPE-Study prove the importance of
co-constructive learning environments in early childhood. They show that early
education settings are effective if they encourage “sustained shared thinking”, which
is defined as “an episode in which two or more individuals ‘work together’ in an
intellectual way to solve a problem, clarify a concept […] etc.” (Siraj-Blatchford,
Sylva, Muttock, Gilden, & Bell, 2002, p. 8). Results of research in developmental
psychology even state that children at an early age have difficulties with explicit and
intentional learning as it is normally practiced in school contexts (Hasselhorn, 2005).
Nevertheless, there are strong-guided programs for early mathematics education
which do not meet the requirement for constructive learning.
One possibility to create and organize early mathematics education – respecting
children’s learning processes and the requirement of the subject mathematics – is to
use natural learning situations, like play and everyday activities (Gasteiger, 2012).
Especially play offers many constructive learning opportunities, which are
appropriate for children in the early years (Fröbel, 1838, Pramling, & Asplund,
2008), and confronts them with central mathematical ideas, consistent with children’s
thinking, and generative of future learning.
EARLY MATHEMATICS LEARNING IN PLAY SITUATIONS
Real play situations follow the theoretically derived guiding principles for the
organisation of early mathematics education, and it can influence mathematical
learning in a positive way, as several scientific findings show:
Moderately mentally handicapped children (mean age 12.3 years) could improve
their counting abilities after a six week intervention of playing dice and card games
which were especially designed for the intervention (McConkey, & McEvoy, 1986).
Another study showed that playing mathematical card and board games in small
groups in the classroom improved low-achieving children’s performance in counting
tasks compared with the performance of children in a control group. The intervention
lasted eight months and the five-year-old children (n=14, control group: n=37) played
once a week with parental support (Peters 1998).
Young-Loveridge (2004) also worked with lower achieving 5-year-old children
(n=23). Over a 7-week period they played daily for 30 minutes modified commercial
dice and card games with a teacher and worked with story books while the control
group (n=83) continued their mathematical lessons. Over time children in the
intervention group could perform significantly better in different tasks, like e. g.
knowledge of numbers, making small collections of objects or adding two collections
of objects, than the control group.
While these studies focused on children at school, there are as well results from
studies with children at pre-school-age. Only four 20-minute sessions playing a linear
board game with a number dice (numbers 1 and 2) in two weeks were sufficient to
improve counting abilities and number line estimation of five-year-old children
(n=68) from low-income background. The control group (n=56) played linear board
games with a dice with colours (Ramani, & Siegler, 2008).
A study in Swiss kindergarten involved 89 children (mean age 6.3 years) in playing
commercial and especially created board and card games three times a week for 30
minutes during an eight-week period. These children performed significantly better in
a posttest than a control group with no intervention (n=125), but not significantly
better or worse than 110 children whom were given a commercial instructional
training in mathematics (Rechsteiner, Hauser, & Vogt, 2012).
In addition to the results of the intervention studies, there is another interesting
finding concerning play in early childhood: Ramani and Siegler (2008) compared
interview data on playing habits of children at home or with friends with children’s
performance in number knowledge items. Board game experience was positively
correlated with the number knowledge of children, while the influence of card or
video games on number knowledge was not so clear.
Many of the here reported intervention studies focus on children with learning
difficulties or on children at school age, and nearly all games used in these studies
were modified or especially designed for the intervention. As learning opportunities
at children’s home seem to have a substantial impact on their mathematical
prerequisites (Anders et al. 2012, Ramani, & Siegler, 2008), play situations especially
for younger children and with games which can be played in normal family or
kindergarten situations should be examined in detail.
PLAYING TRADITIONAL DICE GAMES IN KINDERGARTEN – AN
INTERVENTION STUDY
The intervention study MaBiiS (elementare mathematische Bildung in
Spielsituationen) focuses on the following research question: can children improve
their mathematical knowledge and skills while playing traditional dice games in
“normal” play situations in kindergarten? Moreover, it should be analyzed whether
the intervention has an impact on mathematical learning independent of gender,
migration background, intelligence or day-care center.
Method
Participants of the study were 95 children (43 male, 52 female; 31 with migration
background, 64 without) from five day-care centers in Germany. They had a year and
a half until their school enrolment and were between 4.5 to 6.2 years (mean age 4.8).
They were randomly assigned to intervention group and control group.
For the intervention, children played dice games in small groups (two or three
children) with an adult. They played seven times for 30 minutes over a
three-and-a-half-week period. Children in the intervention group played the
traditional board games “ludo” (Mensch ärgere dich nicht, Schmidt-Spiele) and
“coppit” (Fang den Hut, Ravensburger) and a game called “collecting treasuries”
(Schätze sammeln, ZahlenZauberei, Oldenbourg-Schulbuchverlag), where children
can collect coloured tokens when they pass a square with a shown amount of
treasuries. All these games were played with normal number dice (numbers from 1 to
6).
Children in the control group played games with colour or symbol dice. One game
was very similar to ludo (Der Maulwurf und sein Lieblingsspiel, Ravensburger), but
with a symbol dice. The symbol on the dice shows children to which square their
token should move. The other game (Da ist der Wurm drin, Zoch) with a colour dice
gave no possibility to count, to think about collections of objects or to subitize
amounts.
The adults who played with the children were trained. The most important
information they got was that they should play and not train mathematics to children.
However, the adults were requested to play aware. This means to count loudly while
moving forward the token, to tell the thrown number (“Ah, five!”) or to give verbal
comments like “Oh! Now you can catch someone!” or “Count again. Your token was
here.” Moreover, the adults should give children enough time to manage their play on
their own or to help others.
Children’s mathematical performance was measured with TEDI-Math (Kaufmann,
Nuerk, Graf, Krinzinger, Delazer, & Willmes, 2009). A pretest, a posttest
immediately after the intervention and a follow-up test one year after the intervention
were performed in individual interviews. As in TEDI-Math, there is no item to
measure structure sense or subitizing, a subscale with this content was added
(Gasteiger, 2010). With TEDI-Math, children’s knowledge of numerals and number
words was tested, they were asked to count verbally, to count on or backwards, and to
count, add or subtract amounts. All in all the test instrument had 64 items
(Cronbach’s α=.91, .92 in pre-/posttest). Intelligence was measured with WPPSI
(Petermann, & Lipsius, 2011).
Results
By now, only posttest data have been analysed. An ANCOVA (covariate: pretest,
dependent variable: posttest) is used to examine the development of mathematical
performance and differences between the intervention and the control group. Posttest
score is influenced significantly by the pretest score (F(1,92)=291.88, p<.001) and by
the intervention-condition (F(1,92)=13.57, p<.001) with an effect size of 13% (partial
eta squared). The intervention group shows greater gains in posttest than the control
group, as the solution rates in table 1 show.
N
M (SD)
Pretest
Posttest
Intervention group
48
.60 (.16)
.72 (.14)
Control group
47
.61 (.15)
.67 (.16)
Table 1: Comparison of solution rates.
The intervention condition particularly influences children’s skills to count amounts.
This is not really astonishing, because children train one-to-one-correspondence of
number word and square intensively when they move their token forward during the
game. The effects of the intervention were independent of gender, migration
background, intelligence or the day-care center children attended.
To analyse in detail the mathematical learning processes which occurred during the
play activities, 9 intervention sessions were videotaped. The leading questions are, in
which mathematical contents children are engaged and how much time they spend in
mathematical or non-mathematical activities or dialogues during the play situation.
The period of time of activities and comments of each child was coded in categories
and summed up to get the active time of all children in all intervention sessions.
Mathematical/non-mathematical and verbal/non-verbal activities were differentiated.
Two non-verbal activities were coded as mathematical: moving the token forward
correctly without counting loudly or recognizing the dice-pattern without telling the
right number were seen as mathematical activities.
Children spent most of their active time on “non-verbal play activities” (34%) (figure
1). This category was coded when the game had been prepared, dice was thrown or
passed to the next player, and other similar non-mathematical activities happened. In
42% of their active time, children were mathematically active (all categories except
7, 8, 9, 12), and most of that time was used for enumeration (nearly 25% verbal, 4%
non-verbal) and subitizing (5% verbal, 2% non-verbal).
Fig. 1: Children’s activities/comments (mathematical categories: dark, others: light)
In addition to the mathematical activities enumeration and subitizing, children
showed in nearly 5% of their active time mathematical thinking in their comments on
their own or others’ play activities (category 6). An example for this category is “I
would need a five, to catch you” or “You should move this token” (if it was a tactical
advice, e. g. concerning a number of moves). For all mathematical activities or
comments, sub-categories differentiated between mathematically right or wrong
utterances, but all in all only in less than 5% of the active time children acted
mathematically wrong. Most errors occurred while counting.
DISCUSSION
The results of this study show that traditional board games provide a good
opportunity for mathematical learning in the early years. Children’s mathematical
performance was significantly influenced by the intervention condition. It is often
discussed that early childhood programs should include highly qualitative
mathematics instruction (Cross, Woods, & Schweingruber, 2009, p.3). However,
there are valid arguments against direct instruction, as explicated in the first section:
it seems not to be the best form to learn for children in the early years (Hasselhorn,
2005), it often does not meet the requirement for co-constructive learning or
sustained shared thinking (Siraj-Blatchford, et al. 2002) and the question arises
whether instructive learning can always respect children’s individual learning
processes.
Playing board games – as it was realized in this study – is definitely no direct
instruction. It offers many opportunities for co-constructive learning, as the analyses
of the video data show: children spent much time in mathematical activities while
playing the games, and they reflected and commented their own and others play
activities. Furthermore – as the statistical data shows – it is an effective way to
improve mathematical knowledge and skills which are in line with the big ideas of
early mathematics education and which are predictive for further mathematical
learning (Krajewski, & Schneider, 2009).
The great potential of traditional board games for mathematical learning appears in
another point, too. The adult players gave no explicit instructions. They were urged to
play like in a “normal” play situation as it can occur with parents or with friends.
Nevertheless, the intervention had a significant impact on children’s mathematical
development, independent of gender, migration background or intelligence. This is
highly important regarding the fact that children differ considerably in their
mathematical knowledge and skills – even before they attend kindergarten (Anders et
al. 2012). Playing board games with number dice can be carried out very easily in
family or kindergarten – even with younger children – and it is a child- and
subject-oriented, effective form of early mathematics education.
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