Conference PaperPDF Available

Preschool children's understanding of length and area measurement in Japan

Authors:

Abstract

The purpose of this study is to assess preschool children's understanding of length and area measurement through the activities of a mathematics program in Japan. Japanese preschool children do not formally learn mathematics; thus, we designed and recommended mathematical measurement activities for young, less-experienced preschool teachers to implement for nine five-to six-year-old children. We conducted structured clinical interviews with the children individually before and after the measurement activities, and qualitatively analyzed the results by comparing their answers and connecting them to the activity contents. The results indicated that children learn to understand direct comparison through activities; however, it is more difficult to establish understanding of measurement by arbitrary unit.
Preschool children’s understanding of length and
area measurement in Japan
Nanae Matsuo1 and Nagisa Nakawa2
1Chiba University, Faculty of Education, Chiba, Japan; matsuo@faculty.chiba-u.jp
2Kanto-gakuin University, Japan; nagisa@kanto-gakuin.ac.jp
This study assesses preschool children’s understanding of length and area measurement through the
activities of a mathematics program in Japan. Japanese preschool children do not formally learn
mathematics; thus, we designed mathematical measurement activities for young, less-experienced
preschool teachers to implement for nine five- to six-year-old children. We conducted structured
clinical interviews with the children individually, both before and after the measurement activities,
and qualitatively analyzed the results by comparing their answers and connecting them to the activity
contents. Results indicate that children learn to understand direct comparison through activities;
however, it is more difficult to establish understanding of measurement by non-universal units.
Keywords: Preschool children, mathematical activity, length measurement, area measurement
Research on measurement in preschool
Much of the current research focuses on elementary mathematics education for preschool children
and younger elementary school students (e.g., Brandt, 2013); however, measurement has not been
sufficiently included in these studies (Sarama, Clements, Barrett, Van Dine, & McDonel, 2011;
Smith, van den Heuvel-Panhuizen, & Teppo, 2011), even though it is an important real-world area of
mathematics used in everyday life. In addition, Hachey (2013) states that mathematical conceptual
change in young children is as effective as a substantial change in early childhood mathematical
teaching practice, as a large body of developmental research advocates that young children are born
mathematicians, and therefore, early childhood mathematics education is vital.
It is known that the development of children’s understanding is related to foundational or key ideas
of measurement such as comparison, unit iteration, number assignment, and proportionality (Lehrer,
Jaslow, & Curtis, 2003; Wilson & Osborne, 1992), and several kinds of teaching methods for
measurement have been developed (Battista, 2006; Kamii, 2006). Although Piaget (1968) assumed
that children up to seven years old are unable to consider more than one stimulus dimension in their
judgments, subsequent research has demonstrated that preschoolers can consider two dimensions; for
example, they can consider the width and length of rectangles to estimate their area (Wilkening,
1979). Further, Ebersbach (2009) addressed the question of whether children can also take three
stimulus dimensions into account and showed that preschoolers have the cognitive competencies
required for multidimensional reasoning. There have also been numerous other important studies on
preschool children’s understanding of measurement. Sarama et al. (2011) proposed a learning
trajectory for length and area in the early years and evaluated it. Zöllner and Benz (2013) concluded
that four- to six-year-old children could compare directly and indirectly, but cannot measure using a
non-standardized unit. Tzekaki (2017) indicated that a seven-month intervention helped preschoolers
improve their abilities to reflect on their own activities and express their ideas regarding
measurement. Skoumpourdi (2015) verified that preschoolers do have some strategies for length
Thematic Working Group 13
Proceedings of CERME11
2311
measurement, but confuse the concept of length with perimeter and area. Finally, Kotsopoulos (2015)
showed that a free exploration approach in the context of a play-based learning environment is more
effective in teaching length measurement than guided instruction and center-based learning.
This article examines whether preschool children can improve their understanding of length and area
measurement, especially direct and indirect measurement, as well as measurement by non-universal
units. It does so with an elementary mathematics play-based program, which can even be
implemented by young, less-experienced teachers who have not been trained in mathematics
instruction, thereby fostering children’s understanding of measurement. These play-based activities
are intended as a further contribution to the existing research. The article also seeks to explain the
specific reasons the program is effective. First, we compare pre- and post-interview results for five-
to six-year-old children. Next, we intentionally choose two children with different degrees of change
between their pre- and post-interview results. Finally, we discuss the reason the change occurred,
scrutinizing each child’s actions and attitude during the activities, as well as his/her post-interview.
The early childhood mathematics program in Japan
Elsewhere, we have proposed a framework for constructing an early childhood mathematical
curriculum (Matsuo, 2016). The theoretical and methodological underpinnings of our program are
social constructivism (Ernest, 1994), mathematical guided-play (Weisberg et al., 2013), and the
Structure of the Observed Learning Outcome (SOLO) taxonomy (Biggs, 1982). This program
presents a scope for the content of preschoolers’ activities, while the learning process of those
activities are explained from the viewpoint of sequence, based on the relationship between age and
modes of representation and focusing on activities for five- to six-year-old children to create a
program tailored to these concerns. Based on this framework, we have proposed a mathematical
education program to enable a smooth transition from preschool to elementary school mathematical
education (Matsuo, 2017). Less-experienced preschool teachers should be able to use these activities
to better incorporate mathematical content into children’s play and recognize activities that will make
children think and act mathematically. This practical program was designed for preschools and
elementary schools.
Methodology
Research participants
The preschool kindergarten in an urban area of Kanagawa Prefecture of Japan participated in our pilot
study. This article focuses on nine five- to six-year-old children in the oldest classes. The children are
all Japanese, coming from middle- and upper-middle-class families in Japan. There were six boys
and three girls. The school’s focus was on physical activities, as well as entrance to candidate schools
for the International Baccalaureate (IB). Children started to learn Japanese letters and English to
prepare themselves for primary education in September 2017.
The oldest classes of the preschool had two teachers: Teacher A, a male teacher with three years of
experience teaching preschool, and Teacher B, a female teacher with two years of experience teaching
preschool. Teacher A taught the first three sessions of measurement activities in the first cycle of the
program and Teacher B taught the latter three sessions in the second cycle. For the analysis, we
intentionally chose two children, T1 and T7, based on the results of pre- and post- interviews, to
Thematic Working Group 13
Proceedings of CERME11
2312
intensively examine their outcomes and learning processes, as well as to determine the relations
between the interview results, their actions, and the detailed observations of the teachers during the
activities. T1 was actively engaged in the activities, but provided incorrect answers in the post-
interview. T7 was not actively engaged in the activities, but made correct observations and provided
correct answers in the post-interview. To compare the two children’s understanding, we discuss the
level of understanding of direct and indirect measurement, as well as measurement with non-universal
units, in connection with the young teachers’ instruction.
Data collection and analysis
In the project, we developed six play-based activities related to numbers, measurement, and shapes.
Among these activities, we carefully chose those that were better tailored to relate to numbers,
shapes, and measurements, things that are necessary for our curriculum to focus on and develop. To
evaluate the improvement of children’s mathematical skills after these activities, we developed pre-
and post-interview questions. Pre-interviews were conducted in June 2017 and all six activities were
implemented twice; therefore, twelve activities were implemented after the pre-interview. After all
activities had been completed, a post-interview was conducted with each child in February 2018.
Sixteen questions were developed for the pre-/post-interviews, corresponding with the planned
activities and previous research by the author (Matsuo, 2016), as well as the SOLO taxonomy. Four
of the sixteen questions were in the area of measurement, as shown in Table 1.
Table 1: Description of the interview questions and the problem contexts
Methodologically, the authors conducted structured clinical interviews (Goldin, 1998). To alleviate
children’s anxiety, the teachers conducted the interview themselves, while one of the authors sat next
to the child to record. The assigned teachers rehearsed the interviews to be consistent with
mathematical words and expressions. During the implementation, each child came to the room
randomly and the teacher sat in front of the child. The pre-interviews took fifteen to twenty-five
Q12
Prepare four different pencils, each of a different length, as
shown in the diagram, and ask, “Which do you think is the
longest pencil of the four? And tell me the order of the length,
from longest to shortest.”
Q13
Show four different pencils, each of slightly different
length as shown in the diagram, and ask, “Which do you
think is the longest pencil of the four? And tell me the
order of the length, from longest to shortest.”
Q14
Prepare four different rectangles as shown in the diagram,
and ask, “What do you think is the biggest rectangle of the
four? And tell me the order of the area, from biggest to
smallest.
Q15
Prepare four different rectangles as shown in the diagram,
and ask, “What do you think is the biggest rectangle of the
four? And tell me the order of the area, from biggest to
smallest.
Thematic Working Group 13
Proceedings of CERME11
2313
minutes, and the post-interviews took ten to fifteen minutes. During the interview, the children could
use a ruler, transparent paper, grid paper, and clips to answer questions 12 to 15. The results were
qualitatively analyzed.
Description of measurement activities in the intervention
Teacher A taught the children the first course in how to measure the length of something carefully
using clips as non-universal units. They measured assorted items by connecting clips (e.g., measuring
the edge of the desk, the mat, the height of the teacher). Teacher B taught the second course, in which
the children tried to connect eight clips in order to use them to measure shorter objects than the ones
measured in the first course. After the children used the connected clips to measure a building block
or a box, the teacher asked the children to find objects with lengths equivalent to eight clips. Next,
they looked for objects with lengths equivalent to sixteen clips. They lined up sixteen red magnets
and set out to measure them using the clips.
In the area measurement activity, the children employed indirect comparison and measured using
non-universal units, such as length measurement. The activity was implemented like a code-breaking
game, and when the code was solved, figures whose areas were objects for measurement were
supposed to be arranged in descending order. A teacher directed the children to count the square grids
covered in rectangular paper to measure the total area. First, the children copied the target onto plain
paper, put this paper on grid paper, and counted the number of squares as non-universal units. The
sizes of the three kinds of rectangles (thirty-two, forty, and forty-eight squares) were relatively large,
and the children had been actively counting quite carefully. Finally, the paper with a number written
on it was matched with a corresponding letter, which enabled them to solve the code. Letters with
meaning were arranged in order so that the numbers became larger.
Results and Discussion
Table 2 shows that there is not a significant difference of the correctness toward questions between
length and area measurement.
Looking at the transition of the results from the pre- to post-interview, problems whose answers are
incorrect in both pre- and post-interview cannot be seen except for T1. In the case of Q12 and Q14,
Child
No. Gender Birthday
Q12
Q14
Q15
Pre
Post
Pre
Post
Pre
Post
Pre
Post
T1
F
MAY
1
1
0
0
1
1
1
0
T2
M
JULY
1
1
1
1
1
1
1
1
T3
M
NOVEMBER
1
1
0
1
1
1
0
1
T4
F
NOVEMBER
1
1
0
1
1
1
0
1
T5
F
AUGUST
1
1
1
0
1
1
1
0
T6
M
JUNE
1
1
0
1
1
1
1
0
T7
M
DECEMBER
0
1
0
1
0
1
0
1
T9
M
JULY
1
1
1
1
1
1
0
1
Table 2: Pre- and post-interview results for measurement of length and area
* 1 means correct, 0 means incorrect
Thematic Working Group 13
Proceedings of CERME11
2314
children got the correct answers in the pre- and post-interview, excluding T7. Since the edges of some
part of the figures in these problems are aligned, it is possible to judge the difference in size from the
difference in other parts by visual observation. Children were thought to be mature, developable, and
able to maintain natural demeanor. Regarding Q13 and Q15, about half of the children answered
correctly in the pre-interview, but they had incorrect answers in the post-interview. T2 had correct
answers for everything both pre- and post-interviews. T9 also had correct answers, except for Q15,
in the pre-interview. T8, who was absent from the length and area measurement activity of the second
course, was excluded from analysis.
Comprehensive analysis of eight children’s understanding of measurement
Table 2 indicates that every child understood direct measurement in Q12 and Q14 in the post-
interview, though in Q13 and Q15, a few children did not understand indirect measurement nor
measurement by non-universal units. An erroneous view prevails that the area measurement is harder
to understand than length measurement, as the number of dimensions is higher. As Ebersbach (2009)
stated, not only length and area but also volume can be estimated by a young child, and the rate of
correct answers does not change much between children and adults. It is possible for preschoolers to
tackle activities that are not limited to length, area, and volume measurement, as we inferred from the
results of our survey; however, even though children were working well on the measurement
activities, most of them gave incorrect answers for area measurement.
T3, T4, and T7 answered incorrectly in the pre-interview but correctly in the post-interview for Q13
and Q15. Although they answered visually for Q13 and Q15, they were correct. We infer the program
implemented had a positive effect, especially because the measurement using non-universal units is
not commonly done at home or in preschools. While there were scenes in which they could not be
regarded as proactively working on the program activities, in many cases, it is mentioned that they
were working on collaborative and individual work well and could observe other subjects’ behavior.
Conversely, T1, T5, and T6 had incorrect answers in the post-interview even if they were correct in
the pre-interview, revealing that the influence exerted by the activities of the mathematics program
are strongly related to this fact. Let us consider why they did not have correct answers for Q15, even
if they were trying to answer based on measurements in non-universal units using tools and enjoyed
it in the post-interview. It is probably because children could not distinguish between the
measurement of length and area and did not associate the numerical value measured in non-universal
units with the results of the comparison of length and area, or because they could not understand the
meaning of the work and retain this understanding even if they acquired skills related to measurement.
This can be judged from the state of the post-interview.
Qualitative analysis for two children who had different results and processes of playing
We will now compare and examine the results of two children, T7 and T1, their post-interview
responses, and the teacher’s findings regarding them in Table 3.
T1 was engaged in acting as a group leader in length measurement. After the clip-connecting activity,
she was looking for various items to measure, like building blocks, etc. She was highly interested in
the length-measurement activity. Further, in the area-measurement activity, she actively worked on
counting. She did the measurement work again, but her answer was not correct. She did not seem to
have an opportunity to objectively review the meaning of the work after she had concentrated on it.
Thematic Working Group 13
Proceedings of CERME11
2315
It can be surmised that because the meaning of the work was unclear, she confused length and area.
The result of this study is consistent with previous studies (e.g., Skoumpourdi, 2015). The length and
the area are different in dimension, and in daily life, for a child, the length comparison is familiar and
more understandable than the area comparison.
Table 3: Results of T1 and T7 for Q12-15, post-interview results and teachers’ findings
Conversely, T7 did not do much work and wandered around. He did not work as actively as the other
children, though he observed the other children’s activities well. During area measurement, everyone
was absorbed in drawing figures and counting the squares, while he also worked and looked around.
It is highly likely that his reply was affected by this behavior. As a result, T7 who did not actively
work, gave the correct answer by visual observation. It can be inferred that this is due to the child’s
rich sense of length and area, fostered during the activities (Schoenfeld, 2016). Through these
activities, it seems that the sense of long/short, wide/narrow is more sharpened. Preschool children
may not have this vocabulary, but instead use bigor small to express differences in quantity.
Since children who observed better than doing their own work seemed to grasp the size correctly by
visual observation in the post-interview, rather than just doing work, by sharing the work process of
early mathematical activities, it can be said that the sense of quantity has been enhanced.
Further, even when it seems that a child is not actively working on a problem, or appears not to be
thinking too much, we must accept the fact that the child considers silently and may still answer
correctly. According to Shinohara (1942), since observation is an educationally effective activity, like
experiments, it is important for children to observe othersbehavior and imitate it. In the case of
Q12
She replied visually, pointing with her finger.
Q13
She took the fine-grid tracing paper, and after a little measuring, she answered, "Oh, I
unders tood it."
Q14
She replied visually.
Q15
She laughed and took the clips first, and asked, "Can I measure them sideways?" She joined
the clips, measured the length of the rectangle sideways, adjusting the length of the joined
clips. Next, using the tracing sheet (with a grid), she aligned the edges and measured the
length of the rec tangle.
Post-test
state
She said, "I was looking forward to it today," and "It's fun to find different lengths in
measuring activities."
Teachers'
findings
She is not good at detailed work. She is good at drawing. She loves playing football and
moving her body. S he wo uld rather move her bod y than sit and work. She is interested in the
alphabet.
Q12
He p o inted in a s eq uential order visually.
Q13
He p o inted in the descending o rder visually and quickly without using tools.
Q14
He p o inted in the descending o rder visually and quickly.
Q15
He p o inted in the descending o rder visually and quickly.
Post-test
state
He wo rked s ilently. He did no t us e any too ls at all in the measurement, and answered quickly
and visually. He enjoyed the math activities and espec ially enjoyed using the clips to measure
the room. He did not seem to be an noteworthy type.
Teachers'
findings
He is good at fine work and stud ying. Currently, he is learning division and frac tions
progressively. He is shy and not good at speaking in front of people.
T1
T7
Thematic Working Group 13
Proceedings of CERME11
2316
young children, it is often difficult for them to express in words; therefore, it is important for the
teacher to observe the situation and guess from the usual activities or the activities with other children.
From the countereffect by the program, it appears that by depending on teachers’ questioning,
summary, etc., children concentrate on the work without thinking about its meaning.
Conclusion
Children in this study showed different outcomes of learning in measurement. The children continued
to understand direct comparison in all activities; however, it was difficult to establish understanding
of measurement by non-universal units. The intervention activities seemed to affect their learning
outcomes, although we only focused on how two children’s actions related to their interview results.
Play-based activities offered by the young teachers influenced children’s understanding in both
positive and negative ways. This suggests that play-based activities are effective, but should be
meaningful and substantial mathematical activities, not superficial. This study only analyzed a small
number of children’s understanding of measurement, and therefore cannot be generalized further.
Future research should analyze a larger number of children.
Acknowledgment
This research was supported by a Grant-in-Aid for Science Research (B) (15H02911) in Japan.
References
Battista, M. T. (2006). Understanding the development of students’ thinking about length. Teaching
Children Mathematics, 13(3), 140–146.
Biggs, J., & Collis, K. (1982). Evaluating the quality of learning: SOLO taxonomy. New York,
NY: Academic Press.
Brandt, B. (2013). Everyday pedagogical practices in mathematical play situations in German
“kindergarten.” Educational Studies in Mathematics, 84(2), 227–248.
Ebersbach, M. (2009). Achieving a new dimension: Children integrate three stimulus dimensions in
volume estimations. Developmental Psychology, 45(3), 877–883.
Ernest, P. (1994). Social constructivism and the psychology of mathematics education. In P. Ernest
(Ed.), Constructing mathematics knowledge: Epistemology and mathematical education (pp. 68–
79). Bristol, PA: Falmer.
Goldin, G. A. (1998). Observing mathematical problem solving through task-based interviews. In A.
R. Teppo (Ed.), Qualitative research methods in mathematics education. Monograph 9, Journal
for Research in Mathematics Education (pp. 40–62). Reston, VA: NCTM.
Hachey, A. C. (2013). Early childhood mathematics education: The critical issue is change. Early
Education and Development, 24(4), 443–445.
Kamii, C. (2006). Measurement of length: How can we teach it better? Teaching Children
Mathematics, 13(3), 154–158.
Kotsopoulos, D., Makosz, S., Zambrzycka, J., & McCarthy, K. (2015). The effects of different
pedagogical approaches on the learning of length measurement in kindergarten. Early Childhood
Educational Journal, 43(6), 531–539.
Thematic Working Group 13
Proceedings of CERME11
2317
Lehrer, R., Jaslow, L., & Curtis, C. (2003). Developing an understanding of measurement in the
elementary grades. In D. H. Clements & G. W. Bright (Eds.), Learning and teaching measurement:
yearbook of the national council of teachers of mathematics (pp. 100–121). Reston, VA: NCTM.
Matsuo, N. (2016). Framework for an early mathematical preschool curriculum in Japan. In C.
Csíkos, A. Rausch, & J. Szitányi (Eds.), Proceedings of the 40th International Conference for the
Psychology of Mathematics Education, 3, 299–306.
Matsuo, N. (2017). Shugakuzen sansu kyoiku puroguram no teian: Hirosa kurabe/zukei no hamekomi
katsudou ni tsuite. [The preschool mathematical education program: Focus on the domains of
area measurement and embedding of geometric figures]. Tokyo Gakugei Journal of Mathematics
Education, 29, 63–72. (In Japanese).
Piaget, J. (1968). Quantification, conservation, and nativism. Science, 162, 976–979.
Sarama, J., Clements, D. H., Barrett, J., Van Dine, D. W., & McDonel, S. (2011). Evaluation of a
learning trajectory for length in the early years. ZDM, 43(5), 667–680.
Schoenfeld, A. H. (2016). Learning to think mathematically: Problem solving, metacognition, and
sense making in mathematics (reprint). Journal of Education, 196(2), 1–38.
Shinohara, S. (1942). Kyojyu genron [Teaching principle]. Tokyo, Japan: Iwanamisyoten. (In
Japanese).
Skoumpourdi, C. (2015). Kindergartners measuring length. In K. Krainer and N. Vondrová (Eds.),
Poceedings of the Ninth Congress of the European Society for Research in Mathematics Education
(pp. 1989–1995). Prague, Czech Republic: Charles University in Prague and ERME.
Smith, J., van den Heuvel-Panhuizen, M., & Teppo, A. (2011). Learning, teaching, and using
measurement: Introduction to the issue. ZDM, 43(5), 617–620.
Tzekaki, M., & Papadopoulou, E. (2017). Teaching intervention for developing generalization in
early childhood: the case of measurement. In T. Dooley and G. Gueudet (Eds.), Proceedings of the
Tenth Congress of the European Society for Research in Mathematics Education (pp. 1925–1932).
Dublin, Ireland: Institute of Education, Dublin City University and ERME.
Weisberg, D. S., Hirsh-Pasek, K., and Golinkoff, M. R. (2013) Guided play: Where curricular goals
meet a playful pedagogy. Mind, Brain, and Education, 7(2), 104–112.
Wilkening, F. (1979). Combining of stimulus dimensions in children’s judgments of area: An
information integration analysis. Developmental Psychology, 15, 25–33.
Wilson, P. A., & Osborne, A. (1992). Foundational ideas in teaching about measure. In T. R. Post
(Ed), Teaching mathematics in grades K-8: Research-based methods (pp. 89–121). Needham
Heights, MA: Allyn & Bacon.
llner, J., & Benz, C. (2013). How four to six year old children compare length indirectly. In B.
Ubuz, Ç. Haser, and M. Alessandra Mariotti (Eds.), Proceedings of the Eighth Congress of
European Research in Mathematics Education (CERME 8) (pp. 2258–2267). Ankara, Turkey:
Middle East Technical University.
Thematic Working Group 13
Proceedings of CERME11
2318
ResearchGate has not been able to resolve any citations for this publication.
Conference Paper
Full-text available
Article
Full-text available
Decades of research demonstrate that a strong curricular approach to preschool education is important for later developmental outcomes. Although these findings have often been used to support the implementation of educational programs based on direct instruction, we argue that guided play approaches can be equally effective at delivering content and are more developmentally appropriate in their focus on child-centered exploration. Guided play lies midway between direct instruction and free play, presenting a learning goal, and scaffolding the environment while allowing children to maintain a large degree of control over their learning. The evidence suggests that such approaches often outperform direct-instruction approaches in encouraging a variety of positive academic outcomes. We argue that guided play approaches are effective because they create learning situations that encourage children to become active and engaged partners in the learning process.
Article
Describes assessment tasks and a conceptual framework for understanding elementary students' thinking about the concept of length. Teachers will learn about student difficulties with length and how to differentiate instruction to reach these learners.
Article
Measurement of length is taught repeatedly starting in kindergarten and continuing in grades 1, 2, and beyond. However, during the past twenty-five years, according to the National Assessment of Educational Progress (NAEP), the outcome of this instruction has been disappointing. What is so hard about measurement of length? This article explains, on the basis of research, why instruction has been ineffective and suggests a better approach to teaching. Teachers can use these practical applications for teaching measurement in the classroom.
Article
This research investigated the effects of different pedagogical approaches on the learning of length measurement in kindergarten children. Specifically examined were the pedagogical approaches of guided instruction, center-based learning, and free exploration in the context of a play-based learning environment. This mixed design research was implemented in three different classrooms—with one classroom functioning as a control setting. Results suggest that neither guided instruction nor center-based approaches influenced learning more so than free exploration. Older children did better on the measurement tasks which suggests that age or a developmental progression, rather than the pedagogical approach, is more influential when learning how to measure. More children, regardless of grade, showed a preference for using rulers, albeit the older children were more accurate in their use. Education implications are discussed.
Article
"[Note: This abstract applies to all articles within this issue."] The chapters in this monograph describe qualitative research methods used to investigate students' and teachers' interactions with school mathematics. Each contributing author uses data from his or her own research to illustrate a particular technique or aspect of research design. The different chapters present a wide range of methods, representing a variety of goals and perspectives. Rather than a comprehensive reference manual, this monograph illustrates the diversity of methods available for qualitative research in mathematics education. The monograph begins with a discussion of key elements that contribute to the dynamic and evolving domain of mathematics education research. Background information is then provided that relates to the philosophical and epistemological assumptions underlying all qualitative research. In the chapters that follow, actual studies present the contexts for discussions of research design and techniques. Issues of research design include the importance of making explicit the underlying theoretical assumptions; the selection of an appropriate methodology; the interpretative, intersubjective nature of analysis; and the establishment of reliability and validity. Specific data collection techniques include clinical interviews, stimulated recall interviews, open-ended survey questions, and field notes and video or audio taping to record classroom events. Methods of analysis include participant validation, the categorization of data through constant comparison and software indexing and retrieval, phenomemographic analysis, and the identification of empirical examples of theoretical constructs. The monograph ends with a discussion of general issues, including the role of theory and the establishment of criteria for judging the goodness of qualitative research.
Article
This study describes situations in German daycare facilities (Kindergarten) in which the development of mathematical thinking in children is specifically encouraged through examination of common play objects. Using micro-sociological methods of analysis, the mathematical potential of such interactions between teacher and child is elaborated within the framework of everyday pedagogical practices (Bruner, 1996) and instructional models (Rogoff; Mind, Cult Activ 1(4): 209–229, 1994). It is also considered which concepts of mathematics may be important in these interactions.
Article
Conducted 2 experiments to study developmental changes in the integration of stimulus dimensions in an area judgment task. In Exp I, 24 kindergartners and 10 adults judged 9 rectangles in a 3 (width) × 3 (height) design. In Exp II, 10 each of 5-, 8-, 11-yr-olds, and adults judged 16 rectangles in a 4 × 4 design. Following functional measurement methodology, absolute judgments on a linear graphic rating scale were obtained. Data analyses showed that Ss at all ages based their area judgments on both width and height. The algebraic rule according to which these dimensions were combined, however, changed with age. Whereas the responses of the adults followed a multiplying integration model, the 5-yr-olds combined width and height additively. Between these age groups, there seems to be a gradual increase in the probability that a child will shift from an adding to a multiplying strategy. Implications for possible underlying processes are discussed. (19 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)