24 SES 07 A, Research in Mathematics Education
Students step into school mathematics through making sense of numbers (Jordan, Kaplan, Locuniak, & Ramineni, 2007). Students are expected to have some degree of understanding of numbers and operations to be successful at learning other mathematical concepts (Kaminski, 1997). Therefore, number sense has been an issue debated in the recent mathematics education literature. Among various number sense definitions, we found Howden’s (1989) one more comprehensive: “… exploring numbers, visualizing them in a variety of context, and relating them in ways that are not limited by traditional algorithms” (p.11). Students who developed strong number sense not only understand numbers and operations conceptually but also develop and use flexible strategies for real-life situations involving numbers and operations with numbers (Mcintosh, Reys, & Reys, 1992).
Subitizing was considered as an important component for “the foundation of number sense” (Huinker, 2011, p.7)since it supports children’s development of number sense through providing them with the ability to see quantities in groups quickly (Clements, 1999). As many other mathematical constructs, subitizing was one of the research phenomenon that took experimental psychologist attention before mathematics education researchers. From the eye of experimental psychologists, subitizing is a part of “numerosity estimation” (Mandler & Shebo, 1982), which allows people to determine quantities less than 6 (for infants less than 4) fluently and exactly without using mathematical processes (Hogan & Brezinski, 2003). This description corresponds to Clement’s perceptual subitizing (1999). He also defined conceptual subitizing as another type of subitizing leading to use certain mathematical processes to recognize quantities in larger groups. For conceptual subitizing, students need to recognize a number with different combinations in an exact and rapid way. Therefore, conceptual subitizing leads students to develop strategies contributing to number sense.
Griffin (2004) emphasized that teachers need to address the following five principles while supporting number sense development: (1) Focusing on students’ prior experiences and knowledge, (2) Considering students’ developmental pathway, (3) Improving mathematical competencies such as fluency, (4) Emphasizing problem solving and conceptual understanding, and (5) Constructing a mathematical community. Clements (1999) asserted that subitizing needs to be an integral part of school curriculum. While teaching conceptual subitizing, teachers need to address these principles; allowing students to use their prior knowledge of number and providing discussion on “spatial arrangement of sets” (Clements, 1999). As students become frequent users of subitizing, they flexibly use numbers and recognize numbers greater than four within part-part-whole relationship (Karp, Cardwell, Zbiek, & Bay Williams, 2011).
In recent years, subitizing has started to be investigated thoroughly in the context of different school curricula and cultures in the world. While Clements and his colleagues presented the use of conceptual subitizing from the USA perspective (Clements & Sarama, 2009), Andrews and Sayers (2015) investigated the development of early number sense in European context. They particularly developed activities to enhance conceptual subitizing of Hungarian and Swedish first grade students (Andrews, Sayers, & Boistrup, 2016). In Turkish context, on the other hand, Olkun and Özdem (2015) explored the effects of conceptual subitizing instruction on second and third grade students’ development of subitizing skills.
Considering the importance of this research phenomenon, in this study weaimed to explore how first grade students conceptualize quantities in groups through subitizing activities. This research is a part of a long-term project funded by The Scientific and Technological Research Council of Turkey (TUBITAK). The overarching goal of the parent project is to develop a learning trajectory on number sense for Turkish first graders. Subitizing activities, that aims to develop conceptual subitizing, are an essential part of the hypothetical learning trajectory (HLT), which intends to promote various dimensions of number sense.
The Design of the Study & Participants This study, a part of a long-term project, is a design-based research. The participants involve two first-grade classrooms of a small public school in the capital of Turkey. There were 45 students in total and two first-grade teachers having over 20 years of teaching experience. It is important to note that teachers were willing to collaborate with the research team as a part of design-based research. In this current study, we focused on eight students (four in each classroom) to analyze their development of number sense in detailed. The Data Collection & Data Analysis The data were collected during subitizing activities and semi-structured interviews. . There were eight subitizing activities performed in each classroom and two interviews conducted with the selected students. The first interview was conducted after introducing the numbers from 1 to 5 and the second was after introducing thenumbers from 6-10. Classroom observations and interviews, both video recorded and transcribed, were the main data sources. The dot cards were the main manipulative in subitizing activities. The set of dot cards contains all the numbers that students knew by the time of the activity. For example, while teaching number 5, the shown dot cards included five or less dots. Subitizing activities were conducted after students were introduced number 3. For the context of the study and the grade level being taught, subitizing activities were restricted to teaching numbers between 0 and 10. For classroom discussion, the students were told that subitizing activities require attention; they needed to follow each card and keep three following questions in mind: (1) "How many dots did you see?", (2) "How are the dots arranged in the cards?" and (3) "How do you decide the quantity?". The dot cards were shown only for 2-3 seconds. When the majority of the classroom could not raise their hands, the cards were re-shown. After showing the cards, teacher asked the first question above. To probe correct responses, teachers asked the remaining questions that required more explanations. These activities took 5 to 10 minutes. To explore the communication between students and the teacher during subitizing activities, video records were analyzed. The codes appeared in the videos were thematized with the focus of the changes in students' subitizing skills. These themes were also compared and contrasted with the ones emerged from students' interviews.
Findings In students' first experience with subitizing the number 3, they rapidly recognized the number of dots in the given set. After showing the dot-card briefly (about 3 seconds) and hiding it, teachers demanded explanations based on the locations of dots in the card. Although it was their first subitizing activity, students did not have any difficulty to recognize the arrangement of dots. Before the subitizing activities start, students were engaged with quantifying objects in different arrangements as part of the HLT. This might have helped students in quickly recognizing arrangements of the dots, as well as their cardinality. Furthermore, students engaged in spatial relationships activities before they were introduced numbers, so their familiarity with the spatial arrangement of the dots might have helped them in perceiving the number. Students might still be using perceptual subitizing skills for the numbers 1-5. For 6-10 number group, subitizing activities become more engaging. For example, the teacher asked more probing questions about dot cards, resulting in more student talk about numbers than before. When teacher showed the dot-card of 6, one student's response was "six,… because I saw so". Teacher demanded an explanation from another student for the same card; the other student said "I did addition. I added 3 and 3". For certain cases,teachers warned students "you can decide by grouping, not counting one by one". The teachers promoted conceptual subitizing by demanding part-part-whole relationship. Additionally, some students developed conceptual subitizing. For instance, when the student was shown 6-dot-card, he partitioned the number in two groups ( 4 and 2) he can perceive easily and combined them to conceptualize the number 6, even before he was not formally taught the addition. This type of subitizing activities supported the development of students' number sense and helped them to become more fluent with number relations.
REFERENCES Andrews, P. & Sayers, J. (2015). Identifying opportunities for grade one children to acquire foundational number sense: Developing a framework for cross cultural classroom analyses. Early Childhood Education Journal, 43(4), 257-267. Caldwell, J. H., Karp, K., Bay-Williams, J. M., Rathmell, E., & Zbiek, R. M. (2011). Developing essential understanding of addition and subtraction for teaching mathematics in prekindergarten-grade 2. VA: NCTM. Clements D.H., & Sarama, J. (2009). Learning and teaching early math. New York: Routledge. Clements, D. H. (1999). Subitizing: What is it? Why teach it? Teaching children mathematics, 5(7), 400-408. Griffin, S. (2004). Building number sense with number worlds: A mathematics program for young children. Early Childhood Research Quarterly,19(1), 173-180. Hogan, T. P., & Brezinski, K. L. (2003). Quantitative estimation: One, two, or three abilities? Mathematical Thinking and Learning, 5(4), 259-280. Howden, H. (1989). Teaching number sense. Arithmetic Teacher, 36, 6-11. Huinker, D. (2011). Beyond counting by ones: "Thinking groups" as a foundation for number and operation sense. Wisconsin Teacher of Mathematics, 63(1), 7-11. Jordan, N. C., Kaplan, D., Locuniak, M. N., & Ramineni, C. (2007). Predicting first‐grade math achievement from developmental number sense trajectories. Learning Disabilities Research & Practice, 22(1), 36-46. Kaminski, E. (1997). Teacher education students' number sense: Initial explorations. Mathematics Education Research Journal, 9(2), 225-235. Mandler, G., & Shebo, B. J. (1982). Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General, 111(1), 1-22. McIntosh, A., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense. For the learning of mathematics, 12(3), 2-44. Olkun, S., & Özdem, Ş. (2015). Kavramsal şipşak sayılama uygulamalarının hesaplama performansına etkisi. Başkent University Journal of Education, 2(1), 1-9. Sayers, J., Andrews, P., & Björklund Boistrup, L. (2014). The role of conceptual subitising in the development of foundation al number sense. In T. Meaney (Ed.), A mathematics education perspective on early mathematics learning. Malmö, Sweden: Malmö Högskolan. Sowder, J. T. (1992). Making sense of numbers in school mathematics. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds), Analysis of Arithmetic for Mathematics Teaching (pp. 1-51). Hillsdale, NJ: Erlbaum.
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