# Effects of Physical Manipulative Instructions with or without Explicit Metacognitive Questions on Geometrical Knowledge Acquisition

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## Abstract

In this study, effects of 2 physical manipulative instructions on students’ geometrical knowledge acquisition were compared. Geometry topics’ presentation was conducted via using physical manipulative and the practice stage in two different forms: (1) practice consisting of problems supported with explicit metacognitive questions and (2) practice consisting of problems not supported with explicit metacognitive questions. Participants were 220 6th grade students, studying in 5 classrooms. Students’ knowledge acquisitions were tested by administering knowledge tests before and after the instruction. Furthermore, follow-up interviews were conducted with randomly selected students according to their achievement levels to get their views about the effect of manipulative instruction on their geometrical knowledge acquisition. The results indicated that the two instructions were equally effective in promoting geometrical knowledge acquisition. Metacognitive questions together with manipulative use and group work seemed to help partly students’ knowledge acquisition. Hence, active involvement, both physically and cognitively, seemed to play a crucial role.

## Keywords

Geometry Knowledge acquisition Physical manipulative Metacognitive questions## References

- Aburime, E. F. (2007). How manipulatives affect the mathematics achievement of students in Nigerian schools.
*Education Research Quarterly, 3*(1), 3–16.Google Scholar - Aktaş, Ş., Atalay, A., Aygün, S.Ç., Aynur, N., Bilge, O., Çelik, M., . . . Ünsal, N. (2006).
*İlköğretim 6.sınıf matematik ders kitabı*[Elementary 6. grade mathematics course book]. Ankara, Turkey: Cem Veb Ofset.Google Scholar - Alexander, P. A., & Judy, J. E. (1988). The interaction of domain-specific and strategic knowledge in academic performance.
*Review of Educational Research, 58*(4), 375–404.Google Scholar - Alexander, P. A., Schallert, D. L., & Hare, V. C. (1991). Coming to terms: How researchers in learning and literacy talk about knowledge.
*Review of Educational Research, 61*(3), 315–343.Google Scholar - Baki, A., Kösa, T., & Güven, B. (2011). A comparative study of the effects of using dynamic geometry software and physical manipulatives on the spatial visualization skills of pre-service mathematics teachers.
*British Journal of Educational Technology, 42*(2), 291–310.Google Scholar - Bielaczyc, K., Pirolli, P. L., & Brown, A. L. (1995). Training in self-explanation and self-regulation strategies: Investigating the effects of knowledge acquisition activities on problem solving.
*Cognition and Instruction, 13*(2), 221–252.Google Scholar - Boulton-Lewis, G. M. (1993a). An assessment of the processing load of some strategies and representations for subtraction used by teachers and young children.
*Journal of Mathematical Behavior, 12*(4), 387–409.Google Scholar - Boulton-Lewis, G. M. (1993b). Young children’s representations and strategies for subtraction.
*British Journal of Educational Psychology, 64*, 451–456.Google Scholar - Boulton-Lewis, G., Cooper, T., Atweh, B., Pillay, H., Wilson, L., & Mutch, S. (1997). Processing load and the use of concrete representations and strategies for solving linear equations.
*Journal of Mathematical Behavior, 16*(4), 379–397.Google Scholar - Bruner, J. S. (1986).
*Actual minds, possible worlds*. Cambridge, MA: Harvard University Press.Google Scholar - Butler, D. L., & Winne, P. H. (1995). Feedback and self-regulated learning: A theoretical synthesis.
*Review of Educational Research, 65*(3), 245–281.Google Scholar - Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives.
*Journal of Educational Psychology, 105*(2), 380–400.Google Scholar - Clements, D. H. (1999). ‘Concrete’ manipulatives, concrete ideas.
*Contemporary Issues in Early Chidhood, 1*(1), 45–60.Google Scholar - Cohen, J. (1988).
*Statistical power analysis for the behavioral sciences*(2nd ed.). Hillsdale, NJ: Erlbaum.Google Scholar - Duatepe-Paksu, A. & Ubuz, B. (2009). Effects of drama-based geometry instruction on student achievement, attitudes and thinking levels.
*Journal of Educational Research, 102*(4), 272–286.Google Scholar - Field, A. (2009).
*Discovering statistics using SPSS*(3rd ed.). London, England: Sage Publications.Google Scholar - Flavell, J. H. (1979). Meta-cognitive and cognitive monitoring: A new area of cognitive developmental inquiry.
*American Psychologist, 31*, 906–911.Google Scholar - Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning/teaching approach for first- and second-grade place-value and multidigit addition and subtraction.
*Journal for Research in Mathematics Education, 21*, 180–206.Google Scholar - Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring and mathematical performance.
*Journal for Research in Mathematics Education, 16*, 163–176.Google Scholar - Goetz, T., Hall, N. C., Frenzel, A. C., & Pekrun, R. (2006). A hierarchical conceptualization of enjoyment in students.
*Learning and Instruction, 16*, 323–338.Google Scholar - Hair, J. F., Anderson, R. E., Tatham, R. L. & Black, W. C. (1992).
*Multivariate data analysis with readings*. New York, NY: Macmillan.Google Scholar - Hart, K. (1989). There is little connection. In P. Ernest (Ed.),
*Mathematics teaching: The state of the art*(pp. 138–142). London, England: Falmer Press.Google Scholar - Kamii, C., Lewis, B. A., & Kirkland, L. (2001). Manipulatives: When are they useful?
*Journal of Mathematical Behavior, 20*, 21–31.Google Scholar - Kramarski, B. (2004). Making sense of graphs: Does metacognitive instruction make a difference on students’ mathematical conceptions and alternative conceptions.
*Learning and Instruction, 14*, 593–619.Google Scholar - Kramarski, B., & Gutman, M. (2006). How can self-regulated learning be supported in mathematical E-learning environments.
*Journal of Computer Assisted Learning, 22*, 24–33.Google Scholar - Kramarski, B., & Mevarech, Z. R. (2003). Enhancing mathematical reasoning in the classroom: The effect of cooperative learning and metacognitive training.
*American Educational Research Journal, 40*, 281–310.Google Scholar - Kramarski, B., Mevarech, Z. R., & Arami, M. (2002). The effects of metacognitive training on solving mathematical authentic tasks.
*Educational Studies in Mathematics, 49*, 225–250.Google Scholar - Lane, S. (1993). The conceptual framework for the development of a mathematics performance instrument.
*Educational Measurement: Issues and Practice, 12*(2), 16–23.Google Scholar - Lester, F.K., Garofalo, J. & Kroll, D.L. (1989).
*The role of metacognition in mathematical problem solving*. Final Report. Bloomington, IN: Indiana University, Mathematics Education Development Centre. Retrieved from ERIC database. (ED 314255).Google Scholar - Lucangeli, D., & Cornoldi, C. (1997). Mathematics and metacognition: What is the nature of the relationship?
*Mathematical Cognition, 3*, 121–139.Google Scholar - Martin, T., Lukong, A., & Reaves, R. (2007). The role of manipulatives in arithmetic and geometry tasks [Electronic Version].
*Journal of Education and Human Development*,*1*(1). Retrieved from http://www.scientificjournals.org/journals2007/articles/1073.htm. - Marzano, R. J. (1992).
*A different kind of classroom: Teaching with dimensions of learning*. Alexandria, Egypt: ASCD.Google Scholar - Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment.
*Educational Studies in Mathematics, 38*(1–3), 135–161.Google Scholar - Mevarech, Z., & Fridkin, S. (2006). The effects of IMPROVE on mathematical knowledge, mathematical reasoning and meta-cognition.
*Metacognition and Learning, 1*, 85–97.Google Scholar - Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A multidimentional method for teaching mathematics in heterogeneous classrooms.
*American Educational Research Journal, 34*, 365–394.Google Scholar - Milli Eğitim Bakanlığı-Talim ve Terbiye Kurulu Başkanlığı (MEB) (2005).
*İlköğretim matematik dersi öğretim programı ve kılavuzu (6–8. sınıflar*) [Elementary mathematics course curriculum framework (6–8 grades)]. Ankara, Turkey: Devlet Kitapları Müdürlüğü.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Standards & focal points*. Retrieved from http://www.nctm.org/standards/default. aspx?id_58. - Olkun, S. (2003). Comparing computer versus concrete manipulatives in learning 2D geometry.
*Journal of Computers in Mathematics and Science Teaching, 22*(1), 43–56.Google Scholar - Olkun, S. & Toluk, Z. (2004). Teacher questioning with an appropriate manipulative may make a big difference.
*Issues in the Undergraduate Mathematics Preparation of School Teachers: The Journal, Vol 2 (Pedagogy),*1–10. Retrieved from www.k-12prep.math.ttu.edu. - Özer, H., Budak, M., Altınordu, R. & Çatal, Z. (2001).
*İlköğretim matematik kitabı*[Elementary mathematics book].*Öğretmen kılavuzu*. İstabul, Turkey: Özer Yayıncılık.Google Scholar - Pallant, J. (2003).
*SPSS survival manual: A step by step guide to data analysis using SPSS for windows*(1st ed.). New South Wales, Australia: Allen & Unwin.Google Scholar - Paris, S. G. & Winograd, P. (1990). How metacognition can promote academic learning and instruction. In B. J. Jones & L. Idol (Eds.),
*Dimensions of thinking and cognitive instruction*(pp. 15–51). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.Google Scholar - Perrer-Clermont, A. N. (1980).
*Social interaction and cognitive development in children*. New York, NY: Academic Press.Google Scholar - Piaget, J. (1970).
*Science of education and the psychology of the child*(D. Coltman, Trans.). New York, NY: Orion Press.Google Scholar - Piaget, J. (1976).
*The grasp of consciousness*. Cambridge, MA: Harvard University Press.Google Scholar - Polya, G. (1957).
*How to solve it?*(2nd ed.). Princeton, NJ: Princeton University Press.Google Scholar - Prigge, G. R. (1978). The differential effects of the use of manipulative aids on the learning of geometric concepts by elementary school children.
*Journal for Research in Mathematics Education, 9*(5), 361–367.Google Scholar - Reimer, K. & Moyer, P. S. (2005). Third-graders learn about fractions using virtual manipulatives: A classroom study.
*Journal of Computers in Mathematics and Science Teaching, 24*(1), 5–25.Google Scholar - Schneider, W. & Artelt, C. (2010). Metacognition and mathematics education.
*ZDM: The International Journal on Mathematics Education, 42*, 149–161.Google Scholar - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. San Diego, CA: Academic.Google Scholar - Schoenfeld, A. H. (1987). What’s All the fuss about metacognition? In A. H. Schoenfeld (Ed.),
*Cognitive science and mathematics education*(pp. 189–215). Hillsdale, MI: Erlbaum.Google Scholar - Schraw, G., & Moshman, D. (1995). Metacognitive theories.
*Educational Psychology Review, 7*, 351–371.Google Scholar - Shin, N., Jonassen, D. & McGee, S. (2003). Predictors of well-structured and ill-structured problem solving in an astronomy simulation.
*Journal of Research in Science Teaching, 40*, 6–33.Google Scholar - Skemp, R. (1979).
*Intelligence, learning, and action*. New York, NY: Wiley.Google Scholar - Skemp, R. R. (1987).
*The psychology of learning mathematics*. Hillsdale, NJ: Erlbaum.Google Scholar - Smith, P. L. & Ragan, T. J. (1993).
*Instructional design*. New York, NY: Macmillan.Google Scholar - Steen, K., Brooks, D., & Lyon, T. (2006). The impact of virtual manipulatives on first grade geometry instruction and learning.
*Journal of Computers in Mathematics and Science Teaching, 25*(4), 373–391.Google Scholar - Suh, J., & Moyer, P. S. (2007). Developing students’ representational fluency using virtual and physical algebra balances.
*Journal of Computers in Mathematics and Science Teaching, 26*, 155–173.Google Scholar - Suydam, M. N. & Higgens, J. L. (1977).
*Review and synthesis of studies activity-based approach to mathematics teaching*. Columbus, OH: ERIC/SMEAC.Google Scholar - Ubuz, B. (1994). Problem-solving method with handout material: Max-min word problems.
*International Journal of Mathematical Education in Science and Technology, 25*(3), 367–376.Google Scholar - Ubuz, B. & Ersoy, Y. (1997). The effect of problem solving method with handout material on achievement in solving max-min word problems.
*The Journal of Mathematical Behavior, 16*(1), 75–85.Google Scholar - Ubuz, B. & Duatepe-Paksu, A. (2016). Teaching and learning geometry in drama based instruction.
*European Journal of Science and Mathematics Education, 4*(2), 176–185.Google Scholar - Veenman, M. V. J. (2006). The role of intellectual and metacognitive skills in math problem solving. In A. Desoete & M. Veenman (Eds.),
*Metacognition in mathematics education*(pp. 35–50). Haupauge, NY: Nova Science.Google Scholar - Veenman, M., van Hout-Wolters, B., & Afflerbach, P. (2006). Metacognition and learning: Conceptual and methodological considerations.
*Metacognition and Learning, 1*(1), 3–14.Google Scholar