# A Study on Sixth Grade Students’ Misconceptions and Errors in Spatial Measurement: Length, Area, and Volume

- 2k Downloads
- 3 Citations

## Abstract

The purpose of the present study was to portray students’ misconceptions and errors while solving conceptually and procedurally oriented tasks involving length, area, and volume measurement. The data were collected from 445 sixth grade students attending public primary schools in Ankara, Türkiye via a test composed of 16 constructed-response format tasks. The findings revealed a wide range of misconceptions and errors such as “believing that all rulers are 30 cm long,” “confusing area formula with perimeter formula,” “believing a box has more than one surface area,” “using the volume formula for surface area,” “believing that ruler must be longer than the object measured,” etc. These misconceptions and errors could be considered as the evidences indicating the sixth graders’ lack of comprehension of the fundamental concepts of spatial measurement and their relationships and the procedures and formulas used for measuring length, area, and volume. The possible causes of such misconceptions and overcoming ways were also discussed.

## Keywords

Area Conceptual knowledge Errors Length Mathematical misconceptions Procedural knowledge Spatial measurement Volume measurement## Supplementary material

## References

- An, S. & Wu, Z. (2012). Enhancing mathematics teachers’ knowledge of students’ thinking from assessing and analysis misconceptions in homework.
*International Journal of Science and Mathematics Education, 10*(3), 717–753.CrossRefGoogle Scholar - Anderson, J. (1983).
*The architecture of cognition*. Cambridge, MA: Harvard University.Google Scholar - Ashlock, B. (1990).
*Error patterns in computation*. Columbus, OH: Merrill.Google Scholar - Barrett, J. E., Clements, D. H., Klanderman, D., Pennisi, S.-J. & Polaki, M. V. (2006). Students’ coordination of geometric reasoning and measuring strategies on a fixed perimeter task: Developing mathematical understanding of linear measurement.
*Journal for Research in Mathematics Education, 37*(3), 187–221.Google Scholar - Battista, M. T. (1999). Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquiry-based classroom.
*Journal for Research in Mathematics Education, 30*, 417–448.CrossRefGoogle Scholar - Battista, M. T. (2003). Understanding students’ thinking about area and volume measurement. In D. H. Clements & G. Bright (Eds.),
*Learning and teaching measurement: 2003 Yearbook*(pp. 122–142). Reston, VA: NCTM.Google Scholar - Battista, M. T. (2006). Understanding the development of students thinking about length.
*Teaching Children Mathematics, 13*, 140*–*146.Google Scholar - Battista, M. T. & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes.
*Journal of Research in Mathematics Education, 27*(3), 258–292.CrossRefGoogle Scholar - Ben-Haim, D., Lappan, G. & Houang, R. T. (1985). Visualizing rectangular solids made of small cubes: Analyzing and effecting students’ performance.
*Educational Studies in Mathematics, 16*(4), 389–409.CrossRefGoogle Scholar - Borasi, R. (1996).
*Reconceiving mathematics instruction: A focus on errors*. Norwood, NJ: Ablex Publishing.Google Scholar - Boulton-Lewis, G. M., Wilss, L. A. & Mutch, S. L. (1996). An analysis of young children’s strategies and use of devices for length measurement.
*Journal of Mathematical Behavior, 15*, 329–347.CrossRefGoogle Scholar - Bragg, P. & Outhred, L. (2000).
*What is taught versus what is learnt: The case of linear measurement*. Paper presented at twenty-third annual conference of the Mathematics Education Research Group of Australasia, Fremantle, Western Australia.Google Scholar - Bragg, P. & Outhred, L. (2001). Students’ knowledge of length units: Do they know more than rules about rulers? In M. van den Heuvel-Panhuizen (Ed.),
*Proceedings of the 25th Annual Conference of the PME*(Vol. 1, pp. 377–384). Utrecht, The Netherlands: Program Committee.Google Scholar - Bragg, P. & Outhred, N. L. (2004). A measure of rulers: The importance of units in a measure. In M. J. Høines & A. B. Fuglestad (Eds.),
*Proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 159–166). Bergen, Norway: Program Committee.Google Scholar - Campbell, K. J., Watson, J. M. & Collis, K. F. (1992). Volume measurement and ıntellectual development.
*Journal of Structural Learning, 11*(3), 279–298.Google Scholar - Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P. & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: an experimental study.
*American Educational Research Journal, 26*(4), 499–531.CrossRefGoogle Scholar - Carpenter, T. P. & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennema & T. R. Romberg (Eds.),
*Mathematics classrooms that promote understanding*(pp. 19–32). Mahwah, NJ: Erlbaum.Google Scholar - Chappell, M. F. & Thompson, D. R. (1999). Perimeter or area?: Which measure is it?
*Mathematics Teaching in the Middle School, 5*(1), 20–23.Google Scholar - Cohen, A. & Moreh, A. (1999). Hands-on method for teaching the concept of the ratio between surface area and volume.
*American Biology Teacher, 61*(9), 691–695.Google Scholar - Curry, M., Mitchelmore, M. C. & Outhred, L. (2006). Development of children’s understanding of length, area and volume measurement principles. In J. Novotná, H. Moraová, M. Krátká & N. Stehlíková (Eds.),
*Proceedings of the 30th annual conference of the PME*(Vol. 2, pp. 377–384). Prague, Czechia: Program Committee.Google Scholar - De Jong, T. & Ferguson-Hessler, M. (1996). Types and qualities of knowledge.
*Educational Psychologist, 31*(2), 105–113.CrossRefGoogle Scholar - Dietiker, L., Gonulates, F. & Smith III, J. P. (2011). Enhancing opportunities for student understanding of length measure.
*Teaching Children Mathematics*,*18*(4), 252–259.Google Scholar - Drews, D. (2005). Children’s mathematical errors and misconceptions: Perspectives on the teacher’s role. In A. Hansen (Ed.),
*Children’s errors in mathematics: Understanding common misconceptions in primary schools*. Exeter, England: Learning Matters.Google Scholar - Eames, C. L., Miller, A. L., Barrett, J., Cullen, C., Kara, M., Clements, D. H., Sarama, J. & Van Dine, D. (2014).
*Interactions among hypothetical learning trajectories for length, area, and volume measurement*. New Orleans, LA: NCTM.Google Scholar - Even, R. & Tirosh, D. (2008). Teacher knowledge and understanding of students’ mathematical learning and thinking. In L. English (Ed.),
*Handbook of international research in mathematics education*(2nd ed.). Mahwah, NJ: Erlbaum.Google Scholar - Fennema, E., Carpenter, T., Frankie, M., Levi, L., Jacobs, V. & Empson, S. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction.
*Journal for Research in Mathematics Education, 27*, 403–434.CrossRefGoogle Scholar - Fraenkel, J. R. & Wallen, N. E. (2006).
*How to design and evaluate research in education*. New York, NY: McGraw-Hill.Google Scholar - Furinghetti, F. & Paola, D. (1999). Exploring students’ images and definitions of area. In O. Zaslavski (Ed.),
*Proceedings of PME 23*(pp. 345–352). Haifa, Israel: PME.Google Scholar - Gagatsis, A. & Kyriakides, L. (2000). Teachers’ attitudes towards their pupils’ mathematical errors.
*Educational Research and Evaluation, 6*(1), 24–58.CrossRefGoogle Scholar - Gelman, R. & Williams, E. M. (1998). Enabling constraints for cognitive development and learning: Domain specificity and epigenesis. In D. Kuhn, D. Kuhn & R. S. Siegler (Eds.),
*Handbook of child psychology: Cognition, perception, and language (5th ed.)*(Vol. 2, pp. 575–630). New York, NY: Wiley.Google Scholar - Gilbert, S. (1982). Surface, volume and elephant’s ears.
*The Science Teacher, 49*, 14–20.Google Scholar - Grant, T. J. & Kline, K. (2003). Developing building blocks of measurement with young children. In D. H. Clements & G. Bright (Eds.),
*Learning and teaching measurement: 2003 Yearbook*(pp. 46–56). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Gray, E. & Tall, D. (1993). Success and failure in mathematics: The flexible meaning of symbols as process and concept.
*Mathematics Teaching, 142*, 6–10.Google Scholar - Greeno, G., Collins, M. & Resnick, L. (1996). Cognition and learning. In D. Berli’er & R. Calfee (Eds.),
*Handbook of educational psychology*(pp. 15–46). New York, NY: Macmillan.Google Scholar - Haapasalo, L. & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation.
*Journal fürMathematik-Didaktik, 21*(2), 139–157.CrossRefGoogle Scholar - Hadjidemetriou, C. & Williams, J. S. (2002). Children’s graphical conceptions.
*Research in Mathematics Education, 4*, 69–87.CrossRefGoogle Scholar - Hiebert, J. & Carpenter, T. (1992). Learning and teaching with understanding. In D. Grouws (Ed.),
*Handbook of research on mathematics research and teaching*(pp. 65–100). New York, NY: Macmillan.Google Scholar - Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Erlbaum.Google Scholar
- Hirstein, J. J., Lamb, C. E. & Osborne, A. (1978). Student misconceptions about area measure.
*Arithmetic Teacher, 25*(6), 10–16.Google Scholar - Hook, W. (2004).
*Curriculum makes a huge difference—a summary of conclusions from the TIMSS with California data added*. Unpublished report, March 5, 2004. Retrieved from www.nychold.com/report-hook-040305.pdf. Accessed 13 Apr 2015. - Kadijevich, D. (1999). Conceptual tasks in mathematics education.
*The Teaching of Mathematics, 2*(1), 59–64.Google Scholar - Kamii, C. (1995).
*Why is the use of a ruler so hard?*Paper presented at the 17th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, The Ohio State University, Columbus, OH.Google Scholar - Kamii, C. & Clark, F. (1997). Measurement of length: The need for a better approach to teaching.
*School Science and Mathematics, 97*(3), 116–121.CrossRefGoogle Scholar - Kamii, C. & Kysh, J. (2006). The difficulty of “length x width”: Is a square the unit of measurement?
*Journal of Mathematical Behavior, 25*, 105–115.CrossRefGoogle Scholar - Kembitzky, K. (2009).
*Addressing misconceptions in geometry through written error analyses.*Retrieved from https://etd.ohiolink.edu/. - Kidman, G. & Cooper, T. J. (1997). Area integration rules for grades 4, 6, 8 students. In E. Pehkonen (Ed.),
*Proceedings of the 21st PME*(Vol. 3, pp. 132–143). Lahti, Finland: University of Finland.Google Scholar - Kilpatrick, J., Swafford, J. & Findell, B. (Eds.). (2001).
*Adding it up: helping children learn mathematics*. Washington, DC: National Academy Press.Google Scholar - Kordaki, M. & Portani, D. (1998). Children’s approaches to area measurement through different contexts.
*Journal of Mathematical Behavior, 17*(3), 303–316.CrossRefGoogle Scholar - Kulm, G. (1994).
*Procedural and conceptual learning. Mathematics assessment. What works in the classroom?*San Francisco, CA: Jossey-Bass.Google Scholar - Larsen, B. (2006).
*Math handbook for water system operators: Math fundamentals and problem solving*. Denver, CO: Outskirts Press.Google Scholar - Legutko, M. (2008). An analysis of students’ mathematical errors in the education-research process. In B. Czarnocha (Ed.),
*Handbook of mathematics teaching research: Teaching experiment -a tool for teacher-researchers*(pp. 141–154). Poland: Drukarnia Cyfrowa Kserkop.Google Scholar - Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin & D. E. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*(pp. 179–192). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Lewis, C. & Schad, B. (2006). Teaching and learning measurement: By way of introduction.
*Teaching Children Mathematics, 13*(3), 131.Google Scholar - Light, G., Swarat, S., Park, E. J., Drane, D., Tevaarwerk, E. & Mason, T. (2007).
*Understanding undergraduate students’ conceptions of a core nanoscience concept: Size and scale*. Paper presented at the Proceedings of the International Conference on Research in Engineering Education, Honolulu, Hawaii.Google Scholar - Martin, G. W. & Strutchens, M. E. (2000). Geometry and measurement. In E. A. Silver & P. A. Kenney (Eds.),
*Results from the seventh mathematics assessment of the national assessment of educational progress*(pp. 193–234). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Melis, E. (2004). Erroneous examples as a source of learning in mathematics erroneous examples. In D. G. Sampson & P. Isaias (Eds.),
*International conference: Cognition and exploratory Learning in the Digital Age*(pp. 311–318). New York, NY: Curran Associates.Google Scholar - Moyer, S. P. (2001). Using representations to explore perimeter and area.
*Teaching Children Mathematics, 8*(1), 52.Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and standards for school mathematics*. Reston, VA: The National Council of Teachers of Mathematics.Google Scholar - Novak, J. D. (2009). Foreword. In K. Afamasaga-Fuata’i (Ed.)
*, Concept mapping in mathematics: Research into practice*. Berlin, Germany: Springer. doi: 10.1007/978-0-387-89194-1. - Nührenbörger, M. (2001).
*Children’s measurement thinking in the context of length*. Paper presented at Annual Conference on Didactics of Mathematics, Ludwigsburg, Germany.Google Scholar - Nunes, T., Light, P. & Mason, J. (1993). Tools for thought: The measurement of length and area.
*Learning and Instruction, 3*, 39–54.CrossRefGoogle Scholar - Ohlsson, S. & Rees, E. (1991). The function of conceptual understanding in the learning of arithmetic procedures.
*Cognition and Instruction, 8*(2), 103–179.CrossRefGoogle Scholar - Outhred, L. N. & Mitchelmore, M. C. (2000). Young children’s intuitive understanding of rectangular area measurement.
*Journal for Research in Mathematics Education, 31*, 144–167.CrossRefGoogle Scholar - Outhred, L., Mitchelmore, M. C., McPhail, D. & Gould, P. (2003). Count me into measurement: A program for the early elementary school. In D. H. Clements & G. Bright (Eds.),
*Learning and teaching measurement: 2003 Yearbook*(pp. 81–99). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Owens, K. & Outhred, L. (2006). The complexity of learning geometry and measurement. In A. Gutiérrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present and future*(pp. 83–115). Rotterdam, Netherlands: Sense.Google Scholar - Putnam, R. T. (1987). Mathematics knowledge for understanding and problem solving.
*International Journal of Educational Research, 11*(6), 687–705.CrossRefGoogle Scholar - Radatz, R. (1979). Error analysis in mathematics education.
*Journal for Research in Mathematics Education, 10*(3), 163–172.CrossRefGoogle Scholar - Radatz, H. (1980). Students’ errors in the mathematical learning process.
*For the Learning of Mathematics, 1*(1), 16–20.Google Scholar - Rittle-Johnson, B., Siegler, R. S. & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process.
*Journal of Educational Psychology, 93*(2), 346–362.CrossRefGoogle Scholar - Robinson, E., Mahaffey, M. & Nelson, D. (1975). Measurement. In J. N. Payne (Ed.),
*Mathematics learning in early childhood 37th Year Book*(pp. 228–250). Reston, VA: NCTM.Google Scholar - Ryan, J. & Williams, J. (2007).
*Children’s mathematics 4–15: Learning from errors and misconceptions*. Maidenhead, England: Open University Press.Google Scholar - Sahin, A. E. 2008. A qualitative assessment of the quality of Turkish elementary schools.
*Eurasian Journal of Educational Research, 30*, 117–139.Google Scholar - Sáiz, M. (2003).
*Primary teachers’ conceptions about the concept of volume: The case of volume-measurable objects*. Paper presented at the 27th International PME, Honolulu, HI.Google Scholar - Sarama, J. & Clements, D. H. (2009).
*Early childhood mathematics education research: Learning trajectories for young children*. New York, NY: Routledge.Google Scholar - Schneider, M. & Stern, E. (2006).
*The integration of conceptual and procedural knowledge: Not much of a problem?*Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA.Google Scholar - Schwartz, S. L. (1995). Developing power in linear measurement.
*Teaching Children Mathematics, 1*(7), 412–417.Google Scholar - Skemp, R. R. (1978). Relational understanding and instrumental understanding.
*Arithmetic Teacher, 26*(3), 9–15.Google Scholar - Smith, J. P. (2007).
*Tracing the origins of weak learning of spatial measurement.*Presentation at the Mathematics Education Colloquium, Michigan State University, East Lansing, MI. Retrieved on May 16, 2013 from https://www.msu.edu/~stemproj/presentations/STEM_MSUColloquium_2007.pdf. - Smith, J., diSessa, A. & Rochelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition.
*Journal of Learning Sciences, 3*(2), 115–163.CrossRefGoogle Scholar - Sophian, C. (1997). Beyond competence: The significance of performance for conceptual development.
*Cognitive Development, 12*, 281–303.CrossRefGoogle Scholar - Star, J. R. (2000). Re-“conceptualizing” procedural knowledge in mathematics. In M. Fernandez (Ed.),
*Proceedings of the 22nd Annual meeting of the PME -NA*(pp. 219–223). Columbus, OH: ERIC.Google Scholar - Star, J. R. (2005). Reconceptualizing procedural knowledge.
*Journal for Research in Mathematics Education, 36*(5), 404–411.Google Scholar - Stephan, M. & Clements, D. H. (2003). Linear and area measurement in prekindergarten to grade 2. In D. H. Clements & G. Bright (Eds.),
*Learning and teaching measurement: 2003 Yearbook (3–16)*. Reston, VA: NCTM.Google Scholar - Stipek, D., Givvin, K., Salmon, J. & MacGyvers, V. (2001). Teachers’ beliefs and practices related to mathematics instruction.
*Teaching and Teacher Education, 17*(2), 213–226.CrossRefGoogle Scholar - Tan Sisman, G. & Aksu, M. (2009). Seventh grade students’newapos; success on the topics of area and perimeter.
*İlköğretim-Online*,*8(*1), 243–253.Google Scholar - Thompson, A. G., Philipp, R. A., Thompson, P. W. & Boyd, B. A. (1994). Calculational and conceptual orientations in teaching mathematics. In A. Coxford (Ed.),
*1994 Yearbook of the NCTM*(pp. 79–92). Reston, VA: NCTM.Google Scholar - Van de Walle, J. (2007).
*Elementary and middle school mathematics: Teaching developmentally*. Boston, MA: Pearson.Google Scholar - Voulgaris, S. & Evangelidou, A. (2004). Volume conception in late primary school children in Cyprus.
*Quaderni di Ricerca in Diddattica, 14*, 1–31.Google Scholar - Watson, I. (1980). Investigating errors of beginning mathematicians.
*Educational Studies in Mathematics, 11*(3), 319–329.CrossRefGoogle Scholar - Wilson, P. S. & Rowland, R. (1993). Teaching measurement. In R. J. Jensen (Ed.),
*Research ideas for the classroom: Early childhood mathematics*(pp. 171–194). New York, NY: Macmillan.Google Scholar - Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas.
*Journal of Mathematical Behavior, 25*, 224–239.CrossRefGoogle Scholar