This study explored the relationships between task beliefs about the empty number line (ENL), mathematical ability, gender, and voluntary ENL use in multi-digit subtraction and addition. One hundred twenty-three Dutch third-grade students and nine teachers from six schools participated in this study. The multilevel path analysis showed that task beliefs about the ENL mediated the relationship between students’ mathematical ability and their voluntary ENL use. No gender differences were found in the multilevel path analysis. Finally, the results show that task beliefs about the ENL and voluntary ENL use differed across classrooms. The discussion focuses on the implications of the results for using the ENL in diagnostic assessment.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Alibali, M. W., Phillips, K. M. O., & Fischer, A. D. (2009). Learning new problem-solving strategies leads to changes in problem representation. Cognitive Development, 24(2), 89–101. https://doi.org/10.1016/j.cogdev.2008.12.005
Baroody, A. J., Torbeyns, J., & Verschaffel, L. (2009). Young children’s understanding and application of subtraction-related principles. Mathematical Thinking and Learning, 11(1–2), 2–9. https://doi.org/10.1080/10986060802583873
Baten, E., Praet, M., & Desoete, A. (2017). The relevance and efficacy of metacognition for instructional design in the domain of mathematics. ZDM Mathematics Education, 49(4), 613–623. https://doi.org/10.1007/s11858-017-0851-y
Beishuizen, M. (1993). Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades. Journal for Research in Mathematics Education, 24(4), 294–323. https://doi.org/10.2307/749464
Björn, P. M., Räikkönen, E., Aunola, K., & Kyttälä, M. (2017). Dynamics between student vs. teacher perceptions of mathematics task-orientation and mathematics performance among adolescents. Learning and Individual Differences, 55, 21–28. https://doi.org/10.1016/j.lindif.2017.02.005
Blöte, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10(3), 221–247. https://doi.org/10.1016/S0959-4752(99)00028-6
Bobis, J. (2007). The empty number line: A useful tool or just another procedure? Teaching Children Mathematics, 13(8), 410–413.
Bobis, J., & Bobis, E. (2005). The empty number line: Making children’s thinking visible. In M. Coupland, J. Anderson, & T. Spencer (Eds.), Proceedings of the Twentieth Biennial Conference of The Australian Association of Mathematics Teachers (issue 08, pp. 66–72). Sydney, Australia: The Australian Association of Mathematics Teachers. Retrieved October 28, 2020, from http://aamt.dbinformatics.com.au/index.php/content/download/19063/252036/file/mm-vital.pdf#page=72
Breen, S., & O’Shea, A. (2010). Mathematical thinking and task design. Bulletin of the Irish Mathematical Society, 66, 39–49
Bramald, R. (2000). Introducing the empty number line. Education 3-13: International Journal of Primary, Elementary and Early Years Education, 28(3), 5–12. https://doi.org/10.1080/03004270085200271
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 37–50. https://doi.org/10.2307/749715
Carr, M., Jessup, D. L., & Fuller, D. (1999). Gender differences in first-grade mathematics strategy use: Parent and teacher contributions. Journal for Research in Mathematics Education, 30(1), 20–46. https://doi.org/10.2307/749628
Dowker, A., Bennett, K., & Smith, L. (2012). Attitudes to mathematics in primary school children. Child Development Research, 2012, 1–8. https://doi.org/10.1155/2012/124939
Dowker, A., Sarkar, A., & Looi, C. Y. (2016). Mathematics anxiety: What have we learned in 60 years? Frontiers in Psychology, 7. https://doi.org/10.3389/fpsyg.2016.00508
Fagginger Auer, M. F., Hickendorff, M., & van Putten, C. M. (2016). Solution strategies and adaptivity in multidigit division in a choice/no-choice experiment: Student and instructional factors. Learning and Instruction, 41, 52–59. https://doi.org/10.1016/j.learninstruc.2015.09.008
Fagginger Auer, M. F., Hickendorff, M., van Putten, C. M., Béguin, A. A., & Heiser, W. J. (2016). Multilevel latent class analysis for large-scale educational assessment data: Exploring the relation between the curriculum and students’ mathematical strategies. Applied Measurement in Education, 29(2), 144–159. https://doi.org/10.1080/08957347.2016.1138959
Fives, H., Barnes, N., Buehl, M. M., Mascadri, J., & Ziegler, N. (2017). Teachers’ epistemic cognition in classroom assessment. Educational Psychologist, 52(4), 270–283. https://doi.org/10.1080/00461520.2017.1323218
Fredricks, J. A., Hofkens, T., Te Wang, M., Mortenson, E., & Scott, P. (2018). Supporting girls’ and boys’ engagement in math and science learning: A mixed methods study. Journal of Research in Science Teaching, 55(2), 271–298. https://doi.org/10.1002/tea.21419
Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., … Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130–162.
Geary, D. C., Hoard, M. K., Nugent, L., & Byrd-Craven, J. (2008). Development of number line representations in children with mathematical learning disability. Developmental Neuropsychology, 33(3), 277–299. https://doi.org/10.1080/87565640801982361
Gorin, J. S. (2007). Test construction and diagnostic testing. In J. P. Leighton & M. J. Gierl (Eds.), Cognitive diagnostic assessment for education: Theory and applications (pp. 173–201). New York, NY: Cambridge University Press.
Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128. https://doi.org/10.1207/s15327833mtl0602_3
Gravemeijer, K., Bowers, J., & Stephan, M. (2003). Chapter 4: A hypothetical learning trajectory on measurement and flexible arithmetic. Journal for Research in Mathematics Education, 12, 51–66.
Graven, M., & Heyd-Metzuyanim, E. (2019). Mathematics identity research: The state of the art and future directions. ZDM Mathematics Education, 51(3), 361–377. https://doi.org/10.1007/s11858-019-01050-y
Grootenboer, P., & Marshman, M. (2016). Mathematics, affect and learning: Middle school students’ beliefs and attitudes about mathematics education. Singapore: Springer. https://doi.org/10.1007/978-981-287-679-9
Heritage, M., & Wylie, C. (2018). Reaping the benefits of assessment for learning: Achievement, identity, and equity. ZDM Mathematics Education, 50(4), 729–741. https://doi.org/10.1007/s11858-018-0943-3
Hickendorff, M., van Putten, C. M., Verhelst, N. D., & Heiser, W. J. (2010). Individual differences in strategy use on division problems: Mental versus written computation. Journal of Educational Psychology, 102(2), 438–452. https://doi.org/10.1037/a0018177
Hop, M. (2012). Balans (47) van het rekenwiskundeonderwijs halverwege de basisschool 5 [fifth periodical assessment of mathematics education mid primary school]. PPON report no. 47. Arnhem, the Netherlands: Cito. Retrieved October 28, 2020, from https://www.cito.nl/-/media/files/kennis-en-innovatie-onderzoek/ppon/cito_ppon_balans_47.pdf?la=nl-nl
Hox, J. (2002). Multilevel analysis. Techniques and applications. Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Publishers.
Hox, J., Van De Schoot, R., & Matthijsse, S. (2012). How few countries will do? Comparative survey analysis from a Bayesian perspective. Survey Research Methods, 6(2), 87–93.
Janssen, J., Scheltens, F., & Kraemer, J. M. (2006). Primair onderwijs. Leerling- en onderwijsvolgsysteem. Rekenen-wiskunde groep 5 [Primary education. Pupil and educational monitoring system. Mathematics grade 3]. Arnhem, the Netherlands: Cito.
Janssen, J., Verhelst, N., Engelen, R., & Scheltens, F. (2010). Wetenschappelijke verantwoording van de toetsen LOVS Rekenen-Wiskunde voor groep 3 tot en met 8. [scientific validation report for the LOVS mathematics test for grades 3 through 6]. Resource document. Arnhem, the Netherlands: Cito. Retrieved October 28, 2020, from http://www.toetswijzer.nl/html/tg/14.pdf
Keeley, P., & Tobey, C. R. (2011). Mathematics formative assessment- 75 practical strategies for linking assessment, instruction, and learning. Thousand Oaks, CA: Corwin Press.
Kirk, E. P., & Ashcraft, M. H. (2001). Telling stories: The perils and promise of using verbal reports to study math strategies. Journal of Experimental Psychology: Learning, Memory, and Cognition, 27(1), 157–175. https://doi.org/10.1037//0278-7322.214.171.124
Kraemer, J. M. (2011). Oplossingsmethoden voor aftrekken tot 100 [solution methods for subtraction up to 100] (doctoral dissertation, Eindhoven, the Netherlands: Technical University Eindhoven). https://doi.org/10.6100/IR721544
Leighton, J. P. (2004). Avoiding misconception, misuse, and missed opportunities: The collection of verbal reports in educational achievement testing. Educational Measurement: Issues and Practice, 23(4), 6–15. https://doi.org/10.1111/j.1745-3992.2004.tb00164.x
Leighton, J. P., & Gierl, M. J. (2007). Why cognitive diagnostic assessment. In J. P. Leighton & M. J. Gierl (Eds.), Cognitive diagnostic assessment for education: Theory and applications (pp. 3–18). New York, NY: Cambridge University Press.
Murphy, C. (2011). Comparing the use of the empty number line in England and the Netherlands. British Educational Research Journal, 37(1), 147–161. https://doi.org/10.1080/01411920903447423
Muthén, L. K., & Muthén, B. O. (2019). Mplus User’s Guide (7th ed.). Los Angeles, CA: Muthén & Muthén. Retrieved October 28, 2020, from https://www.statmodel.com/download/usersguide/MplususerguideVer_7_r3_web.pdf.
Nuutila, K., Tuominen, H., Tapola, A., Vainikainen, M. P., & Niemivirta, M. (2018). Consistency, longitudinal stability, and predictions of elementary school students’ task interest, success expectancy, and performance in mathematics. Learning and Instruction, 56, 73–83. https://doi.org/10.1016/j.learninstruc.2018.04.003
Passolunghi, M. C., Cargnelutti, E., & Pellizzoni, S. (2019). The relation between cognitive and emotional factors and arithmetic problem-solving. Educational Studies in Mathematics, 100(3), 271–290. https://doi.org/10.1007/s10649-018-9863-y
Peltenburg, M., van den Heuvel Panhuizen, M., & Robitzsch, A. (2010). ICT-based dynamic assessment to reveal special education students’ potential in mathematics. Research Papers in Education, 25(3), 319–334. https://doi.org/10.1080/02671522.2010.498148
Rupp, A. A., Gushta, M., Mislevy, R. J., & Shaffer, D. W. (2010). Evidence-centered design of epistemic games: Measurement principles for complex learning environments. Journal of Technology Learning and Assessment, 8(4), 1–45.
Rupp, A. A., Templin, J., & Henson, R. A. (2010). Diagnostic measurements. Theory, methods, and applications. New York, NY: The Guilford Press.
Snow, R. E. (1989). Toward assessment of cognitive and conative structures in learning. Educational Researcher, 18(9), 8–14.
Sullivan, P., Clarke, D., & Clarke, B. (2013). Teaching with tasks for effective mathematics learning. New York, NY: Springer. https://doi.org/10.1007/978-1-4614-4681-1
Summers, J. J., Schallert, D. L., & Muse Ritter, P. (2003). The role of social comparison in students’ perceptions of ability: An enriched view of academic motivation in middle school students. Contemporary Educational Psychology, 28(4), 510–523. https://doi.org/10.1016/S0361-476X(02)00059-0
Teppo, A., & van den Heuvel-Panhuizen, M. (2013). Visual representations as objects of analysis: The number line as an example. ZDM Mathematics Education, 46, 45–58. https://doi.org/10.1007/s11858-013-0518-2
Torbeyns, J., De Smedt, B., Ghesquière, P., & Verschaffel, L. (2008). Acquisition and use of shortcut strategies by traditionally schooled children. Educational Studies in Mathematics, 71(1), 1–17. https://doi.org/10.1007/s10649-008-9155-z
Torbeyns, J., Hickendorff, M., & Verschaffel, L. (2017). The use of number-based versus digit-based strategies on multi-digit subtraction: 9–12-year-olds’ strategy use profiles and task performance. Learning and Individual Differences, 58, 64–74. https://doi.org/10.1016/j.lindif.2017.07.004
Treffers, A., van den Heuvel-Panhuizen, M., & Buys, K. (2000). Jonge kinderen leren rekenen. Tussendoelen annex leerlijnen. Hele getallen onderbouw basisschool [Young children learn mathematics. Intermediate goals annex learning trajectories. Whole numbers in lower grades of primary school]. Groningen, the Netherlands: Wolters-Noordhoff.
van den Heuvel-Panhuizen, M. (2008). Learning from “Didactikids”: An impetus for revisiting the empty number line. Mathematics Education Research Journal, 20(3), 6–31.
van den Heuvel-Panhuizen, M., & Peltenburg, M. (2011). A secondary analysis from a cognitive load perspective to understand why an ICT-based assessment environment helps special education students to solve mathematical problems. Research in Mathematics Education, 10(1–2), 23–41.
van der Kleij, F. M., Vermeulen, J. A., Schildkamp, K., & Eggen, T. J. H. M. (2015). Integrating data-based decision making, assessment for learning, and diagnostic testing in formative assessment. Assessment in Education: Principles, Policy & Practice, 22(3), 324–343. https://doi.org/10.1080/0969594X.2014.999024
van der Linden, W. J., & Hambleton, R. K. (1997). Handbook of modern item response theory. New York, NY: Springer.
Verhelst, N. D., & Glas, C. A. W. (1995). The one parameter logistic model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch Models. New York, NY: Springer. https://doi.org/10.1007/978-1-4612-4230-7_12
Verhelst, N. D., Glas, C. A. W., & Verstralen, H. H. F. M. (1994). OPLM computer program and manual. Arnhem, the Netherlands: Cito.
Vermeulen, J. A., & Eggen, T. J. H. M. (2013). Feedback over aftrekmethoden van leerlingen via de lege getallenlijn. Mogelijkheden en uitdagingen [Feedback about students’ subtraction strategies on the empty number line. Possibilities and challenges] In M. van Zanten (Ed.) Rekenen-wiskunde op niveau. [On level mathematics] Proceedings of the 31st Panama conference January 31st and February 1st, 2013 in Utrecht (pp. 93–107). Utrecht, The Netherlands: FIsme, Utrecht University. Retrieved October 28, 2020, from https://www.fisme.science.uu.nl/publicaties/literatuur/panama_cursusboek/pcb_31_93-107_Vermeulen.pdf
Vermeulen, J. A., Scheltens, F., & Eggen, T. J. H. M. (2015). Strategy identification using the empty number line: A comparison between paper-and-pencil and tablets. Pedagogische Studiën, 92(1).
Vermeulen, J. A., Béguin, A., Scheltens, F., & Eggen, T. J. H. M. (2020). Evaluating the characteristics of diagnostic items for bridging errors in multi-digit subtraction. Frontiers in Education, 5. https://doi.org/10.3389/feduc.2020.537531
Zhu, Z. (2007). Gender differences in mathematical problem solving patterns: A review of literature. International Education Journal, 8(2), 187–203.
We are very grateful for the feedback that was given by Herbert Hoijtink on earlier drafts and the analyses of this paper.
This work was supported by the Netherlands Organisation for Scientific Research (NWO) under Grant MaGW/PROO: Project 411–10-750.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Vermeulen, J.A., Béguin, A. & Eggen, T.J.H.M. Task beliefs and the voluntary use of the empty number line in third-grade subtraction and addition. Educ Stud Math 106, 231–249 (2021). https://doi.org/10.1007/s10649-020-10016-x
- Empty number line
- Multi-digit addition and subtraction
- Diagnostic assessment
- Task beliefs