# It’s not a math lesson - we’re learning to draw! Teachers’ use of visual representations in instructing word problem solving in sixth grade of elementary school.

## Main Article Content

## Abstract

*Non-routine word problem solving is an essential feature of the mathematical development of elementary school students worldwide. Many students experience difficulties in solving these problems due to erroneous problem comprehension. These difficulties could be alleviated by instructing students how to use visual representations that clarify the problem structure and the relations between solution-relevant elements (so-called visual-schematic representations). Research shows that instructional effectiveness depends largely on teachers’ mathematical knowledge for teaching. Teachers’ knowledge of visual representations is therefore essential to instructing word problem comprehension in this way. As there is little to no literature investigating teachers’ practices in this area, the goal of the present study is to examine teachers’ use of visual representations to support non-routine word problem solving. Eight mainstream elementary school teachers implemented an innovative approach focused on the use of visual-schematic representations. After a short training, teachers were able to produce these representations during instruction. However, some teachers seemed unclear about what these representations comprise and what function they serve within the word problem solving context. Teachers seemed to base their use of representations on personal preferences rather than on an optimal fit with the word problem characteristics. These aspects need to be addressed in teacher training and professional development programs.*

*This study makes an unique contribution to research in the important and problematic area of word problem solving in regular classrooms. The results of this study are relevant for educational researcher, teachers, and teacher educators who deal with difficulties in instructing mathematical word problems.*

## Article Details

How to Cite

*Frontline Learning Research*,

*4*(5), 34–61. https://doi.org/10.14786/flr.v4i5.245

Section

Articles

FLR adopts the Attribution-NonCommercial-NoDerivs Creative Common License (BY-NC-ND). That is, Copyright for articles published in this journal is retained by the authors with, however, first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.

## References

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Barnes, H. (2005). The theory of Realistic Mathematics Education as a theoretical framework

for teaching low attainers in mathematics. Pythagoras, 61, 42 57.

Beets, M. W., Flay, B. R., Vuchinich, S., Acock, A. C., Li, K.-K., & Allred, C. (2008). School climate and teachers’ beliefs and attitudes associated with implementation of the positive action program: A diffusion of innovations model. Prevention Science, 9, 264 275. doi: 10.1007/s11121-008-0100-2

Bitan-Friedlander, N., Dreyfus, A., & Milgrom, Z. (2004). Types of “teachers in training”: the reactions of primary school science teachers when confronted with the task of implementing an innovation. Teaching and Teacher Education, 20, 607 619. doi: 10.1016/j.tate.2004.06.007

Author (2013).

Author (2014).

Bursal, M., & Paznokas, L. (2006). Mathematics anxiety and preservice elementary teachers’

confidence to teach mathematics and science. School Science & Mathematics, 106, 173 180. doi: 10.1111/j.1949-8594.2006.tb18073.x

Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys, R. E. (1981). National assessment. In E. Fennema (Ed.), Mathematics education research; Implications for the 80's (pp. 22 38). Reston, VA: National Council of Teachers of Mathematics.

Cummins, D. D., Kintsch, W., Reusser, K., & Weimer, R. (1988). The role of understanding in solving word problems. Cognitive Psychology, 20, 405 438. doi: http://dx.doi.org/10.1016/0010-0285(88)90011-4

Depaepe, F., De Corte, E., & Verschaffel, L. (2010). Teachers' approaches towards word problem solving: Elaborating or restricting the problem context. Teaching and Teacher Education, 26, 152 160. doi: 10.1016/j.tate.2009.03.016

Department for Education. (2013). National curriculum in England; Mathematics programmes of study: key stages 1 and 2. Retrieved from www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study.

Dreher, A., & Kuntze S. (2015). Teachers' professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88, 89 114. doi: 10.1007/s10649-014-9577-8

Elia, I., van den Heuvel-Panhuizen, M., & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM, 41, 605-618. doi: 10.1007/s11858-009-0184-6

Evers, W. J. G., Brouwers, A., & Tomic, W. (2002). Burnout and self-efficacy: A study on teachers’ beliefs when implementing an innovative educational system in the Netherlands. British Journal of Educational Psychology, 72, 227 243. doi: 0.1348/000709902158865

Ghaith, G., & Yaghi, H. (1997). Relationships among experience, teacher efficacy, and attitudes toward the implementation of instructional innovation. Teaching and Teacher Education, 13, 451 458. doi: 10.1016/S0742-051X(96)00045-5

Han, S. S., & Weiss, B. (2005). Sustainability of teacher implementation of school-based mental health programs. Journal of Abnormal Child Psychology, 33, 665–679. doi: 10.1007/s10802-005-7646-2.

Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91, 684–689. doi: http://dx.doi.org/10.1037/0022-0663.91.4.68

Hermans, R., Tondeur, J., Van Braak, J., & Valcke, M. (2008). The impact of primary school teachers’ educational beliefs on the classroom use of computers. Computers & Education, 51, 1499 1509. doi: 10.1016/j.compedu.2008.02.001

Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26, 430 511. doi;

10.1080/07370000802177235

Isiksal, M., Curran, J. M., Koc, Y., & Askun, C. S. (2009). Mathematics anxiety and mathematical self-concept: Considerations in preparing elementary-school teachers. Social Behavior and Personality, 37, 631 644. doi: 10.2224/sbp.2009.37.5.631

Jiminez, L., & Verschaffel, L. (2014). Development of children’s solutions of non-standard arithmetic word problem solving. Revista de Psicodidáctica, 2014, 19, 93 123. doi: 10.1387/RevPsicodidact.7865

Jitendra, A.K. (2002). Teaching students math problem-solving through graphic representations. Teaching Exceptional Children, 34, 34 38.

Jitendra, A.K., DiPipi, C. M., & Perron-Jones, N. (2002). An exploratory study of schema based word problem solving instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural understanding. The Journal of Special Education, 36, 23–38. doi: http://dx.doi.org/10.1177/00224669020360010301

Jitendra, A. K., Griffin, C. C., Haria, P., Leh, J., Adams, A., & Kaduvettoor, A. (2007). A comparison of single and multiple strategy instruction on third-grade students' mathematical problem solving. Journal of Educational Psychology, 99, 115 127. doi: 10.1037/0022-0663.99.1.115

Jitendra, A. K., Petersen-Brown, S., Lein, A. E., Zaslofsky, A. F., Kunkel, A. K., Jung, P.-G., & Egan, A. M. (2013). Teaching mathematical word problem solving: The quality of evidence for strategy instruction priming the problem structure. Journal of Learning Disabilities, XX, 1 22. doi: 10.1177/0022219413487408

Jitendra, A. K., & Star, J. R. (2012). An exploratory study contrasting high- and low achieving students' percent word problem solving. Learning and Individual Differences, 22, 151 158. doi: 10.1016/j.lindif.2011.11.003

Jitendra, A. K., Star, J. R., Starosta, K., Leh, J. M., Sood, S., Caskie, G., … Mack, T. R. (2009). Improving seventh grade students’ learning of ratio and proportion: The role of schema-based instruction. Contemporary Educational Psychology, 34, 250 264. doi: 10.1016/j.cedpsych.2009.06.001

Ketelaar, E., Beijaard, D., Boshuizen, H., & Den Brok, P. J. (2012). Teachers’ positioning towards an educational innovation in the light of ownership, sense-making and agency. Teaching and Teacher Education, 28, 273 282. doi: 10.1016/j.tate.2011.10.004

Krawec, J. L. (2010). Problem representation and mathematical problem solving of students with varying abilities (doctoral dissertation, University of Miami). Miami.

Krawec, J. L. (2012). Problem representation and mathematical problem solving of students of varying math ability. Journal of Learning Disabilities, XX, 1 13. doi: 10.1177/0022219412436976

Krippendorff, K. (2004). Reliability in content analysis: Some common misconceptions and recommendations. Human Communication Research, 30, 411 433. doi: http://dx.doi.org/10.1111/j.1468-2958.2004.tb00738.x

Lewis, A. B., & Mayer, R. E. (1987). Students’ miscomprehension of relational statements in arithmetic word problems. Journal of Educational Psychology, 79, 363 371. doi: 10.1037/0022-0663.79.4.363

Montague, M. (2003). Solve It! A practical approach to teaching mathematical problem solving skills. VA: Exceptional Innovations, Inc.

Montague, M., Warger, C., & Morgan, T. H. (2000). Solve It! strategy instruction to improve mathematical problem solving. Learning Disabilities Research & Practice, 15, 110 116. doi: 10.1207/SLDRP1502_7

Mullis, I.V.S. & Martin, M.O. (Eds.). (2013). Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Noteboom, A. (2009). Fundamentele doelen rekenen-wiskunde [Key objectives for arithmetic and math]. Enschede, The Netherlands: SLO, National Expertise Center for Curriculum Development.

OECD (2013). PISA 2012 Assessment and Analytical Framework: Mathematics, Reading, Science,

Problem Solving and Financial Literacy. PISA, OECD Publishing.

Orrill, C. H., Sexton, S., Lee, S.-J., & Gerde, C. (2008). Mathematics teachers’ abilities to use and make sense of drawn representations. In The International Conference of the Learning Sciences 2008: Proceedings of ICLS 2008.

Mahwah, NJ: International Society of the Learning Sciences.

Prenger, J. (2005). Taal telt! Een onderzoek naar de rol van taalvaardigheid en tekstbegrip in het realistische rekenonderwijs. [Language counts! A study into the role of linguistic skill and text comprehension in realistic mathematics education]. Doctoral dissertation, University of Groningen, The Netherlands.

Pantziara, M., Gagatsis, A. & Elia, I. (2009). Using diagrams as tools for the solution of non-routine mathematical problems. Educational Studies in Mathematics, 72, 39 60. doi: 10.1007/s10649-009-9181-5

Rogers, E. M. (2003). Diffusion of innovations (5th ed.). New York: Free Press.

Schoppek, W., & Tulis, M. (2010). Enhancing arithmetic and word-problem solving skills efficiently by individualized computer-assisted practice. The Journal of Educational Research, 103, 239 252. doi: 10.1080/00220670903382962

Sharma, U., Loreman, T., & Forlin, C. (2012). Measuring teacher efficacy to implement inclusive practices. Journal of Research in Special Educational Needs, 12, 12 21. doi: 10.1111/j.1471-3802.2011.01200.x

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1 22.

Simpson, E. H. (1949). Measurement of diversity. Nature.

Swanson, H. L., Lussier, C. M., & Orosco, M. J. (2013). Cognitive strategies, working memory, and growth in word problem solving in children with math difficulties. Journal of Learning Disabilities, XX, 1 20. doi: 10.1177/0022219413498771

Swars, S. L., Daane, C. J., & Giessen, J. (2006). Mathematics anxiety and mathematics teacher efficacy: What is the relationship in elementary preservice teachers? School Science and Mathematics, 106, 306 315. doi: 10.1111/j.1949-8594.2006.tb17921.x

Turner, F. (2008). Beginning elementary teachers’ use of representations in mathematics teaching, Research in Mathematics Education, 10, 209 210. doi: 10.1080/14794800802233795

Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54, 9 35. doi: 10.1023/B:EDUC.0000005212.03219.dc

Van der Veen, I., Smeets, E., & Derriks, M. (2010). Children with special educational needs in the Netherlands: Number, characteristics and school career. Educational Research, 52, 15 43. doi: 10.1080/00131881003588147

Van Dijk, I. M. A. W., Van Oers, H. J. M., & Terwel, J. (2003). Providing or designing? Constructing models in primary math education. Learning and Instruction, 13, 53 72. doi: http://dx.doi.org/10.1016/S0959-4752(01)00037-8

Van Dijk, I. M. A. W., Van Oers, B., Terwel, J., & Van den Eeden, P. (2003a). Strategic learning in primary mathematics education: Effects of an experimental program in modeling. Educational Research and Evaluation, 9, 161 187. doi: http://dx.doi.org/10.1076/edre.9.2.161.14213

Van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. Journal of Learning Disabilities, 39, 496–506. doi: http://dx.doi.org/10.1177/00222194060390060201

Van Garderen, D., & Montague, M. (2003). Visual–spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practice, 18, 246–254. doi: http://dx.doi.org/10.1111/1540-5826.00079

Webb, D. C., Van der Kooij, H., & Geist, M. R. (2011). Design research in the Netherlands: Introducing logarithms using Realistic Mathematics Education. Journal of Mathematics Education at Teachers College, 2, 47 52.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389 407. doi: 10.1177/0022487108324554

Barnes, H. (2005). The theory of Realistic Mathematics Education as a theoretical framework

for teaching low attainers in mathematics. Pythagoras, 61, 42 57.

Beets, M. W., Flay, B. R., Vuchinich, S., Acock, A. C., Li, K.-K., & Allred, C. (2008). School climate and teachers’ beliefs and attitudes associated with implementation of the positive action program: A diffusion of innovations model. Prevention Science, 9, 264 275. doi: 10.1007/s11121-008-0100-2

Bitan-Friedlander, N., Dreyfus, A., & Milgrom, Z. (2004). Types of “teachers in training”: the reactions of primary school science teachers when confronted with the task of implementing an innovation. Teaching and Teacher Education, 20, 607 619. doi: 10.1016/j.tate.2004.06.007

Author (2013).

Author (2014).

Bursal, M., & Paznokas, L. (2006). Mathematics anxiety and preservice elementary teachers’

confidence to teach mathematics and science. School Science & Mathematics, 106, 173 180. doi: 10.1111/j.1949-8594.2006.tb18073.x

Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys, R. E. (1981). National assessment. In E. Fennema (Ed.), Mathematics education research; Implications for the 80's (pp. 22 38). Reston, VA: National Council of Teachers of Mathematics.

Cummins, D. D., Kintsch, W., Reusser, K., & Weimer, R. (1988). The role of understanding in solving word problems. Cognitive Psychology, 20, 405 438. doi: http://dx.doi.org/10.1016/0010-0285(88)90011-4

Depaepe, F., De Corte, E., & Verschaffel, L. (2010). Teachers' approaches towards word problem solving: Elaborating or restricting the problem context. Teaching and Teacher Education, 26, 152 160. doi: 10.1016/j.tate.2009.03.016

Department for Education. (2013). National curriculum in England; Mathematics programmes of study: key stages 1 and 2. Retrieved from www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study.

Dreher, A., & Kuntze S. (2015). Teachers' professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88, 89 114. doi: 10.1007/s10649-014-9577-8

Elia, I., van den Heuvel-Panhuizen, M., & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics. ZDM, 41, 605-618. doi: 10.1007/s11858-009-0184-6

Evers, W. J. G., Brouwers, A., & Tomic, W. (2002). Burnout and self-efficacy: A study on teachers’ beliefs when implementing an innovative educational system in the Netherlands. British Journal of Educational Psychology, 72, 227 243. doi: 0.1348/000709902158865

Ghaith, G., & Yaghi, H. (1997). Relationships among experience, teacher efficacy, and attitudes toward the implementation of instructional innovation. Teaching and Teacher Education, 13, 451 458. doi: 10.1016/S0742-051X(96)00045-5

Han, S. S., & Weiss, B. (2005). Sustainability of teacher implementation of school-based mental health programs. Journal of Abnormal Child Psychology, 33, 665–679. doi: 10.1007/s10802-005-7646-2.

Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91, 684–689. doi: http://dx.doi.org/10.1037/0022-0663.91.4.68

Hermans, R., Tondeur, J., Van Braak, J., & Valcke, M. (2008). The impact of primary school teachers’ educational beliefs on the classroom use of computers. Computers & Education, 51, 1499 1509. doi: 10.1016/j.compedu.2008.02.001

Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26, 430 511. doi;

10.1080/07370000802177235

Isiksal, M., Curran, J. M., Koc, Y., & Askun, C. S. (2009). Mathematics anxiety and mathematical self-concept: Considerations in preparing elementary-school teachers. Social Behavior and Personality, 37, 631 644. doi: 10.2224/sbp.2009.37.5.631

Jiminez, L., & Verschaffel, L. (2014). Development of children’s solutions of non-standard arithmetic word problem solving. Revista de Psicodidáctica, 2014, 19, 93 123. doi: 10.1387/RevPsicodidact.7865

Jitendra, A.K. (2002). Teaching students math problem-solving through graphic representations. Teaching Exceptional Children, 34, 34 38.

Jitendra, A.K., DiPipi, C. M., & Perron-Jones, N. (2002). An exploratory study of schema based word problem solving instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural understanding. The Journal of Special Education, 36, 23–38. doi: http://dx.doi.org/10.1177/00224669020360010301

Jitendra, A. K., Griffin, C. C., Haria, P., Leh, J., Adams, A., & Kaduvettoor, A. (2007). A comparison of single and multiple strategy instruction on third-grade students' mathematical problem solving. Journal of Educational Psychology, 99, 115 127. doi: 10.1037/0022-0663.99.1.115

Jitendra, A. K., Petersen-Brown, S., Lein, A. E., Zaslofsky, A. F., Kunkel, A. K., Jung, P.-G., & Egan, A. M. (2013). Teaching mathematical word problem solving: The quality of evidence for strategy instruction priming the problem structure. Journal of Learning Disabilities, XX, 1 22. doi: 10.1177/0022219413487408

Jitendra, A. K., & Star, J. R. (2012). An exploratory study contrasting high- and low achieving students' percent word problem solving. Learning and Individual Differences, 22, 151 158. doi: 10.1016/j.lindif.2011.11.003

Jitendra, A. K., Star, J. R., Starosta, K., Leh, J. M., Sood, S., Caskie, G., … Mack, T. R. (2009). Improving seventh grade students’ learning of ratio and proportion: The role of schema-based instruction. Contemporary Educational Psychology, 34, 250 264. doi: 10.1016/j.cedpsych.2009.06.001

Ketelaar, E., Beijaard, D., Boshuizen, H., & Den Brok, P. J. (2012). Teachers’ positioning towards an educational innovation in the light of ownership, sense-making and agency. Teaching and Teacher Education, 28, 273 282. doi: 10.1016/j.tate.2011.10.004

Krawec, J. L. (2010). Problem representation and mathematical problem solving of students with varying abilities (doctoral dissertation, University of Miami). Miami.

Krawec, J. L. (2012). Problem representation and mathematical problem solving of students of varying math ability. Journal of Learning Disabilities, XX, 1 13. doi: 10.1177/0022219412436976

Krippendorff, K. (2004). Reliability in content analysis: Some common misconceptions and recommendations. Human Communication Research, 30, 411 433. doi: http://dx.doi.org/10.1111/j.1468-2958.2004.tb00738.x

Lewis, A. B., & Mayer, R. E. (1987). Students’ miscomprehension of relational statements in arithmetic word problems. Journal of Educational Psychology, 79, 363 371. doi: 10.1037/0022-0663.79.4.363

Montague, M. (2003). Solve It! A practical approach to teaching mathematical problem solving skills. VA: Exceptional Innovations, Inc.

Montague, M., Warger, C., & Morgan, T. H. (2000). Solve It! strategy instruction to improve mathematical problem solving. Learning Disabilities Research & Practice, 15, 110 116. doi: 10.1207/SLDRP1502_7

Mullis, I.V.S. & Martin, M.O. (Eds.). (2013). Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Noteboom, A. (2009). Fundamentele doelen rekenen-wiskunde [Key objectives for arithmetic and math]. Enschede, The Netherlands: SLO, National Expertise Center for Curriculum Development.

OECD (2013). PISA 2012 Assessment and Analytical Framework: Mathematics, Reading, Science,

Problem Solving and Financial Literacy. PISA, OECD Publishing.

Orrill, C. H., Sexton, S., Lee, S.-J., & Gerde, C. (2008). Mathematics teachers’ abilities to use and make sense of drawn representations. In The International Conference of the Learning Sciences 2008: Proceedings of ICLS 2008.

Mahwah, NJ: International Society of the Learning Sciences.

Prenger, J. (2005). Taal telt! Een onderzoek naar de rol van taalvaardigheid en tekstbegrip in het realistische rekenonderwijs. [Language counts! A study into the role of linguistic skill and text comprehension in realistic mathematics education]. Doctoral dissertation, University of Groningen, The Netherlands.

Pantziara, M., Gagatsis, A. & Elia, I. (2009). Using diagrams as tools for the solution of non-routine mathematical problems. Educational Studies in Mathematics, 72, 39 60. doi: 10.1007/s10649-009-9181-5

Rogers, E. M. (2003). Diffusion of innovations (5th ed.). New York: Free Press.

Schoppek, W., & Tulis, M. (2010). Enhancing arithmetic and word-problem solving skills efficiently by individualized computer-assisted practice. The Journal of Educational Research, 103, 239 252. doi: 10.1080/00220670903382962

Sharma, U., Loreman, T., & Forlin, C. (2012). Measuring teacher efficacy to implement inclusive practices. Journal of Research in Special Educational Needs, 12, 12 21. doi: 10.1111/j.1471-3802.2011.01200.x

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1 22.

Simpson, E. H. (1949). Measurement of diversity. Nature.

Swanson, H. L., Lussier, C. M., & Orosco, M. J. (2013). Cognitive strategies, working memory, and growth in word problem solving in children with math difficulties. Journal of Learning Disabilities, XX, 1 20. doi: 10.1177/0022219413498771

Swars, S. L., Daane, C. J., & Giessen, J. (2006). Mathematics anxiety and mathematics teacher efficacy: What is the relationship in elementary preservice teachers? School Science and Mathematics, 106, 306 315. doi: 10.1111/j.1949-8594.2006.tb17921.x

Turner, F. (2008). Beginning elementary teachers’ use of representations in mathematics teaching, Research in Mathematics Education, 10, 209 210. doi: 10.1080/14794800802233795

Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54, 9 35. doi: 10.1023/B:EDUC.0000005212.03219.dc

Van der Veen, I., Smeets, E., & Derriks, M. (2010). Children with special educational needs in the Netherlands: Number, characteristics and school career. Educational Research, 52, 15 43. doi: 10.1080/00131881003588147

Van Dijk, I. M. A. W., Van Oers, H. J. M., & Terwel, J. (2003). Providing or designing? Constructing models in primary math education. Learning and Instruction, 13, 53 72. doi: http://dx.doi.org/10.1016/S0959-4752(01)00037-8

Van Dijk, I. M. A. W., Van Oers, B., Terwel, J., & Van den Eeden, P. (2003a). Strategic learning in primary mathematics education: Effects of an experimental program in modeling. Educational Research and Evaluation, 9, 161 187. doi: http://dx.doi.org/10.1076/edre.9.2.161.14213

Van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. Journal of Learning Disabilities, 39, 496–506. doi: http://dx.doi.org/10.1177/00222194060390060201

Van Garderen, D., & Montague, M. (2003). Visual–spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practice, 18, 246–254. doi: http://dx.doi.org/10.1111/1540-5826.00079

Webb, D. C., Van der Kooij, H., & Geist, M. R. (2011). Design research in the Netherlands: Introducing logarithms using Realistic Mathematics Education. Journal of Mathematics Education at Teachers College, 2, 47 52.