Differentiating Instruction for Students Who Fail to Thrive in Mathematics: The Impact of a Constructivist-based Intervention Approach

Authors

Keywords:

inclusion in mathematics . pedagogy . differentiated instruction . mathematics difficulties . intervention approaches

Abstract

In this paper we explore how participation in a constructivist-oriented, classroom aligned mathematics intervention program advanced the learning and positive dispositions of Grade 1 students who were failing to thrive when learning mathematics. Intervention programs are an approach to differentiated instruction that some schools adopt, but there are questions about whether such interventions advance equity and inclusion for students. To provide insight about these issues, we draw on data from the Extending Mathematical Understanding for All study in NSW, Australia (2016-2020) to address two research aims. The first aim was to evaluate the effectiveness of the EMU intervention approach for progressing the mathematics learning of students who were failing to thrive mathematically, and for closing the performance gap with their peers. The second aim was to consider the impact of participating in EMU intervention on students’ dispositions for learning mathematics. The findings confirm the potential of intensive constructivist-based intervention approaches for closing the performance gap for students who were not thriving in mathematics, and for improving the learning dispositions of these students. However, two years after the intervention, the performance gap was again apparent for many students. It is likely that a one-off intervention program is not sufficient for enabling all students to thrive mathematically, and that well-focused classroom-based differentiated instruction and further intervention support may be warranted for some students.

Author Biography

Anne Roche, Monash University

Faculty of Education

Research Fellow

References

Ainley, M. (2012). Students’ interest and engagement in classroom activities. In S. L. Christenson, A. L. Reschly, & C. Wylie (Eds.), Handbook of research on student engagement (pp. 283-302). Springer.

Anghileri, J. (1989). An investigation of young children’s understanding of multiplication. Educational Studies in Mathematics, 20, 367-385. https://www.jstor.org/stable/3482314

Anthony, G., Hunter, J., & Hunter, R. (2019). Working towards equity in mathematics education: Is differentiation the answer? In G. Hine, S. Blackley, & A. Cooke (Eds.). Mathematics Education Research: Impacting Practice: Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia (pp. 117-124). MERGA.

Attard, C. (2012). Engagement with mathematics: What does it mean and what does it look like? Australian Primary Mathematics Classroom, 17(1), 9–13.

Atweh, B. (2011). Quality and equity in mathematics education as ethical issues. In B. Atweh, M. Graven, W. Secada, P. (Eds.), Mapping equity and quality in mathematics education (pp. 63-77). Springer. https://doi.org/10.1007/978-90-481-9803-0_5

Bailey, D. H., Nguyen, T., Jenkins, J. M., Domina, T., Clements, D. H., & Sarama, J. S. (2016). Fadeout in an early mathematics intervention: Constraining content or pre-existing differences? Developmental Psychology, 52(9), 1457-1469. http://dx.doi.org/10.1037/dev0000188

Bannister, N. A. (2016). Breaking the spell of differentiated instruction through equity pedagogy and teacher community. Cultural Studies of Science Education, 11(2), 335-347. https://doi.org/10.1007/s11422-016-9766-0

Berkeley, S., Bender, W. N., Peaster, L. G., & Saunders, L. (2009). Implementation of response to intervention: A snapshot of progress. Journal of Learning Disabilities, 42, 85–95. https://doi.org/10.1177/0022219408326214

Blackwell, L. S., Trzesniewski, K. H., Dweck, C. S. (2007). Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention. Child Development, 78(1), 246-263. https://doi.org/10.1111/j.1467-8624.2007.00995.x

Boaler, J., Wiliam, D., & Brown, M. (2000). Students’ experiences of ability grouping: Disaffection, polarisation and the construction of failure. British Educational Research Journal, 26(5), 631-649. https://doi.org/10.1080/713651583

Boaler, J. (2016). The elephant in the classroom: Helping children learn and love maths. Souvenir Press.

Brooks-Gunn, J. (2003). Do you believe in magic? What can we expect from early childhood intervention programs. Social Policy Report, 17(1), 3-15. https://doi.org/10.1002/j.2379-3988.2003.tb00020.x

Bryant, D. P., Bryant, B. R., Roberts, G., Vaughn, S., Pfannenstiel, K. H., Porterfield, J., & Gersten, R. (2011). Early numeracy intervention for first-grade students with mathematics difficulties. Exceptional Children, 78(1), 7-23. https://doi.org/10.1177/001440291107800101

Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in Grades 1-5. Journal for Research in Mathematics Education, 27(1), 41-51. https://doi.org/10.2307/749196

Clarke, D. (2001). Understanding, assessing and developing young children's mathematical thinking: Research as a powerful tool for professional growth. In J. Bobis, B. Perry, & M. Mitchelmore (Eds), Numeracy and beyond: Proceedings of the 24th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 9-26). MERGA.

Clarke, D. (2013). Understanding, assessing and developing children’s mathematical thinking: Task-based interviews as powerful tools for teacher professional learning. In: A.M. Lindmeier & A. Heinze (eds), Proceedings of the 37th conference of the International Group for the Psychology of Mathematics Education, Mathematics learning across the life span (Vol. 1, pp. 17-30). PME.

Clarke, D. (2021). Calling a spade a spade: The impact of within class ability grouping on opportunity to learn mathematics in the primary school. Australian Primary Mathematics Classroom, 26(1), 3-8.

Clarke, D. M., & Clarke, B. A. (2008). Is time up for ability grouping? EQ, Autumn, 31–33.

Clarke, D. M., Cheeseman, J., Gervasoni, A., Gronn, D., Horne, M., McDonough, A., Montgomery, P., Roche, A., Sullivan, P., Clarke, B. A., & Rowley, G. (2002). Early numeracy research project final report, February 2002. Australian Catholic University.

Clements, D. H., Sarama, J., Wolfe, C. B., & Spitler, M. E. (2013). Longitudinal evaluation of a scale-up model for teaching mathematics with trajectories and technologies: Persistence of effects in the third year. American Educational Research Journal, 50(4), 812–850. https://doi.org/10.3102/0002831212469270

Cobb, P., Jackson, K., Henrick, E. C., Smith, T. M., & MIST Project. (2018). Systems for instructional improvement: Creating coherence from the classroom to the district office. Harvard Education Press.

Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2-33. https://doi.org/10.2307/749161

De Corte, E., & Verschaffel, L. (1996). Empirical test of the impact of primitive intuitive models of operations on solving word problems with a multiplicative structure. Learning and Instruction, 6(3), 219-242. https://doi.org/10.1016/0959-4752(96)00004-7

Downton, A. (2008a). Linking multiplication and division in helpful and enjoyable ways for children. In J. Vincent, R. Pierce, & J. Dowsey (Eds.), Connected maths (Proceedings of the 45th annual conference of the Mathematical Association of Victoria (pp. 68-82). MAV.

Downton, A. (2008b). Links between children’s understanding of multiplication and solution strategies for division. In M. Goos, R. Brown, & K. Makar (Eds.), Navigating currents and charting directions (Proceedings of the 31st annual conference of the Mathematics Education Research Group of Australasia, Mathematics (Vol. 1, pp. 171-178). MERGA.

Downton, A. (2012). Young children’s development of multiplicative thinking: Making the transition from models to symbols in the learning of multiplication and division. [Doctoral dissertation, Monash University]

Downton, A., Giumelli, K., McHugh, B., Roosen, T., Meredith, N., Caleta, G., & Stenning, P. (2018). Pilot study, on the impact of in-situ spaced professional learning on teachers’ mathematical knowledge of multiplicative thinking. In Hunter, J., Perger, P., & Darragh, L. (Eds.). Making waves, opening spaces Proceedings of the 41st annual conference of the Mathematics Education Research Group of Australasia (pp. 274-281). MERGA.

Downton, A., Giumelli, K., McHugh, B., Roosen, T., Meredith, N., Caleta, G., King, M., Salkeld, K., & Stenning, P. (2019). The impact of whole school professional learning on students’ multiplicative thinking. In G. Hine, S. Blackley, & A. Cooke (Eds.). Mathematics education research: Impacting practice Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia (pp. 228-235). MERGA.

Faragher, R. M., & Clarke, B. A. (2020). Inclusive practices in the teaching of mathematics: Some findings from research including children with Down Syndrome. Mathematics Education Research Journal, 32(1), 121-146. https://doi.org/10.1007/s13394-019-00294-x

Faragher, R., Brady, J., Clarke, B., & Gervasoni, A. (2008). Children with Down syndrome learning mathematics: Can they do it? Yes they can! Australian Primary Mathematics Classroom, 13(4), 10–15.

Feigenbaum, R. (2000). Algebra for students with learning disabilities. The Mathematics Teacher, 93(4), 270–276. https://doi.org/10.5951/mt.93.4.0270

Finn, J. D., & Zimmer, K. S. (2012). Student engagement: What is it? Why does it matter? In S. L. Christenson, A. L. Reschly, & C. Wylie (Eds.), Handbook of research on student engagement (pp. 97-131). Springer.

Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74(1), 59–109. https://doi.org/10.3102/00346543074001059

Fullan, M., Hill, P., & Crévola, C. (2006). Breakthrough. Corwin Press.

Gartland, D., & Strosnider, R. (2017). Learning disabilities and achieving high-quality education standards. Learning Disabilities Quarterly, 40(3), 152-154. https://doi.org/10.1177/0731948717696277

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009-4060). National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/.

Gervasoni, A. (2004). Exploring an intervention strategy for six and seven year old children who are vulnerable in learning school mathematics. (Unpublished doctoral dissertation). Bundoora, VIC: La Trobe University.

Gervasoni, A. (2015). Extending mathematical understanding: Intervention. BHS Publishing.

Gervasoni, A., & Lindenskov, L. (2011). Students with ‘Special rights’ for mathematics education. In B. Atweh, M. Graven, W. Secada, P. (Eds.), Mapping equity and quality in mathematics education (pp. 307–323). Springer. https://doi.org/10.1007/978-90- 481-9803-0_22

Gervasoni, A., & Peter-Koop, A. (2020). Inclusive mathematics education. Mathematics Education Research Journal, 32(1), 1-4. https://doi.org/10.1007/s13394-020-00315-0

Gervasoni, A., & Roche, A. (2018). Insights about how to create flexible mathematics classrooms that enable all students to thrive. In Hsieh, F.-J. (Ed.), Proceedings of the 8th ICMI-East Asia Regional Conference on Mathematics Education (Vol 2, pp. 388-397). Taiwan.

Gervasoni, A., & Sullivan, P. (2007). Assessing and teaching children who have difficulty learning arithmetic. Educational & Child Psychology, 24(2), 40-53.

Gervasoni, A., Parish, L., Hadden, T., Turkenburg, K., Bevan, K., Livesey, C., & Croswell, M. (2011). Insights about children’s understanding of 2-digit and 3-digit numbers. In J. Clark, B. Kissane, J. Mousley, T. Spencer, & S. Thornton (Eds.), Mathematics: Traditions and [new] practices: Proceedings of the 23rd biennial conference of the AAMT and the 34th annual conference of the Mathematics Education Research Group of Australasia (Vol 1, pp. 315-323). MERGA/AAMT.

Gervasoni, A., Roche, A., Giumelli, K. & McHugh, K. (2019). Insights about the progress of grade 1 children who are mathematically vulnerable and participate in a mathematics intervention program. In G. Hine, S. Blackley, & A. Cooke (Eds.). Mathematics Education Research: Impacting Practice: Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia (pp. 300-307). MERGA.

Greer, B. (1988). Non-conservation of multiplication and division: Analysis of a symptom. Journal of Mathematical Behaviour, 7(3), 281-298.

Hancock, K. J., & Zubrick, S. R. (2015). Children and young people at risk of disengagement from school: Literature review. Commissioner for Children and Young People.

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In J. Frank & K. Lester (Eds.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Information Age.

Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K.C., Wearne, D., Murray, H., Olivier, A., Human, P. (1997). Making sense. Teaching and learning mathematics with understanding. Heinemann.

Howard, P., & Perry, B. (2005). Learning mathematics: Perspectives of Australian Aboriginal children and their teacher. In H. L Chick & J. L. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 153-160). University of Melbourne.

Hunter, J., Hunter, R., & Anthony, G. (2020). Shifting towards equity: Challenging teacher views about student capability in mathematics. Mathematics Education Research Journal, 32(1), 37-56. https://doi.org/10.1007/s13394-019-00293-y

Jacob L., & Mulligan, J. (2014). Using arrays to build towards multiplicative thinking in the early years. Australian Primary Mathematics Classroom, 19(1), 35-40.

Jitendra, A. K. (2013). Understanding and accessing standards-based mathematics for students with mathematics difficulties. Learning Disabilities Quarterly, 36(1), 4-8. https://doi.org/10.1177/0731948712455337

Kalogeropoulos, P., Russo, J. A., Sullivan, P., Klooger, M., & Gunningham, S. (2020). Re-enfranchising mathematically-alienated students: Teacher and tutor perceptions of the Getting Ready in Numeracy (G.R.I.N- Program). International Electronic Journal of Mathematics Education, 15(1), https://doi.org/10.29333/iejme/588

Lambert, R., & Tan, P. (2020). Does disability matter in mathematics educational research? A critical comparison of research on students with and without disabilities. Mathematics Education Research Journal, 32(1), 5-36. https://doi.org/10.1007/s13394-019-00299-6

Lindenskov, L., & Lindhardt, B. (2020). Exploring approaches for inclusive mathematics teaching in Danish public schools. Mathematics Education Research Journal, 32(1), 57-76. https://doi.org/10.1007/s13394-019-00303-z

Lindenskov, L., & Weng, P. (2008). Specialundervisning I matematik – kan det ikke være lige meget? In: S. Tetler & S. Langager (Eds.), Specialunderisning i skolen (pp. 211–230). Gyldendal.

Markovits, Z., & Forgasz, H. (2017). “Mathematics is a lion”: Elementary students’ beliefs about mathematics. Educational Studies in Mathematics, 96(1), 49-64. https://doi.org/10.1007/s10649-017-9759-2

Marks, H. M. (2000). Student engagement in instructional activity: Patterns in the elementary, middle, and high school years. American Educational Research Journal, 37(1), 153-184. https://doi.org/10.3102/00028312037001153

Massey, S. (2021). Using emojis and drawings in surveys to measure children’s attitudes to mathematics. International Journal of Social Research Methodology

https://doi.org/10.1080/13645579.2021.1940774

McKenna, J. W., Shin, M., & Ciullo, S. (2015). Evaluating reading and mathematics instruction for students with learning disabilities: A synthesis of observation research. Learning Disabilities Quarterly, 38(4), 195-207. https://doi.org/10.1177/0731948714564576

Middleton, J. A. (2013). More than motivation: The combined effects of critical motivational variables on middle school mathematics achievement. Middle Grades Research Journal, 8(1), 77–95.

Mulligan, J. T., & Mitchelmore, M. C. (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28(3), 309-330. https://doi.org/10.2307/749783

Mulligan, J. T., Prescott, A., & Mitchelmore, M. C. (2006). Integrating concepts and processes in early mathematics: The Australian Pattern and Structure Mathematics Awareness Project (PASMAP). In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 209-216). PME.

National Joint Committee on Learning Disabilities. (2016). National Joint Committee on Learning Disabilities. http://www.ldonline.org/njcld

Pogrow, S. (1988). Teaching thinking to at-risk elementary students. Educational Leadership 45, 79–85.

Protzko, J. (2015). The environment in raising early intelligence: A meta-analysis of the fadeout effect. Intelligence, 53, 202-210. https://doi.org/10.1016/j.intell.2015.10.006

Regan, K. S., Berkeley, S. L., Hughes, M., & Brady, K. K. (2015). Understanding practitioner perceptions of responsiveness to intervention. Learning Disability Quarterly, 38(4), 234-247. https://doi.org/10.1177/0731948715580437

Renninger, K. A. (2000). Individual interest and its implications for understanding intrinsic motivation. In C. Sansone & J. M. Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The search for optimal motivation and performance (pp. 375–407). Academic.

Robertson, S., & Graven, M. (2020). Language as an including or excluding factor in mathematics teaching and learning. Mathematics Education Research Journal, 32(1), 77-102. https://doi.org/10.1007/s13394-019-00302-0

Rottmann, T., Haberetti, N., & Krämer, M. (2020). Inclusive assessment of whole number knowledge - Development and evaluation of an assessment interview for children with visual impairments in the primary grades. Mathematics Education Research Journal, 32(1),147-70. https://doi.org/10.1007/s13394-019-00296-9

Russo, J., McCosh, J., & Hewish, T. (2021). Student attitudes towards learning mathematics through problem solving: “I feel good because I make my head sore”. Australian Primary Mathematics Classroom, 26(1), 13-19.

Scherer, P., Beswick, K., Deblois, L., & Optiz, E. M. (2016). Assistance of students with mathematical learning difficulties: How can research support practice? ZDM Mathematics Education, 48(5), 633-649. https://doi.org/10.1007/s11858-016-0800-1

Schnepel, S., Krähenmann, H., Dessemontet, R. S., & Optiz, E. M. (2020). The mathematical progress of students with an intellectual disability in inclusive classrooms: Results of a longitudinal study. Mathematics Education Research Journal, 32(1), 37-55. https://doi.org/10.1007/s13394-019-00295-w

Schukajlow, S., & Krug, A. (2014). Do multiple solutions matter? Prompting multiple solutions, interest, competence, and autonomy. Journal for Research in Mathematics Education, 45(4), 497–533. https://doi.org/10.5951/jresematheduc.45.4.0497

Smith, T. M., Cobb, P., Farran, D. C., Cordray, D. S., & Munter, C. (2013). Evaluating Math Recovery: Assessing the causal impact of a diagnostic tutoring program on student achievement. American Educational Research Journal, 50(2), 397-428. https://doi.org/10.3102/0002831212469045

Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3-40). State University of New York Press.

Strand Cary, M. G., Clarke, B., Doabler, C. T., Smolkowski, K., Fien, H., & Baker, S, K. (2017). A practitioner implementation of a tier 2 first-grade mathematics intervention. Learning Disability Quarterly, 40(4), 211-224. https://doi.org/10.1177/0731948717714715

Sullivan, P., Clarke, D., & Clarke, B. A. (2009). Converting mathematics tasks to learning opportunities: An important aspect of knowledge for mathematics teaching. Mathematics Education Research Journal, 21(1), 85-105. https://doi.org/10.1007/bf03217539

Sullivan, P., Clarke, D., Cheeseman, J., Mulligan, J. (2001). Moving beyond physical models in learning multiplicative reasoning. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (pp. 233-240). Freudenthal Institute.

Sullivan, P., Mousley, J., & Zevenbergen, R. (2005). Increasing access to mathematical thinking. The

Australian Mathematical Society Gazette, 32(2), 105–109.

Suprayogi, M. N., Valcke, M., & Godwin, R. (2017). Teachers and their implementation of differentiated instruction in the classroom. Teaching and Teacher Education, 67, 291-301. https://doi.org/10.1016/j.tate.2017.06.020

Thomson, S., Wernert, N., O’Grady, E., & Rodrigues, S. (2017). TIMSS 2015: Reporting Australia’s results. Australian Council for Educational Research.

Thornton, S., Galluzzo, G., Quinane, M., & Taylor, D. (2010). One on one numeracy intervention: A pilot project in low SES communities. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematical education (Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia (pp. 555-562). MERGA.

Tomlinson, C. A. (2001). How to differentiate instruction in mixed ability classrooms (2nd ed). Association for Supervision & Curriculum Development.

Tomlinson, C. A., Brighton, C., Hertberg, H., Moon, T. R., Brimijoin, K., Conover, L. A., & Reynolds, T. (2003). Differentiated instruction in response to student readiness, interest, and learning profile in academically diverse classroom: A review of literature. Journal for the Education of the Gifted, 27(2/3), 119-145. https://doi.org/10.1177/016235320302700203

Victorian Curriculum and Assessment Authority (2021). Mathematics online interview. https://www.vcaa.vic.edu.au/assessment/f-10assessment/insight/Pages/assessment-tools.aspx

Way, J., Reece, A., Bobis, J., Anderson, J., & Martin, A. (2015), Improving student motivation and engagement in mathematics through one-to-one interactions. In M. Marshman, V. Geiger, & A. Bennison (Eds.), Mathematics education in the margins: Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia (pp. 627-634). MERGA.

Woodward, J., & Montague, M. (2002). Meeting the challenge of mathematics reform for students with LD. Journal of Special Education, 36(2), 89–101. https://doi.org/10.1177/00224669020360020401

Wright, B. (2003). A mathematics recovery: Program of intervention in early number learning. Australian Journal of Learning Difficulties, 8(4), 6–11. https://doi.org/10.1080/19404150309546741

Wright, R., Martland, J., & Stafford, A. (2000). Early Numeracy: Assessment for teaching and intervention. Paul Chapman Publishing.

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2021-08-13