Elementary Preservice Teachers’ Flexible Use of Strategies for Solving Linear Relationship Problems Using Strip Diagrams

Author:
Charles Hohensee University of Delaware

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Teo Paoletti University of Delaware

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Allison L. Gantt University of Delaware

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Srujana Acharya University of Delaware

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Julien Corven Illinois State University

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Volume/Issue:
Volume 13: Issue 2
Page(s):
108–121
DOI:
https://doi.org/10.5951/MTE.2023-0012
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Keywords:
early algebra; algebraic thinking; representation; elementary preservice teachers; equal sign; strategy

Research has shown there are algebra concepts elementary teachers can introduce that help prepare elementary students for the eventual transition to algebra (e.g., the relational interpretation of the equal sign). Early algebra refers to the use of informal approaches to introduce such concepts to elementary students. Strip diagrams, a type of informal diagram, can be used for early algebra. We present an instructional sequence mathematics teacher educators could use to introduce elementary preservice teachers to or reacquaint them with strip diagrams. We also present strategies elementary pre-service teachers used to solve strip diagram problems before and after participating in our instructional sequence. We conclude with a discussion about the strategies and implications for teaching early algebra to elementary preservice teachers.

Contributor Notes

Charles Hohensee, University of Delaware, Newark, DE 19716; hohensee@udel.edu https://orcid.org/0000-0001-5922-6686

Teo Paoletti, University of Delaware, Newark, DE 19716; teop@udel.edu https://orcid.org/0000-0002-1377-1133

Allison L. Gantt, University of Delaware, Newark, DE 19716; agantt@udel.edu https://orcid.org/0000-0001-7160-1231

Srujana Acharya, University of Delaware, Newark, DE 19716; srujana@udel.edu https://orcid.org/0009-0002-7367-560X

Julien Corven, Illinois State University, Normal, IL 61761; jccorve@ilstu.edu https://orcid.org/0000-0002-3139-612X

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  • Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9(3), 221–247. https://doi.org/10.1080/10986060701360902

    • Search Google Scholar
    • Export Citation

  • Allsopp, D., Lovin, L. H., & van Ingen, S. (2017). Supporting mathematical proficiency strategies for new special education teachers. Teaching Exceptional Children, 49(4), 273–283. https://doi.org/10.1177/0040059917692112

    • Search Google Scholar
    • Export Citation

  • Asquith, P., Stephens, A. C., Knuth E. J., & Alibali, M. W. (2007). Middle school mathematics teachers’ knowledge of students’ understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249–272. https://doi.org/10.1080/10986060701360910

    • Search Google Scholar
    • Export Citation

  • Ayieko, R. A., Gokbel, E. N., & Akcay, A. O. (2016). Algebra achievement gaps: A comparative study across the states over the years. In M. B. Wood, E. E. Turner, M. Civil, & J. A. Eli (Eds.), Proceedings of the 38th annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 140–147). The University of Arizona.

    • Search Google Scholar
    • Export Citation

  • Blanton, M., Stroud, R., Stephens, A., Gardiner, A. M., Stylianou, D. A., Knuth, E., Isler-Baykal, I., & Strachota, S. (2019). Does early algebra matter? The effectiveness of an early algebra intervention in grades 3 to 5. American Educational Research Journal, 56(5), 1930–1972 . https://doi.org/10.3102/0002831219832301

    • Search Google Scholar
    • Export Citation

  • Booth, J. L., & Koedinger, K. R. (2012). Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving. British Journal of Educational Psychology, 82(3), 492–511. https://doi.org/10.1111/j.2044-8279.2011.02041.x

    • Search Google Scholar
    • Export Citation

  • Brizuela, B., & Schliemann, A. (2004). Ten-year-old students solving linear equations. For the Learning of Mathematics, 24(2), 33–40. https://www.jstor.org/stable/40248456

    • Search Google Scholar
    • Export Citation

  • Carraher, D. W., & Schliemann, A. D. (2020). Early algebra teaching and learning. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 249–252). https://doi.org/10.1007/978-3-030-15789-0_54

    • Search Google Scholar
    • Export Citation

  • Cohen, J. S. (2013). Strip diagrams: Illuminating proportions. Mathematics Teaching in the Middle School, 18(9), 536–542. https://doi.org/10.5951/mathteacmiddscho.18.9.0536

    • Search Google Scholar
    • Export Citation

  • de Koning, B. B., Boonen, A. J., Jongerling, J., van Wesel, F., & van der Schoot, M. (2022). Model method drawing acts as a double-edged sword for solving inconsistent word problems. Educational Studies in Mathematics, 111(1), 29–45. https://doi.org/10.1007/s10649-022-10150-8

    • Search Google Scholar
    • Export Citation

  • Ding, M. (2018). Modeling with tape diagrams. Teaching Children Mathematics, 25(3), 158–165. https://doi.org/10.5951/teacchilmath.25.3.0158

    • Search Google Scholar
    • Export Citation

  • Ding, M. (2021). Teaching early algebra through example-based problem solving: Insights from Chinese and U.S. elementary classrooms. Routledge.

    • Search Google Scholar
    • Export Citation

  • Fuchs, L. S., Zumeta, R. O., Schumacher, R. F., Powell, S. R., Seethaler, P. M., Hamlett, C. L., & Fuchs, D. (2010). The effects of schema-broadening instruction on second graders’ word-problem performance and their ability to represent word problems with algebraic equations: A randomized control study. The Elementary School Journal, 110(4), 440–463. https://doi.org/10.1086/651191

    • Search Google Scholar
    • Export Citation

  • Fyfe, E. R., & Nathan, M. J. (2019). Making “concreteness fading” more concrete as a theory of instruction for promoting transfer. Educational Review, 71(4), 403–422, https://doi.org/10.1080/00131911.2018.1424116

    • Search Google Scholar
    • Export Citation

  • Gamoran, A., & Hannigan, E. C. (2000). Algebra for everyone? Benefits of college-preparatory mathematics for students with diverse abilities in early secondary school. Educational Evaluation and Policy Analysis, 22(3), 241–254. https://doi.org/10.3102/01623737022003241

    • Search Google Scholar
    • Export Citation

  • Glaser, B. G., & Holton, J. (2004). Remodeling grounded theory. Forum: Qualitative Social Research, 5(2), Article 4. https://doi.org/10.17169/fqs-5.2.607

    • Search Google Scholar
    • Export Citation

  • Goyer, A. (2023). Supporting pre-service elementary teachers’ early algebraic thinking with technology through lesson planning and TPACK [Doctoral dissertation, University of Nevada, Reno]. ProQuest Dissertations & Theses Global. https://www.proquest.com/dissertations-theses/supporting-pre-service-elementary-teachers-early/docview/2832696063/se-2

    • Search Google Scholar
    • Export Citation

  • Guerrero, S. M. (2010). The value of guess and check. Mathematics Teaching in the Middle School, 15(7), 392–398. https://doi.org/10.5951/MTMS.15.7.0392

    • Search Google Scholar
    • Export Citation

  • Haspekian, M., Kieran, C., Drijvers, P., Bråting, K., & Tabach, M. (2023). Algebra education and digital resources: A long-distance relationship? In B. Pepin, G. Gueudet, & J. Choppin (Eds.), Handbook of digital resources in mathematics education (pp. 2–32). Springer. https://doi.org/10.1007/978-3-030-95060-6_16-1

    • Search Google Scholar
    • Export Citation

  • Hohensee, C. (2017). Preparing elementary prospective teachers to teach early algebra. Journal of Mathematics Teacher Education, 20(3), 231–257. https://doi.org/10.1007/s10857-015-9324-9

    • Search Google Scholar
    • Export Citation

  • Hohensee, C., Paoletti, T., Corven, J., & Gantt, A. (2023). Elementary preservice teachers, early algebra, and reasoning deductively with diagrams. Manuscript under review.

  • Huntley, M. A., Marcus, R., Kahan, J., & Miller, J. L. (2007). Investigating high-school students’ reasoning strategies when they solve linear equations. Journal of Mathematical Behavior, 26(2), 115–139. https://doi.org/10.1016/j.jmathb.2007.05.005

    • Search Google Scholar
    • Export Citation

  • Jansen, A. (2020). Rough draft math. Stenhouse Publishers. https://doi.org/10.4324/9781032682327

  • Kaput, J. J. (2008). What is algebra? What is algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). Taylor & Francis Group.

    • Search Google Scholar
    • Export Citation

  • Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. MIT Press.

  • Kasmer, L. A., & Kim, O.-K. (2012). The nature of student predictions and learning opportunities in middle school algebra. Educational Studies in Mathematics, 79(2), 175–191. https://doi.org/10.1007/s10649-011-9336-z

    • Search Google Scholar
    • Export Citation

  • Kieran, C. (2004). Algebraic thinking in the early grades: What is it? Mathematics Educator, 8(1), 139–151. http://math.nie.edu.sg/ame/matheduc/journal/v8_1/v81_139.aspx

    • Search Google Scholar
    • Export Citation

  • Knuth, E. J., Alibali, M. W., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13(9), 514–519. https://doi.org/10.5951/MTMS.13.9.0514

    • Search Google Scholar
    • Export Citation

  • Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312. http://www.jstor.org/stable/30034852

    • Search Google Scholar
    • Export Citation

  • Lim, K. H., Buendía, G., Kim, O.-K., Cordero, F., & Kasmer, L. (2010). The role of prediction in the teaching and learning of mathematics. International Journal of Mathematical Education in Science and Technology, 41(5), 595–608. https://doi.org/10.1080/00207391003605239

    • Search Google Scholar
    • Export Citation

  • Loveless, T. (2013). The algebra imperative: Assessing algebra in a national and international context. Brookings. https://www.brookings.edu/wp-content/uploads/2016/07/Kern-Algebra-paper-8-30_v13.pdf

    • Search Google Scholar
    • Export Citation

  • Matthews, P. G., & Fuchs, L. S. (2020). Keys to the gate? Equal sign knowledge at second grade predicts fourth-grade algebra competence. Child Development, 91(1), e14–e28. https://doi.org/10.1111/cdev.13144

    • Search Google Scholar
    • Export Citation

  • McAuliffe, S., & Lubben, F. (2013). Perspectives on pre-service teacher knowledge for teaching early algebra. Perspectives in Education, 31(3), 155–169. https://hdl.handle.net/10520/EJC145549

    • Search Google Scholar
    • Export Citation

  • McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. E. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and Instruction, 24(3), 367–385. https://doi.org/10.1207/s1532690xci2403_3

    • Search Google Scholar
    • Export Citation

  • Mercer, C. D., & Miller, S. P. (1992). Teaching students with learning problems in math to acquire, understand, and apply basic math facts. Remedial and Special Education, 13(3), 19–35. https://doi.org/10.1177/074193259201300303

    • Search Google Scholar
    • Export Citation

  • Moses, R. P., & Cobb, C. (2001). Organizing algebra: The need to voice a demand. Social Policy, 31(4), 4–12.

  • Murata, A. (2008). Mathematics teaching and learning as a mediating process: The case of tape diagrams. Mathematical Thinking and Learning, 10(4), 374–406. https://doi.org/10.1080/10986060802291642

    • Search Google Scholar
    • Export Citation

  • Nagashima, T., Bartel, A. N., Tseng, S., Vest, N. A., Silla, E. M., Alibali, M. W., & Aleven, V. (2021). Scaffolded self-explanation with visual representations promotes efficient learning in early algebra. In T. Fitch, C. Lamm, H. Leder, & K. Teßmar-Raible (Eds.), 43rd annual meeting of the Cognitive Science Society (pp. 1858–1864). Cognitive Science Society. https://escholarship.org/uc/item/17q6g4db

    • Search Google Scholar
    • Export Citation

  • National Council of Teachers of Mathematics. (2014). Algebra as a strand of school mathematics for all [Position statement]. https://www.nctm.org/uploadedFiles/Standards_and_Positions/Position_Statements/Algebra_2014-04.pdf

  • National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. https://thecorestandards.org/math

    • Search Google Scholar
    • Export Citation

  • Otte, M. (2006). Mathematical epistemology from a Peircean semiotic point of view. Educational Studies in Mathematics, 61(1–2), 11–38. https://doi.org/10.1007/s10649-006-0082-6

    • Search Google Scholar
    • Export Citation

  • Pincheira, N., & Alsina, A. (2021). Teachers’ mathematics knowledge for teaching early algebra: A systematic review from the MKT perspective. Mathematics, 9(20), 1–16. https://doi.org/10.3390/math9202590

    • Search Google Scholar
    • Export Citation

  • Powell, S. R., & Fuchs, L. S. (2018). Effective word-problem instruction: Using schemas to facilitate mathematical reasoning. Teaching Exceptional Children, 51(1), 31–42. https://doi.org/10.1177/0040059918777250

    • Search Google Scholar
    • Export Citation

  • Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26(2), 257–277. https://doi.org/10.1007/s13394-013-0087-2

    • Search Google Scholar
    • Export Citation

  • RAND. (2003). Mathematical proficiency for all students: A strategic research and development program in mathematics education. https://doi.org/10.7249/MR1643

    • Search Google Scholar
    • Export Citation

  • Star, J. R., Rittle-Johnson, B., Lynch, K., & Perova, N. (2009). The role of prior knowledge in the development of strategy flexibility: The case of computational estimation. ZDM Mathematics Education, 41(5), 569–579. https://doi.org/10.1007/s11858-009-0181-9

    • Search Google Scholar
    • Export Citation

  • Stephens, A. C. (2006). Equivalence and relational thinking: Preservice elementary teachers’ awareness of opportunities and misconceptions. Journal of Mathematics Teacher Education, 9(3), 249–278. https://doi.org/10.1007/s10857-006-9000-1

    • Search Google Scholar
    • Export Citation

  • Stephens, A. C. (2008). What “counts” as algebra in the eyes of preservice elementary teachers? Journal of Mathematical Behavior, 27, 33–47. https://doi.org/10.1016/j.jmathb.2007.12.002

    • Search Google Scholar
    • Export Citation

  • Stephens, A. C., Ellis, A. B., Blanton, M., & Brizuela, B. M. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), The compendium for research in mathematics education (pp. 386–420). National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Superfine, A. C. (2021). An asset-based perspective on prospective teacher education. Journal of Mathematics Teacher Education, 24(4), 331–333. https://doi.org/10.1007/s10857-021-09503-6

    • Search Google Scholar
    • Export Citation

  • Suppa, S., & Hohensee, C. (2021). Struggles pre-service teachers experience when taking a pre-symbolic algebra content course. Mathematics Teacher Education and Development, 23(4), 50–73. https://eric.ed.gov/?id=EJ1339841

    • Search Google Scholar
    • Export Citation

  • Van Amerom, B. A. (2003). Focusing on informal strategies when linking arithmetic to early algebra. Educational Studies in Mathematics, 54(1), 63–75. https://doi.org/10.1023/B:EDUC.0000005237.72281.bf

    • Search Google Scholar
    • Export Citation

  • van Garderen, D., & Scheuermann, A. M. (2015). Diagramming word problems: A strategic approach for instruction. Intervention in School and Clinic, 50(5), 282–290. https://doi.org/10.1177/1053451214560889

    • Search Google Scholar
    • Export Citation

  • Van Vooren, A. L. (2016). First graders’ use of the bar model to communicate their understanding of the equal sign [Unpublished master’s thesis]. Boise State University.

    • Search Google Scholar
    • Export Citation

  • Zapatera, A., & Quevedo, E. (2021). The initial algebraic knowledge of preservice teachers. Mathematics, 9, 1–17. https://doi.org/10.3390/math9172117

    • Search Google Scholar
    • Export Citation
Copyright:
The National Council of Teachers of Mathematics, Inc. All rights reserved. 2025
Page Count:
14
Article Category:
Research Article
Received:
10 Apr 2024
Accepted:
21 May 2024
Print Publication Date:
01 Feb 2025
Online Publication Date:
Feb 2025
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