​

Tēnā koutou katoa.

Ko Isle of Man te whakapaparanga mai.

Engari

I tipu ake au ki Pakuranga me Karangahape.

Kei Awaroa tōku kāinga ināianei.

Ko Gail rāua ko Melville ōku mātua.

Ko Rose tōku tamaiti.

Ko Tanya Killip tōku ingoa.

Nō reira, tēnā koutou katoa.

�

Why do my students struggle with fractions?��How can I help them?�

Presentation for AMA online 13/9/25

Tanya Killip

1. Background

​

• Why do students struggle with fractions?

​

• Research findings

​

• How can you help? Some activities from research.

​

What this presentation

will cover�

Background����

• Fractions are a notoriously difficult topic for students
• Fractions are important: a good understanding is essential for secondary school maths and a conceptual understanding of fractions predicts general maths achievement (Lamon, 2001; Resnick et al., 2016; Roesslein & Codding, 2018; Siegler et al., 2011)

​

Fractions: A Weeping Sore in Mathematics Education (Ellerton and Clements, 1994)

“The most important foundational skill not presently developed appears to be proficiency with fractions” (US National Mathematics Advisory Panel, 2008)

Why do students struggle with fractions?

1. Substantial conceptual shift from whole numbers

​

• Five different fraction interpretations

​

• Three different types of physical and visual representations

​

Substantial conceptual shift from whole numbers�

Five different fraction interpretations

​

​

​

(Kieren, 1976, 1980)

+1/4

+1/4

+1/4

Three different types of physical and visual representations of fractions�

​

​

​

Questions for discussion�

Research findings

1. A conceptual understanding is essential
2. Too great an emphasis on part-whole interpretation is unhelpful
3. Greater emphasis on measurement interpretation is helpful
4. An understanding of fraction magnitude is key

​

Conceptual understanding is essential�

​

​

​

• Students need conceptual understanding – algorithms alone insufficient (Hecht & Vagi, 2010; Lamon, 2001; Siegler et al. 2011)
• Physical and visual representations build conceptual understanding (Bruce et al., 2023; Flores et al., 2018; Lamon, 2001)
• However, multiple representations can lead to confusion unless students have good representational competencies, this takes time (Lesh et al., 1987; Rau & Matthews, 2017; Subramaniam, 2019)

​

​

​

​

Too great an emphasis on part-whole interpretation is unhelpful

• Part-whole fraction interpretation linked to area models predominate in fraction instruction

(Berggren, 2023; Dogan & Tertemiz, 2020; Marmur et al., 2020; Zhang et al., 2015)

• Problems with part-whole interpretation:
• Don’t emphasise fractions as a single number and so whole number bias persists,
• Difficult to conceptualise improper fractions,
• Don’t promote a good understanding of magnitude.
• (Sidney, 2019)

Greater emphasis on measurement interpretation is helpful

• Measurement interpretation emphasises fractions as single numbers with magnitude and makes sense of improper fractions
• Number lines also help build an understanding of density, and are more effective in solving fraction comparison and fraction division problems than area models
• Process of constructing common units is the same as the process of finding common denominators for comparing fractions and for fraction addition and subtraction

(Lamon, 2020; Sidney et al., 2019 ; Siegler et al., 2011)

​

An understanding of fraction magnitude is key

• Fraction magnitude understanding predicts fraction arithmetic ability and general mathematics achievement (Siegler et al., 2011)

​

• The unifying property of real numbers is magnitude (Siegler & Braithwaite, 2017)

​

• Iterate unit fractions, understand the inversion property, successively partition the number line, and estimate fractions on unmarked number lines (Braithwaite & Siegler, 2021; Cortina et al., 2014; Hamdan & Gunderson, 2017; Lamon, 2020)

How can you help students?

​

​

• Emphasise the measurement interpretation of fractions and use number lines, more than using the part-whole interpretation of fractions and using area models.

​

• Promote understanding of the magnitude of fractions, including different fraction types such as improper fractions.

​

​

Activities from research�

References

Berggren, J. (2023). Some Conceptual Metaphors for Rational Numbers as Fractions in Swedish Mathematics Textbooks for Elementary Education. Scandinavian Journal of Educational Research, 67(6), 914–927. https://doi.org/10.1080/00313831.2022.2114541

Braithwaite, D. W., & Siegler, R. S. (2021). Putting fractions together. Journal of Educational Psychology, 113(3), 556–571. https://doi.org/10.1037/edu0000477

Bruce, C. D., Flynn, T., Yearley, S., & Hawes, Z. (2023). Leveraging number lines and unit fractions to build student understanding: insights from a mixed methods study. Canadian Journal of Science, mathematics and technology education, 23(2), 322-339. https://doi.org/10.1007/s42330-023-00278-x

Cortina, J. L., Visnovska, J., & Zuniga, C. (2014). Unit fractions in the context of proportionality : supporting students’ reasoning about the inverse order relationship. Mathematics Education Research Journal, 26(1), 79–99. https://doi.org/10.1007/s13394-013-0112-5

Doğan, A., & IŞik Tertemiz, N. (2020). Fraction models used by primary school teachers. Ilköğretim online, 1888–1901. https://doi.org/10.17051/ilkonline.2020.762538

Educational Assessment Research Unit, & New Zealand Council for Educational Research. (2022a).  Thinking about fractions. Ministry of Education. https://nmssa-production.s3.amazonaws.com/documents/NMSSA_2022_Insights_Fractions.pdf

Flores, M. M., Hinton, V. M., & Taylor, J. J. (2018). CRA fraction intervention for fifth-grade students receiving tier two interventions. Preventing School Failure: Alternative Education for Children and Youth, 62(3), 198–213. https://doi.org/10.1080/1045988X.2017.1414027

Hamdan, N. & Gunderson, E. A. (2017). The number line is a critical spatial-numerical representation. Developmental Psychology, 53(3), 587–596. https://doi.org/10.1037/dev0000252

Hecht, S. A., & Vagi, K. J. (2010). Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology, 102 (4), 843-859.  https://doi.org/10.1037/a0019824

Kieren, T. E. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. A. Lesh (Ed.) Number and measurement. Papers from a research workshop (Vol. 7418491, pp. 101-144). ERIC.

Kieren, T. E. (1980). The rational number construct: Its elements and mechanisms. In T. E. Kieren (Ed.), Recent research on number learning (pp. 125-149). ERIC. https://files.eric.ed.gov/fulltext/ED212463.pdf

References

Lamon, S. J. (2001). Presenting and representing: From fractions to rational numbers. In A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 146–165).  National Council of Teachers of Mathematics.

Lamon, S. J. (2020). Teaching fractions and ratios for understanding : essential content knowledge and instructional strategies for teachers (Fourth edition.). Routledge.

Lesh, R., Post, T. R., & Behr, M. J. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp.33–40). Lawrence Erlbaum.

Marmur, O., Yan, X., & Zazkis, R. (2019). Fraction images: the case of six and a half. Research in Mathematics Education, 22(1), 22–47. https://doi.org/10.1080/14794802.2019.1627239

McMullen, J., Laakkonen, E., Hannula-Sormunen, M., & Lehtinen, E. (2015). Modeling the developmental trajectories of rational number concept(s). Learning and Instruction, 37, 14-20. https://doi.org/10.1016/j.learninstruc.2013.12.004

Rau, M. A., & Matthews, P. G. (2017). How to make ‘more’ better? Principles for effective use of multiple representations to enhance students’ learning about fractions. ZDM – Mathematics Education, 49(4), 531-544. https://doi.org/10.1007/s11858-017-0846-8

Sidney, P. G., Thompson, C. A., & Rivera, F. D. (2019). Number lines, but not area models, support children’s accuracy and conceptual models of fraction division. Contemporary Educational Psychology, 58, 288–298. https://doi.org/10.1016/j.cedpsych.2019.03.011

Siegler, R. S., & Braithwaite, D. W. (2017). Numerical development. Annual Review of Psychology, 68(1), 187–213. https://doi.org/10.1146/annurev-psych-010416-044101

Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. https://doi.org/10.1016/j.cogpsych.2011.03.001

Subramaniam, K. (2019) Representational coherence in instruction as a means of enhancing students’ access to mathematics. Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education, 2019, 1, 33–52. https://www.researchgate.net/publication/334443112_REPRESENTATIONAL_COHERENCE_IN_INSTRUCTION_AS_A_MEANS_OF_ENHANCING_STUDENTS'_ACCESS_TO_MATHEMATICS

Zhang, X., Clements, M. A., & Ellerton, N. F. (2015). Conceptual mis(understandings) of fractions: From area models to multiple embodiments. Mathematics Education Research Journal, 27, 233-261. https://doi.org/10.1007/s13394-014-0133-8