Developing Percentage Proficiency

This research overview explores effective strategies for teaching percentages, addresses common student challenges, and provides evidence-based recommendations to help educators improve their instruct

Introduction:

Percentages are a fundamental mathematical concept with widespread applications in everyday life, from calculating discounts to analyzing data. However, research has consistently shown that students often struggle to grasp and fully apply percentage concepts (Lamon, 2007; Van Galen et al., 2008; Moss & Case, 1999). This overview examines key findings and recommendations from studies on teaching and learning percentages, focusing on effective instructional approaches and common challenges students face.

Conceptual Understanding of Percentages

The Nature of Percentages

Percentages, a specialized form of ratio, standardize comparisons between quantities by expressing them relative to a common base of 100. This standardization allows for more intuitive comparisons across different contexts, making percentages a powerful tool in mathematics and everyday life. At its core, a percentage is a proportion—a ratio scaled up or down to maintain its original relationship, with the result always being 100. This concept is deeply rooted in proportional reasoning, which research has shown to be crucial for developing a comprehensive understanding of percentages (Lamon, 2007; Parker & Leinhardt, 1995).

Common Misconceptions

When teaching percentages, it is important to focus on three key ideas: understanding percentages as a ratio to 100, connecting percentages to fractions and decimals, and using percentages for comparisons. However, research shows that students often struggle with certain aspects of percentages. Common difficulties include confusing absolute changes (like adding 50) with relative changes (like a 50% increase), getting mixed up when the reference point (the "whole") changes in a problem, and applying memorized procedures without really understanding why they work (Van Galen & Van Eerde, 2013; Dole et al., 1997). To address these challenges, using visual models, real-world examples, and activities that build conceptual understanding is helpful rather than just teaching calculation methods. By focusing on these areas, we can help students develop a more robust and flexible understanding of percentages.

Grounding in Real-Word Contexts

Research strongly supports introducing and exploring percentage concepts through authentic, real-life situations (Van den Heuvel-Panhuizen, 2003; Freudenthal, 1991). This approach helps students recognize the relevance and applicability of percentages, develop intuitive understanding before formal procedures, and make connections between percentages and other mathematical ideas. Examples of effective contextual learning activities include analyzing sales discounts and tax calculations, exploring data from surveys or experiments, and investigating financial concepts like interest rates.

Representations

Research strongly supports using multiple representations, including iconic (bar model or double number-line), symbolic (ratio table), and verbal forms (amount to a percent compared to the total to 100%), when teaching percentages. This approach encourages students to move flexibly between different representations, create their own, and explicitly discuss connections among them. Students progress through various levels of understanding when exploring these representations: context-connected/informal, referential, general, and formal levels (Gravemeijer, 1994; Van den Heuvel-Panhuizen, 2003). Educators can help students develop a more comprehensive and flexible understanding of percentage concepts by employing multiple representations.

Iconic models play a crucial role in developing percentage understanding. Research highlights key representations, including the Bar Model, 100-grid Charts, and Double Number Lines (Van Galen et al., 2008; Middleton & Van den Heuvel-Panhuizen, 1995). The bar model, in particular, has been shown to be effective in supporting percentage reasoning by visually representing amount-to-percent (part-whole) relationships, facilitating proportional thinking, and supporting problem-solving across various percentage situations.

Visual models can effectively address common misconceptions and challenges in learning percentages. These include distinguishing between absolute and relative changes, confusion when the "whole" changes in situations and struggles with percentages greater than 100% (Van Galen & Van Eerde, 2013; Parker & Leinhardt, 1995). By providing a visual representation that adapts to various problem types, the bar model supports students in developing a more robust and flexible understanding of percentages. This approach ultimately equips them for success in advanced mathematics and real-world applications, helping students connect percentage concepts to other rational number concepts, such as fractions and decimals (Middleton et al., 1998).

Developing Benchmark Understanding

Studies emphasize the importance of establishing strong benchmark percentages as reference points, including 1%, 5%, 10%, 25%, 50%, and 100% (Van den Heuvel-Panhuizen, 2003; Parker & Leinhardt, 1995). Activities that promote flexible thinking with these benchmarks, such as mental math exercises and estimation tasks, can significantly enhance percentage fluency.

Language

Research has identified several conceptual challenges students face when learning percentages, including understanding percentages greater than 100%, grasping the idea of percent change (especially decreases), and flexibly moving between different representations (Parker & Leinhardt, 1995; Dole et al., 1997). While many students can follow rote procedures for percentage calculations, studies show they often struggle with selecting appropriate strategies for different problem types, adapting procedures to non-standard situations, and explaining the reasoning behind their calculations (Van Galen & Van Eerde, 2013; Moss & Case, 1999).

Research on teaching percentages highlights students’ challenges when conceptualizing them as part-whole relationships. A study by Martinez and Dougherty (2020) identified confusion about the whole as the most common misconception about percentage problems. This suggests that traditional part-whole language may be inadequate for understanding percentages comprehensively. This finding aligns with the suggestion to use alternative terminology, such as "amount" instead of part and "base percent" instead of whole, which moves beyond the limitations of part whole descriptions.

To address these challenges, researchers like Van den Heuvel-Panhuizen (2003) propose a shift in language that emphasizes "the change in proportion relative to a base of 100%." This approach, combined with multiple representations and contexts, such as bar models, ratio tables, and real-world situations, can help students develop a more flexible and nuanced understanding of percentages. By moving beyond the limited part-whole conception, students can better grasp the relational nature of percentages and apply this knowledge to a broader range of problem-solving situations, especially understanding percentages greater than 100%.

Conclusion

In conclusion, teaching and learning percentages present both challenges and opportunities. By grounding instruction in real-world contexts, emphasizing visual representations and the ratio table, modifying language, and focusing on conceptual understanding, educators can help students develop robust and flexible percentage thinking (Van den Heuvel-Panhuizen, 2003; Freudenthal, 1991; Lamon, 2007). As percentages continue to play a vital role in many aspects of life and work, effective instruction remains crucial for developing mathematically literate citizens.

References

Dole, S., Cooper, T. J., Baturo, A. R., & Conoplia, Z. (1997). Year 8, 9 and 10 students' understanding and access of percent knowledge. In F. Biddulph & K. Carr (Eds.), People in mathematics education (Proceedings of the 20th annual conference of the Mathematics Education Research Group of Australasia) (pp. 147-154). MERGA.

Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Kluwer Academic Publishers.

Gravemeijer, K. (1994). Developing realistic mathematics education. CD-β Press.

I, J. Y., Martinez, R., & Dougherty, B. J. (2020). Misconceptions on part-part-whole proportional relationships using proportional division problems. Investigations in Mathematics Learning, 12(2), 67-81.

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-667). Information Age Publishing.

Middleton, J. A., & Van den Heuvel-Panhuizen, M. (1995). The ratio table. Mathematics Teaching in the Middle School, 1(4), 282-288.

Moss, J., & Case, R. (1999). Developing children's understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122-147.

Parker, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65(4), 421-481.

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(1), 9-35.

Van Galen, F., & Van Eerde, D. (2013). Solving problems with the percentage bar. Journal of Mathematics Didactics, 34(1), 1-20.

Van Galen, F., Feijs, E., Figueiredo, N., Gravemeijer, K., Van Herpen, E., & Keijzer, R. (2008). Fractions, percentages, decimals and proportions: A learning-teaching trajectory for grade 4, 5 and 6. Sense Publishers.

Social Media

Why are percentages often challenging to teach and learn? Research highlights the critical role of visual representations in developing students' conceptual understanding of percentages. The bar model and ratio table, in particular, have emerged as powerful tools that serve multiple functions, from illustrating ratio relationships to facilitating estimation and calculation.

Moreover, the research suggests a shift in language use, moving beyond traditional part-whole descriptions to emphasizing the change in proportion relative to a base of 100%. This linguistic approach fosters a more nuanced comprehension of percentage situations. By implementing these evidence-based strategies, educators can cultivate flexible thinking about percentages, thereby equipping students for success in advanced mathematics and real-world applications.

Join us in exploring these powerful learning strategies and their impact on mathematical thinking!