According to Van de Walle et al. (2014), teachers have traditionally used significantly different approaches to teach whole number and fraction operations. These differences, they say, help explain why so many students find fraction operations so difficult. The table below outlines some of the major pedagogical disparities they have observed.
Given the differences noted above, it is not surprising that students often struggle with fraction operations. Small (2013) notes that students are more successful with decimal operations because the algorithms used with decimals are similar to those used with whole numbers. Students are less comfortable with fraction operations, in part, because the procedures they have learned in whole number and decimal contexts no longer appear to work. Part of the problem could be students’ lack of readiness. As noted in the table above, students are often made to learn fraction operations before they truly understand fractions themselves. Van de Walle et al. (2014) note that when students have weak understandings of fractions, they tend to over-generalize whole number operational knowledge to fractions (for example, they may add ½ + ½ and determine the sum to be 2/4). When this is the case, it is imperative that educators adjust their teaching to ensure that students obtain the mathematical building blocks upon which success with fraction operations can be built.
Even when students have developed a solid understanding of fractions, the use of such numbers in operations continues to be a traditional point of difficulty in middle school mathematics. Many researchers agree that students tend to lose the sense of the meanings of the operations once fractions are involved, and that a key component to success with fraction operations is to connect them to their whole number counterparts so that the meanings behind them, even in the context of fractions, are made clear to the students (Van de Walle, 2014; Small, 2013; Ontario Ministry of Education, 2014).
Even when students have developed a solid understanding of fractions, the use of such numbers in operations continues to be a traditional point of difficulty in middle school mathematics. Many researchers agree that students tend to lose the sense of the meanings of the operations once fractions are involved, and that a key component to success with fraction operations is to connect them to their whole number counterparts so that the meanings behind them, even in the context of fractions, are made clear to the students (Van de Walle, 2014; Small, 2013; Ontario Ministry of Education, 2014).
The effective use of models is essential to students’ understanding of fraction operations. The area model is often used for this purpose. Van de Walle et al. (2014) remind us that line and set models are also very effective, and that students are best served when they are exposed to all three types. They also make a strong argument against simply sharing one algorithm per operation. They say that algorithms alone do not create conceptual understanding. When students do not understand procedures, they may:
To prevent the problems noted above, Van de Walle et al. (2014) suggest the following strategies when teaching fraction operations:
Please view the videos below for more about Mathematical Knowledge for Teaching in relation to fraction operations.
- easily forget them;
- be unable to determine which contexts are appropriate for their use;
- have difficulty assessing the correctness of their answers;
- accidentally mix procedures (e.g., create common denominators to multiply fractions or change the second addend to its reciprocal for addition); or
- have difficulty dealing with slight changes in presentation (e.g., the introduction of fractions greater than one).
To prevent the problems noted above, Van de Walle et al. (2014) suggest the following strategies when teaching fraction operations:
- Use story-based tasks – These allow students to see what is happening with the operations and build understandings of them.
- Use a variety of models – Area, line, and set models should be used, and connections should be made between the models and symbolic representations.
- Use estimation and informal methods to direct student attention to the meanings of the operations – When students begin fraction operations in this way, they are more likely to think reflectively and build fraction number sense.
- Address misconceptions directly – When teachers are aware of common misconceptions and talk openly about them, they help build correct understandings among their students.
Please view the videos below for more about Mathematical Knowledge for Teaching in relation to fraction operations.
References
Kajander, A. (2014). MB4T... Conceptual building blocks for traditional fraction division procedure: Division by a unit fraction. Gazette - Ontario Association for Mathematics, 52(3), 21-22. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1563633163?accountid=15115
Kajander, A. (2014). MB4T... Fraction division procedures – Not just invert and multiply! Gazette - Ontario Association for Mathematics, 52(4), 23-25. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1563633558?accountid=15115
Kajander, A. (2013). MB4T... The standard procedure for fraction multiplication: Generalizing the area model. Gazette - Ontario Association for Mathematics, 52(2), 21. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1477572074?accountid=15115
Small, M. (2010). Big ideas from Dr. Small: creating a comfort zone for teaching mathematics: grades 4-8. Toronto: Nelson Education.
Small, M. (2013). Making math meaningful to Canadian students, K-8, second edition. Toronto: Nelson Education.
Van de Walle, J. A., Bay-Williams, J. M., Karp, K. S., & Lovin, L. H. (2014). Teaching student-centered mathematics: developmentally appropriate instruction for grades 6-8(Vol. III). Boston: Pearson.
Kajander, A. (2014). MB4T... Conceptual building blocks for traditional fraction division procedure: Division by a unit fraction. Gazette - Ontario Association for Mathematics, 52(3), 21-22. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1563633163?accountid=15115
Kajander, A. (2014). MB4T... Fraction division procedures – Not just invert and multiply! Gazette - Ontario Association for Mathematics, 52(4), 23-25. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1563633558?accountid=15115
Kajander, A. (2013). MB4T... The standard procedure for fraction multiplication: Generalizing the area model. Gazette - Ontario Association for Mathematics, 52(2), 21. Retrieved from https://www.lib.uwo.ca/cgi-bin/ezpauthn.cgi?url=http://search.proquest.com/docview/1477572074?accountid=15115
Small, M. (2010). Big ideas from Dr. Small: creating a comfort zone for teaching mathematics: grades 4-8. Toronto: Nelson Education.
Small, M. (2013). Making math meaningful to Canadian students, K-8, second edition. Toronto: Nelson Education.
Van de Walle, J. A., Bay-Williams, J. M., Karp, K. S., & Lovin, L. H. (2014). Teaching student-centered mathematics: developmentally appropriate instruction for grades 6-8(Vol. III). Boston: Pearson.