Lucie DeBlois, Professor at Université Laval

## Summary

Learning fractions poses a number of challenges for students; for example, they must think about contexts that require multiplication or division. Students learn fractions after learning natural numbers; they often “transfer” what they know about natural numbers to fractions. When they are asked to assign one role to the numerator and another role to the denominator—the terminology of the fraction—students sometimes treat these terms as if they were natural numbers, concluding, for example, that ¾ + ½ = 4/6, instead of 5/4.

In this article, we seek to understand the errors that students make. We offer a number of cautionary notes for creating activities for the acquisition of this mathematical concept. The errors explored in this article come out of research[1] involving students between the ages of 9 years and 12 years, at the moment when they displayed reactions of avoidance, worry or anxiety (DeBlois and Bélanger, 2016, DeBlois, 2014).

## Teaching Fractions

Learning natural numbers and their operations influences the operation of fractions. Students attempt to transfer what they already know, creating what we will refer to as “rules”; instead of drawing on mathematical knowledge, these rules are based on the students’ observations of patterns, most often shaped by the nature of the tasks that they are asked to perform. These rules are based on the logic of actions performed previously and, for the students we encountered, they evoke avoidance, anxiety or worry.

### Understanding why students use subtraction instead of division

We observed that students sometimes resort to subtraction, instead of division, to interpret the fraction-operator (1/10 of 100). An 11-year-old student was asked to determine the value of a gem and the value of a pearl. He was told that a gold ingot was worth 100 points; a gem was worth 1/10 of the value of the ingot; and a pearl was worth 1/5 of the value of the gem. The student determined that 1/10 of 100 = 90, instead of 10. He then determined that 1/5 of 10 = 5, instead of 2. In this context—the fraction-operator—the student first considered the number 100 (the value of the gold ingot), then the number 10 (the value of the gem); he then subtracted them as if they were natural numbers. When asked to illustrate 1/10 of 100 using multi-base materials, he took a board containing 100 blocks, separated it into 10 sticks of 10 blocks each, and then placed 9 sticks on the table, explaining that the answer was 90.

When allowed to continue to find 1/5 of the value of the gem, he took one stick (10) and said, “This is the value of the gem.” This reflects knowledge “in action” of 1/10 of 100. This knowledge is different from what he had said earlier. However, the same reasoning was used because he counted up to five on this stick, saying, “And that’s the value of the pearl [5].” When he was then asked to find ½ of 10, he was able to question his answer and find that 1/5 of 10 is 2. These errors may stem from the first activities in which the student was exposed to fractions.

### Understanding why students see work with fractions as an “identifying activity”

Textbooks generally present geometric figures that have already been divided up (Figure 1) or sets of objects that are already arranged in subgroups (Figure 2), for which students are asked to find ¼, for example. Note that circles are generally avoided, due to the requirements for the diameter when dividing into equal parts.

Figure 1. The fraction ¼ as part of a whole                 Figure 2. The fraction ¼ as part of a whole

When they are simply asked to count the part or parts of a “whole” that has already been divided up, students are deprived of activities that entail dividing the whole into equal parts (by folding, cutting or marking) Counting requires students to identify the parts of a geometric figure or the sub-parts of a whole, instead of requiring them to do the work of dividing it into equal parts.

Unless they are exposed to activities using manipulatives, such as dividing an object or a set of objects into equal parts, students may not grasp the importance of equal parts, used to interpret the relationship between the numerator and the denominator in relation to the unit of reference. An activity using manipulatives involves dividing something which enables students to grasp, for example, that the value ¼ will be the same, no matter how the object is divided  or how big it is. The activities required in Figures 1 and 2 do not lead to this understanding.

The activity of dividing into equal parts also provides an entry into a multiplicative structure, particularly when the students are required to divide a group of objects into equal subgroups. If students are required to perform a counting activity, rather than a dividing activity, their thought process will be in the realm of an additive structure, rather than a multiplicative structure.[2] The perception of students that fractions are an identifying activity may explain why a 9-year-old student asked to represent ¾ of 12 raspberries would correctly divide the 12 raspberries on her sheet of paper into 4 groups, yet colour only one raspberry in 3 of the 4 groups, instead of 3 groups of raspberries (Giguère-Duchesne, 2013).

## Criteria for Precautions

### The choice of numbers

A student may be asked to choose ¾ of the apples in a fruit basket containing 4 apples. This context does not require the student to create groups, because the denominator corresponds to the number of items that make up the whole. The student can respond 3 apples, basing his or her thinking exclusively on the numerator.
Conversely, if the student is asked to take 2/3 of the bananas, knowing that there are 6 bananas in the fruit basket, he or she must first divide the 6 bananas into three subgroups, and then choose two subgroups from the three subgroups created. He or she must then count the items in the subgroups to come up with the answer. Unlike the example with the apples, the student cannot think exclusively in terms of the numerator to arrive at the correct answer.[3]

Thus, the second example requires the student to simultaneously consider the items (bananas), the subgroups, and the whole. Dividing the whole into equal subgroups enables the student to enter the realm of a multiplicative structure, recognizing the relationship between the numerator (two subgroups) and the denominator (three subgroups), rather than only becoming aware of the role of the numerator and the denominator.

### The choice of situations

Analyzing the features of situations is another criterion for cautionary notes; as we have seen, fractions come up in different contexts. The previous examples illustrate the specific requirements for context for the parts of a whole, the parts of a collection of objects, and the operator. The example involving the raspberries helps us to understand the specific requirements for the context of the “parts of a collection of objects”, in contrast to the “parts of a whole”. Once a student is familiar with counting the parts of a geometric figure one-by-one, he or she will transfer this knowledge without adapting it to the need to count complete subgroups. Fractions also come up in other contexts such as measuring (¼ of a cup), ratios (1 unit of red to 4 units of white) which we also find in the context of probabilities, and numbers on a number line or an operation. Mercier and DeBlois (2004) observed that some students use the meaning “part of a whole” to put a fraction on a number line (meaning “number of the fraction”). Educators must take care to familiarize their students with a set of contexts that will enable them to adapt their interpretation of fractions.

### The creation of counterexamples

It may be important to create breaches of the didactic contract (DeBlois, 2010, DeBlois and Larivière, 2012, Larivière and DeBlois, 2013) by looking at a counterexample. Let’s return to the student who is asked to find 1/5 of 10. His answer was 5, which he was able to question when asked to find ½ of 10; his explanation was that there are only two [little blocks]. A counterexample enables a student to become aware of the limits of his or her knowledge, or of the rules that he or she has constructed, and ultimately to recognize that these must be adapted or abandoned, in a context that has different requirements (DeBlois and René de Cotret, 2005).

## Conclusion

The errors that students make are part of the process of transforming knowledge, particularly when they are asked to solve problems, rather than apply learning (DeBlois, 1995). In these conditions, educators need to be curious about their students’ reasoning process in order to interpret their errors and locate them in a process (DeBlois, 2003). For example, at the elementary level, percentages are most often presented as equivalent fractions (CSFL, 2015) at the elementary level. However, at the secondary level, percentages are a part of learning proportional reasoning, which is based on fraction/ratio. Thus, in order for operations on fractions to have meaning, students must develop an understanding not only of the role of the numerator and the denominator, but of the relationships between the numerator, the denominator, and the unit of reference.

## Notes

[1] This research was made possible with funding from the Fond Grégoire-Blouin at l’Université Laval

[2] Additive structures correspond to activities drawing on addition and its opposite, subtraction. Multiplicative structures correspond to activities drawing on multiplication and its opposite, division.

[3] Sincere thanks to Jean-Philippe Bélanger for this example.

## Related Resources on the LD@school Website:

Click here to access the video Math Manipulatives: Tools to Help All Students Become Effective Problem Solvers.

Click here to access the article Concrete – Representational – Abstract: An Instructional Strategy for Math.

Click here to access the video Using Collaborative Teacher Inquiry to Support Students with LDs in Math.

Click here to access the learning module Concrete, Representational, Abstract Strategies for Mathematics Instruction.

## Other Relevant Resources

Click here to access a digital paper entitled Understanding Fractions on the Edugains website.

Click here to access a PDF document entitled Paying Attention to Fractions: K – 12 Support Document for Paying Attention to Mathematics Education, developed by the Ontario Ministry of Education.

Click here to access a digital paper entitled Professional Learning about Fractions: Collaborative Action Research Project Digital Paper on the Edugains website.

## References

Commission du-Fleuve-et-des-Lacs (2015). Les capsules didactiques. Available online at: http://www.csfl.qc.ca/index.php/outils/outils-pedagogiques/capsules-mathématiques

DeBlois L. (1995). La place de l'erreur dans le développement de la compréhension en mathématiques. Instantanés mathématiques. St-Laurent (Québec). XXXI (2), 4-7.

DeBlois L. (2003) Interpréter explicitement les productions des élèves : une piste. Éducation et Francophonie XXXI(2) 176-199. http://www.acelf.ca/c/revue/pdf/XXXI_2_176.pdf

DeBlois L. Larivière, A. (2012) Une analyse du contrat didactique pour interpréter les comportements des élèves au primaire. Colloque Espace Mathématique Francophone 2012. Available online at: http://emf.unige.ch/files/6714/5320/9568/EMF2012GT9DEBLOIS.pdf

DeBlois, L. (2010). Peut-on lire les troubles de comportement autrement ? Bulletin du CRIRES. Nouvelles CSQ. 21-24. Available online at: http://crires.ulaval.ca/sites/crires/files/roles/membre-crires/no_23_2010.pdf

DeBlois, L. (2014) Le rapport aux savoirs pour établir des relations entre troubles de comportements et difficultés d'apprentissage en mathématiques. Dans Le rapport aux savoirs: Une clé pour analyser les épistémologies enseignantes et les pratiques de la classe. Coordinated by Marie-Claude Bernard, Annie Savard, Chantale Beaucher. Available online at: http://lel.crires.ulaval.ca/public/le_rapport_aux_savoirs.pdf

DeBlois, L., René de Cotret, S. (2005). Et si les erreurs des élèves étaient le fruit d'une extension de leurs connaissances. Réussite scolaire : comprendre et mieux intervenir. Presses de l'Université Laval, Ste-Foy, Québec. 135-145.

DeBlois, L., Bélanger, J.-P. (2016). La résolution de problèmes vue par les élèves qui manifestent des réactions d’évitement, d’anxiété ou d’agitation. Vivre le primaire 29 (2). 62-66. Available online at : http://www.fse.ulaval.ca/fichiers/site_fse2015/documents/Actualite/VLP_Vol29No2_Web62_65_2_.pdf

Giguère-Duchesne A. (2013). Une recension des règles et des habitudes des élèves du deuxième cycle du primaire en mathématiques pour favoriser la réussite scolaire. Mémoire de maîtrise. Université Laval. Available online at : http://ariane2.bibl.ulaval.ca/ariane/?wicket:interface=:3.

Larivière, A. DeBlois, L. (2013) Quelles mathématiques font les élèves qui adoptent des comportements d’évitement en mathématiques ? Vivre le primaire 26 (2), 56-61.

Mercier, P., DeBlois, L. (2004). Passage primaire-secondaire dans l’enseignement et l’apprentissage des fractions. Envol 127. 17-24. Available online at: http://grms.qc.ca/wp-content/themes/shamrock/pdfs/articles/revue_127-3.pdf

Ministère de l’éducation des Loisirs et du Sport (2010). Épreuves de mathématique deuxième année du troisième cycle primaire. Québec.

Lucie DeBlois earned a Bachelor’s degree in Remedial Teaching at l’Université de Sherbrooke. After working for over 10 years as a remedial teacher in various school boards in Quebec, she completed a Master’s degree and then a Ph.D. in the teaching of mathematics at l’Université Laval. She has been a professor and a researcher l’Université Laval since 1995. She is currently a full professor in the department of studies in education and learning. Her areas of expertise include the development of mathematical understanding in elementary and secondary students, in-service training for teachers, and pre-service training for elementary and secondary teachers. Through her involvement in a research centre dedicated to student success, Centre de recherche et d’intervention sur la réussite scolaire (CRIRES), she has developed expertise in the factors that affect student success.