This test was given almost 40 years ago, which gave Hugo Lortie-Forgues and me hope that the work of innumerable teachers, mathematics coaches, researchers, and government commissions had made a positive difference. Such difficulties are not limited to fraction estimation problems nor do they end in 8th grade.
On standard fraction addition, subtraction, multiplication, and division problems with equal denominators e. Studies of community college students have revealed similarly poor fraction arithmetic performance. Children in the US do much worse on such problems than their peers in European countries, such as Belgium and Germany, and in Asian countries such as China and Korea.
This weak knowledge is especially unfortunate because fractions are foundational to many more advanced areas of mathematics and science. On the reference sheets for recent high school AP tests in chemistry and physics, fractions were part of more than half of the formulas.
In a recent survey of white collar, blue collar, and service workers, more than two-thirds indicated that they used fractions in their work.
Why are fractions so difficult to understand? A major reason is that learning fractions requires overcoming two types of difficulty: inherent and culturally contingent. Inherent sources of difficulty are those that derive from the nature of fractions, ones that confront all learners in all places. One inherent difficulty is the notation used to express fractions. Another inherent difficulty involves the complex relations between fraction arithmetic and whole number arithmetic.
For example, multiplying fractions involves applying the whole number operation independently to the numerator and the denominator e. A third inherent source of difficulty is complex conceptual relations among different fraction arithmetic operations, at least using standard algorithms.
Why do we need equal denominators to add and subtract fractions but not to multiply and divide them? Why do we invert and multiply to solve fraction division problems, and why do we invert the fraction in the denominator rather than the one in the numerator? They apply procedures that can only be used with whole numbers Nunes and Bryant, Consequently, typical errors appear in addition or subtraction tasks e.
In this case, pupils' reasoning can be resumed as follows: if the number is larger, then the magnitude it represents is larger. But when we think about fractions, a larger denominator does not mean a larger magnitude, but a smaller one. Another difficulty appears in multiplication tasks. Multiplying natural numbers always lead to a larger answer, but it is not the case with fractions e. The inappropriate generalization of the knowledge about natural numbers is even more resistant as it is widely anterior to the one about rational numbers Vamvakoussi and Vosniadou, In order to overcome these mistakes, it would seem necessary for students to perform a conceptual reorganisation which integrates rational numbers as a new category of numbers, with their own rules and functioning Stafylidou and Vosniadou, Furthermore, even in adults, knowledge about natural numbers is often preponderant when processing fractions Bonato et al.
Another major difficulty comes from the multifaceted notion of fractions Kieren, ; Brousseau et al. Kieren was the first to separate fractions into four interrelated categories: ratio; operator; quotient; and measure. The ratio category expresses the notion of a comparison between two quantities, for example when there are three boys for every four girls in a group.
In the operator category, fractions are considered as functions applied to objects, numbers or sets Behr et al. The fraction operator can enlarge or shrink a quantity to a new value. The quotient category refers to the result of a division. In the measure category, fractions are associated with two interrelated notions. Firstly, they are considered as numbers, which convey how big the fractions are.
Secondly, they are associated with the measure of an interval. According to Kieren , the part-whole notion of fractions is implicated in these four categories. That is the reason why he did not describe it as a fifth category. Thereafter, Behr et al. They recommend considering part-whole as an additional category.
They also associated partitioning to the part-whole notion. The part-whole category can then be defined as a situation in which a continuous quantity is partitioned into equal size e.
Other models have been proposed to describe the multiple meanings of fractions Brissiaud, ; Rouche, ; Mamede et al. These models partly overlap, but are not entirely equivalent.
For instance, Mamede et al. In the first stage, the fraction is seen as an operator. This notion refers to sharing situations. The second one is the ratio stage which requires a high level of abstraction because one needs to understand that different fractions can represent the same ratio. This is linked to the notion of equivalent fractions.
The third and last stage is related to the numerical meaning of fractions. Fractions are here conceived as a new category of numbers, with their own rules and properties. Another explanation of children's difficulties when learning fractions lies in the articulation between conceptual and procedural knowledge. Previous studies have shown that children would often perform calculations without knowing why Kerslake, Conceptual knowledge can be defined as the explicit or implicit understanding of the principles ruling a domain and the interrelations between the different parts of knowledge in a domain Rittle-Johnson and Alibali, It can also be considered as the knowledge of central concepts and principles, and their interrelations in a particular domain Schneider and Stern, Conceptual knowledge is thought to be mentally stored in a form of relational representations, such as semantic networks Hiebert, It is not tied to a specific problem, but can be generalized to a class of problems Hiebert, ; Schneider and Stern, Procedural knowledge can be defined as sequences of actions that are useful to solve problems Rittle-Johnson and Alibali, Some authors consider procedural knowledge as the knowledge of symbolic representations, algorithms, and rules Byrnes and Wasik, Moreover, procedural knowledge would allow people to solve problems in a quick and effective way as it can easily be automatized Schneider and Stern, Therefore, it can be used with few cognitive resources Schneider and Stern, However, procedural knowledge is not as flexible as conceptual knowledge and is often bound to specific problem types Baroody, Those two types of knowledge may not evolve in independent ways.
Many theories on knowledge acquisition suggest that the generation of procedures is based on conceptual understanding Halford, ; Gelman and Williams, They argue that children use their conceptual understanding to develop their discovery procedures and adapt acquired procedures to new tasks.
According to this approach, children's difficulties when learning about fractions could be interpreted as a use of mathematical symbols without access to their meaning. Procedural knowledge may also influence conceptual understanding. Using procedures would lead to a better conceptual understanding.
But few studies support this idea. For instance, Byrnes and Wasik argue that many children learn the right procedures to multiply fractions, but they never seem to understand the underlying principles. Other authors support a third point of view.
Both types of knowledge might progress in an iterative and interactive way Rittle-Johnson et al. Conceptual and procedural knowledge might continually and incrementally stimulate each other.
Neither would necessarily precede the other. In mathematics education, teachers seem to focus more on procedural than conceptual knowledge. Children usually learn rote procedures in a repetitive way.
This leads to a misunderstanding of mathematical symbols Byrnes and Wasik, Consequently many computational errors are due to an impoverished conceptual understanding. Taking into account the different theoretical models presented and the issues they arise led us to build our own conceptual framework. In this study exploring the difficulties in learning fractions, two main components were considered: a conceptual component and a procedural component.
Typical tasks used to assess that kind of conceptual knowledge involve shading parts of a figure indicated by a fraction, or the opposite exercise consisting of writing the fraction representing the quantity of a figure that is shaded Hiebert and Lefevre, ; Byrnes and Wasik, ; Ni, Proportion represents the comparison between two quantities.
We used comparison of different expressions of the same ratio e. The numerical meaning of fraction refers to the fact that fractions represent rational numbers that can be ordered on a number line Kieren, Two relevant tasks were used to assess children's understanding of the numerical meaning of fractions: firstly, number lines on which they are asked to place a fraction, and secondly, indicating which of several given fractions represents the largest quantity Byrnes and Wasik, ; Ni, Several variables also held our attention regarding the representation of fractions.
Discrete and continuous quantities were used. Multiple objects and figures, as well as numerical symbols were introduced to assess the possible interference of certain types of representations Coquin-Viennot and Camos, For practical reasons, we did not examine fractions as a measure in this study.
This category is closely related to the metric system. The manipulation of fractions as a measure can be made by splitting units of length, area, volume, time, mass, etc.
Understanding these measuring situations involves several concepts that are not exclusively related to fractions, such as understanding different unit systems or a good grasp of the decimal position system.
Therefore, it is difficult to assess the understanding of this category in isolation from these variables. Procedural items were those that could be easily solved by applying a procedure that could be implemented without checking for meaning outside that particular procedure. The procedural component involved various operations on fractions, namely the addition and subtraction with or without common denominators, multiplication, and simplification of fractions.
Children were given different arithmetical operations to solve as well as simplification exercises. The main aim of this study was to provide empirical data that could explain difficulties encountered by children when they learn fractions. Our first objective was to analyse the mathematics curriculum of the French Community of Belgium, where this study was conducted. Our second objective was to understand the nature of pupils' difficulties through different categories. We addressed several research questions regarding children's difficulties when learning fraction.
First, we wanted to define more precisely the difficulties encountered by primary school children. Second, one of the goals of this study was to clarify the relationship between conceptual and procedural knowledge of fractions.
Does conceptual knowledge of fractions influence procedural knowledge? Or is procedural knowledge sufficient to understand fractions? Our hypothesis is that children's difficulties come from a lack of conceptual understanding of fractions. Their errors would come from the application of routine procedures, but they do not understand the various underlying concepts. Conceptual knowledge of fractions was assessed through tests about the different meanings of fractions part-whole, proportion, number , and the different representations of fractions e.
Procedural knowledge about fractions was evaluated through operations on fractions and simplification tasks. The test was administered to eight Grade 4 classes mean age: 9 years 11 months old , eight Grade 5 mean age: 11 years 1 month old classes and eight Grade 6 classes mean age: 12 years old from five different schools, representing a total sample of participants girls and boys.
The choice of these grades was deliberate, as fraction learning usually starts from Grade 4 in the French Community of Belgium where the study was conducted. Informed consent was obtained from parents and the director of every school, as well as from the 24 teachers involved in this research.
Assent from children was obtained at the onset of both testing sessions. We analyzed 21 mathematics textbooks recognized by the Education Department of the French Community of Belgium. Fraction concepts used in mathematics textbooks in Grade 4—6 were listed.
The goal was to analyse the progression of fraction learning proposed by those textbooks. The most striking observation was that there was a great variety of ways to introduce fractions. In most textbooks, the part-whole concept was considered as the starting point, but in some cases, the measure concept was introduced first. Every concept described in our theoretical framework was represented in the textbooks, but the number of exercises concerning each one of them varied greatly.
We also examined the official mathematics program of the French Community of Belgium. Fractions were divided into two different categories, Numbers and Quantities. Any requirement at the end of primary school Grade 6 is briefly reviewed in this section. In the Number category, pupils should be able count, enumerate and classify fractions as well as decimal numbers.
They should also be able to calculate, identify and solve operations involving fractions and decimal numbers. In the Quantities category, children are supposed to operate and fractionate different quantities in order to compare them. They should be able to add up and subtract two fractions as well as calculating percentages.
The program also mentioned their ability to solve proportionality problems. The official program offers a list of what pupils should know about fractions in primary school.
But what did not appear clearly was a logical progression between all the meanings of fractions. For example, how and when should equivalent fractions be introduced? There was not a clear development for teaching fraction. This situation may be risky as teachers might present fractions as a succession of different independent activities with no real underlying logical progression. In order to complete the information found in the textbooks, we analyzed pedagogical practices about the way teachers introduce and teach fractions.
This investigation revealed the great variety of ways to teach fractions. Our analysis was based on different sources. Firstly, we asked the 24 teachers involved in this study to give us a list of all the activities about fractions conducted in their classrooms. Secondly, teachers gave us a sample of their lessons on fractions as well as pupils notebooks.
Thirdly, we made informal observations during the tests. In Grade 4, pupils learn how to read and represent the value of a fraction. They start placing fractions on a graduated number line. They learn how to simplify fractions i. They learn how to add and subtract of fractions with small and common denominators.
In Grade 5, children learn more about fractions as numbers and how they represent quantities. Pupils are trained to convert fractions into decimal numbers and vice versa. They use addition and subtraction of fractions with different denominators.
Improper fractions are introduced. In Grade 6, multiplication of fractions is introduced. Our analysis highlighted the fact that teachers are more inclined to use procedures than what is recommended by the official program.
The different conceptual meanings are presented successively without any logical progression. The order in which they are introduced depends on the teacher and on the textbook used by the teacher.
Furthermore, fractions seem isolated from mathematics lessons and are taught like a separate topic. A test was designed to answer our research questions. Its construction has been guided by our theoretical framework as well as the primary school curriculum in the French Community of Belgium. The test was split into two parts. Part A was made of 19 questions, Part B of 20 questions.
There were 1 to 8 items for each question. There were 46 items in Part A and 48 in Part B. They will do whatever it takes to help the children they work with understand and possibly love math. August 1, July Brouillette. Mathnasium Team is the best. They know how to keep the kids engaged and interested in math. My daughter no longer struggles. In fact she is so confident in math now that she wants to be math instructor when she grows up. During this pandemic, the Team found a way to keep going.
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