Mastery of fractions is the foundation for many more advanced mathematical and logical reasoning skills, including proportional, probabilistic and algebraic thinking. The degree of early year fraction understanding often correlates with secondary school mathematical achievement (Siegler, Fazio, Bailey, & Zhou, 2013). At the same time, fractions present a wide range of teaching and learning challenges that have been the subject of educational research (Petit, Laird, Marsden, & Ebby, 2015).
In the first part of this post, issues surrounding the teaching and learning of common fractions are described and linked to teaching and learning strategies that can address these. In the second part, implications for the teaching and learning in diverse classrooms are investigated and addressed by the Universal Design for Learning (UDL) framework, with particular reference to opportunities that modern information and communication technology (ICT) can offer. Drawing on both parts, a logical sequence is developed detailing conceptual and procedural steps for teaching and learning the fraction equivalence concept.
Issues surrounding the teaching and learning of common fractions
In primary school, learners move from non-fractional, through early fractional and transitional strategies, to mastery in applying fractional knowledge to magnitude, unit fraction and benchmark reasoning, and in operations (OGAP, 2012). In the Australian Curriculum, teaching and learning of fractions starts in Year 1 with content descriptor ACMNA016 “recognise and describe one-half as one of two equal parts of a whole”, and it progresses to Year 6, where students are expected to have developed procedural fluency in all operations with fractions, decimals and percentages, with the capacity to solve authentic problems (ACARA, 2017).
Fractions, ratios and proportions are the most cognitively challenging concepts encountered in primary school mathematics (Booker, Bond, Sparrow, & Swan, 2015). For students, fractions often mark the transition from concrete to formal operational mathematical thinking (Siegler et al., 2013), where numbers do not anymore relate to whole objects, or the size, shape and arrangements of their parts, but instead to part-whole relationships between two quantities composed of equal parts of a whole (Pantziara & Philippou, 2012). One difficulty in expanding whole-number to rational-number thinking is that both share overlapping cerebral processing areas in the intraparietal sulcus of the prefrontal parietal cortex (Siegler et al., 2013). Additional difficulties are encountered with the notation system used to represent fractions (Brizuela, 2006). Explicit teaching of fraction notation is essential, since “one whole number written above another whole number, do not transparently communicate the meaning of fractions” (Gould, 2013. p.5). The relational action associated with the symbols is not an intrinsic property of the symbols. Learners first need to experience the symbols as an expression of the relational outcomes of actions that they have carried out or observed (Dörfler, 1991). Finally, there is the concept of changing units, where one whole can refer to both multiple objects or composite units within a single object; partition fractions or quantity fractions. Students need to learn to move between different representations, including multiple symbols referring to the same amount (Booker et al., 2015).
In teaching fractions, it is essential to explain and establish fraction terminology first, explicitly addressing language and conceptual misunderstandings that surround rational-number thinking. The links between terminology, symbology, notations and concepts such as whole-number and part-whole relationships must be established before moving on to mathematical operations involving fractions. Mastery requires that students develop both conceptual and procedural knowledge and understanding of fraction concepts (Pantziara & Philippou, 2012). Therefore, teachers need to value and at least initially prioritise deep conceptual understanding over automatic procedural skills (Booker et al., 2015).
Visual models are a central component in teaching fractions at all stages of conceptual development, rational-number thinking, procedural and operational problem solving (Petit et al., 2015). Provision of a variety of visual representations of identical fractions that differ in perceptual features, such as the location and shape of shaded areas (numerator), were demonstrated to be important in the development of a multi-dimensional understanding of fractions. However, it is important that teachers guide learners in developing the knowledge about how visual representations relate to the fraction concept (Rau, 2016).
There are three common visual fraction models: linear, area, and discrete. These can be taught using a variety of representations (e.g. rectangular and circular segments, arrays, object collections) and physical and virtual manipulatives. Recent research into cognitive numerical development highlights the importance of teaching students that fractions represent magnitudes that can be located on a number line. Number lines, where equal parts are defined by equal distance, can serve as a conceptual bridge between whole numbers, proper, improper and mixed fractions, decimals and percentages, and highlight the concepts of equivalence and continuous quantities of fractions (Booth & Newton, 2012; Siegler et al., 2013). Gould recommends focussing on the linear aspects of fraction models as the primary representation of fractions in younger years (2013). Nevertheless, traditional area models, where equal parts are defined by equal area, continue to play an important role in the conceptualisation of numerator and denominator, fraction division, the relationship between unit of measure and reference unit, and equivalence (Lamberg & Wiest, 2015; Booker et al., 2015). Discrete models or ‘set of objects’ arrays, where equal parts are defined by equal number of objects with countable sets and subsets of discrete entities, visualise the mapping of distinct countable sets onto numerators and denominators (Rapp, Bassok, DeWolf, & Holyoak, 2015) and help students to understand equipartitioning (Petit et al., 2015).
All three visual fraction models can be used in different learning modes, including group discussions (verbal, aural), kinesthetic activities, and even through music (Courey, Balogh, Siker, & Paik, 2012). Physical manipulatives are a valuable resource stimulating hands-on learning that can make abstract mathematical ideas more tangible (Petit et al., 2015). Access to a variety of representations and activities support students in building the foundations for solving complex questions and real problems that involve rational-number thinking which cannot be achieved by rote learning alone.
Learners need guidance and practice to expand their conceptual numerical understanding to include rational numbers (Petit et al., 2015). Procedural fluency and algorithmic operational problem-solving skills are best learned by moving back and forth between conceptual and procedural knowledge and practice. Individual students have different learning styles and learning preferences. Student diversity can be accommodated by empowering learners to make choices between different activities and task designs, including group, paired and individual work, different modalities and types of questions, resulting in increased motivation and persistence (Landrum & Landrum, 2016). A degree of choice of tasks, task sequence and stimulus can be introduced into the classroom through blended learning, where students engage part-time with online content and instructions using learning platforms such as Mathletics (see below). Blended learning also provides a degree of flexibility over time, place, path and pace, and can be implemented as station-rotation, flipped classroom, or flex model among others (Staker & Horn, 2012), depending on the opportunities and constraints of individual teaching and learning environments.
There is also a cultural dimension to how students learn mathematics in general and fractions in specific. Mathematics is a cultural construct with its own epistemology. It cannot simply be assumed to constitute a “universal language”. Indigenous Australian mathematician and head of the ‘Aboriginal & Torres Strait Islander Mathematics Alliance’ Chris Matthews developed a model for culturally-responsive mathematics that links students’ perceived reality with curriculum mathematics through a hermeneutic circle of abstraction and critical reflection based on practical problem-solving (Sarra, Matthews, Ewing, & Cooper, 2011).
It has long been argued that Indigenous Australian students prefer kinesthetic learning experiences with physical manipulatives, narrative learning, valuing group discussions and explicit guidance (Kitchenham, 2016). It is therefore important to link formal mathematical concepts to something concrete endowed with real meaning. In reference to the Maths as Storytelling (MAST) pedagogical approach (Queensland Studies Authority, 2011), the fraction concept could for example be learned by acting out, using groups of students to represent fractions in terms of varying parts of the class (e.g. boys vs girls), or perhaps more dynamically by connecting fractions with rhythm and dance (Campbell, 2014).
At the same time, it is important that students also learn that there are differences between everyday colloquial expressions and empirical understanding of fractions, such as in acts of sharing and distributing, and formal mathematical equivalents. Mathematical definitions are developed through theoretical or operative generalisation and abstraction and use symbols (verbal, iconic, geometric or algebraic) to describe the conditions or schemata of actions (Dörfler, 1991). Therefore, explicit teaching of the meaning behind the symbolic mathematical language through exposure to multiple representations and models is essential for student learning of mathematical concepts including rational-number concepts.
Providing a creative and active learning environment, offering choice and variation in learning activities, mathematical representations, and task and assessment modes, will foster student engagement and the development of a positive disposition to mathematics. Similar to the fraction understanding itself (Siegler et al., 2013), a positive mathematical self-belief is another key predictor of middle years students’ mathematics achievement (Dimarakis, Bobis, Way, & Anderson, 2014).
Implications for the teaching and learning in diverse classrooms
Australia is a multicultural country and home to the world’s oldest continuous cultures. Nearly half of the population have an overseas-born parent, 5% identify as Aboriginal and/or Torres Strait Islander, and 20% speak a language other than English at home (Australian Human Rights Commission, 2014; Australian Bureau of Statistics, 2016). This diversity translates to classrooms with diverse social, cultural, religious and linguistic approaches to learning (Shahaeian, 2014). The Australian-wide promotion of an inclusive education policy emphasises the right of students of all abilities to participate in all aspects of the mainstream education, adding an additional dimension of physical, sensory and intellectual diversity (Konza, 2008). According to the Australian Bureau of Statistics, 5% of all primary school-aged children have disabilities resulting in significant core-activity limitations and schooling restrictions (2012). At the other end of the ability spectrum are the 10% of gifted and talented students, often unidentified and significantly underachieving (Parliament of Victoria, Education and Training Committee, 2012).
It is therefore the legal, moral and professional obligation of teachers to embrace all learners in their diversity and make reasonable adjustments to facilitate their full participation towards achieving their best potential (Cologon, 2013; Poed, 2015). There are a number of models for responsive teaching that addresses all learning needs in diverse classrooms. The Universal Design for Learning (UDL) is a set of principles guiding teachers towards developing universally accessible learning environments and instructional practices (Flores, 2008). The fundamental idea is to make the curriculum delivery as accessible as possible to all students, limiting the need for additional modifications and individual support. The design focuses on providing equitable access to the curriculum by offering multiple means of representation, expression and action (Basham & Marino, 2013). Students are offered choice over curriculum content, learning activities and resources to best meet individual skill levels, learning preferences and interests. Assessments offer learners multiple ways of demonstrating acquired skills and knowledge. While UDL can cater for most students in the diverse classroom, preferential intervention and special provisions is given to small groups, including access to resources (e.g. teacher aide) materials (e.g. manipulatives) or equipments (e.g. calculator) for task completion, including additional time or accelerated curriculum, alternative input and response formats (Ashman, 2015). A third level of prevention and intervention offers short-term intensive and explicit instruction for individuals (Fuchs & Fuchs, 2001), for example explicit practice of mathematical terminology and symbols for new EAL/D students.
Utilisation of ICT, including augmented and alternative communication devices that can support students with physical impairments, has great potential to help addressing all individual learning needs in a diverse classroom (Blum & Parete, 2015). Modern teaching and learning devices such as the iPad have been designed with disabilities in mind and can be easily configured to support the visually, hearing and physically impaired (Apple Inc., 2016). The iPad provides quick and simple access to a wide range of mathematics apps. Preliminary research highlights the potential of using iPads in primary school Mathematics classrooms to motivate and engage students (Hilton, 2016). Mathematics teaching and learning software, such as Mathletics developed by the Australian company 3P Learning provides teachers with tools to custom-design learning sequences for any topic in alignment with the Australian Curriculum, even activities with year level and content descriptors, lesson plans and ebooks (3P Learning, 2016). Australian schools that use Mathletics are performing significantly better in NAPLAN numeracy tests irrespective of socio-economic and regional status (Stokes, 2015).
The reported positive outcomes for all students, including students with learning support needs as well as gifted and talented students, could be a result of the combination of design features in the software:
- student-led design that encourages learning at individual pace and at multiple difficulty levels (easier, core, harder)
- instant and encouraging feedback to learners highlighting mistakes and solutions without teacher intervention
- powerful formative assessment capabilities allowing teachers to monitor student progress and to identify learning gaps
- tools that allow teachers to develop individual student learning pathways
- app and web-based access allows Mathletics to be used as a flipped classroom tool and assign individual homework
- gamified character in modules including class, school and world challenges (LIVE Mathletics)
Apps can also provide virtual manipulatives that enable more creative work with objects. For fractions, the educational graphing calculator GeoGebra is discussed below for building fraction bar models (Cooper, 2014).
As powerful as some apps and technology can be, ICT should only complement the teaching and learning of mathematics side by side with explicit teaching and multi-modal activities that encourage verbal and written communication, group discussions and the use of physical manipulatives that encourage kinesthetic learning. Also, apps are not always designed in alignment with UDL and can include barriers for students with disabilities (Smith & Harvey, 2014). Particularly in intervention instruction, it is advised to make use of both virtual and physical manipulatives to teach fractions (Westenskow & Moyer-Packenham, 2016).
Teaching and learning steps for acquisition of the equivalence concept
Fraction equivalence is one of the most important mathematical ideas introduced in primary school and know to cause difficulties for many students (Pantziara & Philippou, 2012). The big idea behind teaching equivalent fractions is for students to understand that fractions of a given size can have an infinite number of different names and corresponding symbols, and to develop efficient procedures for finding equivalent fractions. Finding equivalent fractions enables students to compare, order and operate with fractions (Petit et al., 2015).
The curriculum is the starting point for the design of teaching and learning units by defining the learning objectives and expected outcomes for each year level. The Australian Curriculum (AC) follows a spiral-based approach that gradually builds mastery of skills and concepts by sequentially increasing the cognitive demands (Lupton, 2013). Equivalence is introduced in the AC v8.3 in Year 4, where students are expected to “recognise common equivalent fractions in familiar contexts and make connections between fraction and decimal notations up to two decimal places”. In Year 5, equivalence of fractions is not specifically addressed but students are expected to develop the capacity to “... order decimals and unit fractions and locate them on number lines. They add and subtract fractions with the same denominator”. The equivalence concept is expanded in Year 6, where students are expected to “connect fractions, decimals and percentages as different representations of the same number”, more specifically detailed in content descriptor ACMNA131 “Make connections between equivalent fractions, decimals and percentages”. Full mastery of equivalence of fractions is not expected until Year 8 (ACARA, 2017).
In the learning continuum encountered in diverse classrooms, it is critical to develop an understanding of the sequence of teaching and learning steps of mathematical concepts and establish prior understanding of conceptual knowledge and procedural skills in all students.
- Step One starts with diagnostic assessment to establish existing foundational knowledge of common fractions, notation conventions, the relation between fractions to whole numbers, including proper/improper fractions and mixed numbers. Explicit teaching and practice of terminology and revisiting previously learned concepts might be required to establish critical conceptual understanding without which any further learning would be only procedural and rely on rote learning.
- Step Two explores new concepts and terminology by making use of physical manipulatives and encouraging student discussion. One example would be having students folding paper rectangles that have been vertically subdivided into equal, partially-shaded parts lengthwise in two, three, four bars of equal thickness The shaded fraction remains the same while the total number of equal parts as outlined by the creases increases. Students count shaded and unshaded parts and discuss equivalence (Booker et al., 2015, p.184).
- Step Three elaborates and reinforces equivalence fractions through multiple representations working from the visual-concrete towards the symbolic-abstract. The activities help to develop procedural fluency, the accurate, efficient and flexible use of mathematical skills in renaming equivalent fractions (Petit et al., 2015). Fraction games, ideally focusing on equivalent fraction grouping, are employed using material (Booker et al., 2015) or online virtual resources (e.g. Math Playground Triplets). A “fractional clothesline” can be used to establish the magnitude of fractions, sort and locate equivalent fractions, improper fractions and mixed numbers (Heitschmidt, n.d.). This activity involves kinesthetic and visual learning, and can encourage verbal learning through student discussions. It also serves as a formative assessment tool. Number lines illustrate the big idea that equivalent fractions share the same value (Petit et al., 2015) and are highly recommended as a representation that can conceptually bridge whole-number and rational-number thinking (Booth & Newton, 2012; Gould, 2013).
- Step Four integrates the acquired procedural knowledge and conceptual knowledge by looking for patterns and developing rules, progressing from concrete presentations towards symbolic presentations and abstract algorithms. The focus is on finding the next, rather than any equivalent fraction, making use of “fraction bars” as graphical representations. Fraction bars can be build using Lego blocks and extended by educational dynamic mathematics software (Cooper, 2014). Alternatively, an innovative lesson sequence works with stacks of papers of different thickness (Brousseau, Brousseau, & Warfield, 2014).
- Step Five extends the learned knowledge and understanding of equivalent fractions to real-world scenarios. This includes investigating the relationships between alternative representations of fractions (e.g. decimals, percentages) in wide variety of cross-curriculum contexts (e.g. Science, Economics and Business, Music). At this stage, a summative assessment of learning is important to evaluate the achieved mastery of the concept.
Quality teaching is based on proficient subject-matter and pedagogical knowledge. Teachers need to understand the full spectrum of individual challenges and potential barriers that students can face with cognitively challenging mathematical concepts such as rational-number thinking. It is important to invest the time to allow students to gain deep conceptual understanding before moving on towards procedural fluency. This will require well-sequenced teaching and learning steps, supported by multiple representations, modes and questions, working from physical and visual towards more symbolic and abstract problem-solving activities. Both hands-on manipulatives and appropriate use of ICT can support the learning process, especially at both ends of the ability spectrum. Offering variety and choice will help to engage all learners and establish students’ confidence and positive dispositions towards mathematics.