Although useful to represent children’s cognitive development within a mathematical domain as a series of stages (e.g., see number frameworks in the Numeracy Project, Ministry of Education, 2003b), such a hierarchical description offers incomplete information about the processes through which children acquire increasingly sophisticated strategies. While crucial, developing meaningful understanding of fractions involves more than selecting appropriate tasks that relate to students’ prior knowledge (Empson, 1999; Lamon, 1999; 2001). To explain cognitive change, we need to look further than developmental changes to include consideration of the complexity of the classroom processes and their impact on learning (Cobb, Boufi, McClain, & Whitenack, 1997) – we need to consider the classroom not just as a ‘setting for learning’, but focus on the ‘way’ learning takes place.

Several studies have looked at how better to teach fractions, employing instructional experiences that emphasise realistic and meaningful problem solving situations that allow children to show and discuss their informal knowledge (e.g., Bezerra, Magina, & Spinillo, 2002; Bulgar, Schorr and Maher, 2002; Lamon, 2001; Mack, 1990; Streefland, 1993; Steencken & Maher, 2002). Common to these instructional experience are approaches to fractions based on (a) realistic and meaningful problem solving situations; (b) student’s own fragmentary and informal knowledge; (c) interactive learning environment supporting discussion and collaborative work; and (d) students thinking about their own thought processes when solving problems and communicating verbally their strategies and ideas.

The move to a more student-centred classroom requires teachers to listen to the explanations of their students, probe them for justifications, and encourage them to share their solutions with their peers as they work together to refine, revise and extend their solutions. Bulgar et al. (2002), for example, conducted a year-long teaching experiment based on activities involving the dividing of ribbons. By using responsive questioning to elicit explanations, the teacher was able to assist in students’ development of appropriate justifications and to redirect them when they were engaged in faulty reasoning.

Similarly, Empson’s (1999) exploration into the relation between classroom talk and young children’s evolving fraction understandings suggested that children’s development of ideas about fractions is influenced not only by how their own knowledge is structured but, more importantly, by how the context for thinking about and discussing fractions is structured. Her study of Year 1 students’ engagement with a range of problems involving ‘fair shares’ was more than an investigation into the role of manipulatives and teaching activities: it was an exploration into the ways the teacher created the conditions conducive to learning. In order to engage in instruction which supports mathematical sense-making, the teacher needed to attend to the interactive processes of meaning construction by creating an intellectual community. Using a theoretical framework based on the situative perspective Empson’s study showed how the negotiation of meaning within the classroom played a key role in the process of incorporating and extending students’ informal knowledge. Through classroom talk and negotiation and thought provoking teacher questions, the strategies children employed extended well beyond the partitioning strategies normally exhibited by young children. Empson suggests that with discussion of strategies it is helpful to organise discussion around a particular aspect, such as fraction as quantities, fraction as ratios, equivalence, or the geometric character of fraction.

The way teachers talk with students about the fractional amounts they have created also affects the development of children’s fraction conceptions (Empson, 2002). The establishment of correct fraction terminology needs to be explicitly taught: Results from this study indicated that children have a variety of terms such as ‘fourths’, ‘quarter’ and ‘1 over 4’ for describing fractions, with many of the Year 4 students in preferring to give the answers to “How much…?” as a whole number of pieces. Empson, suggests that teacher questions should focus on “What is that piece called?” in order to frame fractional quantities as something distinct from whole-number quantities, noting that we would not ask a child in reference to a group of five cakes, “What is that called?” Other lines of questioning that draw attention to the quantitative aspects of fraction include asking “How big is that piece?” or “How much pizza does one person get? – both questions emphasising the fractional problem solving as amounts of “stuff”. Another good question for helping children to articulate and conceptualise the relation between a piece and its whole is “How many of that piece would fit into the whole?” Tzur (2002) reports the use of this iteration approach in a classroom teaching experiment involving third graders.

Other studies specifically involving fraction learning (e.g., Ball, 1993; Empson, 2002; Olive, 2002; Warrington & Kamii, 1998) also stress the importance of the teacher creating an appropriate learning climate. The more students are encouraged to contribute the intact products of their own thinking to class discussion the more likely they are to identify themselves as learners of mathematics with understanding. Stipek et al.’s (1998) study, involving 24 teachers, found that teaching practices within an instructional unit on fractions were not significantly associated with student gains on procedural items but were significantly associated with gains on conceptually oriented items. In classrooms in which the teacher emphasised effort and learning, de-emphasised performance, and encouraged autonomy, students made substantially greater gains on the items of the fractions assessment that required conceptual understanding than students in classrooms in which teachers did not focus on the task and learning in these ways. They surmised that the focus on effort and learning promoted the use of active learning strategies such as reviewing material not understood, asking questions, setting goals, monitoring comprehension, which in turn enhanced conceptual learning.

Stipek et al. (1998) also noted effects related to the emotional climate of the classroom and differing formative assessment practices. It was found that the more teachers indicated the number of errors or correct answers on student work, the less students claimed they experience positive emotions while working on fractions. In contrast, providing substantive, constructive feedback on students’ work was associated with positive affective experiences as well as with a mastery orientation to learning fractions. An affective climate that promoted risk-taking was also positively associated with students’ mastery orientation, help-seeking, and positive emotions:

Students in classrooms in which teachers emphasised effort, learning, and understanding rather than performance and in which autonomy was encouraged reported experiencing relatively more positive emotions while doing fractions work and enjoying mathematics relatively more than other students. (p. 483)

There were occasions where students in the NEMP sample volunteered that they did not like fractions, or expressed uncertainty with procedural explanations stating that they did not or could not understand fractions. In order that students do not perceive fractions as the ‘beginning of the end’ of understanding in their mathematics experience the importance of understanding and sense making cannot be underestimated. Quick fixes such as ‘turn it upside down’, while expedient at the time, have serious long-term consequences on students’ beliefs about their role as a mathematics learner.

An algorithm is a “precisely-defined sequence of rules telling how to produce specified output information from given input information in a finite number of steps (Knuth, 1974, p. 323). For fractions, traditional algorithms have proved notoriously difficult for many students, (Knight, 1982), and teachers (Ma, 1999) to comprehend and remember correctly. From our small sample of students who offered an algorithmic explanation to Question 5, we found several examples of students confusing rote-learnt procedures for when attempting addition of fractions (1/2 + 1/8).

While it is agreed that algorithms are important in school mathematics – they can help students understand better the fundamental operations of arithmetic and important concepts such as place value, and pave the way for learning more advanced topics – it is also agreed by educators and researchers alike, that the premature introduction of algorithms, especially those associated with division, has a detrimental effect on children’s performance (Behrend, 2001; Kamii & Dominick, 1998). Rule-based instruction, often detached from children’s informal knowledge, fosters a lack of connectedness both between concepts and procedures and between fractions and students’ everyday lives. When rules and procedures are devoid of personal meaning students are likely to forget them or not always realise when to use them (Hart, 1988; Hiebert & Wearne, 1986; Lamon, 2001). As a result, students lack ‘operation sense’ when manipulating fractions – producing answers that don’t make sense (e.g., answers such as 31/12 for Question 5).

In contrast, studies suggest that students can invent their own strategies for investigating fractional properties (Empson, 2001) and operations with fractions (e.g., Huinker, 1998; Warrington & Kamii, 1998). Huinker claims that allowing students to invent their own algorithms assists students to develop an interest in solving and posing work problems with fractions, with students becoming more proficient in translation among real-work, concrete, pictorial, oral language, and symbolic representations. Moreover, student methods provide a fruitful source for teachers to understand children’s developing fractional ideas and can form the basis for generalising to more traditional algorithms.

However, harnessing children’s invented strategies into developing productive understandings requires careful teacher guidance. Empson (2001), in a study on children’s reasoning about equivalence, claims that students need to do more than invent and talk about their strategies for solving problems if transformation of ideas into generalised, symbolic procures is desired. She suggests that the teacher needs to “maintain a balance between acknowledging and valuing the meaning and variety of student-generated strategies [in equal sharing], and creating a common focus on constructing strategies that work for as many different number combinations and problem contexts as possible” (p. 424). Specifically, for any invented strategy, there are at least two kinds of mathematical questions a teacher can pose (Campbell, Rowan, & Surarez, 1998): Firstly, does the strategy make sense mathematically? Secondly, can the strategy be generalised, and if so to what cases? For example, some strategies may work across all number combinations, and others such as repeated halving may work for only particular number combinations. Determining what strategies generalise, and why, is a critical aspect of classroom activity that supports the development of students’ proficiency in mathematics and ultimately supports the development of students’ identities as mathematically capable (Carpenter, Franke, & Levi, in press).

Extended opportunities, and time, for student exploration is also a crucial factor. Lamon (2001) warns that fraction understanding development is particularly slow at the beginning citing evidence from a classroom study involving delayed formal introduction of algorithms: during the first two - two and a half years, students could not compete in fraction computation with children who had been using the traditional algorithms. However, perseverance with children’s informal strategies resulted in deeper understanding long-term.

Division involving fractions is the most complicated and least understood fraction algorithm. Understanding fractions in this context requires the ability to partition wholes (or units of different kinds); to reconfigure wholes from parts, which is a psychological basis for the concept of inverse; and to subdivide a part and relate the subdivisions to the part as well as to the original whole, which is a geometric basis for composite operations (Kieren, 1993). As was the case in the NEMP findings, children initially use halving processes for continuous material, based on their informal notions of the symmetry of the material, or informal measurement processes involving the estimation of a unit and its reproduction, followed by check and adjustment (Hunting, 1994). Further development of children’s understanding of division of fractions is helped by appreciating different meanings such as measurement division, sharing (partitive division), finding a whole given a part, and missing factors problems (Flores, 2002).

Although the sharing context is extremely useful when sharing among ‘whole’ numbers of people, problems arise for problems involving a fractional divisor – how do we visualise sharing a piece of cake with 1/2 a person? Using the problem: “If 3/5 of a group get 11/2 pizzas, then how much would the whole group get?” Siebert (2002) claims that we can perceive a sharing situation parallel to the way we conceptualised whole-number division from a sharing perspective. Just like sharing for whole numbers, the dividend (11/2) and the divisor (3/5) correspond to the total amount and the number of groups, respectively. The answer to 11/2 ÷ 3/5 tells us how much a single group gets. The action involved sharing the 11/2 between the three 1/5’s so that each 1/5 get 1/3 of 11/2, or 1/2 – therefore the whole group would get 5 x 1/2. Similarly, one can also interpret division of fractions as a problem of finding a whole given a part. “13/4 ÷ 1/2 means that 1/2 of a number is 13/4, the answer will be 31/2 which is exactly the same as 13/4 x 2 (Ma, 1999). Understanding these explanations is not an easy matter and this point will be further elaborated with regard to teacher knowledge in Section 4.4.

Several other interesting examples of children’s invented strategies for division, based on their own or shared explorations, highlight the propensity of children to offer solutions that ‘make sense’ within their emerging understandings of fractions. Flores (2002) reports a teaching episode that built on a shared idea that 1 ÷ 1/n = n that clearly illustrates the productiveness of children’s explorations: Katie had concluded that 1 ÷ 2/3 = 1, (justified by “one two-thirds piece fits one time into 1”). Rather than developing an exposition that would clarify the student’s error, the teacher asked her students to study the situations 2 ÷ 2/3, 3 ÷ 2/3, 4 ÷ 2/3, etc. Katie used this experience to first develop a physical theory – in 3 ÷ 2/3, there are other ‘bits’ that could make up 2/3 pieces – and then used this to develop a successful algorithm as evidenced in her later reasoning: 61/3 ÷ 2/3 means there are 18 one-thirds in 6, so 61/3 contains 19 one-thirds and half that many two-thirds.

Further examples are presented below to illustrate children’s inventiveness and how a student’s solution method may be influenced by the nature of the problem in hand:

• An interesting approach that builds on students’ experience with whole numbers is to apply the distributive law, that is, partitioning the dividend. For example, to calculate 13/4 ÷ 2 write (1 + 3/4) ÷ 2 = (1 ÷ 2) + (3/4 ÷ 2) [this is the approach that was used by most NEMP sample students in Question 5].
• Student solution to the problem 1/2 ÷ 2 using inverse proportionality: “There is only half a two in one, so there is a quarter in half of one” (Pirie, 1988, cited in Flores, 2002). However, such reasoning, while useful for this problem is less productive for solving the pizza problem 5/4 ÷ 2.
• Rowland (1997) reports an unusual method involving division of a whole number by fractions whose numerator is one less than its denominator using a part-complement representation. For example, 100 ÷ 3/4 = 100 + (100 ÷ 3). This process has two steps, firstly we can see that 3/4 goes into 100 units, 100 times, leaving 100 quarter pieces to be divided into lots of 3/4’s, that is (100 ÷ 3).
• A student’s explanation of “How many portions of 3/5 of a cake can you get from 4 cakes? is presented by Sarage (1992) as follows: You’ve got 3/5 of a cake equal to a portion. But you can see that each cake is one portion plus another 2/3 of a portion. That is, each cake is 5/3 of a portion. So when you want to find out how many portions there are in 4 cakes, you can divide by the size of each portion (4 ÷ 3/5) or you can multiply by the number of portions per cake (4 x 5/3).
• Another student in Sarage’s (1992) study explained her reasoning to the problem above as: “She started with 4 cakes. Then she cut each cake into 5 pieces. So she has 4 x 5 = 20 pieces. Then she grouped those pieces by threes since 3 pieces make up a portion. So she got 20 ÷ 3 = 6 ‘2/3 portions’. She multiplied by the denominators and divided by the numerator.”

As these examples illustrate, students’ thinking can be both varied and original and not always easy to understand unless accompanied by an explanation. Just as in the NEMP tasks, students should be encouraged to make use of concrete representations, or empirical evidence and patterns, and properties of numbers and operation to explain their thinking of particular solution strategies.

4.4

The importance of teacher knowledge, both content and pedagogical content, is widely recognised in mathematics education (Ma, 1999). Understanding of fractions, in particular, is an area that requires both multiple connections with a broad range of related but only partially overlapping ideas such as division, multiplicative thinking, and proportion, and a broad understanding of the range of constructs and possible representations. Given the conceptual complexity associated with rational numbers it is not surprising that feedback from New Zealand Numeracy Programmes (e.g., Higgins, 2001, 2002; Irwin & Niederer, 2002) confirms teachers’ lack of confidence in teaching fractions.

Inadequacies of teachers’ knowledge about fractions is of concern given that reform documents urge that students experience multiple ways of representing problems and strategies for solving them. The selection of representational contexts involves conjectures about teaching and learning founded on teachers’ evolving insights about their children’s thinking and their understanding of the mathematics – one must inform the other in the construction and use of presentational contexts. Moreover, in order to maximise the benefits of tasks teachers need to be aware of the advantages and disadvantages and the contexts in which each approach tends to be more helpful. Whereas with whole numbers dual interpretations are sometimes possible, with fractions, depending on the problem, usually one interpretation is more helpful than the other. For example, the following problem suggests a measurement division interpretation: “You have 13/4 cups of flour, and for each batch of cupcakes you need 1/2 a cup. How many batches can you make? Partitive division works particularly well when a fraction is divided by whole numbers (as in Question 5 & 6 in the NEMP study). For example “1/2 of a cake is to be divided among three children, how much will each get.” To solve this, children will divide 1/2 into three equal parts, and express the parts as a fraction of the whole. However, there are some problems that will be interpreted by students from differing perspectives. For example consider the problem: “John ate 1/8 of 16 hot dogs. How many hot dogs did John eat?” Students usually solve this problem by partitive division – sharing out the 16 hot dogs into 8 groups each receiving 2 each. However, occasionally a student may use a ratio conception – that is, John ate 1 in every 8 hot dogs – in this case, they see the denominator as the number in a group rather than the number of groups. Thus, John eats one hotdog from the first group and one from the second group.

As well as a sound content knowledge base, Schifter (2001) argues that teachers need additional mathematics skill in order to be able to:

• attend to the mathematics in students’ explanations;
• assess the mathematical validity of students’ ideas;
• listen for the sense in students’ mathematical thinking, even when something is amiss; and
• identify the conceptual issues that students are working on.

Teachers with such knowledge are not only able to ask children to explain their strategies, they are also able to pose questions that require students to think more deeply about the mathematics underlying both the strategies they were using and the problems they were solving.

The analysis of Pizza problem responses, especially those that required explanations, highlight that children’s thinking is a function of both their informal and instructed knowledge. Using these open-tasks is seen to be a useful way to access children’s thinking. However, in order for teachers to effectively use this thinking to inform the instructional decision-making processes, teachers’ knowledge of both mathematics and of how students learn must be secure: “teachers must have an understanding of how their actions or requirements affect what is going on in the minds of their students” (Nuthall, 2002, p. 45). Simultaneously, more needs to be known about how teachers’ knowledge of rational number influences their instructional decisions in the classroom, especially in relation to the interpretation and presentation of representations.

How to achieve these aims, both in pre-service and in-service professional development, remains an ongoing area of research (e.g., Heaton, 2000; Lubinski, Fox, Thomason, 1998). Within New Zealand, attention of teachers’ learning within professional development programmes associated with numeracy remains a priority. Specifically, research on curriculum development needs to focus on ways to guide teachers’ pedagogical and mathematical decisions rather than make decontextuatlised decisions for them: “teachers need opportunities to learn from new curricula that are similar to the learning that reformers intend for students” (Heaton, 2000, p. 157).

The disjuncture between students’ solution strategies using informal and formal knowledge suggests that we need to develop a greater understanding as to how students can develop understanding of fractions by building on their informal conceptions of partitioning. And while not addressed in the questions in the Pizza problem, we need to more clearly understand how students develop a broader understanding of fractions that integrate the various subconstructs of quotient, measure, ratio number and multiplicative operator.

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