Challenging Traditional Pathways

Abstract

This article is based on one of the main findings from an action research project which investigated how primary school teachers could improve the teaching of fractions. The study focused on what pedagogical strategies might be employed when shifting teaching and learning from procedural application to conceptual understanding. The purpose of the research was to unpack what key elements of lesson design could be utilised in order to raise student achievement and understanding of fractional concepts. Using models, constructs and representations beside the typical area model was deemed a key finding. When students are exposed to a range of models, constructs and representations, they make connections and apply their knowledge in increasingly flexible and generalised ways. The research also sought to develop a strength in teacher pedagogical content knowledge and confidence.

Key words

Linear models, area models, fractions, action research, representation

Being brave: No more area models!

The very definition behind the notion of ‘fractions’ is to fracture – but these rational numbers do not have to break your teaching spirit anymore. We are all guilty of the sorrowful, eye-rolling moan when fractional teaching comes around. We dig at the back of our resource filled cupboards and pull out our trusty fractions circle pieces. Maybe you are lucky enough to have magnetic ones, or even the pretty bars that can be made into eye-catching, colourful fraction walls. We spend hours checking each piece is accounted for in order to make the range of wholes (which of course, are not!); realistically, how many times have you found a twelfth of something. You may even let the students play with, explore, and get a feel for the equipment. Then what? Will you move past equal sharing of equipment? Mathematics is more than a set of procedures, calculations and operations to be completed and appraised; it is aesthetic, presenting many opportunities for creativity, exploration and reasoning (Boaler, 2016). I hope to encourage you to use a variety of drawn models, constructs and representations of fractions – even student made! – to develop a deep and conceptual understanding of this anxiety-ridden, yet crucial, strand.

Part-whole constructs and area models have dominated classroom practices. These are seen to build on students’ understanding of equal sharing and are likely a student’s first experiences with rational numbers (Clarke et al., 2011; Neagoy, 2017). Although an effective starting point, more emphasis on other fractional concepts, representations and models are necessary for in depth understanding. Traditionally, techniques such as folding of paper, shading of different shapes and sharing of counters, for questions such as ‘what is one quarter of 24?’, are used to model ideas and demonstrate understanding. I found that this worked well for juniors or for unit and non-unit fractions under one whole, and I have been guilty of focusing on numbers under 50. These tasks are engaging and hands-on and are a fantastic starting point. Using a variety of models, like number lines, to develop one concept will encourage your students to ‘look for patterns and relationships, make and explore conjectures, and use what they learn from their visual models to generalise concepts’ (Petit et al., 2016, p.4).

Practical issues that are ‘both problematic yet capable of being changed’ are identified (Elliott, 1978, as cited in Cohen, et al., 2007, p.298).

A small-scale, action research study was used to investigate the factors teachers might consider during the teaching and learning of fractions to deepen their students’ understanding and accelerate learning. The research team (two participating teachers and the researcher) wanted a practical, classroom-based approach to find solutions to the identified problematic areas that directly affect student learning. It is important to note that the difference between normal practice and action research is that those who are involved in action research are more careful, systematic and rigorous in their thinking (Kemmis & McTaggart, 1992). As this was a collaborative venture, compromises were made to ensure that the research was relevant and benefited all persons involved. Whitney and Kylie (teacher pseudonyms) acknowledged that their previous learning design of fractional concepts followed no clear structure and they were unsure about the required sequence or scope of learning that would lead to deep understanding of the content.

The research team discovered that when students were exposed to a range of models, constructs and representations, they made connections and applied their knowledge in increasingly flexible and generalised ways. Teacher pedagogical content knowledge was found to be crucial in developing the learning pathways; each teacher exhibited different strengths and weaknesses and, naturally, they often were unaware of what they ‘did not know’. Hence, the teachers in this study decided to implement what Land and Drake (2014) suggested in their research, and that was to collaborate and devise a progression of learning, which is concept based. The research team thus developed an understanding around how models, constructs and representations can support students and their differing abilities.

Research of this type ‘is considered democratic, equitable, liberating, and life-enhancing qualitative inquiry’ (MacDonald, 2012, p.34). The research team focused on transitioning fractional representations using area models (namely circle), to linear models in the form of bars and number lines. Fractions are usually taught through area models, where they are represented as a part of a region and are commonly expressed as rectangles and circles (Petit et al., 2016). The use of different constructs and representations was cognitively challenging for both the teachers and the students, but well worth the effort.

Improper fractions and area models are a confusing mix

In real-life situations not everything is divided into nice, even packages – if only! There are often odd numbers, measures or remainders that need to be shared, which reiterates the need to fracture a unit for accurate measures (Ministry of Education [MoE], 2008). We also cannot carry around a set of fraction circle pieces, ready for equal-sharing opportunities to arise. One key idea when applying fractional understanding is that there can be no remainders or unshared parts; this can then lead to the use of mixed numbers and improper fractions. Unfortunately, the area model, which is used the most regularly in instruction, limits what the student knows and does with fractions as it cannot be applied successfully to a range of fractional contexts and problems.

During the research, students such as S6 would attempt to apply their understanding of area models when working with improper fractions problems. The figures below outline S6’s efforts to solve problems like ‘three tigers ate half a slab of meat each, how much [meat] did they eat altogether’ (MoE, 2020) (see Figures 1 to 3).

_Caption: Figure 1

S6 independent attempt to solve problems using the area model_

Initially, S6 could not understand why her model did not work. The teacher intervened and asked probing questions. For example, ‘How many quarters are in a whole? Can you have five quarters in a whole?’ There was an emphasis around having five quarter-sized pieces, not five pieces of one whole.

_Caption: Figure 2

S6 Fraction representations after the intervention_

The researcher helped S6 connect the ideas used in the area models to other fraction and linear models (see Figure 3), while Kylie observed. The reasoning behind this was that students should be exposed to and use a variety of visual models, so they do not come to rely on or align their understanding to just one source. It is about presenting the same or similar information in different ways; a student may struggle with a concept using fraction circles but may see the pattern and connection through fraction tiles or a number line. Additionally, using a variety of models offers students multiple opportunities to think and rethink about their understanding as they focus on the similarities and differences of the perceptual features (Petit et al., 2016). This is also related to the use of manipulatives. Ball (1992) reminds us that effective learning arises through the ways we – teachers and students together – interact with, talk about and convey meaning through any tool we choose to use, even self-created models. We convey and carry the intended meaning.

_Caption: Figure 3

Making connections between fraction models_

Number lines were used as our main bridging tool to reinforce the understanding that fractions are quantities that can be counted and are numbers that have an ordinal placement. This aligned to previous research of Mills (2018) and Petit et al., (2016). Number lines are deemed useful for fractions greater than one, as it prompts students to think ‘proportionally about a number line, [and] not just sequentially’ (Petit et al., 2016, p.99). Initially, the teachers in this study worked with number lines 0 to 1 and emphasised the proportional distance between fraction units. A common error made by the students was that they would make four marks to represent quarters (for example) instead of showing four equal divisions/partitions. This was addressed and developed during warm up activities, such as skip counting in various fractions. When the students began feeling confident, the teachers built their fractional understanding by using number lines greater than 1 (see Figure 4). This helped the students see that whole numbers can also be written as fractions; a strategy outlined by Fazio & Siegler (2011) which emphasises that fractions are numbers with magnitudes. Kylie and Whitney found that drawing a number line after or below other models, such as an area or bar, made the number line more easily understandable and relatable for the students (see Figure 4).

Caption: Figure 4

Caption: A student’s representation of counting in fractions on a number line

_Caption: Figure 5

S4 connecting area and linear models_

Halfness, non-unit panic and number bias

At the beginning of the study, the teachers had a sneaking suspicion that the students had a limited understanding of fractions, which was confirmed through the results of the pre-unit test and the pre-research, semi-structured interviews. A few examples included that most students could not identify or order unit fractions and no one could label the parts of the fractional number or articulate the related roles. The frustrating part for the teachers was that the students had been marked as competent on many fraction and division related school progressions. This could indicate they had previously only experienced procedural- or instrumental-based learning design and snapshot assessment – demonstrating the skill one time, with potentially leading questions like ‘using equal sharing find 3–4 of 20’. Skemp’s (2006) research cited that instrumental thinking is applying ‘rules without reason’ (p.89), while understanding of the mathematical concept is demonstrated through knowing and using rules, such as formula. However, proficiency and mastery in mathematics is dependent on knowing what to do and why. It quickly became clear that the students had not been presented opportunities to develop their understanding of concepts in sophisticated ways or create what Land and Drake (2014) referred to as an ‘integrated knowledge framework’ (p.111).

As learning progressed beyond using unit fractions, namely halves and quarters, ‘halfness’ and ‘non-unit panic’ (research team terms) set in. ‘Halfness’ referred to the students’ automatically defaulting back to finding halves when other unit fractions were introduced, such as one third. Mills’ (2018) research noted similar findings that students would find half ‘regardless of how many whole objects there were to share and how many people there were to share the objects among’ (p.164). The research team tried to combat this by drawing the students’ attention back to the denominator and its role. There was emphasis on the denominator being understood as the name and number of parts in the whole and the numerator as the number of pieces of that size (Clarke et al., 2011; Mills, 2018). This generalisable rule was particularly useful during improper fractions instruction. Many of the students believed at the beginning of the study that a fraction like 10–10 was the highest you could go. However, by the end of the research they were able to count and identify fractions beyond one whole (see Figure 4). This indicates that the students’ thinking had progressed further than what Fazio and Siegler (2011) referred to as the part-whole approach.

‘Non-unit panic’ (a term coined by the teachers), occurred when the students were asked to find non-unit fractions of a set, for example, 3–4 of 20. The teachers and the students expressed difficulty with word and multi-step problems. Kylie indicated during a research meeting that she had observed that the students exhibited limited number sense and limited problem-solving skills. She noted that her students struggled to understand that they first needed to find one quarter and then iterate the portion the required number of times (in this case three). Additionally, the students automatically defaulted to finding halves and quarters of sets when asked to find other unit fractions (like 1–3) and could not explain why they did so. The students were unable to break down the steps required or often forgot previous connected learning. Kylie explained:

My students can find a quarter easy as, and then we went to 3–4s and they forgot how to find a quarter . . . and then started giving random numbers.

The students continued to show whole number bias throughout the study, particularly viewing the numerator and denominator as two separate whole numbers or confusing the relationship between them, which was also a common problem recognised by researchers such as Aksoy and Yazlik (2017) and Gabriel et al. (2015). Whitney dedicated instructional time to correct a related misconception. Whitney’s students represented the denominator as a whole number rather than a proportional amount, for example, sharing counters into groups of three instead of three groups (thirds).

S2 required help from the researcher during a warmup task. The question had asked the students to compare 7–4 to 5–2. However, S2 had confused the role of the numerator in both numbers and assumed that the numerator in the fraction indicated the number of whole circles that the denominator value would be shared into. For example, for 7–4 S2 drew seven whole circles and divided each into quarters. The student then counted the number of quarters she had made (see Figure 6). The researcher suggested to S2 that she needed to re-look at the fraction number in relation to her drawing. She said to S2, ‘You drew seven wholes cut into quarters, not seven quarter pieces.’

Caption: Figure 6

Caption: S2’s attempt at comparing two improper fractions

An interesting observation was that students were independently able to order unit fractions when given a jumbled set of numbers but reverted to whole number reasoning when asked to create their own number lines, an occurrence that was also observed in Petit et al., (2016) research. For example, placing 1–2 before 1–3 because two is smaller than three. Furthermore, Whitney noted that a lot of instruction time was dedicated to students making the correct number of groups while equal sharing. The students were initially making groups of three when finding one third. Again, interestingly, all the students in this study had previously (prior to this year) been marked as competent (on their school progression record) for equal sharing when carrying out division problems and for finding halves and quarters of sets in fraction problems.

So, moving forward what can educators do?

Bridging the gap between the known and unknown requires teachers to carefully and perceptively decide which images, models and materials will be appropriate to explicitly and concretely represent the mathematical concept (Chinn, 2017; Moyer, 2001). The teachers within this study concluded that understanding mathematical concepts involved in teaching fractions is much more complex than merely implementing a known set of procedures. Boaler (2016) asserted that as teachers explored, reasoned, and justified their own actions and decisions, they became more expert mathematicians. Through working collaboratively, Whitney and Kylie became more confident with what they had to teach, in what order they would teach it, and recognised the depth of understanding the students required to be successful in mathematics.

Below are some points and ideas to consider for learning design and enactment.

Area models have their place.
Area models introduce young students to the concept of equivalence, as well as essential features of visual models – representation of a ‘whole unit’, division of the whole into equal parts (equipartitioning) and the relationship between the parts and the whole (Castro-Rodriguez et al., 2016). Furthermore, area models provide opportunities for equal sharing of materials.
The teachers in this study challenged themselves to learn to use a range of constructs, such as linear and set models, while trying to resist the temptation to rely on the area models and traditional patterns of learning design they had depended on in the past. Recognising the need to upskill their teaching practice will allow teachers to continue to learn how to make connections between the different facets of fractions and to develop more dialogue in their classroom, alongside the need to press for justification and explanation of solution methods.
Students should create their own models.
When planning and enacting instruction it is important to remember that any ‘model is a bridge to the mathematics – it itself, is not the mathematics.’ (Neagoy, 2017, p.53). It should also be noted that many researchers advocate for students to create, share and analyse their own meaningful models (see Figure 7).

_Caption: Figure 7

Example of a student created model_

Connect fractional understanding to real-life.
Aksoy and Yazlik (2017) noted that when fractions are not related to daily life, the teachings and knowledge is easily forgotten in a short stretch of time. When posing or using pre-made word problems, educators should consider where they are teaching and who is sitting in front of them in what Anthony and Walshaw (2007) described as grounding the learning into the life of the learner. For example, using problems that involve sharing cows between paddocks is irrelevant and potentially quite boring for students who have lived in the inner-city their entire lives. Changing vocabulary costs us nothing but a short amount of time.
Variation is key.
Comparing drawn models to manipulatives will help students focus on the features as well as their justifications and conjectures (Petit et al., 2016); good practice also includes students comparing with other peer models (visual and physical). Students need to observe and listen to how visual models connect to each other, and then have time to rehearse and practise themselves (see Figure 8).

_Caption: Figure 8

The researcher and students co-constructed models of 5–4_

Slow down!
Yearley and Bruce (2014) concluded that there is a tendency in classroom instruction to move quickly to fraction symbols, which is not beneficial to sense making.
Consider your modelling structure.
A proposed modelling structure for the teachers to use when teaching new concepts arose from an interaction within Kylie’s second observation (see Table 1). The model consists of three steps – model, assist and supervise, and upon further research, it was found to align with McCoy’s (2011) ‘I do, we do, you do’ instructional model. McCoy (2011) expressed that the model is particularly useful in mathematics as there are often multiple steps and skills to be simultaneously integrated. Additionally, the first two steps equipped the students with a better sense of direction and confidence.

Caption: Table 1

Caption: Proposed Modelling Structure

Step

Teacher Action

Student Action

Who has control?

Model

I do

You watch

Teacher

Assist

I support

You do

Shared

Supervise

I watch

You do

Student

References

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