Introduction.
Over the past years, teaching Mathematics at Grade three (3) has been one of the most difficult tasks to undertake, and it still is to date. This involves particularly teaching fractions. This area of Numeracy has been studied by many Mathematicians and/or authors, and multiple recommendations and strategies to teach fractions have been proposed (Mafenuka N. 2010; Wu H. 2013; Brijlall D. 2011; Moss & Case 1999; Dr J.M. Shaw, 2002; Boggan, M., Harper, S., and Whitmire, A., n.d.; etc.). Nevertheless, issues with regards to teaching this concept in Grade three still persist.
It is a universal and a well-known fact that learners are different in nature, and they receive or conceive information in different fashions (Gardener, H. 1983). This is crucial to consider when preparing to deliver a lesson on fractions. In a foundation phase context, it is important to simplify a concept like fractions for the benefit of the learners. In order to do this, different techniques must be used when teaching fractions to third graders (Brijlall, D. 2011). The techniques include manipulatives and visual aids, such as diagrams, hands-on activities, pictures, counters, base-10 blocks, etc. The paper argues that that the teacher must use different strategies when delivering lessons. Also, examples must be varied so that they can suit the different learning personalities.
The purpose of the following study is to unpack the process of teaching and learning fractions at grade three level using manipulatives and visual aids. Benefits of using manipulatives and visual aids to teach fractions in grade three shall be discussed.
What are manipulatives and visual aids?
Manipulatives have a long history in the area of education and in the general lives of people (Boggan, M., Harper, S., and Whitmire, A., n.d). As far back as the ancient times, people of different philosophies have used tangible objects to count and resolve daily Math problems (Boggan, M. et al., n.d). Romans, Mayans, and Chinese, etc. have all used stones, wood, metal, etc. as means to invent apparatus to use for counting and solving Maths problems. This is proof that Mathematics and manipulatives and/or visual aids are inseparable. This substantiates the claim that people have always been creative and they have improvised in order to overcome Mathematical challenges in their daily lives.
Manipulatives defined.
Manipulatives come in numerous forms. These could be artificial or teacher-made base-ten blocks, two-coloured counters, fraction strips, beans, and geometric solids, etc. (Dr J.M. Shaw, 2002). Manipulatives can also be bought from the shop, brought from home or student-made (Boggan, M. et al., n.d). Manipulatives are defined as “physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics” (“Using manipulatives”, 2009), as quoted by Boggan, M. et al. (n.d.). These are concrete models that students manipulate when solving Math problems. Manipulatives range from the simplest bottle-tops/caps to the most sophisticated Unifix cubes and base-ten blocks (Boggan, M. et al. (n.d.). Hence, even a plain stone is a manipulative if it is able to assist learners solve Math problems in meaningful ways.
As much as manipulatives can be sophisticated, their primary aim is to assist the learner to build understanding and incur conceptual knowledge (Dr J.M. Shaw, 2002). The paper argues that that it should not confuse the child, but it must help elevate their understanding. Manipulatives help learners develop conceptual understanding by representing Mathematical ideas in multiple ways (Dr J.M. Shaw, 2002). Best manipulatives help introduce, practice and remediate Math concepts. A good manipulative does not bring about further complications, but supports the child to make sense of informal knowledge of the concept to the most formal (Boggan, M. et al. (n.d.). As alluded above, best manipulatives bring about further clarity.
In accordance with Smith, (2009, p. 20). As quoted by Boggan, M. et al. (n.d.)., a good manipulative must be at the “developmental level of the child”. It must be attuned to the mathematical ability of the child or it will not be fruitful (Smith, 2009, p. 20 as quoted in Boggan, M. et al., n.d.). Hence, it must be suitable for the child to use and it should correspond to their needs at that point in time. These are crucial considerations one must adhere to if they are to successfully teach fractions using manipulatives in grade three. Study shows that manipulatives have positive effects on learner’s performance and attitudes towards learning Math (Chun-Yi, L. and Ming-Jang, C., 2014).
Visual aids (pictorial models).
Visual aids form an integral part of learning in the foundation phase classroom. Learners must be able to see what you are saying when you teach, and the only way they can see is if you deploy pictorial representations/illustrations, diagrams, graphs, and number line, etc.
Visual aids are a prolific way to teach fractions in grade three. Using visual aids helps the child to visualise Mathematical ideas (Moyer-Packenham P.S. et al., 2012). Arcavi (2003) as quoted by Moyer-Packenham P.S. et al. (2012) defines visualization in mathematics as “the ability to create, interpret, use, and reflect on images in the mind, on paper, or with technological tools”. If a child is able to do this, s/he is able to think Mathematically and solve Math problems. This is important because it can assist the child to bridge the gap between the known and the unknown (or the easy and abstract). In accordance with Moyer-Packenham P.S. et al. (2012), visualisation is key because it affords learners a chance to view a pictorial representation with meaning, and to connect these to abstract concepts.
Visual aids are important in the creation of a print-rich, conducive and stimulating environment that allows learners to autonomously move from illustrations to meaning-making of abstract Mathematical concepts. This is especially the case when teaching fractions in grade three. Fractions can be very tricky to teach, and more so to learn. Hence, the concept should be introduced practically, informally, and visually with the aid of concrete apparatus and visual aids (Moyer-Packenham P.S. et al., 2012). Learners should be able to see what constitutes a whole, half, quarter, etc. in the form of an illustration, faction circles/rectangles, fraction chart, etc.
Visual aids can range from paper-based to computer-based. Paper-based visual aids can refer to anything that is written/printed on paper. In contrast, computer-based visual aids/manipulatives are accessible only on the computer screen (Moyer-Packenham P.S. et al.; 2012). These computer-based visual aids are called virtual manipulatives (Moyer-Packenham P.S. et al., 2012; Moyer, P.S., Bolyard, J.J., and Spikell, M.A. 2002). Virtual manipulatives have two categories, static (non-versatile/flexible) and dynamic (flexible) visual representation (Moyer P.S. et al., 2002). These are alternative teaching and learning aids that one must incorporate when teaching fractions. Visual manipulatives help create imaginary image in the child’s mind, which help them make meaning of the Math concepts.
Using manipulatives correctly.
Manipulatives can at times be non-beneficial if they are not used appropriately or correctly. It is the duty of the teacher to ensure that learners do not perceive manipulatives as toys, but understand that they serve as material to help them improve conceptual understanding. Nevertheless, learners should be allowed time to explore the manipulative and familiarise themselves with it, (Boggan, M., et al., n.d.). According to Smith (2009) as quoted by Boggan, M., et al., (n.d.), after the child has explored the manipulative “it ceases from being just a toy and claims its rightful place in the curriculum”. When the child has learned how to use the manipulative, they use it with purpose and not just to play with it. When a child explores the manipulative, it helps them generate questions and they make certain conclusions about it (Boggan, M., et al., n.d.). As a result, the child engages with the manipulative in a manner that challenges and improve their intellectual growth.
Advantages of using manipulatives to teach fractions.
Many researchers have studied the area of teaching Mathematics using manipulatives in foundation phase and the advantages/benefits it has (Dr J.M. Shaw, 2002; Mafenuka N. 2010; Moyer-Packenham P.S. et al.; 2012; etc.). As alluded before, manipulatives have a long history in Mathematics and they play a vital part in the teaching and learning processes. In accordance with Dr J.M. Shaw (2002), manipulatives are useful when introducing a new concept of Mathematics and they assist the learner to build conceptual understanding without hassles. Manipulatives help the child understand a Math idea by bringing it to life through concrete apparatus.
As much as manipulatives benefit learners, they can a barrier at times. Puchner, Taylor, O’Donnell, & Fick, 2008, n.p. as quoted by Boggan, M., et al., (n.d.), the study discovered that manipulatives delayed learners in some lessons, rather than alleviating student-learning because of the manner in which learners perceived the tool. Learners can be overwhelmed by the introduction of a new manipulative. Hence, the teacher must ensure proper planning beforehand (Boggan, M, et al., n.d.). In case of virtual manipulatives, learners must have had exposure to basic computational skills. In some cases, it is difficult to engage with virtual manipulatives because of the scarcity of resources. Nevertheless, a foundation phase teacher must improvise or invent in the absence of resources, for the benefit of learners.
When teaching fractions in Grade three, it is imperative to deploy manipulatives in order to enhance learners’ understanding of the concept. For example, when introducing a concept of halve, learners must be able to see half of the whole in real life, (Boggan, M, et al., n.d.). It is crucial to demonstrate the concepts in practise so that learners can visualise these in their minds, thereby alleviating their own personal understanding. Manipulatives and visual aids help learners bring abstract mathematical concepts to life, which enhance their understanding.
In Grade three, learners must have fraction stirpes, circles, rectangles and fraction charts on the classroom wall for them to consult continually and remind themselves of the most basic fractions. Learners must have concrete material to manipulate or work with during the lessons (Boggan, M, et al., n.d.). As I have alluded to it before, this helps make the concept concrete or alive. Learners are most likely to remember fractions if they construct them. “Models can help students clarify ideas that are often confusing in a purely symbolic mode.” – Van de Walle, Lovin as quoted in Mathematics Enhanced Scope and Sequence, (2011). The paper argues that that learners use manipulatives to improve their understanding of Math concepts in a much simpler way.
As quoted by Marshall, L. And Swan, P. (2008), a famous proverb, “I hear and I forget, I see and I remember, I do and I understand”, is a substantiation of the importance of manipulatives when teaching mathematics and even other subjects in the elementary phase of schooling. It is much easier for learners to understand if they do what they are learning practically, that way they will better understand the content. Nevertheless, the effectiveness of manipulatives in some cases must be scrutinized for alterations (Marshall, L. And Swan, P.; 2008). As a teacher, it is vital to reflect on practice frequently and examine the effectiveness of manipulatives so that they can be able to use the manipulatives more productively.
Teachers need to plan for when and how to integrate manipulatives in their mathematics lessons. Manipulatives can be a barrier to learning if they are not deployed with proper intention and/or plan. Marshall, L. And Swan, P. (2008) quoted Uttal, et al. (1997), who suggest that “literature is somewhat ambivalent about the use of mathematics manipulatives”. In a study on the effectiveness of mathematics manipulatives conducted by Marshall, L. And Swan, P. (2008), they discovered that there is no consistent advantage of manipulatives over traditional methods of instruction. However, if used appropriately, “models show details that a human eye cannot see”, (Conley, C., et al. n.d.). This helps enhance learners’ understanding. Models are excellent to demonstrate a process, (Conley, C., et al. n.d.). This paper argues that models (manipulatives) are an excellent method to help learners understand fractions better.
Mathematical thinking when teaching fractions.
Mathematical thinking is important when learning Mathematics. According to Mathematician Paul Halmos (1980), as quoted by Stacey, K. (n.d.), “Mathematical thinking is the heart of mathematics”. The paper argues that that without Mathematical thinking, it is almost impossible to learn Mathematics properly. It is key to the processing and understanding of abstract Mathematical concepts from the learner’s perspective.
Definition of mathematical thinking.
After a thorough research on mathematical thinking, I learned that mathematical thinking is a process that one undertakes in attempt to incur explicit understanding of math concepts (Harel, G., and Sowder, L.; 2005). As teachers, our primary goal is to help alleviate learners’ understanding of key concepts. Hence, when teaching Mathematics (Fractions) at grade three, it is crucial to remember this and help learners through the process of thinking and understanding.
Mathematical thinking is a very complex activity, and a great deal of research on this activity has been studied (Stacey, K., n.d.). Stacey, K., (n.d.) wrote that mathematical thinking is a process. Hence, it is highly impossible to agree on one definition because “there are many ‘windows’ through-which mathematical thinking can be viewed” (Stacey, K., n.d.). As argued in the paper, “What is mathematical thinking and why is it important?”, different people can look at mathematical thinking in different ways. In accordance with Stacey, K., (n.d.), the processes of mathematical thinking involve four activities; specialising, generalising, conjecturing and convincing. Specialising includes trying special cases, looking at examples, and generalising is looking for patterns and relationships. Conjecturing is predicting relationships and results. Convincing is finding and communicating reasons why something is true.
Harel, G., and Sowder, L.; (2005) explained that mathematical thinking entails two processes, these are ways of thinking and ways of understanding. Ways of thinking involve learners’ general theories or ideas about a concept (implicit or explicit), whereas ways of understanding refer to a meaning/definition learners give to a specific term, sentence, solutions they provide to a particular problem, or justification they use to validate or refute an answer or opinion.
How learners engage in Mathematical thinking?
As a teacher, you act as a facilitator of learning. Hence, it is the responsibility of the teacher to ensure that learners are provided a platform trough-which they can engage in Mathematical thinking. According to Harel, G., and Sowder, L., (2005), mathematical thinking involves ways of understanding. Each one of the learners may have a unique way of learning and understanding mathematics, (Dr H. Gardener; 1983). However, learners engage in mathematical thinking autonomously. Harel, G., and Sowder, L., (2005), wrote about four activities one engages in as part of mathematical thinking processes. The four activities (specialising, generalising, conjecturing and convincing), explain vividly how learners engage in mathematical thinking.
In respect to specifically the area of fractions in grade three mathematics, learners definitely engage in mathematical thinking. They do this by convincing the teacher or peers if a certain answer is true, they also make generalisations about fractions, (Harel, G., and Sowder, L.; 2005). Learners also conjecture or make assumptions about fraction problems they have to solve during class activities, (Harel, G., and Sowder, L.; 2005). In addition, learners study examples and compare them, hence, specialisation, (Harel, G., and Sowder, L.; 2005). The paper argues that learners participate in the process of thinking mathematically every time they attempt to solve a Math problem.
Recommendations.
After a comprehensive stint with the learners, I got the opportunity to see learners working with manipulatives and visual aids up close. Learners in grade three still require a great deal of guidance when working with fractions, though they have had exposure to fractions before. It is therefore fundamental to make use of fraction rectangles, diagrams and charts as means for learners to remind themselves of basic fractions. Moreover, it is pivotal to have concrete material (e.g. counters, bottle-caps, base-10 blocks, etc.) for utilization when expounding fractions in grade three.
Manipulatives and/or visual aids should be tantamount to all confusions with regards to fractions, but that is possible if the teacher has strategy on how to use them. Hence, prior planning and preparation is inevitable so that the use of manipulatives can be as fruitful. Manipulatives should be used in a meaningful way by the teacher so that they can be meaningful for learners too. As a teacher, it is important to always remain cautious about teaching and learning, using manipulatives so that they do not become a barrier to learning.
School setting and context.
The school site in which the action research was conducted is a primary school situated in Pimville (Soweto). The school offers many services, but education is main priority. The school district in which the school is located is North Gauteng. Grade ranges from Grade R to Grade five (5). The majority of learners enrolled at the school reside in surrounding areas. However, some reside far from school and use transport to get to school. According to the school, they currently cater for a total of 864 learners. 47% Males and 53% female. The ethnic groups that made up the population of learners included 53% Zulu, 17% Sotho, 10% Tswana, 12% Xhosa, and 8% Pedi. Nevertheless, the LoLT (Language of Learning and Teaching) of the school is IsiZulu.
Participants and population.
Thirty-two (32) grade three learners took part in the study. Of the thirty-two (32) learners, 15 were boys and 17 were girls. This study was conducted in March and during the first week of May. The cultural/racial distribution of participants does not fully reflect the make-up of the whole school as there are only Zulu, Sotho and Tswana speaking learners in the Grade three class. In the study, there were no Xhosa and Pedi speaking learners and they make up a big part of the school’s population. The academic Mathematical proficiency of participants ranged from performing below Grade level, on grade level and exceeding grade level in some instances.
Limitations.
The study had few limitations. First up, I experienced difficulties with regards to getting adequate resources/manipulatives to use during action research due to financial constraints. More so, it was not very easy to get material/tools that could be useful when creating/designing man-made manipulatives. This dealt a huge blow as this action research required manipulatives and visual aids in order to determine how they assist learners to learn fractions. Secondly, the school deployed us in various grades and we had to rotate to different grades since there was seven of us from the UJ. Hence, I struggled acquiring sufficient time to carry out the research on a grade three class. This proved to be a challenge because my action research speaks to specifically third graders and how they learn the unit of fractions when visual aids and manipulatives are incorporated. The third challenge the study was faced with is that my prior planning contradicted what the teacher had planned initially in respect to the deliverance of lessons in fraction units. Therefore, lots of changes had to be implemented in order to have congruent plans regarding the lessons on fractions.
Conclusion.
In the foundation phase, learners ought to have concrete apparatus to manipulate. Piaget, J. (n.d.). proposed four stages of development (sensorimotor, preoperational, concrete operational, and formal operational), the third stage corroborates the perspective I support that manipulatives and visual aids are vital when teaching fractions in grade three (3). In the third stage, dubbed concrete operational stage (7 – 11 years), Piaget, J. (n.d.). suggested that “Intellectual development is demonstrated through the use of logical and systematic manipulation of symbols, which are related to concrete objects”. This is valid and relevant for Grade three, since the learners in this grade three are aged between these 7 – 11 years.
As a foundation phase teacher, it is crucial to realize the importance of manipulatives so that we can integrate such in fraction lessons and in other lessons. Manipulatives and visual aids are without question vital to enhance learners’ understanding of fractions in grade three. However, manipulatives and visual aids should be prepared and planned for prior to the lessons. They are not to be used in vain, they must be effectively used.