Manipulatives for Improving Mathematical Understanding

Introduction

Education is always developing and there are new ideas coming into schools every day. Deciding how to implement new ideas and resources can be difficult. Which students will benefit the most? How do you introduce ideas into the classroom? How can teachers change their teaching habits to benefit their students? Manipulatives, such as multi-link cubes and Dienes Blocks, are used more now as schools increasingly introduce Mastery Curriculums, such as programmes run by ARK UK.  Although more popular in primary schools, secondary schools are also beginning to use them. The research focused around the benefit of manipulatives for older students or comparing different ages and attainment levels is however lacking.

This study looks at how effective manipulatives are at improving mathematical understanding at different ages and attainment levels to determine whether they are beneficial for all students or if they are best reserved for certain groups. The study is centred on a single comprehensive secondary school, with a student population made up of 85% White British, 10.5% SEND, 42.5% Disadvantaged and 20.1% on Free School Meals. The study uses focus groups with varying types of student, ranging in ability and attainment level, alongside observations of teachers using manipulatives in lessons and interviews with both students and teachers to try and answer the following overall question; ‘Do Manipulatives have an Age Limit?’

There are two main research questions being considered; firstly ‘How does age and attainment level change the way in which manipulatives are used effectively?’

The second question,

‘Do teachers views on the effective use of manipulatives differ at different levels?’ considers how teacher opinion could have an impact on how manipulatives effectively used at different ages and attainment level.

This research is important as many schools are introducing mastery programmes, which tend to include the use of manipulatives. The use of physical representations to teach mathematical concepts is not a new idea but it is increasingly popular in classrooms. Therefore understanding the effectiveness of manipulatives at all levels and ages is important as it can inform the implementation of manipulatives. The research could help teachers to understand how the age and ability of their students could change their reaction to manipulatives introduction and use.

This research is particularly relevant to schools who are just starting to use manipulatives, so they can understand how different year groups and sets could react to manipulative use and how the opinions of teachers could have an impact on how effectively manipulatives are utilised.

Literature Review

What are Manipulatives?

“If we want children to learn to think deeply and ponder real mathematics as well as to be able to use in depth thinking in real life scenarios we must teach them and assess their knowledge in ways that allow them to show us what they really understand.” (Kelly, 2006, p.185) Getting to the root of true mathematical understanding is where manipulatives have begun to play a big part in the teaching of mathematics.

Manipulatives (or concrete objects), defined as being “any tangible object, tool, model, or mechanism that may be used to clearly demonstrate a depth of understanding.” (Kelly, 2006, p.184) are not a new idea. The idea of learning mathematics by using concrete objects has roots in the theories of Piaget (1952), Bruner (1966) and Montessori (1917). Piaget (1952) believed that children do not have the mental capacity to process mathematics using symbols alone, Bruner (1966) believed children needed a way of representing their environment and Montessori (1917) believed that children learnt best when using their own initiative and natural abilities through physical play.

Behind all of these theories is the idea that “children must understand what they are learning for it to be permanent” (Moyer, 2002, p.175). Kosko & Wilkins (2010) believed that “concrete materials can be used to develop deep understandings of certain mathematical concepts” (p.79) making manipulatives an important part of developing children’s understanding. Uttal et al (1997) found that the concrete nature of manipulatives was the exact reason why children learnt from them, it negated the initial need for them to reason abstractly or symbolically.

Moyer (2002), Kelly (2006) and Uttal et al (1997) all agreed that concrete objects used initially would enable students to effectively reason abstractly later on. Moch (2002) commented that although “manipulatives cannot cure every problem encountered by students” (p.87), many problems could be “avoided altogether by allowing students to work with concrete models prior to dealing with them on a more abstract level.” (p.87). Stein & Bovalino (2001) believed that manipulatives were important in “helping students to think and reason in more meaningful ways” (p.356), building them up to reason abstractly. All of these researchers found that manipulatives were effective as a precursor to abstract mathematics.

Piaget (1952) went further and suggested that children, particularly young children, do not have the capability of reasoning abstractly so the use of concrete objects was paramount to their understanding of mathematical concepts, at least in the early stages of their education. In short, “manipulative use is seen as a way of increasing mathematical understanding” (Kosko & Wilkins, 2010, P.79). Moch (2002) believed that “The use of manipulatives in the classroom is necessary” (p.83), suggesting that manipulatives should be commonplace in every mathematics classroom because it offered “a natural way for children to make sense of the mathematics they are trying to learn.” (p.83).

Communication

“Current research in mathematics education views students as active participants who construct knowledge by reorganising their current ways of knowing and extracting coherence and meaning from their experiences.” (Moyer, 2002, p.176). “Teachers and researchers have suggested that concrete objects allow children to establish connections between their everyday experiences and their nascent knowledge of mathematical concepts” (Uttal et al, 1997, p.38). By using manipulatives to create real situations, teachers can enable students to understand the abstract world of maths bringing the two worlds closer together.

The mere use of manipulatives however, does not guarantee students will immediately be able to understand new mathematical concepts. Clements (1999) found that students might need to use concrete objects initially but that if they were to gain true benefit from the manipulatives they “must reflect on their actions” (p.47) whilst using them. Fennema (1972) and Kosko & Wilkins (2010) both advocated for the need for communication around the concept alongside the use of manipulatives, with Kosko & Wilkins (2010) finding that “students who are using manipulatives to learn mathematics are more likely to be engaged in mathematical communication” (p.88).

Having tangible objects in front of students may encourage discussion around a mathematical concept due to the students having a physical object to talk about, rather than trying to imagine an abstract concept. Kosko & Wilkins (2010) spoke about the students being able to “communicate in a language they are comfortable with” rather than the confusing and abstract language of mathematics. They also found that when using manipulatives students could communicate using an “informal language that makes it easier for them to understand the concepts” (p.81). The manipulatives enable the students to have real conversations around a concept without having to worry about how they are going to describe their ideas.

Connections

To be able to gain understanding from manipulatives students must be able to make connections and see the relationships between manipulatives and the symbols used in written mathematics. “To learn mathematics from manipulatives, children need to perceive and comprehend relations between manipulatives and other forms of mathematical expression” (Uttal et al, 1997, p.38). For example, when working with tens blocks, hand-made or purchased versions such as Dienes blocks, students need to make the connection between how blocks of ten fit together and how place value works with numbers themselves.

Uttal et al (1997) worry that the “sharp distinction that has been drawn between concrete and symbolic forms of mathematics expression” (p.38) means that students are unable to move from concrete materials to written, abstract mathematics effectively. They believe that if manipulatives are to be successful then the manipulatives themselves need to be seen as symbols. They did also find that “Symbols may be difficult to teach to children who have not yet grasped the concept that they represent. At the same time, the concepts may be difficult to teach to children who have not yet mastered the symbols” (Uttal et al. 1997, p.37). So making this initial connection between manipulatives and symbols and vice versa is an important one. However, they found that “although children may learn to perform mathematical operations with manipulatives, they often fail to link this knowledge to more traditional forms of mathematical expression” (p.45), meaning that although the connection is important students struggle to manage it.

We cannot assume that if students do make a connection that it will necessarily be the connection the teacher was aiming for. “There is no direct connection between an external representation and an internal one” (Puchner et al, 2010, p.313). Ball (1992), advocated for the need to think about manipulatives from the view of a child. They believed that one of the reasons why the power of manipulatives might be overstated was the idea that as adults we can see the connections between the concrete and the abstract easily, possibly due to us already understanding the concept. However this does not mean children will see that same things and although “manipulatives are used as a bridge to help children master the abstract” (Uttal et al, 1972, p.48) it cannot be assumed that “children will make this connection on their own” (p.48).

However, Ball (1992) found that teachers guides “often convey the impression that, when students use manipulatives, they will likely draw correct conclusions” (p.16). This is a problem as the majority of the research, as shared above and expanded on below, shows that students do not make these connections easily. Ball (1992) found that this idea stemmed from the belief that “the desired conclusions reside palpably within the materials themselves” (p.16). However, as we will see later on, manipulatives themselves do not hold all the answers in their physical structure. This misconception surrounding manipulatives could mislead teachers “into thinking that mathematical knowledge will automatically arise from their use” (Ball, 1992, p.18).

Holt (1982) described how he and a colleague were excited by their discovery of Cuisenaire rods and the potential they saw in them to make “strong connections between the world of rods and the world of numbers” (p.138). However, they discovered quickly that there was a difference between the connections they could see themselves and the connections the students might make. They knew that the problem they faced was that they “already knew how the world of numbers worked” (p.138). They were concerned that if they “hadn’t known how numbers worked, would looking at the rods have enabled us to find out” (p.138). Puchner et al (2010) found a similar issue when studying various teachers using manipulatives, observing that “often the teacher sees so clearly how the external representation depicts the idea they are trying to teach, they cannot imagine how the student would not easily form an accurate internal representation from the manipulative” (p.314).

Uttal et al (1972) found that young children would not see the relationships between manipulatives and mathematical concepts unless “those relations are specifically highlighted” (p.50). This again means that teachers have to choose their manipulatives very carefully and plan their use so these connections are made clear. “The teachers needs to be constantly aware of the possibility that a child may have a qualitatively different way of perceiving things” (Björklund, 2013, p.474).

We cannot assume “students will automatically draw the conclusions their teachers want simply by interacting with particular manipulatives” (Ball, 1992, p.16). Although there have been positive results from manipulative use, Björklund (2013) found that the “learning effect seems to depend on the teacher’s awareness of how and why to use such aids” (p.473). There needs to be a shift from using manipulatives for the sake of it toward intentional planning that will ensure students gain from their use as “creating effective vehicles for learning mathematics requires more than just a catalog of promising manipulatives” (Ball, 1992, p.18).

As students cannot rely on manipulatives forever, (in exams, they will not have access to them), moving students on from relying on them and being able to understand mathematics abstractly must always be considered when concrete objects are used. Although “concrete models when first used are intriguing and children prefer them because of their novelty and because they make mathematical ideas meaningful” (Fennema, 1972, p.639) there is the hope that eventually they will realise that “concrete models are inefficient and cumbersome to use in problem solving” (p.639). If this occurs then students should naturally progress to using symbolic and abstract models as a faster way of working when they fully understand the concepts.

However, there is a concern that although children may understand a mathematical concept when using the manipulatives they may not make the link between the concrete objects and the abstract and symbolic nature of traditional mathematics (Uttal et al, 1997) making this transition difficult. “Making connection between concrete and abstract representations in mathematics education is not an easy task” (Björklund, 2013, p.473). How manipulatives are utilised in the classroom needs to be carefully considered so the final aim, enabling students to understand mathematics in any form, can be achieved.

Ball (1992), cautioned against “thinking that mathematical knowledge will automatically arise from their use” (p.18). Moyer (2002) explained that physical objects do not carry mathematical meaning themselves and that it is the way in which they are used that give them meaning. Somehow, we must make a link between a student’s internal understanding and the manipulative, “the students own internal representation of ideas must somehow connect with the external representation or manipulative” (Moyer, 2002, p.192). Clements (1999) also spoke about the difference between a manipulatives physical nature and the concept a teacher is trying to teach. They spoke of the need to “guide students to make connections between these representations” (p.56).

Uttal et al (1997) summed it up well by considering the idea that although manipulatives may be interesting, “this is not sufficient to advance their knowledge of mathematics or concepts” (p.38). For manipulatives to be effective teaching tools, students must “see them as representations of something else and understand the nature of the representational relation” (Uttal et al, 1997, p.45).

Purpose

Moyer (2002) looked a great deal at the effect of teachers on manipulative use and effectiveness. They found that the beliefs of teachers “may influence how and why they use manipulatives as they do” (p.178). Björklund (2013) found that although the majority of research has shown the positive effect manipulatives could have on students understanding, “the actual learning effect seems to depend on the teacher’s awareness of how and why to use such aids in the teaching act.” (p.473). Similarly, Fennema (1972) found that there was “an inadequate recognition of the role that concrete and symbolic models can and should play in facilitating the learning of mathematical ideas” (p.635). Understanding the purpose behind manipulatives and how best to use them in the lesson could have more impact on their effectiveness to improve mathematical understanding then the mere objects themselves.

The problem is that “a manipulatives physical nature does not carry the meaning of a mathematical idea” (Clements, 1999, p.56) meaning that when using manipulatives teachers need to carefully plan how they are going to use them and what mathematical concept they are trying to teach. It is easy for students to “overlook the consequences of their actions” (p.55) when working with manipulatives and “even when manipulatives are planned as an integral part of the lesson, student thinking and reasoning can become routine and mechanical” (Stein & Bovalino, 2001, p.356), causing manipulatives to lose their value.

Therefore, it is important to look closely at how manipulatives are selected for use in a lesson as “if not used with careful thought, manipulatives can become little more than window dressing” (Stein & Bovalino, 2001, p.357). If students are given freedom in their choice of representation then they will typically pick objects that mean the ideas make the most sense to them (Fennema, 1972). However, just because the student has chosen their own representation, this does not necessarily mean they will understand the concept being addressed any better than before. Whitin (2004) cautioned against letting students choose their own manipulatives as “too much choice could be overwhelming for the children and difficult to manage” (p.181) which will take away from the positive effect manipulative use can have.

Selection of manipulatives is an area that Clements & McMillen (1996) have looked at closely. They offered a series of guidelines for teachers to follow to make sure that they were using manipulatives for the right reasons. Their first guideline was that manipulatives should primarily be selected for children to use themselves. Although they explained, “teacher demonstrations with manipulatives can be valuable” (p.276) the students themselves should primarily use the manipulatives. Marzola (1987) agreed and stated that “although observing the teacher demonstrate a concept with manipulatives has its place in initial instruction, it is important that students themselves become the active agents as soon as possible.” (p.14) Moyer (2002) however, found that this was not usually that case and that “in most cases the lessons were teacher-directed” (p.184) with the teachers demonstrating the manipulatives only or directing their use instead of allowing students to discover for themselves.

Clements & McMillen’s (1996) second guideline was that manipulatives should be selected so that they “allow children to use their informal methods” (p.276) Manipulatives should not be allowed to limit students by forcing them to complete mathematics in specific ways. Every student should be able to access the mathematics using the given manipulatives regardless of their attainment level. Fennema (1972) also found that for students to learn best they needed to use “models suited to the children’s level of cognitive development” (p.637). When choosing manipulatives teachers need to be aware of their student’s level and the methods they are capable of using independently.

The third guideline is more of a warning; Clements and McMillen (1996) recommend, “caution in selecting pre-structured manipulatives” (p.276-7). For example, using pre-built base-ten blocks rather than letting students create them using multi-link cubes.  They warn that when using manipulatives that have been built for a specific purpose, students can be trapped by the pre-determined ideas they manipulatives are designed for, rather than creating the mathematics themselves. Uttal et al (1997) found that “whether manipulative are effective teaching tools depends upon whether children interpret them as representations of something else and understand the nature of the representational relation.” (p.45). The students’ needs to see the manipulatives as more than objects and having pre-made manipulatives can limit students ability to distance the representation from the physical object.

Moyer (2002) found that although “the physicality of concrete manipulatives does not carry the meaning” (p.177) students gain the most from their use when they “reflect on their actions with the manipulatives to build meaning” (p.177). Moyer also warned that if manipulatives are only used as the manufacturer intended then students would be limited in their potential to learn, as part of the process has been removed.

Another problem that can stem from pre-built manipulatives is that students may link certain mathematical ideas to certain objects, limiting their usefulness in discovering new mathematical ideas. Clements and McMillen’s (1996) fourth guideline suggests that manipulatives are chosen for multiple purposes so students do not link one idea to a single object and instead learn how to use the objects to represent mathematical ideas rather the objects being the ideas themselves. This will allow students to separate the concrete from the abstract and help them to make the connection.

However, Clements and McMillen (1996) also warn against using too many manipulatives for the same idea and that “to introduce a topic, use a single manipulatives instead of many different ones” (p.277). At first, this may seem a contradiction to their previous guideline, which suggested that we do not link one manipulative to one idea. There needs to be a clear distinction here about using the same manipulative to represent different mathematical ideas whilst at the same time not overwhelming one idea with multiple manipulatives. Some students may need different manipulatives to understand an idea but changing the object being used too quickly is likely to confuse students rather than aid them. Clements and McMillen suggest that teachers think very carefully about which manipulative they are planning to use, if they have been used for different ideas before and how they could be used in the future.

Fun and Games

Moyer (2002) found that many teachers were only using manipulatives as “an enrichment activity or a game assigned when there was time at the end of the class period” (p.184). This use of manipulatives, fun rather than instructional, does not create an environment where manipulatives can be effective. Moyer (2002) found that when speaking to teachers “initially the term ‘fun’ seemed to indicate that teachers and students found enjoyment in using the manipulatives during mathematics teaching and learning” (p.185). They found that “embedded in teachers’ use of the word ‘fun’ were some unexamined notions that inhibit the use of manipulatives in mathematical instruction.” (Moyer, 2002, p.185).

Moyer (2002) found that teachers believed that “manipulatives were fun, but not necessary for teaching and learning mathematics” (p.175). This idea of fun at first seems like a good thing; surely, if students are having fun then they are more engaged and will gain more from their lessons. However as Moyer found, ‘fun’ was not necessarily a good thing. They found that when teachers described manipulatives as ‘fun’ they did not believe that manipulatives could be used to further their students learning.

McNeil & Jarvin (2007) discovered the same problem in their study, finding that “teachers may use manipulatives for fun or just to add variety to their math classes, instead of using manipulatives to engage students in mathematics. Herbert (1985) found teachers thought that using manipulatives was the same as playing games in lessons, that they were great for the end of term when you wanted to have fun, but that they did not have a place in the usual classroom routine.

Teacher’s opinions around manipulatives can have a large impact on how effective they are in the classroom; teachers are the ones after all who have to enable their use in the classroom. Scheer (1985) found that when she suggested the use of manipulatives to a colleague her suggestion was rebuffed because they believed that manipulatives were too ‘babyish’ for his middle school students. The opinion of the teachers here was holding them back from using manipulatives that could be helpful to their class. Scheer offered to take on the class for a week and see if she could convince her friend that manipulatives could be useful. After seeing his students flourish with their use, the teacher was convinced and Scheer reported that he would be using manipulatives in the near future to teach fractions. This case shows clearly how teacher opinion around manipulatives can affect their use, in this case their lack of use.

There is also a trend of using manipulatives to reward good behaviour in previous lessons, with manipulatives seen as a “reward system for good behaviour rather than as an important formative tool for routine use in teaching mathematical concepts” (Moch, 2002, p.82) or “as a reward for appropriate student behaviours” (Moyer, 2002, p.186). They are also used as a way of controlling behaviour by using the manipulatives as “highly motivational tools which can greatly increase on-task behaviour and attention span.” (Marzola, 1987, p.11). Both of these uses take away from the main purpose of manipulatives, if they are used to reward good behaviour then those students who struggle with self-control are being limited and if they are used as a motivator then their main purpose is not at the forefront of the lesson.

Teachers must also be careful that if they are using manipulatives for their true purpose, following the guidelines from Clements & McMillen (1996), that they are not skipping steps out and are being careful to try not to force manipulatives to fit traditional teaching methods. Moch (2002) found that the easiest way manipulative lessons went wrong was “if teachers attempt to give the answers too quickly or force students to perform tasks in a step-by-step, prescribed manner” (p.83). McNeil and Jarvin (2007) found that “teachers believe mathematics is best taught by telling. That is, math is best taught by introducing the step-by-step procedures necessary for solving problems, repeating those procedures in clear language, providing students with an opportunity to practice those procedures” (p.313). This is the exact opposite of the purpose of manipulatives. Traditional teaching methods, such as using step-by-step procedures, do not fit together with manipulatives, but teachers are comfortable with these methods so rather than change their whole style they try to make manipulatives fit into their existent model. As McNeil & Jarvin (2007) put it “teachers may use manipulatives for fun or just to add variety to their math classes” (p.312). Using manipulatives to change small parts of their lessons to make them more interesting rather than using them to change their way of teaching.

“Manipulatives seem to offer an attractive alternative to traditional drill and practice procedures” (McNeil & Jarvin, 2007, p.310). Students can find mathematics a ‘boring’ subject and manipulatives are an attractive alternative to this. Although this is not perhaps, the main purpose of manipulatives, Moch (2002) found that the students they worked with found that “by doing the math they felt they were learning a lot more than by just doing worksheets.” (p.85), showing a greater engagement in the learning of mathematics than more ‘traditional’ teaching methods. The “manipulatives afforded students an opportunity to touch and feel mathematics – not just to see or hear it” (Moch, 2002, p.86), allowing students the chance to fully engage with mathematics, rather than see it as an abstract construct that did not relate to their own world. Although they may not be gaining from the manipulatives themselves, the ‘fun’ nature of them makes the mathematics classroom more engaging itself.

Time

For manipulatives to be effective there needs to be an understanding from the teachers point of view of their place in the classroom and how they can be used best in every situation. This requires teachers to be willing to put more initial time and effort into planning with manipulatives and allow sufficient time in their lessons for students to use them. “For children to use concrete representations effectively without increased demands on their processing capacity, they must know the materials well enough to use them automatically.” (Moyer, 2002, p.176). This means that teachers need to be using them often enough in lessons so that students are so used to them being part of their mathematics learning that they can use them without extra input. Stein & Bovalino (2001) found that manipulatives provided “a concrete way for students to link new, often abstract information to already solidified and personally meaningful networks of knowledge” (p. 359). These networks of knowledge had been built up through consistent use of manipulatives, so that the students understood how they could be used to represent different areas of mathematics, enabling them to use them with new topics and “take in new information and give it meaning” (p.359).

“Using manipulatives well takes time and practice. Using them strictly because they are the latest fad or because of some administrative mandate, without teachers investing their time or interest, results in a less than desirable outcomes for students, teachers, and administrators.” (Moch, 2002, p.81). It is very easy for the latest trend to fascinate teachers and cause them to start using a product such as manipulatives without truly understanding the impact they can have if used properly. However, there is a real issue with teachers finding the necessary time to make manipulatives work effectively in their lessons. Uttal et al (1997) found that “extensive instruction and practice may be required before manipulatives become effective” (p.38). Unless the time was put into manipulatives being used then their benefits would never be seen and “in some cases manipulatives seem to do as much harm as good” (p.38). If teachers were not able to give over sufficient time for this extensive work then the use of manipulatives could be detrimental.

“Acquisition of mathematical knowledge through problem solving and with manipulatives has long been considered to be time-consuming and labor intensive” (Kelly, 2006, p.184). Herbert (1985) found that “teachers often claim that not enough time is available to use manipulatives” (p.4) and Moch (2002) found that “Many instructors believe using manipulatives to teach mathematical concepts takes up too much time” (p.82). Moyer (2002) found that “the pressure teachers feel to cover material during mathematics instruction” (p.190) made them decide that manipulatives did not have a place in the day to day classroom because they took up too much time and that manipulatives were reserved for the end of term, ‘fun’ activities. They could be used when the students were winding down and would not miss other ‘normal’ teaching.

This ‘lack of time’ feeling whilst stopping some teachers from using manipulatives all together, can affect how manipulatives are effective even when they are used. Stein & Bovalino (2001) found that some teachers would “shortcut student thinking by jumping in and supplying the ‘way to do it’.” (p.356) instead of giving them the “time and latitude to think it through and make sense of the manipulative activity.” (p.356). Uttal et al (1997) cautioned against “using manipulatives in a classroom without ensuring that students fully understand their relation to the mathematical concepts” (p.47) claiming that in this case their use would be counter-productive. However, making sure students fully understand the concepts requires that sufficient time be given for this to happen. Time again seems to be an important factor to consider when manipulatives are used.

Young Children

As seen there has been much research conducted into the benefit of manipulative use, a lot of it focused on primary or elementary school children. Moyer (2002) found that “students’ early experiences and interaction with physical objects formed the basis for later learning at the abstract level” (p.176). Moyer chose to focus specifically on their ‘early’ use in a student’s education. They also found that the topics the manipulatives were primarily used with were typically concepts that were covered in the early stages of a student’s mathematical education, these topics included, Place value, prime numbers and fraction work. However, Ball (1992) argued that “all learning must proceed from the concrete to the abstract” (p.15) and that manipulatives could have a place in all mathematical topics, regardless of level.

Uttal et al (1997) also focused on young children, noting, “the concrete nature of manipulatives makes them particularly appropriate for kindergartners and young elementary school children.” (p.39). Uttal et al have suggested that the manipulatives are effective because young children learn best from concrete objects. However if manipulatives are considered effective because using concrete objects is best for young children, does this mean manipulatives are not as effective for older children whose reliance on concrete objects may not be the same. As Clements (1999) put it, “teachers of later grades expect students to have a concrete understanding that goes beyond manipulatives.” (p.47).

Young versus Old

There is not as much research focused around older students or comparing the effect of manipulatives between different ages. Uttal et al (1997) found that “research on effectiveness of manipulatives has failed to demonstrate a clear, consistent advantage for manipulatives over more traditional methods of instruction.” (p.38). Is it possible that this lack of consistency is due to manipulatives themselves being less effective for students of certain ages or attainment levels or is there no difference in ages and the inconsistencies are due to other factors?

Fennema (1972) noticed that there was a predominance of concrete models used in early grades but that as children got older there was an increased proportion of abstract and symbolic models used. However, they did also state that “learning environments for children at various development levels should include both concrete and symbolic models” (p.636) and that when new ideas are introduced there might be a need for concrete objects to be utilised again. Stein & Bovalino (2001) were convinced that “the fundamental idea regarding how manipulatives help students learn was the same at any level.” (p.357) whereas Fennema (1972) found that there was a variety in the positive effects of manipulative use when considering older children.

Uttal et al (1972) agreed with Stein & Bovalino and stated, “manipulatives have been recommended as a means of improving performance for all levels of students” (p.38). However, they did find that younger children found it more difficult to relate manipulatives to the underlying concept they were being used to represent. In this case, would older children have more success from using manipulatives if they could perceive the connections faster and better than younger children could? Uttal et al (1972) did not seem to think so as they found that “just as younger children have trouble establishing connections between manipulatives and mathematical concepts and symbols, older children have difficulty making a connection between geometric constructions and proof” (p.46). Although geometric constructions are not quite the same as manipulatives, the ideas behind them are the same, create something real that students can relate to. Uttal et al (1972) found the same issues here in that the students did not realise that the two things were “intended to teach a similar concept” (p.46).

Clements (1999) looked into the difference perceived to exist between younger and older children. They theorised that “younger children possess the relevant knowledge but cannot effectively create a mental representation of the necessary information” (p.49). They claimed that although it usually seems like older children have a better understanding of mathematical concepts the only reason for this is their greater ability to explain their understanding. If younger children were given the tools to explain their thinking, would it become clear that they understand more than is usually apparent? Would the use of manipulatives, as an avenue to help students explain their thinking, show that students understand mathematical concepts earlier than is usually recognised.

Although much of the research conducted around manipulatives focusses on younger children, “there is little disagreement about the utility of manipulatives through the primary grades” (Marzola, 1987, p.11), there have been studies that have advocated for the use of manipulatives at all ages. Fennema (1972) believed that “learning environments for children at various development levels should include both concrete and symbolic models” (p.636). They did find however, that in the early grades manipulative use was effective in deepening understanding, in the middle-grades learning was better when manipulative use was allowed but not enforced and for older children the use of manipulatives neither “improve nor hamper learning” (Fennema, 1972, p.637). This conclusion surrounding the effectiveness of manipulative seems to support the idea that whilst manipulatives are great at the beginning of a mathematical education, their effectiveness lessens as students’ progress through their education. Clements (1999) disagreed and found that students who used manipulatives “across grade level, ability, level and topic” (p.45) performed better than students who did not use them.

This reference to ability is rare amongst the research conducted with manipulatives, the majority of studies focus on the manipulatives overall effectiveness, regardless of age or attainment level. Marzolo (1987) did find that “low achievers may experience greater success with activities requiring use of manipulatives than with other math activities” (p.9), showing that low achievers or students with low prior attainment gained more from manipulative use than more traditional activities. However, they did not compare the success with higher attaining students.

This lack of research around older children and the comparison across attainment level is where this research will take place. It has been established throughout the literature that overall manipulatives are an effective tool for improving mathematical understanding and now it is important to discover whether this holds true for students of all ages and attainment levels or whether manipulatives are more effective for specific groups of students.

References

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