In what ways can concrete resources support the learning and progress of children in maths, who are lower ability or have SEN?

Simon Willoughby

BA (Hons) Primary Teaching Studies 5-11 with recommendations for QTS.

Abstract

The purpose of this research paper was to investigate how children with Special Educational Needs (SEN) and children who are lower ability use concrete resources to support their learning and progress in mathematics. The small-scale survey was carried out in a large, three-form entry junior primary school in Lincolnshire. Three separate, nonparticipant observations were conducted of 6 children – 2 children with Education, Health and Care Plan (EHCP) and 4 children on the SEN register for cognition and learning and social, emotional, and mental health difficulties– whilst they completed pure calculation tasks without concrete resources, with the use of concrete resources and concrete resources with support from the class teaching assistant (TA). This was triangulated with an online questionnaire which gathered the views of Key Stage 2 (KS2) teachers on the use of concrete resources and whether they thought it supported children’s learning. The research findings show that concrete manipulatives are an effective tool to support learning. Though successful, instructional variables showed differences in children’s performance, suggesting that this could cause inconsistencies with the study. Most questionnaire responses showed that concrete resources are highly valued as a tool to support children’s mathematical learning, but also recognised that they could cause a distraction.

Introduction

Mathematical manipulatives are considered to increase attention of children with SEN (Jimenez & Stanger, 2017) if they are introduced in a sequence framework that is gradual (Bouck, Satsangi & Park, 2018). There is a strong connection between the use of manipulatives with a framework of instruction, across the literature; however, limited research exists of the use of manipulatives in isolation. Virtual manipulatives are mentioned consistently throughout literature, with studies that compare virtual and concrete manipulatives to support children with SEN (Burns & Hamm, 2011; Satsangi et al., 2016; Hunt et al., 2011). My research question asked: In what ways can concrete resources support the learning and progress of children in maths who are lower ability or have SEN? The word ‘concrete’ caused difficulties in the search for literature and perhaps this was because concrete resources were defined as tools, paper or everyday materials, used to provide a teaching aid (Drews & Hansen, 2007) which suggests it could apply to other areas of the curriculum such as line guides in English and a word bank in Science, and not objects designed to show abstract mathematical problems (Moyer, 2001). Therefore, the search was altered to include the word ‘manipulative’ as this term was used commonly in literature focusing on the use of concrete resources within a framework such as Concrete, Pictorial and Abstract (CPA). This was possibly to avoid confusion between concrete resources and the concrete step of instruction. Griffiths et al., (2017) define manipulatives as items that can be touched and moved to develop understanding in mathematical concepts. This definition starts to distinguish the possible difference between concrete resources and manipulatives, with concrete resources being used in a set way and maths manipulatives being used in different contexts. This small-scale project involved children and staff in a large, three-form entry junior academy in Lincolnshire.

Literature Review

The literature review explores and examines existing literature on the use of manipulatives to support the learning and progress of children in maths, with a particular focus on those who are lower ability or have SEN. Manipulatives are objects that can be touched and used for instruction (Liggett, 2017) Despite Ofsted (2012) emphasising greater focus on children’s mathematical concepts to rely on less teaching by rote, coupled with a requirement for learners to move fluently between representations of mathematical ideas, using manipulatives (Department for Education, 2013), some schools have been criticised for not using manipulatives well enough to aid conceptual understanding (Ofsted, 2012). Guidance in the National Curriculum (DfE, 2013) mentions the use of concrete objects in Key Stage One (KS1) which differs from the KS2 recommendations, with the use of arrays and number lines as representations, but not manipulatives (Griffiths et al., 2017). This shows that the guidance provides little clarity for the use of manipulatives for children in KS2. Furthermore, the UK government supports the Shanghai approach to mathematics which uses a higher number of visualisations, as opposed to the Singapore method which relies heavily on the CPA method. This conflicts with current policies influenced by Teaching for Mastery (2015) with funding from the National Centre for Excellence in Teaching Mathematics (NCETM). This possibly suggests the use of manipulatives is not new and perhaps the heightened exposure of using them is due to influence by other high-performing countries such as Shanghai, who use the CPA method successfully (Lindorff et al., 2019).

The correlation between using mathematical manipulatives and the CPA approach is very common across the literature (Bouck et al., 2014; Bouck, 2015; Bouck, Park & Nickell, 2017; Cass et al., 2003; Flores, 2009; Flores et al., 2014; Root et al., 2017; Satsangi & Bouck, 2015; Satsangi et al., 2016). The CPA model helps to bridge the gap between conceptual and procedural understanding, and this can be achieved by teaching a procedure such as addition with regrouping using manipulatives like place value counters and fading these scaffolds until children can fluently use numbers only (Flores et al., 2014). The ‘Mastery’ approach to teaching mathematics became central to UK policy in 2015, which was influenced by the Singaporean approaches to maths involving the CPA approach (Boylan et al., 2018). However, this approach was informed heavily by Bruner’s work (1966), as he believed the abstract ways of learning mystified children; therefore, used the CPA method, as a means of scaffolding learning, so the children were able to build on each stage to reach a better understanding of the concepts used. At first glance it seems like the theory is quite recent owing to the increased coverage; however, this shows that this theory was already well established (Hoong et al., 2015).

Virtual manipulatives are representations of concrete manipulatives in digital form that can be used on handheld devices, interactive whiteboards, and computers (Bouck et al., 2018) and are commonly used on technologies that are readily available for primary school children to use (Satsangi & Miller, 2017). A systematic review into literature on mathematics manipulatives to support children with disabilities highlighted an emerging trend in the use of virtual manipulatives (Bouck & Park, 2018).However, it is worthy of note that four out of the six studies on virtual manipulatives, referred to by Bouck and Park (2018), were written by Bouck and associates (Bouck et al., 2014; Satsangi & Bouck, 2015; Bouck et al., 2015). Furthermore, Bouck (2012) teaches assistive technology and is active in the Technology and Media Division of the Council for Exceptional Children, which suggests the use of virtual manipulatives is of interest to her, potentially affecting the reliability of this claim. This outlook is now more enhanced due to more than 1.3 million devices being provided to schools in the UK, to aid with online learning delivery during COVID 19 (DfE, 2020) together with children’s increased confidence in using technology (Organisation for Economic Co-operation and Development, 2020), highlighting that there could be potential scope for this within the primary classroom. It is argued the use of virtual manipulatives offers children with learning difficulties options to differentiate an application to suit their needs (Satsangi & Bouck, 2014) such as changing the colour of the screen which is suitable for children with Irlen Syndrome, for example. This, combined with built in scaffolds in applications such as Maths Bot, allows the children to drag and drop place value counters, exchange, and regroup, without the support of an adult to enable independence (Satsangi & Miller, 2017).

There is an exploratory link between the use of virtual manipulatives combined with the CPA approach (Strickland, 2017; Satsangi et al., 2016; Moyer-Packenham & Suh, 2012); however, for the virtual manipulatives to take on a dual role, within the CPA process, children must be able to physically manipulate an object, by moving it with a mouse or using a finger on a touch screen device, in addition to being provided as a visual representation on the screen (Bouck, 2017). It is argued, however, guided practice and explicit instruction is needed to help embed the use of virtual manipulatives (Shin, 2017) which corroborates with a study by Meyer et al., (2019) who claim pretraining of manipulatives significantly reduced cognitive load, when it came to using them, because the children had already had experiences with them alongside adult instruction.

Carbonneau et al., (2013) argue there is a significant inconsistency in the literature based on children with learning difficulties, using manipulatives to support their learning, due to instructional variables that could possibly affect the overall outcome of the research. For example, the type of manipulative used, the sample size and the amount of adult support. Lafay (2019) conducted a similar systematic review of literature however, focussing on children with mathematical learning disabilities (MLD) instead of children with a range of disabilities, a criticism expressed by Lafay (2019) towards the work of Bouck and Park (2018), which may question the generalisability of the research. However, on further inspection, the studies looked at by Carbonneau et al., (2013) were outdated, questioning whether the inconsistency claim is reliable. A recent study by Bouck et al., (2020) showed explicit modelling by the teacher was used with lots of opportunities to practise application-based manipulatives, whereas in a study by Satsangi, Hammer and Hogan (2018) scaffolds and prompts were only given to children who struggled with problem solving during the session, highlights inconsistent methods were used across two comparable studies.

It is reasoned that mathematics manipulatives are sometimes not as effective due to contextual variables (Donovan & Alibali, 2021) with some lessons not connecting the manipulatives and the concepts properly when taught without a framework such as CPA (Furham, 2017). It is thought manipulatives that are as simple as possible, such as coloured unifix cubes without unnecessary mentions of the real world, promotes the best learning (Laski et al., 2015), as research shows if the manipulative is interesting to play with, it creates a distraction and prevents children from making relevant connections with the manipulative and represented mathematical concept (Donovan & Alibail, 2021). This was shown in a study by McNeil et al., 2009 who set up an experiment with two conditions, one group were assigned play notes and coins and the other group did not have any resources, and both groups were asked to answer some money problems. It was found that the realistic money notes, despite being aimed to activate real-world knowledge, hindered performance. It was argued that the notes and coins looked like play money, which could hinder performance in schoolwork, as the children’s knowledge of play activities would be active (McNeil et al., 2009).

Despite the steady increase in literature advocating children’s use of finger-counting over the last 20 years, Stegemann and Grunke (2014) are cautious in their assessment of this because it causes children to complete arithmetic at a slower pace. The importance of finger-counting in calculations is thought to be emphasised by neuroscience studies; however, using fingers to calculate can become a habit and be relied on too much (Moeller et al., 2011). A qualitative study asked 34 teachers for their views on finger-counting in which participants reported that the children in their classes, who used finger-counting as a tool for learning, had poor memory and high anxiety, but also recognising that fingercounting should be used until children have understanding behind arithmetic operations (Mutlu et al., 2020). The generalisability could be questioned in this study, as most of the participants were special educational needs teachers, but this provided an interesting insight into why some teachers encourage or discourage children to use their fingers. After closely reviewing the literature regarding lower ability and SEN children using maths manipulatives to support their learning, there are several factors involved which possibly creates complexity. The strong correlation between the use of manipulatives as part of the CPA approach was evident in research, with limited research looking at maths manipulatives in isolation, which influenced my own research project. The emergence of virtual manipulatives perhaps gives teachers’ alternatives in their classrooms, as the research shows they have benefits for SEN children. However, it was found that contextual and instructional variables could affect the reliability of some studies into the use of manipulatives, because the instruction from adults, type of manipulatives and the sample sizes varied considerably.

Methodology

This research project was a small-scale survey which concentrated on two samples in one key stage in the same school. The decision was influenced by Blaxter et al., (2010); Bell and Waters (2018) because I wanted to collect quantitative data by analysing the task completed by the children and qualitative data from my observation notes and the questionnaire completed by sample two, which provided me with the same information about all the cases in one sample. I did consider a case study as my methodology, as it shared some similarities with a survey, in that it seeks to study an individual or a group (Cohen et al., 2018); however, a case study often focusses on an existing problem (Blaxter et al., 2010) whereas the research question was designed to identify the current situation (Flick, 2015) and this shows how methodology approaches can differ. A survey methodology enabled standardisation, by asking the same questions in precisely the same way and provided the same task for participants to complete (Sapsford, 1999). This is a clear strength because it provided consistent answers, even though this limited the generalisability of my overall findings (Samuel et al., 2001), I was able to look at the use of maths concrete resources in KS2 in greater depth.

Setting

The project was carried out in one, large junior academy in Lincolnshire, where a mastery maths approach has been used since September 2019. The setting consisted of thirteen classes, with threeform entry in three-year groups and four-form entry in Year 6. The number of children with special educational needs (SEN) is above average and the proportion of children who are supported by pupil premium funding is well above average, at 51% (Ofsted, 2018).

Sample

Sample one of the project comprised of the KS2 teachers with thirteen teachers in total. Three teachers within the sample held a Teaching and Learning Responsibility (DfE, 2020), in addition to their class responsibility. One teacher in the sample had taught in the school since 2001, three teachers joined the school in 2019 and the rest of the sample had taught at the school for five years or more. Sample one were approached through an email to their work email address, with an explanation of the project, in line with Ethical Guidance for Educational Research by The British Educational Research Association (BERA) (2018) and an attached link to the questionnaire.

Sample two of the project consisted of six, year five children of varying ability. Child P and Child C had Education, Health and Care Plans (EHCP) and were entitled to twenty hours of support, as stipulated in their personalised plans. The two children were working significantly below aged related expectations for year five. The remaining four children in the sample were on the school SEN register for cognition and learning and social, emotional, and mental health difficulties. The children were secure at Year 2 and working towards Year 3 expectations in maths.

Methods

Triangulation of two methods was considered and applied in the research in an attempt to show validity and reliability (Guthrie, 2013; Bell & Waters, 2018; Daniel & Harland, 2017). Sample one were provided with the same questions to answer and sample two were given the same tasks to complete at the same time. This agrees with Alridge and Levine (2001) who argue that a survey methodology has a distinct strategy, as it is used to collect the same data, on the people in a sample. This shows that my chosen methodology fitted within my two methods; non participant observation and questionnaire.

Non-participant observation

Firstly, I had planned to conduct a participant observation, where I would take an active part in the activity, whilst observing at the same time. My initial thoughts were shaped by Sharp (2012) who believes participant observations provides the observer richer opportunities to gain additional insights, by experiencing things first-hand. However, Newby (2014) offers part criticism to participant observations as he believes when the observer is actively involved, the way the observer views an issue could change and influence the interpretation of collected data. The decision to conduct a third observation, as the children completed a task with support from a teaching assistant (TA) was to collect some qualitative data without taking an active part myself. Nonetheless, the viewpoint shared by Newby (2014) was balanced with his suggestion of a non-participant observation, which proved to be a viable alternative, as the group already knew me and were aware they were being observed; however, this idea differed from Sharp (2012) because the observation was conducted at a distance from the group. As I knew the group well and already had a bond and trust with them (Newby, 2014), I decided to conduct three, small-scale nonparticipants observations across three consecutive days. By observing in this way, I was able to be socially invisible, as I did not engage with any of the participants during the activity (Denscombe, 2014); however, during the first observation two participants put their hand up to tell me they had finished. This suggested that the participants were seeking approval and still viewed me as an active part of the activity, despite being sat away from the group (Denscombe, 2014). A limitation of collecting qualitative data in observational research was the difficulty of collecting data in real time (Queiros et al., 2017); thus, I found writing notes, whilst observing and listening, more of a challenge than I first anticipated. 

Questionnaire

Due to the questionnaire being accessible online, participants were able to complete it at a time that suited them (Graham, 2008), which was a clear strength when comparing this method against an interview, as this would require a specific time for it to take place. Interestingly, Denscombe (2014) emphasises the strength of a questionnaire that has been created on a website, due to its simplicity, as once completed, the participant presses submit and it is finished. Despite highlighting this as a strength, Denscombe (2014) does raise a valid limitation that the participant must come to the website themselves, instead of completing a questionnaire directly in an email response. This possible limitation was addressed by sending a website link in an email to the participants, so they could click on the link and be taken directly to the questionnaire. Another limitation of an online questionnaire is potential bias for those participants without the use of a computer (Arthur et al., 2012); but a device at the setting was readily available for the participants to use, so this was not an issue. A further limitation was the low number of responses to my questionnaire, which had potential to affect the validity of my research (Arthur et al., 2012).  Upon reflection, allocating more time to the data collection phase of the research could have potentially elicited more responses. In the initial email sent to participants, I asked for it to be completed within 5 days; therefore, with teachers’ heavy workload, perhaps more time would have been an advantage. Possibly another limitation, which is closely linked with the response rate, is anonymity. Without knowing who had responded to the questionnaire, I was unable to send an individual reminder to participants who had not yet responded (Graham, 2008). However, attempts were made to overcome this limitation by sending a polite reminder email to all participants and managed to receive one more response. Counteracting this limitation is the ethical considerations I have made to ensure participants remained anonymous. Providing anonymity allowed the participants to share their experiences in confidence (Curtis et al., 2014) which was an overall strength of the questionnaire. 

Ethical considerations

As the non-participant observation involved observing children in the classroom, it was important to be compliant with the research by (BERA, 2018), particularly being sensitive to children who may have wanted to opt out the study at any time. I reassured sample two, in child friendly terms, about their right to withdraw from the task at any time. Permission was gained from my research tutor before starting the project to ensure it followed the guidelines contained within the Bishop Grosseteste University (2019) Research Ethics Policy. The Executive Headteacher granted me permission to carry out the research within the setting, without the need for parental permission.

Analysis and discussion of findings

Pupil non-participant observation

During the first observation, where no manipulatives were made available, four children used their fingers to support them to answer the pure calculations – this consisted of two 3-digit additions and two 3-digit subtraction problems and two children used their 15cm ruler as a number line to count on and back. The use of a ruler was interesting because it was an example of an everyday item (Drews & Hansen, 2007) suggesting this is a concrete resource because it can be used across other subjects and not just maths. Comparisons between finger counting and using a ruler showed that the finger counting was in fact a quicker process. Child C and Child E, who both used their fingers to count, were the first to finish the task. This differed from the literature, as finger counting was seen as restrictive, as it slowed down children’s ability to complete arithmetic (Stegemann & Grunke, 2014).

The children were provided with a set of manipulatives (laminated number squares, counters, number beads, cubes and a Google Chromebook with the Maths Bot website loaded) but with no adult instruction. Interestingly, Child J still used his fingers, and when children reminded him about the manipulatives, he said ‘Fingers’ whilst wiggling them. Child P immediately grabbed a handful of counters and started to stack them on the table. During wet play activities, I have observed children using the counters to draw around and stacking them up into large towers. Although the counters did not have connotations of the real world (Laski et al., 2015) the counters were a distraction towards Child P’s learning because she was interested in playing with them. As the counters had dual uses in the classroom, Child P was possibly unable to make connections with the counters and what they represented as a mathematical concept (Donovan & Alibail, 2021) and consequently, she attempted two questions and answered both incorrectly. The children did not interact with the virtual observation, opting to use concrete manipulatives instead. This was surprising as research showed that built in scaffolds within virtual manipulatives like Maths Bot, supported independent learning (Satsangi & Miller, 2017) which suited the conditions of the observation. Research suggests that virtual manipulatives need to be introduced alongside explicit instruction and opportunities for guided practice (Shin, 2017), which could suggest why the children opted to use the concrete manipulatives instead because they already have experiences of using them, thus reducing cognitive load (Meyer et al., 2019).

Table 1 shows the children’s scores when completing a set of four pure calculations using manipulatives, compared with their scores when answering another set of pure calculations with manipulatives and TA support.

The literature review debated the idea that instructional variables could result in inconsistencies – something that was argued by Carbonneau et al., (2013). This agrees with my observation, as Table 1 shows five out of six children in the group performed better in the task when they received support from the TA – ranging from scaffolding through modelled steps and verbal feedback.  Furthermore, Child O used cubes in the first observation and scored 2 marks, whereas Child E used a number square and scored 2 marks also. This possibly highlights another difference in instructional variables, adding inconsistency to the results

Table 2 shows a comparison of data between Child J, who consistently used finger-counting across the three observations, and Child P and Child O who used a range of manipulatives.

The findings agree in part with Mutlu et al., 2020 because the observation looked at the use of manipulatives in isolation and it is argued that finger-counting should be used until children have understanding behind the arithmetic operations used; however, it is argued that a framework such as CPA can help to bridge the gap between conceptual and procedural knowledge (Flores et al., 2014); however, this claim was not tested due to the nature of the observation. The results do not draw conclusive thoughts about finger-counting being more effective than manipulatives; however, the fact Child J still used finger-counting during observations when manipulatives were available implies that it has possibly become a habit (Moeller et al., 2011) and the reliance on fingercounting has not produced better results.

Teacher questionnaires

Table 3 shows the types of manipulatives used in teachers’ classrooms to support learning in maths lessons. It was reported that manipulatives designed specifically for maths were used the most by teachers’ (number squares, number lines, dienes, and Numicon) in comparison to counters and Lego which have dual uses (Drews & Hansen, 2007) which could explain why these options were chosen less. The sample for this questionnaire was in a school where a Mastery Maths approach was followed and the teachers are expected to incorporate manipulatives within their lessons; therefore, this data could be questioned as the teachers may be answering in line with school expectations.

Three teachers opted to provide alternative manipulatives that are used in their classrooms which included items such as: fraction walls, straws, pasta, wooden sticks, smarties, buttons, lollypop sticks and food items. I was surprised to see no mention of virtual manipulatives, despite research suggesting that it can support children with SEN (Bouck & Park 2018; Satsangi & Miller, 2017) However, this is perhaps because teachers at my setting use virtual manipulatives to model concepts on the Interactive Whiteboard and the children replicate this with concrete manipulatives, as they are readily available for all classes to use.

When asked how the children used manipulatives in maths, the comments were mixed. One teacher commented that the maths resources were kept next to the maths learning wall for the children to use as and when; however, interestingly, this teacher currently teaches in Year 6 which possibly suggests that the older children could have autonomy when choosing manipulatives because they are confident in using them. In some cases, there was a correlation between how the manipulative was used by the children and the teachers’ view of manipulatives in general. For example, two Year 5 teachers stated that manipulatives do distract children’s learning but when answering how the children use manipulatives, they both said the children did this independently. Although the generalisability could be questioned here due to it being views from the same year group, it does possibly suggest that the distraction could be because the children need more opportunities for guided practice, before using the manipulatives independently (Shin, 2017).

Conclusions and recommendations

Including existing literature with analysis of my research findings, provides quite a clear view that concrete manipulatives can have a positive impact on SEN and lower ability children’s learning and progress. The findings from the research project have highlighted several implications for my own teaching practice in the future. I will strive to ensure children have opportunities to use manipulatives during maths but being mindful that opportunities for guided practice are planned in, so that children can learn how to use them effectively. With the hope that, once the children have used the manipulatives for a sustained period, they will be able to use these with increasing independence, something that a Year 6 teacher stated in the questionnaire. The findings showed that concrete resources, which have dual purposes such as counters, could act as a distraction. I am now aware of this and would possibly replace counters with Cuisenaire rods. Despite low engagement with the virtual manipulative during my non-participant observation, I would trial alternative virtual manipulatives, as research suggests there is scope to differentiate this, and I would be interested to test this further.

As the literature review highlighted, there were strong links between the use of manipulatives and an instructional framework such as CPA. Although the activity did not follow a framework, I think the children were competent at using the manipulatives because manipulatives are a key component to Mastery Maths lessons which is taught throughout the setting. From experience at two placement schools, where a Mastery Maths approach was not used, manipulatives were not actively used by children. Therefore, introducing manipulatives at a setting which does not use a Mastery Maths approach would help to explore this topic further.

Upon reflection, this research could have been improved by increasing the focus group to possibly include children with a range of abilities, to make comparisons between children who were working at age related expectations, against lower ability children, Currently I can only tentatively suggest that manipulatives are effective for children who are lower ability or have SEN, due to the sample size of my study. I could have explored and compared KS2 with KS1 by asking teachers at the infant side to complete the questionnaire to make further comparisons.

Despite the questionnaire generating positive comments on the use of manipulatives, it is important to point out that only six members of teaching staff completed the questionnaire. Teachers are expected to use manipulatives in the setting, as part of each maths lesson, so their answers may have been influenced by the expectations set by the school. Although the questionnaire provided anonymity, perhaps the question which asked for the current year group you teach may have prompted teachers to write what they think I wanted to hear. If I was to do this research again, I could possibly just ask the participants to name the Key Stage they worked in. Nevertheless, my research has emphasised the need to consider the use of manipulatives carefully, ensuring that the children know how to use them effectively, with the possibility of introducing virtual manipulatives to support individual needs.

References

Aldridge, A. & Levine, K. (2001). Surveying the Social World: Principles and Practice in Survey Research. Open University Press.

Arthur, J., Waring, M. & Coe, R. (2012). Research Methods and Methodologies in Education. SAGE Publications. Bell, J. & Waters, S. (2014). Doing your research project: A guide for first time researchers. McGraw-Hill Education.

BERA (2018). Ethical Guidelines for Educational Research. Retrieved from https://www.bera.ac.uk/wp-content/uploads/2018/06/ BERA-Ethical-Guidelines-for-Educational-Research_4thEdn_2018. pdf?noredirect=1

Bishop Grosseteste University (BGU). (2017). Research Ethics Policy. Retrieved from http://www.bishopg.ac.uk/wp-content/ uploads/2017/06/Research-Ethics-Policy-2017.pdf

Blaxter, L., Hughes, C. and Tight, M. (2010). How to Research. Open University Press. Bouck, E. and Park, J. (2018). A Systematic Review of the Literature on Mathematics Manipulatives to Support Students with Disabilities. Education and Treatment of Children, 41 (1), 65-106. https:// doi:10.1353/etc.2018.0003

Bouck, E., Flanagan, S, Miller, B. and. Bassette, L. (2012). Technology in Action, Journal of Special Education Technology, 27 (4), 47-57. https://doi:10.1177/016264341202700404

Bouck, E., Flanagan, S. and Bouck, M. (2015). Learning area and perimeter with virtual manipulatives. Journal of Computers in Mathematics and Science Teaching, 34 (4), 381-393.

Bouck, E., Park, J. and. Nickell, B. (2017). Using the concrete-representational-abstract approach to support students with intellectual disability to solve change making problems. Research in Developmental Disabilities, 60, 24–36.

Bouck, E., Park, J., Levy, K. and. Cwiakala, K. (2020). App-base manipulatives and explicit instruction to support division with remainders. Exceptionally, 28 (1), 45-59. https://doi.org/10/1080/0 9362835.2019.1586709

Bouck, E., Satsangi, R. & Park, J. (2018). The Concrete-Representational- Abstract Approach for Students With Learning Disabilities: An Evidence-Based Practice Synthesis, Remedial and Special Education, 39 (4), 211-228. https://doi. org/10.1177/0741932517721712

Bouck, E., Satsangi, R., Doughty, T. & Courtney, T. (2014). Virtual and concrete manipulatives: A comparison of approaches for solving mathematics problems for students with autism spectrum disorder. Journal of Autism and Developmental Disabilities, 44 (1), 180–193. https://doi:10.1007/s10803-013-1863-2

Boylan, M., Maxwell, B., Wolstenholme, C, Jay, T. & Demack, S. (2018). The Mathematics Teacher Exchange and ‘Mastery’ in England: The Evidence for the Efficacy of Component Practices. Education Sciences, 8, 202. https://doi:10.3390/educsci8040202

Burns, B. & Hamm, E. (2011). A Comparison of Concrete and Virtual Manipulative Use in Third-and Fourth-Grade Mathematics, School Science and Mathematics, 111 (6), 256-261. https://doi. org/10.1111/j.1949-8594.2011.00086.x

Carbonneau, K., Marley, S. and. Selig, J. (2013). A Meta-Analysis of the Efficacy of Teaching Mathematics with Concrete Manipulatives. Journal of Educational Psychology, 105 (2), 380-400. https:// doi:10.1037/a0031084

Cass, M., Cates, D., Smith, M. & Jackson, C. (2003). Effects of manipulative instruction on solving area and perimeter problems by students with learning disabilities. Learning Disabilities Research & Practice, 18 (2), 112–120. https://

Cohen, L., Manion, L. &  Morrison, K. (2017). Research Methods in Education. Taylor & Francis Group. Curtis, W., Murphy, M. & Shields, S. (2014). Research and Education. Routledge.

Daniel, B. & Harland, T. (2017). Higher Education Research Methodology: A Step-by-Step Guide to the Research Process. Routledge.

Denscombe, M. (2014). Good Research Guide. McGraw-Hill Education.

Denscombe, M. (2014). The Good Research Guide: For Small-scale Research Projects. McGraw-Hill Education

Department for Education. (2012). Mathematics: made to measure. HMSO.

Department for Education. (2013). National curriculum in England: Primary curriculum. HMSO.

Department for Education. (2020). Guidance: Get help with teaching during coronavirus (COVID-19). HMSO.

Department for Education. (2020). School teachers’ pay and conditions document 2020 and guidance on school teachers’
pay and conditions. HMSO. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/
file/920904/2020_STPCD_FINAL_230920.pdf

Donovan, A. & Alibali, M. (2021). Toys or Math Tools: Do Children’s Views of Manipulatives Affect Their Learning, Journal of Cognition and Development, 22 (2), 281-304. https://doi.org/10.1080/152483372.2021.1890602

Drews, D. (2007). Using Resources to Support Mathematical Thinking, Primary and Early Years. Learning Matters Limited.

Flores, M. (2009). Teaching subtraction with regrouping to students experiencing difficulty in mathematics. Preventing School Failure, 53 (3), 145–152.

Flores, M., Hinton, V. and Schweck, K. (2014). Teaching multiplication with regrouping to students with learning disabilities. Learning Disabilities Research & Practice, 29 (4), 171-183. https://doi:10.1111/1drp.12043

Furham, C. (2017). Making Sense with Manipulatives: Developing Mathematical Experiences for Early Childhood Teachers, Education and Culture, 33 (2), 67-86.

Graham, B. (2008). Developing a Questionnaire. Bloomsbury Publishing Plc.

Guthrie, G. (2013). Basic Research Methods: An Entry to Social Science Research. SAGE.

Hoong, Y. L., Kin, W. H.& Pien, L.C. (2015). Concrete-Pictorial-Abstract: Surveying its origins and charting its future, The Mathematical Educator, 16 (1), 1-18. Retrieved from http://math.nie.edu.sg/ame/matheduc/tme/tmeV16_1/TME16_1.pdf

Hunt, A., Nipper, K. & Nash, L. (2011). Virtual vs. Concrete Manipulatives in Mathematics Teacher Education: Is One Type More
Effective Than the Other, Current Issues in Middle Level Education, 16(2), 1-6. Retrieved from https://files.eric.ed.gov/fulltext/EJ1092638.pdf

Jimenez, A. & Stanger, C. (2017). Math manipulatives with severe intellectual disability: A survey of special education teachers, Physical Disabilities Education and Related Services, 36, 1-12. https://doi:10.14434/pders.v36i1.22172

Lafay, A., Osana, H. & Valat, M. (2019). Effects of Interventions with Manipulatives on Immediate Learning, Maintenance and Transfer in Children with Mathematics Learning Disabilities: A Systematic Review. Education Research International, 2019 (1). https://doi:10.1155/2019/2142948

Laski, E., Jordan, J., Daoust, C. & Murray, A. (2015). What Makes Mathematics Manipulatives Effective? Lessons From Cognitive
Science and Montessori Education, SAGE Open, 5 (2), 1-8. https://doi.org.10.1177/2158244015589588

Liggett, R. (2017). The Impact of Use of Manipulatives on the Math Scores of Grade 2 Students, Brock Education Journal, 26 (2), 87-101. Retrieved from https://files.eric.ed.gov/fulltext/EJ1160704.pdf

Lindorff, A., Hall, J. & Sammons, P. (2019). Investigating a Singapore Based Mathematics Textbook and Teaching Approach in Classrooms in England. Frontiers in Education, 37 (4). https://doi:10.3389/
feduc.2019.00037

Manca, F. & Meluzzi, F. (2020). Strengthening Online Learning When Schools Are Closed: The Role of Families and Teachers
in Supporting Students During the COVID-19 Crisis. OECD.

McNeil, N., Uttal, D, Jarvin, L. & Sternberg, R. (2009) Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19, 171-184. https://doi.org/10.1016/j.learninstruc.2008.03.005

Meyer, A., Omdahl, K. & Makransky, G. (2019). Investigating the effect of pre-training when learning through immersive virtual reality and video: A media and methods experiment. Computers and Education, 140 (103603). https://doi.org/10.1016/j.compedu.2019.103603

Moeller, K., Martignon, L., Wessolowski, S., Engel, J. & Nuerk, C-H. (2011). Effects if Finger Counting on Numerical Development – The Opposing Views of Neurocognition and Mathematics Education. Front Psychol 328 (2). https://doi.org/10.3389/fpsyg.2011.00328

Moyer-Packenham, S. and Suh, M. (2012). Learning mathematics with technology: The influence of virtual manipulatives on different achievement groups. Journal of Computers in Mathematics and Science Teaching, 31 (1), 39-59.

Moyer, P. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational
Studies in Mathematics, 47 (2), 175-197. https://doi.org/10.1023/A:1014596316942

Mutlu, Y., Akgun, L. & Akkusci, Y. (2020). What Do Teachers Think About Finger-Counting. International Journal of Curriculum and
Instruction, 12 (1), 268-288. Retrieved from https://www.ijci.wcciinternational.org

Newby, P. (2014). Research Methods for Education. Routledge.

Ofsted. (2018). Green Field Primary School. https://files.ofsted.gov.uk/v1/file/2761513

Queriros, A., Faria, D. & Almeida, F. (2017). Strengths And Limitations Of Qualitative And Quantitative Research Methods. European Journal of Education Studies, 3 (9), 369-387. https://doi:10.5281/zenodo.887089

Root, J., Browder, D., Saunders, A. & Lo, Y.-Y. (2017) Schema- based instruction with concrete and virtual manipulatives to teach problem solving to students with autism. Remedial and Special Education, 38 (1), 42–52.

Samuel, J., Krueger, B., Hubbard, C. & Smith, A. (2001). An Assessment of the Generalizability of Internet Surveys. Social
Science Computer Review, 19(2), 131-145. https://doi. org/10.1177/089443930101900201

Sapsford, R. (1999). Survey research. Sage.

Satsangi, R. and Bouck, E. (2015). Using virtual manipulative instruction to teach the concepts of area and perimeter to secondary students with learning disabilities. Learning Disability Quarterly, 38, 175–186.

Satsangi, R.& Miller, B. (2017). The case for adopting virtual manipulatives in mathematics education for students with disabilities, Preventing School Failure: Alternative Education for Children and Youth, 61 (4), 303-310. https://doi:org/10.1080/1045988X.2016.1275505

Satsangi, R., Bouck, E., Taber-Doughty, T., Bofferding, L. & Roberts, C. (2016). Comparing the effectiveness of virtual and concretemanipulatives to teach algebra to secondary students with learning disabilities. Learning Disability Quarterly, 39 (4), 240-253. https://doi.10.1177/0731948716649754

Satsangi, R., Hammer, R. & Hogan, C. (2018). Studying virtual manipulatives paired with explicit instruction to teach
algebraic equations to students with learning disabilities. Learning Disability Quarterly, 41 (4), 227-242. https://doi:org.10.1177/0731948718769248

Sharp, J. (2012). Success with Your Education Research Project. Learning Matters.

Shin, M. & Bryant, D. (2015). A Synthesis of Mathematical and Cognitive Performances of Students with Mathematics Learning
Disabilities. Journal of Learning Disabilities, 48 (1), 96-112. https://10.1177/0022219413508324

Strickland, T. (2017). Using the CRA Strategy to Develop Conceptual and Procedural Knowledge of Quadratic Expressions,
Teaching Exceptional Children, 49 (2), 115-125. https://doi. org/10.1177%2F0040059916673353

Tournaki, N., Bae, Y. and Kerekes, J. (2008). Rekenrek: A manipulative used to teach addition and subtraction to students with learning disabilities. Learning Disabilities: A Contemporary Journal, 6 (2), 41-59.