How can mathematics learning in primary school be facilitated?
A recent study conducted by the University of Geneva (UNIGE), Switzerland, had shown that our everyday knowledge strongly influences our ability to solve problems, sometimes leading us into making errors.
This is why UNIGE, in collaboration with four research teams in France, has now developed an intervention to promote the learning of maths in school. Named ACE-ArithmEcole, the programme is designed to help schoolchildren surpass their intuitions and informal knowledge, and rely instead on the use of arithmetic principles.
And the results are surprising. More than half (50.5%) of the students who took part in the intervention were able to solve difficult problems, as compared to only 29.8% for pupils who followed the standard course of study.
The corresponding study can be found in the journal ZDM Mathematics Education..
From the age of 6 or 7, schoolchildren are confronted with mathematical problems involving additions and subtractions.
Instinctively, they use mental simulations of the situations described by the problems in order to come up with solutions. But as soon as a problem becomes complex, recourse to this representation using imagery becomes impossible or leads the student into error.
“We reflected on a method that would enable them to detach themselves from these initial representations and that would foster the use of abstract principles of arithmetic,” explains Katarina Gvozdic, a researcher at the Faculty of Psychology and Education (FPSE) at UNIGE.
This approach, based on semantic re-encoding, spurs students to achieve knowledge in arithmetic at an early age.
It was put into practice by teachers in a primary school arithmetic course called ACE-ArithmEcole that substituted the standard arithmetic curriculum.
So that intuitive mental representations will give way to mathematical representations
At the end of the school year, the UNIGE team evaluated ten classes of children aged 6 to 7 in France (second grade of primary school).
In five classes, known as the control classes, the teachers had taught maths in a conventional way.
In the other five classes, they had implemented the ACE-ArithmEcole intervention which encouraged students to favour abstraction.
“To get the students to practice semantic re-encoding, we provided them with different tools such as line diagrams and box diagrams,” says Emmanuel Sander, professor at the Department of Education of the FPSE at UNIGE.
The idea is that when they read a problem, such as “Luke has 22 marbles, he loses 18.
How many marbles does he have left?,” the pupils should detach themselves from the idea that subtraction always consists in a search for what remains after a loss, and should instead manage to see it as the calculation of a difference, or a distance that has to be measured. It’s all about showing students how to re-encode this situation.”
After a year of teaching based on this practice, the UNIGE researchers evaluated their intervention by asking the pupils to solve problems that were divided into three main categories: combine (“I have 7 blue marbles and 4 red marbles, how many do I have in all?”), comparison (“I have a bouquet with 7 roses and 11 daisies, how many more daisies do I have than roses?”) and change problems (“I had 4 euros and I earned some more.
Now I have 11. How much did I earn?”). In each of these categories, there were some problems that were easy to represent mentally and to solve using informal strategies, and others that were difficult to simulate mentally and for which it was necessary to have recourse to arithmetic principles.
At the conclusion of the tests, researchers observed undeniable results. Amongst students who had learned to solve mathematical problems with the ACE-ArithmEcole method, 63.4% gave correct answers to the problems that were easy to simulate mentally, and 50.5% found the answers to the more complex problems.
“In contrast, only 42.2% of the pupils in the standard curriculum succeeded in solving simple problems, and only 29.8% gave the right answer to the complex problems,” exclaims Katarina Gvozdic. “Yet their level measured on other aspects of maths was exactly the same,” adds Emmanuel Sander.
Amongst students who had learned to solve mathematical problems with the ACE-ArithmEcole method, 63.4% gave correct answers to the problems that were easy to simulate mentally, and 50.5% found the answers to the more complex problems. Image is credited to UNIGE.
This discrepancy can be explained by the frequent recourse to the use of mathematical principles rather than to mental simulations by the students who had taken part in the ACE-ArithmEcole intervention.
“Thanks to the representational tools that had been offered to them and to the activities they had recourse to in class, the students learned to detach themselves from informal mental simulations and avoid the traps they lead to,” comments Katarina Gvozdic enthusiastically.
The results are promising and they provide a foundation for promoting abstraction and breaking away from mental simulations.
“Now we want to extend this teaching method to higher classes, incorporating multiplication and division as well,” continues Gvozdic.
“Moreover, the method could be applied to other subjects — such as science and grammar — for which intuitive conceptions constitute obstacles,” adds Sander.
The idea is to put semantic re-encoding to widespread use in schools and to incorporate it more amply into teaching methods. in the gut-brain connection by influencing how the brain responds to injury.
We investigated if a novel and innovative technology-based education intervention is suitable for supporting the development of early mathematical skills in pupils with Special Educational Needs and Disabilities (SEND).
The intervention is currently being trialed across Malawi, a low-income country in Sub-Sahara Africa, and has been shown to be highly effective at supporting the acquisition of basic mathematical skills with mainstream children in Malawi and the UK (Pitchford, 2015; Outhwaite et al., 2017; Pitchford and Outhwaite, 2017).
As the project scales within Malawi to reach all primary schools (Hubber et al., 2016), in line with the United Nation’s Sustainable Development Goal 4 to “ensure inclusive learning and equitable quality education and promote lifelong learning for all” (United Nations, 2016, p. 5), it is now timely to consider if this innovative technology can also enhance the education of children with SEND to help them reach their full potential.
Pitchford (2015) conducted a pupil-level randomized control trial in a primary school in Malawi with mainstream children attending the first 3 years (standards) of compulsory education, to establish proof of concept that this digital education technology intervention could be effective at raising learning outcomes in Malawi—a country with a history of poor educational attainment and high innumeracy and illiteracy rates (Kadzamira and Rose, 2003). By the end of primary school, <50% of Malawi children have achieved basic competency in mathematics and reading (Milner et al., 2011).
The technology intervention utilizes hand-held tablets to deliver a series of child-centered, self-paced, interactive apps designed to support the acquisition of basic mathematical skills in mainstream pupils. Pupils received the technology intervention on a daily basis in a purpose-built Learning Center—a small classroom separate from the rest of the school. Class teachers implemented the technology intervention to groups of 30 children at a time.
Technical support was provided by the Voluntary Service Overseas (VSO) to teachers implementing the intervention but teachers were instructed to offer no pedagogical support to children whilst interacting with the apps. Children were assessed on psychometric measures of mathematical ability before and after the intervention period.
This enabled performance gains to be quantified over the intervention period and compared across groups of children receiving the technology intervention (intervention group) or regular teacher-led mathematics tuition (control group). Performance gains were taken as a direct measure of learning.
Results showed after just 8 weeks of intervention with the technology, pupils in the first 3 years of primary school in Malawi significantly outperformed those following standard teacher-led mathematics instruction.
This could not be attributed to novelty effects of using the tablet technology as the study included a placebo group of pupils from the same school who accessed the touch-screen tablets with the same dosage as the intervention group but the placebo group interacted with design apps rather than maths apps. Results showed the placebo group made similar gains in mathematics over the duration of the intervention as pupils in the control group, who received standard teacher-led practice.
This demonstrates that the tablet technology alone did not contribute toward the higher maths performance found at post-test in the intervention group, but rather the maths apps that the pupils interacted with throughout the intervention were responsible to improving learning outcomes.
As the intervention scales across Malawi for mainstream pupils it is also being delivered pupils with SEND. It is therefore necessary to evaluate the effectiveness of this intervention for pupils with SEND as this intervention was not designed specifically for SEND pupils, many of who have specific motor and sensory processing difficulties that might make interacting with this technology problematic.
The National Special Needs Education Policy of Malawi describes learners with special educational needs as those “who require special service provision and support in order to access education and maximize the learning process” (Ministry of Education, 2007, p. 6). According to the Ministry of Education in Malawi (2007), pupils with SEND include children who fall into any of the following categories: sensory impairment (vision, hearing, deaf-blind); cognitive difficulties (intellectual, specific disabilities, and gifted and talented); socio-emotional and behavioral difficulties (autism, hyperactivity, and other vulnerable children); and physical and health impairments (spina bifida, hydrocephalus, asthma, and epilepsy).
Approximately 2.3% of children at primary school in Malawi are registered with SEND (Munthali, 2011). The intervention being evaluated in this study requires a high level of sensory processing (visual, auditory, and kinaesthetic) and manual coordination and dexterity to interact with the apps, so some SEND pupils with difficulties in these domains might find interacting with the apps problematic, without modification of the software. As the intervention has been designed to support development of numeracy skills in mainstream pupils, it is therefore vital to assess its suitability for SEND pupils, as software modification may be required for maximize learning potential for SEND pupils, especially those with sensory processing and/or motor difficulties.
High-quality education is deemed critical in helping individuals with SEND to maximize their full potential. However, as the Convention on the Rights of Persons with Disabilities (United Nations, 2007) articulates, there is a complex interaction between individuals with SEND and attitudinal as well as environmental barriers that hinders their full participation in society on an equal basis with others.
In some circumstances, education can be a barrier for children with SEND participating in society if the country’s educational system is not sufficiently geared toward supporting these individuals. Attitudes of teachers can impact both positively and negatively on how pupils with SEND progress through school, and attitudes toward disability in society can influence who gets access to education. Demonstrating that pupils with SEND can learn could be influential in changing attitudes which, in turn, could encourage more SEND pupils to attend and stay in school.
The technology intervention evaluated in this study enables learning to be monitored directly within the apps as children are required to pass a quiz at the end of each topic which requires application of knowledge taught to solve a series of novel questions. Progression throughout the app is not possible unless the quizzes are passed, so passing the quiz for a particular topic is a direct measure of learning. Quantifying progress, especially in relation to mainstream pupils accessing the same intervention, could be instrumental in changing attitudes toward SEND pupils’ capacity to learn. This is particularly poignant in low-income countries, such as Malawi, where there are additional hurdles to accessing education, which result in many children with SEND being kept at home.
Within Malawi, barriers to quality education for children with SEND include poverty, large pupil-to-teacher ratios, distance from home to school—which for families of children with mobility difficulties can be particularly problematic as many do not have wheelchairs or other mobility aides, inadequate learning materials (Hughes et al., 2016), and discrimination within the community (Kelly et al., 2012).
If parents of children with SEND are ashamed of their child’s disability, or do not believe their child is capable of learning, and if there are few opportunities for employment of individuals with SEND upon completion of schooling, parents may opt to withdraw their child from the education system (Mutua and Dimitrov, 2001). It is therefore critical to demonstrate that children with SEND can learn, so they are given the opportunity to access primary education that is provided by the state and teaches fundamental skills, such as numeracy and literacy, which form the bedrock of later learning.
Mobile technologies that deliver apps designed to support the acquisition of basic numeracy and literacy skills in a personalized, self-paced, manner, with minimal need for specialized adult support, could be particularly useful for children with SEND in accessing quality instruction, as they can be deployed in the home as well as the school environment (Melhuish and Falloon, 2010). Such technologies could be particularly advantageous for children with severe mobility difficulties that prevent them from attending school on a regular basis.
Inclusive education interventions are needed to address these barriers and help change attitudes toward the learning capabilities of children with SEND. Within Malawi, to embrace inclusive education, children with SEND are enrolled in mainstream schools, as advocated by international standards and frameworks such as the Salamanca Statements and Framework for Action on Special Needs and Education (UNESCO, 1994) and the Convention on the Rights of Persons with Disabilities (United Nations, 2007).
Although the Malawi government requires all teacher training colleges to include SEND training in their courses for pre-service trainee teachers, few institutions offer specialized SEND training programmes (Chitiyo et al., 2015). There are therefore currently insufficient specialist teachers to meet the growing number of pupils identified with SEND entering the school system (Hughes et al., 2016) and there is no specific curriculum for SEND pupils (Chitiyo et al., 2015). As a result, pupils with SEND face many challenges to learning in mainstream schools when there are no specialist teachers or materials to support their needs.
The intervention evaluated in this study requires little specialist adult support to implement, as children work on a one-to-one basis with the technology and a virtual teacher within the apps demonstrates “how to” perform certain tasks before children practice a particular topic. Instructions can be repeated upon demand, as often as needed, so this affords personalized, self-paced, learning, without the need for specialist adult support.
Tablet technology could help to address some of the shortfalls within the current education system within Malawi for pupils with SEND. Touch-screen tablets, such as iPads, are light-weight and portable, have a long battery life and screen size appropriate for young children, and do not rely on additional dexterity devices, such as keyboard and mouse (Kucirkova, 2014), making them particularly suited to pupils with SEND.
In addition, they can store multiple child-friendly educational apps hosting software features that place the child in active control of their learning. Specifically, apps that include multiple representations of information, such as pictures, video, and animation, varying or progressive levels of task difficulty, clear goals and rules, learner control, task feedback, and repetition, can serve to create an individualized learning environment (see also Rose et al., 2005; Condie and Munro, 2007), enabling children to progress at their own pace. Whilst compelling, few studies to date have investigated systematically if this technology can assist pupils with SEND in acquiring basic skills, such as numeracy or literacy.
Tablet technology can also offer a vehicle for engendering an inclusive learning environment when used with SEND pupils alongside their mainstream peers. Kagohara et al. (2013) reported that SEND pupils willingly adopt this technology, such as iPads, which is perceived as socially more acceptable and less stigmatizing than previous forms of assistance technology, such as adjustable keyboards, which are bulky and awkward to use (Flewitt et al., 2014). Moreover, due to their portability and long battery life, touch-screen tablets can offer a bridge between the school and home environment, which could be highly beneficial for a low-income country like Malawi, where many pupils with SEND experience difficulties in getting to school and grid supply of electricity is often lacking or unreliable in villages and homes.
The current intervention utilizes solar panels to charge the devices overnight, providing a sustainable and reliable electricity source, which could impact on communities in Malawi by providing dependable home-school links. Strnadova and Cumming (2013) studied the use of iPad-based interventions across home-school settings and reported pupils with SEND experienced positive teaching and learning, higher engagement with educational tasks, and closer home-school links. This raises the possibility that tablet technology could act to breakdown some of the sociocultural barriers concerning attitudes toward the educational capability of pupils with SEND, especially in low-income countries such as Malawi.
Despite the potential for tablet technology coupled with high-quality educational apps to support the learning of pupils with SEND, there is little empirical evidence demonstrating that this technology is an effective means of instructional delivery for raising learning standards (Chmiliar, 2017). A recent systematic review of the existing evidence of the effectiveness of touch-screen tablets in primary and secondary schools for raising learning outcomes of mainstream pupils is limited and fragmented (Haβler et al., 2016).
Studies that have measured learning outcomes of pupils with SEND, by tracing progress quantitatively whilst working through educational apps, are scarce. Most studies exploring the utility of touch-screen tablets with SEND pupils are qualitative in nature (Khoo, 2017) or report on testimonials from parents and educators as to their effectiveness (Shah, 2011). These exploratory studies have indicated that tablet technology can provide an effective means of learning support for pupils with SEND, especially in the early years (see Chmiliar, 2017), but this needs to be validated by studies that quantify learning gains.
Thus, it remains to be determined if educational apps can be utilized effectively by pupils with SEND to raise attainment in basic skills.
This type of instruction may pose particular challenges to pupils with physical and sensory difficulties, such as difficulties with manual coordination and/or hearing and visual impairments, as multisensory, interactive, apps require visual, auditory, and kinaesthetic processing (including hand-eye coordination and manual dexterity) to interact successfully with the software.
In addition, some apps, such as those evaluated in this study, require pupils to attend to visual demonstrations and understand verbal instructions so pupils with attentional, language, and/or intellectual difficulties may struggle to engage with the software. Although most previous research reports SEND pupils have positive experiences of using touch-screen tablets and high levels of engagement (Strnadova and Cumming, 2013; Chmiliar, 2017; Khoo, 2017) if they find interacting with the technology difficult, because of their disabilities, they may not enjoy this type of instruction so could disengage with the learning process. Assistive aids, either built into the tablet technology or external to the technology, can be used to overcome some of these difficulties (Dell et al., 2017), but these are not always available in low income countries, such as Malawi.
University of Geneva
Katarina Gvozdic – University of Geneva
The image is credited to UNIGE.
Original Research: Closed access
“Learning to be an opportunistic word problem solver: going beyond informal solving strategies”. Katarina Gvozdic, Emmanuel Sander.
ZDM Mathematics Education doi:10.1007/s11858-019-01114-z.