Methodological Guide

Chapter 1 – An introduction to the concept of non-formal Mathematics

Chapter 2 – European Museums of non formal Mathematics

Chapter 3 – Adoption of digitized methodologies that appeal to kindergarten educators to broaden and enrich math experiences in kindergarten classrooms

Chapter 4 – Alternative pedagogical methodologies and synchronous interdisciplinary bestpractices to approaching simple math concepts and reasoning for preschool ages

Chapter 5 – The recreaMATHS approach

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Chapter 4 - Alternative pedagogical methodologies and synchronous interdisciplinary best-practices to approaching simple math concepts and reasoning for preschool ages

4.1 – What is interdisciplinary learning and why is it important in preschool?

With the term interdisciplinary in education and training pedagogies we describe the use of methods and insights of several disciplines or fields of study. The adjective interdisciplinary is most often used in educational circles when researchers from two or more disciplines pool their approaches and modify them so that they are better suited to the specific situation, including the case of the team-taught course where students or children are required to understand a given subject in terms of multiple traditional disciplines.

Interdisciplinary teaching and learning are maximized when several professionals and experts from different disciplines work together to serve a common purpose to help students integrate and make the connections between different disciplines or subject areas, along with their specific perspectives. By doing so, students can apply the knowledge gained in one discipline to another discipline. Repeated exposure to interdisciplinary thought can help learners develop more advanced knowledge, enhance critical thinking ability and metacognitive skills, and increase the understanding of the relations among perspectives derived from different disciplines.

It is common in scientific research, that the provision of choice-making is an indicator of developmentally appropriate practice for young children with and without disabilities; however, there is little empirical evidence regarding the rate of delivery of choices within the preschool classroom. The introduction of intervention strategies by a team, that is based on an interdisciplinary classroom, also is not well documented, which makes it necessary to increment the exposure of children to this kind of strategy.

The new curriculum for preschool education allows the engagement of the young pupil in as many experiential fields, through learning experiences and this “allows the interdisciplinary integrated approach of the suggested contents and provides freedom to the teacher in planning the daily activity with young pupils. Therefore, the interdisciplinary approach is required even by the curriculum for preschool education, as a logic requirement of modeling the young pupil and as a natural modality of action with the contents of several experiential fields” (Dinuță, 2015).

Since the last few years, interdisciplinary learning has been an issue but a necessity in preschool education, yet the traditional nature of many institutions and kindergarten has barriers that in many ways discourage or prevent such activities from happening. For example, teaching in teams from various disciplines to children is commonplace, but there are often tensions when combining experts from multiple fields, and this always helps to consider chemistry and fit when talking about team building.

Moreover, new and emerging technology skills are an essential part of standard curricula in many disciplines today because technology shifts occur at a rapid rate that it is virtually impossible to keep up with them through the traditional learning strategies. (Lorenzen-Huber et al., 2010; Loewer, 2012).

On the other hand, one of the clearest advantages for students and kindergarten children is the reality that multiple instructors enrich a student’s learning experience through diversity exposure and multiple points of view.

For instance, we have loads of examples of themes that cross over disciplinary boundaries in literature, art, and history or science and mathematics. The implementation of Science, Technology, Engineering, and Mathematics (STEM) at school or preschool is one of the challenges of education in the twenty-first century for many countries (Sanders 2008), especially concerning the development of critical thinking during argumentative interactions. The literature on scientific activities in education and in particular in early childhood is relatively recent (Impedovo et al. 2017). Most of the studies about how to implement and analyse STEM in education deal with elementary, middle, or high school (Smyrnaiou et al. 2015). Regardless of the specific STEM field, argumentation is in fact a transversal process of knowledge construction about the natural world (Erduran and Jiménez-Aleixandre 2008). It is also a process of critical reasoning that is important for the development of young citizens (Schwarz and Baker 2017). As the mind is argumentative by nature (Moshman 2004), young children are naturally inclined to explore the environment and to ask questions about scientific phenomena (Danish and Enyedy 2015; Ravanis 1994).

Another aspect is the student’s high motivation due to interesting topics and the interest that these topics can create in them. As a result, one of the most effective ways of presenting the content is often to connect activities in life experiences, giving an authentic purpose for the learning and connecting it to a real-world context. Consequently, the learning becomes meaningful, purposeful and deeper resulting in learning experiences that stay with the student for a lifetime.

As students look across disciplinary boundaries, critical thinking skills and metacognitive skills are used and developed to consider other points of view. Worthwhile topics of research can fill the gaps between the traditional disciplines. Children also begin to compare and contrast concepts across subject areas, which helps them develop their personal inner world and thoughts.

Moreover, students begin to consolidate learning by synthesizing ideas from many perspectives and consider an alternative way of acquiring knowledge. Of course, the consequence of the repeated exposure and exploration of topics across a range of subject boundaries motivates students to pursue new knowledge in different subject areas, which results in transferable skills of critical thinking, synthesis and research. These skills are developed and apply to future learning experiences, which will be applicable not only in the specific topic handled, but also to transversal subjects across disciplines. Therefore, interdisciplinary knowledge and application of different disciplines can lead to greater creativity.

4.2 – Approaching Mathematics in Kindergarten as a multifaceted and interdisciplinary field

With raising research results the knowledge of how children develop mathematical skills is gaining consistency. It is recognised that “children require significant amounts of time to develop the foundational mathematical skills and understandings.” (NRC, 2009, p. 124). It is therefore crucial that enough time is dedicated to mathematics in preschool programs so that children develop foundational mathematical skills and knowledge. But it should not only focus on the more formal parts of mathematics instruction and discussions but also include non-formal education like exploring, creating, and playing. (NRC, 2009, p. 124)

The potential of everyday activities such as cooking, playing with mathematical shapes and telling the time is recognised and harnessed in order to empower the children learning program. Also, everyday experiences using mathematical words and phrases is one of the key elements for children to induce them to talk about their mathematical thinking.

Essential is the children’s engagement in what is interesting and relevant to them and the experiences that show the usefulness of mathematics for solving everyday problems. Children’s eagerness to participate in everyday activities such as cooking (Vandermaas-Peeler et al, 2012), or shopping is an effective way of fostering a positive disposition. As a result of her study of the number sense of 4-year-old children, Dunphy (2006) concluded that the ways in which children are engaged with mathematics, how they view mathematics, and the contexts in which mathematics are presented to them are what shape their dispositions towards mathematics. In the same study, children with a positive disposition also demonstrated a strong number sense. For instance, young children starting school may already have developed a liking or enthusiasm for numbers, based on experiences during the preschool period, i.e., their disposition towards number is already developing (Dunphy, 2006).

Effective teachers use a variety of ‘worthwhile mathematical tasks’ and help learners ‘make connections’ across mathematics, between different solution paths in problem-solving, and between mathematics and everyday life. Effective teachers of mathematics carefully choose ‘tools and representations’ to stimulate and support learners’ thinking. From a teaching and learning perspective, projects are a valuable approach to organising mathematical activities for young children (Katz & Chard, 2000; Ginsburg & Golbeck, 2004). Listed in the table below, is a variety of possible applications of mathematics in everyday life and in activities that can be developed in early childhood.

DIGITAL TOOLSUsing technology is an increasingly important avenue of learning and expression for children. For example, Kalas (2010) describes children’s engagement with technology and digital tools and exploration of direction and locationExploring spatial concepts Develop the language of spatial relations (e.g., besides, towards, short / shortest) Develop algorithmic thinking (processes or rules for calculating)
COOKINGA study of water (adapted from Dixon, 2001)Document processes through diagrams, drawings, charts, photographs, data and models Explain mathematical processes

Teaching Practices Making apple sauce (adapted from Ginsburg & Golbeck, 2004)Decide how many jars of apple sauce are required They count, ‘read’ a pictorial recipe Discuss the itinerary to the supermarket Weigh ingredients, compare size, shape, colour, and price of fruits

The pizza project (adapted from Gallick & Lee, 2009)The sequence of making pizza Estimate, measure and cut circles of paper to represent pizza slices
MUSICShilling (2002) identifies a strong link between the order, timing, beat and rhythm of music and attributes of mathematics such as counting, sequencing and understanding time and order. This enables teachers to make learning both music and mathematics more meaningful for the children (Kim, 1999; McGrath, 2010; Montague-Smith & Price, 2012; Pound, 1999; Shilling, 2002).Development of other skills and attitudes that are important for mathematics such as concentration, creativity, perseverance, self-confidence, and sensitivity towards others (Fox & Surtees, 2010) Develop children’s mathematical language and concepts
VISUAL ARTSPattern and shape are key features of both the visual arts and mathematics. In the visual arts, children encounter colour, form, texture, pattern and rhythm, and shape (Government of Ireland, 1999c). In mathematics, they discover patterns of number and shape, symmetry, tessellation, and the properties of a range of 2D and 3D shapes.
Drawings provide a forum in which children’s confusions about particular aspects of mathematics can be addressed (e.g., orientation can vary).
Develop the child’s awareness of the visual and spatial qualities in the environment Enhancing children’s ability to apply mathematical knowledge in the environment 70 Identifying 2D shapes in fabrics Symmetry in pictures Perimeters Convey their growing awareness of number and quantity Develops abilities to translate mathematics from one language (verbal) to another (graphic) (Worthington & Carruthers, 2003).
DRAMADrama and PE Role-play offers many opportunities for children to engage with mathematical concepts and skills. Story contexts such as ‘The Three Little Pigs’ can give rise to a range of mathematically-related play, especially if appropriate props are provided to stimulate mathematical thinking (e.g., Pound, 2008).Phrases such as ‘just enough’ (equality), ‘not enough’ (less than) and ‘too many’ (greater than) can be used and their meaning explored in the context of the play Forming groups for games, representing basic processes such as addition or subtraction, by combining or separating groups of children Partitioning of numbers can be explored
SPORTParticipating in swimming or athletics. Very young children can be exposed to mathematical vocabulary through everyday discourses such as swimming lessons (Davies et al, 2012).Creating 2D shapes using children’s bodies, and discussing the properties of such shapes Calculating times and distances
MEASUREMENTMeasurement is an important mathematical topic because of its applicability to everyday activity, its connection with other subject areas, and it can serve as the basis of other content areas in mathematics (Clements, 2003).Vocabulary for quantity or magnitude of a certain attribute Comparing two objects Equality or inequality Overcoming perceptual cues
NATUREObservation of plants, vegetables and fruits geometric forms.Geometric pattern observation Develop the child’s awareness of the visual and spatial qualities in the environment

4.3 – Inclusive Math Approaches

To ensure that children have a rounded and fulfilling educational experience, an inclusive math approach is a crucial part in achieving this goal. The opportunity should be given to engage in a wide, balanced and rounded curricular experience that supports all aspects of their development – not only the formal part and content acquisition, but the social, emotional, imaginative, aesthetic, and physical dimensions as well.

Children need to be assisted in using the newly acquired mathematical language in their descriptions and explanations. Good mathematics pedagogy recognises that some children (e.g., children living in disadvantaged circumstances; children who speak a language that is different from the language of instruction) may experience difficulty with problems presented in verbal format and there may be a need to adjust the presentation accordingly (Ginsburg et al, 2006).

Most important is that children play an active role in the development of their knowledge and to be developmentally appropriate and avoid premature formality.

Other contexts in which preschool educators can promote mathematical concepts and language include, for example, playing games, reading books with a mathematical theme, using computers, and constructing objects (e.g., block building).

Just as mathematics is learned in context, so it is used in context to achieve some worthwhile purpose.

Dan Finkel, a Ph.D. in mathematics from the University of Washington, says that mathematical miseducation is one of the most common things and we expect lessons to be about memorization and repetition of disjointed technical facts. Therefore, children need to learn that mathematics is not about following rules but about playing and exploring.

Classrooms are filled with all types of learners and the teachers’ role in kindergarten is critical because it will set a base for the learners’ future and have an impact on how they will perceive different topics later on. It is also a time to identify certain difficulties and help children strengthen their problem-solving capabilities. Therefore, a kindergarten curriculum must be well thought of and cover different styles of learning steering positivity into topics that have proved to be challenging in later school years for example mathematics. The teacher in kindergarten can stimulate the interest of the pupils towards learning and creativity through different pedagogical approaches. This phase in a child’s education is where educators can start understanding their individuality in learning and how they adapt to different activities.

4.3.1 – Learning Dys

All types of “dys” may affect the learning of mathematics but the most common Math related Specific Learning Disorder (SLD) is Dyscalculia. Dyscalculia affects a person’s ability to understand numbers and learn mathematical facts. Students with this SLD often lose track while counting, mistake numbers during operations, have trouble memorizing and recalling mathematical procedures and rules. Students with Dyscalculia typically perform poorly in tests and become easily overwhelmed and develop math anxiety.

Math related disorders can start to be evident from preschool age according to the article “Specific learning disability in mathematics: a comprehensive review” in the Translational Pediatrics Journal. Toddlers may start showing difficulty learning to count, sorting, corresponding numbers to objects, memorizing numbers by hearing them. In order to have the earliest diagnosis it is critic that children are exposed to mathematics as early as possible. The earlier the diagnosis is made, the earlier the children may get help to develop good foundational skills.

Rochelle Kenyon, a consultant and trainer in education and disabilities topics, lists a series of strategies to teach students with math related learning disabilities. The first strategy is to not overload students’ memory and assign tasks in manageable amounts as skills are understood. The premise of recreaMATHS is to shift the emphasis from “numeracy skills” and ‘skill and drill practices’ to teaching the ‘language of math’ focusing on the comprehension instead of the fast memorization of math procedures. This strategy profits all types of learners because it proposes an adapted timetable for task assignment. The division of activities in clear, short steps, allow enough time for students with SLD to grasp the concepts. Memorization is not the strong suit of all students, so this method also favors those who need to understand mathematical concepts in order to learn math. It is easier for students with SLD to focus more on logic than memory. Therefore, one of the approaches is to challenge critical thinking to boost problem solving within learners with SLD using real life problems to render mathematics more tangible. Using real-life situations changes the point of view on mathematics making problems functional and applicable to everyday life.

The provision of supervised activities to ensure the good practice of mathematical concepts and the proper application of rules is another approach that can benefits students with SLD that is covered by recreaMATHS for example with the module on 3D modelling. During the supervised activities it is important to give constant feedback to learners and also have them track their progress. Another strategy that aligns with recreaMATHS approach is the use of manipulatives and technology specially with the help of the 3D modeling module.

Rochelle Kenyon advises to help students visualize math problems by drawing, this is an approach that helps learners to understand better using visual elements to illustrate concepts. She also advices the use of auditory examples by focusing on a multisensory method. The project will be applying this approach in most of its intellectual outputs, for instance mathematical e-books for children between 4 and 5 years old and a version for children between 6 and 7 years old, an entire collection of 12 hands-on plus 2 Virtual interactive Mathematical exhibits for kindergartens and finally 3D modeling with a 3D printer. These intellectual outputs also correspond to the Rochelle Kenyon’s strategy of using age-appropriate games as motivational materials. If possible, it is better to make creative, constructive activities, rather than exclusion or competition-based activities.

Distractions and unnecessary information in classes with learners with SLD must be avoided, for this reason cluttered worksheets and too much visual information is not helpful specifically for students with ADD, ADHD and Dyslexia, structuring papers with clearly distinguishable titles and subtitles is a good beginning to have a better structure.

4.3.2 – Deaf and Blind

Mathematics is simple, but the language used to explain it is complicated. The difficulty in decoding the language determines the limits of understanding for most of hard of hearing, blind and deaf students, who find themselves, considering it abstruse and complicated, not to like the discipline. The ability to extract relevant information from a text, with respect to the resolution of the same, is influenced by the mode of presentation of the text. Mathematics is a communication code that, although different from other codes such as language or sign language, can be compared to them.

In an article, from the Journal of Research on Technology in Education (vol.45 no.4) “Teaching Mathematics Vocabulary with an interactive Signing Math Dictionary”, Judy Vesel and Tara Robillard documented the five most common problem areas among deaf students: words with more than one meanin; technical language; specific words in mathematics; the presence of varied but related forms; and specific abbreviations and symbols. Mathematical language, the simplest mathematical symbols, (+, -, x, :), numbering in base 10, are, for the deaf, an ideal language because each sign, each digit, each rule has a meaning for itself and for the position they occupy. In the early stages of learning, the deaf person seems to have an easier time with mathematics than with language. But when, through language, we have to solve problematic situations, they have enormous difficulties.

Teaching mathematics to a blind person is not entirely different from teaching it to a sighted person, but it presupposes knowledge of the ways in which concepts are formed and the world explored that are typical of the blind. In fact, blind people need a longer time to approach reality, as such is the tactile exploration of objects. In addition, it is important to assess whether what the blind express in words are learned concepts, or are the result of verbalism, repetition of terms empty of their meaning. It is always necessary to follow an order that goes from concrete to abstract and therefore at the basis of the formation of any concept there is the manipulation of reality, then the representation, and finally the symbolization.

When talking about deafness, children and integration, we are talking about ‘special’ inclusion. Indeed, the quality of inclusion depends to a large extent on the capacity of the school or classroom to become a supportive community. It requires a high degree of flexibility on the part of the teachers and the pupils themselves in a spirit of mutual support and coeducation. A redefinition of the school context is therefore the main task of the educational institution and should focus on being aware that the hearing and visual deficit only limits the ability to hear and see but does not cause any cognitive damage.

To be able to overcome difficulties and win the attention of the student, it is necessary to aim at effective communication, since the main obstacle to overcome is precisely at the communicative level. Therefore, one of the first fundamental suggestions is to reduce the time of frontal teaching and adopt strategies and modes of explanation that make it easy to understand, considering the specific needs of each deaf and blind student. Moreover, it is preferable to privilege a dialogic style rather than a tutorial one. Also, a kind of peer-to-peer adaptation is needed, during which all, blind, hard of hearing and deaf people should try to cooperate for a good communication strategy.

Specifically, for deaf children in this regard, it is necessary to stop talking in class when one is turned to write or draw on the blackboard; speak in turn, one at a time, and signal with the hand when someone interrupts and intervene in the conversation; touch the child lightly on the arm to call his attention, never suddenly and behind his back; make him participate in everything that happens in class and that he/she may miss.

Listening in a noisy environment is a source of difficulty and stress: for almost all blind, deaf, hard of hearing and students with traditional hearing aids or cochlear implants, the presence of noise means that they cannot refer to the ear canal. Even for deaf students who do not use any aids, background noise can be a source of discomfort. To improve the ecology of the school, making the environment more conducive to listening and/or communicative exchange we make the following suggestions:

  • Provide for the presence of deaf and blind students by placing them in a non-noisy classroom, use soundproofing materials (curtains on windows, carpeting, rugs) and anti-noise materials (grommets on chairs, bumpers on doors).
  • Verify the appropriateness of using frequency-modulated (FM) systems that improve the signal-to-noise ratio (25 dB).
  • Arrange desks in a semicircle or otherwise so that the student can easily see and hear both the teacher and his or her peers.
  • In cases where sign language is the student’s preferred language, ensure that the communication assistant or teacher has an easily visible accommodation.
  • If the child uses a cochlear implant or traditional hearing aid, the teacher should familiarize him or herself with the aid.
  • Use good lighting, making sure the light source is not glaring.
  • In order for the deaf person to lip-read well, the optimal speaking distance should not exceed one and a half meters, and the speed of speech should be moderate. Lip-reading is based on correct pronunciation. If possible, use short, simple but complete sentences.
  • When using names of people, places or unusual terms, lip-reading is very difficult. If the deaf person is unable to do so, the word can be written in block letters.

Other inclusive approaches, supplies and supports that can be given by the teacher to help and improve the experience of deaf and blind children are:

  • Make students experience problematic situations also through graphic representation, dramatization, to encourage the association of actions to mathematical symbols. For the deaf and blind students, it is fundamental to learn by doing associated with mathematical symbols and then visualize it.
  • The student should know in advance the topic of the lesson and have access to instructional materials and resources provided through textbooks, audiobooks, the Web, and electronic media.
  • A certain flexibility in the time needed to achieve the objectives identified and interventions that allow the transition from concrete to abstract are appropriate, using a wide range of representation models such as concrete experiences, structured material, images, laboratory activities and computer and multimedia technologies.
  • In addition, discovery can give satisfaction and therefore motivation to learn and initiate autonomous learning.
  • The use of technological tools such as interactive whiteboards and tablets. Among the possible teaching technologies, which are found to be a useful tool for breaking down communication barriers, are multimedia tools.
  • Instruction should be given through simple language.
  • Use of rubrics and specialized content vocabulary.

Specific for deaf children:

  • The teacher can learn to read from the face and posture whether the deaf child is following the explanation or not.
  • Computer and subtitles, because they use the visual channel and not the acoustic one.
  • Self-correction decreases the deaf student’s sense of humiliation due to continuous corrections which, if continued over time, lowers the sense of self-esteem necessary for the construction of autonomy.
  • Using concrete and directly visible exercises helps the child to form the first mathematical concepts that will be the basis for later learning. With the deaf child it is therefore very important to use the visual channel as well, because this channel can clarify ambiguities of the oral code. In addition to being the preferred channel of communication for the deaf, the visual channel is a cognitive resource.

Specific for blind children:

  • The way in which numbers are written must also be considered. The blind student will acquire knowledge of graphic signs only when he/she has finished studying the alphabet. In fact, to represent numbers, the first ten letters of the alphabet are used, preceded by a special sign called a “number sign”.
  • It is necessary that the teacher, together with the word, allows the blind student to manipulate the objects taken into consideration, because it is important that the child has the opportunity to make experiences, not to fall into the risk of empty verbalism.
  • According to Del Campo (2000, p. 216-217), it is possible to establish characteristics that can be found in all educational materials used by the blind. These characteristics are Transportability, Adequacy to perceptual characteristics, Simplicity, Cost-effectiveness.
  • Very often, even in the normal dynamics of teaching mathematics, there is recourse to particularly useful aids, such as logic blocks, because the child, manipulating, internalizes concepts that would otherwise be difficult to understand.
  • With the use of sophisticated computer programs and Software for Learning Mathematics we can support the child in acquiring knowledge

Today, teachers benefit from excellent multimedia materials that enable students to personally interact with the computer, even converse with it.

4.4 – Alternative pedagogical approaches

Ensuring that the learning experience for all learners is enjoyable and satisfying is one of the aspects to take into account when developing an approach. The opportunity to experience and appreciate the fun of exploring mathematical problems and the satisfaction of arriving at a solution is a unique situation that should not be ignored. These experiences and activities should arise from children’s interests, questions, concerns and everyday experiences.

One of the foundations of alternative pedagogical approaches is to use approaches in the development of emergent literacy and numeracy skills that complement learning in other areas and be child-centred, broadly-based, prioritise play and reinforce the concept of the child as an active learner, the importance of treating all children as if they already have knowledge and experience, taking account of the child’s strengths, interests and previous experiences and use these as contexts for new learning. All learners should have the opportunity to engage with learning approaches, including cooperative learning, differentiated learning, active learning, and problem-solving activity, which we know not only contribute to more effective learning, but increase learner’s participation and enjoyment of learners in the process.

In addition, it demands that in schools there is an ‘ethic of everybody: teachers have both the opportunity and responsibility to work to enhance the learning of all’ (Florian & Linklater, 2010, p. 372).

Learning by problems, for example, which is part of inductive teaching, consists of starting from a problem and, through the observation of a certain finite number of facts or events or particular experiences, arriving at a resolution. The method of teaching by problems allows students to learn to solve, gradually, increasingly complex problems that allow them to acquire cognitive skills at a higher and higher level. The learner is, therefore, at the centre of the process;

With this method you can also develop some fundamental aspects of personality such as:

  • Responsibility
  • Autonomy
  • Self-confidence
  • Self-esteem
  • Cooperation with others
  • Solidarity
  • Decision-making skills

Research in the field of Children with Hearing Impairment has demonstrated that ‘deaf children have different knowledge, learning styles and problem-solving strategies than hard of hearing children. Teachers need to know how their deaf students think and learn if they are to accommodate their needs and utilise their strengths’ (Marschark & Spencer, 2009, p. 210). Recommendations for deaf and hard-of-hearing children include recognising their visual-spatial orientation, which they do not always apply, and their relative lack of confidence in problem-solving. ‘It is clear that modifications in curricula and teaching strategies are required if deaf and hard-of-hearing students are to develop to their potential in the important areas of maths. Interventions that have shown promise include those which focus on building problem-solving skills through producing schematic illustrations emphasising visual-spatial over verbal activities (Nunes, 2004).

Access to a mathematics curriculum for children with visual impairment often hinges on specialist teacher knowledge of the unique aspects of mathematics education for such children. This includes the use of calculation with abacus or braillewriter, talking calculator, concrete materials and tactile displays and teaching of the Nemeth Code (Kapperman et al, 2000). A focus on mathematical language and its accuracy by the educator is also stressed.

The conception of a Universal Design for Learning (UDL) is a step towards the expansion of inclusion in education for all removing barriers created by CAST, an American organization. It is not a unique universal solution to all students’ problems, but it is a flexible approach that can be personalized by educators and adapted to the classroom that they have. It helps educator to understand the needs of different students by planning lessons with some flexibility to allow adaptation, instead of the traditional planification that assumes that all pupils learn in the same way, in a “one size fits all” manner. The creation of class profiles helps the lesson planning in order to know the weaknesses and strengths of the group, it is a student-centered approach. To develop the UDL approach it is easier if teachers collaborate and provide feedback in order to share the effects of the UDL approach.

The first UDL principle is representation, which means the provision of multiple ways to represent content instead of relying upon only on one type which is often the case with textbooks, it is important to use video and audio representation as well. The second principle is expression, the way that the student might express themselves is important because traditionally it is most frequently focused on a paper exam but students should also be able to make a presentation for example. The third and last principle concerns the student’s engagement in learning and this principle can be used by adding gamification, experience approach, or by relying upon multisensorial activities. The main benefit of using multisensorial activities is the possibility of including blind and deaf children. Multisensorial activities help students with SLD to be more motivated and engaged in learning, children will sustain their attention longer if they are doing something that interests them.

The collaborative learning approach and Inquiry-Based Learning Approach can be complimentary to UDL. The collaborative learning approach is based on the premise of creating profile groups that can work together based on their strengths and likes, it is important to pair children with the same level and pace of development and according to Vygotsky, learning is a social experience (Cooperative Learning and Support Strategies in Kindergarten, n.d.). The educator assumes a very important role as the person responsible for preparing the environment for the children to learn (Ibid, n.d.). The educator is the figure who encourages children to develop their problem-solving abilities and critical thinking. As a result, this method should help develop interpersonal, social, and communicational skills. This approach is inclusive because it takes into account the profile of all pupils and respect their barriers while being part of the same class. While working together children can help one another and compare their knowledge, this way children are active participants in their learning process (Ibid, n.d).

Inquiry-Based Learning Approach is a flexible and dynamic method based on the scientific learning process that follows the scheme: orientation, conceptualization, investigation and conclusion. It promotes a deeper understanding of scientific concepts promoting and at the same time a spirit of self-direction and independence, while pupils learn at their own pace. This method helps children learn how to ask scientific questions while being guided by the educator (Inquiry-Based Science Learning, n.d.). Inquiry-Based Learning Approach allows students to struggle positively since the process is more important than the answer in a scientific context. This method is particularly inclusive for students that cannot learn with the traditional method of memorization-reproduction. Educators might use tools to help visualize the questions generated by students for example idea boards, posters, and scientific notebooks where they can record drawings and photographs. These tools might motivate students to use scientific concepts through discussions, observation and experiences.

UDL, Collaborative learning approach, and Inquiry-Based approach are not exclusive, educators should mix and match alternative pedagogical approaches to traditional teacher-centered learning to accommodate all types of learners, paying close attention to students with disabilities. Therefore, it is important to have some flexibility while planning classes in order to get to know the students’ profiles and adapt the lessons to include all learners.

A student-centered approach is often welcomed by students with SLD because it is engaging and empowering. This approach helps avoid future negative psychological effects on students and the earlier it starts the better because they are not passive learners. This approach allows students to understand their interests and needs because they take part in the planning and implementation of activities with the teachers steering the activities.

Dan Finkel’s work is focused on bringing a positive perception of mathematics to ensure children fall in love with it. He therefore, created five principles to teach mathematics, of which the fifth and most important principle is play.

Introducing play in mathematics is beneficial to all learners. It helps to avoid future math anxiety and self-confidence difficulties while dealing with problem-solving. The neuroscientist Norman Doidge has written that our brains reorganize every day and it is highly adaptable, being capable of developing different pathways. If children with SLD have an early diagnosis and adapted intervention they can do well in school decreasing overall difficulties (There is a Better Way to Teach Students with Learning Disabilities, n.d.).

4.5 – Best Practices to approach simple math concepts

For an equitable curriculum that is inclusive to all children, it is important to consider the needs of children with intellectual, cognitive, and developmental difficulties but also children that are talented in math. It can be quite challenging to have an inclusive curriculum for such a large spectrum of learners varying from children with difficulties and/or mental delays that need constantly to catch up to others to mathematically talented children that need to have their potential met to not become disaffected by mathematics (Mathematics in Early Childhood, n.d.).

Multisensorial strategies that grab pupil’s attention, such as using an object to a point, giving emphasis on gesture and using rhythm are beneficial while teaching a child with moderate intellectual or developmental difficulties (Mathematics in Early Childhood, n.d.).

Children with hearing impairment vary from hard of hearing children, the first step is to understand how they learn and to tackle self-confidence while problem-solving. Typically, hard-of-hearing children have difficulties relating bits of information and identifying relationships, also with visual-spatial orientation. These difficulties do not apply to all hard-of-hearing children; therefore, educators must understand the learning style of the child and adapt. The teaching of mathematics to children with visual impairment often relies on specialists and supporting technology such as tactile displays, talking calculators, and others (Ibid, n.d.).

Teaching mathematics for all also means considering cultural and social backgrounds. For example, the child’s first language might differ from the one spoken in class and that can therefore have an impact at the beginning of schooling. To counterbalance math language barriers, the educator can expose the children to formal and informal mathematics contexts, emphasize in the teaching of mathematical language and concepts, plan the use of mathematical concepts in problem-solving situations and also in other areas (Mathematics in Early Childhood, n.d.).

In the Mathematics Instruction for Students with Learning Disabilities or Difficulty Learning Mathematics guide for teachers there are seven recommendations that can be effective for students with SLD:

  1. Teach students using explicit instruction regularly verbalizing the instructions introducing mathematical reasoning.
  2. Teach students using multiple instructional examples, teachers should plan out thoroughly effective instructions with multiple examples.
  3. Have students verbalize decisions and solutions to a math problem, this is to encourage students to think-aloud helping to solidify skills and strategy.
  4. Teach students to visually represent the information in the math problem, graphic representations mixed with explicit instructions often achieve better results.
  5. Teach students to solve problems using multiple/heuristic strategies, this strategy helps organize the problem and gives freedom to the student to choose their strategy.
  6. Provide ongoing formative assessment data and feedback to teachers, this strategy can help teachers understand the students’ rhythm, difficulties, and disabilities in order to adapt the planning.
  7. Provide peer-assisted instruction to students, a collaboration between students is beneficial but cross-age tutoring seems to be a better choice when dealing with students with SLD than within-class tutoring (Jayanthi et al, 2008).

Ladson-Billings, an American pedagogical theorist and teacher educator, lists six ways in which a curriculum can be inclusive to teach simple math concepts for all children:

  • The importance of treating all children as if they already have knowledge and experience that can be used as a foundation for teaching.
  • The creation of a learning environment that allows children to move from what they do not know to what they do know.
  • A focus on high-quality mathematics learning rather than on ‘busy’ work.
  • The provision of challenging tasks to all children.
  • The development of in-depth knowledge of children and subject matter.
  • The fostering of strong teacher-child relationships” (Mathematics in Early Childhood, n.d.).

The shift of focus in the teaching of mathematics towards the normalization of struggle and freedom to think can be implemented from a young age, this allows children to have challenging tasks that will help them better develop problem-solving skills and have an in-depth knowledge of the process behind a mathematical problem. For this to happen it is important to change the pace and the idea that students should have answers as fast as possible because mathematical thinking is a process, and this struggle can lead to brilliant and creative ideas instead of memorization and repetition.