Tactile and acoustic teaching material in inclusive mathematics classrooms

Abstract

In this article, touch and hearing are analysed regarding their use in inclusive mathematics education of children who are blind, focussing on the development of the number concept in primary school. Relevant publications from different psychological and educational disciplines were compiled resulting in a research synthesis. This review shows that acoustic teaching material is a viable option to supplement tactile material for representing number. In practice, decisions have to be made about the most valuable resources regarding the individual needs of the child with visual impairment, the situation in the inclusive classroom and the concrete learning objectives for all children. To facilitate decisions in choosing or designing teaching material, an evaluation procedure is proposed and explained with examples.

Keywords

Education inclusion mathematics teaching material visual impairment

Introduction

External representations of mathematical concepts, for example, manipulatives, are indispensable in mathematics education (Carbonneau, Marley, & Selig, 2013; National Council of Teachers of Mathematics [NCTM], 2000). They support the development of internal representations and help children to make the connection between mathematics inside and outside the classroom (Goldin & Shteingold, 2001). For sighted children, many mathematical concepts are represented through pictures and concrete manipulatives, for example, the number line, counters, or graphs. Often, colour is used to represent mathematical structures or highlight important features. Thus, many of these resources are inaccessible or challenging for children who are blind or partially sighted and need to be adapted or replaced. The focus of this article will be on these imagistic representations. Verbal or notational representations (as described by Goldin & Shteingold, 2001, p. 5) for students who are blind entail different issues and will not be discussed here.

The development and production of adaptations for visuospatial representations require time and funding; therefore, this effort should be based on substantiated criteria. Especially in inclusive classrooms, teaching materials need to be adequate and available, so that blind and sighted students can work together (Whitburn, 2014). One way to ensure quality would be to do empirical research on the effect the specific materials have on learning. While this is certainly a useful approach, it is hard to achieve, given the sheer number of materials to be tested, the relatively small number of students with visual impairments, and the strong heterogeneity of this group (Andreou & McCall, 2010, p. 116).

Thus, it might be more feasible to bring together existing knowledge: Research in different scientific disciplines such as neuroscience, cognitive and developmental psychology, mathematics education, and special education could be used to yield preliminary criteria for good teaching material consistent with current scientific knowledge. The aim of this theoretical review is to provide access to this information for researchers and practitioners.

Methods

A theoretical review should include the current state of research; the discussion of theoretical, empirical, and methodical strengths and weaknesses of the studies included; the description of relevant experiments; and the reformulation and synthesis of aspects of different theories (Cooper, 2005, p. 3). The full review is available from Leuders (2012).

Given the broadness of the different scientific fields to be addressed in this review, some constraints in literature research were necessary. The issue of mathematical teaching material was narrowed down to material regarding the key competences of number and operations in elementary and primary school. The focus was on children who are blind, and thus on tactile and acoustic material with the potential to replace visuospatial representations for sighted children.

Relevant publications from the above-mentioned psychological and educational disciplines were identified through a database search (ERIC, PubMed, MathEduc, PSYNDEX, ScienceDirect), and the references cited in suitable articles were used to identify additional literature (for a full list of references, see Leuders, 2012). Altogether, 515 books and articles were used. The distribution across disciplines was as follows (multiple selections possible):

Neuropsychology – 82

Cognitive psychology – 160

Perceptional psychology – 141

Developmental psychology – 80

Mathematics education – 121

Education of students with visual impairments – 124

Many different keywords were used in the search, including blindness, mathematics, number, visual, auditory, haptic, manipulatives, teaching, imagery, education, and their synonyms and German translations.

As suggested by Cooper (2005), these publications were then treated as data and processed to yield a research synthesis. For example, studies from perceptional psychology were sorted regarding sensory modality, number, visual faculties, and age of participants, relevance to mathematical learning and publication date. On this basis, the results of the studies were summarized and analysed in relation to each other and to studies from other disciplines.

Publication dates range from 1874 to 2012. While in some fields of research (such as neuropsychology and cognitive psychology) recent publications were preferred, older publications were included in other fields (such as education of students with visual impairments) to include historical points of view.

Below, tactile and acoustic material is analysed concerning positive and negative aspects for teaching number and operations, based on the theoretical review. Since acoustic material is used much more rarely, a practical example for its use is given and discussed.

Tactile material

The mathematical content of visual teaching material is usually represented by its spatial structure, for example, an array of counters or a number line. Since touch and vision both enable spatial perception, visual teaching material can be adapted for touch.

Haptic perception is characterized as an expert system for the identification of objects, implying that identification by touch is a very efficient process (Lederman & Klatzky, 1997). This efficiency is achieved through focussing on object attributes such as texture, surface temperature, and outstanding features like points, edges, or holes. Such features are by default in the focus of attention when children who are blind explore tactile objects (Simpkins, 1979; see also Argyropoulos, 2002), enabling them to identify objects in a fast and reliable manner. However, mathematical manipulatives often require processing of the exact spatial structure (e.g. dot patterns in arithmetic, or graphs in algebra), in contrast to the perception of outstanding features needed for identification. Perceiving the existence of edges or holes is usually sufficient for recognizing an object, but it is not enough when the exact relations of these object parts are required to understand a mathematical concept, for example, in a dot pattern. In that case, the whole structure has to be recognized. Texture, another crucial attribute in identification, is often irrelevant for understanding mathematical content – it does not matter whether the dots in a spatial pattern are wooden or plastic.

The task of perceiving and processing exact spatial relations through active touch requires attention and planning. A blind child aiming to count the dots in a tactile pattern needs to make sure that she found all dots and did not count any dot twice. This is difficult because touch does not provide a preliminary overview of the pattern the way vision does. Sicilian (1988) has analysed the way children who are blind count in such situations, and found that they need to develop their abilities on three dimensions:

Preliminary scanning – Children learn to deliberately scan the structure before counting.

Count organizing – Children learn to follow given structures like dot lines or circles.

Partitioning – Children develop strategies to keep track of elements already counted by moving them aside or by using one hand to indicate a partition.

Even if the children already learned these strategies, touch still needs more planning and controlling than vision does; it is considerably slower, requires more working memory, and increases the processing load (Andreou & McCall, 2010; Ballesteros, Bardisa, Millar, & Reales, 2005; Millar, 2000). It has been shown for sighted children in primary school that they need to be well accustomed to their material in order to handle it automatically and reduce the processing load. Otherwise, their processing capacity is too small to focus on the concrete material and the formal mathematical concept at the same time (Boulton-Lewis, 1998). Hence, it can be hypothesized that this is a common problem for children who are blind. Accordingly, Ahlberg and Csocsán (1994) found that the development of sophisticated strategies for active touch seems to be correlated with the development of number concept in children who are blind. Three implications can be drawn from these results:

Children who are blind need to learn strategies for active touch to enhance their understanding of spatial structures.

Teaching material needs to be well adapted to touch. It should be clearly structured and emphasize the features representing the mathematical content.

Perceiving spatial relations by touch, as required when using manipulatives, absorbs concentration and time, and thus might interfere with the processing of the mathematical concepts – even if (1) and (2) are fulfilled.

In conclusion, tactile materials are indispensable in mathematics education for students who are blind, but their design and use in the classroom need to be well deliberated. Also, a closer look at acoustic materials appears to be useful.

Acoustic material

This group of teaching materials encompasses pre-produced recordings (e.g. music or natural sounds, but not speech) and sounds produced by students (e.g. by clapping or singing). It has to be questioned whether acoustic teaching material can foster number development in young students. Csocsán, Klingenberg, Koskinen, and Sjöstedt (2002) propose using sound and rhythm in mathematics education. Research from this group shows that children who are blind can determine the number of beats in a rhythm very effectively. They even use this ability in counting and calculating spontaneously, without any formal instruction (Ahlberg & Csocsán, 1994, 1999). In addition, while memory for haptic-spatial structures has a low capacity in blind as well as sighted children (Ballesteros et al., 2005; Millar, 2000), the memory of children who are blind for auditory, sequential information is above average (e.g. Dekker, 1993; see below). Thus, acoustic material, such as clapping numbers, might be useful for teaching number and operations to them.

On the other hand, there are concerns about using acoustic representations of number. It has been found that children with dyscalculia often do not develop beyond counting strategies for simple calculations (Geary & Hoard, 2005). Counting strategies can be an impediment to the understanding of calculation because they keep children from thinking about the quantities and operations, and have them focus on the sequence of number words instead. Auditory representations could have the unwanted effect of prompting counting strategies because of their temporal structure: Countable sounds must be played one after the other. This problem is exacerbated by the fact that tactile material can also prompt counting strategies because the counted objects are often touched one by one. A material that enables children who are blind to perceive number simultaneously would be very valuable.

These concerns can be addressed based on research from perceptional and cognitive psychology. First, a summary of findings on mathematical cognition will help to elucidate number processing and the role of different perceptual channels for the input of number.

Number processing and perception

Research has yielded several models for number processing (e.g. Dehaene, 1992; McCloskey, 1992; Noël & Seron, 1997). The triple-code model developed by Dehaene, Piazza, Pinel, and Cohen (2005) will be described in further detail because it has been very influential, and, even more important, because there is research with blind participants regarding this model (see below).

It predicts that three functionally and anatomically distinct systems of representation are involved in numerical processing (Dehaene et al., 2005): In the verbal system, numerals are represented by number words in speech or text. This system is activated when a number word is heard or read, but also in recalling number facts (e.g. the times table). In the visual system, numbers can be encoded as strings of Arabic numerals. It is activated when numerals are read or needed for calculation (e.g. in multi-digit addition). No research regarding Braille numbers could be found. Thus, it can only be hypothesized that an analogous tactile system for Braille readers might exist. The quantity system, also called the ‘approximate number system’, or ANS (e.g. Feigenson, Dehaene, & Spelke, 2004), is described as a nonverbal semantic representation of the size and distance relations between numbers. It enables the estimation of dot quantity in a picture, and it adds meaning to the numbers we read or hear. The quantity system can be conceptualized as a mental number line, but it is not an exact, digital representation. The activated range on the mental number line usually encompasses several neighbouring numbers, resulting in a certain fuzziness (Nieder & Miller, 2003). The range of this fuzziness grows with number size: For example, it is larger for 556 than for 5 (Piazza, 2010). Also, the fuzziness is generally larger in children and is reduced with age (Halberda & Feigenson, 2008).

The quantity system is of special interest because it has important implications for number processing in people who are blind. It has been shown that the representation of number in the quantity system is abstract in the sense that it does not depend on the sensory channel through which the numerosity is perceived (e.g. Barth, Kanwisher, & Spelke, 2003; Meck & Church, 1983). Subsequently, these results have been reproduced for adults who are blind, proving that they also possess and use the quantity system and a mental number line (Castronovo & Seron, 2007; Szücs & Csépe, 2005). Apparently, there is no general problem with number processing in blind adults on this basic and crucial level. However, this result spawns another question: How is it possible that the brain handles simultaneously perceived visuospatial patterns of dots in the same way as consecutively perceived, temporal sounds? To answer this question, a closer look at auditory perception and processing is necessary.

The specifics of auditory perception

On the most basic level, sensory input is stored in sensory memory, also called ‘echoic’ memory in the case of hearing. This means that raw, unprocessed auditory information can be stored for a short time (a few seconds, depending on the nature of the input). Auditory information cannot be ‘reviewed’ like visual information: While a picture is still there for another look, sound fades away when the sound production ceases. Thus, storage is needed in order to compare and link new auditory input with already received input (Winkler & Cowan, 2005).

Research on sighted adults shows that this is possible on surprisingly abstract levels: Studies found a very stable reaction in the human auditory cortex to input that is in some way different from previous input. This reaction is called mismatch negativity (MMN) and is measured through event-related potentials in the auditory cortex. It can be triggered by many different kinds of mismatches, such as changes in the duration of sounds, melody contour or rhythm (Näätänen, Tervaniemi, Sussman, Paavilainen, & Winkler, 2001), and even by changes in the number of tones in a rhythm (Zuijen, Sussman, Winkler, Näätänen, & Tervaniemi, 2003). These results can also be interpreted based on Gestalt psychology: sounds are grouped according to their features, following the laws of proximity, similarity, and continuity (Deutsch, 1999; Koffka, 1935; Kubovy & Valkenburg, 2001). In a rhythm or melody, these groups contain motifs of a few consecutive tones.

For further processing, stimuli must be held in short-term memory. This is crucial to the synthesis of perceived information over time, including spoken sentences or whole melodies (as opposed to single motifs). The capacity of working memory is limited to five to seven chunks (Miyake & Shah, 1999), but the chunks can be of different sizes – they might consist of a single auditory event, or a group of sounds, like a multi-syllable word or a rhythmic motif. Thus, the processing that occurs on the level of sensory memory has an important impact because the pre-processed groupings (e.g. motifs) now serve as single chunks and raise the capacity of the short-term memory manifold.

Thus, cognitive processing of consecutive sounds can be described as very efficient. Compared to other sensory channels, auditory perception is the most useful modality for the processing of temporal information (Conway & Christiansen, 2005; Kubovy & Valkenburg, 2001; Lechelt, 1975). Regarding mathematical cognition of adults, it has been shown that temporally presented stimuli can be quantified more exactly if they are presented acoustically (tones) in comparison to visual stimuli (flashes) (Barth et al., 2003).

The research on auditory perception cited earlier in this chapter has been performed with adult, sighted participants. Naturally, research concerning children who are blind is much scarcer. Some studies have shown that these children can memorize temporal auditory information better than sighted children. This result is a by-product of research on intelligence assessment (Dekker, 1993; Tillman & Osborne, 1969) and Piaget experiments (Hatwell, 1985). Ahlberg and Csocsán (1994, 1997) found that many children who are blind use auditory representations for mental calculations: They were spontaneously counting rhythmically in ones or in groups, although they report that auditory representations for number were not used in school. This may be caused by the fact that children who are blind usually do not use finger-counting (Ahlberg & Csocsán, 1997; Crollen, Mahe, Collignon, & Seron, 2011). Presumably, the proprioceptive perception of the fingers is not salient enough for representing number without visual control.

Since auditory perception efficiently uses memory processes to synthesize information over time, the cognitive result is relatively similar to simultaneous perception offered by vision: As described above, quasi-simultaneous wholes (e.g. rhythms) are created through the cooperation of auditory cortex and short-term memory. Thus, auditory representation does not necessarily prompt counting strategies in calculation. Also, basic processing of quantities is abstract and works just as well for temporal and spatial patterns and for auditory and visual stimuli. Linking this knowledge with the observations made by Ahlberg and Csocsán (1994, 1997, see above) and others, it has to be concluded that auditory representations of number are a useful addition for teaching number to children who are blind. They might serve the same purpose as dot patterns for sighted children: creating mental representations of number that include important structures such as part-whole relations (as described by Resnick, 1989). Thus, tactile materials, such as raised-line drawings and manipulatives, should be supplemented with acoustic material for children who are blind. An example of acoustic material will be described and evaluated in the following section.

Evaluation procedure for the design of teaching material

The previously mentioned results of the review support the use of acoustic material in the mathematics classroom. This has already been proposed regarding the sonification of functions in secondary school (Brown, Brewster, & Ramloll, 2003; Flowers, 2005; Van Scoy, McLaughlin, & Fullmer, 2005). Below, an example from primary school will be discussed to fathom the usefulness of acoustic material as a representation for number and operations. Based on the theoretical review, criteria for good teaching material in inclusive classrooms and an evaluation procedure have been developed. Several pages from a frequently used German textbook (Wittmann & Müller, 2006) for first-graders were adapted for blind students. This textbook contains the problem shown in Figure 1.

Figure 1.

Linear patterns in a first-grade textbook.

The students are supposed to reproduce and continue patterns that are represented by blue and red dots (e.g. r, b, b, r, b, b, . . . or b, r, r, b, b, b, r, r, r, r, . . .). The first pattern is repetitive, the following two patterns contain growing numbers, and the fourth pattern combines both structures.

What would be the best way to adapt this task for a blind student in an inclusive classroom? The answer to this question requires bringing together empirical research results with criteria for good teaching material from both mathematics education and education of children who are blind, and including the information on the specific child and class. To make sure that everything is taken into account, an evaluation procedure was developed (Leuders, 2012). Based on a task analysis (Step 1), it is asked how the competences that are supposed to be fostered by the task can be supported in the adaptation (Step 2). In the next step (Step 3), the usability in an inclusive classroom is ensured. Finally (Step 4), the material is analysed regarding criteria for good teaching material from mathematics education. These steps are described below regarding the textbook example.

Task analysis

Which mathematical competences is the task focussing on? In addition, are there any specific competences a blind student could learn from this task? In the given task, the students should develop their counting ability and learn to look for patterns and structures. A specific competence of children who are blind could be the use of tactile or auditory counting strategies.

Adaptation outline

How can these competences be supported in children who are blind? They could use counters with different textures instead of different colours. But according to the research on haptic perception cited above, this might lead to processing overload: Blind students would need to count the textured dots one by one, than to remember the number of each group (‘red’ and ‘blue’), while planning and enacting tactile strategies at the same time.

Because of the linear structure of these patterns, they could alternatively be represented acoustically by using different sounds, for example, by ‘body music’: Clapping various body parts, stomping, or knocking on the desk (Cslovjecsek, 2001). Patterns with a repetitive structure (like the first one in the given task) create a rhythm when presented acoustically. Thus, they are easier to notice and to reproduce than patterns with growing numbers, which prompt a stronger focus on counting activities. As described above, the auditory system is very efficient in processing rhythm, so the step to non-rhythmic patterns could be difficult. This should be taken into account by slowing down in the case of growing numbers, or by using more repetitive patterns (depending on the children’s abilities).

Usability for inclusive settings

Can the material be used by all children in the inclusive classroom? When using the dots and counters with different textures to represent the colours, children who are blind would need to learn the texture-colour combinations (e.g. blue = smooth) in order to communicate with other children, and they would have no access at all to the sighted children’s printed visual patterns. Also, understanding and continuing a pattern by touch takes much longer and requires more concentration than by sight, so blind students would likely work slower than their sighted peers. In body music, on the other hand, all children can work together effortlessly, and the children who are blind may even be more successful than the sighted in recognizing the patterns because they are more familiar with the analysis of auditory input.

Criteria from mathematics education

The acoustic version seems to be more useful, but one question remains: Does the material adhere to criteria for good teaching material in general mathematics education? Or would the sighted children still be better off with their visual material?

One important aspect for good material is the usability in varied classroom situations and the continuability throughout at least one school year. Using and understanding new material needs to be learned: There is a bridge to gap between the intuitive understanding good material can foster and the formal mathematical descriptions of the same mathematical concept (McNeil & Jarvin, 2007; National Research Council, 2001). Thus, if a new material is introduced, it should be worth the effort; if clapping is used to represent number, it also should be able to represent the important structure of fives and tens in the decimal system. For fives, this can be achieved by rhythmic grouping (Figure 2).

Figure 2.

Rhythmic pattern using groups of five.

Groups of five are best represented using six beats per bar, with a rest on the last beat of each bar. First-graders can use this pattern to clap numbers from 5 to 20. For larger units like tens or hundreds, it seems sensible to use a rattling sound, which can be understood as a representative of many knocks or claps (Cslovjecsek, 2001).

Good teaching material should also support the documentation of the learner’s results. This is important because it enables the students to discuss their results (Boulton-Lewis, 1998) and stores them for later review. Also, if sighted students make a sketch of a structure they found using manipulatives, this prompts a switch from enactive to iconic representation (Bruner, 1966), fostering abstract thinking. Is this possible for clapping numbers? Sounds can be recorded by technical devices, but using this technology might be difficult for young students. In addition, recordings are not iconic in the sense of Bruner. To address this issue, a change in modality can be helpful: Acoustic patterns can be written down, represented by dots of different sizes (for volume), with gaps of different sizes in between (for duration or rests). Children who are blind can use Braille dots, they can do raised-line drawings or use felt stickers.

Moreover, the documentation of auditory results in tactile form is more than just a way of dealing with the limitations of acoustic material: Another important aim in the use of teaching material is helping children to focus on the abstract mathematical structure instead of the concrete perception (Boulton-Lewis, 1998; National Research Council, 2001). The translation between different sensory channels can be very effective for this purpose. Modal transfer can foster children’s understanding of the abstract meaning of number beyond dot patterns (Bauersfeld & O’Brien, 2002) – after all, numbers describe all kinds of sets, not just visual or tactile ones. Bauersfeld and O’Brien (2002) actually suggest using haptic and acoustic material for sighted children. In the textbook example, the mathematical content to be understood is the abstract structure of the pattern. From this perspective, it makes no difference whether the pattern is represented by colour, texture, or sound.

In this particular case, the evaluation of body music as a material for a specific textbook task affirms its usefulness. In other situations, tactile material might be favoured, for example, in geometry. Often there are competing goals – for the blind student to develop tactile strategies and also mathematical understanding, for the class to work with similar material and discuss their results. Decisions have to be made based on the specific situation, but the evaluation procedure can help to organize the decision process.

Conclusion

Given the results of the research synthesis, it can be concluded that acoustic teaching material can support mathematical learning, both in education of children who are blind and in general mathematics education. The use of auditory representations as a supplement to tactile and visual versions may be beneficial whenever a content can be understood in a linear or temporal manner because auditory perception is the most suitable sensory channel for the processing of temporal information. Examples range from number and linear patterns to fractions or functions.

However, these are preliminary conclusions based on implications from the research synthesis. More research and practical experience is needed to develop and evaluate other uses for the sense of hearing in the mathematics classroom. Also, its limitations for mathematical learning remain unclear. The evaluation procedure can be used in the design process to ensure that all necessary perspectives on teaching material are being covered. In future research, it will have to be adjusted according to new research results. Its feasibility could be ascertained in a design research project. Further research should also analyse the use of acoustic material with sighted children, which is not necessarily limited to inclusive classrooms.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Ahlberg

Csocsán

(1994). Grasping numerosity among blind children. Göteborg: Department of Education and Educational Research, University of Gothenburg.

Ahlberg

Csocsán

(1997). Blind children and their experience of numbers: Specialpedagogiska rapporter No. 8. Göteborg: Göteborgs Universitet, Institutionen för Specialpedagogik.

Ahlberg

Csocsán

(1999). How children who are blind experience numbers. Journal of Visual Impairment and Blindness, 9, 549–561.

Andreou

McCall

(2010). Using the voice of the child who is blind as a tool for exploring spatial perception. British Journal of Visual Impairment, 28, 113–129.

Argyropoulos

V. S.

(2002). Tactual shape perception in relation to the understanding of geometrical concepts by blind students. British Journal of Visual Impairment, 20, 7–16.

Ballesteros

Bardisa

Millar

Reales

J. M.

(2005). The haptic test battery: A new instrument to test tactual abilities in blind and visually impaired and sighted children. British Journal of Visual Impairment, 23, 11–24.

Barth

Kanwisher

Spelke

E. S.

(2003). The construction of large number representations in adults. Cognition, 86, 201–221.

Bauersfeld

O’Brien

(2002). Mathe mit geschlossenen Augen: Zahlen und Formen erfühlen und erfassen [Maths with closed eyes: Touching and understanding numbers and shapes]. Mülheim an der Ruhr: Verl. an der Ruhr.

Boulton-Lewis

G. M.

(1998). Children’s strategy use and interpretations of mathematical representations. The Journal of Mathematical Behavior, 17, 219–237.

10.

Brown

Brewster

Ramloll

(2003). Design guidelines for audio presentation of graphs and tables. In Brazil

Shinn-Cunningham

(Eds.), Proceedings of the 2003 International Conference on Auditory Display, July 6–9, 2003 (pp. 284–287.). Boston, MA: Boston University Publications Production Department.

11.

Bruner

J. S.

(1966). Towards a theory of instruction. Cambridge, MA: Harvard University Press.

12.

Carbonneau

K. J.

Marley

S. C.

Selig

J. P.

(2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105, 380–400.

13.

Castronovo

Seron

(2007). Numerical estimation in blind subjects: Evidence of the impact of blindness and its following experience. Journal of Experimental Psychology: Human Perception and Performance, 33, 1089–1106.

14.

Conway

C. M.

Christiansen

M. H.

(2005). Modality-constrained statistical learning of tactile, visual, and auditory sequences. Journal of Experimental Psychology: Learning, Memory and Cognition, 33, 24–39.

15.

Cooper

(2005). Synthesizing research: A guide for literature reviews. Thousand Oaks, CA: SAGE.

16.

Crollen

Mahe

Collignon

Seron

(2011). The role of vision in the development of finger–number interactions: Finger-counting and finger-montring in blind children. Journal of Experimental Child Psychology, 109, 525–539.

17.

Cslovjecsek

(Ed.). (2001). Mathe macht Musik: Impulse zum musikalischen Unterricht zum Zahlenbuch 1 und 2 [Maths makes music: Impulses for musical lessons with the textbook “Zahlenbuch” for first and second grade] (Vol. 1/2). Zug: Klett und Balmer AG.

18.

Csocsán

Klingenberg

Koskinen

K.-L.

Sjöstedt

(2002). Maths ‘seen’ with other eyes: A blind child in the classroom. Esbo: Schildts.

19.

Dehaene

(1992). Varieties of numerical abilities. Cognition, 44, 1–42.

20.

Dehaene

Piazza

Pinel

Cohen

(2005). Three parietal circuits for number processing. In Campbell

J. I. D.

(Ed.), Handbook of mathematical cognition (pp. 433–454). Hove: Psychology Press.

21.

Dekker

(1993). Visually impaired children and haptic intelligence test Scores: Intelligence test for visually impaired children (ITVIC). Developmental Medicine and Child Neurology, 35, 478–489.

22.

Deutsch

(1999). Grouping mechanisms in music. In Deutsch

(Ed.), The psychology of music (pp. 299–348). London, England: Academic Press.

23.

Feigenson

Dehaene

Spelke

E. S.

(2004). Core systems of number. Trends in Cognitive Sciences, 8, 307–314.

24.

Flowers

J. H.

(2005). Thirteen years of reflection on auditory graphing: Promises, pitfalls, and potential new directions. In Brazil

(Ed.), Proceedings of ICAD 05-Eleventh Meeting of the International Conference on Auditory Display (pp. 406–409). Atlanta, GA: Georgia University of Technology.

25.

Geary

D. C.

Hoard

M. K.

(2005). Learning disabilities in arithmetic and mathematics: Theoretical and empirical perspectives. In Campbell

J. I. D.

(Ed.), Handbook of mathematical cognition (pp. 253–268). Hove: Psychology Press.

26.

Goldin

Shteingold

(2001). Systems of representation and the development of mathematical concepts. In Cuoco

A. A.

Curcio

F. R.

(Eds.), The roles of representation in school mathematics (pp. 1–23). Reston, VA: National Council of Teachers of Mathematics.

27.

Halberda

Feigenson

(2008). Developmental change in the acuity of the ‘number sense’: The approximate number system in 3-, 4-, 5-, and 6-year-olds and adults. Developmental Psychology, 44, 1457–1465.

28.

Hatwell

(1985). Piagetian reasoning and the blind. New York, NY: American Foundation for the Blind.

29.

Koffka

(1935). Principles of Gestalt psychology. New York, NY: Harcourt.

30.

Kubovy

Valkenburg

(2001). Auditory and visual objects. Cognition, 80, 97–126.

31.

Lechelt

E. C.

(1975). Temporal numerosity discrimination: Intermodal comparisons revisited. British Journal of Psychology, 66, 101–108.

32.

Lederman

S. J.

Klatzky

R. L.

(1997). Relative availability of surface and object properties during early haptic processing. Journal of Experimental Psychology: Human Perception and Performance, 23, 1680–1707.

33.

Leuders

(2012). Förderung der Zahlbegriffsentwicklung bei sehenden und blinden Kindern. Empirische Grundlagen und didaktische Konzepte [Fostering the development of number concept in sighted and blind children: Empirical foundations and educational conceptions]. Wiesbaden: Vieweg+Teubner.

34.

McCloskey

(1992). Cognitive mechanisms in numerical processing: Evidence from acquired dyscalculia. Cognition, 44, 107–157.

35.

McNeil

Jarvin

(2007). When theories don’t add up: Disentangling the manipulatives debate. Theory Into Practice, 46, 309–316.

36.

Meck

W. H.

Church

R. M.

(1983). A mode control model for counting and timing processes. Journal of Experimental Psychology: Animal Behaviour Processes, 9, 320–334.

37.

Millar

(2000). Modality and mind: Convergent active processing in interrelated networks as a model of development and perception by touch. In Heller

M. A.

(Ed.), Touch, representation, and blindness (pp. 99–135). Oxford: Oxford Univ. Press.

38.

Miyake

Shah

(1999). Toward unified theories of working memory: Emerging general consensus, unresolved theoretical issues, and future research directions. In Miyake

Shah

(Eds.), Models of working memory (pp. 442–481). Cambridge: Cambridge University Press.

39.

Näätänen

Tervaniemi

Sussman

Paavilainen

Winkler

(2001). ‘Primitive intelligence’ in the auditory cortex. Trends in Neurosciences, 24, 283–288.

40.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (4th ed.). Reston, VA: Author.

41.

National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

42.

Nieder

Miller

E. K.

(2003). Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex. Neuron, 37, 149–157.

43.

Noël

M.-P.

Seron

(1997). On the existence of intermediate representations in numerical processing. Journal of Experimental Psychology: Learning, Memory and Cognition, 23, 697–720.

44.

Piazza

(2010). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14, 542–551.

45.

Resnick

L. B.

(1989). Developing mathematical knowledge. American Psychologist, 44, 162–169.

46.

Sicilian

S. P.

(1988). Development of counting strategies in congenitally blind children. Journal of Visual Impairment and Blindness, 82, 331–335.

47.

Simpkins

K. E.

(1979). Tactual discrimination of shapes. Journal of Visual Impairment and Blindness, 73, 93–101.

48.

Szücs

Csépe

(2005). The parietal distance effect appears in both the congenitally blind and matched sighted controls in an acoustic number comparison task. Neuroscience Letters, 384, 11–16.

49.

Tillman

H. M.

Osborne

R. T.

(1969). The performance of blind and sighted children on the Wechsler Intelligence Scale for Children: Interaction effects. Education of the Visually Handicapped, 1, 1–4.

50.

Van Scoy

McLaughlin

Fullmer

(2005). Auditory augmentation of haptic graphs: Developing a graphic tool for teaching precalculus skill to blind students. In Brazil

(Ed.), Proceedings of ICAD 05-Eleventh Meeting of the International Conference on Auditory Display. International Community for Auditory Display.

51.

Whitburn

(2014). ‘A really good teaching strategy’: Secondary students with vision impairment voice their experiences of inclusive teacher pedagogy. British Journal of Visual Impairment, 32, 148–156.

52.

Winkler

Cowan

(2005). From sensory to long-term memory: Evidence from auditory memory reactivation studies. Experimental Psychology, 52, 1–17.

53.

Wittmann

E. C.

Müller

G. N.

(2006). Das Zahlenbuch: 1. Schuljahr, Lehrerband [The number book: First grade textbook, teacher’s edition]. Leipzig: Klett-Grundschulverl.

54.

Zuijen

T. L.

Sussman

Winkler

Näätänen

Tervaniemi

(2003). Grouping of sequential sounds: An event-related potential study comparing musicians and nonmusicians. Journal of Cognitive Neuroscience, 16, 331–338.