Arithmetisches Lernen in neurologischen Patienten

The number processing and calculation system

Humans have developed different “tools” for the daily use of the number processing and calculation system. The most basic tool is the representation of numbers in a specific code such as Arabic (5), Roman (V), phonological (/five/), alphabetic (five), etc. This tool is needed for tasks such as counting. An important aspect of using codes is the transformation (i.e. “transcoding”) from one code to another. Transcoding is a skill required by many numerical tasks. For example, reading Arabic numerals aloud requires converting an Arabic code into a “spoken” phonological code. Calculation requires also the use of more complex tools such as arithmetic signs (e.g., +, −, ×, :, = ), arithmetic facts (e.g., 6 × 3 = 18, 6 + 2 = 8, 6 − 3 = 3), rules (e.g., 3 × 0 = 0, 3 × 1 = 3), procedures (multi-digit addition, subtraction, multiplication, and division problems with carrying and borrowing procedures) or principles (e.g., commutativity principle: 5 × 6 = 6 × 5). As shown by many neuropsychological studies, the use of these tools requires different skills, which can be selectively disrupted or spared in conditions that go under the generic name of acalculia. The high degree of modularity of the number processing and calculation system is perhaps the main reason why the rehabilitation of number competences is very difficult. Indeed, rehabilitating one numerical skill does not necessarily imply a positive effect on other skills.

The number processing and calculation system can be described from different points of view, that is, according to different theoretical models. These models are useful not only to understand how the brain/mind processes and uses numbers, but also to identify which components of the system are no longer working in a patient. They can also help in the development of rehabilitation strategies and in measuring the treatment effects. Here, we briefly summarize the main hypotheses underlying two of the most influential theoretical models on number cognition. In particular, we highlight assumptions on calculation, as this is the focus of the intervention studies presented below (for a review on number transcoding and its rehabilitation, e.g., Willmes, 2008). Each of these two models has been extended in subsequent studies: for the sake of simplicity, these extensions are considered here as variants or additions to the main models. The first model is that of McCloskey and colleagues (McCloskey, 1992; McCloskey, Caramazza & Basili, 1985). It emphasizes the use of numbers in various codes and is purely cognitive, i.e. it does not make any reference to anatomy. The second model is that of Dehaene and Cohen (Dehaene, 1992; Dehaene, Piazza, Pinel & Cohen, 2003; Dehaene & Cohen, 1995, 1997) and is commonly known as the “Triple Code” model. This model makes assumptions about the anatomical basis of each of its components.

In McCloskey's model (McCloskey, 1992; McCloskey et al., 1985), a central semantic system is accessed during all numerical and calculation processes, independently from the input (and output) format of the numbers involved (Fig. 1). According to this model, all number formats (spoken number words, written number words, Arabic digits) are converted into abstract representations, which specify the magnitude of the number. These abstract representations serve as input and output for the calculation system (see below). The transformation (or “transcoding”) from one code to another as well as calculation necessarily occurs through the central semantic system (for “asemantic” points of view, see Cipolotti & Butterworth, 1995; Dehaene, 1992; Deloche & Seron, 1982).

In addition to number processing and transcoding, the model by McCloskey provides assumptions on the calculation system. This system entails three main components: “arithmetic signs”, “arithmetic fact knowledge”, and “procedures”. Arithmetic signs constitute an independent component, in which each sign (e.g., +, −, ×, :, =, etc.) can be selectively lost. Arithmetic fact knowledge includes information on simple calculation that is stored in semantic memory as a result of overlearning and can be accessed directly from there. As demonstrated by its preservation in severe cases of semantic dementia (Zamarian, Karner, Benke, Donnemiller & Delazer, 2006), this information constitutes a relatively independent portion of semantic memory. Examples of arithmetic facts are the multiplication tables (e.g., 2 × 3) and single-digit operations (e.g., 3 + 4, 7 − 3), whose result can be retrieved from memory quickly, without actually calculating it. According to Dagenbach & McCloskey (1992), there are segregated memory networks specific for each operation. The so-called “arithmetic rules” are also learned by heart and are part of this semantic knowledge. Arithmetic rules are operations whereby one of the elements (e.g., the addend or the multiplier) can be replaced by any number (e.g., n × 0 = 0, n × 1 = n, n + 0 = n, n + 1 = next number, n − 1 = previous number). Procedures define how to perform multi-digit addition, subtraction, multiplication, and division. They define not only the specific steps to be done (e.g., carrying or borrowing), but also the spatial arrangement and alignment of partial results in multi-digit operations.

Dehane and Cohen proposed the so-called “Triple Code” model of number processing and calculation, which has strongly influenced research and clinical practice in the following decades (Dehaene, 1992; Dehaene et al., 2003; Dehaene & Cohen, 1995, 1997). Importantly, this model makes also assumptions about the underlying anatomical correlates of its components. The model specifies three different codes (Fig. 2). The visual Arabic number code represents numbers as sequences of simple digits. This code is used for number reading and writing, in multi-digit operations and also for parity judgments. The visual Arabic code is implemented in the bilateral posterior superior parietal lobe (PSPL). The auditory verbal number code regards verbal sequences of words. This code is used in counting, reading aloud, writing verbally expressed numbers, and reciting arithmetic fact knowledge (e.g., “two-times-three-is-six”). This code depends on the perisilvian language regions of the left hemisphere, particularly on the left angular gyrus (AG). The analogue-magnitude code represents the quantity associated with a number as local distributions of activation on an oriented mental number line. It is used in number comparison, subitizing, quantitative estimation, back-up strategies, and approximate calculation. Subitizing refers to the ability to immediately appreciate small quantities, typically from 1 to 4, without apparent serial processing. The analogue-magnitude code is supported by the horizontal portion of the interparietal sulcus (HIPS) bilaterally.

The “Triple Code” model also postulates a “two-way” pathway for simple calculation (Dehaene & Cohen, 1997). A direct, asemantic verbal-route is used for overlearned calculations (in particular, multiplication tables and small additions). This direct route is supported by a cortico-subcortical circuit located in the left hemisphere, including the thalamus and the basal ganglia. An indirect, semantic route is employed in answering simple calculation problems when no verbal association is available (typically, subtraction and division problems). This indirect route requires access to quantity representations and is supported by the parietal areas bilaterally (Dehaene et al., 2003). Further extensions of the “Triple Code” model and updates on the cerebral correlates at the basis of number processing and calculation can be found in recent meta-analysis studies (Arsalidou, Pawliw-Levac, Sadeghi & Pascual-Leone, 2018; Arsalidou & Taylor, 2011; Hawes, Sokolowski, Ononye & Ansari, 2019).

Other models have been proposed as well. The “preferred-entry code” hypothesis emphasizes individual differences (Noel & Seron, 1995; Noel & Seron, 1993). Each person may use an individually preferred code to access number meaning and perform numerical tasks. Accordingly, some people would perform a calculation using a verbal entry code, while others would prefer a visual entry code. It follows that impairments of the transcoding from one specific notation (e.g., Arabic) to the preferred-entry code (e.g., verbal) will impair all numerical tasks (e.g., number comparison or calculation) presented in this notation (Arabic). In sum, some models on calculation assume that arithmetic facts are stored as verbal sequences, while others suggests that they are stored as abstract memory representations. As a consequence, models differ in their explanations of the typical effects observed in acquired calculation disorders.

Acquired impairments of calculation and number processing

The number processing and calculation system is, to a high degree, modular, i.e. its components are anatomically and functionally independent. As a consequence this system can be damaged in many ways and, therefore, acalculia is actually an extremely variable syndrome that may consist of very different and often independent symptoms. The study of single cases, conducted with the methods of cognitive neuropsychology, has indeed highlighted a great variety of very selective deficits (for a recent and exhaustive discussion, see Semenza, 2019).

In the clinical setting, the traditional classification of numerical disorders is still regarded for. Lewandowski & Stadelmann (1908) are generally considered to be the first authors who described a patient with a severe calculation disorder without linguistic disturbances. The first systematic observations of numerical disorders led to a first classification (Bergen, 1926; Henschen, 1919, 1926), later revised by Hecaen and colleagues (Hecaen, Angelergues & Houillier, 1961). This classification was based on the distinction between acalculia secondary to language disorders (due to a damage to language areas, primarily to the perisilvian areas of the left hemisphere), primary acalculia (“anarithmetia”, consequent to a left parietal damage), and spatial acalculia (secondary to generic or specific spatial disturbances, typically due to unilateral spatial neglect, generally following a damage to the right hemisphere). However, this simple classification does not account for the great variety of numerical disorders reported in the literature, whose explanation definitely profits from the theoretical models described above. Here, we shortly describe the main forms of number processing and calculation disorders.

Clinical assessment of number processing and calculation

To detect and precisely characterize numerical disorders in neurological patients, the clinical evaluation of numerical skills should consider separately the different components of the number processing and calculation system described above. Not only the patient's response times and accuracy but also the type of errors committed may be very informative of the type of numerical disorder presented. It is advisable that the evaluation of numerical skills takes place within a comprehensive neuropsychological assessment, in order to exclude a possible influence of more general cognitive deficits such as deficits in attention, memory or language. Different standardized test batteries including normative data have been proposed (Arcara et al., 2019; Claros-Salinas, 1993; Delazer, Girelli, Granà & Domahs, 2003; Dellatolas, Deloche, Basso & Claros-Salinas, 2001; Deloche et al., 1994; Ibrahimovic, Bulheller & Häcker, 2002; Kalbe, Brand & Kessler, 2002; Semenza et al., 2014). In general, these are (more or less extensive) screening batteries, i.e. typically, it is necessary to administer more items of the type that the patient has failed, once a selective numerical deficit is identified, in order to fully understand the nature and severity of the deficit. One example of numerical test batteries that is also standardized for German-speaking patients is the Number Processing and Calculation (NPC) Battery (Delazer et al., 2003). This battery assesses:

1. 1.
   Counting skills: Counting sequences (tested in oral and written format; e.g., counting backward starting from 15, counting forward two by two starting from 3); dot counting (“how many dots are there?”; dot sets of different numerosity are presented).
2. 2.
   Numerical understanding: Number comparison (“which of these two numbers is larger?”, tested in three formats: Arabic numerals, written number words, and spoken numerals); Parity judgments (“is this number odd or even?”); Analogue number scale (the position corresponding to an Arabic numeral has to be indicated on two analogue scales, one from 1 to 100, the other from 1 to 50, choosing from among three alternatives); Transcoding from Arabic numerals into tokens (the tokens are of value 1, 10 or 100, and have to be used to indicate the value of a given Arabic numeral: e.g., 33 = three tokens of 10 and three tokens of 1).
3. 3.
   Numerical transcoding: Reading aloud Arabic numerals or written number words; Writing Arabic numerals to dictation; Transcoding from written number words or from tokens into Arabic numerals.
4. 4.
   Calculation skills and arithmetic principles: Arithmetic facts (simple additions, subtractions, multiplications, and divisions, presented in Arabic format and answered orally); Multiplication multiple choice (the answer to a simple multiplication problem has to be selected among four alternatives, one of which is correct, one belonging to the same multiplication table, one belonging to another multiplication table, one not belonging to the multiplication tables); Approximate calculation (multi-digit additions, subtractions, multiplications and divisions, where the value that comes closest to the correct answer has to be chosen among four alternatives); Text problems (involving the use of one of the four operations presented in a concrete context; e.g., “A fire truck costs € 232,000 and four municipalities share the costs. How much does each pay?”); Mental calculation (multi-digit operations that must be solved orally); Written calculation (multi-digit operations that must be solved in written format); Arithmetic principles (e.g., “32 + 56 = 88, 56 + 32 = ?”).

In addition to the “classic” test batteries, one can also use more sophisticated, experimental computer tasks. Different paradigms have been proposed, including production, verification, matching, and bisection tasks. In computerized verification tasks, healthy subjects are typically slower in rejecting related results (e.g., 5 × 6 = 35) than in rejecting unrelated results (e.g., 5 × 6 = 32). Such tasks may be revealing about the degree of automaticity in arithmetic fact retrieval from LTM (Lefevre, Bisanz & Mrkonjic, 1988). In “number-matching” tasks, pairs of digits (e.g., 5 9) are presented on the screen and followed by a target number (e.g. 59). The subject has to decide whether the first two numbers are identical to the second pair of numbers presented. Healthy subjects are slower in rejecting false target numbers that match the sum (14) or product (45) of the previously presented pair than in rejecting unrelated target numbers (e.g., 47) (Galfano, Rusconi & Umiltà, 2003). Number matching tasks have been used in the assessment of arithmetic fact knowledge in patients with Alzheimer's dementia (Jurado & Rosselli, 2017) and mild cognitive impairment (Zamarian, Stadelmann, et al., 2007), as well as in a single case with semantic dementia (Zamarian et al., 2006). Table 1 gives an overview of numerical and calculation skills together with typical tasks assessing these skills.

Table 1 Examples of numerical and calculation skills together with typical tasks for their assessment

	Skills & Knowledge	Tasks
Numerical skills	Counting	Counting sequences (in oral or written format)
		Dot counting
	Number understanding	Quantity estimation
		Number comparison
		Parity judgements
		Positioning numbers on an analogue scale
		Transcoding from Arabic numerals into tokens
	Number transcoding	Reading aloud Arabic numerals or written words
		Writing Arabic numerals to dictation
		Transcoding from written number words or from tokens into Arabic numerals
Calculation skills	Arithmetic fact knowledge	Simple addition, subtraction, multiplication, and division problems (e.g., 2 × 6 = ?)
		Arithmetic rules (e.g., 3 × 0 = ?)
	Procedural knowledge	Written multi-digit addition, subtraction, multiplication, and division problems (e.g., 23 × 6 = ?)
	Conceptual knowledge	Arithmetic principles (e.g., “32 + 56 = 88, 56 + 32 = ?”)

Table 1 Examples of numerical and calculation skills together with typical tasks for their assessment

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As seen above, the components of the number processing and calculation system can be selectively disrupted or spared following brain damage. Before starting a treatment, a comprehensive neuropsychological assessment is necessary to characterize the pattern of disrupted/spared numerical abilities. In this work, we aim to present evidence that systematic rehabilitation can lead to improvement of numerical competence in adult neurological patients, even in very unfavorable circumstances (e.g., older age, chronic stage or severe impairments).

Rehabilitation methods

Most rehabilitation studies on simple calculation in neurological patients have adopted the “drill” approach, i.e. intensive repetition. Usually, in this approach, the patient is presented repeatedly with specific problems and receives immediate feedback on the correctness of the answer. This approach is based on the principles of associative learning, shared by the most current cognitive models on the representation of arithmetic fact knowledge in LTM (Ashcraft, 1995; Campbell, 1987; Siegler, 1988; for a review, Domahs & Delazer, 2005). According to these models, arithmetic facts, once they are learned by heart, are stored in an associative network in declarative memory. Through intensive repetition, the association between a problem and its correct solution should be (re-)strengthened. Immediate feedback is given to prevent the formation or strengthening of incorrect associations. Similarly, to foster error-free learning, one can present simultaneously a problem together with its correct answer. Some studies have combined intensive repetition with the use of a sort of help or “cue” (see below; Domahs,Lochy, Eibl & Delazer, 2004, Domahs, Zamarian & Delazer, 2008). In principle, simple calculation relies not only on the retrieval of arithmetic facts from LTM but also on the execution of procedures and the application of conceptual knowledge (Delazer, 2003). It follows that the (re-)learning of arithmetic facts can be based not only on intensive repetition but also on the improvement of procedural or conceptual knowledge (Girelli, Bartha & Delazer, 2002). For example, one can teach the patients backup strategies that are based on the use of arithmetic principles (e.g., decomposition principle; e.g., 6 × 3 = 6 + 6 + 6) (Girelli et al., 2002). Some studies have also shown positive training effects by adopting both conceptual and drill approaches (Domahs, Bartha & Delazer, 2003) (see Table 2 for a summary of the main rehabilitation approaches on simple calculation).

Table 2 Rehabilitation approaches on simple calculation

Rehabilitation approach	Training method
“Drill” approach	Intensive repetition of selected arithmetic facts
“Conceptual” approach	Use of back-up strategies, reference to arithmetic principles
“Mixed” approach	A combination of the two above mentioned approaches
“Unspecific” approach	A combination of reading and arithmetic tasks to improve domain-general functions (e.g., executive functions)

Table 2 Rehabilitation approaches on simple calculation

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Rehabilitation of arithmetic fact knowledge through intensive repetition

As described above, most rehabilitation studies on simple calculation have used the “drill” approach (Miceli & Capasso, 1999). For example, Kashiwagi, Kashiwagi and Hasegawa (1987) described eight aphasic patients having difficulty with multiplication and division but not with addition and subtraction. Patients underwent a month or two of daily intensive “drill” training of multiplication tables. After training, all patients showed an improvement in multiplication facts retrieval. This improvement was, however, very specific as it was evident only in tasks where the presentation and response modes were identical to those used during training.

Hittmair-Delazer and colleagues (1994) also adopted intensive training of multiplication facts. They reported on a 45-year-old accountant (patient BE), who had suffered from a cerebral embolism in the left basal ganglia and had a highly selective deficit with multiplication and division facts. BE could retrieve from memory only the 2 times table and the so-called “ties” (e.g., 4 × 4). Other problems were solved through strategies based on arithmetic principles (e.g., 8 × 4 = 8 × 2 + 8 × 2), which led him to the correct solution but were complicated and time-consuming. Problems that could not be solved through direct retrieval from memory were divided into two sets and presented separately according to an experimental design of the ABACA type (where A indicates a test session, B the training with the first set, C the training with the second set) (Hittmair-Delazer et al., 1994). After an intensive training of four weeks with each set, BE showed highly specific performance improvements. That is, he could quickly solve the problems exercised but not, for example, problems complementary to those practiced during the training (e.g., problem practiced 3 × 4; complementary problem 4 × 3). BE was, however, able to apply the learned multiplication facts to solve complex written calculations and divisions.

Girelli, Delazer, Semenza and Denes (1996) also used a very similar training with their patients TL and ZA, who had severe difficulties in retrieving multiplication facts from memory. After an intensive training of eight weeks, which had an experimental design of the ABACA type (see above), TL and ZA showed significant improvements by responding correctly to more than 90% of the trained problems. Improvements of both patients were greater with the trained problems than with untrained problems, although they also showed some generalization effects. Through training, accuracy increased but also the type of errors changed. For example, before training, TL mostly committed mistakes of the type 2 × 9 = 44, where the answer given did not belong to any tables. After training, however, TL's errors were mostly of the type 6 × 3 = 21, where the answer given belonged to the same table as the correct one. ZA's errors before training were similar to those of TL. However, after training, ZA mostly committed errors that were numerically close to the correct answer (e.g., 5 × 6 = 31). According to the authors (Girelli et al., 1996), the type of errors after training reflected the type of strategies adopted. Likely, TL recited the series of multiplications belonging to a table, stopping one answer before or after the correct one. ZA used repeated addition, sometimes committing calculation errors (e.g. 4 × 6 = 6 + 6 + 6 + 6 = 25). Patients ZA and TL could also apply the acquired knowledge in new contexts such as, for example, with divisions or written problems.

Difficulties in the retrieval of multiplication facts were also presented by patient MC, a 42-year-old computer programmer, who had a brain tumor in his left parietal lobe surgically removed (Whetstone, 1998). During training, MC intensely repeated three different sets of multiplications that were presented separately in a single specific format (phonological, graphemic or Arabic code). Problems were presented together with their correct answer to avoid the formation or strengthening of incorrect associations. Each training session was, however, preceded and followed by a test in which MC had to produce the solution himself. After training, MC responded highly accurately (97% correct) but tended to give faster responses with the problems presented in the exercised code than in a different one.

The next two studies adopted both intensive repetition and the use of some sort of help (or “cue”). Domahs and colleagues (2004) described ME, a 38-year-old mechanic who had suffered a head injury three years prior to their investigation. ME had difficulties in particular with multiplication and division. He could answer most problems correctly (98.0% and 93.3% correct answers, respectively) but was extremely slow (he needed up to 90 seconds), which suggests the use of backup strategies rather than direct retrieval from LTM. During training (12 sessions), problems were presented in nine different colors based on the unit of the result (e.g., the 4 × 3, 6 × 2, 8 × 4, 7 × 6 and 9 × 8 problems were all presented in yellow since the unit “2” of the result, i.e. 12, 12, 32, 42 and 72, was associated with the color yellow). After ten “colored” training sessions, ME could respond fast to the trained problems, even when these were presented in black. ME also showed improvements with complementary multiplications and untrained multiplications.

In a subsequent study, Domahs and colleagues (2008) aimed to understand whether this facilitating effect was due to a special relationship between colors and numbers in the brain. In their new study, Domahs and colleagues (2008) administered a “sound”-based training to their patient EK, an 81-year-old accountant who had suffered a cerebral infarction a few years earlier. During training (15 sessions), two sets of different multiplications (set A and set B) were repeated intensively. While problems of set A were presented without cue, those of set B were presented together with a cue (a sound). According to the same principle used in the previous study, problems of set B were presented together with different sounds, associated with the second digit of the result. Unlike the previous study, however, the authors did not explicitly explain to the patient the function of the sounds and the underlying association principle. After training, EK showed significantly faster reaction times with both sets. Improvements in accuracy were, however, higher with problems of set B (with cue) than with those of set A (without cue). Considering the effectiveness of the “sound”-associated training, the results reported by Domahs et al. (2004) cannot be simply explained by a “special relationship” between numbers and colors. Results of both studies demonstrate the effectiveness of a multimodal rehabilitation approach and that “implicit” cues can help in the rehabilitation of simple calculation in adult patients whose numerical abilities are partly preserved.

Rehabilitation of arithmetic fact knowledge through a conceptual training

A conceptual approach to the rehabilitation of simple calculation in adult neurological patients was adopted, for example, by Girelli et al. (2002) and by Domahs et al. (2003). In Girelli et al. (2002), patient FS, a 64-year-old retired bank employee, had severe difficulties to retrieve multiplication facts and to use independently back-up strategies to find out solutions. The authors focused the training on the strategic use of FS's residual knowledge and on the explicit reference to arithmetic principles (for example, commutativity or decomposition principle). Aim of the training was to help FS deriving unknown solutions from problems that he could retrieve automatically and, at the same time, reducing the amount of information to be stored. FS was taught a specific backup strategy for each problem to be learned (e.g., 2 × 4 = 4 × 2; 3 × 4 = 3 × 3 + 3; 3 × 9 = 3 × 10 − 3). Problems were divided into two sets, one set practiced a first week, a second set on the following week. After the first training week, FS showed greater accuracy with both trained and untrained sets. Improvements were, however, greater with the trained set. After the second week, FS achieved a perfect performance. After training, FS also showed to be able to use different strategies than the ones explicitly taught.

Domahs et al. (2003) used both a conceptual and a drill approach. Patient HV, an year-old former manager, had a severe calculation deficit. He was unable to retrieve multiplication tables from memory and showed little conceptual knowledge of the underlying operation. HV had, however, less difficulty with addition. Therefore, he underwent first a “conceptual” training, where he learned to solve a set of multiplication problems through repeated additions. After five “conceptual” training sessions, he was presented with six sessions of “drill” training, where he intensely repeated the same problems. Accuracy with trained multiplications improved to 63.6% after the conceptual training and to 70.0% after the “drill” training. Improvements with both, trained and untrained problems, were only found after the conceptual training but not after the drill training. As a result of conceptual training, the type of errors also changed qualitatively, becoming more plausible.

Overall, these two studies demonstrate the effectiveness of a short conceptual rehabilitation approach to simple calculation. Furthermore, they show that the newly acquired conceptual knowledge proves to be flexible, as it can be also applied to new contexts. As the studies suggest, repetition training alone is not successful in cases of conceptual deficits. Indeed, in these cases conceptual training should precede any form of repetition or drill. Although in some cases after a conceptual training performance fails to reach perfection, errors become more plausible, due to the type of strategies used.

Neural correlates of performance improvements following a rehabilitation of arithmetic fact knowledge

There are also studies that have addressed which brain activation changes occur following the rehabilitation of calculation skills in neurological patients (Claros-Salinas et al., 2014; Zaunmüller et al., 2009). Zaunmüller and colleagues (2009) administered an intensive training of multiplication facts (40 sessions over 4 weeks) to WT, a 49-year-old patient, who suffered a left brain damage 29 months earlier. Both before and after training, WT was assessed with functional magnetic resonance imaging (fMRI). Post-training fMRI results showed an activation increase in the right AG in the contrast trained versus untrained problems. An increase was, therefore, found in the hemisphere contralateral to the side of the lesion, in an area homologous but contralateral to that typically activated by healthy subjects during arithmetic fact retrieval (for a review see Zamarian, Ischebeck, & Delazer, 2009). Claros-Salinas and colleagues (2014) described seven patients with dyscalculia following differently located brain lesions. Patients underwent a computer training on the four basic operations (14 sessions over 4 weeks). As in the study by Zaunmüller and colleagues (2009), they were assessed through fMRI both before and after training. Post-training fMRI results showed an association between performance improvements and a reduction in activation of prefrontal areas, specifically in the middle and superior prefrontal gyrus. According to the authors, these results suggest that performance improvements following training reflect an increased effectiveness in the functioning of the prefrontal lobes.

The results of these two studies are indeed very interesting, but they must be interpreted with some caution. In fact, the great variability among the participants, regarding both the localization and etiology of the lesion and the type of disorders shown, calls for further confirmation.

“Unspecific” arithmetic training in older people with or without cognitive decline

The studies reported above should be differentiated from those using calculation as training material to maintain or improve cognitive functioning in older age. For example, Kawashima et al. (2005) administered a 6-months cognitive training to patients with Alzheimer's disease, who were required to perform 20min a day (2–6 days a week) both reading and arithmetic tasks. Compared to patients in the control group, who did not perform any training, patients in the experimental group showed at follow-up improvements in a screening test battery of executive functions (Kawashima et al., 2005). Similar results were reported for healthy older people (Nouchi et al., 2016) and older postoperative patients who underwent general anesthesia (Kulason et al., 2018). Although interesting, these studies cannot be considered as direct evidence of training-related arithmetic improvements. However, they demonstrate the indirect positive effects of an intensive, although unspecific training on general cognitive functioning in older adults.

Limitations

Although the literature search used various databases, we cannot rule out that individual relevant works were not included in the study. In addition, the review includes only works written in English. It is possible that further research written in other languages would have contributed to a more comprehensive picture of the existing literature. This review does not claim to be systematic or exhaustive.

Relevance for the practice

This study shows that improvements of calculation deficits are possible through a targeted training program even in very unfavorable circumstances. Although tailored to the patient's needs and characteristics, the clinical assessment and the training program should be systematic and possibly based on well-established theoretical assumptions and evidence. For the professionals involved (psychologists, therapists …), this implicates an ad-hoc preparation that includes both theoretical and practical knowledge on number cognition. Thus far, very few Universities and post-gradual training institutes, at least in German-speaking countries, offer seminars or practical opportunities in this area. Professionals are therefore mostly required to rely on their own initiative and self-education. This makes the rehabilitation of number deficits something “elitist”, if compared to the rehabilitation of other cognitive deficits (e.g., language, memory, executive functions, attention, or visual exploration), which cannot be offered to all the patients needing it. Specific training for professionals in number cognition should be posit among the priorities of public health policies, strategies, and plans.