According to Baddeley's model (Baddeley and Hitch, 1974; Baddeley, 1986) and subsequent adaptations (Gathercole and Alloway, 2006), memory may be divided into two components, namely a more passive one, also known as short-term memory (STM), which is dedicated to the retention of transient information stored in the phonological loop and visuo-spatial sketchpad; and a more active one, also known as working memory (WM), which corresponds to the central executive and is involved in information elaboration. Memory plays a relevant role in the development of math-related skills (Alloway and Passolunghi, 2011; Friso-van den Bos et al., 2013; Peng et al., 2016), and, more specifically, in the following two main aspects: (1) elaboration of phonological and visuo-spatial components, and (2) level of information manipulation (passive vs. active processes). Although the distinction between phonological and visuo-spatial components of memory has been largely investigated in the literature with regard to typical development (Baddeley, 1986, 2000, 2002), the volume of data on the active vs. passive distinction is still limited (Cornoldi and Vecchi, 2003; Gathercole and Alloway, 2006; Passolunghi and Cornoldi, 2008), especially in relation to autism. It is our belief that a WM evaluation for each component is crucial, as it allows researchers to separately investigate the phonological and visuo-spatial component, on the one hand, and the active manipulation and transformation of data while recalling them, on the other. The latter model, evaluating passive and active component of WM, is observed through tasks that require recalling data in the same format in which they are presented (i.e., passive tasks: the subject is asked to recall a list of numbers, patterns, or positions in the same order they were presented), or through tasks that require multi-level management of incoming data (e.g., The Listening Span Test, Daneman and Carpenter, 1980).

Effects on memory at componential and manipulation level were tested through an analysis tool specifically developed to investigate the arithmetic skills displayed by a child with autism spectrum disorder without intellectual impairment. Our goal was to yield results able to provide a fine-grained evaluation of the relevant component- and process-related aspects underlying WM performance. We found that children with autism spectrum disorder without intellectual impairment are far from numerous among our patients, and that high-level cognitive tests are quite demanding for autistic children with average intellectual abilities.

Research on WM and math abilities (MA) in children with autism still accounts for a limited share of studies in the literature, although investigations carried out on typical development clearly show a significant correlation between these two variables (Alloway and Passolunghi, 2011; Friso-van den Bos et al., 2013; Peng et al., 2016). This knowledge gap may be due to (1) reported inconsistencies in studies on WM in autism (Kercood et al., 2014), (2) divergent results in studies on MA in autism (Meyer and Minshew, 2002; Mayes and Calhoun, 2006; Iaculano et al., 2014), and (3) an overall paucity of scientific work on componential aspects of MA in relation to this pathological condition (Titeca et al., 2015).

A recent review of the literature on memory in autism shows that research published so far has focused mainly on componential aspects: children with autism seem to be as likely as typical children to activate articulatory rehearsal, despite performing poorly in spatial tasks (Kercood et al., 2014). However, the active/passive paradigm in autism remains largely understudied. Our case report aims at affording new insight to the scientific community that may help fill this knowledge gap, by providing a comprehensive picture of calculation abilities in an individual with autism spectrum disorder without intellectual impairment (hereafter M.N.) through both a set of math-related tasks, specifically aimed at testing arithmetic skills, and a set of WM tasks covering different processes (passive vs. active) and components (phonological vs. visuo-spatial). Our testing method aims at yielding results that may indicate a relation—if any—between the type of material or process and the child's performance. On the basis of the results obtained by Kercood et al. (2014), our hypothesis was that M.N. would perform better in tasks involving the verbal component and worse in tasks involving the visuo-spatial component. Moreover, as far as the passive vs active process debate is concerned (Gathercole and Alloway, 2006), we expected M.N. to perform very well in WM tasks requiring a passive manipulation of data (e.g., Digit Span Forward test), and to perform averagely well in tasks requiring an active manipulation of data (e.g., Listening Span test), independently of the componential level of analysis. This hypothesis is based on previous studies carried out on children with typical as well as atypical development and their manipulation of memorized pieces of information (Passolunghi and Cornoldi, 2008).

As regards arithmetic skills, we expected M.N. to solve simple oral arithmetical calculations, like subtractions or additions, quite effortlessly, as such tasks largely depend on a passive component of memory, i.e., effective memory storage or Short Term Memory (Gathercole and Alloway, 2006). Conversely, M.N. was expected to score sensibly lower in active manipulation of data, such as the one involved in oral division and multiplication operations, given the cognitive challenge of successfully performing complex semantic representations of data (Gathercole and Alloway, 2006). Finally, written operations, which allow for a lower level of memory load regardless of the complexity of multi-digit figures, were expected to yield average test scores.

At the time of the study M.N. was a fourth-grader of 10 years and 9 month of age. He was diagnosed with autism at 3 years of age, on the basis of criteria illustrated in the DSM IV-TR (American Psychiatric Association, 2000), resulting from two structured research diagnostic instruments administered by a team of specialized professionals including a neuropsychologist, a psychologist, and an educator. Autism Diagnostic Interview Revised tool (Lord et al., 1994) showed a qualitative impairments in (1) Reciprocal social interaction (score 20, cut-off-10), (2) Communication (verbal score 16, cut-off-8, non-verbal score 9 cut-off-7), (3) Repetitive behaviors and stereotyped patterns (score 4, cut-off-3), the child showed the abnormality development before 36 months. The Autism Diagnostic Observation Schedule (ADOS) showed an algorithm score 20 (Lord et al., 2000). General intelligence was tested tool trough the WISC-IV tool (Wechsler, 2003), which resulted in a synthetic score of 89¹. M.N.'s profile displays significant discrepancies, particularly between verbal comprehension (lowest score) and perceptual reasoning (higher score). Furthermore, M.N. obtained average scores in relation to Working Memory and Processing Speed Indexes. Indeed, he performed averagely in all tasks, except for Block Design and Picture Completion (above average), Vocabulary and Matrix Reasoning (below average), and Comprehension (extremely below average).

We put together a control sample comprising twenty 10-year-old children (mean age: 122.25 months; 11 boys and 9 girls) with Typical Development (T.D.). Evaluation was carried out at the end of fourth grade. Parents (both mothers and fathers) were asked to sign an informed consent to (1) authorize their children's participation in the study and (2) authorize the publication of the research report.

M.N.'s memory abilities were evaluated through tasks testing WM-related componential and manipulation levels using Digit Span Forward, Listening Span, Visual Pattern Test, and Counting Span Test. M.N.'s performance can be described as follows: compared to the typical development (T.D.) group, M.N. was better at remembering numbers in the Digit Span Forward test [t₍₁₉₎ = 3.67, p = 0.002], scoring a 99.92 percentile (95% CI: 99.36–100.00). In the Listening Span test M.N.'s performance was within the average [t₍₁₉₎ = 0.15, p = 0.880]. M.N.'s score was approximately 55.98 percentile (95% CI: 38.71–72.42), with respect to the T.D. distribution.

The visuo-spatial sketchpad was carried out via the Visual Pattern test, whose indexes yielded significant results: M.N. scored a 99.87 percentile (95%: CI 99.04–100%) in the matrix test [t₍₁₉₎ = 3.46, p = 0.003], and a 99.77 percentile (95% CI: 98.44–100%) in the box test [t₍₁₉₎ = 3.21, p = 0.005], with respect to the T.D. group. M.N. obtained an average score with respect to the control group in the Counting Span Task [t₍₁₉₎ = 0.39, p = 0.18] as shown in Table 1.

The MA profile of M.N. was outlined using the AC-MT tool (Cornoldi et al., 2002) and Evaluation of Math Abilities tool for fourth grade (Giovanardi Rossi and Malaguti, 1998). M.N.'s performance was subsequently compared with the Italian population norms. The scores obtained by M.N. are in line with his grade's average. More specifically, M.N. reached the 50 percentile in “Written calculation in class”, 50 percentile in “Numerical knowledge”, 50 percentile in “Accuracy”, and between 50 and 60 percentile in terms of “Total time” as shown in Table 2.

M.N.'s arithmetic written and oral skills were assessed using the Evaluation of Math Abilities tool (Giovanardi Rossi and Malaguti, 1998): M.N. performed well in addition and subtraction in oral calculation (total score 13/13), and insufficient in oral multiplication and division (total score 6/10). In Written calculation M.N. performed well in addition and subtraction (total score 8/8) and discretely in multiplication and division (total score 6/8). All results are shown in Table 3.

WM and MA have been widely investigated in typical development, but only sporadically in subjects with autism. The present study is an attempt at shedding some light on this topic, by providing an account of both mathematical-ability- (Titeca et al., 2015) and working- memory-related (Gathercole and Alloway, 2006) components, and the processes that lie at the basis of the connection between the two. In this study we analyzed the calculation abilities of a child with autism spectrum disorder without intellectual impairment (M.N.) and an average intelligence level, with the goal of testing his arithmetic abilities in connection with WM performance.

SP designed the study, drafted a first version of work and provided a final approval of the manuscript. MP supervised a critical revision of the work and proposed the final approval of the version to be published.