原著
発達性計算障害1成人例における数概念と計算の障害
2026 年 46 巻 2 号 p. 24-50
詳細
2026 年 46 巻 2 号 p. 24-50
Background：There have been many reports indicating that individuals with developmental dyscalculia show difficulties in acquiring arithmetic facts（e.g., 2 ＋ 3 ＝ 5). In contrast, few studies have examined impairments in numerical concepts, a more basic level of numerical representation. This study presents an adult case of developmental dyscalculia in which numerical magnitudes up to five can be understood from Arabic numerals, whereas those for single-digit numbers greater than five and multi-digit numbers cannot. The symptoms observed in this case may provide insights into the structure of numerical concepts and the abilities required for their acquisition.
Methods：The participant was a right-handed woman in her 20s and a fourth-year university student. Numerical and calculation abilities were assessed using subtests examining basic number manipulation, numerical concepts, reading and writing of numbers, knowledge of numbers and calculations, calculation ability, and left-right orientation and finger gnosis.
Results：The participant showed impaired understanding of numerical magnitude for single-digit numbers above approximately five and difficulty grasping the meaning of multi-digit numerals. In number decomposition and composition tasks involving zero, both errors and self-corrections were observed. She had not acquired basic arithmetic facts for single-digit addition and subtraction and had difficulty recalling multiplication tables. Although she understood procedures for multi-digit written multiplication and division, she was unsure where to write the numbers. Cardinality（understanding that the last counted number represents the total quantity）and perceptual subitizing were preserved, whereas conceptual subitizing（the rapid grasp of small quantities as structured wholes）was impaired.
Conclusions：This case, showing impaired numerical concepts above five, together with previous reports, suggests that numerical concepts differ between single- and multi-digit numbers and vary even within single-digit numbers, particularly below versus above about five. Furthermore, cardinality alone may be insufficient for acquiring numerical concepts, and conceptual subitizing may also play a role.
発達性計算障害1成人例における数概念と計算の障害について報告した。本例はおおよそ5を超える1桁の数について数字から大きさを判断できず，2桁以上の数については数字から量的な感覚をつかめないという特徴を示し，数概念に障害があると考えた。また，1桁同士の加減算の算術的事実が獲得できておらず，九九の想起にも困難を示した。複数桁の筆算では，計算手続きは理解しているが数字を書く位置がわからないと述べ，位の概念の障害が疑われた。本例と過去の報告例から，数概念は1桁と複数桁の数では異なる構造を持つこと，1桁の数であっても数概念は一様ではなく，おおよそ5までの数とそれより大きい数では異なる可能性があることが示唆された。また，基数性は数概念獲得の基礎であると考えられているが，数概念の獲得には基数性だけでは十分ではなく，概念的サビタイジングのような数をパターンとして認識する能力が関与するのではないかと考えられた。
Acalculia refers to acquired pure calculation impairments not caused by other cognitive dysfunctions such as aphasia, attentional deficits, or visuospatial impairments. Some reports on acalculia show impaired number concepts (Cipolotti, et al., 1991, Furumoto, et al., 1993, Suzuki, et al., 2005, Nagatomo, et al., 2009), impaired recall of simple addition/subtraction or multiplication facts―including kuku, the Japanese term for memorizing basic multiplication table (1×1 to 9×9)(Warrington, 1982, Delazer, et al., 1997, Hittmair-Delazer, et al., 1994, Hirayama, et al., 2002）―, and impaired calculation procedures in multi-digit written calculations (Lucchelli, et al., 1993).
Developmental dyscalculia is classified as a learning disability within the field of education, specifically as a disorder involving “impairment in calculation or reasoning abilities” (Ministry of Education, Culture, Sports, Science, and Technology, 2013). In the medical field, it is listed as a “specific disorder of arithmetical skills” under “specific developmental disorders of scholastic skills” in the International Classification of Diseases, 10th Revision (ICD-10)(World Health Organization, 2005), which states that “The deficit concerns mastery of basic computational skills of addition, subtraction, multiplication, and division rather than more abstract mathematical skills involved in algebra, trigonometry, geometry, or calculus.” In the Diagnostic and Statistical Manual of Mental Disorders, 5th Edition (DSM-5)(American Psychiatric Association, 2014), “number sense,” “memorization of arithmetic facts,” “accurate or fluent calculation,” and “accurate math reasoning” are listed as specific symptoms under “Specific Learning Disorder.”
The “mathematical facts” mentioned are also called “arithmetic facts,” referring to knowledge about simple single-digit calculations stored as long-term memory (Ashcraft, 1992, Butterworth, 2005). Once arithmetic facts are acquired through learning arithmetic, individuals can respond by retrieving the answer from long-term memory；therefore, it is possible to derive answers for simple single-digit calculations without much thought. By contrast, in cases of developmental dyscalculia, it is reported that children often derive answers by counting on fingers, counting aloud, or counting in their heads, even as they advance to the grade level. Difficulty in acquiring arithmetic facts is considered a characteristic of developmental dyscalculia (Jordan, et al., 2003, Geary, et al., 2004, 2012).
Additionally, regarding characteristics related to “number sense” in developmental dyscalculia cases, difficulties in estimating large numbers (Decarli, et al., 2020) and number line tasks (Lamb, et al., 2024, Ruiz, et al., 2024) have been reported. Regarding subitizing (instantly grasping the number of dots, etc., without counting individually)(Starkey, et al., 1980, Dehaene, 2010), some reports state that it is preserved (Decarli, et al., 2020, Lamb, et al., 2024), whereas others report differences in reaction times (Landerl, et al., 2004).
The author had the opportunity to assess the case of an adult who had been aware of difficulties with grasping numbers and calculations since early elementary school, at the participant’s own request. No prior detailed reports have described impairments in number concepts and calculations in adult cases of dyscalculia, making this case valuable. In addition to reporting and discussing the impairments regarding numbers and calculations that this individual faces daily and the results of the assessment conducted, I consider the structure of number concepts through previously reported cases and the symptoms in this case.
Case Presentation. The participant is a right-handed woman in her 20s. She is a fourth-year university student. She offered to participate in the study after learning that the author had a research interest in difficulties with numbers and calculations. She has never received any formal assessment or support for difficulties with numbers or calculations since childhood.
Episodes Regarding Numbers and Calculation. The participant has been aware of her difficulties with numbers and calculations since early elementary school. She reported that during math class, she would leave the classroom and read books in the library. She also mentioned that when she used her fingers to think, teachers would frequently point it out, asking, “Are you still calculating with your fingers?” or saying “Don’t use your fingers.” This caused her such distress that she came to dislike the title “Sensei (teacher）” from elementary school all the way through high school. When asked about her difficulties in elementary school, she stated, “I understand fractions. I don’t understand decimals at all. I don’t understand percentages either. I don’t understand number lines. I can’t convert numbers written in Kanji―logographic characters of Chinese origin used in Japanese―into Arabic numerals. I also can’t convert numbers written in Arabic numerals into Kanji.” Although she struggles with simple addition and subtraction and recalling Kuku (Japanese multiplication tables), she stated that she was good at systems of equations, proofs, and geometry problems learned in junior high school, and remarked, “My math grades in junior high were good”.
Difficulties in grasping numbers and performing calculations have persisted into adulthood, hindering various aspects of daily life. For example, she tends to overestimate monetary amounts；therefore, she notes that when shopping, the payment amount is often considerably lower than her expectations. Additionally, because it is difficult for her to estimate the amount to pay, she sometimes manages by paying with the largest bill in her wallet at staffed registers, and by inserting all her coins at self-checkout registers. Regarding time, she finds analog clocks easier to understand than digital ones, but because she cannot grasp the time without numeric markings, she manages to check the position of the hands on an analog clock and confirms the exact times using a digital clock. Regarding age notation, she stated that expressions such as “X years and Y months” are hard to understand, and if it said “36 months,” she did not know how old that is.
Neuropsychological Findings. She participated in tests for concentration. On the Wechsler Adult Intelligence Scale-Fourth Edition (Wechsler, 2018), her scores were as follows：Full Scale IQ 109, Verbal Comprehension Index (VCI) 128, Perceptual Reasoning Index (PRI) 95, Working Memory Index (WMI) 88, and Processing Speed Index (PSI) 111 (Table 1). Among the subtests, the scaled scores for Arithmetic (scaled score 4；range：1-19) and Visual Puzzles (6) were lower than those for the other items. Regarding reading and writing characters, she was able to take notes during university lectures and reported feeling no hindrance. When dictation of short sentences (about 4 phrases) was conducted, she was able to dictate sentences with a mixture of Kanji and Kana (Kana are phonograms that represent sounds, whereas kanji are logographic characters that represent meaning). As for neuropsychological symptoms, impairments in number concepts and calculation (described later) and difficulty distinguishing between the left and right sides were observed.
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The test items, contents, and results are presented in Table 2. The assessment was conducted from October to November 2022 in the training room of the Division of Speech-Language-Hearing Therapy at Kumamoto Health Science University. M.N., a speech-language therapist, conducted the following detailed assessment. Our investigation encompassed basic number manipulation；assessment of number concepts；reading and writing of numbers；knowledge of numbers and calculations (including linguistic knowledge and operation symbols）；assessment of calculation ability；and right-left orientation and finger gnosis.
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The subtests were created based on the numerical abilities associated with the four-stage developmental model of numerical cognition (von Aster, et al., 2007) and the subtests included in the assessment of numerical activities in daily life (Semenza, et al., 2014, Burgio, et al., 2021), with some modifications to the content and methods of the tasks. We also added some original subtests to this.
The section of Basic Operations of Number has four subtests aimed at assessing the abilities that are fundamental to the acquisition of number concepts：Counting, Backward counting, Subitizing, and Magnitude comparison (dots). Counting, Subitizing, and Magnitude comparison (dots) were selected from items related to steps 1 and 2 of the four-stage developmental model of numerical cognition (von Aster, et al., 2007), and Backward Counting (dots) was newly added. The subitizing tasks were designed based on the ideas of perceptual and conceptual subitizing (Clements, 1999).The section of Assessment of Number Concepts consisted of eight subtests aimed at determining whether the participant had acquired number concepts, particularly whether she understood multi-digit numbers.：Magnitude Comparison (digits), Number Composition, Number Decomposition, Questions Regarding the Decimal System , Number line tasks, Quantitative Sense of Numbers, Estimation, Digits Alignment. The number line task was created based on Lamb et al. (2024) with the correct answer criteria changed. The other subtests were originally created for this study. The section of Reading and writing numbers was designed to assess the ability to transcode Arabic numerals and number words and consisted of two subtests：Reading numbers aloud and Writing numbers. The section of Knowledge regarding numbers and calculations consisted of two subtests aimed at assessing knowledge of numbers and calculations required in everyday life situations.：Linguistic knowledge regarding numbers, Understanding of operators. The section of Assessment of calculation ability consisted of six subtests designed to assess the ability to perform simple single-digit calculations and multi-digit calculations.：Simple calculations include Single-digit addition, Subtraction with single-digit subtrahend and answer, Multiplication tables (Kuku), Division answerable by recalling Kuku multiplication table. Complex calculations include Multi-digits addition/subtraction (vertical written calculation), Multi-digits multiplication/division (vertical written calculation). For the simple calculation tasks, the way the tasks were presented was changed, with the problems presented on paper instead of orally and the participants being asked to write down the answers. In addition, to assess the calculation strategies used when solving the problems, we asked participants how she get the answer for each problem, and classified the calculation strategies according to Siegler (1987) and Geary, et al. (2004, 2012).
Out of the test items, stimuli for Magnitude Comparison (digits), Number Composition, Number Decomposition, Questions Regarding the Decimal System, Quantitative Sense of Numbers, Estimation, reading numbers aloud, understanding of operators, single-digit addition, subtraction with single-digit subtrahend and answers, multiplication (Kuku), and division answerable by recalling Kuku, were presented on A4-sized papers using 14 pt. font size text. The stimuli for multi-digit addition and subtraction problems (vertical written calculations) and multi-digit multiplication and division problems (vertical written calculations) were presented on A4-sized papers using 18 pt. font size text. And the stimuli for the subitizing task and the size comparison (dot) task were presented on a card measuring 6 cm in height and 7.5 cm in width.
3.1. Basic Manipulation of Numbers 3.1.1. CountingThe task of counting numbers from 1 to 20. She passed this test.
3.1.2. Backward CountingThe task of counting numbers from 20 to 1. In this task, although her response was correct,she expressed that “I had to think a little for 15, 14, and 13.”
3.1.3. SubitizingDots with a diameter of 4 mm were printed on a card measuring 6 cm in height and 7.5 cm in width. Tasks included randomly arranging 1-5 dots (Figure 1a) and 1-10 dots at equal intervals of 6 mm（Figure 1b). In the task with equal intervals, for 1-5, the dots were arranged in a single horizontal row；for 6-10, 5 dots were arranged in a lower single horizontal row；and the remaining dots were arranged above in a single horizontal row from the left. For example, in the case of 8, 5 dots were arranged below, and 3 dots were arranged above, each in a horizontal row (Figure 1b). These cards were presented individually and the participant was asked to quickly count the number of dots. Additionally, for each question, she was asked whether she knew immediately or if she counted. As a result, in the task with randomly arranged dots, she reported “I knew immediately” and could instantly grasp the numbers for 1-3 dots, but reported “I counted them” for 4 dots. For 5, she reported “I knew immediately. Because it’s the same arrangement as the face of the dice.” In the task where 1-10 dots were arranged at equal intervals in a single horizontal row or 2 vertical rows based on 5, she reported “I knew immediately” for 1-4 dots, but reported “I counted them all” for 5-10.
a. Stimulus with 1-5 dots arranged randomly. b. Stimulus with dots arranged in row（s）；1-5 dot（s) are arranged in a single horizontal row, and 6-10 are arranged in 2 rows.
3.1.4. Magnitude Comparison （dots）
Cards with 1-10 dots arranged at equal intervals were used in the subitizing task. Three cards were placed randomly on the desk and she was asked to rearrange them in descending order. The 5 questions were (2, 3, 5), (3, 6, 9), (1, 3, 8), (4, 6, 7), and (5, 7, 10). She answered all 5 questions correctly.
3.2. Assessment of Number Concepts 3.2.1. Magnitude Comparison （digits）The participant was asked to choose the larger of 2 numbers ranging from 1 to 4 digits (9 questions with the same number of digits；1 question with different numbers of digits). The 10 questions were (8, 6), (3, 5), (24, 31), (62, 59), (91, 78), (97, 103), (475, 512), (689, 791), (4032, 4029), and (9034, 8967). She answered these questions correctly. When asked about her particular impressions, she reported “I thought a little and got it” for the comparison of (8, 6). For (24, 31), (62, 59), and (91, 78), she reported “I compared at the tens place.” For (91, 78), she was observed representing “7” and “9” with her fingers to think. For (97, 103), where the number of digits differed, she could easily judge magnitude, stating “Because there are 3 digits and 2 digits.” For (689, 791）and (9034, 8967), she reported “I looked at the first digit and counted forward from 5,” and for (4032, 4029), she reported “I looked only at the third digit and gave the answer.”
3.2.2. Number CompositionComposition tasks of 2- to 3-digit numbers were conducted. For questions such as “What do you get with 5 10s and 7 1s?,” the participant was asked to answer by writing Arabic numerals. Consequently, 3 out of 5 were correct, and errors were seen in questions involving “0” ：for the question “What do you get with 4 100s and 2 1s?,” she wrote “42,” and for “What do you get with 7 100s and 5 10s?,” she wrote “75.”
3.2.3. Number DecompositionThe participant was asked to fill in the blanks with Arabic numerals for questions such as “45 is [ ] 10s and [ ] 1s.” Five questions were posed and all were correct, but self-correction was observed in tasks involving “0.” In the question “308 is [ ] 100s, [ ] 10s, and [ ] 1s,” she first wrote “8,” then she drew a double line through it and corrected it to “0” for the tens digit.
3.2.4. Questions Regarding the Decimal SystemThe participant was asked to orally answer the following 5 questions presented visually： “What does it become when 10 1s combine?” ； “What does it become when 10 100s combine?” ； “What does it become when 10 10s combine?” ； “What does it become when 10 10000s combine?” ；and “What does it become when 10 1000s combine?”. She answered 4 out of 5 questions correctly and did not provide any answer for 1 question. For 3 of the 4 correctly answered questions, she wrote some numbers on the question sheet to think, and then answered orally.
3.2.5. Number Line TaskOn an A4 size paper with a 20 cm straight line printed in the center, with the left end as 0 and the right end as 100, the participant was asked to estimate where a certain number would be on the line and to indicate by drawing a vertical line. Five questions, “30, 87, 56, 12, 25,” were asked. If a vertical line was drawn within 1 cm of the correct position, it was considered correct. Consequently, 1 question was answered correctly. When asked for her impressions of the task, she introspected, “I don’t get it at all. I judged whether it was greater than 50 and then did it randomly.” (Figure 2).
The thick vertical line shows the patient’s response to the question “About where is 30?.” The line indicated by the arrow was drawn by the author at the correct position after the patient’s response.
3.2.6. Quantitative Sense of Numbers
From the 2 choices of numbers presented visually, the participant was asked to circle the number closer to a number presented orally. The 5 questions were as follows： “Closer to 300” for (289, 355), “Closer to 50” for (45, 61), “Closer to 700” for (697, 721), “Closer to 100” for (89, 120), and “Closer to 200” for (195, 209). She only answered correctly for “Closer to 50” for (45, 61). When asked about her insight, she stated, “I don’t get it at all.”
3.2.7. EstimationThe participant was asked to choose the approximate answer for 3-digit additions from the 3 choices placed to the right of the question. For example, the participant was asked to choose and circle the closest answer for “158＋443” from (500, 600, 700). Five questions were answered and none were correct. For example, for “598＋201,” she chose “900” from (700, 800, 900) and reported “Because it’s 600 and 300.” When asked to choose the answer for “202＋520” from (600, 700, 800), she stated “The answer is not in here” and wrote “900.”
3.2.8. Digit AlignmentThree numbers were listed horizontally (numbers including decimals and those not including them), and the participant was asked to write them in the blank space below, aligning the numbers so that digits of the same place value (ones, tens, tenths, etc.) were vertically aligned. The 5 questions were (346, 1035, 35.9), (0.463, 0.816, 17), (5, 690, 0.11), (9310, 783, 74.9), and (0.953, 28.0, 7). Consequently, she was unable to answer any of the questions correctly. She wrote all numbers right-aligned, regardless of whether they had a decimal point or where it was located, and reported “I don’t get it at all.” (Figure 3).
When 3 numbers were presented and she was instructed to write them while vertically aligning the digits, she wrote them aligning the rightmost digits, regardless of the decimal point.
3.3. Reading and Writing Numbers 3.3.1. Reading Numbers Aloud
The participant was asked to read aloud numbers ranging from 1 to 4 digits. The task consisted of 20 items：7 1-digit numbers, 3 2-digit numbers, 5 3-digit numbers, and 5 4-digit numbers. As a result, she read all 20 items correctly, but for some items, a delay of more than 3 s was observed before she started reading.
3.3.2. Writing NumbersA dictation task was conducted for 2-4-digit numbers. The examiner presented numbers orally and the participant was asked to write them in Arabic numerals. The task consisted of 10 items：4 2-digit numbers, 3 3-digit numbers, and 3 4-digit numbers；among these, 7 numbers included “0” in their digits. She correctly answered 5 out of 10 items. She correctly answered all 2-digit numbers that did not contain “0.” Among numbers containing “0,” whereas she correctly answered “hachiju (80）” and “gohyaku-sanju (530),” she wrote “720” for “nanahyaku-ni (702),” “384” for “sanzen-hachiju-yon (3084),” “9610” for “kyusen-roppyaku-ichi (9601),” and “403” for “yonsen-san (4003).”
3.4. Knowledge Regarding Numbers and calculations （Linguistic Knowledge, Operation Symbols） 3.4.1. Linguistic Knowledge Regarding NumbersThe patient was asked to answer the following 10 questions orally： “How many days in a week?” ； “How many days in a year?” ； “How many months in a year?” ； “How many hours in a day?” ； “How many seconds in a minute?” ； “How many meters in 1 km?” ； “How many cm in 1 m?” ； “How many mm in 1 cm?” ；and “How many mL in 1 liter?.” She answered the 7 questions correctly. She answered “40” for the question “How many minutes in 1 h?” ；“10” for “How many meters in 1 km?” ；and “50” for “How many cm in 1 m?”
3.4.2. Understanding of OperatorsShe was asked to fill in the correct operation symbol in basic arithmetic equations where the symbols for addition, subtraction, multiplication, or division had been replaced with squares, such as “3▢3＝9”. Four questions were posed, and she answered all the questions correctly.
3.5. Assessment of Calculation Ability 3.5.1. Single-Digit AdditionTasks with 5 additions without carrying and 5 additions with carrying were conducted. In addition to the number of correct answers, the calculation strategy used to derive answers was identified for each question. Strategy classification followed a two-step procedure, with overt behavior prioritized and self-report used only when no overt behavior was observed；counting with fingers was classified based on either observable finger movements or reported finger imagery (Siegler, 1987, Geary, et al., 2004, Geary, et al., 2012). First, behavior during the calculation was observed；if hand movements such as finger counting were seen, it was classified as “Counting with fingers” ；if mouth movements were seen, it was classified as “Verbal counting.” If no hand or mouth movements were observed, the examiner required a self-report by asking, “How did you arrive at your answer?” Reports were classified as： “Retrieval” if the answer came without thought (e.g., “It came out immediately,” “I knew it”）； “Decomposition” if it was judged that the participant decomposed the addend or augend (e.g., “I split it,” “I made it 10 first”）； “Verbal counting” if it was judged the participant counted in her head (e.g., “I counted,” “I counted in my head”）；and “Counting with fingers” if it was judged she used fingers (e.g., “I used my fingers,” “I counted with my fingers”). If none of the 4 strategies applied, it was classified as “Other.”
She answered all 10 questions correctly. The calculation strategies used were： “Retrieval” for 2 questions, “Verbal counting” for 4 questions, and “Counting with fingers” for 4 questions. For the additions without carrying, she reported “It came out instantly” for “3＋2” and “2＋1”, judged to have answered by “Retrieval,” but for others, she derived answers by “Verbal counting” or “Counting with fingers.” Even when finger movements were not observed, sometimes she answered “I counted with my fingers” when asked about her strategy. When asked how she counts with her fingers without finger movements, she reported “Because I get scolded for using fingers, I acquired a technique to count with fingers without moving them. I count by putting force into them.”
3.5.2. Subtraction with Single-Digit Subtrahend and AnswerTasks with 5 subtractions without borrowing and 5 subtractions with borrowing were used. Similar to single-digit addition, the calculation strategies were identified for each question through observation and questioning. She answered all 10 questions correctly. Regarding the strategies, for subtractions without borrowing, only for “4-1,” she answered by “Retrieval,” while answers for the other 4 questions were derived by “Counting with fingers.” For all 5 subtractions with borrowing, she used “Other” methods. Specifically, she used a method of subtracting the subtrahend from 10 and counting up the remainder；for example, for “12-5,” she thought “10 minus 5 is 5... 5, 6, 7”.
3.5.3. Multiplication Table （Kuku）Tasks with 10 multiplication questions answerable by recalling the Kuku were used. When she was not confident in the answer recalled from the table, she would recall the reverse phrasing；if the answers were the same, she judged correctly, and if the recalled answers did not match, she adopted another strategy. For example, for “4x7”, after writing “4x7＝32, 7x4＝28”, she wrote “3x7＝21, 21＋7＝28” to derive the answer. Thus, she answered all 10 questions correctly.
3.5.4. Division Answerable by Recalling Kuku Multiplication TableTasks with 10 division questions were used. She answered 3 out of 10 questions by recalling Kuku, and used addition for the others. For example, for “42÷7”, she wrote “7, 14, 21, 28, 35, 42” to derive the answer.
3.5.5. Multi-Digit Addition/Subtraction （Vertical Written Calculation）Tasks presented in 18 pt. font size on A4 paper were used. For addition, a total of 5 questions (“1 digit ＋ 2 digits” (2), “2 digits ＋ 2 digits” (1), “2 digits ＋ 3 digits” (2)) were presented, and she answered all correctly, but she calculated after drawing a line between the ones place and the tens place（Figure 4a). When asked for the reason, she explained, “It curves... or rather, so I know where to add, because I might add diagonally.” For subtraction, a total of 5 questions (“2 digits - 1 digit” (2), “2 digits - 2 digits” (1), “3 digits - 2 digits” (2)) were asked, and she answered all correctly. She derived answers while writing about the calculation process in the margins (Figure 4b).
a. Addition, b. Subtraction, c. Multiplication, d. Division. She solved multiplication problems when one of the multipliers was a single digit. For division, she derived the answer by adding the divisor.
3.5.6. Multi-Digit Multiplication/Division （Vertical Written Calculation）
The task was presented on an A4-sized paper with a font size of 18 pt. For multiplication, a total of 5 questions were posed, consisting of 2 “2 digits x 1 digit” questions, 1 “2 digits x 2 digits” question, 1 “3 digits x 1 digit” question, and 1 “3 digits x 2 digits” question. She answered 3 questions correctly. She could solve problems if 1 of the multipliers was a single digit. However, for “2 digits x 2 digits,” she could only stare at the problem without writing anything. For “3 digits x 2 digits,” she could not solve it, reflecting “I understand the order of multiplication, but I don’t know where or how to write the answer and the process” (Figure 4c). For division, a total of 5 questions were presented, consisting of 2 “2 digits ÷ 1 digit” questions and 3 “3 digits ÷ 2 digits” questions. She answered 4 of the 5 questions correctly. In division, she used addition to derive the answers（Figure 4d).
3.6. Right-Left Orientation and Finger Gnosis 3.6.1. Assessment of Right-Left OrientationThe participant was asked to follow instructions such as “Touch your left eye with your left hand” and “Touch your left eye with your right hand.” Although she passed all 5 trials correctly, she expressed her insights that “I can’t tell which is left and which is right in an instant. I imagine myself standing in my elementary school classroom with my back to the blackboard, and I identify the direction where the teacher’s desk stands as ‘right’.”
3.6.2. Finger GnosisFinger-naming tasks and auditory comprehension tasks were conducted, in which the participant was asked to point to the corresponding finger upon hearing the finger name, using 5 questions each. She answered all questions correctly in both tasks.
This patient shows no decline in intellectual ability or attentional deficits, and no difficulty in reading and writing texts. The difficulties she has experienced in daily life and learning situations from childhood to the present have all been related to number concepts, such as estimating prices and grasping time, and to the acquisition of arithmetic facts, including simple addition, subtraction, multiplication, and division. In addition, the persistent course of difficulties with numbers and calculations from early elementary school through adulthood, without a history of brain injury, led to the conclusion that this case corresponds to developmental dyscalculia.
We acquire number concepts and calculation abilities through arithmetic learning. Cardinality refers to the fact that when counting objects by reciting numerals such as “1, 2, 3...” in sequence, the last number spoken represents the total quantity；this is considered the foundation of number concepts. The decimal system refers to a place-value numeral system where 10 is the base (unit of grouping) and the position of a digit indicates its place value；through learning this system, we acquire the concept of multi-digit numbers. Regarding simple calculations, arithmetic facts are acquired by repeatedly solving addition, subtraction, multiplication, and division problems.
Based on these basic characteristics of numbers and calculations, I discuss the impairments in number concepts and calculations observed in the participant based on the results. Furthermore, I consider the structure of number concepts based on previously reported cases of impaired number concepts and the current case.
4.1. Impairments of Number Concept and Calculation in the Current CaseIn basic number operations, the participant was able to perform “counting” and “backward counting.” However, for “backward counting,” she reported “I had to think a little for 15, 14, and 13,” suggesting that the sequence of numbers is somewhat unstable when imaging them in reverse.
Regarding subitizing, we can instantly grasp the number of objects, up to 3 or 4, regardless of the arrangement (Ashcraft, 1992, Revkin, et al., 2008, Starkey, et al., 1980, Starkey, et al., 1995). This ability is said to be present even in infants before they acquire knowledge of numbers or digits, enabling a 2-year-old child to distinguish quantities of less than 4 items (Starkey, et al., 1980). Furthermore, subitizing is thought to include perceptual subitizing, recognizing numbers without using other mathematical processes, and conceptual subitizing, recognizing number patterns as a composite of parts and further recognizing them as a whole (Clements, 1999). According to Clements (1999), perceptual subitizing is considered close to the original definition of subitizing, whereas conceptual subitizing refers to cases such as instantly grasping a number without counting individually when, for example, 7 dots are divided into 3 and 4 and arranged at equal intervals in 2 rows. Additionally, the ability to instantly grasp the number of pips on a die, such as 5 or 6, is also considered to be due to conceptual subitizing.
In the present case, she was able to instantly grasp the numbers for 3 randomly arranged dots and 4 dots arranged at equal intervals without counting them individually；therefore, perceptual subitizing appears possible. By contrast, in tasks where 5-10 dots were arranged in a single horizontal row or 2 vertical rows based on 5, the participant could not grasp the number immediately and answered by counting individually；thus, conceptual subitizing is considered difficult. In addition, in the randomly arranged task, she instantly grasped the number of 5 dots by likening them to the arrangement on a die. I inferred that while this response falls under conceptual subitizing, it is a response limited to the generally familiar dot arrangement of a die.
In magnitude comparison, when asked for her insights after answering each question, she reported “I compared at the tens place” for the comparison of 2-digit numbers, and stated “Because there are 3 digits and 2 digits” for the comparison of 3-digit and 2-digit numbers. This suggests she possesses the knowledge that “for numbers with the same number of digits, compare the digit at the largest place value” and “for numbers with different numbers of digits, the one with more digits is larger.” Examining her reactions and self-reports, for (3, 5), (24, 31), (62, 59), (475, 512), and (4032, 4029), she was able to judge the magnitude without counting or using fingers, so it was inferred that she could grasp the magnitude from Arabic numerals for small numbers of up to 5 or 6. By contrast, for the comparison of (8, 6), she reported “I thought a little and understood,” and for (91, 78), she was observed representing the numbers with fingers to think. Further, for (689, 791) and (9034, 8967), she reported “I looked at the first digit and counted from 5,” suggesting that she cannot grasp the magnitude from Arabic numerals for numbers from 6 to 9. Based on inferences from her reactions and reports, although I did not confirm with her strictly what number she can grasp from Arabic numerals, it is considered that she can grasp the magnitude of small numbers from 1 to 5 or 6 by looking at Arabic numerals, and for numbers larger than that, it is difficult to grasp the magnitude from Arabic numerals.
She also showed difficulties with number composition, number decomposition, and questions regarding the decimal system. In number composition, she made errors with numbers containing “0,” and in number decomposition, self-correction was seen in tasks containing “0.” The confusion regarding the position of “0” in multi-digit numbers was also observed in the cases of Furumoto, et al. (1993) and Nagatomo, et al. (2009), suggesting an impairment in the concept of place value. In addition, she inserted vertical lines between places in multi-digit addition/subtraction, and she stated that the reason was “because I might add diagonally,” which suggests the fragility of her notion of place value. Regarding questions regarding the decimal system, she was seen writing numbers on paper to think before answering. It was assumed that while she cannot directly derive the answer to the question “What happens when 10 100s combine?,” she has the knowledge that “0” increases by 1 when 10 of them collected；she may have derived the answer by adding “0” to “100” to write “1000” and then reciting the sequence “ichi, ju, hyaku, sen (one, ten, hundred, thousand).” From these errors and reactions in number decomposition/composition and questions regarding the decimal system, she likely has an impairment in the concept of the decimal system. Furthermore, the number line task, the quantitative sense task of choosing the closer number, and estimation were very difficult for her, and she seemed halfway resigned stating “I don’t get it at all” ；it was inferred that for numbers of 2 digits or more, she cannot grasp the quantitative sense from numbers expressed in the decimal system.
In digit alignment, she wrote numbers aligning the rightmost digits vertically and reported “I don’t understand at all.” Consequently, she was able to align the digits for integers, but errors occurred with numbers containing decimals. A decimal is a concept that extends the idea of the decimal system for integers to numbers smaller than 1 (Ministry of Education, Culture, Sports, Science and Technology, 2018), where digits representing places smaller than 1 (e.g., 10ths, 100ths, and 1000ths) appear to the right of the decimal point. It is thought that because she could not grasp the meaning represented by the decimal point and digit positions for numbers containing decimals, it was difficult for her to align the digits, and the cause is likely the impairment of the decimal system concept.
Based on the above analysis, I summarize the impairment of number concepts in this case. While cardinality, which is considered the basis of number concepts, was preserved, the participant’s ability to grasp magnitude from Arabic numerals was limited to very small numbers (up to 5 or 6), indicating severe impairment in number concepts. She also presented with an impairment in the concept of the decimal system, which affected the quantitative grasp and estimation of multi-digit numbers and the understanding of numbers containing decimals.
Regarding reading and writing of numbers, errors were seen in the dictation of numbers in tasks containing “0.” In positional notation, “0” indicates that a certain place is empty. As errors in dictation did not occur for numbers 1-9 but only for “0,” this error was considered to stem from an inability to understand the meaning of place value.
In tasks of linguistic knowledge regarding numbers, she answered “40” for “How many minutes in 1 h?” ；“10” for “1 km is how many m?” ；and “50” for “1 m is how many cm?” These tasks require knowledge of numbers, and the involvement of number concepts is considered low. In the case of Nagatomo, et al. (2009), who reported abnormalities in the use of the base-12 and base-60 systems in addition to the base-10 system due to cerebral infarction after subarachnoid hemorrhage, the patient was able to correctly answer the number of hours in a day and minutes in an hour verbally, indicating that linguistic knowledge and number concepts are dissociated. Whether a severe impairment in number concepts, as seen in this case, affects linguistic knowledge regarding numbers cannot be judged solely from the errors in this case, and further study is necessary.
Regarding calculation, in single-digit addition and subtraction, when the calculation strategy used to derive the answer was identified individually based on her report in addition to correctness, she reported “It came out instantly” for “3＋2,” “2＋1,” and “4－1” ；thus, it was judged that she answered by recalling arithmetic facts. However, for other problems, she derived answers by verbal counting or counting with fingers. Regarding Kuku, there were cases in which she could recall answers and cases in which she recalled incorrect answers. From these facts, it is considered that she has not acquired arithmetic skills for addition, subtraction, or multiplication, except for very simple calculations. In addition, in multi-digit calculations, she calculated by drawing lines between places in addition, and in multiplication, she looked bewildered, stating, “I understand the order of multiplication, but I don’t know where or how to write the answer and the process.” From these reactions, it is inferred that although she has acquired knowledge regarding calculation procedures, confusion occurs in the position of writing numbers during the process of written calculation, which is thought to stem from an inability to understand the meaning of place value.
Summarizing these calculation errors, regarding single-digit addition/subtraction and multiplication (Kuku) in this case, there are underlying difficulties in acquiring arithmetic facts. Errors in written calculations could be stemming from a lack of understanding of place values, presumably arising from difficulties with the concept of the decimal system.
4.2. Structure of Number ConceptsAccording to the triple-code model by Dehaene, et al. (1995), numbers are represented in the brain using 3 codes：a verbal framework (number words), a visual number form (Arabic numerals), and an analog magnitude representation (number concepts). Regarding the structure of number concepts, various reports exist, such as the concept of single-digit numbers and the concept of the decimal system operating independently (Suzuki, et al., 2005）；there are 2 systems, one representing large approximate numerical values and another precisely representing small numbers of individual objects (Feigenson, et al., 2004）；and number recognition is established through the intersection of 2 systems of meaning：cognition of digits 0-9 and place value (Furumoto, et al., 1993）―but this is not yet fully clarified.
In the case of an individual reported by Furumoto, et al. (1993), who presented with an impairment of number concepts due to right hemisphere damage, the person was capable of magnitude comparison of single-digit numbers and reading/writing of digits 0-9, but showed confusion in place value, such as misplacing “0” in the reading aloud and dictation of multi-digit numbers. In addition, Nagatomo, et al. (2009) reported regarding an individual with impairment of the decimal system due to bilateral frontal lobe infarction following subarachnoid hemorrhage. The patient preserved the meaning of numbers 1-9 but showed errors in the decomposition, composition, and magnitude comparison of multi-digit numbers, suggesting an impairment in the decimal system. From these cases, it is considered that number concepts have different structures for at least single-digit and multi-digit numbers.
Case C.G., reported by Cipolotti, et al. (1991), presented an impairment of number concepts regarding single-digit numbers. C.G. exhibited a state that could be described as the loss of number concepts due to cerebral infarction in the left frontoparietal region. While she could process numbers 4 or less, she lost recognition of numbers beyond 4. Even reading aloud and reciting numbers were difficult for numbers exceeding 4. Case C.G. also showed impairment in perceptual subitizing；it was reported that she could instantly grasp only up to 2 items, and when counting, could answer correctly up to 4.
In the case of the present participant, she can grasp numbers from digits up to about 5；for single-digit numbers larger than that, she cannot grasp the magnitude from the digits, indicating a severe impairment in number concepts, similar to C.G. What differs from C.G. in the current case is that even for numbers where she cannot grasp the magnitude from the digits, she recognized them as numbers；she could grasp the magnitude of numbers larger than 5 if she counted them individually, and perceptual subitizing was possible.
From the characteristics regarding numbers shown by these previously reported cases and the present case, it is suggested that number concepts have different structures for single-digit and multi-digit numbers. Furthermore, the structure of number concepts for single-digit numbers is not uniform, suggesting that a distinction may exist between smaller and larger numbers around the threshold of 5. Additionally, although cardinality has been considered the foundation for acquiring number concepts, the present case presented with an impairment of number concepts regarding single-digit numbers exceeding 6, despite having acquired cardinality. The characteristics of the present case suggest that cardinality alone is not sufficient for the acquisition of number concepts, and that abilities such as conceptual subitizing―recognizing and manipulating numbers as patterns, that is, a collection with a certain structure―may be involved.
Finally, I consider the concepts of place value and right-left orientation. In the evaluation of right-left orientation, the participant stated, “I can’t tell which is left and which is right in an instant,” and reported that she still judges left and right by imagining the blackboard and teacher’s desk in her elementary school classroom. Although she answered all tasks correctly, because right-left orientation relies on her own mental compensatory means, it is considered that she has difficulty distinguishing between the left and right. The concepts of place value and right-left orientation are common in that they both give meaning to spatial positions extending the left and right. There may be a common neural basis for the co-occurrence of these 2 impairments, which merits further research in the future.
4.3. Study Limitations and Future Research DirectionsBecause the current case was of a “healthy volunteer” with developmental dyscalculia who offered to cooperate in this study, I could not perform imaging tests such as head computed tomography, magnetic resonance imaging, or single photon emission computed tomography due to various constraints. Therefore, I cannot directly discuss the relationship between the symptoms and the cerebral lesion in this case, constituting a limitation of this study. However, calculation impairment and difficulty distinguishing between the left and right were observed in this case, suggesting dysfunction of the left parietal lobe. A low score was also observed on the Visual Puzzles subtest of the WAIS-IV, suggesting possible difficulties in spatial perception and processing, although this could not be fully examined in the present study. The relationship between the low score in WAIS-IV Visual Puzzles performance and the numerical concept impairment in this case remains a topic for future research.
According to a survey by the Ministry of Education, Culture, Sports, Science and Technology (2022), the percentage of students in regular classes who show significant difficulties in “calculating” or “reasoning” is estimated to be 3.4％ in elementary and junior high schools. Despite the fact that a certain number of students face difficulties with numbers and calculations, similar to the current case, understanding regarding the use of compensatory means and reasonable accommodations in educational settings has not progressed. By accumulating research evidence of cases such as the current one, I believe it is necessary to clarify impairments in number concepts and difficulties in conceptual subitizing seen in developmental dyscalculia, as well as the mechanisms that cause them.
I express my heartfelt gratitude to the participant who voluntarily offered to report her symptoms and willingly participated in this study.
The purpose of the study was explained to the participant, and written informed consent for journal submission was obtained.
The authors declare no conflicts of interest related to this study. No companies, organizations, or groups had any role that could constitute a conflict of interest.
This study was previously presented at the 47th Annual Meeting of the Japan Society for Higher Brain Dysfunction (October 2023).
失語症や注意障害，視空間障害など他の要因によらない後天性の純粋な計算障害は失計算（acalculia）とよばれる。失計算には，数概念に障害をきたした症例（Cipolottiら 1991，古本ら1993，鈴木ら 2005，永友ら 2009）や，簡単な加減算や九九の想起に障害をきたした症例（Warrington 1982，Delazerら 1997，Hittmair-Delazerら 1994，平山ら 2002），複数桁の筆算における計算手続きに障害をきたした症例（Lucchelliら 1993）などの報告がある。
一方，発達性の計算障害は，教育の領域では学習障害のなかの「計算する又は推論する能力」（文部科学省 2013）の障害として位置づけられている。医療の領域ではICD-10（World Health Organization 2005）の「学力の特異的発達障害」のなかに「特異的算数能力障害」として記載されており，「この障害は（代数学，三角法，幾何学あるいは微積分学のような，より抽象的な数学力よりもむしろ）加減乗除のような基本的な計算力の習得に関係している」と記されている。また，DSM-5（American Psychiatric Association 2014）においては，「限局性学習症 / 限局性学習障害」のなかに，具体的な症状として「数の感覚」「数学的事実の記憶」「計算の正確さまたは流暢性」「数学的推理の正確さ」が挙げられている。
このなかにある「数学的事実」は，算術的事実（arithmetic fact）ともよばれるもので，長期記憶のなかに貯蔵されている1桁同士の簡単な計算に関する知識を指す（Ashcraft 1992，Butterworth 2005）。我々は，算数の学習を通して算術的事実を獲得すると，長期記憶から答えを検索することで解答できるようになるため，1桁同士の簡単な計算については，あまり考えずに答えを導くことができる。一方，発達性の計算障害例においては，学年が上がっても指を使って数えたり，声に出して，あるいは頭のなかで数えたりして答えを出す場合が多いことが報告されており，算術的事実の獲得の困難さは発達性計算障害の特徴の1つとされている（Jordanら 2003，Gearyら 2004，Gearyら 2012）。
このほか，発達性計算障害例では，「数の感覚」に関連する特徴として，大きい数の見積もり（Decarliら 2020）や数直線課題（Lambら 2024，Ruizら 2024）で困難を示すことが報告されている。また，1つひとつを数えずにドットなどの数を瞬時に把握するサビタイジング（Starkeyら 1980，Dehaene 2010）については，保たれているとする報告（Decarliら 2020，Lambら 2024）がある一方で，反応時間に差がみられるとする報告（Landerlら 2004）もある。
今回，筆者は症例自らの申し出により，小学校低学年の頃から数の把握や計算の苦手さを自覚していた成人例に対して評価を行う機会を得た。発達性計算障害の成人例における数概念や計算の障害について詳細に検討した報告は見当たらず，本例は貴重な症例であると考えられる。本例が日常的に抱えている数や計算に関する障害と今回実施した評価結果について考察を加え報告するとともに，過去の報告例と本例の症状を通して数概念の構造について考える。
【症例】20歳代，女性，右手利き。大学4年生。筆者が数や計算の障害について興味を持っていることを知り，協力を申し出てくれた症例である。幼少期からこれまで，数や計算の障害について評価や支援を受けたことはない。
【数や計算に関するエピソード】本例は小学校低学年の頃から，数や計算の苦手さを自覚しており，算数の時間は教室を出て図書室で本を読んでいたとのことであった。また，指を使って考えていると，先生から「まだ指を使って計算しているの？」「指を使わないように」とたびたび指摘されるのが苦痛であり，小学校から高校に至るまで，「先生と名のつく職業が大嫌いだった」と話した。本例に，小学校のときに困っていたことについて尋ねると，「分数はわかる。小数はまったくわからない。パーセントもわからない。数直線もわからない。漢字で書いてある数をアラビア数字に変換できない。アラビア数字で書いてある数を漢字に変換することもできない」と述べた。本例は，簡単な足し算や引き算，九九の想起が苦手であるにもかかわらず，中学校で学習する連立方程式や証明，図形の問題は「得意だった」と述べ，「中学校の数学の成績は良かった」と話した。
成人した現在も数の把握や計算に関する困難さは持続しており，日常生活のさまざまな場面で支障がみられる。たとえば，本例は金額を高く見積もる傾向があるため，買い物では支払い額が自分の予想よりもかなり安くなると話した。また，金額を推測して支払うことが難しいため，有人レジでは財布のなかで一番大きい紙幣で支払い，セルフレジでは小銭を全部入れることで対処しているとのことであった。時間については，デジタル時計よりアナログ時計がわかりやすいが，数字の表記がないと時間を把握できないため，アナログ時計で針の位置を確認し，さらにデジタル時計で数字を見ることで把握していた。年齢の標記については，〇歳〇ヵ月という年齢表現がわかりづらく，36ヵ月と書いてあっても何歳かわからないと述べた。
【神経心理学的所見】検査には集中して取り組んだ。WAIS-IV成人知能検査（Wechsler Adult Intelligence Scale-Fourth Edition：WAIS-IV）では，FIQ109，言語理解（VCI）128，知覚推理（PRI）95，ワーキングメモリー（WMI）88，処理速度（PSI）111であった（表1）。下位検査では，算数（評価点4）とパズル（評価点6）の評価点が他の項目に比べて低かった。文字の読み書きに関しては，大学の講義のノートテイクが可能であり，症例本人も読み書きに関しては支障を感じていないと報告した。試しに，4文節程度の短文の書き取りを行ったところ，漢字交じりの文で書き取りが可能であった。神経心理学的症状としては，後述する数概念と計算の障害，左右の識別困難が認められた。
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検査項目と内容，結果を表2に示す。評価は2022年10～11月にかけて熊本保健科学大学言語聴覚学専攻の訓練室にて実施した。検査項目のうち，大小比較（数字），数の合成，数の分解，十進法に関する質問，数の量的感覚，概算，数字の音読，演算記号の理解，1桁同士の足し算，引く数と答えが1桁の引き算，九九，九九の想起で解答できる割り算については，A4サイズの用紙にフォントサイズ14ptで作成した課題を用いた。
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1． 数の基礎的な操作 1） 数かぞえ
1～20までの数を数えてもらう課題は可能であった。
2） 逆数かぞえ20～1までの数を数えてもらう課題では，「15，14，13の部分はちょっと考えた」との感想がきかれた。
3） サビタイジング縦6 cm，横7.5 cmのカードに直径4 mmのドットを印刷した。1～5個のドットをランダム配置した課題（図1a）と，1～10個のドットを6 mmの等間隔で配置した課題（図1b）を用いた。等間隔で配置した課題は，1～5はドットを横1列に，6～10は下段に5個のドットを横1列に配置し，上段に残りのドットを左から横1列に配置した。たとえば，8の場合，下段に5個，上段に3個のドットをそれぞれ横1列に配置した（図1b）。これらのカードを1枚ずつ呈示し，ドットの数をすばやく数えてもらった。また，1問ごとにすぐにわかったのか，数えたのかを質問した。結果，ドットをランダムに配置した課題では，1～3は「すぐにわかった」と瞬時に数を把握できたが，4個のドットは「数えた」と報告した。また，5は「すぐにわかった。サイコロと同じ並びだから」と報告した。1～10個のドットを，5を基準として横1列または上下2列に等間隔に並べて配置した課題では，1～4までは「すぐにわかった」と報告したが，5～10は「（ドットを）全部数えた」と報告した（図1）。
a：1～5個のドットをランダムに配置した課題，b：5を基準に上下に配置した課題。1～5は横1列に，6～10は上下2列にドットを配置。
4） 大小比較（ドット）
サビタイジング課題で用いた1～10個のドットを等間隔に配置したカードを用いた。3枚のカードを机上にランダムに並べ，数の大きい順に並べ変えてもらった。課題は，（2，3，5），（3，6，9），（1，3，8），（4，6，7），（5，7，10）の5問とした。結果，5問すべて正答した。
2． 数概念の評価 1） 大小比較（数字）1～4桁の2つの数（桁数を揃えたもの9問，桁数が異なるもの1問）について，大きい方を選んでもらった。課題は，（8，6），（3，5），（24，31），（62，59），（91，78）（97，103），（475，512），（689，791），（4032，4029），（9034，8967）の10問とした。結果，すべて正答した。1問ずつ感想を問うと，（8，6）の比較では「ちょっと考えてわかった」と報告し，（24，31），（62，59），（91，78）は，「10の位で比較した」と報告した。ただし，（91，78）では「7」と「9」を指で表して考える様子が観察された。桁数の異なる（97，103）は，「数字が3つと2つだから」と述べ，桁数が異なる場合には，容易に大小の判断ができていた。（689，791）と（9034，8967）では，「一番最初の数を見て5から数えた」と報告し，（4032，4029）は「3番目だけ見てわかった」と報告した。
2） 数の合成2～3桁の数の合成課題を実施した。「10が5個と1が7個では何になりますか？」などの問題に対して，アラビア数字を書いてもらった。結果，5問実施し，正答は3問で，「0」が含まれる問題で誤りがみられた。たとえば，「100が4個と1が2個では何になりますか？」という問題には「42」，「100が7個と10が5個では何になりますか？」という問題には「75」と記入した。
3） 数の分解「45は，10が 個と，1が 個」などの問題の空欄にアラビア数字を記入してもらった。5問実施しすべて正答したが，「0」が含まれる課題で自己修正がみられた。「308は，100が 個と，10が 個と，1が 個」の問題の「10が 個」の所に，「8」と記入したあと，二重線で訂正し「0」と修正した。
4） 十進法に関する質問問題を見て口頭で答えるよう求めた。問題は，「1が10個集まると何になりますか？」 「100が10個集まると何になりますか？」 「10が10個集まると何になりますか？」 「10000が10個集まると何になりますか？」 「1000が10個集まると何になりますか？」の5問とした。結果，5問中4問正答し，1問は無反応となった。正答した4問中3問は問題用紙に数字を書いて考えてから口頭で解答した。
5） 数直線課題A4サイズの用紙の中央に印刷した20 cmの直線の左端を0，右端を100とし，ある数が直線のどのあたりになるか推測し，縦線を記入してもらった。「30，87，56，12，25」の5問を実施し，正しい位置から左右1 cm以内に縦線が記入された場合を正答とした。結果，正答は1問であった。課題に対する感想を問うと，「全然わからない。50より大きいかを考えて，あとは勘」と内省した（図2）。
太い縦線は「30はどのあたりですか？」という課題に対する症例の反応を示す。矢印が示す線は症例の反応後に筆者が正答の位置を記入したもの。
6） 数の量的感覚
2つの数を提示し，口頭で，ある数に近い方を丸で囲むよう求めた。問題は（289，355）に対して「300に近い方」，（45，61）に対して「50に近い方」，（697，721）に対して「700に近い方」，（89，120）に対して「100に近い方」，（195，209）に対して「200に近い方」の5問とした。正答は（45，61）に対して「50に近い方」の1問であった。課題に対する感想を問うと，「全然わからない」と内省した。
7） 概算3桁同士の横書きの足し算のおおよその答えを，問題の右に示した3つの数から選んでもらった。たとえば，「158＋443」の答えを（500，600，700）のなかから選んで丸で囲むよう求めた。5問実施し正答はなかった。たとえば，「598＋201」では（700，800，900）のなかから「900」を選び，「600と300だから」と報告し，「202＋520」の答えを（600，700，800）のなかから選ぶよう求めると「このなかには答えがない」と述べ「900」と記入した。
8） 数の桁合わせ横に並べた3個の数（小数点を含む数と含まない数）の下の空欄に，これらの数を，桁を揃えて縦に並べて書くよう求めた。問題は，（346 1035 35.9），（0.463 0.816 17），（5 690 0.11），（9310 783 74.9），（0.953 28.0 7）の5問とした。結果，正答はなかった。本例は少数点の有無や位置にかかわらず，右端の数字を縦に揃えて書き，「全然わからない」と報告した（図3）。
3つの数を提示し，桁を揃えて縦に並べて書くよう指示したところ，小数点に関係なく右端の数字を縦に揃えて書いた。
3． 数の読み書き 1） 数字の音読
1～4桁までの数を音読してもらった。1桁の数7問，2桁の数3問，3桁の数5問，4桁の数5問とした。結果，20問すべて正答したが，読み始めるまでに3秒以上の遅延がみられるものもあった。
2） 数の書き取り2～4桁の数の書き取り課題を行った。検査者が口頭で提示した数をアラビア数字で書き取るよう求めた。課題は2桁の数4問，3桁の数3問，4桁の数3問とし，このうち7問は桁に「0」を含む数とした。結果，10問実施し5問正答した。2桁の数で桁に「0」が含まれない数はすべて正答した。桁に「0」が含まれる数のうち，1の位に「0」が含まれる「はちじゅう」と「ごひゃくさんじゅう」は正答したが，「ななひゃくに」は「720」，「さんぜんはちじゅうよん」は「384」，「きゅうせんろっぴゃくいち」は「9610」，「よんせんさん」は「403」と表記した。
4． 数や計算に関する知識（言語的知識，演算記号） 1） 数に関する言語的知識以下の10個の課題に口頭で答えてもらった。課題は，「1週間は何日か」「1年は何日か」「1年は何ヵ月か」「1日は何時間か」「1時間は何分か」「1分は何秒か」「1 kmは何mか」「1 mは何cmか」「1 cmは何mmか」「1 Lは何mLか」とした。正答は7問であった。「1時間は何分か」という問いに「40」，「1 kmは何mか」という問いに「10」，「1 mは何cmか」という問いに「50」と解答した。
2） 演算記号の理解「3▢3＝9」のように演算記号を四角で示した足し算，引き算，掛け算，割り算の計算式に正しい演算記号を記入してもらった。4問実施しすべて正答した。
5． 計算能力の評価 1） 1桁同士の足し算繰り上がりのない足し算5問，繰り上がりのある足し算5問を印刷した課題を用いた。正答数に加えて，解答を導くために用いた計算方略を1問ずつ把握した。計算方略は，SieglerとGearyらの方法（Siegler 1987，Gearyら 2004，Gearyら 2012）に準じて以下のように分類した。まず，1問ずつ計算場面での行動を観察し，指折りなどの手の動きがみられた場合は「指での計数」に，口の動きがみられた場合は「口頭での計数」に分類した。手の動きや口の動きがみられなかった場合は，「どのような方法で答えを出しましたか？」と質問し，報告を求めた。報告は，「すぐに出た」「知っていた」などと考えずに答えが出た場合は「検索」に，「分けた」「まず10にして」などと加数や被加数を分解して考えていると判断できる場合は「分解」に，「数えた」「頭の中で数えた」と頭のなかで数えていると判断できる場合は「口頭での計数」に，「指を使った」「指で数えた」など指を使って数えたと判断できる場合は「指での計数」に分類した。4つの方略のいずれにも当てはまらない場合は「その他」とした。結果，10問すべて正答した。使用した計算方略の内訳は，「検索」2問，「口頭での計数」4問，「指での計数」4問であった。繰り上がりのない足し算の「3＋2」と「2＋1」は「ぱっと出た」と報告し「検索」によって解答したと判断したが，その他は「口頭での計数」や「指での計数」で答えを出した。本例は，指の動きが観察されない場合にも，計算方略について質問すると「指で数えた」と答えた。どのようにして指で数えているのかを問うと「指を使うと怒られるので，動かさずに指で数える方法を身につけた。指を動かさないで力を入れることで数えている」と報告した。
2） 引く数と答えが1桁の引き算繰り下がりのない引き算5問，繰り下がりのある引き算5問を印刷した課題を用いた。正答数に加えて，1桁同士の足し算と同様に，計算場面の観察と計算方略に関する質問によって，解答を導くために用いた計算方略を1問ずつ把握した。結果，10問すべて正答した。使用した計算方略の内訳は，繰り下がりのない引き算のうち，「4-1」は「検索」で解答したが，その他の4問は「指での計数」で答えを導いた。繰り下がりのある引き算は，5問とも「その他」の方法を用いた。具体的には，減数を10から引いて残りを数えあげる方法，たとえば「12-5」は「10引く5は5…5，6，7」で考えた。
3） 九九九九の想起で解答できる掛け算10問を印刷した課題を用いた。本例は，九九を想起して答えに自信がないときは，逆の言い方でも想起し，同じ答えならば正解と判断，想起した答えが異なる場合は，他の方法を用いて考えた。たとえば，「4×7」では，「4×7＝32，7×4＝28」と書いた後に，「3×7＝21，21＋7＝28」と書いて答えを出した。結果的には10問すべて正答した。
4） 九九の想起で解答できる割り算割り算10問を印刷した課題を用いた。10問中3問は九九の想起で解答し，その他は足し算を用いた。たとえば，「42÷7」では，「7，14，21，28，35，42」と書いて答えを出した。
5） 複数桁の加減算（縦書きの筆算）A4サイズの用紙にフォントサイズ18ptで作成した課題を用いた。加算は「1桁＋2桁」2問，「2桁＋2桁」1問，「2桁＋3桁」2問の計5問行いすべて正答したが，一の位と十の位の間に線を入れてから計算した（図4a）。理由を問うと「曲がるというか…足すところがわかるように，斜めに足したりするから」と説明した。減算は「2桁－1桁」2問，「2桁－2桁」1問，「3桁－2桁」2問の計5問行い，すべて正答した。余白に計算過程を書きながら答えを出した（図4b）。
乗算は掛ける数の片方が1桁であれば解くことができた。除算は割る数を足していくことで答えを出した。
6） 複数桁の乗除算（縦書きの筆算）
A4サイズの用紙にフォントサイズ18ptで作成した課題を用いた。乗算は「2桁×1桁」2問，「2桁×2桁」1問，「3桁×1桁」1問，「3桁×2桁」1問の計5問を行い，3問正答した。掛ける数の片方が1桁の場合は解くことができたが，「2桁×2桁」は問題を見つめるものの何も記入することができず，「3桁×2桁」では，「掛ける順序は理解しているが，答えとそのプロセスをどこに，どう書けばいいかわからない」と内省し解答に至らなかった（図4c）。除算は「2桁÷1桁」2問，「3桁÷2桁」3問の計5問行い4問正答した。除算では足し算を用いて答えを出した（図4d）。
6． 左右認知と手指認知 1） 左右認知「左手で左目を触ってください」「右手で左目を触ってください」などの指示に応じてもらった。5問行いすべて正答したが，「ぱっと左右がわからない。小学校の教室をイメージし，黒板を背にして立って，先生の机がある方が『右』と認識している」と内省した。
2） 手指の認知手指の呼称課題と手指名を聞いて対応する自分の指を指示する課題をそれぞれ5問行った。いずれもすべて正答した。
本例は知的能力の低下や注意障害はなく，文字の読み書きについても支障はなかった。本例が幼少期からこれまで，生活場面や学習のなかで感じていた困難さは，お金の見積もりや時間の把握など数概念に関連するものと，簡単な加減算や乗除算ができないという算術的事実の獲得に関するものに集約されていた。また，数や計算の障害は小学校低学年から成人した現在まで持続していること，脳損傷の既往はないことから，発達性の計算障害に該当すると考えられた。
我々は数概念や計算の能力を算数の学習を通して獲得する。基数性とは，数える対象物に「いち，に，さん…」と数詞を対応させて唱えた場合に，最後にくる数が全体の個数を表すことをさし，数概念の基礎とされる。また，十を底（てい：まとまりのこと）とし，数字の位置が位を表す十進位取り記数法（十進法）の学習を通して，複数桁の数概念を獲得していく。簡単な計算については，加減算や乗除算の問題を繰り返し解くことで，算術的事実を獲得すると考えられている。
これらの数や計算に関する基本的な特徴をふまえたうえで，本例にみられた数概念や計算の障害について，評価結果をもとに考察する。また，数概念に障害を呈した過去の報告例と本例の症状から，数概念の構造について考える。
1． 本例の数概念や計算の障害について本例は数の基本的な操作における「数かぞえ」と「逆数かぞえ」は可能であったが，「逆数かぞえ」では，「15，14，13はちょっと考えた」と報告しており，数詞を逆からイメージする場合には，数詞の並びがやや不安定であることがうかがえた。
サビタイジングに関して，我々は対象物の個数が3～4個であれば，どのような配置であっても瞬時にその数を把握できると考えられている（Ashcraft 1992，Revkinら 2008，Starkeyら 1980，Starkeyら 1995）。この能力は数詞や数字を獲得する前の幼児にも備わっており，2歳児でも4未満の数の区別ができるとされる（Starkeyら 1980）。また，サビタイジングには，ほかの数学的プロセスを使わず数を認識する知覚的サビタイジングと，数のパターンを部分の複合体として認識し，さらに全体として認識する概念的サビタイジングがあると考えられている（Clements 1999）。Clementsによる知覚的サビタイジングは，サビタイジングの元の定義に近いとされ，概念的サビタイジングは，たとえば，7個のドットを3個と4個に分け上下2段に等間隔に配置すると，1つひとつを数えなくとも瞬時に数を把握できる場合などをさす。そのほか，サイコロの5や6の目の数が瞬時に把握できるのも概念的サビタイジングによると考えられている。
本例の場合，ランダムに配置した3個のドットと等間隔に配置した4個のドットは，1つひとつを数えなくても瞬時に数を把握できたことから，知覚的サビタイジングは可能であると考えられた。一方で，5～10個のドットを，5を基準として横1列もしくは上下2列に配置した課題では，即座に数を把握できず，1つひとつを数えて解答したことから，概念的サビタイジングは困難であると考えられた。また本例はランダムに配置した課題で，5個のドットをサイコロの並びに見立てて瞬時に数を把握した。この反応は概念的サビタイジングに含まれるが，サイコロという一般的になじみのあるドットの配置に限定された反応であると推察した。
数の大小比較では，解答した後に1問ずつ過程を問うと，2桁同士の数の比較では「10の位で比較した」と報告し，3桁と2桁の数の比較では，「数字が3つと2つだから」と述べた。このことから本例は「同じ桁数の数では，一番大きい位の数を比較する」「桁数が異なる数は，桁数が多い方が大きい」という知識を持っていることがうかがえた。本例の反応や報告をみると，（3，5），（24，31），（62，59），（475，512），（4032，4029）では，数えたり指を使ったりすることなく大小を判断できていたことから，5～6までの小さい数はアラビア数字から大きさを把握できていると推察された。一方，（8，6）の比較では，「ちょっと考えてわかった」と報告し，（91，78）では，数を指で表して考える様子が観察された。また，（689，791）と（9034，8967）では，「一番最初の数を見て5から数えた」と報告したことから，6～9の数はアラビア数字から大きさ把握できないことがうかがえた。厳密にいくつまでの数をアラビア数字から把握できるかを，本人に確認していないため，反応や報告からの推察にはなるが，本例は，1～5ないし6までの小さい数は，アラビア数字を見てその大きさを把握できるが，それより大きい数は，アラビア数字から大きさを把握することが困難であると考えられた。
本例は，数の合成や数の分解，十進法に関する質問で困難を示した。数の合成では「0」が含まれる数で誤りを示し，数の分解では「0」が含まれる課題で自己修正がみられた。この複数桁の数における「0」の位置の混乱は，古本らや永友らの症例でも観察されており（古本ら 1993，永友ら 2009），位の概念の障害を示唆するものと考えた。また，複数桁の加減算で，位と位の間に縦線を入れる理由を「斜めに足したりするから」と述べていることからも，位の概念の不安定さがうかがえる。十進法に関する質問では，紙に数字を書いて考えてから解答する様子がみられた。これは，「100が10個集まると何になりますか？」という問いに対して直接解答を導くことができないが，10個集まると「0」が1つ増えるという知識はあるため，「100」に「0」を加え「1000」と書き，「いち，じゅう，ひゃく，せん」と系列語として唱えることで解答を導いていた可能性があると推測した。これら数の分解・合成，十進法に関する質問での誤りや反応から，本例は十進法の概念に障害があると考えられた。さらに，数直線課題やある数に近い方を選ぶ数の量的感覚の課題，概算は非常に困難で，「全然わからない」と半ば諦めた様子がみられ，2桁以上の数については，十進法で表記された数から量的な感覚をつかめないことが推察された。
数の桁合わせでは右端の数字を縦に揃えて書き，「全然わからない」と報告した。結果的に整数については桁を揃えることができたが，小数点を含む数で誤りが生じた。小数は，整数の十進法の考えを1より小さい数に拡張して用いたものであり（文部科学省 2018），小数点の右側に10分の1の位，100分の1の位，1,000分の1の位と小数が並ぶ。本例は小数点を含む数について，小数点や数字の位置が表す意味を理解できないために，桁を揃えることが困難であったと思われ，その原因として十進法の概念の障害があることが推察された。
以上より本例の数概念の障害について整理する。本例は，数概念の基礎とされる基数性は保たれていたが，アラビア数字から大きさを把握できるのは，1～5ないし6までのごく小さい数に限られており，重篤な数概念の障害が認められた。また，十進法の概念にも障害を呈しており，複数桁の数の量的な把握や概算，小数点を含む数の理解にも影響が及んでいた。
数の読み書きに関しては，数の書き取りにおいて，「0」を含む課題で誤りがみられた。「0」は，位取り記数法において，ある位が空であることを示す。書き取りでの誤りが1～9までの数には生じず，「0」でのみ生じていることから，この誤りは位の意味が理解できないことに起因すると考えられた。
本例は数に関する言語的知識の課題で，「1時間は何分か」に「40」，「1 kmは何mか」に「10」，「1 mは何cmか」に「50」と答えた。これらの課題は，数に関する知識を問うものであり，数概念の関与は低いと考えられる。くも膜下出血後の脳梗塞により，十進法の使用に加え12進法や60進法にも異常を認めた永友らの症例（永友ら 2009）においては，1日の時間数や1時間の分数に言葉で正しく答えることができており，言語的知識と数概念は独立していることを示している。本例にみられるような重篤な数概念の障害が，数に関する言語的知識に影響を及ぼすのかどうかについては，本例の誤りのみからは判断できず，さらに検討していく必要があると思われた。
計算に関しては，1桁同士の加減算において，正誤に加えて，解答を導くために用いた計算方略を本人の報告により1問ずつ把握したところ，「3＋2」「2＋1」「4－1」は「ぱっと出た」と報告したことから，算術的事実の想起で解答したと判断したが，その他の問題は口頭での計数や指での計数で解答を導いていた。九九については，想起できる場合と誤って答えを想起してしまう場合があった。このことから本例は，ごく簡単な計算を除いて，加算，減算，乗算に関する算術的事実が獲得できていないと考えられた。また，複数桁の計算では，加算で位と位の間に線を入れて計算し，乗算では「掛ける順序は理解しているが，答えとそのプロセスをどこに，どう書けばいいかわからない」と途方に暮れる様子がみられた。これらの反応から本例は，計算手順に関する知識は習得しているものの，筆算の過程で数字を書く位置に混乱が生じていると推察され，位の意味が理解できないことに起因するものと考えられた。
以上，計算の誤りについて整理すると，1桁同士の加減算と九九に関しては，算術的事実の獲得に障害があると考えられた。一方，筆算における誤りは位の意味が理解できないために生じており，十進法の概念の障害に起因するものと思われた。
2． 数概念の構造についてDehaeneらのtriple-code-modelでは，数は言語的枠組み（数詞），視覚的な数字形式（アラビア数字），アナログ的な数量表現（数概念）の3つのコードによって脳内に表象されているとされる（Dehaeneら 1995）。数概念の構造に関しては，1桁の数概念と十進法の概念が独立して営まれている（鈴木ら 2005），大きな近似数値を表すシステムと少数の個々の対象物を正確に表現するシステムの2つがある（Feigensonら 2004），数の認識はdigitとしての0-9の認知と位という2つの意味の体系が交差する形で成立する（古本ら 1993）などの報告があるが，いまだ十分に明らかではない。
右半球の損傷により数概念の障害を呈した古本らの症例は，1桁同士の数の大小比較や0-9の数字の読み書きは可能であるが，複数桁の数の音読や書き取りにおいて，「0」の位置を誤り，位取りに混乱がみられた（古本ら 1993）。また，くも膜下出血後の両側前頭葉梗塞により十進法に障害を呈した永友らの症例は，1-9までの数の意味は保たれているが，複数桁の数の分解や合成，大小比較に誤りがみられ，十進法に障害があると考えられた（永友ら 2009）。これらの症例から，数概念は少なくとも1桁の数と複数桁の数では異なる構造を持つと考えられる。
1桁の数に関して数概念の障害を呈した症例として，Cipolottiらの症例C.G.がある（Cipolottiら 1991）。C.G.は左前頭頭頂葉領域の脳梗塞により4以下の数の処理は可能だが，5を超える数については数という認識すら失われており数概念の喪失といえる状態を呈した。5を超える数については，音読や数唱も困難であった。知覚的サビタイジングにも障害を示し，C.G.が瞬時に把握できたのは2つまでであり，数えた場合には4まで正答したと報告されている。
本例の場合，数字から数を把握できるのは，5程度までであり，1桁の数に関してもそれ以上大きい数は数字から大きさを把握できず，C.G.と同様，数概念に重篤な障害を呈した。症例C.G.と異なるのは，数字から大きさを把握できない数に関しても，それが数であるという認識はあった点と，1つひとつ数えれば5より大きい数でも大きさを把握できた点，知覚的サビタイジングは可能であった点である。
これら過去の報告例と本例が示した数に関する特徴から，数の概念は，1桁の数と複数桁の数では異なる構造を持ち，さらに，1桁の数においても数概念の構造は一様ではなく，5程度の小さい数とそれ以上の大きい数では異なる可能性があることが示唆された。また，基数性は数概念獲得の基礎と考えられてきたが，本例は基数性が獲得できているにもかかわらず，5を超える1桁の数に関して数概念の障害を呈した。本例のこの特徴をみる限り，数概念の獲得には基数性だけでは十分ではなく，概念的サビタイジングのような数をパターン，すなわち，一定のまとまりを持った集合体として認識し，操作する能力が関与している可能性があると考えられた。
最後に位の概念と左右認知について考える。本例は，左右認知の評価で，「ぱっと左右がわからない」と述べ，現在も小学校の教室の黒板や先生の机をイメージし左右を判断していることを報告した。課題にはすべて正答したものの，左右の認知は本人の心的な補助手段に頼ってなされていることから，左右識別困難があるものと考えられた。位の概念と左右の認知は，どちらも左右に広がる空間的な位置に意味を与えるという点で共通している。この2つの障害が生じる背景には，何らかの共通した神経基盤が存在する可能性があり，今後さらに検討していく必要があると思われた。
3． 本研究の限界と今後の課題本例は，自ら本研究への協力を申し出た発達性の計算障害例であることから，頭部CTやMRI，SPECTなどの画像検査を行うことはできなかった。したがって，本例の症状と大脳の局在について直接的に論じることはできない。その点が本研究の限界である。しかしながら，本例には，計算障害や左右識別困難が認められており，左頭頂葉の機能低下を示唆しているのではないかと考えられる。WAIS-IVのパズルでも低下がみられており，空間関係の把握や処理に困難を呈している可能性があるが，今回は十分に検討できておらず，WAIS-IVのパズルでの成績低下と本例の数概念の障害との関連に関しては今後の課題としたい。
文部科学省の調査によると，通常学級のなかで「計算する」又は「推論する」に著しい困難を示す児童生徒の割合は，小学校・中学校で3.4％と推定されている（文部科学省 2022）。本例のように，数や計算に困難を抱えている児童生徒が一定数存在しているにもかかわらず，教育の場での代償手段の使用や合理的配慮についての理解は進んでいない現状がある。本例のような純粋例の検討を重ねることで，発達性計算障害にみられる数概念の障害や概念的サビタイジングの困難さ，それらを引き起こすメカニズムについて明らかにしていく必要があると考える。
自らご自身の症状について申し出て本研究に快くご協力頂きました症例に心より感謝申し上げます。
症例本人に本研究の目的を説明し，書面にて雑誌投稿についての同意を得た。
本論文に関連し，開示すべき利益相反状態にある企業，組織，団体はない。
本研究は第47回日本高次脳機能障害学会学術総会（2023年10月）にて発表した。
