EDUCATION AND TREATMENT OF CALCULATION ABILITIES OF LOW-ACHIEVING STUDENTS AND STUDENTS WITH...

Cognition and Learning in Diverse SettingsEducation and Treatment of Calculation Abilities of Low-Achieving Students andStudents with Dyscalculia: Whole Class and Individual ImplementationsDaniela Lucangeli, Patrizio Tressoldi, Chiara De Candia

Article information:To cite this document: Daniela Lucangeli, Patrizio Tressoldi, Chiara De Candia."Education and Treatment of Calculation Abilities of Low-Achieving Students andStudents with Dyscalculia: Whole Class and Individual Implementations" In Cognitionand Learning in Diverse Settings. Published online: 2005; 199-223.Permanent link to this document:http://dx.doi.org/10.1016/S0735-004X(05)18009-4

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http://dx.doi.org/10.1016/S0735-004X(05)18009-4

EDUCATION AND TREATMENT

OF CALCULATION ABILITIES OF

LOW-ACHIEVING STUDENTS AND

STUDENTS WITH DYSCALCULIA:

WHOLE CLASS AND INDIVIDUAL

IMPLEMENTATIONS

Daniela Lucangeli, Patrizio Tressoldi and

Chiara De Candia

ABSTRACT

Italian epidemiological data reveal a large discrepancy between the in-

cidence of learning disabilities in mathematics and simple difficulties in

mathematics. The incidence of dyscalculia (specific learning disability in

mathematics) is about 2%, whereas the incidence of students with dif-

ficulties in arithmetic is surprisingly greater, estimated by teachers to be

about five students out of 25 (that is, 20%). This unexpectedly high

number of students with difficulties invites serious consideration of its

cause and remedy. In this chapter, we try to answer to these questions

in the light of two educational studies aimed at improving calculation

Cognition and Learning in Diverse Settings

Advances in Learning and Behavioral Disabilities, Volume 18, 199–223

Copyright r 2005 by Elsevier Ltd.

All rights of reproduction in any form reserved

ISSN: 0735-004X/doi:10.1016/S0735-004X(05)18009-4

199

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abilities and the outcome of an intervention with three single cases with

dyscalculia using the model and the materials of an innovative arithmetical

curriculum. The results suggest that for most children, their arithmetic

difficulties are simple consequences of their math instruction, which may

be remediated by integrating traditional math curricula with information

derived from the research on the cognitive arithmetical architecture and

its development. There are also implications that even the arithmetical

difficulties of dyscalculic children may be improved with special training

focused on their specific impairments revealed after a detailed assessment.

Developmental dyscalculia is a learning disability. The International Clas-sification of Diseases (ICD–10, 1992) and the Diagnostic and StatisticalManual (DSM-IV, 1996) describe this disability with the following thesymptoms:

� difficulty understanding base concepts of arithmetical operation;� difficulty processing operational symbols;� difficulties in manipulating standard arithmetic concepts;� difficulties in recording data in mathematical problem solving;� spatial difficulties in calculation; and� difficulties in learning basic arithmetic facts.

It is evident that this disability represents a constellation of very differentcharacteristics: from knowledge of arithmetic signs to the knowledge ofquantity related to Arabic numbers; from the capacity to choose the relevantdata for problem solution, to the ability to correctly align numbers, to thesimple rote memory of number facts to the knowledge of the different al-gorithms to execute calculations.

Do these different symptoms derive from a single or multiple dysfunc-tions? The answer to this question is at the base of the comprehension ofthe nature of this learning disability and of the implementation of efficienteducational and treatment curricula. The literature on the arithmetical abil-ity has its roots in numerical knowledge (see Butterworth, 1999), and tendsto emphasize automatized specific competencies that can be affected byspecific difficulties in children (e.g., Temple, 1991), and specific impairmentsin adults (e.g., McCloskey, 1992). Temple’s model (1989, 1991, 1997) ofdevelopmental dyscalculia has received positive consensus in the literature(Butterworth, 1999). This model is in accord with McCloskey, Caramazza,

DANIELA LUCANGELI ET AL.200

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and Basili’s (1985) neuropsychological model, which differentiates the nu-meral processing system and the calculation system, as shown in Fig. 1.

These authors consider number comprehension and number productionto be distinct mechanisms. The number comprehension mechanism is usedto convert numerical inputs into central semantic representations to use insubsequent processing, such as calculation. The numeral production mech-anism translates central semantic representation of numbers into specificforms for output. Within these subsystems (comprehension and produc-tion), the processing of Arabic numbers is distinct from the processing oforal numbers. Thus, for example, reading a price tag involves the Arabiccomprehension mechanisms, whereas writing a check involves both Arabicand verbal production mechanisms. Within each of the numeral compre-hension and numeral production components, the model includes differentlexical and syntactic processing. Lexical processing refers to the com-prehension or production of individual elements in a number (e.g., the digit2 or the word two), whereas syntactic processing involves the processing of

Fig. 1. General Architecture of McCloskey et al. (1985) Model of Numerical and

Calculation Abilities.

Education and Treatment of Calculation Abilities 201

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spatial relations among elements in order to comprehend or produce anumber as a whole (e.g., 2 in the number 12). A final distinction in lexicalprocessing is made between verbal numeral comprehension and production.There is a distinction between the phonological processing mechanisms (i.e.,spoken numbers) and the graphemic processing mechanisms (i.e., writtennumbers). For example, the comprehension of the spoken number word‘‘five’’ is distinct from the process of writing the word ‘‘five’’.

The model comprises three components for calculation: the processing ofoperation symbols (e.g., +, –, � ) and words (e.g., plus, minus, times), theretrieval of arithmetic facts (e.g., 3� 7 ¼ 21) and the execution of calcu-lation procedures (e.g., in multi-digit addition, start at the right column,retrieve the sum of the digits in the column, write the ones digit of the sum atthe bottom of the column, carry the tens digit, if any, etc.).

Studying the errors made by dyscalculic children, Temple (1997) describedthree types of developmental dyscalculia. The first, digit dyslexia derivesfrom a lexical processing impairment in both comprehension and produc-tion systems. The second type, procedural dyscalculia regards the executionof calculation procedures. Temple described a 17-year-old boy without im-pairments in numeral processing or in retrieval of arithmetic base facts, butwith impairment in the execution of the calculation procedure. The thirdtype is the arithmetic facts dyscalculia characterized by a difficulty in theacquisition of arithmetic facts within the calculation system. Temple de-scribed a 19-year-old girl with an impairment in this area, and intactprocessing of number and of calculation procedures.

Analyzing the arithmetical performances of Italian students in primaryschool (from first to fifth grade), epidemiological data demonstrate a largediscrepancy between the incidence of mathematical learning disabilities andsimple mathematical difficulties. Cornoldi and Lucangeli (2004) reportedthat the incidence of dyscalculia as a specific learning disability is about 2%,a very small number of children. However, the incidence of students withmathematical difficulties was much greater, although this was no surprise tothe teachers. Lucangeli (2003, in press) reported that five students out of 25(that is, 20%) have some mathematical difficulties since the first years ofelementary school. Why is there such a high number of students with dif-ficulties, if only 2% are expected to be dyscalculic? And furthermore, whatshould be done in school and clinical practice?

Our hypothesis for explaining this discrepancy could be that most arith-metical instruction in the current mathematical curricula, at least inItaly, ignores methods for supporting specific domain processes of numer-ical knowledge. In the last 3 years, our research group has developed an


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arithmetical program to improve numerical intelligence, intended to supportnormal activities at school with cognitive strategies based on recent scientificmodels of mathematical cognition.

DESCRIPTION OF THE MATHEMATICAL PROGRAM

‘‘NUMERICAL INTELLIGENCE’’

The program is prepared as a didactic curriculum, but with some specificcharacteristics. Its focus is not the acquisition of concepts at the base of eachmathematical curriculum, but it is intended as a program intended to fosterthe numerical abilities from the preschool to the fifth-grade level accordingto the models of numerical intelligence as derived from cognitive develop-mental science.

The entire curriculum, entitled ‘‘Numerical Intelligence’’, consists of fourlevels. Level 1 includes the cognitive and metacognitive abilities neededfor construction of numerical knowledge at the preschool level (3–6-years-old; Lucangeli, Poli, & Molin, 2003a); level 2 develops cognitive and meta-cognitive abilities needed for construction of the numerical knowledgefrom the first to the second grade (6–8-years-old; Lucangeli, Poli, & Molin,2003b); level 3 develops cognitive and metacognitive abilities implied inthe construction of the numerical knowledge from the third to the fifthgrade (9–11-years-old; Lucangeli, De Candia, & Poli, 2003). Furthermore, theprogram contains materials for the analysis and the treatment of develop-mental dyscalculia.

The aim of the program is to give teachers specific educational strategies,which are useful for improving specific cognitive processes, on which nu-meric knowledge is based. The volumes are articulated in didactic opera-tional modules, referring to the following six areas:

� counting;� lexical processes;� semantic processes;� syntactic processes;� oral calculation;� written calculation.

The Counting area refers to the counting capacity and is guided bythe knowledge of principles based on non-verbal numerical expertise: bi-univocal correspondence, stable order/sequence and cardinality (Gelman &


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Gallistel, 1978, Gallistel & Gelman, 1992; Gelman & Meck, 1983; Fuson,1988; Wynn, 1990). Exercises are inspired by everyday life experiences; for-ward and backward numerations +1 and +n are introduced.

The Lexical Processes area refers to the ability to assign names to num-bers. According to its position, each digit assumes a different name (McC-loskey et al., 1985; Grewel, 1952, 1969; Deloche & Seron, 1982a, 1982b;Seron & Deloche, 1983, 1984). These processes are practised through trans-coding exercises, where both number reading and number writing are in-volved. At a more complex level (i.e., level 3) reconstructive strategies areinvolved, as well as metacognitive processes.

The Semantic Processes area refers to the capacity to comprehend themeaning of numbers as a measurable mental representation, and hasnumber–quantity correspondence as a final objective (McCloskey, 1992).The quantity representation is practised by analogic means using quantitycomparison and counting. At a more complex level, reconstructive strategiesare involved, as well as decompositions and composition strategies.

The Syntax Processes area refers the particular spatial relationship amongdigits representing numbers: the position of digits defines their value withina number (units, tens, hundreds, etc.) according to an ordered organizedsystem (Grewel, 1952, 1969; Deloche & Seron, 1982a, 1982b; Seron &Deloche, 1983, 1984).

The general objective in the Oral Calculation area is to learn differentstrategies to practice oral calculations to reach a stable automaticity (Ba-roody, 1983, 1987; Siegler & Mitchell, 1982; Geary, 1990, 1993; Geary,Brown, & Samaranayake, 1991; Beishuizen, 1993; Ashcraft, 1994). Differentstrategies are suggested, such as n+1, reconstructive strategies, decompo-sitions/roundings to ten and recovering of arithmetic facts.

TheWritten Calculation area is presented as a necessary means to performvery complex calculations that require paper to facilitate memory. Verbaland visual routines and algorithms are introduced, and the metacognitivecomponents involved by strategic control.

Each of the above six areas is composed of different didactic units de-veloped to clarify for the teacher the learning process, which is the focus ofthe lesson. Every part of the program includes a sequence of specific goals,intended to translate the general process under consideration into didacticpractices. Every goal is made explicit presenting/demonstrating the activityby written instructions to help the teacher teach pupils the exercises withmetacognitive mediators. Within each unit, there are also some ‘‘guidingicons’’ aimed at helping children recognize different proposed activities:to carry out an issue, to learn strategies, to think from a metacognitive point


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of view and self-evaluation. Each unit is interdependent with the other,following a principle of synergy and integration among the different proc-esses under instruction.

APPLICATION STUDIES

In this part of the chapter, we present the results obtained in three differentstudies using our program. The first two studies were intended to verify theefficacy of the application of the program in normal educational activities atschool. We expected that children in the experimental groups would performbetter in calculation and arithmetical competencies, compared with thecontrol groups educated with traditional curricula. The third application is amultiple single-case study aimed at verifying the efficacy of our programwith children diagnosed as developmental dyscalculics.

Study 1: Effects of the Numerical Intelligence Program with

Preschool Children

The main objective of this study was to determine if it is possible to traincognitive processes involved in the arithmetical competencies in the earlyphases of education.

Participants

Forty-two children attending kindergarten participated in this experiment.Twenty-two children (15 females and 7 males) were included in the exper-imental group, and 20 children (12 females and 8 males) were includedin the control group. Mean age was, respectively, 5.5 (SD ¼ 0:9) and 5.7(SD ¼ 0:7) years. All children had normal intelligence, medium socio-economic status (SES), and no known sensory or neurological deficits.

Materials and Procedures

All children were pre- and post-tested using the Italian original test PRCRNumber (Criterial Tasks related to Numerical Knowledge; Lucangeli,Molin, Poli, & Cortesi, in press). The test is composed of nine subtests,including:

� Writing Arabic numbers. Children were asked to write five numbers inrandom order, 2, 1, 4, 3, and 5 (score 0–5).


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� Counting forward from 1 to 20 and backward from 10 to 1 (score 1–30).� Reading Arabic numbers. Children were asked to tell the names of ninenumbers presented in random order, 3, 9, 2, 4, 7, 1, 5, 8, and 6 (score 1–9).

� Name–Arabic number correspondence. Children were asked to indicateamong three numbers presented in a paper the one named by the exam-iner, e.g., ‘‘Which of these is the number 2?’’ (score 1–9).

� Arabic numbers comparison. Children are asked to indicate the larger oftwo Arabic numbers presented simultaneously on paper (score 1–10).

� Dot quantities identification. Children were asked to indicate the quantityof dots corresponding to the Arabic number presented, among three pre-sented in a paper, e.g., ‘‘Show me where there are as many dots as thisnumber’’ (score 1–10).

� Pre-syntax knowledge. Children were asked to complete part-whole sen-tences such as: ‘‘A hand is formed by many y .’’ (score 1–11).

� Arabic number seriation. Children were asked to order in ascending valuethe following Arabic numbers, 1, 2, 3, 4, and 5, and to insert the correctnumbers in an interrupted series such as, 1,y,3, 4 (score 1–10).

Training Phase. Children of the experimental group were trained in fourmain domains, lexical knowledge, quantity (semantic) knowledge, syntacticknowledge and counting, following a special dedicated curriculum for pre-schoolers. For each area, different materials and objectives were preparedand trained during the normal school time instead of the traditional pre-mathematical curriculum taught to the control group. The details of eachtraining components are given in the Appendix. The training was appliedfrom January to April, 2004, and consisted of 17 weekly sessions lastingapproximately 90min. An example unit is provided in the Appendix.

Results

Total, pre–post treatment differences of both the experimental and thecontrol group are presented in Fig. 2a, whereas differences related to thenine subtests are presented in Fig. 2b.

The differences were statistically significant, tð40Þ ¼ 2:1; p ¼ 0:04; with amoderate effect size: d ¼ 0:67; demonstrating the overall positive effect ofthe training on learning of the cognitive strategies of arithmetical abilities.

Analyzing the effects of the training on the nine subtests, we observedstatistically significant difference in Reading, tð40Þ ¼ 2:2; p ¼ 0:032; d ¼

0:67; Number comparison, tð40Þ ¼ 1:92; p ¼ 0:06; d ¼ 0:59; and Syntaxsubtests tð40Þ ¼ 3:01; p ¼ 0:004; d ¼ 0:95:


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Fig. 2. (a) Post- Minus Pre-Scores Total Mean Differences, Including Confidence

Intervals, of the Experimental and the Control Group. (b) Post- Minus Pre-Scores

Total Mean Differences, Including Confidence Intervals, of the Experimental and the

Control Group in the Nine Different Treatment Components.


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Discussion

Based on the observed results, we can draw three interpretative hypotheses.First, it is possible to improve early arithmetical abilities using appropriateand scientifically based instruction. Second, these abilities derive from mul-ti-dimensional processes involving, e.g., writing, reading, counting, andcomparison, all of which are necessary for numerical knowledge. Third, thefact that program efficacy was observed only on the processes less empha-sized in the traditional educational curricula (such as reading, numbercomparison, and numerical syntax), may be interpreted as evidence of therole of cognitive instruction on specific domains of learning, such as nu-merical knowledge.

Study 2: Effects of the Numerical Intelligence Program with

Elementary Children

The main objective of this study was to verify the efficacy of the applicationof the program in normal educational activities at school. We expected thatchildren in the experimental groups would obtain better performances incalculation and arithmetical competencies with respect to the control groupseducated with traditional curricula.

Participants

Thirty-nine students took part in the experiment, 18 in the experimentalgroup and 21 in the control group. Their mean chronological age was sim-ilar, with means of 7.8 (SD ¼ 1:3) and 7.6 (SD ¼ 1:6), respectively. Allchildren had normal intelligence, medium SES, and no known sensory orneurological deficits.

Materials and Procedures

All students were pre- and post-tested with the Italian AC-MT test (Corn-oldi, Lucangeli, & Bellina, 2002). This standardized test includes four parts:written calculation, numerical knowledge, general accuracy, and generalspeed.1 Training focused on five areas: counting, lexical knowledge, seman-tic knowledge, mental calculation, and written calculation. Each area in-cludes different components (see Appendix), trained in sequence from thesimple to the complex.

The training in the experimental group was carried out from January toMarch in ten weekly sessions lasting one hour, during their normal math-ematics period. During the same sessions, the control group students were


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trained by teachers in the routine arithmetical activities planned accordingto their curriculum.

Results

The pre- and post-treatment score differences of both the experimental andthe control group are presented in Fig. 3. Only the pre–post difference inarithmetic calculation reached a statistically significant level, tð37Þ ¼ 2:37;p ¼ 0:023; with a moderate effect size, d ¼ 0:77:

Discussion

Considering our hypotheses, with only ten sessions it was possible to obtaina statistically significant difference in calculation. With respect to the othermeasures on numerical competencies, we emphasize that the lack of statis-tically significant differences may be due to the fact that students receivingthe traditional curriculum were also taught numerical knowledge and ac-curacy. Regarding speed, we think that the time spent with our program (tenhours in total) was probably not sufficient to automatize the competenciestaught, more so than the control curriculum.

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Fig. 3. Mean Differences with 95% Confidence Intervals of Post- Minus Pre-Test

Scores at the Four Areas Tested with the Standardized Test AC-MT.


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Study 3: Effects of the Numerical Intelligence Program with Discalculic

Children

This investigation was a multiple single-case study aimed to verify the ef-ficacy of our program with children diagnosed as dyscalculics.

Participants

Three students, two girls (S.M., 10.5-years-old, and D.V., 11-years-old) anda boy (L.B. 11-years-old), all attending the fifth grade, were trained with theNumerical Intelligence Program.

S.M. was diagnosed as dyscalculic according to the DSM-IV criteria, thatis, normal IQ, normal educational and socio-cultural opportunities, noneurological or sensorial deficits; D.V. has a chromosomal syndrome(CATCH22) due to an interstitial deletion of the long arm of chromosome22 (region 22q11.2), this syndrome is associated with delayed developmentsin cognitive and motor areas, learning disabilities and attentional deficits;L.B. was affected by a ventricular haemorrhage at birth with subsequentsmall ventricular enlargement. His calculation abilities met a clinical level ofdeficit, with marked difficulties in carrying out school requirements. Apartfrom their school marks, all participants were tested with the ABCA2 test, astandardized Italian test for assessment of calculation abilities (Lucangeli,Tressoldi, Fiore, 1998).

Subject S.M. S.M. was a ten-year-old girl with normal IQ and reading dif-ficulties. As can be seen from Fig. 4a and 4b, she manifests difficulties inmental and written calculation (in both accuracy and speed), in comparisonof quantities, and in number dictation and tables production (in both ac-curacy and speed).

In mental calculation, S.M. used primitive and slow strategies such ascounting on fingers. In written calculation, there were procedural errors.Comprehension mechanisms were preserved in both accuracy and speed, butthere was a drop in auditory judgement quantities. In the production sys-tem, there were difficulties in the dictation of numbers (probably correlatedto the deficit in lexical retrieval), and in recovering tables.

The training program. For S.M. the training was focused on improving:

� the lexical production mechanism;� mental calculation strategies;� basic arithmetical facts;� written calculation procedures.


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(a)

(b)

Fig. 4. (a) Pre ( ) and Post ( ) Treatment Percentile Scores of Subject S.M. Related to the Calculation and Comprehension

Systems on the Standardized Test ABCA. (b). Pre ( ) and Post ( ) Treatment Percentile Scores of Subject S.M. Related to the

Production System on the Standardized Test ABCA.

Educatio

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The rehabilitation program is based upon the book ‘‘Numerical Intelli-gence’’ (vol. 3) (Lucangeli, De Candia, & Poli, 2003), characterized by ametacognitive approach. For the automatization of tables a dedicated com-mercial software program (‘‘Tabelline

TM

’’) was also employed. The compar-ison between pre-test and post-test after 6 months of training, 40min twice aweek, is presented in Fig. 4a and 4b.

The post-test assessment evidences great improvement in both accuracyand speed in written calculation. Oral calculation was improved in accuracy,but speed was still slow. These results in the calculation system evidencegreat improvement in the use of procedures that now are automatizated. Inoral calculation, S.M. now uses efficacious strategies, even if they are not yetautomatized. In the production system, we observed great improvement intables. Minor improvement was observed in numeral dictation, because thisperformance may be aggravated by the difficulty in lexical retrieval asso-ciated with her reading deficit.

Subject D.V. D.V. is an 11-year-old girl. She has a borderline IQ, FullScale ¼ 75, good reading and writing abilities, but poor text comprehension.D.V. has widespread arithmetic difficulties: oral and written calculation (inboth accuracy and speed), ordering by quantity, and in dictation and re-covery of numerical facts in the production systems (see Fig. 5a and 5b).

In mental calculation D.V. use inefficacious and slow strategies likecounting on fingers and mental algorithm (she images the calculations usingthe written calculation procedures). In written calculation there are proce-durals and arithmetical fact errors. The comprehension mechanisms showsemantic and syntactic errors: in the ordering from minor to major and frommajor to minor. In the production system, D.V. fails in dictation of numbersand in the recovery of numerical combinations, she is slow but accurate inrecovering table combinations. In summary, her difficulties involve: mentalcalculation strategies, written calculation procedures, recall of arithmeticalfacts, lexical, and semantics mechanisms.

For D.V. the training was directed toward improving:

� lexical mechanisms;� syntactic mechanisms;� semantic mechanisms;� mental calculation strategies;� arithmetical base facts;� written calculation procedures.


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(a)

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Fig. 5. (a) Pre ( ) and Post ( ) Treatment Percentile Scores of Subject D.V. Related to the Calculation and Comprehension

Systems on the Standardized Test ABCA. (b) Pre ( ) and Post ( ) Treatment Percentile Scores of Subject D. V. Related to the

Production System at the Standardized Test ABCA.

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Also in this case, the rehabilitation program was based upon the book‘‘Numerical Intelligence’’ (vol. 3) (Lucangeli et al. 2003), plus the use of thecommercial software program ‘‘Tabelline

TM

’’ to automatize tables retrieval.The comparison between pre-test and post-test after eight months of in-struction, for 40min twice a week, is presented in Fig. 5a and 5b.

Post-test scoring reveals great improvement in accuracy in oral cal-culation. Now D.V. uses more elevated strategies such as composing,decomposing, and rounding to multiples of ten. The training is not yetcompleted, and we are still working to improve written calculation. Re-garding arithmetical facts, great improvement can be seen in accuracy; nowD.V. is able to use efficacious mental strategies to retrieve results. Theseabilities are still not automatized. In recall of tables, speed has increasedbecause the process is now automatized. In lexical, syntactic, and semanticmechanisms, we see improvement in accuracy (see ordering, quantity com-parison, and dictation).

Subject L.B. The third participant is an 11-year-old boy (L.B.), with a nor-mal IQ. His difficulties involved calculation, comprehension (quantity com-parison), and production (dictation and numerical facts) systems (seeFig. 6a and 6b), with a general deficit in the automatization of both systems.

In written calculations, L.B. used incorrect procedures (he confused pro-cedures, e.g., addition instead of multiplication) and his rate of production isslow. In oral calculations, the strategies are very elementary such as themental algorithm (he images the calculations using the written calculationprocedures) and not automatized as in written calculation. Comprehensionmechanisms are correct except for difficulty with quantity comparison, butwith a slow rate overall. In production, there were difficulties in lexicalprocesses (the dictation of numbers), and in retrieving numerical facts. As forthe comprehension system the rate of production system is also very slow.

For L.B., the training was focused upon improving:

� semantic mechanisms;� syntactic mechanisms;� lexical mechanisms;� oral calculation;� written calculation.

In this case also, the rehabilitation program was based upon the samebook and the same commercial software as the previous cases. Comparisonbetween pre-test and post-test after 8 mon, during which time the studentreceived instruction for 40min, twice a week, is presented in Fig. 6a and 6b.


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(a)

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Fig. 6. (a) Pre ( ) and Post ( ) Treatment Percentile Scores of Subject L.B. Related to the Calculation and Comprehension

Systems at the Standardized Test ABCA. (b) Pre ( ) and Post ( ) Treatment Percentile Scores of Subject L.B. Related to the

Production System at the Standardized Test ABCA.

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In the post-test, L.B. demonstrated improvement in accuracy in oral cal-culation. Like D.V., L.B. now uses more elevated strategies such as compos-ing, decomposing, and rounding to multiples of ten. The training is not yetcompleted, and we are still working to improve written calculation. We cansee great improvement in accuracy and less in speed in areas of lexical, se-mantic, and syntactic difficulty. The results in the area of arithmetical factsdemonstrate improvement in accuracy, but this ability is still not automatized.The next objective in our training regards procedures in written calculation.

GENERAL DISCUSSION

This chapter provides a synthesis of three arithmetical intervention trainingprocedures on cognitive components of numerical intelligence; two intendedto improve the development of numerical knowledge in normally function-ing children, and one intended to improve functioning in three single casesof developmental dyscalculia.

In the last few years, our research group has developed a program toimprove numerical intelligence. This program is intended to support normalactivities at school with cognitive strategies based on recent scientific modelsof mathematical cognition aimed at fostering numerical abilities from thepreschool to the fifth-grade level, according to the models of numericalintelligence as derived from cognitive developmental science. Furthermore,the program contains materials for the analysis and the treatment of de-velopmental dyscalculia.

The results obtained using the Numerical Intelligence Program are dis-cussed in the chapter analyzing three different experimental investigations.

In the first experimental investigation, preschool children were trained infour main domains – lexical knowledge, quantity (semantic) knowledge, syn-tactical knowledge, and counting. Analyzing the effects of the training, weobserved statistically significant effects in Reading, Number Comparison, andSyntax. In light of the observed results, we suggest the following hypotheses:

� it is possible to improve precocious arithmetical abilities using appropriateand scientifically based instruction;

� these abilities depend on multi-dimensional processes necessary for nu-merical knowledge;

� the role of cognitive instruction on specific domains, such as numericalknowledge, can determine not only good performance but also goodcognitive competencies in numerical abilities.


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In the second experimental investigation, second-grade children weretrained in four areas: counting, lexical knowledge, semantic knowledge,mental calculation, and written calculation. The main objective of this studywas to verify the efficacy of the application of the program in normal ed-ucational activities at school.

The results suggest that for arithmetical written calculation only, ten ses-sions were sufficient to obtain statistically significant improvement in ac-curacy. Regarding speed, we think that the time spent with our program(10 h in total) was probably not sufficient to automatize the trained com-petencies, beyond that of the control curriculum. The results reinforce theevidence obtained in the first study that it is possible to improve arithmeticalabilities and reduce the number of children showing arithmetical difficultiesusing a teaching program focused on the cognitive processes involved onnumerical knowledge.

The third experimental application was a multiple single-case study aimedat verifying the efficacy of our program with children with dyscalculia. Themethod we used is based on an accurate assessment of their numerical andcalculation abilities according to a precise cognitive model that permits us toobtain an individual profile where strengths and deficit in the components ofcalculation, comprehension, and number production are presented. Foreach of the difficulty areas, we applied instructional units based on thenumerical competencies needed. In light of the results we can suggest quan-titative and qualitative observation.

Our primary conclusion is that for developmental dyscalculia, the use ofspecific training focused on observed cognitive difficulties should reduce thedeficit. This reduction could be interpreted not as a normalization of thedeficit but as a concrete reduction of the impairment.

Among the characteristics of our curriculum that seem important to itsefficacy are not only its contents (the six core areas) and its structure inhierarchical modules, but also the emphasis on promoting active learning ofstudents, fostering metacognitive knowledge of what is being taught, inparticular when it is necessary to change inefficient calculation strategieswith more efficient ones. This metacognitive emphasis does not excludethe necessity to automatize factual recall. However, in light of our evidence,we find it necessary to integrate the drill and practice approach with ametacognitive approach that offers more opportunities to help the studentbecome more active and autonomous.

In conclusion, the present results suggest that for most children, theirmath difficulties are consequences of inadequate instruction. We believethat the traditional math curriculum should be integrated, placing more


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emphasis on the cognitive architecture implied in numerical knowledge withboth perspectives:

� to foster the development of these competencies;� to focus the intervention on the specific impairments.

Even for children with dyscalculia, there are promising suggestions that itis possible to reduce their symptoms and to promote a more active andstrategic approach to mathematics learning.

NOTES

1. The AC-MT test is a standardized Italian original test for individual and col-lective assessment of calculation abilities from the first to the eighth grade. At eachgrade, different tasks are proposed. As part of total score, it is possible to derive awritten calculation score, an accuracy and a speed score summing up the data ob-tained from the following subtests: ordering by magnitude, recover of automaticnumbers combination and number dictation.2. The ABCA test is inspired by McCloskey’s model (1985), to evaluate the dif-

ferent components of the numeral and calculation systems. The test is addressed tochildren from 8 to 10-years-old (3, 4, 5, elementary), and it is divided in two parts. Inthe first competencies in written and oral calculation, in all four arithmetical cal-culations are assessed. If the results are below the 10th percentile with respect to thenorms, we proceed to the second part. In this part numbers comprehension andproduction is assessed with different subtests. Comprehension: Denomination anduse of calculation symbols; ordering digit in increasing or decreasing value; insertionof calculation symbols; judgment of numerousness; positional value. Production:Backward enumeration; numbers dictation; tables knowledge; counting of dots;aligning digits; recover of numerical combinations. For each subtest we record errorsand speed, the first measure give information about the accuracy of the process andthe second about its automatization.

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APPENDIX

Details of the curriculum for preschoolers.

Components of Lexical Processes Objectives

1. To use a linguistic sequence to learn the base of counting.2. To use a linguistic sequence to enhance the auditory memory.3. To learn the names of numbers.4. To automatize the number sequence.5. To start to learn the rule +1.6. To write the Arabic numbers.7. To read the Arabic numbers.

Components of Semantic Processes Objectives

1. To estimate the weight of objects in direct relationship and independ-ently of their volume.

2. To estimate the space dimension in relation to objects.3. To estimate numerosity independently from the objects dimension.4. To formulate an hypothesis on the results of an increment +1.5. To recover quantitative experiences.6. To estimate space in relationship to quantity.7. To recover the quantity ‘‘one’’ and introduce the concept of ‘‘zero.’’


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8. To define the quantity ‘‘one’’ and apply it by subtracting and addingunits.

9. To demonstrate the quantity ‘‘two’’ by increment.10. To know the quantities 3, 4, and 5 by the procedure +1.11. To help in representing quantity using a visuo-spatial code.12. To start to reflect on the concept ‘‘equivalent.’’13. To represent quantity using an analogical code.14. To reflect on grouping.15. To use subitizing.16. To use the quantities 6, 7, 8, and 9 by increment +1.17. To learn the quantity representation according to the personal cognitive

style.18. To reflect in metacognitive terms.

Components of Syntactic Processes Objectives

1. To select objects according to different characteristics.2. To select objects according to different functions.3. To select objects according to different dimensions.4. To differentiate a quantity from the whole of elements of which it is part.5. To introduce the concept of ‘‘ordinality.’’6. To differentiate the dimensions, ‘‘big,’’ ‘‘little,’’ ‘‘medium.’’

Counting

1. To acquire and master one to one correspondence.2. To build a progressive and ordered sequence.3. To retrieve the name of numbers and their semantic correspondence.4. To start to count using Arabic numbers.5. To subitize recognizing the quantity ‘‘five.’’6. To increase a given quantity by 1.7. To count and introduce the concept of ‘‘no one’’ as precursor of the

concept of ‘‘zero.’’


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See an example


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Details of the curriculum for second graders.The curriculum is made up of six different components:

1. Counting forward and backward.2. Lexical processes: to recognize, read, and write the Arabic numbers.3. Semantic processes: to understand the quantity underlying the numbers.4. Syntactical processes: to consider different classes of quantity and their

position in number representation.5. Mental and written calculation.

See an example:


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EDUCATION AND TREATMENT OF CALCULATION ABILITIES OF LOW-ACHIEVING STUDENTS AND STUDENTS WITH...

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