Metacognition and errors: the impact of self-regulatory trainings in children with specific learning disabilities

Abstract

Even in primary school, mathematics achievement depends upon the efficiency of cognitive, metacognitive and self-regulatory processes. Thus, for pupils to carry out a computation, such as a written calculation, metacognitive mechanisms play a crucial role, since children must employ self-regulation to assess the precision of their own thinking and performance. This assessment, in turn, can be helpful in the regulation of their own learning. In this regard, a body of literature suggests that the application of psychoeducational interventions that promote the development of mathematics-related metacognitive (e.g., control) processes, based on the analysis of the students’ errors, can successfully influence mathematics performance. The main objective of the current study was to investigate the impact of a metacognitive and cognitive training program developed to enhance various arithmetic skills (e.g., syntax, mental and written calculation), self-regulatory and control functions in primary and secondary school students exhibiting atypical mathematical development. Sixty-eight Italian children, 36 male and 32 female (mean age at pretest = 9.3 years, SD = 1.02 years), meeting the criteria for the diagnosis of dyscalculia or specific difficulties in mathematics, took part in the study. Of these, 34 children (i.e., experimental group) underwent the cognitive and self-regulatory intervention enhancing mathematics skills training for 16 weekly sessions. The remaining students were assigned to the control group. For a pre-test and post-test, a battery of standardized mathematical tests assessing different mathematics skills, such as written and mental operations, digit transcription and number ordering skills, was administered and provided a series of measures of calculation time and accuracy (i.e., number of errors). In the post-test, the experimental group exhibited better accuracy in written calculation and in digit transcription. Overall, the current outcomes demonstrate that psychoeducational interventions enriching metacognitive and mathematical achievements through error analysis may be an effective way to promote both the development of self-regulatory and control skills and mathematical achievement in children with atypical mathematical development.

Full text

Vol.:(0123456789)
1 3
ZDM (2019) 51:577–585
https://doi.org/10.1007/s11858-019-01044-w
ORIGINAL ARTICLE
Metacognition anderrors: theimpact ofself-regulatory trainings
inchildren withspecific learning disabilities
DanielaLucangeli1· MariaChiaraFastame2· MartinaPedron3 · AnnamariaPorru2· ValeriaDuca3·
PaulKennethHitchcott4· MariaPietronillaPenna2
Accepted: 14 March 2019 / Published online: 21 March 2019
© FIZ Karlsruhe 2019
Abstract
Even in primary school, mathematics achievement depends upon the efficiency of cognitive, metacognitive and self-regula-
tory processes. Thus, for pupils to carry out a computation, such as a written calculation, metacognitive mechanisms play a
crucial role, since children must employ self-regulation to assess the precision of their own thinking and performance. This
assessment, in turn, can be helpful in the regulation of their own learning. In this regard, a body of literature suggests that
the application of psychoeducational interventions that promote the development of mathematics-related metacognitive (e.g.,
control) processes, based on the analysis of the students’ errors, can successfully influence mathematics performance. The
main objective of the current study was to investigate the impact of a metacognitive and cognitive training program developed
to enhance various arithmetic skills (e.g., syntax, mental and written calculation), self-regulatory and control functions in pri-
mary and secondary school students exhibiting atypical mathematical development. Sixty-eight Italian children, 36 male and
32 female (mean age at pretest = 9.3 years, SD = 1.02years), meeting the criteria for the diagnosis of dyscalculia or specific
difficulties in mathematics, took part in the study. Of these, 34 children (i.e., experimental group) underwent the cognitive
and self-regulatory intervention enhancing mathematics skills training for 16 weekly sessions. The remaining students were
assigned to the control group. For a pre-test and post-test, a battery of standardized mathematical tests assessing different
mathematics skills, such as written and mental operations, digit transcription and number ordering skills, was administered
and provided a series of measures of calculation time and accuracy (i.e., number of errors). In the post-test, the experimental
group exhibited better accuracy in written calculation and in digit transcription. Overall, the current outcomes demonstrate
that psychoeducational interventions enriching metacognitive and mathematical achievements through error analysis may
be an effective way to promote both the development of self-regulatory and control skills and mathematical achievement in
children with atypical mathematical development.
Keywords Self-regulation training· Psycho-educational intervention, primary school, specific learning disabilities·
Mathematics· Dyscalculia
* Martina Pedron
martina.pedron@gmail.com; info@poloapprendimento.it
Daniela Lucangeli
daniela.lucangeli@unipd.it
Maria Chiara Fastame
chiara.fastame@unica.it
Annamaria Porru
annamariaporru@gmail.com
Valeria Duca
duca.valeria@gmail.com
Paul Kenneth Hitchcott
pkhitch@gmail.com
Maria Pietronilla Penna
penna@unica.it
1 Università degli Studi di Padova, Padua, Italy
2 Università degli Studi di Cagliari, Cagliari, Italy
3 Polo Apprendimento, Padua, Italy
4 Università di Pisa, Pisa, Italy

578 D.Lucangeli et al.
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1 Introduction
Mathematics is one of the most important subjects learned
in school, since mathematic achievement is one of the
strongest predictors of later academic success (Duncan
etal. 2007; Wang etal. 2013). Moreover, mathematical
competence in adult learners is the consequence of math-
ematics skills learnt in childhood (Cargnelutti etal. 2016).
However, around 3–8% of school-age children experience
mathematical difficulty (MD) (Nelson and Powell 2017)
in several cognitive tasks such as processing number sets,
counting and basic fact retrieval (Geary etal. 2012).
Specific learning disorder in mathematics is one exam-
ple of a MD that is the result of a specific neural functional
organization, present from birth, that is most evident dur-
ing calculating tasks. Although MDs can be differenti-
ated from non-specific difficulties or delays in learning
mathematics, which may be temporary or secondary to
other disorders (Cornoldi 2007), both conditions can
be improved with targeted interventions (Lucangeli and
Mammarella 2010). However, MDs are more problem-
atic since they are associated with life-long impairment
of academic and work success or performance. Crucially,
MDs have been shown to affect employment, income, and
occupational productivity more than intelligence and read-
ing performance (Rivera-Batiz 1992). Although MDs are
typically persistent, two key factors can reduce the risk
of poor outcomes in secondary school and in adulthood:
early recognition of mathematical difficulty and targeted
interventions (Nelson and Powell 2017). While MD can
also be reduced at later ages, with appropriate and specific
interventions students with MD can improve their math-
ematics performances at different grade levels (Geary etal.
2012; Re etal. 2014). Such evidence points to the need
to investigate and develop effective methods to promote
mathematics knowledge and reasoning.
The outcome of interventions for MD can be influenced
by different variables, as shown in a recent meta-analysis
by Dennis etal. (2016). The authors examined 25 studies
published between 2000 and 2014 and analyzed differ-
ences in participant characteristics (e.g., age, grade level,
etc.), intervention structure (instructional grouping, inter-
vention agent, duration of the intervention), domain of
intervention, intervention approaches, quality of research
methodology, and intervention components. Whereas
interventions had an overall positive impact on mathemat-
ics performances of students with MD, grade level was a
significant factor: interventions at kindergarten level were
less effective than those at elementary level. The charac-
teristics of the trainer were also significant: interventions
were much more effective when provided by researchers
or a researcher-trained graduate assistants compared to
teachers or paraprofessionals. No significant differences
were found between training provided to small groups or
on a one-to-one setting. Lastly, controlling task difficulty
significantly and positively predicted effect-size estimates.
Additional research, comparing specific empowerment
mathematical training with general academic training, has
highlighted the importance of task difficulty (Re etal. 2014).
Participants were children with different levels of MD,
including children with diagnosed dyscalculia. The train-
ing tasks were personalized to the children’s mathematical
profile, and task difficulty was controlled: in fact, all the
training activities were sequenced from easy to difficult. The
post-training results showed that both students with dyscal-
culia and with mild mathematical difficulties in the instruc-
tion condition outperformed the control group. These results
were maintained also at a later follow-up assessment.
Other factors, also influence intervention outcomes.
For example, multi-component intervention, e.g., combin-
ing visual models with verbal instructions, appears to be
a key variable (Jitendra etal. 2018). Furthermore, in the
same meta-analysis, it was observed that a minimum of
10h of instruction was required. Recent evidence also sug-
gests that interventions involving a digital component may
be effective. For example, Re etal. (submitted) evaluated
the effectiveness of a specific and digitally supported train-
ing program, for improving numerical skills in primary and
secondary school children with MD. The training tasks were
implemented on a Web App i.e., “I bambini contano” and
the 57 participants were randomly assigned to two groups.
For the experimental group the tasks were differentiated and
adapted to each student’s learning profile and performed on
the Web App; for the control group the difficulty of the activ-
ities was adapted to the school curriculum and did not use
the Web App. Pre- to post-training measurements showed
that children of the experimental group had a significantly
higher improvement than the control group, in particular in
arithmetic facts and written calculation. Moreover, a follow-
up evaluation showed that the efficacy of the experimental
training program lasted up to 2months after the intervention.
Mathematical training certainly has several cognitive
components, but also relates to metacognitive domains.
Reflecting this aspect, the NCTM (National Council of
Teachers of Mathematics 2000): as underlined by Mevar-
ech (2006), has emphasized that meaningful mathematical
teaching must enhance not only knowledge construction via
problem solving, a skill that involves cognitive and math-
ematical tasks, but also metacognition. Metacognition is
defined as “thinking about thinking” (Flavell 1979) and
has two components: knowledge of cognitive processes
and products, and the ability to control, monitor, and evalu-
ate cognitive processes, namely self-regulation processes.
Knowledge of cognition has various inter-related compo-
nents: metacognitive knowledge about self, the task and

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strategies; knowledge about how to use the strategies; and
metacognitive experience, that is, the feeling about being
successful (or unsuccessful) in performing the task. These
components are related, as metacognitive knowledge leads
to self-regulated strategy use relevant to the objective. This
self-regulation process affects the metacognitive experience
and shapes the basis for the acquisition of more metacogni-
tive knowledge, and so on.
In their review, Schneider and Artelt (2010) noted that the
first researchers to examine the relationship between meta-
cognition and mathematics education, were Garofalo and
Lester (1985), who
claimed that a pure cognitive analysis of mathemati-
cal performance is inadequate. They emphasized the
importance of metacognition for the analysis and
understanding of mathematical performance. Refer-
ring to the distinction between knowledge about and
regulation of cognition, they argued that not only regu-
latory metacognitive behaviors but also person, task,
and strategy categories of metacognitive knowledge
are important in mathematical performance (Schneider
and Artelt 2010, p.153).
According to Garofalo and Lester (1985), metacognitive
knowledge in the domain of mathematics includes one’s
assessment of one’s own capabilities and limitations in this
domain and beliefs about the nature and the difficulty of the
mathematical task. Garofalo and Lester (1985) reformulated
a four phase cognitive–metacognitive framework for math-
ematical performance, to be incorporated in evaluation and
training activities, as cited by Schneider and Artelt (2010):
[A]n orientation phase (strategic behavior to assess and
understand a problem), an organization phase (plan-
ning of behavior and choice of actions), an execution
phase (regulation of behavior to conform to plans), and
a verification phase (evaluations of decisions made and
of outcomes of executed plans), each being filled with
cognitive as well as metacognitive activities. (p.6).
How do metacognitive processes improve mathemat-
ics learning and performance? Metacognitive knowledge
includes individual attitudes, knowledge and emotions about
mind functioning. Such knowledge can help children in
understanding both how different cognitive processes work
and which cognitive processes are involved in learning math-
ematics, as in mathematical problem solving (Cornoldi etal.
2015). Furthermore, metacognitive knowledge affects the
selection and use of specific strategies and control processes
directly affecting performance in mathematics. Emotions
such as anxiety can exert a pervasive detrimental effect on
cognitive performance, by precipitating negative thoughts
and concern, which impair key memory resources, such as
working memory (Ashcraft 2002). In contrast, developing
metacognitive control allows children to implement self-
regulatory control processes that enhance mathematical
performance. Accurate judgment of task difficulty allows
for the selection of appropriate sub-actions and strategies
as well as checking for possible errors in their thinking and
performances. It is likely that metacognitive mechanisms
are involved in most mathematical tasks, every time children
employ self-regulation to assess the accuracy of their own
thinking and performance. Self-regulation is an important
part of metacognitive control processes (Dinsmore etal.
2008) and self-regulatory mechanisms are involved in the
analysis of the students’ errors, a key process that can suc-
cessfully influence mathematics performance.
Baten etal. (2017) explored the efficacy of metacognition
within theory-based instructional techniques in mathemat-
ics, as a means to provide “‘opportunities’ to focus explic-
itly and reflect on what, how and when they learn” (p.620).
They found positive outcomes even in kindergarten children.
Metacognitive mathematical training yields substantial ben-
efits for children with low and regular mathematics perfor-
mance (Desoete etal. 2003; Schneider and Artelt 2010) and
also in older children, even in college students. In a 2006
study, Mevarech and Fridkin (2006) examined the effects
of a metacognitive instructional method, called IMPROVE
(Mevarech and Kramarski 1997), on students’ mathematical
knowledge and reasoning, as well as on general and domain
specific metacognitive knowledge. Participants were stu-
dents from a pre-college course in mathematics, randomly
assigned to two groups. Both groups were exposed to the
same learning materials and were taught by the same expe-
rienced teacher. However, IMPROVE students were also
explicitly trained to activate metacognitive processes during
the solution of mathematical problems. Crucially, the results
indicated that these students
significantly outperformed their counterparts on both
mathematical knowledge and mathematical reasoning.
In addition, the IMPROVE students attained signifi-
cantly higher scores then the control group on the three
measures of metacognition: (a) general knowledge of
cognition; (b) regulation of general cognition; and (c)
domain-specific metacognitive knowledge (Mevarech
and Kramarski 1997, p.389).
These conclusions are supported by Trainin and Swan-
son (2005), who underlined how “One of the ways college
students with LD may compensate for their cognitive dif-
ficulties is by relying on metacognition; that is, consciously
controlling actions that are too complex to be controlled
automatically” (p.262).
Collectively, research within this field has provided
various recommendations concerning the development of
effective training practices for children with dyscalculia or
MD. One key theme that has emerged is the need to develop

580 D.Lucangeli et al.
1 3
cognitive and metacognitive abilities in tandem. In relation
to this requisite, the main aim of the current study was to
investigate the effect of a metacognitive and cognitive train-
ing program developed to enhance different arithmetic skills
(e.g., digit transcription, mental and written calculation) and
self-regulatory and metacognitive control functions, in Ital-
ian students exhibiting atypical mathematical development.
It was hypothesized that if the training was effective, the
group of participants undergoing the intervention would
outperform the control group in the mathematical measures
proposed.
2 Method
2.1 Participants
Sixty-eight Italian students (mean age at pre-test = 9.3 years,
SD = 1.02 years, age range = 7–12years of age), attending
the 2nd, 3rd, 4th and 5th grades of primary school and 1st
grade of secondary school, were recruited in one of the edu-
cational centers affiliated to Polo Apprendimento, a scientific
institution where the assessment, diagnosis and interven-
tion for children with specific Learning Disabilities (LD)
or transitory learning difficulties are conducted. The whole
sample was composed of 36 males and 32 females diagnosed
with dyscalculia (i.e., they met the criteria for the diagnosis
of dyscalculia in accordance with the current Italian law
number 170, 8 October 2010 concerning the diagnosis of
Specific Learning Disabilities) or that presented transitory
difficulties in mathematics (i.e., their performance on stand-
ardized mathematics tests validated for the Italian school
population in the low-average range, defined as performance
between one and two standard deviations below the mean
score in each measures used in the clinical practice). Ion
the pre-test 34 participants (18 males and 16 females, mean
age = 9.3 years, SD = 1.01years) were matched for age, gen-
der- and diagnosis with a further subsample composed of
34 children (mean age = 9.5 years, SD = 1.07). After that,
the former group was presented with a psychoeducational
intervention to promote the empowerment of the deficient
mathematical processes (i.e., they comprised the experimen-
tal group), whereas the latter group (i.e., control group) was
involved in the assessment phases only. In order to exclude
the occurrence of possible cognitive deficits especially in
the youngest participants or children showing specific math-
ematics difficulties only, in the pre-test each participant was
screened using the Raven Coloured Progressive Matrices
test (Raven and Court 1998; Italian version Belacchi etal.
2008) or using the Wechsler Intelligence Scale for Chil-
dren IV (WISC IV Wechsler 2003; Italian version; Orsini
etal. 2011). All students were Caucasian, had no physical,
sensory, or neurological impairments and spoke Italian
fluently. None of the participants showed signs of cognitive
deficits (i.e., IQ < 70). Gender was counterbalanced across
the participants (χ2 = 0.23, df = 1, p = .63) and between the
groups (χ2 = 0, df = 1, p = 1). Furthermore, no age differences
were found between the experimental and control groups
[t(66) = − 0.675, p = .50]. Table1 illustrates the character-
istics of the sample.
2.2 Instruments
For the aim of the current study, before the training phase
(i.e., pre-test) and after it (i.e., post-test), each participant
of the experimental and control groups was presented with
a set of mathematical tests included in the AC-MT 6–10
(Cornoldi etal. 2012) and AC-MT 11–14 (Cornoldi and
Cazzola 2004) batteries. These tests have been developed
for the assessment of calculation and problem-solving skills
of Italian students attending primary (i.e., AC-MT 6–11)
and secondary (i.e., AC-MT 11–14) schools, respectively.
Specifically, the whole sample was asked to complete the
following tests:
1. Written calculation test: this is a pencil-and-paper task
assessing procedural knowledge and the degree of auto-
maticity necessary to complete a set of operations (i.e.,
2 additions, 2 subtractions, 2 multiplications, and 2 divi-
sion), as fast and accurately as possible. For each algo-
rithm, children were instructed to perform the calcula-
tion in agreement with the procedure that they learned at
school. Performance was scored in terms of total number
of correct responses, that is, 1 score was given for each
correct computation (i.e., maximum score = 8).
2. Mental calculation test: this requires the computation of
3 additions and 3 subtractions as fast as possible without
using any external aid (e.g., pencil and paper). The task
Table 1 Education, gender and diagnostic status of the participants
enrolled in the study
Males Females Dyscalculia Difficulties in
mathematics
Primary school
2nd grade 0 2 0 2
3rd grade 6 6 4 8
4th grade 11 9 5 15
5th grade 17 15 8 24
Secondary school
1st grade 2 0 1 1
Experimental group
n = 34 18 16 9 25
Control group
n = 34 18 16 9 25

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Metacognition anderrors: theimpact ofself-regulatory trainings inchildren withspecific…
1 3
assesses the strategies underpinning mental calculation.
Specifically, after the examiner read aloud the compu-
tation that had to be carried out, the child was given
30s to compute the operation. After this time interval,
the response was considered incorrect and 1 score was
assigned. Performance was assessed in terms of accu-
racy (i.e., number of total errors, maximum score = 6)
and speed (i.e., time spent to complete the whole task).
3. Written digits transcription test: this is a pencil-and-
paper task assessing the capacity to process the syn-
tactic properties of numbers and its positional value.
Specifically, students were verbally presented with the
quantitative information relating to the units, tens and
hundreds comprising a 3-digit integer and, immediately
after the presentation, required to convert that informa-
tion to the corresponding written digit number (e.g.,
“write the number corresponding to 3 tens, 8 units, and
2 hundreds”). Performance was scored in terms of accu-
racy, that is, 1 score was given to each correct response
(i.e., maximum total score = 8).
4. Number ordering test: this is a pencil-and-paper task
assessing the semantic representation of numbers by
comparing different quantities. In order to perform the
task correctly, quantitative information presented by the
examiner must first be processed. After that, a compari-
son among those quantitative representations is neces-
sary to arrange and then write down the corresponding
digit numbers in ascending or descending fashion. Five
series of numbers were presented (e.g., 15, 58, 36, 7 for
second grade, or 36, 15, 576, 154 for fourth grade) and
1 score was given to each set of numbers arranged in the
correct order (maximum score = 5).
Before and after the training phase eight questions from
the Mathematics and Metacognition questionnaire by Cor-
noldi etal. (1995) were administered to the children of the
experimental and the control group to evaluate their meta-
cognitive skills in mathematics tasks. The first questions
assess the general level of awareness about the main char-
acteristics of mathematics (possible causes of error, strategic
attitude, awareness of the cognitive functions involved in
learning mathematics). The last three questions are about
planning, monitoring and evaluation skills. The following
are some examples of questions about control and self-
regulation processes: “When you perform mathematical
operations how do you avoid mistakes?”, “When you finish
a mathematical task do you review how you have done it,
especially in the more critical parts?”, “When you perform
a mathematical exercise do you try to make sure you under-
stand what is asked of you?”. The responses were included
in a qualitative analysis.
After the assessment of mathematical and metacogni-
tive skills of the whole sample in the pre-test, participants
of the experimental group were presented with an indi-
vidualized cognitive and self-regulatory training program
that was provided by a team of psychologists specialized
in the treatment of specific learning disabilities. The psy-
chologists were supervised in the use of psychoeducational
techniques to enhance mathematical skills by the first and
third authors. During delivery of this program, the psy-
chologists met with supervisors every 2 weeks.
Based on the pre-test assessment of mathematics skills,
the complexity of self-regulatory training was adapted to
the learning profile of each child. During each (approxi-
mately 60min) training session, a series of pencil-and-
paper and computer-assisted tasks were presented to
empower children in the development of their mathemati-
cal performance (i.e., mental and written calculation,
semantic and syntactic representation of numbers) and
related self-regulatory and control processes (e.g., fill-
ing questionnaires concerning the processes necessary
to carry out the proposed activity, analyzing the actual
performance with the educator and then discussing and
correcting possible mistakes, encouraging reflection about
the errors or the progress made after completing each type
of mathematic task). The latter metacognitive activities
were selected firstly to develop specific knowledge about
the processes underpinning the specific mathematics tasks
and, secondly, to enhance the acquisition of strategies and
procedures necessary to monitor, control and evaluate (i.e.,
self-regulation) mathematics performance at school. The
analysis of errors was conducted by each child under the
supervision of the trainer, to make students more aware
about the quality (i.e., strengths and weak points) of their
performance in solving mathematical tasks and enhanc-
ing their speed of processing too. Children were presented
with time-limited, computer-assisted and pencil-and-paper
tasks for this purpose. In addition, children had the oppor-
tunity to reflect about each proposed task, with empha-
sis on how their emotional state and motivational beliefs
affected performance (Hultsch etal. 1988; Borkowski and
Muthukrishna 1994). As suggested by Re etal. (2014),
each training activity was organized as follows:
1. the task was presented and its goal was explained by
the trainer, followed by a discussion of the methods for
accessing and coding the number (e.g., using phonologi-
cal, visual, and analogical pathways, giving students the
chance to use the approach they find most congenial to
their comprehension of the task);
2. a discussion followed, of the strategies that the students
could adopt or modify to carry out the task;
3. the child was invited to practice each learned strategy to
complete the task; during this phase, particular emphasis
was given to the integration of the procedural aspects of
the learning tasks with the reasoning and metacogni-

582 D.Lucangeli et al.
1 3
tive processes necessary to monitor and regulate one’s
performance;
4. after comparing the usability and utility of the strategies
discussed earlier, the activity performed by the child was
first verbally repeated by him/her and then summarized
again by the operator, in order to emphasize some cru-
cial information missing in the presentation of the stu-
dent;
5. finally the student was invited to self-assess his/her per-
formance, highlighting his/her cognitive, metacognitive
achievements and discussing the motivational aspects
related to the task performance.
In brief, this method derives from Vygotsky’s construct
concerning the zone of proximal development (Vygotskij
1931; Veggetti 1974), which he defined as the distance
between the actual developmental level (e.g., determined
by the actual mathematics skills) reached by the student and
the potential development that the child could achieve (e.g.,
control of mental calculation) with the aid of adult guidance
or by peer tutoring experience.
It is crucial to highlight that the activities were selected
to strengthen the specific skills that each child needed to
improve and to gradually remove the amount of scaffolding
that he/she received. This process was aimed at enhancing
mathematics achievement and promoting children’s self-
regulation and control during the execution of the tasks.
2.3 Procedure
The participants took part in the study after both of their
parents and the head of the school provided written informed
consent to participate. In the pre-test and post-test partici-
pants were individually tested in a quiet room by a LD spe-
cialized psychologist at one of the centers affiliated to the
clinical center Polo Apprendimento. Before the presentation
of each test, the examiner read aloud the instructions and
provided several practice trials to ensure that all the children
understood how to carry out the task. If required, the exam-
iner also recorded the time spent to perform the test. Finally,
all participants were presented with eight questions from
the Mathematics and Metacognition questionnaire (Cor-
noldi etal. 1995). After the pre-test phase, the participants
of the experimental group were individually presented with
the psychoeducational training for approximately 4 months.
The training design involved the following phases:
1. Learning level assessment: this phase involved defining
each child’s learning profile, emphasizing the main areas
of mathematical learning difficulties, their general cog-
nitive abilities and emotional and motivational aspects
related to learning.
2. Baseline (analysis of individual profiles and treatment
planning): All the individual learning profiles were
assessed to select the areas of greatest deficit on which
to focus the training. In this phase an accurate error
analysis was also conducted.
3. Training. Sessions were planned for 16 weeks. The
cycle of training was provided once a week for at least
4 months. The sessions lasted 60min each.
4. Post-training assessment (efficacy analysis) to evaluate
any improvement in the different areas of calculation.
At post-test, after the conclusion of the training phase,
both the experimental and control groups were asked to per-
form the battery of pencil-and-paper tasks that was already
administered in the pre-test. In the pre-test and post-test, the
presentation order of the tasks was counterbalanced across
the participants. Each experimental session lasted approxi-
mately 2h.
3 Results
First, response distributions were inspected and where nec-
essary transformed to achieve normality. Then, statistical
analyses were performed separately for each administered
test.
In order to investigate the impact of group (i.e., experi-
mental versus control) at pre-test, a series of Analyses of
Variance (ANOVAs) were computed for each mathematic
test (i.e., accuracy and speed measures or accuracy index
only). As expected, no differences between the experimen-
tal and control groups were found at pre-test in terms of
mathematical skills. Specifically, the effect of group was
not significant for errors in the written calculation test
[F(1,66) = 0.83, p = .366, η2p = .01] and in the mental cal-
culation test [F(1,66) = 1.91, p = .172, η2p = .02], the speed
in performing the mental calculation test [F(1,66) = 2.14,
p = .148 η2p = .03], the number of correct responses in the
digit transcription test [F(1,66) = 1.851, p = .178 η2p = .03]
and in the number ordering condition [F(1,66) = 0.459,
p = .50, η2p = .007].
A different pattern was observed when the same tests
were repeated on the post-test data. Here, the experimental
group outperformed the control group in some, but not all,
measures. Specifically, the main effect of group was signifi-
cant when correct responses were computed in the written
calculation test [F(1,65) = 6.600, p = .013, η2p = .09] and in
the digit transcription condition [F(1,65) = 5.824, p = .019,
η2p = .08]. In contrast, no differences between the groups
were found in terms of speed [F(1,65) = 1.155, p = .286,
η2p = .01] and accuracy [F(1,65) = 1.961, p = .166, η2p = .03]
in the mental calculation test and the correct responses in

583
Metacognition anderrors: theimpact ofself-regulatory trainings inchildren withspecific…
1 3
the number ordering condition [F(1,65) = 1.690, p = .198,
η2p = .02].
Finally, in agreement with previous studies (Caviola
etal. 2009; Fastame and Callai 2015), the gain scores were
computed to more deeply investigate the possible benefits
related to the treatment in the mental calculation and number
ordering conditions. Specifically, the formula [(post-training
scores–pre-training scores)/pre-training scores] was used to
compute the gain scores. Thus, a second series of ANOVAs
was conducted on the post-test data. This revealed significant
main effects of group on the speed [F(1,65) = 5.561, p = .02,
η2p = .08] and errors [F(1,65) = 6.112, p = .016, η2p = .09] in
the mental calculation test. However, no significant differ-
ence for number of correct responses in the number order-
ing condition [F(1,65) = 0.32, p = .574, η2p = .005] was
observed. Overall, in the post-test the experimental group
reported fewer errors and used less time to conduct the men-
tal calculation task than the control participants (Table2).
Regarding the metacognitive component of the training,
the children answered some of the metacognitive ques-
tionnaire items before and after the treatment, and their
responses were analyzed using qualitative descriptors. Dur-
ing the pre-training phase, children recognized the value and
usefulness of the metacognitive control processes; however
they did not use them when they worked on a mathemati-
cal task. In other words, they were able to understand what
it would be useful to do in terms of control but they were
less accurate in practice. During the post-training phase, the
children from the experimental group were more involved
and active in the control and implementation of self-regula-
tion strategies. For example, they recognized when the tasks
were more difficult and required more attention, and they
were more autonomous in commenting and sharing with
the operator what they were doing to complete the task and
which strategy they were using. When they made a mistake,
children appeared more willing and involved in the search
for the cause and the possible solution.
4 Discussion
The main aim of the current study was to explore the impact
of the combined computer-assisted and pencil-and-paper
training programs to enhance mathematics skills and self-
regulatory metacognitive processes in primary and second-
ary school students.
Substantial benefits of intensive training on mathemat-
ics performance were documented. Results were positive
in most of the areas of the intervention: digit transcription,
mental calculation and written calculation. Specifically,
teaching calculation strategies and metacognitive reflection
about task-specific strategy selection produced greater pre-
vs post-test gains in the experimental group compared to the
control one both in terms of accuracy and speed of calcula-
tion. Similarly, gain in digit transcription was achieved by
the trained participants.
These results are in line with a recent meta-analysis show-
ing positive effects of “well-design interventions” in LD and
MD children (Jitendra etal. 2018). Specifically, the meta-
analysis indicates that when secondary students with MD
receive more than 10h of well-design mathematical inter-
ventions, one can expect an increase of 22 percentile points
after intervention.
The use of some questionnaires was useful, in particular
at the beginning of the intervention: these offered an overall
estimate of the child’s ability to reflect on the activity of
his or her own mind while performing mathematical tasks,
and children who gave more adequate answers also achieved
greater success in mathematics (Lucangeli etal. 1991). From
the qualitative analysis of the collected questionnaires, pro-
cesses such as monitoring, but in particular self-assessment,
require additional time and specific interventions to become
more effective. The development of effective self-regulation
appears more effective when children are both aided in rec-
ognizing the necessary skills for carrying out the calcula-
tions and encouraged to choose and to apply more effective
operational strategies.
The training had also the objective of improving other
aspects of learning including the following:
• the ability to reflect on one’s attitude in mathematics,
Table 2 Mean number of errors and time spent to complete selected
AC-MT 6–11 and AC-MT 11–14 tests
Mean scores at pre-test and post-test in the digits transcription, num-
ber ordering condition, written calculation, mental calculation speed
and mental calculation errors are reported
Pre-test Post-test
MSD MSD
Digits transcription
Experimental group 3.7 1.862 4.67 1.472
Control group 3.09 2.036 3.62 1.518
Number ordering
Experimental group 6.97 1.96 7.73 1.941
Control group 6.41 2.017 7.15 1.708
Written calculation
Experimental group 4.09 2.006 5.82 1.286
Control group 4.5 1.745 4.88 1.665
Mental calculation speed
Experimental group 126.42 86.386 90.85 53.013
Control group 100.41 47.165 104.15 48.215
Mental calculation errors
Experimental group 2.67 1.963 1.58 1.582
Control group 2.03 1.605 2.12 1.684

584 D.Lucangeli et al.
1 3
• the acquisition of a greater awareness of one’s strengths
and difficulties,
• the development of a greater ability in self-control and
self-assessment errors, defining the area of difficulty on
which to focus a child’s attention regarding strengths and
improvements, and promoting his/her capacity to reflect
and self-assess the strategies used in mathematics tasks,
consciously activating metacognitive processes of control
while performing these tasks and when they were com-
pleted.
The study has two important limitations: the first is the
lack of statistical analysis with pre- and post-training com-
parison for the questionnaires used to investigate some meta-
cognitive aspects. The questionnaires were analyzed only
using qualitative descriptors therefore it was not possible
to measure statistically the extent of the change in these
aspects. However, these questionnaires were qualitatively
analyzed, particularly in the pre-training phase, in order to
prepare personalized training on calculation with the addi-
tion of metacognitive reflection elements. The second limita-
tion is the lack of a follow-up evaluation.
To summarize, our results lead us to conclude that a spe-
cific training program adapted to the cognitive profile of
each child is the best solution to achieve effective results,
on condition that it improve both cognitive and metacogni-
tive skills, in particular guiding the students to analyze and
reflect on their errors and the most effective strategies to
overcome them.
The long-term goal, in the perspective of future studies,
will consist in helping the child to actively direct his own
cognitive processes, empowering the internalization and
use of reflection and control of error strategies through the
development of a strategic thinking style.
Acknowledgements The authors thank the children who participated
in the study and their families and the Polo Apprendimento Centres in
Como, Fermo, Monselice, Padova, Porto Viro, Prato, Sardegna, Treviso
and Vicenza.
References
Ashcraft, M. A. (2016) Math anxiety: personal, educational, and cogni-
tive consequences. Current Directions in Psychological Science
11(5):181–185
Baten, E., Praet, M., & Desoete, A. (2017). The relevance and efficacy
of metacognition for instructional design in the domain of math-
ematics. ZDM Mathematics Education, 49, 613–623. https ://doi.
org/10.1007/s1185 8-017-0851-y.
Belacchi, C., Scalisi, T. G., Cannoni, E., & Cornoldi, C. (2008). Manu-
ale CPM. Coloured progressive matrices. Standardizzazione Itali-
ana [CPM manual. Coloured progressive matrices. Italian valida-
tion]. Firenze: Giunti O.S. Organizzazioni Speciali.
Borkowski, J. G., & Muthukrishna, N. (1994). Lo sviluppo
della metacognizione nel bambino: un modello utile per
introdurre l’insegnamento metacognitivo in classe. Insegnare
all’Handicappato, 8, 229–251.
Cargnelutti, E., Tomasetto, C., & Passolunghi, M. C. (2016). How is
anxiety related to math performance in young students? A longi-
tudinal study of Grade 2 to Grade 3 children. Cognition and Emo-
tion, 31, 755–764. https ://doi.org/10.1080/02699 931.2016.11474
21.
Caviola, S., Mammarella, I., Cornoldi, C., & Lucangeli, D. (2009). A
metacognitive visuospatial working memory training for children.
International Electronic Journal of Elementary Education, 2(1),
122–136.
Cornoldi, C. (Ed.). (2007). Difficoltà e disturbi dell’apprendimento
[Learning difficulties and disabilities]. Bologna: Il Mulino.
Cornoldi, C., Caponi, B., Focchiatti, R., Lucangeli, D., Todeschini, M.,
& Falco, G. (1995). Matematica e metacognizione [Mathematics
and metacognition]. Trento: Edizioni Erickson.
Cornoldi, C., Carretti, B., Drusi, S., & Tencati, C. (2015). Improving
problem-solving in primary school students: the effect of a train-
ing program focusing on metacognition and working memory.
British Journal of Educational Psychology, 85, 424–439. https ://
doi.org/10.1111/bjep.12083 .
Cornoldi, C., & Cazzola, C. (2004). AC-MT 11–14. Test di valutazi-
one delle abilità di calcolo e problem solving dagli 11 ai 14 anni
Gruppo MT [AC-MT 11–14. Test for the assessment of calcula-
tion and problem solving skills from 11 to 14 years of age MT
Group]. Trento: Edizioni Erickson.
Cornoldi, C., Lucangeli, D., & Bellina, M. (2012). AC-MT 6–11. Test
di valutazione delle abilità di calcolo e soluzione dei problemi
Gruppo MT [AC-MT 6–11. Test for the assessment of calculation
and problem solving skills from 6 to 11 years of age Gruppo MT].
Trento: Edizioni Erickson.
Dennis, M. S., Sharp, E., Chovanes, J., Thomas, A., Burns, R. M.,
Custer, B., & Park, J. (2016). A meta-analysis of empirical
research on teaching students with mathematics learning diffi-
culties. Learning Disabilities Researche and Practice. https ://doi.
org/10.1111/ldrp.12107 .
Desoete, A., Roeyers, H., & De Clercq, A. (2003). Can off-line
metacognition enhance mathematical problem solving? Jour-
nal of Educational Psychology, 95, 188–200. https ://doi.
org/10.1037/0022-0663.95.1.188.
Dinsmore, D. L., Alexander, P. A., & Loughlin, S. M. (2008). Focus-
ing the conceptual lens on metacognition, self-regulation, and
self-regulated learning. Educational Psychology Review, 20(4),
469–475.
Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A.
C., Klebanov, P., etal. (2007). School readiness and later achieve-
ment. Developmental Psychology, 43, 1428–1446. https ://doi.
org/10.1037/0012-1649.43.6.1428.
Fastame, M. C., & Callai, D. (2015). Empowering visuo-spatial ability
in primary school: Results from a follow-up study. Educational
Psychology in Practice, 31, 86–98. https ://doi.org/10.1080/02667
363.2014.98931 5.
Flavell, J. H. (1979). Meta-cognitive and cognitive monitoring: A new
area of cognitive developmental inquiry. American Psychologist,
31, 906–911.
Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive moni-
toring, and mathematical performance. Journal of Research in
Mathematics Education, 16, 163–176.
Geary, D. C., Hoard, M. K., & Bailey, D. H. (2012). Fact retrieval
deficits in low achieving children and children with mathematical
learning disability. Journal of Learning Disabilities, 45, 291–307.
https ://doi.org/10.1177/00222 19410 39204 6.
Hultsch, D. F., Herzog, C., Dixon, R. A., & Davidson, H. (1988). Mem-
ory self-knowledge and self-efficacy in the aged. In M. L. Howe e
C. J. Brainerd (Eds.), Cognitive development in adulthood: Pro-
gress in developmental research (pp.65–92). New York: Springer.

585
Metacognition anderrors: theimpact ofself-regulatory trainings inchildren withspecific…
1 3
Jitendra, A. K., Lein, A. E., Im, S. H., Alghamdi, A. A., Hefte, S. B., &
Mouanoutoua, J. (2018). Mathematical interventions for second-
ary students with learning disabilities and mathematics difficul-
ties: A meta-analysis. Exceptional Children, 84(2), 177–196. https
://doi.org/10.1177/00144 02917 73746 7.
Lucangeli, D., Cornoldi, C., & Tessari, F. (1991). Bambini con dis-
turbi di apprendimento in lettura e matematica: Aspetti comuni
e specificità nei deficit cognitivi e di conoscenza metacognitiva.
Psichiatria dell’Infanzia e dell’Adolescenza, 58(5–6), 629–664.
Lucangeli, D., & Mammarella, I. (2010). Psicologia della cognizione
numerica: Approcci teorici, valutazione e intervento [Numerical
knowledge psychology: Theory, assessment and intervention].
Milan: Franco Angeli.
Mevarech, Z., & Fridkin, S. (2006). The effects of IMPROVE on
mathematical knowledge, mathematical reasoning and meta-
cognition. Metacognition Learning. https ://doi.org/10.1007/s1140
9-006-6584-x.
Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A multidimen-
sional method for teaching mathematics in heterogeneous class-
rooms. American Educational Research Journal, 34, 365–395.
https ://doi.org/10.3102/00028 31203 40023 65.
National Council of Teachers of Mathematics (NCTM) (2000). Prin-
ciples and standards for school mathematics. National Council of
Teachers of Mathematics (NCTM): Reston.
Nelson, G., & Powell, S. R. (2017). A systematic review on longitu-
dinal studies of mathematics difficulty. Journal of Learning Dis-
abilities. https ://doi.org/10.1177/00222 19417 71477 3.
Orsini, A., Pezzuti, L., & Picone, L. (2011). WISC-IV. Contributo alla
taratura Italiana. [WISC-IV Italian edition]. Firenze: Giunti, OS.
Raven, J. C., & Court, J. H. (1998). Raven’s progressive matrices and
vocabulary scales. Oxford: Oxford Pyschologists Press.
Re, A. M., Benavides-Varela, S., Pedron, M., De Gennaro, M. A., &
Lucangeli, D. (submitted). Response to specific and digitally sup-
ported training for students with mathematical difficulties.
Re, A. M., Pedron, M., Tressoldi, P. E., & Lucangeli, D. (2014).
Response to specific training for students with different levels
of mathematical difficulties: A controlled clinical study. Excep-
tional Children, 80(3), 337–352. https ://doi.org/10.1177/00144
02914 52242 4.
Rivera-Batiz, F. L. (1992). Quantitative literacy and the likelihood of
employment among young adults in the United States. Journal of
Human Resources, 27, 313–328. https ://doi.org/10.2307/14573 7.
Schneider, W., & Artelt, C. (2010). Metacognition and mathematics
education. ZDM The International Journal on Mathematics Edu-
cation, 42, 149–161. https ://doi.org/10.1007/s1185 8-010-0240-2.
Trainin, G., & Swanson, H. L. (2005). Cognition, metacognition, and
achievement of college students with learning disabilities. Learn-
ing Disability Quarterly, 28, 261–272.
Veggetti, M. S. (1974). Storia dello sviluppo delle funzioni psichiche
superiori. Firenze: Giunti-Barbera.
Vygotskij, L. S. (1931). Istorija razvitija vysših psichi eskih funkcuj,
in Sobrani so inenij, vol.3 Moskva: Pedagogijka.
Wang, A. H., Shen, F., & Byrnes, J. P. (2013). Does the opportunity-
propensity framework predict the early mathematics skills of
low-income pre-kindergarten children? Contemporary Educa-
tional Psychology, 38, 259–270. https ://doi.org/10.1016/j.cedps
ych.2013.04.004.
Wechsler, D. (2003). Wechsler intelligence scale for children: Fourth
edition. San Antonio: Psychological Corporation.
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