Perspectives to Technology-Enhanced Learning and Teaching in Mathematical Learning Difficulties

Full text

International
Handbook of
Mathematical
Learning Di culties
Annemarie Fritz
Vitor Geraldi Haase
Pekka Räsänen Editors
From the Laboratory to the Classroom
Foreword by
Brian Butterworth

Annemarie Fritz • Vitor Geraldi Haase
Pekka Räsänen
Editors
International Handbook
of Mathematical Learning
Difculties
From the Laboratory to the Classroom
Foreword by Brian Butterworth

ISBN 978-3-319-97147-6 ISBN 978-3-319-97148-3 (eBook)
https://doi.org/10.1007/978-3-319-97148-3
Library of Congress Control Number: 2018959436
© Springer International Publishing AG, part of Springer Nature 2019
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microlms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specic statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the
editors give a warranty, express or implied, with respect to the material contained herein or for any errors
or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims
in published maps and institutional afliations.
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Editors
Annemarie Fritz
Faculty of Education
Centre for Education Practice Research
University of Johannesburg
Johannesburg, South Africa
Pekka Räsänen
Niilo Mäki Institute
Jyväskylä, Finland
Vitor Geraldi Haase
Departamento de Psicologia
Faculdade de Filosoa e Ciências Humanas
Universidade Federal de Minas Gerais
Belo Horizonte, Brazil

1© Springer International Publishing AG, part of Springer Nature 2019
A. Fritz et al. (eds.), International Handbook of Mathematical Learning
Difculties, https://doi.org/10.1007/978-3-319-97148-3_1
Chapter 1
Introduction
AnnemarieFritz, VitorGeraldiHaase, andPekkaRäsänen
Twenty years ago, the main global educational issue was how to arrange schooling
for all children in the world (Dakar, 2000; World Conference on Education for all,
Jomtien, 1990). Still, in 1997, more than 100 million children did not have access to
education (Roser & Ortiz-Ospina, 2017). The recent UNESCO report (2017) states
that we have managed to half that gure, but at the same time a new issue has been
raised: even though children would go to school, over 600 million (56%) do not
reach even the basic level of skills in reading and mathematics. Globally, six out of
ten children and adolescents are not able to read or handle mathematics with pro-
ciency by the time they are in the age to complete primary education (UNESCO
Institute for Statistics, 2017).
The aim of our book is to offer a global view of mathematical learning difcul-
ties and their different causes, whether they are connected to quality of education or
other reasons. In this book, these difculties are covered from genetic as well as
cognitive, neuroscientic and pedagogical perspectives. We also describe the cur-
rent situation of mathematical learning difculties in different parts of the world.
This trip around the world provides the reader with a unique insight in how difcul-
ties in mathematics are approached in different cultural settings. The amount of
A. Fritz (*)
Faculty of Education, Centre for Education Practice Research, University of Johannesburg,
Johannesburg, South Africa
e-mail: fritz-stratmann@uni-due.de
V. G. Haase
Departamento de Psicologia, Faculdade de Filosoa e Ciências Humanas,
Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
P. Räsänen
Niilo Mäki Institute, Jyväskylä, Finland
e-mail: pekka.rasanen@nmi.

2
research on mathematical learning difculties has doubled every decade, but the
question of how the lessons learned from research and laboratories can be applied
to everyday practice at schools still remains. Many teachers struggle with the ques-
tion of how to identify those children who face problems in learning, how to support
them and how to select the best methods to intervene.
In the current information society, machines can do calculations faster and more
accurately than humans. Thus, does it appear necessary for each and every one to
learn the basics of mathematics? The answer can be found all around us. Like
colour, quantity is in everything. Whatever we do, whatever we see, there always is
a number of something. Being numerate means that we are able to communicate
about these numbers. Children start this journey to numeracy by comparing “more”
and “less”. They learn to talk about amounts, changes and differences in quantities
when we teach them the quantitative concepts and the number system. It has taken
thousands of years for mankind to develop the current efcient system, shared by
the whole world. It was developed to describe the exact number of something with
words and symbols. These are tools that we can use in almost any kind of context:
to inform others about amounts or a change in an amount; to build different kinds of
scales and measurement systems like time, length, weight or money; to describe
amounts or ratios; and to share, multiply or divide. Acquiring basic mathematical
knowledge means to acquire basic competencies for participating in our human
culture.
Worldwide the number of children who do not learn these basic competencies
during the primary education varies from 15% in North America and Europe to
about 85% in sub-Saharan Africa (UNESCO, 2017). These are clearly higher g-
ures than usually described in studies on developmental dyscalculia, a persistent
difculty in learning arithmetic, where the prevalence estimates vary from 2% to
7% (Devine, Soltész, Nobes, Goswami, & Szűcs, 2013; Rapin, 2016). While in
dyscalculia the difculties in learning are considered to stem from more or less
specic neurocognitive factors of the child’s brain, the high number of low-
performing children tells us that the majority of the difculties are related to an
interplay of environmental factors like quality of education and opportunities to
learn, not to forget the early development and home learning environment. Children
in different parts of the world have very unequal opportunities to learn
mathematics.
However, we can also nd a strong heterogeneity among the children even within
developmental dyscalculia (Fias, Menon, & Szucs, 2013; Rubinsten & Henik, 2009;
Rykhlevskaia, Uddin, Kondos, & Menon, 2009), and researchers continue the quest
of depicting the key variables behind the individual variation in numerical abilities
and difculties. Large steps in understanding this variation have been taken recently.
This book gives an overview of the current state of the art about the mechanisms
behind the difculties, about recognition and diagnostics at school or in clinical
practice as well as about the effectiveness of different types of interventions.
The book has been divided into ve parts. Each part provides multiple perspec-
tives to the topic area with a summarising discussion chapter for four sectionsat the
end. The rst part (Part I: Development of Number Understanding: Different
Theoretical Perspectives) covers the different theoretical perspectives on the devel-
A. Fritz et al.

3
opmental issues. In the last three decades, research on mathematical difculties and
developmental dyscalculia has boomed. Difculties in mathematical learning have
been approached from different theoretical perspectives. Research has been carried
out most actively in neuropsychology and cognitive neurosciences. The rapid devel-
opment in technology and the drop of price in using these technologies have surged
the amount of research using brain imaging or genetic analyses. After the initial
excitement, the discussion turned to the applicability of neuroscientic ndings to
educational practice (Ansari & Coch, 2006; De Smedt & Grabner, 2016; Goswami,
2006; Howard-Jones, 2014).
At the same time, developmental psychologists conducted pioneering research
with infants and toddlers to understand the development of the complex construct of
number.
Learning difculties cannot be traced back to a monocausal explanatory model.
Such models have proven to be generally insufcient for the conception of learning
difculties. Instead, complex interaction models are to be favoured. Among the
multitude of inuencing factors, socioeconomic factors have to be emphasised. Also
important to consider are the complex interactions between individual differences
and contextual factors, such as public policies, poverty, culture, school as well as
classroom effects and, of course, the quality of the pedagogy in the classroom.
Pedagogical models have also seen many revisions. The goal of learning is not any-
more considered to be studying specic subjects in order to reach curricular goals,
but education is seen via broader concepts of acquiring competencies like “thinking
skills” (Marope, Grifn, & Gallagher, 2017).
In the second part (Part II: Mathematical Learning and Its Difculties Around
the World), we focus on learning difculties around the world. The authors draw a
picture of the similarities and differences in research, education and public policies.
Progress and obstacles in translating basic cognitive research to the classroom are
discussed from the perspectives of different countries and areas. The quality of
available diagnostic instruments and intervention programmes is evaluated.
Based on the international comparison, the levels of learning even the basic skills
vary extensively in different parts of the world. This variation shapes the topics
discussed at the local level. When in the welfare societies, learning motivation has
risen to be one of the topics of discussion, in developing countries, the questions of
social factors causing low performance are urgent problems. The chapters offer us
the view of scientic denitions of learning disabilities being universal but at the
same time point out how the school systems react, recognise and support those who
struggle with learning that varies from one country to another. Likewise, there are
large differences in how much evidence-based tools, like assessment materials stan-
dardised and normed in the country or eld-tested intervention programmes, are
available for practice. We would be glad if these chapters would encourage research-
ers in different countries to engage cross-country collaborations in these efforts.
Part III discusses the cognitive, motivational and emotional underpinnings of
mathematical learning difculties. The development of arithmetic can be approached
from different perspectives: the neurobiological, the cognitive and the behavioural
level. The development of arithmetic skills is based on the complex interplay of
these different levels, which are dependent on each other. In this part, authors
1 Introduction

4
present the recent development in research, trying to uncover the relationship
between neurocognitive, motivational and emotional variables and learning or
learning difculties in mathematics.
Part III starts with a chapter about genetics, which is one of the most rapidly
developing research elds. From the perspective of genetics, developmental dyscal-
culia is a heterogeneous phenotype. There are multiple methods to evaluate the
effects of genetic factors. Different methods show varying levels of genetic impact
on learning mathematics, especially about the role of genetics in developmental
dyscalculia and comorbid disorders like the common overlap between dyscalculia
and dyslexia. The interplay of genetic and environmental factors is an important
issue. One way to analyse the role of genetics to dyscalculia has been to focus on
specic syndromes where mathematical learning difculties have been found fre-
quently. The cognitive disorders within these syndromes illustrate the different
sources of dyscalculia, most typically difculties in language, working memory and
spatial skills. The following chapters deepen our understanding about these rela-
tionships between cognition and mathematical learning. To get a full picture, sepa-
rate chapters are dedicated to describing how the motivational and emotional factors
are tied to learning and learning disabilities.
A considerable body of fMRI studies with healthy adults has increased our knowl-
edge about the brain networks which are relevant for mathematical performance.
However, there is less information on how this network develops during learning and
on how the ndings of the differences between the dyscalculic and typically develop-
ing brain could inform us about the interventions needed. Children with dyscalculia
show functional as well as structural abnormalities in this network. However, this
information only gives us very little knowledge to guide education. As described in
the chapter about the comorbid disorders, a detailed analysis at cognitive and behav-
ioural levels is needed for designing the support at individual level.
The multilingual classroom is an understudied topic, considering that the major-
ity of countries in the world are multilingual by nature and that immigration has
increased, reaching almost 250 million persons (United Nations, 2016). Migratory
movements on a mass scale have brought various new languages to other countries
and continents; the Internet has dramatically affected the way in which language
and languages are used for communication and indeed for learning (Education in a
Multilingual World, 2003). One of the key ideas of this international handbook was
to make this diversity in the classrooms visible, whether we are talking about the
teaching and learning languages of mathematics in the classroom or of how educa-
tional policies take diversity into account.
Part IV turns the discussion to development, learning and teaching (Part IV:
Understanding the Basics: Building Conceptual Knowledge and Characterising
Obstacles to the Development of Arithmetic Skills). High-quality education requires
our understanding of the basic steps children take in learning and of how the educa-
tion should be designed to support them in their progress to more advanced and
complex representations. Likewise in this part, we ask for the specic mathematical
obstacles which make learning and understanding mathematics so difcult.
Implications for teaching and learning will be discussed in all papers.
A. Fritz et al.

5
One of the main problems of students with mathematical difculties is that most
of them have not been able to develop more advanced calculation strategies than
counting by one. This is disadvantageous in many ways: counting takes a long time,
is error prone and burdens the working memory. But worst of all is that these chil-
dren do not represent numbers as sets (cardinal aspect) but only as a single number
on the mental number line (ordinal aspect). For the understanding of effective addi-
tion and subtraction strategies as well as multiplication and division, it is imperative
to understand numbers as sets which can be decomposed.
While the very basics of the number system can also be learnt outside of school,
more advanced skills require explicit education starting from calculating with multi-
digit or decimal numbers and especially grasping the idea of the rational number
system. In mathematics, a rational number is any number that can be represented as
a fraction. As most of the research has focused on fractions in typically achieving
students, the specic challenges that fractions pose to learners with mathematical
difculties are less well understood. Learning to solve problems using calculation
skills adds one more demand for pedagogy.
The denitions of dyscalculia usually do not even mention geometry. However,
geometrical knowledge has extensive practical implications and creates the basis for
understanding more advanced mathematics.
Part V describes different approaches to recognition and intervention, elaborat-
ing ways of how to assess mathematical learning difculties and focusing on differ-
ent types of interventions and how the research on learning difculties could guide
education.
Research has opened our eyes to multidimensional diversity of skills in the class-
room, which asks for improved methods of recognising the individual differences.
Likewise, the discussion about how to dene and assess the mathematical learning
difculties is vivid. The old diagnostic models of discrepancy between math and
other skills have been challenged with new ideas, ideas that try to connect educa-
tion, remediation and interventions more tightly to assessments. In this last part, we
hence want to give an overview about the range of diagnostic procedures as well as
intervention approaches and their theoretical bases.
All around the world, countries have plans on how to digitalise education. In a very
near future, the access to technologies like the Internet will revolutionise assessment,
e.g. national assessments which are carried out more and more often with computer-
assisted tests. Likewise, there are increasingly more electronic educational and inter-
vention materials available. Technology will provide teachers and practitioners with
new tools for different ways to assess skills and progress of learning.
Research has demonstrated that children from low-resource communities who
experience gaps in opportunities for learning may also have lower executive func-
tioning (EF), and this risk is exacerbated for children who are second-language learn-
ers. Differences in EF between groups raise important equity issues that we must
address to meet the needs of all children and thus the entire community of learners in
a fair way. As such, children with special needs likely require special interventions.
The authors introduce what we know about group-based interventions as well as
about how the rapidly increasing technology changes the educational world.
1 Introduction

6
This last chapter also focuses on the title of the book by elaborating on how the
ndings from the labs turn into effective practice.
With this book, we would like to help in bridging the research on numerical
cognition and learning to practical applications in the classrooms, schools and clinics.
There are a lot of national and international initiatives and actions underway on
improving learning and teaching numeracy globally. We, together with our almost
100 authors, hope that this book will encourage you to walk over that bridge.
References
Ansari, D., & Coch, D. (2006). Bridges over troubled waters: Education and cognitive neuroscience.
Trends in Cognitive Sciences, 10(4), 146–151.
De Smedt, B., & Grabner, R.H. (2016). Potential applications of cognitive neuroscience to math-
ematics education. ZDM, 48(3), 249–253.
Devine, A., Soltész, F., Nobes, A., Goswami, U., & Szűcs, D. (2013). Gender differences in devel-
opmental dyscalculia depend on diagnostic criteria. Learning and Instruction, 27, 31–39.
Education in a Multilingual World. (2003). UNESCO Education Position Paper. Retrieved from:
http://unesdoc.unesco.org/images/0012/001297/129728e.pdf
Fias, W., Menon, V., & Szucs, D. (2013). Multiple components of developmental dyscalculia.
Trends in Neuroscience and Education, 2(2), 43–47.
Goswami, U. (2006). Neuroscience and education: From research to practice? Nature Reviews
Neuroscience, 7(5), 406–413.
Howard-Jones, P.A. (2014). Neuroscience and education: Myths and messages. Nature Reviews
Neuroscience, 15(12), 817–824.
Marope, M.. Grifn, P., & Gallagher, C. (2017). Future competences and the future of curriculum:
A global reference for curricula transformation. International Bureau of Education. Retrieved
from: http://www.ibe.unesco.org/sites/default/les/resources/future_competences_and_the_
future_of_curriculum.pdf
Rapin, I. (2016). Dyscalculia and the calculating brain. Pediatric Neurology, 61, 11–20.
Roser, M., & Ortiz-Ospina, E. (2017). Primary and Secondary Education. Published online at
OurWorldInData.org. Retrieved from: https://ourworldindata.org/primary-and-secondary-edu-
cation. [Online Resource].
Rubinsten, O., & Henik, A. (2009). Developmental dyscalculia: Heterogeneity might not mean
different mechanisms. Trends in Cognitive Sciences, 13(2), 92–99.
Rykhlevskaia, E., Uddin, L.Q., Kondos, L., & Menon, V. (2009). Neuroanatomical correlates of
developmental dyscalculia: Combined evidence from morphometry and tractography. Frontiers
in Human Neuroscience, 3(51), 1–15.
UNESCO Institute for Statistics. (2017). More than one-half of children and adolescents are not
learning worldwide. Fact Sheet No. 46, September 2017. UNESCO Institute for Statistics.
Retrieved from: http://uis.unesco.org/sites/default/les/documents/fs46-more-than-half-
children-not-learning-en-2017.pdf
United Nations, Department of Economic and Social Affairs, Population Division (2016).
International Migration Report 2015: Highlights (ST/ESA/SER.A/375).
World Conference on Education for all (2000); Education for all: Meeting our Collective
Commitments. The Dakar framework for action. Retrieved from: http://unesdoc.unesco.org/
images/0012/001211/121147e.pdf
World Conference on Education for all (1990). Meeting basic needs learning. Vision for 1990.
Jomtien 1990. Retrieved from: http://unesdoc.unesco.org/images/0009/000975/097552e.pdf
A. Fritz et al.

107© Springer International Publishing AG, part of Springer Nature 2019
A. Fritz et al. (eds.), International Handbook of Mathematical Learning
Difculties, https://doi.org/10.1007/978-3-319-97148-3_8
Chapter 8
Mathematical Learning andIts Difculties:
TheCase ofNordic Countries
PekkaRäsänen, EspenDaland, ToneDalvang, ArneEngström,
JohanKorhonen, JónínaValaKristinsdóttir, LenaLindenskov,
BentLindhardt, EddaOskarsdottir, ElinReikerås, andUlfTräff
The Nordic countries are located in the North corner of Europe and consist of
Denmark, Finland, Iceland, Norway, and Sweden. They form a culturally and politi-
cally isomorphic group with tight relationships. These welfare societies share the
ideology of a strong responsibility of the state on the well-being of the members of
the society. The strong economies (World Bank, 2013) and high levels of taxation
(see KPMG International, 2013) have been the guarantees for that the states have had
the assets to organize the welfare including health, social services, and education.
During the current millennium, the Nordic countries have consistently been at the top
P. Räsänen (*)
Niilo Mäki Institute, Jyväskylä, Finland
e-mail: pekka.rasanen@nmi.
E. Daland · T. Dalvang
Statped, Oslo, Norway
e-mail: Espen.Daland@statped.no; Tone.Dalvang@statped.no
A. Engström
Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden
e-mail: arne.engstrom@kau.se
J. Korhonen
Faculty of Education and Welfare Studies, Åbo Akademi University, Vaasa, Finland
e-mail: jokorhon@abo.
J. V. Kristinsdóttir
School of Education, University of Iceland, Reykjavík, Iceland
e-mail: joninav@hi.is
L. Lindenskov
Department of Educational Theory and Curriculum Studies, Aarhus University,
Aarhus, Denmark
e-mail: lenali@edu.au.dk

108
of different international comparisons on welfare, health, quality of living, economic
competitiveness, and even happiness of the citizens (Helliwell, Layard, & Sachs,
2013). Likewise, these countries are similar in a high expenditure on education, rela-
tively small class sizes in schools, and long academic teacher education (see OECD,
2012). Investing in education has been one of the core features of the success of these
countries. Despite many similarities, there are differences how the educational sys-
tems work and how education is conceptualized. For example, Finland and Denmark
have compulsory learning, while Iceland, Norway, and Sweden have compulsory
schooling. Compulsory schooling means that a pupil is obliged to attend school,
while compulsory learning means that the educational authorities are obliged to
ensure that pupils acquire the knowledge laid down in the curriculum (Tomas, 2009).
One of the marked differences between the Nordic countries has been the results
of the OECD PISA studies, where Finland since the rst study in 2000 has been
among the top performers in mathematics and Denmark signicantly above OECD
average, while the other Nordic countries have been close to the OECD average
(OECD, 2013a). In PISA 2015, Finland’s and Denmark’s students performed
equally high and signicantly higher than students in the other three countries.
However, in Denmark, Finland, Sweden, and Iceland, the trend of performance
level since 2003 has been declining (OECD, 2016). The percentage of low perform-
ers (dened as below Level 2) was as low as 6in Finland in 2006 but has raised to
14% in the 2015 assessment. In other Nordic countries, the percentage of low per-
formers in mathematics has varied from 14% in Denmark in 2006 and 2015 up to
27% in Sweden 2012. The latest TIMMS studies for fourth- and eighth-grade stu-
dents have shown similar trends (Mullis, Martin, Foy, & Arora, 2012) (Fig.8.1).
Despite these differences at school age, the Nordic countries reach the world’s high-
est levels of numeracy in adulthood. In the recent study on the numeracy prociency in
adulthood (16–65years of age), all Nordic countries topped the list together with Japan
and the Netherlands (Iceland did not participate) (OECD, 2013b).
Likewise, the participation rates in adult lifelong learning and training have been the
highest in the world in the Nordic countries (Eyridice, 2012). The high levels of
basic skills in adulthood may be connected to the dynamic nature of the work life.
B. Lindhardt
The Competence Center for Mathematics Didactics KomMat, University College Sjælland,
Roskilde, Denmark
e-mail: bli@pha.dk
E. Oskarsdottir
Faculty of Education Studies, University of Iceland, Reykjavík, Iceland
e-mail: eddao@hi.is
E. Reikerås
The Reading Centre, The University of Stavanger, Stavanger, Norway
e-mail: elin.reikeraas@uis.no
U. Träff
Department of Behavioural Sciences and Learning, Linköping University, Linköping, Sweden
e-mail: ulf.traff@liu.se
P. Räsänen et al.

109
The Nordic countries had the highest percentage of workers who reported changes that
affected their work environment (substantial restructuring or reorganization and an
introduction of new processes or technologies) in their current workplace during the
previous 3years (OECD, 2013b). Continual changes in work life require continuous
training of employees to be successful. At the same time, the workers need to have
strong basic skills to be able to assimilate new skills and ways to work and to adjust to
the changes what digitalization and automatization bring. Those with low achievement
a
b
Fig. 8.1 (a) Percentage of low performers (below Level 2) in Nordic countries in PISA studies
2003–2015, (b) percentage of low performers in fourth grade in TIMSS 2012 and 2016
8 Mathematical Learning andIts Difculties: TheCase ofNordic Countries

110
Table 8.1 Comparison of the countries in different features of education and education policy
DK SE FI NO IS
Compulsory education (age)
6–16 7–16 7–16 6–16 6–16
Public expenditure on education, % of GDP (2007, source: Eurostat, UOE) (EU average 4,98%)
7,83 6,69 5,91 6,76 7,36
Ratio of pupils to teachers in ISCED 1 (2008, source: Eurostat, UOE)(EU average 16)
10,1 12,2 14,4 10,8 10,0
Decision-making authorities involved in developing and approving the principal steering
documents for mathematics teaching
Curriculum (a) Central Central Central Central
Guidelines for teachers Central Central Regional/
school
All levels –
School plans Schools Schools Schools Regional/
schools
Schools
Evaluation of the effectiveness of curriculum implementation
External Yes Yes No No No
School self-evaluation No Yes Yes Yes (b)
Assessment criteria prescribed
Learning objectives/outcomes Yes/yes Yes/yes Yes/no Yes/no Yes/yes
Recommended minimum taught time compared to total time
Primary 15,3 13,5 17,5 17,2 15,1
Compulsory secondary 12,9 13,5 11,8 11,0 13,5
Total time primary, estimation
for (9years)
1200 900 (from
2016 1125
912 1092 1200
Textbooks
Autonomy Yes Yes Yes Yes No
Monitoring of consistency No No No No No
Central level guidelines for teaching methods
Prescribed or recommended Yes No Yes Yes No
Types of grouping Yes No No No No
Low achievement
Surveys or report on low
achievement
Yes Yes No Yes No
Central level support Yes (c) No Yes (d) Yes (e) No
Differentiation of curriculum
content according to ability
No (f) No Yes Yes No (f)
Support for low achievers
Standardized tests Yes No Yes Yes No
Intervention of a specialized
teacher
No Yes
Small group tuition Yes No
Compulsory diagnostic tests at
grades
Third,
sixth,
ninth
Third,
sixth, ninth
Second
(continued)
P. Räsänen et al.

111
in mathematics are vulnerable in work life and often among the rst to suffer from
economic turbulences, or as Parsons and Bynner (2005, p.7) state it, “Poor numeracy
skills makes it difcult to function effectively in all areas of modern life.”
There are large differences how the educational systems in the Nordic countries
(see Table8.1) have responded to low achievement in mathematics at school age
and how the educational system provides support both to the children with low
achievement and to the teachers in the schools to work with these children.
Therefore, it is reasonable to look at the similarities and differences how low
achievement in mathematics is treated in these countries. To enlighten the similari-
ties and differences in the educational support systems on learning disorders in
mathematics between the Nordic countries, we presented ve questions.
Table 8.1 (continued)
DK SE FI NO IS
National surveys on motivation Yes No Yes Yes No
Strategy to increase motivation No Yes Yes Yes No
Lack of qualied teachers in
upper primary education (%
reported by principles)
2 2,6 2,9 17,8 7,6
Teacher training (advocated by central authorities)
Differentiating teaching for pupils
with different abilities and
motivation levels
No No Yes Yes No
Detecting and tackling pupils’
difculties in mathematics
No No Yes No Yes
Source: Eurydice (2011)
(a) Denmark: National authorities develop and publish a document entitled Fælles Mål which
includes central guidelines and objectives for mathematics teaching, but this is not dened as a
curriculum in national regulations
(b) Iceland: School self-evaluation is obligatory, but schools do not have to focus on the curriculum
(c) In Denmark, the Ministry of Education has produced a special document that contains several
recommendations on how to address learning difculties in mathematics. It recommends that
mathematics teachers carefully observe low achievers, engage in a dialogue with them, and focus
on what they can do, rather than on what they cannot do. Beyond assigning such students easier
tasks, teachers should also guide them toward new strategies to cope with their difculties
(d) In Finland, the core curriculum contains guidelines on general support for students. The most
common approach is early detection and support. The Ministry of Education organizes targeted
in-service teacher training and maintains a website (10) with information on the most common
learning problems in mathematics in the early school years. The site provides access to computer-
assisted instruction methods for mathematics (Number Race, Ekapeli-Matikka, and Neure). In
addition, specic tests for the diagnosis of learning problems are available for purchase from pri-
vate companies
(e) In Norway, the main elements of the national policy to reduce low achievement are based on
early intervention, national tests and mapping (diagnostic) tests, and the integration of basic math-
ematics skills in all subject curricula. The national strategy, Science for the future: Strategy for
strengthening mathematics, science and technology (MST) 2010–2014, and the National Centre
for Mathematics Education (see Annex) are important agents in promoting mathematics education
(f) Same content but at different levels of difculty
8 Mathematical Learning andIts Difculties: TheCase ofNordic Countries

112
1. How are special needs in mathematics education (mathematical learning dis-
abilities, MLD) dened?
2. What kind of support do children get at school for severe MLD?
3. Who gives the support, and what qualications they have for this work?
4. Are the evidence-based assessment tools and intervention methods available?
5. What are the key issues and current trends in MLD at the moment?
Sweden
In Sweden, the legislative text does not use the terms MLD, “dyscalculia,” “math
disabilities,” or even special needs in math education. Instead, the legislative text
says that all children that are in risk of not attaining the national knowledge goals in
a school subject have the right to special support, that is, some form of special edu-
cation. The legislative text says nothing about what kind of or how much support the
children have the right to receive. It only says that the schools must have a compe-
tence for special education. It is the responsibility of a school to provide the child
individually adapted and adequate support.
In practice, there is large variability how support is organized at the school level.
For example, some municipalities/schools require that the child has an ICD-10-
based (KSH, 2011; WHO, 2005) medical diagnosis for mathematical disabilities to
receive any special support, whereas other municipalities/schools focus on the func-
tional level, which is in line with the legislation. Unfortunately, not all children
receive the support that they need and have the right for it according to the law,
because the schools do not have the required nancial resources or the special edu-
cation competence. In principle, there are two types of support: individual support
with special needs teacher (one-to-one teaching) and level-groupings in small
groups (2–5 children) with special needs teacher or regular class teacher.
An additional problem in Sweden is that not all schools acknowledge the con-
cept or term mathematical disabilities (dyscalculia). Accordingly, it is very dif-
cult to estimate the prevalence of children with mathematical disabilities. About
15% of the students usually do not get a pass on the national test in mathematics
for the nal grade (Skolverket, 2013). Immigrant students or students whose par-
ents have low levels of education are overrepresented among those who do not
reach the goals. There is great variation between schools and municipalities in
performance levels. There are suburban schools in metropolitan areas with large
numbers of immigrant students where the majority of students do not get a pass
on the national test.
In Sweden, there is a considerable lack of special needs teachers on mathemati-
cal difculties, because the university-based special needs teacher program started
as late as 2008. This program is a 1.5-year-long training program for teachers.
Therefore, most of the teachers responsible for helping children with MLD are not
specialized to these pedagogical questions. The legislative text does not specify that
the schools should have special needs teachers and/or special education teachers.
P. Räsänen et al.

113
It only says that the schools must have special educational competence in some form.
Likewise, there are no organized systems for continuing education or further training
for special needs teachers or special education teachers.
The schools do not use any evidence-based assessment tools or intervention
methods because there are none available. Furthermore, there are very few experts
in Sweden who do assessments on MLD.However, at Danderyds Hospital in
Stockholm, which is one of the few places where this kind of assessments is done,
they use the British Dyscalculia Screener (Butterworth, 2003), and recently they
have started to use the Panamath test (Halberda, Mazzocco, & Feigenson, 2008).
The new Swedish Education Act from 2010 stipulates that the education and
instructions used in Swedish schools must be founded on scientic evidence and
established experience. Thus, in the future, the Swedish school authorities will
probably put more emphasis on matters regarding evidence-based teaching methods
and evidence-based assessment tools. There is, however, some skepticism about the
“evidence movement” developed in Anglo-Saxon countries.
Norway
The Norwegian educational policy is founded on the principles of inclusion and
adapted education. However, to develop educational practices that achieve these
overarching principles is a continuous challenge (Haug, 2010; Mathisen &
Vedøy, 2012).
Laws and regulations in Norway do not dene or apply the terms dyscalculia and
mathematical disabilities. The term learning difculties is used. According to the
Educational Act, the focus is on pupils who do not benet satisfactorily from ordi-
nary teaching and thereby have the right to be assessed for being in some kind of
special needs (See section “A Lack of Certain Arithmetical Abilities or a Certain
Way of Doing Arithmetic?” in Chap. 6). Pupils should be referred to educational
and psychological counseling service (EPS) for an expert assessment. The expert
assessment shall consider and determine the following:
• The pupil’s learning outcome from the ordinary educational provisions
• Learning difculties the pupil has and other special conditions of importance to
education
• Realistic educational objectives for the pupil
• Whether it is possible to provide help for the pupil’s difculties within the
ordinary educational provisions
• What kind of instruction is appropriate to provide (See section “Evidence on the
Impact of Instructional Efforts Focused on Non-counting Strategies” in Chap. 6)
In 2013 an amendment became in force that describes more details about admin-
istrative procedures in connection with decisions concerning special education.
“Before an expert assessment is undertaken, the school must have considered and
tested out, if relevant, measures within the ordinary education facilities that might
8 Mathematical Learning andIts Difculties: TheCase ofNordic Countries

114
make the pupil benet satisfactorily” (See section “Overcoming Computing by
Counting as a Didactic Challenge” in Chap. 6).
This can be interpreted as pointing toward a more systematic problem-solving
approach in line with recent response to intervention models (Glover & Vaughn,
2010). Further descriptions or guidelines regarding how to assess satisfactory
learning outcome and/or the substance of the local schools’ investigations are not
provided. However, obligatory standardized national test (grades 2, 5, 8, and 9) is
a part of the assessment of the children’s mathematics at school. The tests aim to
be a tool for the teachers to adapt the teaching to each child.
An emerging use of the term dyscalculia is taking place in Norway, and related
diagnostic practices evolve. There is, however, no unied and agreed upon deni-
tion overall related to mathematics difculties. On this grounding, it is not straight-
forward to nd the extent of pupils with MLD.If difculties are dened as getting
a low grade in mathematics in school (low achievers), the results from the exam of
Norwegian 15-year-old pupils show that 35–40% got 1 or 2in mathematics (the
grading system 6–1, with 6 as the highest). In 2012–2013 the percentage of pupils
with individual decisions about special needs education was 8.6% in total (The
Ministry of Education, 2013). How many of them with special needs in mathemat-
ics is not known. In research, e.g., Ostad (1997) used the term mathematical disable
for the lowest performing 10% of children in Norwegian schools and found this
level of low performance to be stable through all school years.
The support provided by schools varies. Lessons can be given in smaller groups
or individually, outside or inside the regular classes and classrooms. The quality of
the support also varies in line with the helper’s background, from adequate support
from a teacher in special needs with competence in mathematics to an assistant
without teacher training at all. The use of assistants in special education increased
from 2001 to 2008 (Bonesrønning, Iversen, & Pettersen, 2010).
Due to a lack of research-based knowledge about what goes on in segregated and
inclusive special education in Norway, a joint research project was carried out from
2012 to 2015. The project had as main research questions: “What special education
is about, and what is its function?” (http://www.hivolda.no/speed).
One ofthe main points from the researchis to build education for all on a profes-
sional ground, to understand the complexity of the challenges, and to make institu-
tions responsible, not only individuals (Haug, 2016).
Laws and regulations in education put emphasis on identifying pedagogical
needs and developing supportive actions. Categorizing students or groups by diag-
nostic labeling is subordinate. However, this question of diagnosis and labeling
causes a tension in the public and is a constant topic of the educational debate.
New practices of assessment in contexts (Nielsen, 2013) are being developed and
tried out by Statped and EPS (Daland & Dalvang, 2009, 2016). It adopts a stance
toward curiosity on how mathematical learning situations can be understood and
further developed. This assessment approach seeks to investigate and analyze relations
between three main dimensions: developing as a person, learning mathematics, and
participating in learning communities.
P. Räsänen et al.

115
Iceland
Like in other Nordic countries, laws and regulations in Iceland do not dene dif-
culties with mathematics or dyscalculia at any school level. Schools set their own
targets of competency in mathematics in coherence with the national curriculum
guidelines, and pupils are offered support based on them, as well as on outcomes
from standardized testing in mathematics in grades 4, 7, and 10. On those tests,
between 17% and 24% of pupils score 0–22 points on a normal scale and fall into
the category of poor performance (Sverrisson & Skúlason, 2012).
Support in schools for pupils with difculties in mathematics is either in the
hands of special education needs (SEN) teachers or mathematics (or other) teachers.
In a survey from 2010 (Óskarsdóttir, 2011), different approaches to grouping and
teaching were evident. In some schools, the tradition is that the SEN teachers work
with pupils that need support in small groups of two to four pupils two to four les-
sons a week usually in a separate room. In other schools, pupils in the same year
group are tracked into groups in mathematical classes depending on their level of
performance, and the low-performing pupils work in small groups often with a
mathematics teacher (or other experienced teachers) up to six lessons a week. In a
minority of schools, SEN teachers or mathematics teachers go into classrooms and
assist pupils that need support.
SEN teachers, according to the survey, map pupils’ abilities before they begin
working with them and tend to work with tailor-made assignments. They use
manipulatives and physical models in their teaching and do not necessarily follow
the textbook that is used for mainstream mathematics teaching. The focus in their
teaching is on how to learn algorithms as a means of solving problems and to estab-
lish ways of working with mathematics. Mathematics teachers on the other hand use
the textbooks and other teaching materials used by the year group and tend to follow
the curriculum guidelines. The emphasis in teaching is placed on basic algorithms,
teaching pupils how to calculate but less on how to use manipulatives other than
computers and calculators.
In Iceland teachers and SEN teachers have a university degree. There is one course
aimed at preparing SEN teachers to teach mathematics, and it is called “Mathematics
for all.” The focus of this course is on mathematics learning and how children develop
mathematical thinking. The participants of the course also work to develop their own
understanding of mathematics and discuss their different ways of approaching math-
ematical problems. The aim is to be able to understand children’s diverse ways of
developing mathematical thinking. The main goal of the course is to prepare teachers
to map pupils’ abilities and to learn how to support children to overcome their dif-
culties in learning. Also, there is a discussion about diverse pupils’ difculties and
how SEN teachers need to collaborate with mathematics teachers in assisting pupils.
The course is based on research on how children learn mathematics as well as on
research on learning difculties in mathematics and teacher development in teaching
mathematics in inclusive settings (Guðjónsdóttir, Kristinsdóttir, & Óskarsdóttir,
2007, 2009, 2010).
8 Mathematical Learning andIts Difculties: TheCase ofNordic Countries

116
One standard-based assessment tool is available to SEN teachers as well as
mathematics teachers. This test, Talnalykill (Guðmundsson & Arnkelsson, 1998), is
standardized and criterion-referenced in Iceland. Those who want to use it must be
licensed. The test is made up of two main test components, written group tests, and
individual oral testing. Some schools in Reykjavik and other places have used the
written part of the test to screen third grades for mathematics difculties. The test
has been criticized for focusing mainly on children’s uency in using traditional
algorithms and not screening for other mathematical competencies such as the
ability to deal with mathematical language and tools. Many teachers in schools also
nd it too time-consuming, both regarding assessing the pupil and the time it takes
to calculate the results. School psychologists also assess pupils for difculties with
mathematics using tests such as WISC-IV (Guðmundsson, Skúlason, & Salvarsdóttir,
2006), which has been standardized and localized for the Icelandic context.
In the new national curriculum (Ministry of Education, Science and Culture,
2011), the emphasis is on equal opportunities for all pupils regardless of their abili-
ties or circumstances. At the compulsory school level, all pupils have the right to
compulsory education in their inclusive neighborhood school. The focus in the
mathematics chapter is on the right of all children to develop their mathematical
thinking and get the support they need to develop mathematical competencies
(Mennta-og menningarmálaráðuneytið, 2013).
Finland
The Finnish educational system is state governed and funded but municipally orga-
nized. The private school sector is practically non-existing. The leading principle of
the educational policy has been to offer free, high-quality education to all inlocal
schools. There are no standardized or national assessments in primary education,
but every school and teacher have a freedom to decide how they monitor the devel-
opment and learning of their pupils. Typically, teachers use a lot of formal and
informal exams to follow the progress of their students.
The number of pupils in special education increased rapidly in Finland during
the last two decades from less than 3% up to 8.5% in 2010. At the same time, the
number of children receiving part-time special education peaked at 23.3% (The
Finnish Centre for Statistics, 2013) resulting in about one-third of children at the
early-grade education to receive some individualized support. Even though support-
ing reading skills was clearly the largest subject, special needs education in mathe-
matics showed the largest growth (Räsänen & Koponen, 2011). In 2010 about
one-fourth of part-time special education was devoted to mathematics. All these
gures were world records at their time.
This unexpected growth in special education caused the Finnish special educa-
tion system to be reformed. It started to be a too expensive solution for treating
individual differences in learning. The changes in the Basic Education Act were
P. Räsänen et al.

117
passed in 2010. The new strategy emphasized inclusion over segregation and
stressed the importance of a pedagogical approach over medical and psychological
approaches. The aim was to change the old diagnostic terminology to a more peda-
gogically oriented language. According to the “new educational talk,” medical or
psychological terms like mathematical learning disorders or dyscalculia were not
recommended. Instead, the focus should be given to identifying pedagogical needs
and taking supportive measures (Thuneberg etal., 2013).
The new support system is divided into three levels of intensity. General support,
targeted to all children, is for temporary needs in learning. The second level, con-
cerning about 20% of children with needs for more regular support, is referred to a
pedagogical assessment and to an intensied support with a time limit. Main tools
are part-time special needs education, individual guidance counseling, and use of
exible teaching groups, as well as home-school cooperation. The third level, tar-
geted to about 5% of the children, special support, is provided for those who cannot
adequately achieve their growth, development, or learning objectives through other
support measures. The most serious cases, dened in the previous system, as having
severe mathematical learning disorders, go through a broad pedagogical evaluation
and if needed may study according to an individual learning plan (ILP). The peda-
gogical evaluation is coordinated by the school teachers and typically contains a
consultation of a child psychologist who has many options for standardized tests of
mathematical achievement to be used as part of the assessment.
Even though the system reminds the descriptive conventions of the response to
intervention (RtI) model (Fuchs & Fuchs, 2007), it was not the foundation for the
new model in Finland. The key differences between the RtI and the Finnish models
are the absence of standardized assessments and structured evidence-based inter-
ventions in the Finnish model. In the Finnish model, the teachers are at the helm,
and they are given freedom and responsibility to tailor the needed processes to
support each child. This requires a well-organized system at the school level and
continuous further training for teachers. In larger cities, there are “Mathlands,”
which are support and learning centers for teachers. Likewise, there is a government-
funded web service (lukimat.) run by Niilo Mäki Institute, a research center on
learning disorders. The service offers information and free tools for early interven-
tions and assessment of reading and mathematical difculties in early primary
education.
In Finland, practically every school has qualied special education teachers with
a university degree. The majority of them give part-time special education in col-
laboration with the classroom and subject teachers. Likewise, every school has a
student welfare group for multi-professional collaboration. However, even though
the school system offers a lot of individualized support, there are still a lot of chal-
lenges to meet. According to the two recent analyses from the national assessments
on mathematical achievement from sixth (Räsänen, Närhi, & Aunio, 2010) and
ninth grades (Räsänen & Närhi, 2013), close to half of the children identied having
a low achievement in mathematics (about 5–6% in total) get only a little attention
from the school or teachers.
8 Mathematical Learning andIts Difculties: TheCase ofNordic Countries

118
These results were a surprise because the criterion for low achievement in
these evaluations was a combination of assessment and teacher’s identication.
Therefore, the reason for ignoring these children with low achievement from support
was not due to non-identication. The biggest challenge in Finland is not whether
the pupils will be identied having mathematical learning disabilities but how to
guarantee that they all are offered the support and care they need.
Denmark
The Danish educational system is free and publicly funded. Even private schools get
public funding for as much as 73% of the amount given to public schools, while the
rest is paid by the parents (per private school student around 130 Euro per month).
Private schools are getting more popular. While in 2000 the percentage was 12% in
private schools, in 2016 the percentage was 18%.
All public schools prepare the students for national exam at the end of grade 9.
From 2017, national compulsory assessments also include 14 digital, adaptive tests
from grades 2 to 8, including 3 mathematics tests in grades 3, 6, and 8. Most private
schools offer these national tests, too. For teachers, the aims of the national testing
program are to provide a tool for teachers’ own formative assessment of their stu-
dents’ progress and a tool for monitoring their own teaching. Nevertheless, many
teachers nd it difcult to use the national tests according to these aims. Other assess-
ment tools are provided by publishers or the teachers themselves, and every school
has the freedom to decide how to act upon test results. Besides, some schools and
adult learning centers use the British Dyscalculia Screener (Butterworth, 2003).
In the present national curriculum guidelines for mathematics (Common Goals,
2016), no student characteristics (i.e., special needs students) are described. But for
some specic skills and knowledge, eight “attention points” are described: they
refer to the level of basic skills that are a prerequisite in order to acquire sufcient
skills later on.
The political agreement in the Parliament June 2013 on improving Danish school
children’s performance in school subjects included initiatives for “students with
dyscalculia.” On this background, a test for dyscalculia for grade 4in Danish schools,
guidelines for test takers, and ideas for follow-up assistance are being developed in
2015–2018. A proposed denition of dyscalculia serves as a starting point:
“Dyscalculia is an impairment that may inuence education and work. Weak calcula-
tion skills are not matched by corresponding weak skills in other elds” (SFI, 2013).
Expected percentage of students to be identied by this future dyscalculia test is as
low as 1%. Many more students than 1% are facing mathematics difculties and in
need for focused support, either just in mathematics or also in other subjects, draw-
ing on social, psychological, physical, and didactical perspectives. Support in math-
ematics is needed for students in segregated special schools and classes as well as in
regular school and classes.
P. Räsänen et al.

119
Since the Salamanca Declaration (UNESCO, 1994), professionals and politi-
cians have argued for increasing efforts for inclusion. The number of students in
special education and the costs of special education have been steadily growing in
Denmark. Data from public schools showed that in 2008–2009 support organized in
special classes and special schools was provided for 5,6% of the students, while
special education in ordinary classes was provided for another 8,7%. When these
and other data were brought up and analyzed (Finansministeriet etal., 2010), politi-
cal efforts were intensied to include more students in ordinary classes and schools
and to replace special education with another instrument. Economistic arguments
were put forward but also humanistic arguments for better learning and well-being
when “special students” would be more along with “regular students.”
As several special schools have been closed the last years, also some students
with diagnoses as autism, Tourette syndrome, conduct disorders, or general learning
difculties are now being included in regular classes and schools. However, it has in
many cases proven to be problematic, as several teachers have not been trained, are
not knowledgeable, or are not given sufcient resources to create the needed inclu-
sive learning environment.
After the law “No 379– 28 April 2012,” less than 9 specialized lessons of 60min
(equivalent to 12 lessons of 45min) per week are not seen or regulated as a special
education program. Support less than nine lessons is given as part of mainstream
education. In instruction, can be used, inter alia, differentiated teaching, tracking for
shorter periods, two teachers in class, teaching assistants who can both help each
student and the class as a whole, or supplementary teaching and other kinds of sup-
port (www.uvm.dk 2015). Some programs for supplementary teaching are devel-
oped and used as an early intervention in mathematics; see, for instance, Lindenskov
and Weng (2014).
Available data on mathematics in special education in special and regular schools
is extremely sparse (Lindenskov, 2012). Nevertheless, the interest in special needs
in mathematics has been growing since 2000 among school teachers in public and
private schools, adult educators, high school teachers, school psychologists, special
education teachers, consultants, teacher educators, and researchers. To increase the
overall quality of school mathematics, 10years ago a diploma program was set up
for mathematics teachers in service to become “math tutors.” The 1-year program
includes six modules, and one module focuses on students in mathematics difcul-
ties. Several seeds have been sowed for continuous interest and for development
projects and initiatives at school and municipality levels. The educated math tutors
have organized a national network covering about 1000 tutors spread over 800 out
of 1400 schools.
In 2010, the association DanSMa (Danish Special Mathematics) was founded as a
common meeting place for these professionals to discuss typical issues concerning
people with special needs in mathematics in order to improve offers for children, ado-
lescents, and adults. DanSMa initiates public debates, disseminates the latest research
on the character and background of mathematics difculties, as well as on identica-
tion and interventions, and arranges seminars with invited speakers (dansma.dk).
8 Mathematical Learning andIts Difculties: TheCase ofNordic Countries

120
Summing Up
We presented ve questions to analyze the similarities and differences between the
Nordic countries how children with MLD are recognized and how their learning is
supported. To summarize our ndings, we go through the replies question by
question.
The rst question concerned how special needs in mathematics education and
mathematical learning disabilities (MLD) are recognized and dened in each coun-
try. In all countries, the legislations recognize low achievement as a special ques-
tion, but none of those take any stance on ICD or other clinical diagnostic systems.
There are no commonly accepted criteria for diagnosing MLD.The assessment
procedures used in Iceland, Denmark, and Norway are rather close to those dened
in ICD, namely, combining standardized achievement tests and cognitive assess-
ment. In Finland, standardized tests and a psychological assessment are a common
practice in a case with persistent learning disabilities, but giving a diagnostic label
for MLD is exceptional. The educational reform in 2010 pushes Finland closer to
the Swedish approach where there is a strong aim to avoid assessments and diagnos-
tic labeling and to concentrate on methods of inclusive education.
The Finnish and Icelandic schools have been extremely sensitive to dene a child
as having special needs in education (SNE). In Iceland, about 24% of children are
dened as having special needs, while in Finland about 8% of children are dened
as pupils with SNE, and an additional 20% receive a part-time special education.
Denmark and Norway are in the middle, but a striking contrast is Sweden, where
only 1.5% of children are dened having SNE (see Table8.1, NESSE, 2012).
We can also contrast the Nordic models against the response-to-instruction mod-
els of special education. In the RtI modelsthe extremes of a continuum could be
called as “a standard protocol” at one and “a problem-solving approach” at the other
end (Fuchs, Fuchs, & Stecker, 2010). In the standard protocol, assessment means an
evidence-based intervention with standardized measures of improvement before
and after the intervention to be able to dene those with MLD and needs for more
intensied and individualized special educational intervention. The problem- solving
approach sees the assessments as a tool for a non-categorical evaluation of skills
mastered and yet to be mastered and is used primarily to inform classroom instruc-
tion, rather than to guide decision-making on a diagnosis or for a more individual-
ized intervention. In other words, while the rst stresses the importance of special
education as a separate process, the latter sees that the special education should be
blurred inside the regular instruction (for more about this discussion, see, e.g.,
Fuchs, Fuchs, & Compton, 2012).
If we try to put the Nordic models into this discussion and continuum, none of
the countries follow the standards approach. The success of the RtI model in the
USA has not attracted the policy-makers in Nordic countries to formalize the sup-
port systems or increase the usage of standardized tests. The general discussion has
been more about howto develop inclusive models and lessen the needs for separate
special needs education (e.g., Statped model in Norway). Finland is the only country
where SNE has been formally structured to levels of support with dened procedures
P. Räsänen et al.

121
how the evaluation should be done when moving between the levels. This mimics
vaguely the standards approach with pedagogical evaluations, but without specica-
tions of assessment procedures. At the same time, there is an aim to push forward
the inclusive problem-solving RTI model. Sweden has been an extreme on its reluc-
tance toward assessments and diagnostics with a strong inclusive ideology and aims
to apply the problem-solving approach.
One of the largest differences between the Nordic countries lies in the details how
children’s progress in learning is monitored. In Norway, Denmark, and Iceland, there
are standardized assessments at specic grade levels, which are absent from the
Swedish and Finnish systems. In Finland, the evaluation is totally in the hands of
the teachers, who typically use a lot of formal and informal examinations to monitor
the children’s development in their own classroom. The specic feature in the Swedish
discussion on education has been the reservations against assessments, especially
standardized assessments and the evidence-based, “quantitative” approaches.
The second and third questions were: what kind of support do children with
MLD get at school, and what are the qualications for the support personnel? In all
countries, the importance of inclusive education is stressed, but still, a common way
of dealing with MLD is still taking the child out of a classroom to individualized or
to a selected small group receiving special education. In none of the countries, there
are ofcially recommended or recognized intervention programs to be used. In
Finland, there are research centers on learning disorders, which have developed
widely used programs on learning disorders. According to a recent analysis (Sabel,
Saxenian, Miettinen, Kristensen, & Hautamäki, 2011), these research centers have
had a large role in shaping the Finnish special education. In Norway, a state-funded
Statped is developing models for special education. However, their aim is not to
produce evidence-based intervention programs but to guide teachers in professional
development (cf. problem-solving approach in RtI). In Denmark, the development
work is concentrated around the large network of diploma-trained teachers.
In Sweden, there has been a lack of specialized teachers, and the university train-
ing of special educators started as late as in 2008, while in Finland it started in 1959,
and nowadays the majority of the Finnish universities have units of special educa-
tion offering studies up to the Ph.D. level. Therefore, it is not a surprise that from
Nordic countries, what kind of, and who gives extra support to children with MLD,
varies the most in Sweden. The Swedish educational ofce (Skolverket, 2009) has
also raised concerns over the inuence of increasing segregation in the Swedish
school system after it transformed itself from one of the most centralized school
systems into one of the most decentralized (Tomas, 2009). Even though the variance
between schools in mathematics has increased in Sweden, the Nordic countries still
have the smallest between schools variance in mathematics achievement in the
world (Gaber, Cankar, Umek, & Tasner, 2012).
Our fourth question concerned the role of research and evidence-based approaches
in interventions on MLD.Following the international trends, research interest toward
MLD has been raising in all Nordic countries. There is a biannual Nordic Congress on
special needs education in mathematics (NORSMA, The Nordic Research Network
on Special Needs Education in Mathematics) where experiences on different types
of assessment and interventions and on the effectiveness of special education are
8 Mathematical Learning andIts Difculties: TheCase ofNordic Countries

122
shared. However, none of the educational systems require that special educational
approaches should be evidence-based. Therefore, research-based tools, even though
welcomed at schools, are not a standard, and it depends on teacher’s own activity, if
they apply any of the models or instruments.
In all Nordic countries, an increasing number of researchers are pushing toward
more research-based assessment and intervention procedures. The increasing under-
standing of the dyscalculic brain and changes in the diagnostic denitions encour-
age the researchers. At the same time, new questions emerge for the interplay
between research and educational practice. The new competency-based curriculums
redene the learning aims and bring new colors to the practices at school and new
challenges and research questions for studies on learning disabilities. It seems that
the gap between everyday activities and aims in classrooms and the neuroscientic
research is not getting narrower in the near future.
Our last question was about the future challenges. We can see a perennial battle
between different views on the role of individualized special needs education and
inclusive education. The puzzle how to teach the whole classroom effectively but at
the same time individualize education within and outside of the classroom is an
open question asking for scientic efforts. Neuroscientic research on learning and
learning disorders gets the headlines (e.g., Coughlan, 2014) but still gives a little to
the actual educational practices in classrooms. A lot of different views are presented,
and the only thing where all parties agree is the lack of scientic evidence for any
of the opinions.
According to the latest TIMMS study (Mullis etal., 2012), low motivation toward
mathematics learning is more apparent and concerning feature of current Nordic stu-
dents than low achievement. However, in the international assessments, there has been
interesting feature: Within each participating country, there is a positive correlation
between students’ learning motivation and achievement; but when aggregating the
data at a country level, the correlation between motivation and achievement becomes
negative (He & Van de Vijver, 2016). High-performing countries show lower averages
in motivation than lower-performing countries. From the Nordic countries particu-
larly Finland, together with the many Asian top performing countries, they show this
strong achievement paradox of high achievement and low motivation. Despite high
general well-being of youth in Nordic countries, enjoyment of learning mathematics,
especially in the upper primary education, has not been a part of it. The equation how
to combine efcient learning, self-efcacy, and motivation in mathematics education
is a big challenge for both research and practice to solve.
References
Bonesrønning, H., Iversen, J.M. V., & Pettersen, I. (2010). Kommunal skolepolitikk etter
Kunnskapsløftet. Med spesielt fokus på økt bruk av spesialundervisning. Trondheim, Norway:
Senter for økonomisk forskning (SØF).
Butterworth, B. (2003). Dyscalculia screener. London: nferNelson.
Coughlan, S. (2014). BBC News Education and Family: Brain scientists to work with schools on
how to learn. January 7th, 2014. Available at http://www.bbc.co.uk/news/education-25627739
P. Räsänen et al.

123
Common Goals (2016). Om Fælles Mål. København: Undervisningsminiteriet. Retrieved from
https://uvm.dk/folkeskolen/fag-timetal-og-overgange/faelles-maal/om-faelles-maal
Daland, E., & Dalvang, T. (2009). The compass model– A possible tool for dialogue, reasoning
and understanding of situations in which the learners experience difculties in their mathemati-
cal eduaction. In K.Linnanmäki & L.Gustafsson (Eds.), Different learners– Different math?
(pp.173–180). Vasa, Finland: The Faculty of Education, Åbo Akademi University.
Daland, E., & Dalvang, T. (2016). Matematikkompasset. Statped. Læringsressurs.
Eurydice (2011). Key data on Education in Europe. Eurydice network. Retrieved from http://www.
eurydice.org
Eurydice. (2012). The European higher area in 2012. Brussels, Belgium: Eurydice.
Finansministeriet, Undervisningsministeriet og KL. (2010). Specialundervisning i folkeskolen–
veje til en bedre organisering og styring. København, Denmark: Rosendahls-Schultz Grask.
Available at http://www.ft.dk/samling/20091/almdel/udu/bilag/244/859480/index.htm
Fuchs, D., & Fuchs, L.S. (2007). A model for implementing responsiveness to intervention.
Teaching Exceptional Children, 39, 14–20.
Fuchs, D., Fuchs, L.S., & Stecker, P.M. (2010). The “blurring” of special education in a new con-
tinuum of general education placements and services. Exceptional Children, 76(3), 301–323.
Fuchs, L.S., Fuchs, D., & Compton, D.L. (2012). The early prevention of mathematics difculty
its power and limitations. Journal of Learning Disabilities, 45(3), 257–269.
Gaber, S., Cankar, G., Umek, L.M., & Tasner, V. (2012). The danger of inadequate conceptuali-
sation in PISA for education policy. A Journal of Comparative and International Education,
42(4), 647–663. https://doi.org/10.1080/03057925.2012.658275
Glover, T.A., & Vaughn, S. (2010). The promise of response to intervention– Evaluating current
science and practice. NewYork: The Gilford Press.
Guðjónsdóttir, H., Kristinsdóttir, J.V., & Óskarsdóttir, E. (2007). Mathematics for all: Preparing
teachers to teach in inclusive classrooms. In L. Østergaard Johansen (Ed.), Mathematics teach-
ing and inclusion. Proceedings of the 3rd Nordic research conference on special needs educa-
tion in mathematics (pp.123–136). Aalborg, Denmark: Aalborg University.
Guðjónsdóttir, H., Kristinsdóttir, J.V., & Óskarsdóttir, E. (2009). Mathematics for all: Working
with teachers. In K.Linnanmäki & L.Gustafsson (Eds.), Different learners– different math?
Proceedings of the 4th Nordic research conference on special needs education in mathematics
(pp.273–284). Vasa, Finland: Åbo Akademi University.
Guðjónsdóttir, H., Kristinsdóttir, J.V., & Óskarsdóttir, E. (2010). Teachers’ development in math-
ematics teaching through reective discussions. In B.Sriraman, C.Bergsten, S.Goodchild,
G.Pálsdóttir, B.D. Søndergaard, & L.Haapasalo (Eds.), The rst sourcebook on Nordic
research in mathematics education (pp.487–494). Charlotte, NC: Information Age Publishing.
Guðmundsson, E., & Arnkelsson, G.B. (1998). Talnalykill: staðal-og markbundið próf í stærð-
fræði 1.-7. bekkur/Einar Guðmundsson og Guðmundur B.Arnkelsson. Reykjavík, Iceland:
Rannsóknastofnun uppeldis-og menntamála.
Guðmundsson, E., Skúlason, S., & Salvarsdóttir, K. (2006). WISC-IV is: mælifræði og túlkun.
Reykjavík, Iceland: Námsmatsstofnun.
Halberda, J., Mazzocco, M., & Feigenson, L. (2008). Individual differences in nonverbal number
acuity predict maths achievement. Nature, 455, 665–668.
Haug, P. (2010). Approaches to empirical research on inclusive education. Scandinavian Journal
of Disability Research, 12(3), 199–209.
Haug, P. (2016). Ein likeverdig skule I framtida? Nordisk tidsskrift for pedagogikk og kritikk, 2,
64–77.
He, J., & Van de Vijver, F.J. (2016). The motivation-achievement paradox in international educa-
tional achievement tests: Toward a better understanding. In The psychology of Asian learners
(pp.253–268). Singapore, Singapore: Springer.
Helliwell, J.F., Layard, R., & Sachs, J.D. (Eds.). (2013). World happiness report 2013. NewYork:
UN Sustainable Development Solutions Network.
KPMG International. (2013). http://www.kpmg.com/GLOBAL/EN/SERVICES/TAX/Pages/
default.aspx
8 Mathematical Learning andIts Difculties: TheCase ofNordic Countries

124
KSH. (2011). Klassikation av sjukdomar och hälsoproblem (ICD-10-SE). Stockholm:
Socialstyrelsen.
Lindenskov, L. (2012). Matematik i specialundervisning og andre særlige Foranstaltninger.
København, Denmark: DPU.
Lindenskov, L., & Weng, P. (2014). Early mathematics intervention in a Danish municipality:
Theory and teachers’ reections in the pilot project. In A.B. Fuglestad (Ed.), Special needs
education in mathematics: New trends, problems and possibilities (pp.64–74). Kristianssand,
Norway: Portal.
Mathisen, I.H., & Vedøy, G. (2012) Spesialundervisning– Drivere og dilemma: Rapport IRIS
2012/017. International Research Institute of Stavanger.
Mennta-og menningarmálaráðuneytið. (2013). Aðalnámskrá grunnskóla: greinasvið. Reykjavík,
Iceland: Mennta-og menningarmálaráðuneyti. Retrieved 03.09.2013 from http://www.mennta-
malaraduneyti.is/utged-efni/namskrar/adalnamskra-grunnskola/
Ministry of Education (2013). Meld. St. 20 (2012–2013). På rett vei. Kvalitet og mangfold i
fellesskolen.
Ministry of Education, Science and Culture. (2011). The Icelandic national curriculum guide for
compulsory school: General section. Reykjavík, Iceland: Ministry of education, science and
culture. Retrieved 03.09.2013 from http://www.menntamalaraduneyti.is/utged-efni/namskrar/
adalnamskra-grunnskola/
Mullis, I.V. S., Martin, M.O., Foy, P., & Arora, A. (2012). PIRLS 2011 international results in
reading. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.
NESSE (2012). Education and disability/special needs: Policies and practices in education, train-
ing and employment for students with disabilities and special educational needs in the EU,
an independent report prepared for the European Commission by the NESSE network of
experts. As of 30 March 2013: http://www.nesse.fr/nesse/activities/reports/activities/reports/
disability-specialneeds-1
Nielsen, J.(2013). Utredning i kontekst. Utredninger– i vor tid. Psykologi i Kommunen, (1),
39–48.
OECD. (2012). Education at a Glance 2012: OECD indicators. OECD Publishing. https://doi.
org/10.1787/eag-2012-en
OECD. (2013a). PISA 2012 results: What students know and can do– Student performance
in mathematics, reading and science (volume I). PISA, OECD Publishing. https://doi.
org/10.1787/9789264201118-en
OECD. (2013b). OECD skills outlook 2013: First results from the survey of adult skills. OECD
Publishing. https://doi.org/10.1787/9789264204256-en
OECD. (2016). PISA 2015. Results in focus. OECD Publishing. https://www.oecd.org/pisa/pisa-
2015-results-in-focus.pdf [22.01.2018].
Ostad, S. (1997). Developmental differences in addition strategies: A comparison of mathemati-
cally disabled and mathematically normal children. British Journal of Educational Psychology,
67, 345–357.
Óskarsdóttir, E. (2011). Sérkennsla í stærðfræði: rannsókn á framkvæmd og skipulagi. Glæður,
21(1), 57–62.
Parsons, S., & Bynner, J.(2005). Does numeracy matter more? National Research and
Development Centre for Adult Literacy and Numeracy. Available at http://www.nrdc.org.uk/
download.asp?f=2979&e=pdf
Räsänen, P., & Koponen, T. (2011). Matemaattisten oppimisvaikeuksien neuropsykologisesta tut-
kimuksesta. NMI-Bulletin, 20(3), 39–53. [About neuropsychological research on mathematical
disorders].
Räsänen, P., & Närhi, V. (2013). Heikkojen oppijoiden koulupolku. Teoksessa JMetsämuuronen
(toim.), Perusopetuksen matematiikan oppimistulosten pitkittäisarviointi vuosina 2005–2012.
Koulutuksen seurantaraportit 4/2013. Helsinki, Finland: Opetushallitus. (pp.173–224)
[Pathways to poor performance in mathematics. A book chapter in the report on the longitu-
dinal study of the National Learning Assessment of Mathematical skills, National Board of
Education].
P. Räsänen et al.

733© Springer International Publishing AG, part of Springer Nature 2019
A. Fritz et al. (eds.), International Handbook of Mathematical Learning
Difculties, https://doi.org/10.1007/978-3-319-97148-3_42
Chapter 42
Perspectives toTechnology-Enhanced
Learning andTeaching inMathematical
Learning Difculties
PekkaRäsänen, DianaLaurillard, TanjaKäser, andMichaelvon Aster
Today, technology is a part of almost every aspect of life of those living in a devel-
oped country. People are constantly “online” and have an easy access to information
and services. The speed of change has been high. Therefore, predicting how our
digitalized life will change within the next 5 to 10years couldonlygo wrong. One
new innovation in battery or processor chip technology will change totally how the
future will look. Innovations appear nearly every day. Education is one of the areas
where this rapid development of technologies has opened up a lot of new possibili-
ties, but it has also raised fears in the same way as when schoolbooks were intro-
duced 100years ago. There were fears that introducing study books at school would
destroy children’s abilities to memorize (Wakeeld, 1998). But despite the fears,
now that itis cheaper to buy tablets for children instead of printing books for them,
the discussion is changing more toward questions about the key elements of the
pedagogies inside the technologies.
It would be difcult to cover all possible ways in which technology-enhanced
learning (TEL)is and could be used at schools or homes for learning, collabora-
tively or individually, and therefore we only take the reader on a short global trip to
P. Räsänen (*)
Niilo Mäki Institute, Jyväskylä, Finland
e-mail: pekka.rasanen@nmi.
D. Laurillard
UCL Knowledge Lab, Institute of Education, London, UK
e-mail: D.Laurillard@ioe.ac.uk
T. Käser
AAALab, Graduate School of Education, Stanford University, Stanford, CA, USA
e-mail: tkaeser@stanford.edu
M. von Aster
Kinderspital Zürich, University of Zurich, Zürich, Switzerland
e-mail: m.aster@drk-kliniken-berlin.de

734
consider the barriers and possibilities there are at the moment in using TEL in
mathematics education and learning. Our focus is especially on using TEL tools to
support children with low performance in mathematics. Technology is not only used
for educational applications; there are plenty of computer-assisted tools being used
to organize and plan the education system, as well as tools for assessment, and in
research on mathematical skills and disabilities. In addition, there are more and
more computer-assisted tests, tools for brain imaging, and technologies affecting
brain activations that have increased our understanding about the mathematical
brain. However, these methodological and technical advancements are out of the
scope of this chapter. Hence, we concentrate on technologies directed toward
advancing learning and the pedagogies of mathematics in the classrooms (Fig.42.1).
The majority of the chapters in this book tell us that the mathematical learning
disabilities (MLD) are by denition something special: standard classroom educa-
tion is not enough for these children (see Landerl, Chap. 2, Santos Carvalho & Vitor
Geraldi Haase, Chap. 22, this volume), and alternative pedagogies should be used.
However, teachers have a limited amount of knowledge and information about
effective and research-informed pedagogies on MLD. Additional knowledge is
needed to understand the phenomena, how to identify these children in the class-
room, what kinds of alternative ways there exist, and how to use them to support the
children in this special group. Technology can offer globally accessible media to
inform teachers and other professionals about MLD on a level never seen before.
Therefore, we will raise the issue of teacher professional education as one of the key
global issues where TEL tools will play a signicant role.
Fig. 42.1 Current questions in usage of TEL at schools
P. Räsänen et al.

735
The second story in this book is that there is a huge variety in mathematical skills
in general and also within the group of children with MLD.There is a variance in
both domain-general and domain-specic skills, both affecting learning and teach-
ing and requiring very individual pathways for development. Partly, this variation in
individual skills is connected to differences in brain functioning (see, De Smedt,
Peters, & Ghesquière, Chap.23, this volume) and partly to learning environments
and pedagogies (see Gaidoschik, Chap. 6, this volume). The brain development of
the frontoparietal numerical network of the children with MLD seems to function
differently than in typically developing children (McCaskey etal., 2017). However,
two recent studies with computer-assisted training showed that abnormally func-
tioning connectivity in MLD will be normalized on the neuronal level by rather
short, intensive training (Iuculano etal., 2015; Michels, O’Gorman, & Kucian,
2017). This indicates that at least under the age of 10, children with MLD do not
develop compensatory mechanisms to reach the same level of prociency, butthe
training offers them the means to build representations that are similar to those of
their typically performing peers.
Technology offers many possibilities for this kind of veryindividualized inten-
sive training that has not been possible to conduct in large classrooms. However,
before we discuss the different types of interventions, we need to look at the global
questions in ICT in classrooms. One of them is access to technologies.
Global Inequalities inAccess toLearning Technologies
An access to electricity is one of the key issues for educational equality today. The
United Nations Department of Economic and Social Affairs reported that even
though the access to electricity has more than tripled from 1990s still “about 90
percent of children in Sub-Saharan Africa go to primary schools that lack electricity,
27 percent of village schools in India lack electricity access, and fewer than half of
Peruvian schools are electried. Collectively, 188 million children attend schools
not connected to any type of electricity supply” (UNDESA, 2014).
According to UNDESA the educational benets of electrication are clear.
Lighting extends the studying hours by enabling longer school days, more reading
time, and possibilities to do homework. It also allows teachers to prepare learning
materials after the school days. Electricity enables both students and teachers to use
modern mass media tools such as radio, television, computers, and the Internet.
Likewise, it improves the quality of the basic circumstances, such as sanitation and
health. Electricity in schools is needed for many of the basic tools used daily in
developed and developing countries: audiotapes, projectors and slide projectors,
printers and copy machines, digital cameras, radios and television, phones, and the
ICT technologies– computers, tablets, mobile phones, and the Internet. Therefore,
it is not a surprise that the electried schools outperform non-electried schools on
crucialeducational indicators and that electricity enables broader social and
economic development of the communities (see, e.g., UNESCO, 2014a, 2014b;
Zhang, Postlehwaite, & Grisay, 2008).
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

736
As always when there is development in access to technologies, inside the big
positive picture, the data from individual cases show contradictory results. While
some studies on access to electricity do nd a positive effect, some nd no effect,
and some even negative effects. For example, when we look at the effects of bring-
ing electricity to the area, some studies do nd an effect in an increase in time spent
studying (Barron & Torero, 2014; Khandker, Samad, Ali, & Barnes, 2012), but
some do not (Bensch, Kluve, & Peters, 2011). Squires (2015) actually found that
access to electricity in rural areas in Honduras increased the school dropout and
produced less attendance to the school due to increased “need for child labour” at
homes. Whatever the question with technology, just having it does not guarantee
positive outcomes.
From 2015in developing countries, more households had a mobile phone than
they had electricity or running water (World Bank, 2016). Mobile devices, with
their increasing affordability and storage, can contain a vast amount of educational
content, including reading and learning materials and games targeted to on a range
of ages. In addition, unlike computers, handheld mobile devices require substan-
tially less electricity or infrastructure. Due to the advantages in solar power, mobile
devices are capable of reaching even the most marginalized communities, and
research has shown mobile learning devices have the potential to widen access and
supplement education in remote and underserved areas of the world (Kim etal.,
2012; Ling, 2004). In their meta-analyses Sung, Chang, and Liu (2016) showed that
the effect size of implementing mobile devices into classroom education was sig-
nicantly more effective than teaching methods that only use pen and paper or desk-
top computers. For mathematics the effect size was 0.34 including different types of
approaches from cooperative learning to games. That is about the same level as the
other meta-analyses have given to using TEL in mathematics education (summa-
rized in Räsänen, 2015).
Still only one in seven persons in the world has access to a high-speed Internet
connection. High-speed connections are needed for the rich educational contents
already available on the Internet. The situation in access to TEL tools via the Internet
is changing rapidly. However, there is only a limited amount of up-to-date world-
wide information about the current situation of ICT at schools. UNESCO has
recently started a project to collect such information (http://uis.unesco.org/).
Online Learning, Virtual Worlds, andSocial Learning
Environments
All teachers, most parents, and some students know that poor mathematical skills
will affect chances in life. In fact, poor mathematical skills are more of a handicap
in life than poor reading (Bynner & Parsons, 1997; Parsons & Bynner, 2005). The
problem for teachers and parents is to identify the causes of the MLD.It is often
assumed that children who cannot learn even simple arithmetic must be low intel-
ligence (just as it used to be assumed that children who were unable to learn to read
P. Räsänen et al.

737
must be stupid). In fact, intelligence has little to do with the ability to learn arithme-
tic. Almost any child can. However, there is a small proportion of children who are
unable to learn arithmetic in the normal way. These children are dyscalculic. That
is, they show quite early and specic cognitive decits, just as we now know, as well
as dyslexics with different specic decits making it more difcult to acquire read-
ing skills (Vanbinst, Ansari, Ghesquière, & De Smedt, 2016). We can now easily
distinguish children with dyslexia from children with other difculties that prevent
the normal acquisition of reading, and we can now distinguish dyscalculia from
other causes of poor math attainment. However, very few teachers or parents or
educational psychologists, not to mention education authorities, have heard about
dyscalculia and therefore have never been trained to identify it or to provide the
specialist help needed for the people concerned.
One way of spreading the word about the MLD and dyscalculia is to use a tech-
nology that is accessible to most of the key professionals in many developing coun-
tries, namely, the Internet. Technology works best by responding to the most
challenging problems, and education has plenty to offer. By 2025, the global demand
for higher education will double to ~200m per year, mostly from emerging econo-
mies (NAFSA, 2010). It has been estimated that there is a need for millions of new
teaching posts for universal primary education (UNESCO, 2014b), the largest
growth being in sub-Saharan Africa.
One of the big challenges is how to reach the children who need good primary
education and, especially, how to train teachers to spot MLD as early as possible.
There are straightforward tests that have already been standardized in many coun-
tries for identifying the children at risk and models of how to build personalized
learning plans, including the use of interactive games that could be applied. Many
of these games are digital, adaptive, and available widely at low cost. This is an
example of how teachers– and other professionals– globally could make use of an
educational technology resource that has been created and developed in one place.
A different technology-enhanced method, for developing teachers on the large
scale, is to create massive open online courses (MOOC), and a start has been made
on this, aimed at primary teachers (Laurillard, 2016a, b).
An international course team working in partnership with UNESCO developed a
Coursera MOOC on “ICT in Primary Education,” which reached ~10,000 teachers
around the world, over 1200 of whom were located in low-income countries (see
Laurillard, 2016a, b), showing that such an approach is not conned to reaching
only the rich countries. A more niche course on Primary Education, dyscalculia and
other mathematics disabilities, with targeted marketing, could certainly provide col-
laborative professional development for teachers and leaders in most of the coun-
tries of the world.
To generate the network of development in the most underdeveloped areas, one
possibility could be that each of these MOOC-participating teachers could work
locally to engage 25 teachers in collaborating on using the course resources to
develop improved localized classroom methods at regional level. To reach the chil-
dren in need, each of those teachers could then set up support groups of eight adults
in villages, townships, and communities, working together to train them to become
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

738
more able teachers. This multiplies up to hundreds of thousands of teachers. The
large-scale technological capability is only needed at the rst stage. After that the
local systems can be used, making use of the cascaded digital resources and innova-
tive ideas. However, for the collaborative approach to be preserved, it is important
for the teachers experimenting with their localized solutions to pass their ideas and
experiences back up the chain.
Increasingly, some of the most challenging contexts– remote rural areas, urban
slums, and border cities– are beginning to have access to mobile devices and con-
nectivity. It is not the technology that makes it difcult, but the organization and
support for the human systems in the network. In the urban slum areas, for example,
adults set up their own private schools where there are too few government schools,
but they are unofcial, so they have no support or access to professional develop-
ment (Oketch, Mutisya, Ngware, & Ezeh, 2010). Providing this kind of support
could now be affordable but would still have to overcome the political barriers.
This is where digital technology could make the critical difference by offering
the means for collaborative professional development. The two-way communica-
tion and sharing of designs, products, and localized solutions is a way of building
professional knowledge of effective practice. This is not the typical trajectory of
pilot– rollout– fade. For example, Khan Academy, an educational website with
thousands of free video lessons on various topics, especially illustrating ways to
present mathematical content from basics to upper classes, has more than ten mil-
lion users monthly. And it is not only students using it, but also teachers, to improve
their pedagogical skills.
It is worth asking, for any big challenge, “how can technology help?” because
digital tools and environments operate on the very large scale and vastly increase
efciency and scope. MOOCs are an opportunity for the academic communityto
think through how such technologies could serve our moral imperative to achieve a
wider reach and greater contribution to society. A new model of collaborative pro-
fessional development is one way to do that.
In the long run, we can expect that the Internet access will harmonize the peda-
gogies used in mathematics teaching and interventions on MLD globally. Some of
the models of effective, research-informed methods used to support children with
MLD in top performing schools can be localized and mimicked in rural schools and
in less advanced schools. Most of the pedagogical methods do not require additional
resources; the critical issue is how concepts are presented and what kinds of ele-
ments do the interactions between the teachers and students contain. As important
as it is to use TEL to support the students, the effects can be multiplied via support-
ing teachers in their work.
Availability: TheSurge ofLearning Games
Shuler, Levine, and Ree (2012) calculated 6years ago that there were approxi-
mately half a million apps available on the Appstore. In the education category of
the apps, general early learning was the most popular subject (47%) and
P. Räsänen et al.

739
mathematics the second (13%). From that the total amount of apps has raised to
over two million with educational apps reaching soon 200,000 individual apps or
games, and the comparable store for Android machines shows even larger gures.
However, there is no clear criteria what an educational app means in these distribu-
tion channels. In their discussion of mathematics apps, Pelton and Pelton (2012)
noted that “while some are commendable, almost all of the rest are simple ash-
cards, numeric procedures, or mobile textbooks. Very few currently available apps
have engaged best practices by integrating visual models to support sense-making.”
The ease of access and the fact that the majority of these apps and games are low
cost, often totally free, mean that they are readily available to the general popula-
tion, but the question remains, what quality they have and what is being learned by
using these apps. There are relatively few applications that are built on research or
have an evidence-based background (Doabler, Fien, Nelson-Walker, & Baker, 2012;
Young etal., 2012). The What Works Clearinghouse (WWC, see https://ies.ed.gov/
ncee/wwc/FWW/) collects information about promising intervention programs
including TEL applications, but only a few of them have any research that could be
used to evaluate the program efcacy.
Usage: Does Using TEL Tools Help toProduce Better
Learning?
With increased use of TEL at schools, the question of effectiveness has been in focus
of discussions during the last years. The answer to the question depends on the data
and the design used. The often-found result from individual controlled studies (e.g.,
Carter, Greenberg, & Walker, 2017) to the international comparison datasets (OECD,
2015) has been that increasing computer usage in studying at schools does not per se
produce better learning. This discussion is actually old. Already Clark (1983; Kozma,
1994) stressed that the content is more important than the media that is used to deliver
it. They argued that separating media from educational method is an unnecessary
schism which does not produce real new insights in education.
Nevertheless, the question is still important to educational policy-makers.
Technologies require large investments but age very quickly. Are these investments
worth doing, or could it be more benecial to invest in something else? In this ques-
tion, we often turn to the international datasets. However, those have not shown
promising results on these investments, but the complexity of interpreting these
results is clear. If we look at the results from international datasets, like PISA
( illustrated in Fig.42.2), and compare the country averages in mathematics and the
number of students that have used computers in the classroom to do mathematics
during the last month before the PISA assessment, we nd a negative correlation
(r=−0.36). The more students there were who had used computers in a math class,
the lower the respective country’s average score. The negative correlation seems to
have two roots. Firstly the majority of the high-performing countries (e.g., Korea,
Japan, or Finland) have taken a slow start in moving to digitalized education, and
secondly, there are some below average-performing countries which have invested a
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

740
-35
-30
-25
-20
-15
-10
-5
0
5
10
15
20
01020304050607
08
0
Change in country average PISA mathematics 2012
-
2015
The use of comp uters in mathematics lessons (% of stud ents, 2012)
Change in Math average score from 2012 to 2015 and the
percentage of students who reported the use of
computers in mathematics lessons in 2012
350
400
450
500
550
600
01020304050607
080
Country average in PISA mathematics 2015
The use of comp uters in mathematics lessons (% of students, 2012)
PISA Math average 2015 and the percentage of students
who reported the use of computers in mathematics
lessons in 2012
a
b
Fig. 42.2 The percentage of students who had used computers in mathematics lessons during the
month prior to the PISA test in 2012 contrasted against (a) the country average in PISA 2015
(trend line r=−0.36, p<0.03) and (b) change in the average score from 2012 to 2015 (trend line
r=0.41, p<0.01). (Source of the data: OECD, 2015)
P. Räsänen et al.

741
lot to using TEL in education. If we remove these extremes from the equation, the
negative effect disappears. Also, the within-country data shows a nonlinear effect:
students not using or using a lot of computers in classrooms do not perform as well
as those who use computers moderately. Moderate usage seems to indicate better
pedagogical considerations when and where the TEL is used. An alternative look on
exactly the same data leads to another result. If we change the level of performance
to improvement in average performance during the last 3years, the correlation turns
into positive (r=+0.41). It means that more usage of computers in a math class turns
to better results 3years later. Why is this? When we look at the gain score from the
latest PISA studies (i.e., here the change from 2012 to 2015), we notice that many
countries that had invested a lot in using ICT in education (e.g., Denmark, Sweden,
Uruguay) have shown low average scores in 2012 but rapid improvement and at the
same time many high-performing countries (e.g., Korea, Taipei, Finland) showed the
largest declines. How much these changes are connected to ICT or whether they are
connected to other changes in educational cultures and to other investments in educa-
tion require more detailed analyses from various national and international datasets.
The effectiveness of computer-assisted interventions has been studied since the
1960s. Thisoffers usanother set of data to look at the value of investment to TEL
tools in mathematics. There are several meta-analytic studies (summarized in, e.g.,
Räsänen, 2015) showing that there have been three most-gaining subgroups across
the years of using TEL: (1) younger children show a larger gain than older children,
(2) children with special needs seem to show more benet than children in studies
with more heterogeneous samples, and (3) studies where TEL has been used as
supplementary education instead of replacing the teacher have shown better results
(e.g.Lavin & Sanders, 1983; Li & Ma, 2010; Niemiec & Walberg, 1987; Slavin &
Lake, 2008). In addition, the studies conducted in developing countries tend to show
higher effectiveness than those done in developed countries. These results indicate
that it seems to be easier to produce better results when the starting level is lower,
especially if the reasons for a lower starting level have been poor access to educa-
tion or low SES and not cognitive factors. Children with cognitive decits or math-
ematical learning disabilities have only recently become a focus of research.
Chodura, Kuhn, and Holling (2015) looked specically at interventions for children
with MLD.They found a similar level of effectiveness for computer-assisted and
face-to- face interventions. These two are often contrasted:While some stressthe
importance of direct contact and social interaction in the learning process, one of
the most commonly presented reasoning on usingTEL tools has been that the gami-
cation brings engagement and motivation into the learning that the standard class-
room or special education lacks.
Affective andMotivational Factors
Educational TEL programs are typically designed in a format of a game. The serious
games, as educational games are often called, are hypothesized to address both the
cognitive and the affective dimensions of learning (O’Neil, Wainess, & Baker, 2005),
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

742
to enable learners to adapt learning to their cognitive needs and interests and to
provide motivation for learning (Malone, 1981). However, the assumption underly-
ing the motivational appeal of serious games is based on the addictive nature the
commercial computer games have. However, the results of a meta-analysis show
that serious games are not more motivating than other instructional methods
(ES=0.26, a nonsignicant difference; Wouters, Van Nimwegen, Van Oostendorp,
& Van Der Spek, 2013).
An essential difference between leisure computer games and serious games is
that playing for entertainment is chosen by the players and played whenever and
for as long as the player wants, whereas with the serious games, the playing and
playing time are dened by someone else (e.g., teacher, game developer,
researcher). In addition, logic of effort needed and cognitive load in entertainment
and educational games are different. Educational games typically aim to increase
the difculty and cognitive load systematically to match the players current skill
level to boost performance (about the adaptive logic inside math games, see, e.g.,
Räsänen etal., 2015), while in entertainment games, the cognitive load varies
more freely and does not aim to maximize the performance level. Therefore, there
is no strong evidence that gamication of TEL in mathematics education would
produce by itself stronger and long-lasting internal motivation and thatit would
producebetter learning that way.
Contents: What Is Insidethe Intervention Games forMLD?
There have been two content areas that have dominated the studies with TEL games
targeted to children with MLD or low performance. The rst has been the key symp-
tom area in MLD, the lack of arithmetic uency. This approach dominated the rst
decades of research with ashcard-type of training and its variants to improve mem-
ory retrieval of arithmetic facts. The rise of ideas of number sense as a core dif-
culty behind the difculties to learn the basic arithmetic skills (e.g., Piazza etal.,
2010) has led to focus the interventions to either symbolic or nonsymbolic number
sense and to games combining arithmetic to number line representation to illustrate
the distances and relations between numbers. The latter approach has dominated the
research during the last 10years, and we will concentrate on these ndings. A third
raising hypothesis has been that training domain-general skills strongly connected
to numerical cognition might likewise boost learning mathematics. Especially
working memory (Passolungi & Costa, Chap. 25, this volume) and increasingly
also, spatial skills (Resnick etal., Chap. 26, this volume) have recently grasped the
attention of the researchers.
P. Räsänen et al.

743
Training Number Sense
There is an ongoing debate about the roles of symbolic and nonsymbolic number
sense in the development of MLD.Dehaene (1997) suggested that an evolutionarily
grounded analogue magnitude representation, also called an approximate number
system (ANS) or “number sense,” underliesthenumerical understanding. After this
suggestion many studies have aimed to train the ANS with the intention of transfer-
ring improvements to symbolic arithmetic. There are some grounds for this idea.
The ANS, typically measured by requiring participants to choose which of two dot
arrays contains more dots, correlates with measures of symbolic math in both adults
and children (e.g., DeWind & Brannon, 2012; Halberda, Ly, Wilmer, Naiman, &
Germine, 2012; Halberda, Mazzocco, & Feigenson, 2008; Lyons & Beilock, 2011)
and also predicts from preschool to math achievement tests at school age (Gilmore,
McCarthy, & Spelke, 2010; Mazzocco, Feigenson, & Halberda, 2011a). Likewise,
children with MLD perform less well in ANS tasks compared with typically per-
forming children (Mazzocco, Feigenson, & Halberda, 2011b).
For example, Park and Brannon (2013, 2014) showed that adults after a very
short training with nonsymbolic addition and subtraction tasks improved the perfor-
mance in symbolic addition tasks. In their training two clouds of dots were pre-
sented to the participant, who needed to estimate without counting, which one of the
two new clouds of dots would match with the answer of the calculation presented.
According to Park and Brannon this illustrates a causal link between these two rep-
resentations. Wang, Odic, Halberda, and Feigenson (2016) got similar results with
5-year-old children (see however Merkley, Matejko, & Ansari, 2017 about the crit-
ics). Park and colleagues (2016) gave a similar tablet game with nonsymbolic
approximate addition and subtraction of large arrays of items to 3–5-year-old chil-
dren nding selective improvements in math skills after multiple days of playing
compared with children who played a memory game. Khanum and others (Khanum,
Hanif, Spelke, Berteletti, & Hyde, 2016) have replicated these results using similar
training tasks with Pakistani children of school-age, demonstrating that there is a
cultural invariance in these results.
In addition to these experimental games, Wilson and colleagues (Wilson, Revkin,
Cohen, Cohen, & Dehaene, 2006) have developed a free adaptive computer learn-
ing game for children with MLD (http://www.thenumberrace.com/nr/). The Number
Race game is reminiscent of traditional board games in which one throws a dice and
advances by that number of steps. The dice throwing has been replaced with a com-
parison task (nonsymbolic and symbolic). The game is constantly adapting based
on an articial intelligence algorithm. This algorithm represents the learner’s cur-
rent skill level (“knowledge space”) in three dimensions, and it is programmed to
ensure an average accuracy of 75%. The three dimensions of the model are the ratio
of the quantities presented (the distance effect), the time allowed to respond, and
the conceptual complexity of the format in which the quantities are presented (from
dot patterns to arithmetic). Several studies have used this game module to test if this
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

744
kind of training would produce benets to learning (Obersteiner, Reiss, & Ufer,
2013; Räsänen, Salminen, Wilson, Aunio, & Dehaene, 2009; Sella, Tressoldi,
Lucangeli, & Zorzi, 2016; Wilson, Dehaene, Dubois, & Fayol, 2009). Szűcs and
Myers (2017) critically analyzed these studies and concluded that there is no con-
clusive evidence that specic ANS training improves symbolic arithmetic. They
found many problems in these studies, not limited to the fact that it was unclear
whether the game directly focused on ANS or on some other numerical processes
more important for learning arithmetic. In their study,Sella etal. (2016) divided
4–6-year- old children into two groups, one playing Number Race and a control
group playing with a drawing program. There were clear effects of the Number
Race game compared to drawing activities to boost numerical skills of typically
performing young children. However, the result does not directly point to the ben-
ets of computer- assisted number sense training in early development, because it
was a comparison between math and non-math training. In the study of Obersteiner
etal. (2013), exact numerical representations were contrasted against approximate
training, and he found no difference in learning between these two trainings.
Räsänen etal. (2009) used the Number Race training with 6-year-old children with
a risk of MLD and contrasted this against a training with a game with explicit train-
ing of number symbols, where the latter, according to a reanalysis of Szűcs and
Myers (2017), seemed to produce slightly better results. In a similar fashion,
Honoré and Noël (2016) contrasted symbolic and nonsymbolic training. Both train-
ings produced signicant learning effects compared to control conditions, but sym-
bolic training led to a signicantly larger improvement in arithmetic than did
nonsymbolic training.
Maertens and his colleagues (2016) used another type of approach to train the
relations between numbers. They contrasted the above-described comparison tasks
to number line training. Performance on tasks where the child is asked to estimate
the position of numbers in the number line has been shown to be related to chil-
dren’s mathematical achievement (e.g., Booth & Siegler, 2008; Friso-van den Bos
etal., 2015; Muldoon, Towse, Simms, Perra, & Menzies, 2013; Siegler & Booth,
2004). Moreover, interventions that have focused on improving numerical represen-
tations through game-based number linetasks have shown transfer to arithmetic
learning and mathematical performance (Fischer, Moeller, Bientzle, Cress, &
Nuerk, 2011; Link, Moeller, Huber, Fischer, & Nuerk, 2013; Siegler & Ramani,
2008). Maertens etal. (2016) found that both comparison and number line estima-
tion trainings had a positive effect on arithmetic. However, there were no transfer
effects from one task to another. Thissuggests that comparison and number line
estimation rely on different mechanisms and probably inuence arithmetic through
different mechanisms.
Another game that uses number line as a way to present numbers and calculations
is Calcularis (Käser, Baschera, etal., 2013). Because it is one of the few research-
informed games developed for children with MLD, we look at it in more detail. The
model of the game is based on the theory of a hierarchical development of mental
number representations (von Aster & Shalev, 2007): The game builds up on early
available concrete number representations (number as a set of objects) and the verbal
P. Räsänen et al.

745
symbolization (spoken number) that develops during preschool age followed by the
development of the Arabic symbolization taught in school. At the last level, the men-
tal number line is gradually built over the rst years of elementary school. Children
with MLD often exhibit problems in constructing and accessing this mental number
line representation (Kaufmann etal., 2009; Kucian, Loenneker, Dietrich, Martin, &
von Aster, 2006; Mussolin etal., 2010). The scientic evaluation of the precursor
version of Calcularis (called Rescue Calcularis) demonstrated that children with and
without MLD benet from a number line training. Kucian and others (2011) showed
that the neuronal changes observed after playing the game indicated a rened mental
number representation as well as more efcient number processing.
Calcularis turns these ndings on number processing and numerical cognition
into the design of different instructional games, which are hierarchically structured
according to number ranges and can be further divided into three areas (a content
model: numerical understanding and representations, addition and subtraction, mul-
tiplication and division). The rst area focuses on different number representations
as well as number understanding in general. Transcoding between alternative repre-
sentations is trained, and children learn the three principles of number understand-
ing: cardinality, ordinality, and relativity. The rst area is exemplied by the
LANDING game illustrated in Fig.42.3a. In this game, children need to indicate the
position of a given number on a number line. To do so, a falling cone has to be
steered using a joystick or the right and left arrow key. The second and third areas
cover cognitive operations and procedures with numbers. In this area, children train
the concepts and automation of arithmetic operations. In the PLUS-MINUS game
(see Fig.42.3b), children solve addition and subtraction tasks using blocks of tens
and ones to model them.
Fig. 42.3 In the
LANDING game (a), the
position of the displayed
number (16) needs to be
indicated on the number
line. In the PLUS-MINUS
game (b), the task
displayed needs to be
modeled with the blocks of
tens and ones
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

746
To offer optimal learning conditions, the training program adapts to the knowl-
edge state of a specic child (Käser etal., 2012; Käser, Busetto, etal., 2013). All
children start the training with the same game. After each item, the program esti-
mates the knowledge state of the child and displays a new task adjusted to this state.
In order to adapt the difculty level and the task selection to the needs of a spe-
cic child, the training program needs to represent and estimate the mathematical
knowledge of the child. This knowledge is modeled with a dynamic Bayesian net-
work representing different mathematical skills and their dependencies as a directed
acyclic graph. The model used for Calcularis consists of more than 100 different
skills. A small excerpt of the network is displayed in Fig.42.4. The skills are sorted
into different number ranges. Within a number range, they are ordered according to
their difculties. The difculty of a task depends on the magnitude of the numbers
involved in the task, the complexity of the task, and the means allowed to solve the
task. Modeling “46+33=79” with one, ten, and hundred blocks (Support Addition
2,2) is easier than calculating it mentally (Addition 2,2). Furthermore, tasks includ-
ing a carry such as “46+37=83” (Addition 2,2 with bridging to ten) are more
complex to solve than tasks not requiring carrying. In order to also be able to adapt
to specic problems of a child, the program contains a bug library storing typical
error patterns. If a child commits a typical error several times, the controller system-
atically selects actions for remediation.
The effects of the training program have been assessed in a pilot study with 41
children conducted in Switzerland (Käser, Baschera, etal., 2013) and a following
comprehensive study with 138 children in Germany, where children were randomly
assigned to 1 of 3 conditions (Calcularis training group, waiting control group,
spelling control training group) with 6 and 12weeks of training time (Rauscher
etal., 2016, 2017). The results largely conrmed those of the pilot study: Compared
to the two control conditions, children of the Calcularis training group demonstrated
signicant improvements with regard to arithmetic performance and spatial number
processing abilities. These effects were already present after 6weeks of training
and became even larger after 12weeks. In addition, the positive effects in math
performance were accompanied by a signicant decrease of math anxiety, which is
Fig. 42.4 Addition skill net in the number range 0–100 with example tasks
P. Räsänen et al.

747
known to substantially contribute to developmental dyscalculia. Due to its adaptive
nature, Calcularis is widely used in inclusive classroom settings to achieve intra-
class differentiation. It is also suitable for intervention, in which children practice at
home without direct supervision. The supervisors can monitor their students’ work
with the coaching application, which, as Calcularis itself, is browser-based.
Another game that has used the number line representation toillustrate the num-
bers and calculations is a free tablet game Vektor (http://cognitionmatters.org).
However, it differs from the other new research-informed games in a criticalway that
it combines numerical and cognitive training. In the Vektor game, the numbers and
calculations are presented bothas ve- and ten-pals (see also “number bonds to ten”
in Butterworth, Varma, & Laurillard, 2011) and in symbolic calculation tasks with a
number line representation. The cognitive training in Vektor is based on WM training
with predominantly visuospatial tasks that have previously been shown to be effec-
tive in increasing WM efciency (Bergman-Nutley & Klingberg, 2014; Melby-
Lervåg & Hulme, 2013). The newest version of the game also contains visuospatial
and visuospatial reasoning tasks, because it has been shown that visuospatial WM
predicts later mathematical skills and that especially the number line representation
is tied to visuospatial skills (Simms, Clayton, Cragg, Gilmore, & Johnson, 2016).
However, even though there is a lot of evidence on the connections between
visuospatial and numerical skills (Resnick etal., Chap. 26, this volume), we still
lack studies about direct transfer effects from spatial training to arithmetic skills
(however, see Lowrie, Logan, & Ramful, 2017).
Passolunghi and Costa (2016) have shown that working memory (WM) training
signicantly enhances children’s numeracy abilities involving concepts of compari-
son, classication, correspondence, seriation, counting, and general knowledge of
numbers (see also Holmes, Gathercole, & Dunning, 2009; Kroesbergen, van’t
Noordende, & Kolkman, 2014; Kuhn & Holling, 2014; St Clair-Thompson, Stevens,
Hunt, & Bolder, 2010; Witt, 2011). In their study with the Vektor game on combin-
ing working memory and arithmetic training with 6-year-old children, Nemmi and
his colleagues (2016) found that a combined training of cognitive skills and arith-
metic was more effective than either WM or arithmetic training alone. However,
they also found that when going beyond these group effectsto a more individual
level, there is a whole new world for researchers to tackle.
Typically, theeffectiveness of atraining isanalyzed at a group level. Effectiveness
of an intervention is considered to be good when childrenin the experimental group
improvesignicantly more than children in the control groups.Theoretically, the
education is the samein all subjectsin the experimental group of the TEL interven-
tion study. In a similar fashion asthe education is the same to all children in the
classroom. However,in the classroom the teacher shouldfocus on individual perfor-
mances. In reality,some children learnmore, and someless, irrespective of the
method used. This is a challenge to the teacher on how to raise the level of learning
of the children who learned less. The same method and pedagogy is not benecial
to all. And every teacher knows this.
Likewise, this is a challenge to researchers. Instead of concentrating on groups,
more research is needed about the factors behind the individual gains than about the
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

748
effects at the group level. In the study of Nemmi and others (2016), they divided the
children into subgroups based on the baseline level of WM and mathematical skills
measured before the intervention. The main nding of the effectiveness of the com-
bined WM and numerical training over numerical training alone was nonexistent in
children with low WM and in children with below average skills in mathematics.
The impact of an intervention varied by a factor of 3 between the subjects, depend-
ing on their baseline performance. Therefore, while one intervention can be
extremely benecial to some, another child with different prole of numerical and
cognitive skills may not benet from that specic training at all. Focusing on this
question would bring researchers closer to the educational practices within the
classroom. What do we need to know about the child to learn what kind of interven-
tion is benecial? As soon as we have more understanding about this question, we
can build individualized, adaptive, and effective interventionswith TEL tools.
From theClassrooms totheLab
The majority of the noncomputerized interventions for children with MLD and pro-
grams for learning the basic number concepts recommend using manipulatives:
small collections of objects to be ordered, categorized, compared, and counted
(Clements & Sarama, 2011; Samara & Clements, 2009). This is a common knowl-
edge for well-trained special needs teachers, who use a wide variety of activi-
tieswith manipulatives in their classes (Dowker, 2004; Emerson & Babtie, 2014).
They help children with MLD to learn the meaning of numbers by using concrete
materials as well as by articulating their practice in multiple representations of dia-
grams and number lines and then building up to symbols and equations (Emerson &
Babtie, 2014). These kinds of activities are also offered as computerized tasks in
some TEL programs, such as the Building Blocks early educational program
(Sarama & Clements, 2004), and in the NumberBeads game targeted to children and
adults with severe MLD (Laurillard, 2016b).
Investigations using computerized manipulatives for geometry and fractions
show that these can lead to statistically signicant gains in learning new concepts
(Reimer & Moyer, 2005). Olson (1988) found that students who used both physical
and software manipulatives demonstrated a greater sophistication in classication
and logical thinking than did a control group that used physical manipulatives alone.
A computer environment offers students greater control and exibility over the
manipulatives, allowing them to, for example, duplicate and modify the computer
bean sticks (Char, 1989; Moyer, Niezgoda, & Stanley, 2005).
Digital entertainment games are more and more combining the realities of virtual
and real worlds. Games happen in 3D worlds, and the players can more and more
realistically manipulate objects in these worlds. The educational applications are
slowly moving to this direction, and most probably the next wave of intervention
research on supporting children with MLD using TEL will concentrate on bringing
in the effective traditions used by the experienced and well-informed special need
P. Räsänen et al.

749
teachers. Studies like Iuculano etal. (2015) show that an intensive face-to-face
intervention is very effective in helping children with MLD to built up numerical
skills and, thus, will provide a good starting point for researchers to think about the
key features of the effective interactions and pedagogies needed in TEL tools. Do
we have virtual manipulatives in virtual classrooms only, or do we see virtual teach-
ers as well? Most probably yes, but rst,there is still a lot to learn from the best
teachers.
Final Word
Technologies are spreading fast, and almost all children in OECD countries have
computers at home, and cheaper mobile technologies are reaching even the most
underdeveloped areas. Even though technologies have a promise of advancing the
education, the OECD report on technology usage at schools gives a serious warn-
ing: “perhaps the most disappointing nding of the report is that technology is of
little help in bridging the skills divide between advantaged and disadvantaged stu-
dents. Put simply, ensuring that every child attains a baseline level of prociency in
reading and mathematics seems to do more to create equal opportunities in a digital
world than can be achieved by expanding or subsidising access to high-tech devices
and services” (OECD, 2015).
The question of access to technologyis easier to solve than the question ofeffec-
tivecontents for learning, especially when we aim to improve the skills of those
children with learning disabilities. In this chapter we have tried to introduce some
new ideas from research during the last decade on how the question of contenthas
been approached.Advances in basic neuroscientic research will uncover more
about the mechanisms of learning, raising new contents to be implemented in seri-
ous games and even to electronic school books.
Technology is making education a joint global issue. It offers teachers new
sources for collaboration as well as for professional development and training. It
also gives access everywhere to the same TEL tools. Innovations connecting online
systems with adaptive learning systems can easily create extensive datasets from
tens if not hundreds of thousands of children to uncover individual learning path-
ways and mechanisms. Usage of big data is opening up new possibilities for using
technology in educational research.
In the end, the successful solutions for TEL on MLD depend on building bridges
between the best educational practices and basic research on the mathematical
brain. These are at best combined on intervention studies that can inform us both
about the mechanisms of numerical learning and about theeffective methods of
using TEL tools in education and remediation. While the paper books are more and
more changingto net-connectede-books, the researchers will have a totally new
possibility to build such studies in collaboration with teachers as part of children’s
daily school and homework.
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

750
References
Barron, M., & Torero, M. (2014). Short term effects of household electrication: experimental
evidence from northern El Salvador, MPRA Paper No. 63782, 2014.
Bensch, G., Kluve, J., & Peters, J.(2011, December). Impacts of rural electrication in Rwanda,
IZA Discussion Papers 6195, Institute for the Study of Labor (IZA).
Bergman-Nutley, S., & Klingberg, T. (2014). Effect of working memory training on working mem-
ory, arithmetic and following instructions. Psychological Research, 78(6), 869–877.
Booth, J.L., & Siegler, R.S. (2008). Numerical magnitude representations inuence arithmetic
learning. Child Development, 79(4), 1016–1031.
Butterworth, B., Varma, S., & Laurillard, D. (2011). Dyscalculia: from brain to education. Science,
332(6033), 1049–1053.
Bynner, J., & Parsons, S. (1997). Does numeracy matter? London: The Basic Skills Agency.
Carter, S.P., Greenberg, K., & Walker, M.S. (2017). The impact of computer usage on aca-
demic performance: Evidence from a randomized trial at the United States Military Academy.
Economics of Education Review, 56, 118–132.
Chodura, S., Kuhn, J.T., & Holling, H. (2015). Interventions for children with mathematical
difculties: A meta-analysis. Zeitschrift für Psychologie, 223(2), 129.
Char, C.A. (1989). Computer graphic feltboards: New software approaches for young children’s
mathematical exploration. San Francisco: American Educational Research Association.
Clark, R.E. (1983). Reconsidering research on learning from media. Review of Educational
Research, 53(4), 445–459.
Clements, D.H., & Sarama, J.(2011). Early childhood mathematics intervention. Science,
333(6045), 968–970.
Dehaene, S. (1997). The number sense. NewYork: Oxford University Press.
DeWind, N.K., & Brannon, E.M. (2012). Malleability of the approximate number system: Effects
of feedback and training. Frontiers in Human Neuroscience, 6, 68.
Doabler, C.T., Fien, H., Nelson-Walker, N.J., & Baker, S.K. (2012). Evaluating three elementary
mathematics programs for presence of eight research-based instructional design principles.
Learning Disability Quarterly, 35(4), 200–211.
Dowker, A. (2004). What works for children with mathematical difculties? (Vol. 554).
Nottingham, UK: DfES Publications.
Emerson, J., & Babtie, P. (2014). The dyscalculia solution: Teaching number sense. London, UK:
Bloomsbury Publishing.
Fischer, U., Moeller, K., Bientzle, M., Cress, U., & Nuerk, H.C. (2011). Sensori-motor spatial
training of number magnitude representation. Psychonomic Bulletin & Review, 18(1), 177–183.
Friso-van den Bos, I., Kroesbergen, E.H., Van Luit, J.E., Xenidou-Dervou, I., Jonkman, L.M.,
Van der Schoot, M., & Van Lieshout, E.C. (2015). Longitudinal development of number line
estimation and mathematics performance in primary school children. Journal of Experimental
Child Psychology, 134, 12–29.
Gilmore, C.K., McCarthy, S.E., & Spelke, E.S. (2010). Non-symbolic arithmetic abilities and
mathematics achievement in the rst year of formal schooling. Cognition, 115(3), 394–406.
Halberda, J., Ly, R., Wilmer, J.B., Naiman, D.Q., & Germine, L. (2012). Number sense across the
lifespan as revealed by a massive Internet-based sample. Proceedings of the National Academy
of Sciences, 109(28), 11116–11120.
Halberda, J., Mazzocco, M.M., & Feigenson, L. (2008). Individual differences in non-verbal num-
ber acuity correlate with maths achievement. Nature, 455(7213), 665.
Holmes, J., Gathercole, S.E., & Dunning, D.L. (2009). Adaptive training leads to sustained
enhancement of poor working memory in children. Developmental Science, (4), 12.
Honoré, N., & Noël, M.P. (2016). Improving preschoolers’ arithmetic through number magnitude
training: The impact of non-symbolic and symbolic training. PLoS One, 11(11), e0166685.
Iuculano, T., Rosenberg-Lee, M., Richardson, J., Tenison, C., Fuchs, L., Supekar, K., & Menon, V.
(2015). Cognitive tutoring induces widespread neuroplasticity and remediates brain function in
children with mathematical learning disabilities. Nature Communications, 6, 8453.
P. Räsänen et al.

751
Käser, T., Baschera, G.-M., Kohn, J., Kucian, K., Richtmann, V., Grond, U., etal. (2013). Design
and evaluation of the computer-based training program Calcularis for enhancing numerical
cognition. Frontiers in Psychology, 4. https://doi.org/10.3389/fpsyg.2013.00489
Käser, T., Busetto, A.G., Baschera, G.-M., Kohn, J., Kucian, K., von Aster, M., & Gross, M.
(2012). Modelling and optimizing the process of learning mathematics. Proceedings of ITS,
7315, 389–398. https://doi.org/10.1007/978-3-642-30950-2_50
Käser, T., Busetto, A.G., Solenthaler, B., Baschera, G.-M., Kohn, J., Kucian, K., etal. (2013).
Modelling and optimizing mathematics learning in children. International Journal of Articial
Intelligence in Education, 23, 115–135. https://doi.org/10.1007/s40593-013-0003-7
Kaufmann, L., Vogel, S.E., Starke, M., Kremser, C., Schocke, M., & Wood, G. (2009).
Developmental dyscalculia: Compensatory mechanisms in left intraparietal regions in
response to nonsymbolic magnitudes. Behavioral and Brain Functions, 5(1), 1–6. https://doi.
org/10.1186/1744-9081-5-35
Khandker, S.R., Samad, H.A., Ali, R., &, Barnes, D.F. (2012). Who benets most from rural
electrication? Evidence in India. World Bank, Working Paper, 2012.
Khanum, S., Hanif, R., Spelke, E.S., Berteletti, I., & Hyde, D.C. (2016). Effects of non-symbolic
approximate number practice on symbolic numerical abilities in Pakistani children. PLoS One,
11(10), e0164436.
Kim, P., Buckner, E., Kim, H., Makany, T., Taleja, N., & Parikh, V. (2012). A comparative anal-
ysis of a game-based mobile learning model in low-socioeconomic communities of India.
International Journal of Educational Development, 32(2), 329–340.
Kozma, R.B. (1994). Will media inuence learning? Reframing the debate. Journal of Educational
Technology Research and Development, 42(2), 7–19.
Kroesbergen, E.H., van’t Noordende, J.E., & Kolkman, M.E. (2014). Training working
memory in kindergarten children: Effects on working memory and early numeracy. Child
Neuropsychology, 20(1), 23–37.
Kucian, K., Loenneker, T., Dietrich, T., Martin, E., & von Aster, M. (2006). Impaired neural net-
works for approximate calculation in dyscalculic children: A fMRI study. Behavioral and
Brain Functions, 2(1), 31.
Kucian, K., Grond, U., Rotzer, S., Henzi, B., Schönmann, C., Plangger, F., etal. (2011). Mental
number line training in children with developmental dyscalculia. NeuroImage, 57(3), 782–795.
Kuhn, J.T., & Holling, H. (2014). Number sense or working memory? The effect of two computer-
based trainings on mathematical skills in elementary school. Advances in Cognitive Psychology,
10(2), 59.
Laurillard, D. (2016a). The educational problem that MOOCs could solve: Professional develop-
ment for teachers of disadvantaged students. Research in Learning Technology, 24(1), 29369.
Laurillard, D. (2016b). Learning number sense through digital games with intrinsic feedback.
Australasian Journal of Educational Technology, 32(6).
Lavin, R.J., & Sanders, J.E. (1983). Longitudinal Evaluation of the Computer Assisted Instruction,
Title I Project, 1979–82.
Li, Q., & Ma, X. (2010). A meta-analysis of the effects of computer technology on school students’
mathematics learning. Educational Psychology Review, 22(3), 215–243.
Ling, R. (2004). The mobile connection. Amsterdam, Netherlands: Morgan Kaufmann Publishers.
Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary students’
mathematics performance. British Journal of Educational Psychology, 87(2), 170–186.
Lyons, I.M., & Beilock, S.L. (2011). Numerical ordering ability mediates the relation between
number-sense and arithmetic competence. Cognition, 121(2), 256–261.
Link, T., Moeller, K., Huber, S., Fischer, U., & Nuerk, H.C. (2013). Walk the number line – An
embodied training of numerical concepts. Trends in Neuroscience and Education, 2(2), 74–84.
Malone, T. (1981). What makes computer games fun? ACM, 13(2–3), 143.
Maertens, B., De Smedt, B., Sasanguie, D., Elen, J., & Reynvoet, B. (2016). Enhancing arithmetic
in pre-schoolers with comparison or number line estimation training: Does it matter? Learning
and Instruction, 46, 1–11.
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

752
McCaskey, U., von Aster, M., Maurer, U., Martin, E., Tuura, R.O. G., & Kucian, K. (2017).
Longitudinal brain development of numerical skills in typically developing children and chil-
dren with developmental dyscalculia. Frontiers in Human Neuroscience, 11.
Mazzocco, M.M., Feigenson, L., & Halberda, J.(2011a). Impaired acuity of the approximate
number system underlies mathematical learning disability (dyscalculia). Child Development,
82(4), 1224–1237.
Mazzocco, M.M., Feigenson, L., & Halberda, J.(2011b). Preschoolers’ precision of the approxi-
mate number system predicts later school mathematics performance. PLoS One, 6(9), e23749.
Melby-Lervåg, M., & Hulme, C. (2013). Is working memory training effective? A meta-analytic
review. Developmental Psychology, 49(2), 270.
Merkley, R., Matejko, A.A., & Ansari, D. (2017). Strong causal claims require strong evidence:
A commentary on Wang and colleagues. Journal of Experimental Child Psychology, 153,
163–167.
Michels, L., O’Gorman, R., & Kucian, K. (2017). Functional hyperconnectivity vanishes in chil-
dren with developmental dyscalculia after numerical intervention. Developmental Cognitive
Neuroscience, 30, 291–303.
Moyer, P.S., Niezgoda, D., & Stanley, J.(2005). Young children’s use of virtual manipulatives and
other forms of mathematical representations. Technology-Supported Mathematics Learning
Environments, 67, 17–34.
Muldoon, K., Towse, J., Simms, V., Perra, O., & Menzies, V. (2013). A longitudinal analysis of
estimation, counting skills, and mathematical ability across the rst school year. Developmental
Psychology, 49(2), 250.
Mussolin, C., De Volder, A., Grandin, C., Schlögel, X., Nassogne, M.C., & Noël, M.P. (2010).
Neural correlates of symbolic number comparison in developmental dyscalculia. Journal of
Cognitive Neuroscience, 22(5), 860–874.
NAFSA. (2010). The Changing Landscape of Global Higher Education, Association of
International Educators. Washington, DC.
Nemmi, F., Helander, E., Helenius, O., Almeida, R., Hassler, M., Räsänen, P., & Klingberg, T.
(2016). Behavior and neuroimaging at baseline predict individual response to combined math-
ematical and working memory training in children. Developmental Cognitive Neuroscience,
20, 43–51.
Niemiec, R., & Walberg, H.J. (1987). Comparative effects of computer-assisted instruction: A
synthesis of reviews. Journal of Educational Computing Research, 3(1), 19–37.
Obersteiner, A., Reiss, K., & Ufer, S. (2013). How training on exact or approximate mental repre-
sentations of number can enhance rst-grade students’ basic number processing and arithmetic
skills. Learning and Instruction, 23, 125–135.
OECD. (2015). Students, computers and learning: Making the connection. Paris: PISA, OECD
Publishing.
Oketch, M., Mutisya, M., Ngware, M., & Ezeh, A.C. (2010). Why are there proportionately more
poor pupils enrolled in non-state schools in urban Kenya in spite of FPE policy? International
Journal of Educational Development, 30(1), 23–32.
Olson, J.K. (1988). Microcomputers make manipulatives meaningful. Budapest, Hungary:
International Congress of Mathematics Education.
O'Neil, H.F., Wainess, R., & Baker, E.L. (2005). Classication of learning outcomes: Evidence
from the computer games literature. The Curriculum Journal, 16(4), 455–474.
Park, J., & Brannon, E.M. (2013). Training the approximate number system improves math pro-
ciency. Psychological Science, 24(10), 2013–2019.
Park, J., & Brannon, E.M. (2014). Improving arithmetic performance with number sense training:
An investigation of underlying mechanism. Cognition, 133(1), 188–200.
Park, J., Bermudez, V., Roberts, R.C., & Brannon, E.M. (2016). Non-symbolic approximate
arithmetic training improves math performance in preschoolers. Journal of Experimental Child
Psychology, 152, 278–293.
Parsons, S., & Bynner, J.(2005). Does numeracy matter more? London: National Research and
Development Centre for Adult Literacy and Numeracy, Institute of Education.
P. Räsänen et al.

753
Passolunghi, M.C., & Costa, H.M. (2016). Working memory and early numeracy training in
preschool children. Child Neuropsychology, 22(1), 81–98.
Pelton, T., & Pelton, L.F. (2012, March). Building mobile apps to support sense-making in math-
ematics. In Society for Information Technology & Teacher Education International Conference
(pp.4426–4431). Chesapeake, VA: Association for the Advancement of Computing in
Education (AACE).
Piazza, M., Facoetti, A., Trussardi, A.N., Berteletti, I., Conte, S., Lucangeli, D., etal. (2010).
Developmental trajectory of number acuity reveals a severe impairment in developmental
dyscalculia. Cognition, 116(1), 33–41.
Räsänen, P. (2015). Computer-assisted interventions on basic number skills. In A.Dowker &
R.Cohen Kadosh (Eds.), The Oxford handbook of numerical cognition. Oxford, UK: Oxford
University Press.
Räsänen, P., Käser, T., Wilson, A., von Aster, M., Maslov, O., & Maslova, U. (2015). Assistive
technology for supporting learning numeracy. In B.O’Neill & A.Gillespie (Eds.), Assistive
technology for cognition: A handbook for clinicians and developers. Current issues in neuro-
psychology (pp.112–117). NewYork: Psychology Press.
Räsänen, P., Salminen, J., Wilson, A., Aunio, P., & Dehaene, S. (2009). Computer-assisted inter-
vention for children with low numeracy skills. Cognitive Development, 24, 450–472.
Rauscher, L., Kohn, J., Käser, T., Kucian, K., McCaskey, U., Wyschkon, A., etal. (2017). Effekte
des Calcularis-Trainings. Lernen und Lernstörungen, 6, 75–86.
Rauscher, L., Kohn, J., Käser, T., Mayer, V., Kucian, K., McCaskey, U., etal. (2016). Evaluation
of a computer-based training program for enhancing arithmetic skills and spatial number rep-
resentation in primary school children. Frontiers in Psychology, 7, 913.
Reimer, K., & Moyer, P.S. (2005). Third-graders learn about fractions using virtual manipulatives:
A classroom study. The Journal of Computers in Mathematics and Science Teaching, 24(1), 5.
Sarama, J., & Clements, D.H. (2004). Building blocks for early childhood mathematics. Early
Childhood Research Quarterly, 19(1), 181–189.
Samara, J., & Clements, D.H. (2009). “Concrete” computer manipulatives in mathematics educa-
tion. Child Development Perspectives, 3(3), 145–150.
Sella, F., Tressoldi, P., Lucangeli, D., & Zorzi, M. (2016). Training numerical skills with the adap-
tive videogame “The Number Race”: A randomized controlled trial on preschoolers. Trends in
Neuroscience and Education, 5(1), 20–29.
Shuler, C., Levine, Z., & Ree, J.(2012). iLearn II An analysis of the education category
of Apple’s app store. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/summary?
doi=10.1.1.362.6454
Siegler, R.S., & Booth, J.L. (2004). Development of numerical estimation in young children.
Child Development, 75(2), 428–444.
Siegler, R.S., & Ramani, G.B. (2008). Playing linear numerical board games promotes low-
income children's numerical development. Developmental Science, 11(5), 655–661.
Simms, V., Clayton, S., Cragg, L., Gilmore, C., & Johnson, S. (2016). Explaining the relationship
between number line estimation and mathematical achievement: the role of visuomotor inte-
gration and visuospatial skills. Journal of Experimental Child Psychology, 145, 22–33.
Slavin, R.E., & Lake, C. (2008). Effective programs in elementary mathematics: A best-evidence
synthesis. Review of Educational Research, 78(3), 427–515.
St Clair-Thompson, H., Stevens, R., Hunt, A., & Bolder, E. (2010). Improving children's working
memory and classroom performance. Educational Psychology, 30(2), 203–219.
Squires, T. (2015). The impact of access to electricity on education: Evidence from Honduras.
Job Market Paper, Brown University. Available at https://economics.ucr.edu/seminars_collo-
quia/2014-15/applied_economics/Squires_JMP_Electricity.pdf
Sung, Y.T., Chang, K.E., & Liu, T.C. (2016). The effects of integrating mobile devices with
teaching and learning on students’ learning performance: A meta-analysis and research syn-
thesis. Computers & Education, 94, 252–275.
Szűcs, D., & Myers, T. (2017). A critical analysis of design, facts, bias and inference in the
approximate number system training literature: A systematic review. Trends in Neuroscience
and Education, 6, 187–203.
42 Perspectives toTechnology-Enhanced Learning andTeaching inMathematical…

754
UNESCO. (2014a). Institute of Statistics. A view inside schools in Africa: Regional education
survey. Paris.
UNESCO. (2014b). EFA Global Monitoring Report: Teaching and learning– Achieving quality
for all. Paris.
UNDESA. (2014). Electricity and education: The benets, barriers, and recommendations for
achieving the electrication of primary and secondary schools. Retrived from https://sustain-
abledevelopment.unorg/content/documents/1608Electricity%20and%20Education.pdf
von Aster, M.G., & Shalev, R.S. (2007). Number development and developmental dyscalculia.
Developmental Medicine and Child Neurology, 49(11), 868–873.
Vanbinst, K., Ansari, D., Ghesquière, P., & De Smedt, B. (2016). Symbolic numerical magnitude
processing is as important to arithmetic as phonological awareness is to reading. PLoS One,
11(3), e0151045.
Wakeeld, J.F. (1998). A Brief History of Textbooks: Where Have We Been All These Years? A
paper presented at the Meeting of the Text and Academic Authors (St. Petersburg, FL, June
12–13).
Wang, J.J., Odic, D., Halberda, J., & Feigenson, L. (2016). Changing the precision of preschool-
ers’ approximate number system representations changes their symbolic math performance.
Journal of Experimental Child Psychology, 147, 82–99.
Wilson, A.J., Revkin, S.K., Cohen, D., Cohen, L., & Dehaene, S. (2006). An open trial assessment
of “The Number Race”, an adaptive computer game for remediation of dyscalculia. Behavioral
and Brain Functions, 2(1), 20.
Wilson, A.J., Dehaene, S., Dubois, O., & Fayol, M. (2009). Effects of an adaptive game inter-
vention on accessing number sense in low-socioeconomic-status kindergarten children. Mind,
Brain, and Education, 3(4), 224–234.
Witt, J.K. (2011). Action’s effect on perception. Current Directions in Psychological Science,
20(3), 201–206.
World Bank. (2016). World Development Report 2016: Digital Dividends. Washington, DC: World
Bank. https://doi.org/10.1596/978-1-4648-0671-1. License: Creative Commons Attribution
CC BY 3.0 IGO
Wouters, P., Van Nimwegen, C., Van Oostendorp, H., & Van Der Spek, E.D. (2013). A meta-
analysis of the cognitive and motivational effects of serious games. Journal of Educational
Psychology, 105(2), 249.
Young, M.F., Slota, S., Cutter, A.B., Jalette, G., Mullin, G., Lai, B., etal. (2012). Our princess is
in another castle: A review of trends in serious gaming for education. Review of Educational
Research, 82(1), 61–89.
Zhang, Y., Postlehwaite, T.N., & Grisay, A. (2008). A view inside primary schools: A World
Education Indicators (WEI) cross-national study. Paris: UNESCO.
P. Räsänen et al.