Mathematics teacher education and the flexible use of technologies in digital age
KHOA HỌC, GIÁO DỤC VÀ CÔNG NGHỆ
MATHEMATICS TEACHER EDUCATION AND THE FLEXIBLE
USE OF TECHNOLOGIES IN DIGITAL AGE
Hans-Georg Weigand
University of Wuerzburg, Germany
Email:
dvantages and disadvantages of the use of digital
A
technologies (DT) in mathematics lessons are worldwide
dissussed controversially. Many empirical studies show the benefit
of the use of DT in classrooms. However, despite of inspiring
results, classroom suggestions, lesson plans and research reports,
the use of DT has not succeeded, as many had expected during the
last decades. One reason is or might be that we have not been able
to convince teachers and lecturers at universities of the benefit of
DT in the classrooms in a sufficient way. However, to show this
benefit has to be a crucial goal in teacher education because it will
be a condition for preparing teachers for industrial revolution 4.0.
In the following we suggest a competence model, which classifies
– for a special content (like function, equation or derivative) –
the relation between levels of understanding (of the concept),
representations of DT and different kind of classroom activities.
The flesxible use of digital technologies will be seen in relation
to this competence model, results of empirical investigations will
be intergrated and examples of the use of technologies in the up-
coming digital age will be given.
Received: 16/5/2019
Reviewed: 20/5/2019
Revised: 27/5/2019
Accepted: 10/6/2019
Released: 21/6/2019
DOI:
Keywords: Digital technologies; Industial revolution 4.0;
Computer algebra system; Teacher education; Mathematics
education.
Concerning the Use of Digital Technologies
2010a, Weigand & Bichler 2010b). The main result
of hese projects and investigations can roughly be
summarized as follow: DT (and especially CAS).
- Allow a greater variety of strategies in the
frame of problem solving processes;
- Are a catalyst for individual, partner and group
work;
- Do not lead to a deficit in paper-and-pencil
abilities and mental abilities (if these abilities are
regularly supported in the teaching process)
- Allow more realistic modelling problems in
the classroom (but also raise the cognitive level of
understanding these problems);
- Do not automatically lead to charged or
modified test and examination problems (compared
to paper-and-pencil tests);
- Demand and foster advanced argumentation
strategies (e.g if equations are solved by pressing
only one button).
- Overall, Drijvers et al. (2016) concluded from
a meta-study-survey of quantitative studies that
there are “significant and positive effects, but with
small average effect sizes” (p. 6), if we ask for the
(DT) in Mathematics Classrooms
There are many theoretical considerations,
empirical investigations and suggestions for the
classroom concerning the use of DTin mathematical
learning, and teaching (e.g Guin et al. 2005, Zbiek
2007, Drijvers and Weigand 2010). There are also
some empirical studies concerning the integration
of DT and of computer algebra systems (CAS)
especially into regular classroom teaching and
covering longer periods of investigation, e. g. the
e-CoLab1 (Aldon et al. 2008), RITEMATHS2.
CALIMERO3 (Ingelmann and Bruder 2007),
M3-Project4 (Weigand 2008, Weigand and Bichler
1. e-CoLab = Expérimentation Collaborative de Laboratoires
mathématiques. See:
collaborative-de-laboratoires-mathematiques. Accessed 29 Oct 2018
2. RITEMATHS = The project is about the use of real problem (R)
and information technology (IT) enhence (E) students’ commitment
to, and achievement in, mathmatics (MATHS),
http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS.
Accessed 29 Oct 2018.
3. CALIMERO = Computer Algebra in Mathematics Lessons:
Discovering, Calculating, Organizing (translate title).
4. M3 = Model Project New Media in Mathematics Education
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benefits of integrating DTin mathematics education. contents - and we have not been able to convince
Moreover, there is also a broad consensus that teachers, lecturers at university and parents of the
gainful changes in classroom teaching and learning benefit of DT in the classrooms.
need didactic and methodic considerations and a
thorough reflection about the goals of teaching and
learning.
Worldwide, the current situation concerning
the use of DT is very varied. There are countries
(like the Scandinavian countries in Europe) that are
intensively using laptops, tablets (with the programs
GeogebraorMaple)orsymboliccalculators(likethe
TI-Nspire or the Casio Classpad). These countries
even allow using such tools in examinations. There
are other countries (like the UK or France) that
allow “only” symbolic calculators in examinations,
there are countries - especially in Asia - which
are very skeptical about the use in examinations,
and there are countries (like Germany) where
there are different situations about the use of DT
depending on the state. However, the situation in
all these countries is not stable, but very flexible.
There are changes from examinations with DT to
examinations without DT, from the use of graphing
calculators to the use of calculators with CAS
and vice versa. There always has to be a critical
evaluation of recent developments.
Visions and Disillusions
The first ICMI study in 1986 “The Influence of
Computers and Informatics on Mathematics and its
Teaching” (Churchhouse) was started because of
a creat eathusiasm concerning the perspectives of
mathematics education in view of the availability of
new technologies. Many mathematics educators, for
instanceJimKaput,forecastedthatnewtechnologies
would change all fields of mathemantics education
quite quickly.
“Technology in mathematics education might
work as a newly active volcano – the mathematical
mountain is changing before our eyes” (1992,
p.515).
The NCTM standards of 1989 (and in the
revised version of 2000) have been visionary -
concerning the field of mathematics education – by
representing a vision for the future of mathematics
education. This is especially true for the use of new
technologies in mathematics classrooms, expressed
in the ‘Technology Principle’:
Thesis 2: Changes and new developments in
mathematics education – especially in relation to
digital technologies – have only terminable validity
and have to be re-evaluated continuosly.
Reflecting the developments of the use of DT
in the last decade, we started to rethink the results
concerning the possibilities of supporting students’
learning processes and them especially raised the
question: What is the benefit of using DT in the
classroom? (Weigand 2017) More specifically, we
asked:
In relation to which mathematical contexts does
the use of DT make sence and which (mathematical)
conpetencies are supported ang developed?
Which mathematical ang tool competencies are
necessary, or at least helpful, when working with
DT specific mathematics content?
In the following, we will try to give some
answers to these questions by constructing a model
that show the relation between working with DT,
different levels of understanding and specifics
activities in the classroom and will illustrate the
model by giving some example.
“Technology is essential in teaching and
learning mathematics; it influences the mathematics
that taught and enhances students’ learning” (p.24)
And:
“Calculators and computers are reshaping the
mathermatical landscape… Students can learn
more mathematics more deeply with the appropriate
and responsible use of technology” (p.25)
However, in the ICMI Study 17 “Mathematics
Education and Technology - Rethinking the terrain”
(Hoyles & Lagrange 2010) and the OECD Study
(2015) “Students, computers and learning. Making
the connecton” disappointment is quite offen
expressed about the fact that - despite the countless
ideas, classroom suggestions, lesson plans and
research reports - the use of DT has not been as
successful as many had expected at the beginning
of the 1990s. In her closing statement concerning
“the Future of Teaching and Learning Mathematics
with DT” Michèle Artigue summarizes in the ICMI
Study:
“The situation is not so brillant and no one would
claim that the expectations expressed at the time of
the first study (20 years ago) have been fulfilled”
(p.464)
This brings us to the first thesis concerning the
development of DT in the next decade:
Competence models - Theoretical Foundations
The concepts of competence and competence
(level) models have aroused interest in mathematics
education in the past years. Starting with the NCTM
Standark (1989) and especially the PISA studies,
competence and competencies are expressions often
used in the context of standards and that substituted
the “old expression” goats which envisaged
knowledge and abilities in mathematies education.
“Mathematical competence means the ability to
understand, judge, do, and use mathematics in a
Thesis 1. We underestimated the difficulties of
DT-use in a technical sense and in relation to the
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variety of intra-and extra-mathematical contexts Example: (a + b)3 = a3 +…
and situations in which mathematics plays or could
play a role…” (Niss 2004, p. 120). In the PISA
studies, competencies are on the one hand related
to content, e.g. numbers, space and shape, change,
ete., and on the other hand – in a more general way -
related to processes like problem solving, modeling
and the use of mathematical language. In order to
evaluate or operationalize the competencies through
the construction of items and tests, it is helpful to
organize these competencies in levels, categories or
classes. In the PISA studies, each of the possible
pairs (content, process) can be divided into three
different levels or competence classes (OECD
1999, p.43). This leads to a three-dimensional
competence-model with the dimensions content,
basic or process competencies and cognitive
activation5
In Weigand & Bichler (2010a) a competence
model for the use of symbolic calculators with
CAS in mathematics lessons within the context of
working with functions was developed. Different
levels of understanding the function concept have
been seen in relation with the representations and -
as a third dimension - with cognitive activation. The
ability or the competence to use the tool adequately
requires technical knowledge about the handling
of the tool. Moreover, it requires the knowledge of
when to use which features and representations and
for which problems it might be helpful.
•
Control: DT as a controller of hand-written
solution, suggestions and ideas on a graphical,
numeric or symbolic levels.
•
Explain: DT are catalysts for the
communication between the user (student, learner)
and someone who has to interpret or understand the
DT-solutions (e.g. a teacher). DT are sources for
explanations and argumentations.
•
Discover: DT as a tool for evaluating and
testing suggestions and strategies in a problem
solving process.
This classification may be seen as a hierarchy
while moving from procedural knowledge
(Culculate) to conceptual knowledge (explain,
discover).
This URA-competence-model has three
categories and gives us 4 x 4 x 5 = 80 cells. If each
cell is again subdivided into three levels of cognitive
activation, this makes a tatal of 240 cells and this is
only for a special concept. This already shows that it
is very difficult or even impossible to create special
examples for each of these cells. This competence
model is more for pointing out the directions and
goals concerning understanding, the kinds of used
representations and the kinds of activities DT might
be used for to adequately develop a special concept.
Thesis 3. The URA-competence-model puts
understanding in relation to activities and working
with DT-representations. It is helpful for diagnostic
reasons and for creating learning-strategies in a
DT-supported classroom.
Competence model for DT-use in the
classroom
The model above is promising if tasks and
problems have to be classified, e.g for tests
and examinations. It does not adequately fit if
activities in the classroom are to be integrated
and evaluated. In the following model the second
dimension “Representation” was changed due
to the well-established theory of representation
which emphasizes the reasoning with multiple and
dynamic representations (Bauer 2013 or Ainsworth
1999). Moreover, understanding and working with
representations is seen in relation to classroom
activities. We introduced a third dimension with the
following activities:
Tool Competencies
What do we mean by tool competence? A tool
as “something you use to do somethin” (Monaghan
et.al. 2016, p. 5) is a quite general definition.
Mathematical tools allow us to create, to operate
with and to change mathematical objects. DT and
CAS are digital tools. We use the word tool instead
of instrument because we focus on the facilities of
DT, reflect these in relation to mathematics aspects
in the classroom, and we leave the development
of the user-tool-relationship in the frame of an
strumental orchestration, which is the heart of the
instrumental genesis (see Artigue 2002, Drijvers
et.al. 2010), to the user oe learner. Tool-competence
is the ability to apply the competence-model to a
special concept. Tool-competence describes the
development of the understanding of a concept in
relation to the tool-representation in the frame of
classroom-activities.
•
Calculate: DT as a tool for (numeric and
symbolic) calculations, and DT, especially CAS, are
tools which allow calculation on a symbolic level
in notations close to the mathematical language.
Example: Seeing parameter dependent functions
as functions of several variables allows efficient
problem solving processes.
•
Consult: DT - especially CAS - as a
While empirical competence models - like the
PISA model - help answer the question whether
students or learners do benetit from special learning
or teaching interventions, the models is more
process-oriented and should give reasons why and
consultant in the sense of using a formulary.
5. In PISA, these dimensions are called “Overarching ideas”
(content), “Competencies” (process) and “Competence Clusters”
(cognitive activation)
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how this might be the case.
This concept of the Z-function can be transferred
to polynomials of a higher degree. For example. for
the Z-polynomial with
Thesis 4: Tool competencies describe the
relation of working with (representations of) a tool,
the levels of understanding of a concept and the
kind of activity in the classroom and can only be
evaluated in the frame of this relationship.
f (z) = az3 + bz2 + cz + d
you receive the ditference-Z-polynomial
Df (z) = 3az2 + (3a + 2b)z + a + b + c
In the following, we are going to give examples
of tasks and problems and their solutions in a DT-
supported environment. We refer to the long-term
empirical M3-project6, which was initiated by
the Bavarian Ministry of Education in Germany.
Students were allowed to use symbolic calculators
(with CAS) - TI-Nspire, Casio ClassPad, notebooks
- In mathematics lessons, for homework and
in examinations, especially also in the final
baccalaureate examinations, which is a statewide
(final) examination (Weigand 2008, Weigand and
Bichler 2010b).
Concerning teacher education in the Digital Age
we think that teacher students should get familiar
with examples like these and that they should be able
to see the relationship to goals and competencies to
be developed in the classroom. The URA - model
is a help for classifying and evaluating classroom
activities with DT.
This can easily be calculated with a CAS or
at least be verified with the programme. The use
of computer algebra systems is especially useful
and helpful when difference-Z-polynomials of
Z-polynomials of a higher degree have to be
calculated.
This leads to the hypothesis that a Z-polynomial
of degree n (€N) has a difference-Z-polynomial of
degree n-1.7
Here we can see that there are different aspects
or representations that are connect to the CAS. It is
a tool for multiple and dynamic representations, it is
an experimentation tool and reflects mathematical
expressions on a symbolic level with nonation
that is close to mathematical notation. Concerning
the competence model for CAS-use, all kinds of
representations are used and also nearly all kinds of
activities are used in these examples.
Especially the following mathematics of
calculus competencies are supported:
1. Example: Difference sequences
Difference sequences (∆a ) N with ∆ak = ak+1
–
a and a given sequence (ak)Nk are well suitable for
ak discrete introduction of the difference quotient.
Based on sequences or functions that are defined on
N, we will now take a look on functions
Learners
•
Understand the definition of difference
sequences of sequences and Z-functions and realize
the relation between a Z-function and its difference
sequence;
f : Z → R, the so called Z-functions that are
defined on Z and their relation with difference-Z-
function. Df : Df (z) = f (z+1) – f (z), pe. f (z) =
z2 – 2z + 3
The dependence of Df, on the used parameters
of f with f(z) = az2 + bz +c can be graphically
depicted. The dynamics of the representation can
be induced by the “Slide bars”.
•
Interpret the relation between a Z-function
and its difference sequence in
•
•
different representations;
Can determine the difference-Z-function of
a Z-function on the symbolic level, basic examples
by hand and more complex examples using a CAS.
Z-functions and difference-Z-functions are
a conceptual basis for the understanding of
the difference- and differential quotient and
subsequently the derivative of real functions.
These two graphs of f and Df already suggest
that the graph of the difference-Z-function is linear.
This can be explained on a symbolic level: With
f(z) = az2 + bz + c you receive the difference-
2. Example: An examination problem
Z-function
Examinations influence the way content is
taught in the classroom. Only if DT are allowed
in examinations they will also be used for
classroom work. Moreover, examination problems
set standards in the way of teaching. DT show
advantages especially if you work on open problems
and if modeling of realistic problems or discovery
learning are the focus of the classroom work.
However, these problems are not very suitable in
D(f) = f (z + 1) – f (z) = a (z +1)2 + b (z +1) +
c - (az2 + bz + c) = 2az + a + b
Therewith, the changes of the graph by varying
a and b and the independence of Df can be
explained. Moreover, it has to be noticed, that the
mathematical communication - concerning used
expressions and gestures - is different in static and
dynamic environments. You will get more dynamic
verbalizations if you use dynamic representations
(Ng 2016).
7. For G. W. Leibniz (1646 – 1715) sequences and their differences
sequences have been a source for the development of the derivative
and the calculus.
6. M3: Model project Modern Media in Mathematics classrooms
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- traditional – examinations because they require if the CAS solves the following equation. The Casio
time and patience to try different solution strategies ClassPad, for example, can also solve equations
or to allow pursuing dead-ead streets in the problem that depend on parameters.
solving process. If we stick to traditional ways of
examination forms, new kinds of problems are
necessary. Moreover, the question of the relation
between expected knowledge, abilities and skills
in examinations has to be discussed under new
aspects.
The following is an example of a problem from
the final examination 2014 in Baviria. The crossing
of the two motorways A92 and B299. One length
unit in the coordinate system corresponds to 20
metters. The models fo the strees A92 is the straight
line with y = -0.5 x and this one of B299 is y = 0.45
x. If you come from Munich and leave the A92 at P
you will come to the point Q. The course of the exit
from P to Q is modeled by a graph of a polynomial
function of grade 3.
However, a constructive handling of these
solutions asks for futher knowledge concerning the
solution formulas or these solutions can encourage
dealing with solution formulas – here the Cadano
formulas.
It is, however, only possible to solve more
complex equations with a CAS when were is
already a basic knowledge of the solution variety of
the considered equiations present. Furthermore, one
needs strategies for the handling of a representation
type especially, with regard to necessary changes
of the representation types, because, if an approach
that had been used did not lead to a successful
solution, a strategy is needed.
An example is the solution of the equation 1 +
sin (x) = 2x.
Geogebra-CAS cannot solve the equation on a
symbolic level. The Casio ClassPad offers several
numerical solutions, although these are hard to
understand for (almost) every user.
The exit at P should be without any kinks and
should be perpendicular in Q to the road B299.
A solution of this problem had been showed. Of
course, this is not a real modelling problem. The
coordinate system does not exist in reality, the streets
are not straight lines and the graph of a polynomial
of grade 3 is not a good fit of the course of the exit.
If you consider the real situation in above, you can
imagine the potential of this problem, if you discuss
it in a common classroom-learning environment
rather than in an examination.
The examination situation does not give you
the time needed for planning, for testing different
solutions and for checking and evaluating, and it
does not offer the possibility of making mistakes
while working with such a complex situation.
Thesis 5. The construction of - good or
meaningful - test and examination problems - if
we think about a traditional oral or written exam
— is even more challenging if DT are allowed.
Nevertheless, we should always consider that
examinations are guidelines for classroom activites.
In the future, new ways of examination forms
are possible, e.g. (digital) portfolios, project work
and oral presentations.
Auseful strategy would be switching to a graphic
representation and zooming in the intersection point
of the graphs. Therefore, mathematical knowledge
about basic properties of the two functions is
absolutely necessary. Tonisson (2015) gives a good
verview of the solution variety of equations, as he
has solved and compared 120 quations of school
mathematics with eight different CAS.
A last examples: x7 – 4x5 + 4x3 = 0.
The CAS gives the solutions of a polynomial
of grade 7, but only because the expression can
be factorized. The - surprising - solution has to be
interpreted with the graphic representation.
An efficient use of a CAS when solving equations
that are a bit more complex is only possible with a
mathematical knowledge conceming the solution
of equations, the characteristics of the underlying
functions of the equations and the possibilities of the
solution varieties. For calculations, the CAS is used
within the static isolated symbolic representation,
but it is possible to add graphic representations for
interpreting or explaining symbolic results and use
dynamic representations to change parameters. This
3. Example: CAS and the solving of complex
equations
CAS can be used to calculate the zeros of a kind of extended CAS is a prototype of a flexible
function by only pressing one button, but moreover, digital tool. The advantage of using CAS is the
it serves as visualization. Furthermore, the notation of solutions on a symbolic level, especially
relationship of function and equation is fundamental while working with equations with parameters.
for the mutual representation in the CAS and the Like in the case of working with functions, the
graphic window.
communication with the tool is possible in a
language close to the traditional mathematical
language. The CAS is a consultant in the sense of
a formulary for symbolic solutions especially for
polynomial equations of order 2 or 3.
A CAS can solve equations of higher degree,
particularly equations of degree 3, in a symbolic
form, but the solution depends on the equation.
However, there will be different forms of solutions
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Thesis 6: The efficient working with DT needs “If you do this..., you have to care for this..., you
mathematical knowledge, strategies in working have to be carefully with... and you can expect
with different kinds of representations, and the this...”. The results of the empirical research and
flexible use of the interrelationship between these the future-oriented considerations provide a basis
two dimensions and the kind of activity, quite in the for this kind of advice; no more, but also no less.
sense of the URA-competence-model.
Two important aspects for the future:
Connectivity and Visions
Looking ahead
A key question to ask is, what do we know
nowadays about technology integration in
Students’ access to mobile technology, the mathematics teaching and learning and what is the
availability of online mathematics learning basis of knowledge we could take for granted when
resources and the existence of social networks developing ideas for the (teacher) education in the
will open up the classroom, there will be no fixed up-coming digital age (see Trgalová et al. 2017 and
“inside” and “outside the classroom”, it will open Weigand 2018)?
the learning time and will multiply the access to
alternative learning materials (see Borba et al.
2017, p. 230). Education with DT has to be flexible.
We know, that it takes a significant amount
of time for learners and teachers to besome fully
instrumentalised, that is to learn to use and apply
Moreover, an intergrated global concept of the the technology for their relevant mathematical
use of DT has to follow different aspects. It concerns purpose, which for teachers includes important
the interaction of different digital components didactic considerations and the development of thei
such as laptops, netbooks, the Internet and pocket resource systems. In the last decades, the focus of
computers under technical aspects; it concerns the research was on the effects of using technology on
use of classroom materials like digital schoolbooks students’ learning and teachers’ practices. Now, as
and it should support the cooperation between we know more about these effects, our attention has
the teachers of a school, the parent and of course shifted to be concerned with researching how we
the students. Finally, the coopenation of teachers can scale ‘successful’ innovations in mainstream
of different schools, between schools and school education systems. Assessment is and will be a
administration and the university are important.
crucial point while integrating technologies into the
classroom. If DT are not allowed to use in tests and
examinations in high schools, they will not be used
in the classroom. If we think about scaling-up, we
also have to think about formative and summative
assessment in schools. Moreover, we have to see
both, assessment through technology and with
technology (Drijvers et al., 2016).
Finally, with a focus on emergent technologies,
touch screens and human-computer interaction will
get more important, gestures will help visualising
and, hopefully, understanding better mathematical
concepts. There will be an emphasize conceming 3D
technology, including the use of 3D printers within
mathematics education, virtual and aumented
reality, artificial intelligence features to include
intelligent tutoring and support systems that take
account of large data sets. Finally, DT will support
individuality, for example, the creation of portfolios
and personalised e-textbooks.
Thesis 7: Connectivity and interconnectedness
will be key words in the future. The acceptance of
DT and their profitable use require a global concept
of teaching and learning.
Above all, visions will be important in the future,
in all fields of scientific and public life. Without
visions, there are no further developments. We need
visions which are based on empirical results and
theoretical considerations, but we also need visions
which are based “only” on new and creative ideas,
and we need to have the courage to also discuss
visions which - nowadays - look like illusions.
Teachers expect specific answers to their
questions concerning why and how they shall use
DT in their classes. These questions are at the
heart of mathematics education: We are - as a
mathematics educators - in the situation of advisors
or consultants, who can “only” give some advice:
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ĐÀO TẠO GIÁO VIÊN TOÁN VÀ SỬ DỤNG HIỆU QUẢ
CÁC CÔNG NGHỆ TRONG THỜI ĐẠI KỸ THUẬT SỐ
Hans-Georg Weigand
Đại học Wuerzburg, Germany
Email:
Tóm tắt: Những lợi thế và bất lợi của việc sử dụng công nghệ
kỹ thuật số trong các bài toán đang gây tranh cãi trên toàn thế giới.
Nhiều nghiên cứu thực nghiệm cho thấy lợi ích của việc sử dụng
công nghệ số trong lớp học. Tuy nhiên, mặc dù những kết quả đầy
cảm hứng, những đề xuất trong lớp học, những giáo án và những
báo cáo nghiên cứu, việc sử dụng công nghệ số đã không thành
công như nhiều người mong đợi trong những thập kỷ qua. Một lý
do có thể là do chúng tôi đã không thể thuyết phục các giáo viên
và giảng viên tại các trường đại học về lợi ích của công nghệ số
trong các lớp học một cách đầy đủ. Tuy nhiên, để cho thấy lợi ích
này phải là một mục tiêu quan trọng trong đào tạo giáo viên bởi vì
nó sẽ là điều kiện để chuẩn bị giáo viên cho cuộc cách mạng công
nghiệp 4.0. Sau đây chúng tôi đề xuất một mô hình năng lực, phân
loại - cho một nội dung đặc biệt (như hàm, phương trình hoặc đạo
hàm) - mối quan hệ giữa các cấp độ hiểu biết (về khái niệm), biểu
diễn của công nghệ số và các loại hoạt động khác nhau trong lớp
học. Việc sử dụng linh hoạt các công nghệ số sẽ liên quan đến mô
hình năng lực này, kết quả điều tra thực nghiệm sẽ được tích hợp
và các ví dụ về việc sử dụng các công nghệ trong kỷ nguyên số
sắp tới sẽ được đưa ra.
Ngày nhận bài: 16/5/2019
Ngày gửi phản biện: 20/5/2019
Ngày tác giả sửa: 27/5/2019
Ngày duyệt đăng: 10/6/2019
Ngày phát hành: 21/6/2019
DOI:
Từ khóa: Công nghệ kỹ thuật số; Cách mạng công nghiệp 4.0;
Hệ thống đại số máy tính; Đào tạo giáo viên; Giáo dục toán học.
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