Mathematics teacher education and the flexible use of technologies in digital age

 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.
A useful 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 
kind of extended CAS is a prototype of a flexible 
digital tool. The advantage of using CAS is the 
notation of solutions on a symbolic level, especially 
while working with equations with parameters. 
Like in the case of working with functions, the 
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.
KHOA HỌC, GIÁO DỤC VÀ CÔNG NGHỆ 
82 JOURNAL OF ETHNIC MINORITIES RESEARCH
Thesis 6: The efficient working with DT needs 
mathematical knowledge, strategies in working 
with different kinds of representations, and the 
flexible use of the interrelationship between these 
two dimensions and the kind of activity, quite in the 
sense of the URA-competence-model. 
Two important aspects for the future: 
Connectivity and Visions
Students’ access to mobile technology, the 
availability of online mathematics learning 
resources and the existence of social networks 
will open up the classroom, there will be no fixed 
“inside” and “outside the classroom”, it will open 
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. 
Moreover, an intergrated global concept of the 
use of DT has to follow different aspects. It concerns 
the interaction of different digital components 
such as laptops, netbooks, the Internet and pocket 
computers under technical aspects; it concerns the 
use of classroom materials like digital schoolbooks 
and it should support the cooperation between 
the teachers of a school, the parent and of course 
the students. Finally, the coopenation of teachers 
of different schools, between schools and school 
administration and the university are important.
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: 
“If you do this..., you have to care for this..., you 
have to be carefully with... and you can expect 
this...”. The results of the empirical research and 
the future-oriented considerations provide a basis 
for this kind of advice; no more, but also no less.
Looking ahead 
A key question to ask is, what do we know 
nowadays about technology integration in 
mathematics teaching and learning and what is the 
basis of knowledge we could take for granted when 
developing ideas for the (teacher) education in the 
up-coming digital age (see Trgalová et al. 2017 and 
Weigand 2018)?
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 
the technology for their relevant mathematical 
purpose, which for teachers includes important 
didactic considerations and the development of thei 
resource systems. In the last decades, the focus of 
research was on the effects of using technology on 
students’ learning and teachers’ practices. Now, as 
we know more about these effects, our attention has 
shifted to be concerned with researching how we 
can scale ‘successful’ innovations in mainstream 
education systems. Assessment is and will be a 
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.
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KHOA HỌC, GIÁO DỤC VÀ CÔNG NGHỆ 
84 JOURNAL OF ETHNIC MINORITIES RESEARCH
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: 
weigand@mathematik.uni-wuerzburg.de
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:
https://doi.org/10.25073/0866-773X/304
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.
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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of technologies in the digital age. To appear in: Proceedings for the 8th ICMI-East Asia Regional 
Conference on Mathematics Education 2018, Taipeh. 
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in Mathematics Lessons - The Case of Functions, ZDM - The International Journal on Mathematics 
Education 42(7), 697-713. 
Weigand, H.-G., Bichler, E. (2010b). Symbolic Calculators in Mathematics
Education - The Case of Calculus. International Joumal for Technology in Mathematica Education, No. 
1 (17), 3-15. 
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Information Age Publishing.