=Paper=
{{Paper
|id=Vol-2733/paper7
|storemode=property
|title=Design of a Smart ABN Device for Early Math Education
|pdfUrl=https://ceur-ws.org/Vol-2733/paper7.pdf
|volume=Vol-2733
|authors=Ana Martín Díaz,Iria Estévez-Ayres,Carlos Alario-Hoyos,Carlos Delgado Kloos
|dblpUrl=https://dblp.org/rec/conf/siie/DiazEAK20
}}
==Design of a Smart ABN Device for Early Math Education==
https://ceur-ws.org/Vol-2733/paper7.pdf
Design of a Smart ABN Device for Early Math
Education
Ana Martı́n Dı́az Iria Estévez-Ayres Carlos Alario-Hoyos
Dpto. Ingenierı́a Telemática Dpto. Ingenierı́a Telemática Dpto. Ingenierı́a Telemática
Universidad Carlos III de Madrid Universidad Carlos III de Madrid Universidad Carlos III de Madrid
Leganés, Madrid, Spain Leganés, Madrid, Spain Leganés, Madrid, Spain
amdiaz@pa.uc3m.es ayres@it.uc3m.es calario@it.uc3m.es
Carlos Delgado Kloos
Dpto. Ingenierı́a Telemática
Universidad Carlos III de Madrid
Leganés, Madrid, Spain
cdk@it.uc3m.es
Abstract—New methodologies are emerging in the current development in a short time, becoming one of the alternatives
educational system as an alternative to traditional teaching, some to the conventional method known as CBC, “Closed Based on
of which are related to the area of basic mathematics for primary Cipher” (Cerrado Basado en Cifras) [5].
school students. One of them is the Open Calculation method
Based on Numbers, ABN (Abierto Basado en Números). This The main purpose of ABN method is to help children to
method is based on showing what the meaning of the number know the meaning of the number [6]. This is typically worked
is. Thus, the manipulation of objects is its base. The aim of this with the manipulation of objects, especially using chopsticks.
paper is to present the design and development of a smart device Thus, it is easy to use it in the classroom although it has
for teaching ABN method. An electronic device has been designed some limitations. The student’s progress cannot be followed
to facilitate the learning of the students in a simple and physical
way of the basic operations, trying not to lose the essence of the to help him to improve in the future and to offer him a more
ABN method. In addition, the device saves any interaction that individualised education.
the student has when performing an operation to allow showing There are already some web sites to support ABN, but they
analytics of the gathered data in the future. are not tangible. All the tools that exist are web applications.
Index Terms—early math education, primary school, educa- Therefore, the aim of this paper is to present the design and
tional systems, ABN method, data gathering, technological tool
development of a smart device that allows the student to learn
the basic operations in a simple and physical way using ABN.
This smart device saves all the interaction that the student has
I. I NTRODUCTION
with it during performing the operation so that the teacher can
Technologies today are causing a great transformation, and have a follow up of the child learning in the future.
education is one of the areas that has the greatest impact The rest of the paper is organised as follows: Section
[1]. The way in which education is taught is constantly II presents the principles on which the ABN method is
evolving, pursuing to improve its quality, and revolutionising based, Section III defines the requirements that the device
the way in which the student obtains, processes, and interprets must fulfil, Section IV describes the physical design for the
information. implementation of the device, Section V presents the physical
An analysis of the current situation in the classroom reveals device appearance, and an example of an addition and add-
certain weaknesses in the field of mathematics, especially subtraction; and, finally, Section VI concludes and presents
in some countries, such as Spain, Ukraine or Argentina, the future work of this research.
according to the results obtained in the evaluation of the ed-
ucational system, PISA (Programme for International Student II. T HE ABN METHOD
Assessment) [2]. This area is fundamental for the develop- ABN stands for Open Calculation method Based on Num-
ment of STEAM, Science, Technology, Engineering, Arts and bers, and it was born to help the children with the simple
Mathematics skills, which are currently the most demanded arithmetic expressions [7]. The resolution of operations with
ones for future works [3], and even present ones, where there the traditional models prevents the adequate development of
is already a lack of professionals to cover [4]. the mental arithmetic [8]. The ABN method ensures that the
New ways of teaching mathematics have appeared in the student does not learn the contents in a mechanical way, mem-
current educational situation, among them, the ABN method, orizes rules and works on the operations in a single way, but
“Open Calculation method Based on Numbers” (Abierto gives children the freedom to experiment by themselves and
Basado en Números). This method is undergoing a great to make their own experiences the source that gives meaning
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
to mathematics [6]. The student works with manipulative and B. Addition
motivating materials to achieve this. The essence of addition is to remove an amount from one
The principles that support ABN method are based on the operand and add it to the other. Therefore, the solution will
evidence of the MRE, “Realistic Mathematics Education”, be reached when one of the two summands is zero. The
approach [9] and are as follows [10]: children represent the terms with chopsticks, each group of
• Principle of equality: all students can achieve an accept- chopsticks in a “tray” (see Fig. 1). Thus, they have two groups
able mathematical competence with the corresponding of chopsticks representing each number. The addition consists
help, the existence of a “mathematical gen” is rejected. of passing chopsticks from one tray to another, until one is
• Principle of experience: manipulation of objects is es- empty.
sential for the child to build his own learning; experi- In Fig. 1, there is an example of how a student performs an
mentation is necessary to acquire the abstract concepts addition step by step. The steps are detailed as follows:
of mathematics. • First, the child writes the terms of the addition in the tray
• Principle of using whole numbers: when a transaction is and grid (Fig. 1.A).
complex it is divided into smaller whole numbers, never • Then, the child removes 3 tens from the right tray. The
into meaningless units. The student always manipulates, amount that the student decides to “remove” or “put”
operates, calculates and estimates based on numbers. must be placed in the left column of the grid. Thus, (s)he
• Principle of transparency: all the steps and processes writes 30 and updates the terms of the sum, 87 + 8 (Fig.
carried out can be visualized and the symbolic materials 1.B).
used help to reflect reality. • After that, (s)he passes 3 units to be able to group 1 ten in
• Principle of adaptation to the individual rhythm of each the left tray with the 7 single units. The sum is updated,
subject: it is an open calculation method and does not 90 + 5 (Fig. 1C).
follow any pattern, the students have flexibility to perform • Finally, the student moves the remaining 5 units and now,
the operation. (s)he has one empty tray. Therefore, the solution has been
• Principle of self-learning and self-control: there is the found: 95 (Fig. 1D).
possibility of controlling the intermediate steps carried
out, the student verifies the accuracy of what he is doing.
The following operations can be performed by means of
ABN: addition, subtraction, multiplication, and division, but
also “new operations”: double addition, double subtraction,
and addition-subtraction.
A. Characteristics of an operation in ABN
The order by which an operation is started is irrelevant
in this method, ABN does not follow the strict order of the
magnitudes that constitute the number [12]. In addition, the
numbers can be decomposed into smaller parts to facilitate
the operation.
The possibility of operating more than one order of mag-
nitude simultaneously exists. The calculations in ABN can be
done recursively in one direction or another, depending on the
strategy of each student.
The students work a lot the “friends of 10” in this method,
later extended to “friends of 100, 1000. . . ”. It consists of filling
in quantities until a higher order is reached., e.g., having 10
units and replacing them with 1 ten.
Fig. 1. Steps to calculate the addition of “57 + 38” using chopsticks and a
Thus, two actions appear when performing operations: grid in the ABN method.
“group” and “ungroup”. “Group” is mandatory if an order of
magnitude is completed when performing an addition, e.g.,
10 is reached. The set of 10 is grouped to a higher order of C. Subtraction
magnitude. The terms of the operation are represented in the same way
The action of “ungroup” appears with the subtractions. If in the case of subtraction. But in this case, the toothpicks that
there is an order of magnitude in one term of the operation are removed from one tray are removed in the other as well.
and not in the other one, a higher order is opened in the last The operation ends when one tray is empty, just as in the
one, e.g., 1 ten is decomposed into 10 units. addition.
Fig. 2. Example in the design prototype showing an intermediate step of the add-subtraction 123 + 142 – 134
III. S MART DEVICE REQUIREMENTS ten LEDs that light up in blue to represent the units, a
The smart device is designed as a tool to make it easier for second strip of ten LEDs that light up in red representing
students to learn basic operations. The essence of ABN cannot the tens, and a last strip of ten LEDs that light up in
be lost. Therefore, tangible and motivating materials must be green for the hundreds.
• Each grid has its own LCD display on which the number
used to capture the student’s attention.
The device must fulfil the following requirements: in decimal will be represented by the LEDs on the tray
at any given time (Fig. 2.D).
• To allow the student to perform the following operations:
• Each grid has two buttons at the bottom (Fig. 2.E). One
addition, subtraction, double addition, double subtraction,
is for “remove” and the other is for “put”.
and addition-subtraction.
• The LCD screen of the tray on which something is being
• To help the student with the learning of the operations
modified will remain illuminated (Fig. 2.F). This way the
• The user must be able to perform two- and three-digit
student knows where (s)he is at any moment. All screens
operations.
turn on again at the end of the step.
• The number must be physically and visually represented.
• It also has a button to indicate when the initial operands
• The complete interaction of the student with the device
are being set and when the operation begins (Fig. 2.H).
must be able to be saved.
• In the centre is the “control unit” (Fig. 2.I). It consists
• The teacher will be able to keep track of the operations
of a button for the units, another one for the tens an a
performed by his students.
third one for the hundreds. Once the student has decided
IV. P HYSICAL DESIGN whether (s)he wants to put or remove from each of the
grids, these buttons are the ones the child has to press
The physical smart device has been designed according to
to indicate whether (s)he wants to modify units, tens or
the following requirements.
hundreds.
• The following operations can be performed: addition,
• There is a button (Fig. 2.J) to indicate the end of the step.
subtraction, double addition, double subtraction, and It must be pressed when the student finishes modifying
addition-subtraction. A strip of 5 LEDs will be used to the quantities. The LEDs of the grids appear in yellow
choose which of the operations to perform (Fig. 2.A). until the moment of verifying if the changes made are
• A different LED will turn on when pressing the button
correct (Fig. 2.G). If the button (Fig. 2.J) is pressed and
(Fig. 2.A) to determine the chosen operation. The sym- the movements made are right, the new quantities of
bols next to each LED represent the operations that can LEDs on are established with their corresponding colour.
be performed (see Table I). The button will move the If there is an error, the LEDs remain on yellow.
LED that is illuminated one position on each pulse. The • The concepts of “group” and “ungroup” are important
chosen operation will be the one that has the lit LED in ABN, so two buttons will also be used to perform
fixed. these actions (Fig. 2.K). An order of magnitude has been
• The traditional grid is equivalent to a set of three LED
completed when one of the LED strips is fully lit. Thus,
strips (Fig. 2.B) and each LED represents a toothpick a LED located between these buttons will light up to
(Fig. 2.C). Therefore, each grid is composed of a strip of
alert the user of this. If the “group” button is pressed,
the ten LEDs will turn off and one LED of the higher
order will light up. This action is mandatory, that is, the
student cannot have a complete order and not “group” to
a higher one. However, “ungroup” is a voluntary action
and the user is not notified. The “ungroup” button should
be pressed if the user needs it.
• Six LEDs will be placed on the top of the grids to indicate
progress (Fig. 2.L). It has been considered important
because in this way the teacher will be able to identify
immediately if there are any students who need help. The
progress is given by the number of steps in which the
operation is made. For each step, a progress light turns
on; if more steps than the “optimal” ones are performed,it
lights up in a different color. The approach for this
depends on the operation to be performed.
TABLE I
S YMBOLS AND THEIR O PERATION
Symbol Operation
+ Addition
- Subtraction
++ Double addition
-- Double subtraction
+- Addition-subtraction
V. I MPLEMENTATION
The device has been implemented in compliance with all
requirements, as shown in Fig. 3.
Fig. 4. Flowchart with the addition operation
A. Addition example
A student should perform the addition 51 + 183. In order
to add two numbers, the student should follow the flowchart
shown in Fig. 4. One possible solution implemented by the
student could be as follows:
• First, as shown in Fig. 4, the terms of the operation are
established. The child puts 5 tens, and 1 unit in the middle
tray; and 1 hundred, 8 tens, and 3 units in the right
one. The device changes to the “operate” mode once the
operands are correct and it will continue in that mode
while the remainder of the operation is different from 0.
• Then, the student decides to remove one unit from tray 2.
Now there is only one option, (s)he must put that unit in
tray 1. As the step is correct, the terms of the operation
are updated to 50 + 184 (see Fig. 3).
Fig. 3. Example of an intermediate step of the addition 51 + 183 by means
• The student realizes that (s)he needs two tens to be able
of the smart ABN device. In the Figure, the number 50 is represented in the
middle tray, and the number 184 is shown in the right tray. to group one hundred in tray 1, so for the next step the
child decides to remove two tens from tray 2 and put
An example of how the device works in an addition accord- them in tray 1. (S)he presses the group button, so the
ing to its flowchart is shown after its implementation to help strip of ten LEDs turns off and a hundred LED is lit up.
understand it better (see Fig. 4.). • Finally, the student removes the remaining three tens from
tray 2 and (s)he puts them into tray 1. The step is correct, LEDs the units. This colour code is because teachers working
and the remainder is 0. Thus, the addition finishes (see with ABN identify with red rubber the tens and green one the
Fig. 4). hundreds.
The use of this smart device contributes to the development
B. Addition-subtraction example
of the capacity of abstraction of the student, as it is a further
Addition-subtraction is the union of an addition and a step in the transition to the representation of the different
subtraction. If the operation is supposed to be A + B - C, orders of units in the decimal numbering system. At the same
A + B must be greater than C for the operation to be properly time, it simplifies the algorithm of the basic operations when
performed. replacing the chopsticks with the use of the smart device,
The following steps can be performed once the operation is maintaining the essence of the manipulative idea of ”group”
in progress: or ”ungroup” the different orders of magnitude.
• If the student starts removing from tray 1, that is from C, Once the smart device has been developed, the aim is to
(s)he will be able to remove from A, B or remove from carry out an evaluation of the device with real users in a school
A and B the sum of what (s)he has removed from C. that uses the ABN method in its classrooms. The intention is
• If the student starts removing from B, the only option is to use this smart device in the classes for several weeks and
to put that amount in A; since these are the terms that with different Primary Education courses.
are between the “+” sign. In addition, a web application is going to be developed as a
• If the student starts removing from tray A, the only option complement of the smart device to help the teacher. To do this,
is to continue adding the corresponding amount in B. the application requires that the teachers can see the interaction
This operation is best understood by stating a problem. For of the student in real time, and they can send operations
example, Olivia has 32 C and her uncle gives her 44 C . She directly to each student’s device. Also, the web application
spends 28 C . How much money does Olivia have left? requires sequencing the contents by levels, saving the record
• First, the student sets the operands on the device: 32 + of each class, and saving all the interaction of the students to
44 – 28. observe their evolution.
• (S)he decides that Olivia spends 20 C and (s)he removes
ACKNOWLEDGMENT
2 tens from C and B. Now the operands are 32 + 24 - 8.
• Then, the child decides that Olivia spends 6C and (s)he The authors would like to acknowledge the support from
removes 6 units from A. Now, the student must remove FEDER/Ministerio de Ciencia, Innovación y Universidades
that amount from the other trays. In this case, (s)he – Agencia Estatal de Investigación through Project Smartlet
decides to remove 4 units from B and 2 from C. The (TIN2017-85179-C3-1-R). This project has also received par-
terms of the operation are: 30 + 20 - 2. tial support from the eMadrid Network, which is funded by the
• The student only has 2 units in one tray. Thus, (s)he Madrid Regional Government (Comunidad de Madrid) with
presses the “ungroup” button. The smart device will open grant No. P2018/TCS-4307.
the ten of the term that is greater, in this case, 30. Now, It has also received partial support from the European
a red LED turns off and a strip of 10 blue LEDs lights Commission through Erasmus+ projects LALA (586120-EPP-
up. The student can remove the 2 units from A and C. 1-2017-1-ES-EPPKA2-CBHE-JP), InnovaT (598758-EPP-1-
The terms are now: 28 + 20 - 0. 2018-1-AT-EPPKA2- CBHE-JP) and PROF-XXI (609767-
• The child already knows that Olivia has already spent the EPP-1-2019-1- ES-EPPKA2-CBHE-JP). This publication re-
28 C , but (s)he does not know what Olivia has in total. flects the views only of the authors and funders cannot be held
Therefore, (s)he decides to remove the 2 tens from B and responsible for any use which may be made of the information
put them in C. Finally, the smart device has the terms: contained therein.
48 + 0 – 0 and the student knows that Olivia has 48 C
R EFERENCES
left.
[1] Kushwaha, R. C., & Singhal, A. (2017). Impact study of teaching math-
VI. C ONCLUSIONS AND FUTURE WORK ematics using ICT enabled learning. International Journal of Advanced
Research in Computer Science, 8, 333-336
The ABN method works on acquiring the sense of the [2] Schleicher, A. (2019). PISA 2018: Insights and Interpretations. OECD
number from the earliest ages to understand the meaning of Publishing.
basic operations. In this process, it is essential to use a model [3] Dell’Erba, M. (2019). Preparing Students for Learning, Work and Life
through STEAM Education. Policy Brief. Education Commission of the
that helps the students to make tangible something as abstract States.
as the orders of units established by the decimal numbering [4] Klar, J. (2018). Mentoring teachers in STEAM improves likelihood of
system. Using chopsticks, the student is able to manipulate the application. (Master thesis) University of Wisconsin-Platteville, Plat-
teville, Wisconsin.
concepts that the decimal system requires. Once the student [5] Cerda, G., Aragón, E., Pérez, C., Navarro, J. I., & Aguilar, M. (2018).
acquires greater degree of abstraction, the toothpicks can be The Open Algorithm Based on Numbers (ABN) method: an effective
replaced by models with different figurative content. The smart instructional approach to domain-specific precursors of arithmetic de-
velopment. Frontiers in psychology, 9, 1811.
device presented in this paper is within this model. The green [6] Martı́nez, J. & Sánchez, C. (2011). Desarrollo y mejora de la inteligencia
LEDs represent the hundreds, red LEDs the tens and blue matemática en Educación Infantil. Madrid; Wolters Kluwer.
[7] Montero, J. M. (1995). Los problemas aritméticos elementales verbales
de una etapa, desde el punto de vista de las categorı́as semánticas, en
los cursos 3º, 4º y 5º de EGB/Primaria (Doctoral dissertation, UNED.
Universidad Nacional de Educación a Distancia (España)).
[8] Montero, J. M. (2001). Los efectos no deseados (y devastadores) de
los métodos tradicionales de aprendizaje de la numeración y de los
algoritmos de las cuatro operaciones básicas. Epsilon: Revista de la
Sociedad Andaluza de Educación Matemática “Thales”, (49), 13-26.
[9] Freudenthal, H. (1986). Didactical phenomenology of mathematical
structures (Vol. 1). Springer Science & Business Media.
[10] Montero, J. M. (2011). El método de cálculo abierto basado en números
(ABN) como alternativa de futuro respecto a los métodos tradicionales
cerrados basados en cifras (CBC). Bordón. Revista de pedagogı́a, 63(4),
95-110.
[11] Aragón, E., Canto, M. C., Marchena, E., Navarro, J., & Aguilar, M.
(2017). Cognitive profile in learning mathematics with open calculation
based on numbers (ABN) algorithm. Revista de Psicodidáctica, 22(1).
[12] Martı́nez, J. & Sánchez, C. (2019). Enriquecimiento de los aprendizajes
matemáticos en Infantil y Primaria con el Método ABN. Madrid;
Pirámide.