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. 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