Spanish primary school students’ engagement with a non-traditional method when adding and subtracting: ritualised versus exploratory participation

1 Spanish primary school students’ engagement with a non-traditional method when adding and subtracting: Ritualised versus exploratory participation Inés Gallego-Sáncheza*, Verónica Martín-Molinaa, Isabel Caro-Torróa and José María Gavilán-Izquierdoa aDepartamento de Didáctica de las Matemáticas, Universidad de Sevilla, Sevilla, Spain Correspondence details: Departamento de Didáctica de las Matemáticas, Facultad de Ciencias de la Educación, Universidad de Sevilla, Calle Pirotecnia, s/n, 41013, Sevilla, Spain. *Email: [email protected] ORCIDs Inés Gallego-Sánchez https://orcid.org/0000-0002-1088-3172 Verónica Martín-Molina http://orcid.org/0000-0002-6359-5246 Isabel Caro-Torró https://orcid.org/0000-0002-1539-7333 José María Gavilán-Izquierdo https://orcid.org/0000-0002-3369-5377 Short Biographical Notes of the authors: Inés Gallego-Sánchez is an assistant professor of mathematics education at the Universidad de Sevilla, Spain. She is a mathematician and has a Ph.D. in applied mathematics. She is currently researching mathematics teaching and learning through the lens of the commognitive framework, as well as graph theory education through the van Hiele framework. Verónica Martín-Molina is an associate professor of mathematics education at the Universidad de Sevilla, Spain. She is a mathematician and has a Ph.D. in mathematics. She is currently researching mathematics learning through the lens of the commognitive framework, as well as mathematics teachers’ practices and mathematicians’ research practices. 2 Isabel Caro-Torró is a primary school English teacher. After obtaining her undergraduate degree in primary education at the Universidad de Sevilla, she studied a master in bilingual education and she is now investigating the ABN method with primary school kids. José María Gavilán-Izquierdo is an associate professor of mathematics education at the Universidad de Sevilla, Spain. He is a mathematician and has a Ph.D. in mathematics education. He is currently researching mathematics learning through the lens of the commognitive framework, as well as mathematics teachers’ practices and mathematicians’ research practices. 3 Spanish primary school students’ engagement with a non-traditional method when adding and subtracting: Ritualised versus exploratory participation Our work investigated how six primary school students used a non-traditional method for adding and subtracting: the ABN method, a Spanish acronym for Open (method) Based on Numbers. Commognitive theory (Sfard 2008) was employed to study the students’ mathematical routines. In particular, we studied whether their routines were exploratory or ritualised and found that four students had a ritualised use of the ABN method, while two others showed some signs of incipient exploratory use (manifested as an increase in flexibility, applicability, performer’s agentivity, and substantiability). The results show some of the problems that students have when applying this method. Keywords: ABN method; algorithms; commognition; discourse; primary education Introduction Algorithms play a relevant role in mathematics since they are linked to problem solving (Möller and Collignon 2019; Rasmussen et al. 2005). The first encounter with algorithms usually occurs in primary education, where an important part of the curriculum consists of teaching algorithms to add, subtract, multiply, and divide (Morrow and Kenney 1998). These algorithms are tools to solve problems in daily life and they constitute the first step towards awareness regarding the mathematical practice of building algorithms (algorithmatising, in terms of Rasmussen et al. 2005). Despite their importance in the curriculum, algorithms are sometimes taught inadequately. For instance, Plunkett (1979) pointed out that conventional (traditional) algorithms are frequently learnt by heart without knowing the mathematical principles and properties that support them. Years later, Fuson and Briars (1990) stated that many children carry out algorithms correctly, but they cannot explain why they work. Later, 4 Thomas (2008) stated that the step-by-step nature of algorithms leaves no room for deviation and thus cannot foster versatile thinking. More recently, Laudano, Tortoriello, and Vincenzi (2020) added that, if teachers teach only the purely procedural aspects of algorithms, then they run the risk of offering a reductionist and distorted view of mathematics, which would foster disaffection towards this discipline. The National Council of Teachers of Mathematics (NCTM 2000) highlighted the need to employ complementary or alternative methods to traditional algorithms, and there are several studies that state that employing these methods is more beneficial than using the standard algorithms. For example, Ulu and Özdemir (2018) concluded that students who used algorithmic strategies to estimate results in arithmetic operations had less approximate results than those who used other strategies, which reaffirmed the thesis of Kamii and Dominick (1997). Other mathematics education researchers (e.g., Ashlock 2010; Carpenter et al. 1998; Thompson 1998) found that invented methods improved self-confidence, favoured critical assessment, and produced a greater development of number sense. Furthermore, Nemeth et al. (2019) compared the learning that occurred when there were different strategies to choose to solve an arithmetic problem with the learning that occurred when that variety was unavailable. In the first case, there were more gains in flexibility (use of multiple strategies) and adaptivity (making appropriate strategy choices), in the sense of Verschaffel et al. (2009). Indeed, several authors highlight flexibility and adaptivity in the use of strategies as desirable aspects in the teaching and learning of elementary school arithmetic (Baroody 2003; Carpenter et al. 1998; NCTM 2000; Torbeyns and Verschaffel 2016; Ulu and Özdemir 2018; Verschaffel et al. 2009). In Spain, the primary education stage (6-12 years) is compulsory. It is preceded by childhood education (0-6 years), which is voluntary, and followed by secondary 5 education (12-16 years), which is the last compulsory educational stage. The national and regional curricula for primary education contain a “numbers and operations” section which emphasises that the construction of the different classes of numbers and the decimal system should be contextualized. Moreover, this construction should be done with the help of manipulatives in the first stages and progressively evolve into abstract thinking, in parallel and related to arithmetic operations. The curricula also mention that the students should be able to do operations with pencil and paper algorithms, with the calculator and mentally, choosing the best strategy in each case. One of the alternative methods to the use of conventional algorithms that is gaining popularity in Spain is the ‘Open (method) Based on Numbers’ proposed by Martínez Montero (2011), called ABN for the Spanish ‘Abierto Basado en Números’. According to Martínez Montero (2011), students who master the ABN method learn to calculate mentally at a high level, solve problems more effectively, and can explain what they are doing. Furthermore, he indicates that this method is also beneficial in the long term, since he found that students who used it in primary education tended to have fewer conceptual errors in secondary education. There have been several studies focusing on examining the differences in mathematical competence or abilities between primary students who were taught only the ABN method and students who were taught only the traditional algorithms (Adamuz-Povedano and Bracho-López 2014; Aragón-Mendizábal et al. 2017; BrachoLópez et al. 2014; Cerda et al. 2018; Martínez Montero 2011; Mendoza Velazco et al. 2020). The results of all these studies found that the first showed a higher level of mathematical competence. However, none of these works investigated qualitatively how the students use the ABN method or their explanations of why they carry out each step. 6 Our objective is to explore, through the commognitive framework, how primary school students have learned to use the ABN method. The article is organised as follows: we first present the ABN method and then introduce the theoretical framework used to analyse our data. Next, we present the methodology of the study, the results, and the section of conclusions and discussion. ABN method The ABN method was devised by the teacher and researcher Jaime Martínez Montero (2011) as an alternative way of solving problems without recurring to the traditional closed methods based on digits. The Spanish acronym ABN stands for Open (method) Based on Numbers. It is called open because there is free choice in each step of the arithmetic operation and based on numbers because, in each step, complete numbers are manipulated (and not just digits as in the case of traditional algorithms). Therefore, the ABN method is not an algorithm in the traditional sense, which Thomas (2020) defines as a step-by-step set of instructions arranged in logical order to solve a specific task. Since ABN is an open method, Martínez Montero (2011) claims that it adapts to different learning paces because each student can carry out the steps they need and in the way that best suits them. Moreover, the ABN method is always introduced in the context of solving word problems derived from real situations, which allows the student to associate the arithmetic operation with the action or relation expressed in the word problem. This work focuses on problems that can be solved by adding or subtracting. Several examples of addition and subtraction by the ABN method are shown below. Since the ABN method is not an algorithm, there are several ways to perform an addition. As an example, we show how to compute 645+178 in two ways. The smallest addend (178) can be divided into parts (as many as the student wishes) to be added to 7 645 in successive steps. This method uses the fact that 645+178=(645+x)+(178-x) for any number x. The calculations are usually written in a grid with three columns (normally without labels), like the one in Table 1. The first column indicates the quantity that is going to be added, the second one has the partial sums, and the third one the quantities that remain to be added. When there is a zero in the third column, the last number in the second column is the sum. Table 1. Example of addition using ABN. The number in bold indicates the final result. 645+178 100 745 78 5 750 73 50 800 23 23 823 0 The example in Table 1 took four steps. In the first step, 100 has been added, leaving as a partial result in the second column 745 (= 645+100) and in the third column 78 (= 178-100). This removed all the hundreds from the second addend. With the sums of the second and third steps (745+5=750 and 750+50=800), there is 800 as a partial result in the second column, and then it is not difficult to add the remaining 23, which amounts to a total of 823. When using the above method, each student can decide which quantities to add in each step and in what order. Therefore, there are other possible grids, each one with as many rows as steps are needed. For instance, Table 2 shows how to add 645+178 in three steps instead of the four steps of Table 1. The difference is that, in the grid of Table 2, the second and third steps of the previous grid have been merged in a single 8 step. This grid would be appropriate for more advanced students since some of its steps would be more difficult. Table 2. Another example of addition using ABN. The number in bold indicates the final result. There are also various methods for performing subtraction. In the following, we show how to compute 335-246 in several ways. The first way would be by successive subtractions, that is, by subtracting the same quantities from both minuend and subtrahend until one of them disappears. The remaining number would be the result. This method uses the fact that 335-246=(335-x)-(246-x) for any number x. The calculations are also written in a grid with three columns, like the one in Table 3. The first column in this case indicates the quantity that is going to be subtracted from both minuend and subtrahend, and the second and third columns have the minuend and subtrahend that are becoming smaller and smaller until the subtrahend is zero. When that happens, the last minuend is the difference. Table 3. Example of subtraction using ABN. The number in bold indicates the final result. 645+178 100 745 78 55 800 23 23 823 0 9 The example in Table 3 took five steps. In the first step, 200 has been subtracted, leaving as a partial result in the second column 135 (= 335-200) and in the third column 46 (= 246-200). This removed all the hundreds from the subtrahend. In the second and third steps, the tens and units are removed from the minuend, leading to the subtraction 100-11. To finish, a ten is first removed and finally just one. Each student can choose the quantities to subtract from the minuend and subtrahend. For instance, Table 4 shows how to do the subtraction 335-246 in four steps instead of the five steps of Table 3. The difference with the grid of Table 3 is that the second and third steps of that grid have been merged in one step. Students can also reorder the steps depending on the order in which they subtract units, tens, hundreds, etc. Table 4. Another example of subtraction using ABN. The number in bold indicates the final result. 335 - 246 200 135 46 35 100 11 335 - 246 200 135 46 30 105 16 5 100 11 10 90 1 1 89 0 16 The proposed problems are presented below. The names of the schools, towns, and the game console are all blinded. (1) There are 673 students in SCHOOL A and 594 students in SCHOOL B. They want to go to the San Pedro Theatre together. How many students will go to the theatre? (2) The distance from CITY 1 to CITY 2 is 996 kilometres and the distance from CITY 1 to CITY 3 is 321 kilometres. How many more kilometres are there from CITY 1 to CITY 2 than from CITY 1 to CITY 3? (3) Our refrigerator broke down and we had to buy a new one. If my father had saved 734 euros to buy a new refrigerator and it finally cost only 372 euros, how much money do we have left? (4) There were 643 trees in the forest and 268 have been cut down. How many trees are left in the forest? (5) My cousin Maria and I are saving to buy the new GAME CONSOLE because it is on sale for 537 euros. Maria has 438 euros and I have 277 euros. How much money do we have together? Are we short of money to buy the CONSOLE, or do we have any money left? How much? The first four problems can be solved with a single arithmetic operation (the first with a sum and the second, third and fourth problems with a subtraction). The fifth one needs an addition and a subtraction. Data sources and data An audio recorder and a video camera were used for data collection. The participants’ parents gave their consent to record the students. The audio recordings were completely transcribed by one of the researchers, and some still images from the videos were 17 selected. To preserve the identities of the students, for this work, those images in which only the grid or the manipulatives appear were selected. Students’ written responses were also collected. To facilitate reading, clarifications about actions that can be seen in the video have been included in the transcripts in square brackets. Analysis procedure The data was analysed individually by the investigators, then joint sessions were held to discuss discrepancies until a consensus was reached. Specifically, the focus was on identifying the routines that each student used to solve the problems. Those routines were inferred by examining the transcripts and images of the students employing visual mediators (grids, hundreds charts, toothpicks, etc.) to find recurring patterns. Some students solved some of the problems mentally and remained silent throughout the resolution process; however, others addressed the teacher or the other student and used visual mediators, which helped us determine how they performed specific steps that are visible on the grids. We then searched for characteristics of the use of routines (flexibility, applicability, performer’s agentivity, or substantiability) that may indicate de-ritualisation. Results There were noticeable differences among the routines that the six students used to solve the proposed problems. While four of them (Anna, Daisy, Elsa, and Flora) seemed to engage with the ABN method as if performing a ritual, two others (Beth and Carla) appeared to have partially undergone a process of de-ritualisation because they had more flexibility when using a routine, they were able to choose the most appropriate routine without the teacher’s help and they could explain the steps of their procedure. 18 To better illustrate these differences in performance among the students, we will highlight the cases of two students, one whose engagement with the method took the form of a ritual (Flora) and one whose explorations showed that she was in the process of de-ritualising the use of the ABN method (Carla). Flora’s use of the ABN method was ritualised Flora always tried to use the same procedure when doing the computations: decomposing the smallest number into units, tens, and hundreds and then adding or subtracting those units, tens, and hundreds. For instance, when solving the fourth problem, she decided to do the subtraction 643-268 by decomposing 268 as 200+60+8 and then subtracting those three smaller numbers in three different steps, as can be seen in Figure 2. Figure 2. Flora’s grid for problem 4. This created difficulties for her in some of the partial subtractions. For example, in the second step, when subtracting 60 from 443, she hesitated when she had to move from the 400s to the 300s: Flora (F): Now minus 60. 43, 33, 23… 83. Preservice teacher (PT): Then it is... F: 683… no, 383. 19 In the first line, we see that she counted backwards by 10 from 443. To facilitate this, she decided to operate with only the units and tens, ignoring the hundreds. Instead of saying ‘443, 433, 423, …, 383’, she first said ‘43, 33, 23…83’, which caused her problems later when she had to ‘recover’ the hundreds digit. Indeed, she first said ‘683’ instead of ‘383’. She had lost track of the complete number, which contradicts one of the principles of the ABN method. Moreover, subtracting 60 in the second step is not the most efficient choice for beginners in mental arithmetic because this forced her to change hundreds. Flora repeated this procedure when solving problems 2, 3, and 5. It is worth mentioning that she confused a unit with a ten in problem 2 (she says that 676-1 = 666), although she used the hundreds chart as a visual mediator on several occasions. We can deduce that her use of the ABN method was ritualised because she used the same procedures in all the problems, even when they were not efficient. Therefore, her use of routines was not flexible. The only problem she solved differently was the first one, where she used a different routine due to the suggestion of the teacher. In that problem, which can be solved by adding 673 and 594, she began the same way, by decomposing 594 as 500+90+4. However, in the second step, when trying to add 90+1173, this student said: F: 11 plus 9 are 19, aren’t they? [She wants to add 11, which are hundreds in the number 1173, and 9, which are tens in 93] The teacher then suggested that Flora use the toothpicks as a visual mediator to represent the addends. However, Flora was unable to add those numbers in this way, even using the hundreds chart (Figure 3). The teacher finally proposed: PT: Why don’t you take smaller numbers instead of 90, for example, 20 by 20, or how much is it from 73 to one hundred? 20 F: I don’t know… PT: Let’s take a smaller number. How much is it to one hundred? Flora also had some problems with this step (both with 1173+27 and with 9427), but with the help of the teacher and the hundreds chart managed to get the result (see Figure 4). Figure 3. Flora using the hundreds chart. Figure 4. Flora’s grid for problem 1. More evidence has been found that Flora had problems with the change of hundreds. For example, in the second step of the third problem, she had to use the hundreds chart for 434-70. She counted backward by tens: 434, 424, … but when she reached 404, she again needed the help of the teacher. Moreover, in the fifth problem, to do 638+70, she needed again the hundreds chart. In the same problem, Flora said that 21 215-30 equals 385 and the teacher had to correct her, and also when she said that 185-7 equals 778 instead of 178. In summary, Flora lost track of what she was doing when computing with large numbers. Flora did not seem to have taken any steps in the process of de-ritualisation because she showed no flexibility (she always tried to employ the same procedure to solve the tasks), she had little agentivity (since she depended on the teacher to decide what to do and usually needed help with the computations), and she did not know how to substantiate her choices. For example, when solving the third problem, she did not even know how to justify her choice of operation: F: Add. PT: Tell me why. F: Subtract, I don’t know. PT: But tell me why. I don’t mean that it’s wrong. F: A subtraction, it is a subtraction, I don’t know why. Three other students (Anna, Daisy, and Elsa) engaged with the ABN method in a similar way to Flora. For instance, on some occasions, those students confused the columns of the grid in a step of the method, or they did a sum when the correct operation was a subtraction, or vice versa. This way of doing operations on a grid that has no indications of what to do in each column led some of the students to lose track of what they were doing, that is, they did not make sense of their partial additions or subtractions. In addition, there were also situations in which those students chose partial operations that they were not able to perform without the teacher’s help because they worked with very large numbers instead of dividing them into smaller parts. In conclusion, their (ritualised) use of the ABN method was not advantageous for them, since they had not learned to use it exploratively to unleash its full potential. 22 Carla sometimes took advantage of the potential of the ABN method and adapted the procedure to the problem she was solving None of the students showed signs of having completely de-ritualised their routines when using the ABN method, but both Beth and Carla had taken steps in that direction. In fact, Carla sometimes tried to adapt her procedures to the different calculations she had to make instead of always applying the same sequence of instructions. Sometimes, she seemed to show signs of flexibility in her use of the ABN method, although it led her to hesitate when writing down some of the numbers (such as 1094) or needed the help of the teacher when doing some sums (such as 1094+100). However, in other computations, she made mistakes that made us question if she was engaging with the ABN method as if it were a ritual. In the first problem, which can be solved by adding 673 and 574, the arrow that Carla drew indicates that she added 673+574, rather than adding 574+673 as the rest of the students did (Figure 5). Therefore, she swapped two columns of the grid. However, when adding 673+574, she began by adding the same number to both 673 and 574, instead of adding a number to one of them and subtracting it from the other. When she became aware of her mistake, Carla erased her work until then and started again. This leaves some doubts about whether Carla’s routine was process-oriented rather than outcome-oriented. After starting the computation for the second time, Carla decided to add 673 by decomposing that number as 500+100+73. She did not explain why she decided on that particular decomposition, which gave her problems in the first two steps. In the first one, when adding 500+594, she correctly said that it is 1094, but she consulted the teacher to check if she had written it down correctly. In the second one, when adding 1094 and 100, she again asked the teacher for help: 23 Carla (C): [Preservice teacher’s name], if I pass 100, is it two thousand? PT: No. If this is one thousand plus hundred, it is one thousand and… if these are hundreds and these are thousands, how many are one thousand plus one hundred? C: One thousand one hundred? PT: Very good! […] PT: Be careful, the [number] 4 is missing. Figure 5. Carla’s grid for problem 1. Carla forgot that she was adding 1094+100 instead of only 1000+100. When she wrote 1100 instead of 1194, the teacher warned her about the missing ‘4’ but not about the missing ‘9’. Carla’s answer may have happened because the ABN method advocates for computing using the complete numbers, which is more difficult for students when the numbers are big for them. In the rest of the problems, Carla did not ask for help again and showed signs of using more advanced procedures. Indeed, in the third and fourth problems, she did the subtractions by first subtracting the maximum number of units, tens, and hundreds that she could from both numbers without regrouping. In the third problem, she subtracted 332, thus removing all the units and hundreds from 372 and all the tens from 734, and then subtracted the rest of the subtrahend (see Figure 6). That first step is certainly very 24 efficient, although not the second, which forced her to do 402-40. When Carla realised that her partner, Daisy, had difficulty with the partial subtraction 402-40, she tried to help her by telling her how she could do it with the aid of her fingers: Daisy (D): Yes, I know how to do it, but with the [hundreds] chart. PT: Come on, think, 402 minus 10 is... D: three hundred... and two PT: Remember that they are tens, not hundreds. Recall the hundreds chart. C: If you take away ten from 400, how much is it? Hold up 10 fingers of your hand and think that there are 10 of them. This could be characterised as a further step in Carla’s agentivity, since she not only makes her own decisions, but can help her classmate by adding to the teacher’s explanation. In the fourth problem, Carla first subtracted all the hundreds from 268 and all the units and tens from 643 (Figure 7), thus showing applicability in that routine. The difficulty came when performing some of the subtractions. In Figure 7, in the first row of the grid, she was subtracting 243 from both the minuend and subtrahend, so the other two numbers should be 400 and 25, instead of 40 and 225. Continuing with the numbers that she had, instead of subtracting 40 in one step (which would mean subtracting 22540), she decided to first subtract 20 and then the other 20. Figure 6. Carla’s grid to solve problem 3. Figure 7. Carla’s grid to solve problem 4. 25 This last choice and the choices she made in the first step of both the third and fourth problems show that Carla was able to adapt her procedure to the numbers that she had to work with so that the subtractions did not force her to regroup. Carla’s solution of problem 5 was again a mixture of explorations and rituals. She began by incorrectly identifying the arithmetic operations that she needed to perform. She said that she needed to do an ‘addition-subtraction’, which she proceeded to do on a single grid (Figure 8). Figure 8. Carla’s grid to solve problem 5. In that grid, Carla wrote in the first column the number that she was adding or subtracting (depending on the operation she was doing) and in the other three columns she wrote the numbers that remained after operating on 537, 438, and 277. In the first row of the grid, Carla again showed that she could adapt her procedure to the situation by removing all the units and tens from the minuend and all the hundreds and tens from 32 Mathematical Thinking and Learning 7 (1): 51–73. doi:10.1207/s15327833mtl0701_4. Sfard, A. 2008. Thinking as communicating: Human development, the growth of discourses, and mathematizing. New York: Cambridge University Press. doi: 10.1017/CBO9780511499944. Thompson, I. 1998. “Written subtraction algorithms.” Education 3-13 26 (3): 55–58. doi:10.1080/03004279885200331. Thomas, M. O. 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