The emergent models in linear algebra: an example with spanning set and span

The emergent models in linear algebra: an example with spanning set and span Abstract A continuing challenge in mathematics education at the university level is how to create learning environments that give students the opportunity to progress from their informal ways of reasoning to more formal ways of reasoning. This article shows how the emergent models heuristic (Gravemeijer, 1999, Math. Think. Learn, 1, 155–177) provides this opportunity to first-year university students, specifically for them to construct the Linear Algebra concepts of spanning set and span. For this, a design-based research methodology was used. A hypothetical learning trajectory (HLT) was applied, created within the framework of this heuristic. The results provide evidence that the emergent models heuristic support the construction of these concepts of Linear Algebra. Moreover, useful tips are given for using this heuristic to construct mathematical concepts. 1. Introduction For some time, a growing amount of literature in mathematics education has indicated that students have difficulties with advanced mathematical thinking. This is due, on the one hand, to the complexity and abstract nature of advanced mathematical topics, but on the other hand, to the difficulties that they encounter due to the shift from less formal to more formal modes of reasoning. However, despite this, the systematic descriptions of the means by which and the stages through which students advance from informal reasoning to formal reasoning are still in initial research (Dawkins, 2012). These investigations include those that use the levels of activity described by Gravemeijer (1999) to detail the transition of students from an informal mathematical reasoning of their activity towards a more formal type of mathematical reasoning. This research has been carried out in the university courses of: differential equations (Rasmussen & Blumenfeld, 2007), geometry (Zandieh & Rasmussen, 2010), and Linear Algebra (Wawro et al., 2013). In their study, Rasmussen and Blumenfeld (2007) exemplify the four levels of activity of the emergent models as a framework for the analysis of the mathematical activity of the students observed for the study in constructing the solutions to systems of two differential equations. They conclude that their work contributes to the emergent models heuristic, as they ‘demonstrated how analytic expressions can underpin the entire model-of/model-for transition’ (p. 208). This is due to the fact that, in previous research, most of which has been done at the primary and secondary (school) levels, the model-of phase (situational and referential level) has not involved analytical expressions; instead, it usually includes concrete tools, graphs or diagrams. Only later, in the model-for phase (general and formal level) do students work with formal and symbolic expressions. In addition, Rasmussen & Blumenfeld (2007) consider that their study in the course of differential equations offers a prototypical example of a work inspired by the Realistic Mathematics Education (RME) and the emergent models heuristic at the undergraduate level to support students in creating new mathematical realities. They also note that these ideas about ‘student thinking can be useful for teachers in their difficult task of making sense of what students say and, do for being proactive in supporting their intellectual growth’ (p. 208). Wawro et al. (2013) describe the reasoning used by students in constructing the concepts of span, linear independence and linear dependence through the four levels of activity of the emergent models. These authors indicate how the emergent models heuristic reports the creation of an instructional sequence for these concepts of Linear Algebra. Moreover, they describe how the activity levels of this heuristic are manifested in the sequence. They give evidence of how students give indications of being in the first three levels of activity. However, they argue that the last level of activity (the formal level) does not occur during the sequence, but rather is observed in other tasks unrelated to this sequence. In addition, they indicate that the refinement of their instructional sequence, ‘and its framing within the four levels was made possible by the cyclical process of design research’ (p. 911). Our research is grounded in the emergent models heuristic. In particular, our research describes how student reasoning has progressed through each of the four levels of activity in relationship to the tasks of the hypothetical learning trajectory (HLT) that students worked on. This is the case because we want to contribute to this design heuristic that, according to Dawkins (2012), needs further exploration due to the fact that many mathematical topics do not immediately suggest contexts in which students can begin their learning. Among these mathematical topics are several contents of the Linear Algebra course, including the concepts of spanning set and span, which we considered in this study. These concepts are chosen because of their relationship with important contents of this course, such as: base and dimension (Stewart & Thomas, 2010). The objective of this study is to show how the emergent models heuristic supports students in the construction of the concepts of spanning set and span through an HLT that has an initial task that is associated with the context of secure passwords. 2. Theoretical background The theoretical perspective of our research is based on the instructional design heuristic of emergent models that guide the design of the HLT and the analysis of the data. The emergent models heuristic is one of the key instructional design heuristics that may help designers or researchers to develop local instructional theories. This heuristic is part of the domain-specific instruction theory for RME. The value of the emergent models heuristic is that it helps students in constructing a mathematical reality by themselves (Gravemeijer, 2007) through a process of gradual growth, in which formal mathematics assume primary importance as a natural extension of their experiential reality (Gravemeijer, 1999). The emergent models heuristic is an alternative to instructional approaches that focus on teaching ready-made representations, this is, the approaches that only transmit knowledge to the student (Gravemeijer, 2002). The term ‘model’ should not be taken literally (Gravemeijer, 1997). In general, models are defined as ways generated by students for organizing their mathematical activity with both observable and mental tools. Observable tools are things in their environment, such as: graphs, diagrams, explicitly defined definitions, physical objects, etc. Meanwhile, mental tools allude to the ways in which students think and reason as they solve problems (Zandieh & Rasmussen, 2010). Such models are emergent ‘in the sense that various ways of creating and using tools, graphs, analytic expressions, etc. emerge together with increasingly sophisticated ways of reasoning’ (Rasmussen & Blumenfeld, 2007, p. 198). At this first level, the model must allow for informal strategies that correspond with solution strategies about the context. From that point onwards, the role of the model begins to change. Then, while the students gather more experience with similar problems, their attention can shift towards mathematical relations and strategies. As a consequence, the model obtains a more object-like character and becomes more important as a basis for mathematical reasoning than as a way to represent a contextual problem. In this way, the model begins to become a reference base for the level of formal mathematics. In short, model-of informal mathematical activity becomes model-for, more formal mathematical reasoning (Gravemeijer, 2007). For the transition from model-of to model-for, Gravemeijer (1999) details four levels of activity that do not involve any strictly ordered hierarchy, known as: situational, referential, general and formal. The situational activity involves students working towards the mathematical objectives through an experience that is real to them. The referential activity involves models of descriptions, concepts and procedures that relate to the problem of the situational activity. The general activity involves models to explore, to reflect upon and to generalize about what appeared at the previous level, but with a mathematical focus on strategies, without making any reference to the initial problem. The formal activity leads students to reflect the emergence of a new reality in mathematics; therefore, it involves working with procedures and conventional notations. Regarding activity levels, Zolkower & Bressan (2012) point out that at the situational activity, the real experiential problem is organized by students through strategies that arise spontaneously from the problem. At the referential activity level (model-of), students make graphs, notations and procedures that outline the problem, but refer to the initial problem. At the general activity (model-for) level, students explore, reflect and generalize what has appeared at the previous level, but no reference is made to the initial context. At the formal activity level, students work with conventional procedures and notations that are disconnected from the situations that gave them their initial meaning. Dawkins (2015) argues that the levels of activity mentioned by Gravemeijer (1999) are quite general, so it is necessary for instructional designers to elaborate the levels in the mathematical context in which they wish to work. On the other hand, Dawkins (2015) points out that the teacher guides the construction of mathematical knowledge of students and supports them in three basic processes that are involved with the four levels of activity. These are: when students engage in an experientially real activity, when they reflect on that activity to understand the structure and when they adapt that structure to other experiences. The last two processes allow students to create the model-of and model-for mathematical activities, respectively. 3. Methods The methodology of our study is the design-based research. This methodology aims to investigate the possibilities of educational improvement through the creation and study of new forms of learning (Gravemeijer & van Eerde, 2009). In design-based research, experiments are designed to transform classrooms into learning environments that foster reflective practices among students, teachers and researchers (Brown, 1992). These experiments involve continuous cycles of design, review and redesign until all errors are resolved (Collins et al., 2004). The design-based research distinguishes three phases: (1) the preparation for the experiment, (2) the teaching experiment and (3) the retrospective analysis (Gravemeijer & van Eerde, 2009). In the first phase, a HLT was elaborated (Simon, 1995) based on emergent models and mathematical modelling. In the experimental teaching phase, three design experimental cycles were developed in which the initial HLT was refined. In Table 1, we present a summary of the difficulties and modifications that were made in cycles 1 and 2 of this research (for more details of these cycles, see the articles Cárcamo et al., 2016, 2017). Table 1. Summary of the difficulties and modifications that were made in cycles 1 and 2 of this research Cycle 1   Cycle 2   Difficulty  Modification to cycle 2  Difficulty  Modification to cycle 3  Use of mathematical language.  The teacher promotes the use of correct mathematical language.  The mathematical notation of spanning set.  The method for delivering the definitions of the concepts was modified once again, with emphasis to their names and notations.  Obtain a span of R2 given a certain spanning set of this span.  Questions are added in task 3, about spanning set and span properties.  The mathematical notation of set.  Graph a span of R2.  This question is deleted.  Designate the same name to the span and to a spanning set that generates it.  The different ways to denote span.  It modifies the way that definitions of the concepts are given, with emphasis to their names and notations.      Associate the spanning set and span names to your notations in an interchangeable way.  Name another spanning set by associating it with its mathematical notation.  Cycle 1   Cycle 2   Difficulty  Modification to cycle 2  Difficulty  Modification to cycle 3  Use of mathematical language.  The teacher promotes the use of correct mathematical language.  The mathematical notation of spanning set.  The method for delivering the definitions of the concepts was modified once again, with emphasis to their names and notations.  Obtain a span of R2 given a certain spanning set of this span.  Questions are added in task 3, about spanning set and span properties.  The mathematical notation of set.  Graph a span of R2.  This question is deleted.  Designate the same name to the span and to a spanning set that generates it.  The different ways to denote span.  It modifies the way that definitions of the concepts are given, with emphasis to their names and notations.      Associate the spanning set and span names to your notations in an interchangeable way.  Name another spanning set by associating it with its mathematical notation.  Table 1. Summary of the difficulties and modifications that were made in cycles 1 and 2 of this research Cycle 1   Cycle 2   Difficulty  Modification to cycle 2  Difficulty  Modification to cycle 3  Use of mathematical language.  The teacher promotes the use of correct mathematical language.  The mathematical notation of spanning set.  The method for delivering the definitions of the concepts was modified once again, with emphasis to their names and notations.  Obtain a span of R2 given a certain spanning set of this span.  Questions are added in task 3, about spanning set and span properties.  The mathematical notation of set.  Graph a span of R2.  This question is deleted.  Designate the same name to the span and to a spanning set that generates it.  The different ways to denote span.  It modifies the way that definitions of the concepts are given, with emphasis to their names and notations.      Associate the spanning set and span names to your notations in an interchangeable way.  Name another spanning set by associating it with its mathematical notation.  Cycle 1   Cycle 2   Difficulty  Modification to cycle 2  Difficulty  Modification to cycle 3  Use of mathematical language.  The teacher promotes the use of correct mathematical language.  The mathematical notation of spanning set.  The method for delivering the definitions of the concepts was modified once again, with emphasis to their names and notations.  Obtain a span of R2 given a certain spanning set of this span.  Questions are added in task 3, about spanning set and span properties.  The mathematical notation of set.  Graph a span of R2.  This question is deleted.  Designate the same name to the span and to a spanning set that generates it.  The different ways to denote span.  It modifies the way that definitions of the concepts are given, with emphasis to their names and notations.      Associate the spanning set and span names to your notations in an interchangeable way.  Name another spanning set by associating it with its mathematical notation.  The data collected in each design experimental cycles were as follows: the written responses of the students to the tasks proposed in the HLT, audio and video recordings of group work and an individual interview at the end of the experimentation. The data of each cycle were analyzed comparing the HLT with the actual learning trajectory (ALT) through the matrix proposed by Dierdorp et al. (2011), looking for background to support or rebut the conjectures of the HLT. Also, the emergent models were used to interpret students’ progress in constructing the concepts of spanning set and span. The results that we presented correspond to the third cycle of this design-based research. The participants in the study were from a course group composed of 7 first-year engineering students from a public university in Spain. This course group, like the other first-year engineering groups, had not previously worked with mathematical modelling, nor had they studied the concepts of spanning set and span. The choice of this course group was based on the interest of its students in participating in the teaching experiment and for their communicative competence. This experiment was carried out over 5 h, divided into three class sessions, in which the students worked in two groups (group A of four students and group B of three students). The teacher guided the students in the resolution of the tasks of the HLT, and he led the discussion of topics that he considered relevant to favour learning. For example, in task 1, the teacher guided the students towards the solutions of the problem using the mathematical modelling cycle proposed by Blum and Leiss (2007). In addition, the teacher provided guidelines for constructing the mathematical model for generating passwords (suggesting the use of vectors or linear combinations) and create a coding for his password generator (suggesting the use of tables and giving them two examples). The HLT was composed of four tasks. The first task made use of an experientially real problem setting to support the construction of the concepts of spanning and span. This setting is experientially real for students in that it utilizes both their mathematical knowledge and their experience with passwords as a foundation on which to build more formal mathematics. In this task, mathematical modelling serves as a tool to help the study of mathematics (Julie & Mudaly, 2007). We consider the modelling cycle proposed by Blum and Leiss (2007) to guide students in solving this task. On the other hand, in this study, we emphasize how the four levels of activity linked to the emergent models heuristic can manifest themselves within the tasks of HLT. Table 2 summarizes the tasks and major conjectures of the learning trajectory of the HLT and how students are expected to manifest their activity levels in each of the tasks of HLT. Table 2. Summary of the tasks and major conjectures of the learning trajectory of the HLT and how students are expected to manifest their activity levels in each of the tasks of HLT Task and its major conjectures of learning trajectory  Manifestation of activity levels  Task 1: create a password generator based on vectors. Major conjectures: (1) Students read information from the secure passwords; (2) students created a password generator by following the steps of the modelling cycle and using their previous knowledge of vectors and passwords.  Situational activity. Students begin the construction of the concepts of spanning set and span in the context of passwords. Students explore different ways of using vectors and linear combinations to propose a mathematical model that generates passwords.  Task 2: make an analogy table between their password generator and the concepts of spanning set and span. Major conjectures: (1) Students find two sets from their password generator (one which has all the vectors that allow for the generation of the numerical passwords and the other which contains the vectors that, after creating the linear combination, is obtained by the vector for each numeric password with them); (2) students identify common features between two sets connected to their password generator and the concepts of spanning set and span; (3) students distinguish between spanning set and span with the analogy table.  Referential activity. Students use the set notations along with the definitions of spanning set and span to find two sets linked to their password generator that in mathematical notation correspond to the concepts of spanning set and span, respectively. These two sets function as models-of with respect to their previous activity with passwords and with vectors.  Task 3: Conjecture what the rank of the matrix should be so that its rows generate R2. Major conjectures: (1) Students explore particular cases and identify some regularity; (2) students relate spanning set and span with other concepts; (3) students conjecture the number of vectors in a set needed to generate R2; (4) students determine that the number of vectors is not sufficient to indicate if a set generates the span of R2.  General activity. Students explore, reflect and make new conjectures about sets of vectors in Rn that correspond to spanning sets or spans and that do not refer to sets within the context of passwords. The two sets linked to the context of the referential level passwords function as models-for reasoning about properties of this type of set, but without hinting at the initial situation.  Task 4: Indicate if the set C = {(1,0,0,1),(0,1,0,0)} is a spanning set for span W = {(x,y,z,w)/x=w}. Major conjectures: Students to pose a solution: (1) they explore possible routes for resolution; (2) they find a spanning set or span (according to the resolution they decided); (3) they verify if the set C is a generator of W.  Formal activity. Students work with the conventional notations of spanning set and span as well as their properties in situations that differ from those presented in previous levels, allowing them to demonstrate an overall understanding of these concepts.  Task and its major conjectures of learning trajectory  Manifestation of activity levels  Task 1: create a password generator based on vectors. Major conjectures: (1) Students read information from the secure passwords; (2) students created a password generator by following the steps of the modelling cycle and using their previous knowledge of vectors and passwords.  Situational activity. Students begin the construction of the concepts of spanning set and span in the context of passwords. Students explore different ways of using vectors and linear combinations to propose a mathematical model that generates passwords.  Task 2: make an analogy table between their password generator and the concepts of spanning set and span. Major conjectures: (1) Students find two sets from their password generator (one which has all the vectors that allow for the generation of the numerical passwords and the other which contains the vectors that, after creating the linear combination, is obtained by the vector for each numeric password with them); (2) students identify common features between two sets connected to their password generator and the concepts of spanning set and span; (3) students distinguish between spanning set and span with the analogy table.  Referential activity. Students use the set notations along with the definitions of spanning set and span to find two sets linked to their password generator that in mathematical notation correspond to the concepts of spanning set and span, respectively. These two sets function as models-of with respect to their previous activity with passwords and with vectors.  Task 3: Conjecture what the rank of the matrix should be so that its rows generate R2. Major conjectures: (1) Students explore particular cases and identify some regularity; (2) students relate spanning set and span with other concepts; (3) students conjecture the number of vectors in a set needed to generate R2; (4) students determine that the number of vectors is not sufficient to indicate if a set generates the span of R2.  General activity. Students explore, reflect and make new conjectures about sets of vectors in Rn that correspond to spanning sets or spans and that do not refer to sets within the context of passwords. The two sets linked to the context of the referential level passwords function as models-for reasoning about properties of this type of set, but without hinting at the initial situation.  Task 4: Indicate if the set C = {(1,0,0,1),(0,1,0,0)} is a spanning set for span W = {(x,y,z,w)/x=w}. Major conjectures: Students to pose a solution: (1) they explore possible routes for resolution; (2) they find a spanning set or span (according to the resolution they decided); (3) they verify if the set C is a generator of W.  Formal activity. Students work with the conventional notations of spanning set and span as well as their properties in situations that differ from those presented in previous levels, allowing them to demonstrate an overall understanding of these concepts.  Table 2. Summary of the tasks and major conjectures of the learning trajectory of the HLT and how students are expected to manifest their activity levels in each of the tasks of HLT Task and its major conjectures of learning trajectory  Manifestation of activity levels  Task 1: create a password generator based on vectors. Major conjectures: (1) Students read information from the secure passwords; (2) students created a password generator by following the steps of the modelling cycle and using their previous knowledge of vectors and passwords.  Situational activity. Students begin the construction of the concepts of spanning set and span in the context of passwords. Students explore different ways of using vectors and linear combinations to propose a mathematical model that generates passwords.  Task 2: make an analogy table between their password generator and the concepts of spanning set and span. Major conjectures: (1) Students find two sets from their password generator (one which has all the vectors that allow for the generation of the numerical passwords and the other which contains the vectors that, after creating the linear combination, is obtained by the vector for each numeric password with them); (2) students identify common features between two sets connected to their password generator and the concepts of spanning set and span; (3) students distinguish between spanning set and span with the analogy table.  Referential activity. Students use the set notations along with the definitions of spanning set and span to find two sets linked to their password generator that in mathematical notation correspond to the concepts of spanning set and span, respectively. These two sets function as models-of with respect to their previous activity with passwords and with vectors.  Task 3: Conjecture what the rank of the matrix should be so that its rows generate R2. Major conjectures: (1) Students explore particular cases and identify some regularity; (2) students relate spanning set and span with other concepts; (3) students conjecture the number of vectors in a set needed to generate R2; (4) students determine that the number of vectors is not sufficient to indicate if a set generates the span of R2.  General activity. Students explore, reflect and make new conjectures about sets of vectors in Rn that correspond to spanning sets or spans and that do not refer to sets within the context of passwords. The two sets linked to the context of the referential level passwords function as models-for reasoning about properties of this type of set, but without hinting at the initial situation.  Task 4: Indicate if the set C = {(1,0,0,1),(0,1,0,0)} is a spanning set for span W = {(x,y,z,w)/x=w}. Major conjectures: Students to pose a solution: (1) they explore possible routes for resolution; (2) they find a spanning set or span (according to the resolution they decided); (3) they verify if the set C is a generator of W.  Formal activity. Students work with the conventional notations of spanning set and span as well as their properties in situations that differ from those presented in previous levels, allowing them to demonstrate an overall understanding of these concepts.  Task and its major conjectures of learning trajectory  Manifestation of activity levels  Task 1: create a password generator based on vectors. Major conjectures: (1) Students read information from the secure passwords; (2) students created a password generator by following the steps of the modelling cycle and using their previous knowledge of vectors and passwords.  Situational activity. Students begin the construction of the concepts of spanning set and span in the context of passwords. Students explore different ways of using vectors and linear combinations to propose a mathematical model that generates passwords.  Task 2: make an analogy table between their password generator and the concepts of spanning set and span. Major conjectures: (1) Students find two sets from their password generator (one which has all the vectors that allow for the generation of the numerical passwords and the other which contains the vectors that, after creating the linear combination, is obtained by the vector for each numeric password with them); (2) students identify common features between two sets connected to their password generator and the concepts of spanning set and span; (3) students distinguish between spanning set and span with the analogy table.  Referential activity. Students use the set notations along with the definitions of spanning set and span to find two sets linked to their password generator that in mathematical notation correspond to the concepts of spanning set and span, respectively. These two sets function as models-of with respect to their previous activity with passwords and with vectors.  Task 3: Conjecture what the rank of the matrix should be so that its rows generate R2. Major conjectures: (1) Students explore particular cases and identify some regularity; (2) students relate spanning set and span with other concepts; (3) students conjecture the number of vectors in a set needed to generate R2; (4) students determine that the number of vectors is not sufficient to indicate if a set generates the span of R2.  General activity. Students explore, reflect and make new conjectures about sets of vectors in Rn that correspond to spanning sets or spans and that do not refer to sets within the context of passwords. The two sets linked to the context of the referential level passwords function as models-for reasoning about properties of this type of set, but without hinting at the initial situation.  Task 4: Indicate if the set C = {(1,0,0,1),(0,1,0,0)} is a spanning set for span W = {(x,y,z,w)/x=w}. Major conjectures: Students to pose a solution: (1) they explore possible routes for resolution; (2) they find a spanning set or span (according to the resolution they decided); (3) they verify if the set C is a generator of W.  Formal activity. Students work with the conventional notations of spanning set and span as well as their properties in situations that differ from those presented in previous levels, allowing them to demonstrate an overall understanding of these concepts.  4. Results In this section, we describe how the reasoning used by the student groups progressed through each of the four levels of activity using the tasks of the HLT that they developed. In accordance with Rasmussen and Blumenfeld (2007) ‘these different levels offer a journey through students’ mathematical thinking without imposing a strict hierarchy’ (p. 201). 4.1 Situational activity With respect to the first of the four levels of activity, ‘situational activity involves students working toward mathematical goals in an experientially real setting’ (Zandieh & Rasmussen, 2010, p. 58). The context of generating secure passwords serves as an experientially real starting point for initiating the construction of spanning set and span. In their development of task 1, student groups showed they were at the level of situational activity because they explored different ways of using vectors and linear combinations towards the goal of creating a numerical password generator based on vectors. This is exemplified by the following dialogue of group A where it was observed that they made use of their previous conceptions of vectors and that the students S3 and S1 of this group proposed different ways of using the vectors to propose a solution: S3: First, what are we going to use? What operations are we going to do with the vectors? Are we going to add them up, multiply them or something like that? S1: A number that multiplies each vector and then, add the vectors. S3: How many vectors will we make? S4: Three. S2: Three vectors. S1: Three vectors. Student S3 started questioning how they would use the vectors to propose a solution for task 1. Student S3 asked to his companions ‘What operations are we going to do with the vectors? Are we going to add them up, multiply them or something like that?’. Student S1 suggested ‘A number that multiplies each vector and then, add the vectors’. That is, implicitly, student S1 proposes to make a linear combination. Then, student S3 asked about the number of vectors their model would contain and the other students in the group answered ‘three’ vectors. The ideas proposed by the students of group A led them to propose their mathematical model to generate the numerical passwords presented in Table 3. Table 3. Some of the written work that the groups A and B carried out to propose a solution for the task 1 Group A  Group B  Mathematical model      Codification      Example of how their password generator operates      Group A  Group B  Mathematical model      Codification      Example of how their password generator operates      Table 3. Some of the written work that the groups A and B carried out to propose a solution for the task 1 Group A  Group B  Mathematical model      Codification      Example of how their password generator operates      Group A  Group B  Mathematical model      Codification      Example of how their password generator operates      Table 3 presents some of the written work that the groups A and B carried out to propose a solution for the task 1, which they carried out following the steps of the modelling cycle (Blum & Leiss, 2007), but adapted to task 1 (Cárcamo et al., 2016). Table 3 shows that the mathematical model for generating numerical passwords for group A corresponded to a linear combination of vectors of R3 while that of group B was a linear combination of vectors of R4. To develop passwords encoded from numerical passwords, group A designed a mixed coding set, unlike group B, which created their own coding set. Group A created a two-line code: the first line of ‘code’ contains letters and symbols designated by them, whereas the second row ‘ASCii’ consists of characters taken from the ASCII code. The coding of group A operated as follows: those numerical password numbers that were in an even position were replaced by ‘code’ and those in an odd position were replaced by ‘ASCii’. On the other hand, group B designed their own code composed of numbers, letters and symbols that is divided into two rows: ‘code 1’ and ‘code 2.’ To transform a numeric password to a coded one, group B changed the number of the second position of the numeric password using a character from ‘code 1’ and in addition, it replaced both the first and last number of it using ‘code 2’. The groups showed how they operated their password generator by giving an example (Table 3). Group A gave specific values to the letters of its mathematical model and obtained as a numerical password the vector (3,5,4), corresponding to the numeric password 354. Consequently, they used their own code and had the encrypted password ETXZEOT. Group B followed a process similar to Group A and generated the numeric password 52302166, which was transformed into the encrypted password//430216\\. The solution given by the two groups for task 1 allowed for these students’ level of situational activity to be shown. Also, their solutions allowed for the observation that they made use of their previous conceptions of vectors, and used the steps of the mathematical modelling cycle to propose a mathematical model that generates numeric passwords. They will use this vector-based mathematical model to obtain two sets that are related to the context of the passwords, but also with the concepts of spanning set and span. In this way, they are given an initial familiarization with these concepts of Linear Algebra. 4.2 Referential activity After students established their mathematical models to create a numerical password generator based on vectors, the teacher introduced the definitions of generating set and generated space in relation to the work and ideas established by the students. The introduction of these definitions served to support students in the transition from situational activity (work done within the context of passwords) to referential activity (work done with sets within the context of passwords that are examples of spanning set and span). With respect to the four levels of activity, Task 2 led the students towards the referential activity that, according to Gravemeijer (1999), implies the use of descriptions, concepts and procedures that relate to the problem of the situational activity; in this case, of task 1. Task 2 asked the students to make an analogy table between their password generator that they created in task 1 and the concepts of spanning set and span. In this task, the students showed that they were at the reference level, due to the fact that they used set notations and definitions to find two sets linked to their password generator: one representing a spanning set and another describing a span. To find these two sets, the students used the mathematical model they had designed in task 1. In the following dialogue between students S1 and S3 of group A, we observe how their ideas for relating the span with a set from their password generator came about: S1: Okay, span. S3: It’s the formula. S1: Describes all the vectors that can be made. S3: Describes the operations to find the numerical vector that generates a password. S1: All the passwords that could exist. To respond to the relationship between the password generator and span, student S3 made a very general comment, noting that ‘it’s the formula’, in response to which student S1, said that the span ‘describes all the vectors that can be made’. This comment from student S1 is general because it does not specify what the purpose of those vectors is and does not mention the link with the passwords. Perhaps, for this reason, student S3 specified that the span ‘describes the operations to find the numerical vector that generates a password’. Student S3 mentioned that the span describes operations. He linked this concept to the context of passwords when he specified that these ‘operations’ served to find the numeric vector that generates a password. From the written response of group A (Table 4), it follows that when student S3 talked about operations, he was referring to those that have a linear combination (addition and multiplication). The answer was the set {(a,b,c) ε R3: a(1,0,1) + b(1,1,0) + c(0,1,1)}. However, student S1 disagreed with the last statement made by student S3, and he pointed out that the span has ‘all the passwords that could exist’. This statement made by student S1 allows it to be inferred that he is thinking of the span as the set that contains all the vectors that allow for numerical passwords to be generated. Table 4. Analogy tables of task 2 of groups A and B Group A  Mathematical name for this concept  Name given in your password generator  How it is written in mathematical language  Spanning set  Vectors that generate numeric passwords.  {(1,0,1),(1,1,0),(0,1,1)}  (a) Span  Describes the operations to find the numerical vector that generates a password  {(a,b,c)∈R3:a(1,0,1)+b(1,1,0)+c(0,1,1)}  Group B  Spanning set  Spanning set V allows to generate from a pattern different numerical passwords  V={(5,0,0,0),(0,−10,0,0),(0,0,−18,0),(0,0,0,3)}  (b) Span  Span V provides the characteristics of said pattern while creating our passwords.  V={(5a,−10b,−18c,3d)∈R4}  Group A  Mathematical name for this concept  Name given in your password generator  How it is written in mathematical language  Spanning set  Vectors that generate numeric passwords.  {(1,0,1),(1,1,0),(0,1,1)}  (a) Span  Describes the operations to find the numerical vector that generates a password  {(a,b,c)∈R3:a(1,0,1)+b(1,1,0)+c(0,1,1)}  Group B  Spanning set  Spanning set V allows to generate from a pattern different numerical passwords  V={(5,0,0,0),(0,−10,0,0),(0,0,−18,0),(0,0,0,3)}  (b) Span  Span V provides the characteristics of said pattern while creating our passwords.  V={(5a,−10b,−18c,3d)∈R4}  Table 4. Analogy tables of task 2 of groups A and B Group A  Mathematical name for this concept  Name given in your password generator  How it is written in mathematical language  Spanning set  Vectors that generate numeric passwords.  {(1,0,1),(1,1,0),(0,1,1)}  (a) Span  Describes the operations to find the numerical vector that generates a password  {(a,b,c)∈R3:a(1,0,1)+b(1,1,0)+c(0,1,1)}  Group B  Spanning set  Spanning set V allows to generate from a pattern different numerical passwords  V={(5,0,0,0),(0,−10,0,0),(0,0,−18,0),(0,0,0,3)}  (b) Span  Span V provides the characteristics of said pattern while creating our passwords.  V={(5a,−10b,−18c,3d)∈R4}  Group A  Mathematical name for this concept  Name given in your password generator  How it is written in mathematical language  Spanning set  Vectors that generate numeric passwords.  {(1,0,1),(1,1,0),(0,1,1)}  (a) Span  Describes the operations to find the numerical vector that generates a password  {(a,b,c)∈R3:a(1,0,1)+b(1,1,0)+c(0,1,1)}  Group B  Spanning set  Spanning set V allows to generate from a pattern different numerical passwords  V={(5,0,0,0),(0,−10,0,0),(0,0,−18,0),(0,0,0,3)}  (b) Span  Span V provides the characteristics of said pattern while creating our passwords.  V={(5a,−10b,−18c,3d)∈R4}  In Table 4 we present the analogy tables made by each group, in which it is shown that they correctly associated the sets in mathematical notation that arise from their mathematical model to generate passwords using the concepts of spanning set and span. However, we observed that group B designated the letter V for both sets. From this, it could be inferred that the students from this group considered that the two sets are equal, but they give a description of these in the context of their password generator that shows that they differentiate between them, and that the fact they assign the letter V to both set is due only to a notation error. On the other hand, in the second column of the analogy table that asks for the name or what corresponds in their password generator. Group B wrote that the spanning set ‘allows to generate from a pattern different numerical passwords’ while the span ‘provides the characteristics of said pattern while creating our passwords.’ We deduced that group B used the term ‘pattern’ to refer to the vectors of the spanning set, since it was these vectors that allowed the generation of numerical passwords. It was important for the students to make the analogy table and to link the real context with the mathematical one. From this form, they continued to expand their grasp of spanning set and span, given that they created a preliminary example of both concepts in analytical notation, and established a distinction between these concepts when they were required to relate them to sets derived from a real situation. In the referential activity, the two sets linked to the context of the passwords, which were also used as examples of spanning set and span, functioned for the students as a model-of their previous mathematical activity with the passwords and the vectors. In the next section, students are presented with a more challenging question that allows them to progress towards a general level of activity. In other words, the students begin to change the model they use: from thinking of two sets that are examples of spanning set and span to thinking in terms of properties of these types of sets, but without alluding to the scenario with the passwords. 4.3 General activity One of the goals of task 3 was to shift students away from situational and referential activity with respect to the context of the passwords, and towards a general level of activity with respect to the concepts. General activity involves exploring, reflecting and generalizing the material that appeared at the previous level, but without referencing the initial context (Zolkower & Bressan, 2012). The students showed they were at this level of activity when they explored, reflected and made conjectures about sets of vectors in Rn that corresponded to spanning sets or spans and that did not refer to the scenario in task 1. For example, for question (a) of task 3, students conjectured: what should be the rank of the matrix that has a set of R2 as vector rows such that this set generates R2? The groups, after calculating the rank of four matrices (M1, M2, M3, and M4), correctly conjectured that the rank of the matrix having a set of R2 as vector rows must be two in order for it to generate R2. This is shown in Table 5. Table 5. Written answer of groups A and B to question (a) of task 3 Group A  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: If the rank is 2, the set can generate R2.  A={(0,−3)}  M1=(0−3)  1  B={(5,0),(7,0)}  M2=(5070)  1  C={(1,0),(1,−1)}  M3=(1   01−1)  2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  2  Group B  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: The rank of the matrix must be equal to 2.  A={(0,−3)}  M1=(0−3)  RM1=1  B={(5,0),(7,0)}  M2=(5070)  RM2=1  C={(1,0),(1,−1)}  M3=(1   01−1)  RM3=2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  RM4=2  Group A  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: If the rank is 2, the set can generate R2.  A={(0,−3)}  M1=(0−3)  1  B={(5,0),(7,0)}  M2=(5070)  1  C={(1,0),(1,−1)}  M3=(1   01−1)  2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  2  Group B  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: The rank of the matrix must be equal to 2.  A={(0,−3)}  M1=(0−3)  RM1=1  B={(5,0),(7,0)}  M2=(5070)  RM2=1  C={(1,0),(1,−1)}  M3=(1   01−1)  RM3=2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  RM4=2  Table 5. Written answer of groups A and B to question (a) of task 3 Group A  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: If the rank is 2, the set can generate R2.  A={(0,−3)}  M1=(0−3)  1  B={(5,0),(7,0)}  M2=(5070)  1  C={(1,0),(1,−1)}  M3=(1   01−1)  2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  2  Group B  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: The rank of the matrix must be equal to 2.  A={(0,−3)}  M1=(0−3)  RM1=1  B={(5,0),(7,0)}  M2=(5070)  RM2=1  C={(1,0),(1,−1)}  M3=(1   01−1)  RM3=2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  RM4=2  Group A  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: If the rank is 2, the set can generate R2.  A={(0,−3)}  M1=(0−3)  1  B={(5,0),(7,0)}  M2=(5070)  1  C={(1,0),(1,−1)}  M3=(1   01−1)  2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  2  Group B  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: The rank of the matrix must be equal to 2.  A={(0,−3)}  M1=(0−3)  RM1=1  B={(5,0),(7,0)}  M2=(5070)  RM2=1  C={(1,0),(1,−1)}  M3=(1   01−1)  RM3=2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  RM4=2  To answer question (a) of task 3, in group A, student S1 indicated that the rank of the matrix M1 ‘is 1’ and then, referring to the other matrices, noted that ‘this is also 1 (showing M2). Yes, this is two (indicating M3). One, one, two, two’. When student S1 mentions one, one, two, two, he alludes to the ranks of the matrices M1, M2, M3, and M4, respectively, for that is what appears in the written answer (Table 5). From this, the students of group A surmised that if the rank is 2, the set can generate R2. The fact that group A has put in its conjecture the word ‘may’ indicate to us that they left open the possibility for there to be a set for which the associated matrix has rank 2, but does not generate R2. The students from group A did not express any idea about the characteristics of the vectors that are contained in each matrix. However, student S3 from group B did comment on it, as he pointed out to his group that the vectors of set B = {(5,0),(7,0)} ‘form a linear combination between them’ and then also indicated that ‘they are not independent of each other.’ Therefore, the rank of the matrix associated with this set is one. The affirmations of student S3 from group B allowed for us to infer that he was able to identify when the vectors are linearly dependent, as was the case of the set B = {(5,0),(7,0)}. From the written response of both groups (Table 5), we infer that they noticed that a spanning set of R2 can have more than two vectors (such as set D), but that the minimum must be two vectors that are linearly independent (such as set C). That is, it is not enough that the set has two vectors (such as set B). From this it follows that the students observed that the number of vectors is not sufficient to indicate if a set generates R2, because apart from the fact that they are two vectors, they must be linearly independent. The formation of this conjecture was fundamental, because it made the students relate spanning set and span with the concepts of the rank of a matrix and linear independence. In addition, they identified a property that characterizes the sets that generate the span of R2, which indicates that the number of vectors is not sufficient to indicate if a set generates R2. At a general level, the two sets related to the context of the referential-level passwords functioned for students as models-for, to reason about properties of this type of set, but without referring to the scenario of task 1. 4.4 Formal activity Formal activity, as described by Gravemeijer (1999), involves reasoning with conventional symbolism, which is no longer dependent on the support of models-for mathematical activity. For their part, Zolkower and Bressan (2012) specify that formal activity involves students working with conventional procedures and notations disconnected from the situations that gave them their initial meaning. We consider that the development made by students in task 4 of the HLT gave indications that they may have reached the level of formal activity, because they worked implicitly with the concepts of spanning set and span in a situation different from those presented for the previous levels. In task 4, the students were asked questions different from those they had previously been asked in the tasks of the HLT. For example, they answered the question: indicate if C = {(1,0,0,1), (0,1,0,0)} is a spanning set for the span W = {(x,y,z,w)/x = w}. The students of the groups gave evidence that they were at the formal activity level when they answered this question (see answer in Table 6), because we have inferred that they used the definitions of spanning set and span to solve it, although this was not indicated in their process for finding the solution to this question. Table 6. Written answers of groups A and B to the question: indicate if C = {(1,0,0,1), (0,1,0,0)} is a spanning set for the span W = {(x,y,z,w)/x = w} Group A    Group B    Group A    Group B    Table 6. Written answers of groups A and B to the question: indicate if C = {(1,0,0,1), (0,1,0,0)} is a spanning set for the span W = {(x,y,z,w)/x = w} Group A    Group B    Group A    Group B    Both groups pointed out that the set C is not a spanning set for W (Table 6). In the case of group A, to reach this conclusion, they followed a process to determine a spanning set of W, as shown in the dialogue between students S1 and S2 of this group: S2: (looking to the set W) condition x is equal to w. S1: we are going to replace it. S2: you do the operation. S1: Look at this (S1 shows what he wrote to S2) … there w 1 0 0 1. I was left 0 1 … 0 0. No, it is not because of the z component. S2: the z nothing, it does not influence. S1: this (indicating the set {(1,0,0,1), (0,1,0,0)}) is not a spanning set of this (showing W). Look (showing them w(1,0,0,1) + y(0,1,0,0) + z(0,0,1,0)). Student S2 expressed (looking to the set W) that ‘condition x is equal to w’ and immediately, student S1 told his colleagues ‘we are going to replace it’. Student S1 referred to substitution, the condition that was indicated by student S2, in the generic vector of the span W, since that was what they did and wrote the linear combination w(1,0,0,1) + y(0,1,0,0) + z(0,0,1,0). On seeing it, student S1 stated that the set {(1,0,0,1),(0,1,0,0)} is not a spanning set for W because of ‘component z’ and, although student S2 pointed out that z does not influence, student S1 added that ‘this (indicating the set{(1,0,0,1), (0,1,0,0)}) is not a spanning set of this (showing W). Look (showing them w(1,0,0,1) + y(0,1,0,0) + z(0,0,1,0))’. Student S1 proposed to his colleagues that they compare the expression of the three vectors with the set that gave them two. From this, they concluded that it is not a spanning set of W. On the other hand, group B pointed out in its written answer (Table 6) that C is not a spanning set of W because it lacks the vector (0, 0, 1, 0). From this, it is inferred that these students carried out a process similar to that of group A. This is because the vector they mention is associated with the z component in the linear combination shown by students in group A. In the responses of both groups, we observe that they verify if the set C is generator of W and give a correct answer. This is similar to the description of the THA proposed for task 4 (Table 2). We consider that the students would have given evidence that they were at the formal activity level in this question in task 4 (indicate whether C = {(1,0,0,1), (0,1,0,0)} is a (x, y, z, w)/x = w}) if they had explicitly stated the use of spanning set or span during their attempt to solve the problem. For example, if they had indicated: ‘we are going to determine a spanning set for W to see if it is C’ or ‘for C to be a spanning set, any vector of W must be a linear combination of the vectors of C’. However, this did not occur. It should be noted that emergent models do not specify what happens when students do not give evidence of being at the formal level, but rather that they are in the transition from the general level to the formal level. This may be an indicator of the existence of a new level between the general and formal levels, or the existence of sub-levels within each level. 5. Discussion and conclusions The main contribution of this research is to show how the emergent models can be a good alternative for the design of a HLT that favors the construction of the students of the concepts of spanning set and span through an HLT for which the initial context is the creation of secure passwords. In this way, it is also intended to contribute to the field of application of emergent models at the university level, since few studies have been conducted in this area. In order to respond to the objective of the study, we reconstructed the reasoning groups of students used throughout each of the levels of activity. We observe that students transitioned from their model-of informal mathematical activity (using passwords and vectors) to a model-for formal mathematical reasoning (on the concepts of spanning set and span). Therefore, we can point out that the emergent models supported the students’ development of the concepts of spanning set and span. The level of situational activity implied that students would work towards mathematical goals in an experientially real environment, particularly in the context of creating secure passwords. In task 1, students determined how to create a password generator based on vectors. Each group developed a mathematical model that, in both cases, was a linear combination (Table 3) and that complied with the requested characteristics (create a password generator based on vectors). Consequently, the level of referential activity required students to work with set notations and with the definitions of spanning set and span in the password scenario. This was done to determine two sets that referred to the context of the passwords, but that also were examples of spanning set and span (Table 3). For the students, these two sets were models-of of what worked in the context of passwords and vectors. At the level of general activity, the two sets that were models-of at the referential activity level became models-for for reasoning about properties of this type of set of Rn, but without hinting at the initial situation. For example, Table 5 shows that students gave correct conjecture as to what the rank of the matrix should be such that the rows generate R2. Finally, students gave indications of being at the formal activity level, because, for example, in solving a question from task 4 (Table 6), they worked implicitly with the concepts of spanning set and span in a situation different from those presented at previous levels. The choice of the emergent model that will support students in the transition from the model-of to the model-for is considered to be essential. This information will allow the teacher to better guide the students in their process of constructing mathematical knowledge. In this study, the emergent models were the two sets that initially emerged as models-of of what worked in task 1, that is, with passwords and vectors. Gradually, however, these two sets changed in nature as students began to change their outlook on these two sets; from seeing the two sets as examples of spanning set and span to reasoning about properties of these types of sets. Finally, these two sets became models-for for reasoning about the properties of spanning set and span. These emergent models allowed students to transition from real experience (passwords and vectors) to a new mathematical reality (spanning set and span). We agree with the results obtained by Rasmussen and Blumenfeld (2007), in their research conducted on the differential equations course, in relation to which analytic expressions can support the entire transition from the model-of to the model-for. This occurred in our study with the construction of the concepts of spanning set and span. In our case, we observed that students began their work with vectors and linear combinations (situational activity), then continued with sets of vectors (referential activity and general activity) and finally with conventional mathematical procedures and notations (formal activity). With respect to the tasks applied in this study, Table 2 shows a summary of the HLT and how students were expected to manifest their transition by activity levels. The results provided evidence that students progressed through the different levels of activity by solving the task sequence of the HLT. This indicates to us that there was a close relationship between the HLT and the current trajectory of student learning. However, in order for the teacher to enhance the students’ reasoning in their process of solving tasks 3 and 4, we suggested that the teacher ask the students to justify the steps of their solution to a given question by, for example, the use of the definitions of spanning set or span or its properties, or indicating how these concepts were used in their proposed resolution. We recognize that the tasks and the order in which they are presented to students are critical in facilitating their transition from their informal knowledge to more formal mathematical reasoning. We consider that it is important for each task to offer the student the possibility of fully passing through the level of activity that is intended for this activity, otherwise it will need to be presented differently. In this study, we observed that the sequence of tasks of the HLT tended to lead students to transit through different levels of activity. This was not a coincidence, but rather a result of the two previous refinements of the HLT, that is to say, as indicated by Wawro et al. (2013), it ‘was made possible by the cyclical process of the design research’ (p. 911). On the other hand, we agree with Dawkins (2015) that the teacher should support the students in the basic processes that are involved in the levels of activity. However, we also consider it important that, prior to this, the teacher should do the following: choose an initial activity that is experientially real, conjecture about what the emergent model will be (the model guiding the transition between model-of and model-for) and develop key questions that promote the advancement of students between levels of activity. The results showed that the HLT based on the emergent models contributed to the students’ progression from their informal mathematical activity to a more formal mathematical form of reasoning regarding the concepts of spanning set and span. Therefore, this didactic proposal could be useful as a teaching tool to be applied in another classroom (previously adapted to its context), but also as a tool for teachers to use as an example in their consideration aimed at devising instructional sequences based on this heuristic design. We consider that a limitation of emergent models is that they do not detail possible sub-levels within each level which would be useful to help students progress within each level and bring them to the next level. Another limitation is that it does not detail what happens in the transitions between one level and another. For future research, the challenge will be to design HLTs based on the emergent models for other Linear Algebra concepts with the objective of determining if this approach also contributes to the construction of these. Andrea Cárcamo is a PhD candidate in the area of mathematics education and experimental sciences teaching at the Autonomous University of Barcelona. Professor at the Centre for Teaching Basic Science Engineering at the Austral University of Chile. Her research focuses on teaching Linear Algebra and mathematical modelling. Josep Fortuny is PhD in Mathematics. Professor at the Department of Teaching Mathematics and Experimental Sciences at the Autonomous University of Barcelona. His research focuses on learning and interaction. Claudio Fuentealba is a PhD candidate in the area of mathematics education and experimental sciences teaching at the Autonomous University of Barcelona. Professor at the Centre for Teaching Basic Science Engineering at the Austral University of Chile. Her research focuses on teaching Linear Algebra and Calculus. Acknowledgement We would like to express a gratitude to the participants of the seminar led by Dr. Asuman Oktaç for their suggested improvement about this article. References Blum W., Leiss D. ( 2007) How do students and teachers deal with modelling problems? 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( 2010) Defining as a mathematical activity: a framework for characterizing progress from informal to more formal ways of reasoning. J. Math. Behav ., 29, 57– 75. Google Scholar CrossRef Search ADS   Zolkower B., Bressan A. ( 2012) Educación matemática realista. Educación Matemática. Aportes a la Formación Docente desde Distintos Enfoques Teóricos  ( Pochulu M., Rodríguez M. eds). Argentina: UNGS–EDUVIM, pp. 175– 200. © The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Teaching Mathematics and Its Applications: International Journal of the IMA Oxford University Press

The emergent models in linear algebra: an example with spanning set and span

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Institute of Mathematics and its Applications
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© The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com
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0268-3679
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1471-6976
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10.1093/teamat/hrx015
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Abstract

Abstract A continuing challenge in mathematics education at the university level is how to create learning environments that give students the opportunity to progress from their informal ways of reasoning to more formal ways of reasoning. This article shows how the emergent models heuristic (Gravemeijer, 1999, Math. Think. Learn, 1, 155–177) provides this opportunity to first-year university students, specifically for them to construct the Linear Algebra concepts of spanning set and span. For this, a design-based research methodology was used. A hypothetical learning trajectory (HLT) was applied, created within the framework of this heuristic. The results provide evidence that the emergent models heuristic support the construction of these concepts of Linear Algebra. Moreover, useful tips are given for using this heuristic to construct mathematical concepts. 1. Introduction For some time, a growing amount of literature in mathematics education has indicated that students have difficulties with advanced mathematical thinking. This is due, on the one hand, to the complexity and abstract nature of advanced mathematical topics, but on the other hand, to the difficulties that they encounter due to the shift from less formal to more formal modes of reasoning. However, despite this, the systematic descriptions of the means by which and the stages through which students advance from informal reasoning to formal reasoning are still in initial research (Dawkins, 2012). These investigations include those that use the levels of activity described by Gravemeijer (1999) to detail the transition of students from an informal mathematical reasoning of their activity towards a more formal type of mathematical reasoning. This research has been carried out in the university courses of: differential equations (Rasmussen & Blumenfeld, 2007), geometry (Zandieh & Rasmussen, 2010), and Linear Algebra (Wawro et al., 2013). In their study, Rasmussen and Blumenfeld (2007) exemplify the four levels of activity of the emergent models as a framework for the analysis of the mathematical activity of the students observed for the study in constructing the solutions to systems of two differential equations. They conclude that their work contributes to the emergent models heuristic, as they ‘demonstrated how analytic expressions can underpin the entire model-of/model-for transition’ (p. 208). This is due to the fact that, in previous research, most of which has been done at the primary and secondary (school) levels, the model-of phase (situational and referential level) has not involved analytical expressions; instead, it usually includes concrete tools, graphs or diagrams. Only later, in the model-for phase (general and formal level) do students work with formal and symbolic expressions. In addition, Rasmussen & Blumenfeld (2007) consider that their study in the course of differential equations offers a prototypical example of a work inspired by the Realistic Mathematics Education (RME) and the emergent models heuristic at the undergraduate level to support students in creating new mathematical realities. They also note that these ideas about ‘student thinking can be useful for teachers in their difficult task of making sense of what students say and, do for being proactive in supporting their intellectual growth’ (p. 208). Wawro et al. (2013) describe the reasoning used by students in constructing the concepts of span, linear independence and linear dependence through the four levels of activity of the emergent models. These authors indicate how the emergent models heuristic reports the creation of an instructional sequence for these concepts of Linear Algebra. Moreover, they describe how the activity levels of this heuristic are manifested in the sequence. They give evidence of how students give indications of being in the first three levels of activity. However, they argue that the last level of activity (the formal level) does not occur during the sequence, but rather is observed in other tasks unrelated to this sequence. In addition, they indicate that the refinement of their instructional sequence, ‘and its framing within the four levels was made possible by the cyclical process of design research’ (p. 911). Our research is grounded in the emergent models heuristic. In particular, our research describes how student reasoning has progressed through each of the four levels of activity in relationship to the tasks of the hypothetical learning trajectory (HLT) that students worked on. This is the case because we want to contribute to this design heuristic that, according to Dawkins (2012), needs further exploration due to the fact that many mathematical topics do not immediately suggest contexts in which students can begin their learning. Among these mathematical topics are several contents of the Linear Algebra course, including the concepts of spanning set and span, which we considered in this study. These concepts are chosen because of their relationship with important contents of this course, such as: base and dimension (Stewart & Thomas, 2010). The objective of this study is to show how the emergent models heuristic supports students in the construction of the concepts of spanning set and span through an HLT that has an initial task that is associated with the context of secure passwords. 2. Theoretical background The theoretical perspective of our research is based on the instructional design heuristic of emergent models that guide the design of the HLT and the analysis of the data. The emergent models heuristic is one of the key instructional design heuristics that may help designers or researchers to develop local instructional theories. This heuristic is part of the domain-specific instruction theory for RME. The value of the emergent models heuristic is that it helps students in constructing a mathematical reality by themselves (Gravemeijer, 2007) through a process of gradual growth, in which formal mathematics assume primary importance as a natural extension of their experiential reality (Gravemeijer, 1999). The emergent models heuristic is an alternative to instructional approaches that focus on teaching ready-made representations, this is, the approaches that only transmit knowledge to the student (Gravemeijer, 2002). The term ‘model’ should not be taken literally (Gravemeijer, 1997). In general, models are defined as ways generated by students for organizing their mathematical activity with both observable and mental tools. Observable tools are things in their environment, such as: graphs, diagrams, explicitly defined definitions, physical objects, etc. Meanwhile, mental tools allude to the ways in which students think and reason as they solve problems (Zandieh & Rasmussen, 2010). Such models are emergent ‘in the sense that various ways of creating and using tools, graphs, analytic expressions, etc. emerge together with increasingly sophisticated ways of reasoning’ (Rasmussen & Blumenfeld, 2007, p. 198). At this first level, the model must allow for informal strategies that correspond with solution strategies about the context. From that point onwards, the role of the model begins to change. Then, while the students gather more experience with similar problems, their attention can shift towards mathematical relations and strategies. As a consequence, the model obtains a more object-like character and becomes more important as a basis for mathematical reasoning than as a way to represent a contextual problem. In this way, the model begins to become a reference base for the level of formal mathematics. In short, model-of informal mathematical activity becomes model-for, more formal mathematical reasoning (Gravemeijer, 2007). For the transition from model-of to model-for, Gravemeijer (1999) details four levels of activity that do not involve any strictly ordered hierarchy, known as: situational, referential, general and formal. The situational activity involves students working towards the mathematical objectives through an experience that is real to them. The referential activity involves models of descriptions, concepts and procedures that relate to the problem of the situational activity. The general activity involves models to explore, to reflect upon and to generalize about what appeared at the previous level, but with a mathematical focus on strategies, without making any reference to the initial problem. The formal activity leads students to reflect the emergence of a new reality in mathematics; therefore, it involves working with procedures and conventional notations. Regarding activity levels, Zolkower & Bressan (2012) point out that at the situational activity, the real experiential problem is organized by students through strategies that arise spontaneously from the problem. At the referential activity level (model-of), students make graphs, notations and procedures that outline the problem, but refer to the initial problem. At the general activity (model-for) level, students explore, reflect and generalize what has appeared at the previous level, but no reference is made to the initial context. At the formal activity level, students work with conventional procedures and notations that are disconnected from the situations that gave them their initial meaning. Dawkins (2015) argues that the levels of activity mentioned by Gravemeijer (1999) are quite general, so it is necessary for instructional designers to elaborate the levels in the mathematical context in which they wish to work. On the other hand, Dawkins (2015) points out that the teacher guides the construction of mathematical knowledge of students and supports them in three basic processes that are involved with the four levels of activity. These are: when students engage in an experientially real activity, when they reflect on that activity to understand the structure and when they adapt that structure to other experiences. The last two processes allow students to create the model-of and model-for mathematical activities, respectively. 3. Methods The methodology of our study is the design-based research. This methodology aims to investigate the possibilities of educational improvement through the creation and study of new forms of learning (Gravemeijer & van Eerde, 2009). In design-based research, experiments are designed to transform classrooms into learning environments that foster reflective practices among students, teachers and researchers (Brown, 1992). These experiments involve continuous cycles of design, review and redesign until all errors are resolved (Collins et al., 2004). The design-based research distinguishes three phases: (1) the preparation for the experiment, (2) the teaching experiment and (3) the retrospective analysis (Gravemeijer & van Eerde, 2009). In the first phase, a HLT was elaborated (Simon, 1995) based on emergent models and mathematical modelling. In the experimental teaching phase, three design experimental cycles were developed in which the initial HLT was refined. In Table 1, we present a summary of the difficulties and modifications that were made in cycles 1 and 2 of this research (for more details of these cycles, see the articles Cárcamo et al., 2016, 2017). Table 1. Summary of the difficulties and modifications that were made in cycles 1 and 2 of this research Cycle 1   Cycle 2   Difficulty  Modification to cycle 2  Difficulty  Modification to cycle 3  Use of mathematical language.  The teacher promotes the use of correct mathematical language.  The mathematical notation of spanning set.  The method for delivering the definitions of the concepts was modified once again, with emphasis to their names and notations.  Obtain a span of R2 given a certain spanning set of this span.  Questions are added in task 3, about spanning set and span properties.  The mathematical notation of set.  Graph a span of R2.  This question is deleted.  Designate the same name to the span and to a spanning set that generates it.  The different ways to denote span.  It modifies the way that definitions of the concepts are given, with emphasis to their names and notations.      Associate the spanning set and span names to your notations in an interchangeable way.  Name another spanning set by associating it with its mathematical notation.  Cycle 1   Cycle 2   Difficulty  Modification to cycle 2  Difficulty  Modification to cycle 3  Use of mathematical language.  The teacher promotes the use of correct mathematical language.  The mathematical notation of spanning set.  The method for delivering the definitions of the concepts was modified once again, with emphasis to their names and notations.  Obtain a span of R2 given a certain spanning set of this span.  Questions are added in task 3, about spanning set and span properties.  The mathematical notation of set.  Graph a span of R2.  This question is deleted.  Designate the same name to the span and to a spanning set that generates it.  The different ways to denote span.  It modifies the way that definitions of the concepts are given, with emphasis to their names and notations.      Associate the spanning set and span names to your notations in an interchangeable way.  Name another spanning set by associating it with its mathematical notation.  Table 1. Summary of the difficulties and modifications that were made in cycles 1 and 2 of this research Cycle 1   Cycle 2   Difficulty  Modification to cycle 2  Difficulty  Modification to cycle 3  Use of mathematical language.  The teacher promotes the use of correct mathematical language.  The mathematical notation of spanning set.  The method for delivering the definitions of the concepts was modified once again, with emphasis to their names and notations.  Obtain a span of R2 given a certain spanning set of this span.  Questions are added in task 3, about spanning set and span properties.  The mathematical notation of set.  Graph a span of R2.  This question is deleted.  Designate the same name to the span and to a spanning set that generates it.  The different ways to denote span.  It modifies the way that definitions of the concepts are given, with emphasis to their names and notations.      Associate the spanning set and span names to your notations in an interchangeable way.  Name another spanning set by associating it with its mathematical notation.  Cycle 1   Cycle 2   Difficulty  Modification to cycle 2  Difficulty  Modification to cycle 3  Use of mathematical language.  The teacher promotes the use of correct mathematical language.  The mathematical notation of spanning set.  The method for delivering the definitions of the concepts was modified once again, with emphasis to their names and notations.  Obtain a span of R2 given a certain spanning set of this span.  Questions are added in task 3, about spanning set and span properties.  The mathematical notation of set.  Graph a span of R2.  This question is deleted.  Designate the same name to the span and to a spanning set that generates it.  The different ways to denote span.  It modifies the way that definitions of the concepts are given, with emphasis to their names and notations.      Associate the spanning set and span names to your notations in an interchangeable way.  Name another spanning set by associating it with its mathematical notation.  The data collected in each design experimental cycles were as follows: the written responses of the students to the tasks proposed in the HLT, audio and video recordings of group work and an individual interview at the end of the experimentation. The data of each cycle were analyzed comparing the HLT with the actual learning trajectory (ALT) through the matrix proposed by Dierdorp et al. (2011), looking for background to support or rebut the conjectures of the HLT. Also, the emergent models were used to interpret students’ progress in constructing the concepts of spanning set and span. The results that we presented correspond to the third cycle of this design-based research. The participants in the study were from a course group composed of 7 first-year engineering students from a public university in Spain. This course group, like the other first-year engineering groups, had not previously worked with mathematical modelling, nor had they studied the concepts of spanning set and span. The choice of this course group was based on the interest of its students in participating in the teaching experiment and for their communicative competence. This experiment was carried out over 5 h, divided into three class sessions, in which the students worked in two groups (group A of four students and group B of three students). The teacher guided the students in the resolution of the tasks of the HLT, and he led the discussion of topics that he considered relevant to favour learning. For example, in task 1, the teacher guided the students towards the solutions of the problem using the mathematical modelling cycle proposed by Blum and Leiss (2007). In addition, the teacher provided guidelines for constructing the mathematical model for generating passwords (suggesting the use of vectors or linear combinations) and create a coding for his password generator (suggesting the use of tables and giving them two examples). The HLT was composed of four tasks. The first task made use of an experientially real problem setting to support the construction of the concepts of spanning and span. This setting is experientially real for students in that it utilizes both their mathematical knowledge and their experience with passwords as a foundation on which to build more formal mathematics. In this task, mathematical modelling serves as a tool to help the study of mathematics (Julie & Mudaly, 2007). We consider the modelling cycle proposed by Blum and Leiss (2007) to guide students in solving this task. On the other hand, in this study, we emphasize how the four levels of activity linked to the emergent models heuristic can manifest themselves within the tasks of HLT. Table 2 summarizes the tasks and major conjectures of the learning trajectory of the HLT and how students are expected to manifest their activity levels in each of the tasks of HLT. Table 2. Summary of the tasks and major conjectures of the learning trajectory of the HLT and how students are expected to manifest their activity levels in each of the tasks of HLT Task and its major conjectures of learning trajectory  Manifestation of activity levels  Task 1: create a password generator based on vectors. Major conjectures: (1) Students read information from the secure passwords; (2) students created a password generator by following the steps of the modelling cycle and using their previous knowledge of vectors and passwords.  Situational activity. Students begin the construction of the concepts of spanning set and span in the context of passwords. Students explore different ways of using vectors and linear combinations to propose a mathematical model that generates passwords.  Task 2: make an analogy table between their password generator and the concepts of spanning set and span. Major conjectures: (1) Students find two sets from their password generator (one which has all the vectors that allow for the generation of the numerical passwords and the other which contains the vectors that, after creating the linear combination, is obtained by the vector for each numeric password with them); (2) students identify common features between two sets connected to their password generator and the concepts of spanning set and span; (3) students distinguish between spanning set and span with the analogy table.  Referential activity. Students use the set notations along with the definitions of spanning set and span to find two sets linked to their password generator that in mathematical notation correspond to the concepts of spanning set and span, respectively. These two sets function as models-of with respect to their previous activity with passwords and with vectors.  Task 3: Conjecture what the rank of the matrix should be so that its rows generate R2. Major conjectures: (1) Students explore particular cases and identify some regularity; (2) students relate spanning set and span with other concepts; (3) students conjecture the number of vectors in a set needed to generate R2; (4) students determine that the number of vectors is not sufficient to indicate if a set generates the span of R2.  General activity. Students explore, reflect and make new conjectures about sets of vectors in Rn that correspond to spanning sets or spans and that do not refer to sets within the context of passwords. The two sets linked to the context of the referential level passwords function as models-for reasoning about properties of this type of set, but without hinting at the initial situation.  Task 4: Indicate if the set C = {(1,0,0,1),(0,1,0,0)} is a spanning set for span W = {(x,y,z,w)/x=w}. Major conjectures: Students to pose a solution: (1) they explore possible routes for resolution; (2) they find a spanning set or span (according to the resolution they decided); (3) they verify if the set C is a generator of W.  Formal activity. Students work with the conventional notations of spanning set and span as well as their properties in situations that differ from those presented in previous levels, allowing them to demonstrate an overall understanding of these concepts.  Task and its major conjectures of learning trajectory  Manifestation of activity levels  Task 1: create a password generator based on vectors. Major conjectures: (1) Students read information from the secure passwords; (2) students created a password generator by following the steps of the modelling cycle and using their previous knowledge of vectors and passwords.  Situational activity. Students begin the construction of the concepts of spanning set and span in the context of passwords. Students explore different ways of using vectors and linear combinations to propose a mathematical model that generates passwords.  Task 2: make an analogy table between their password generator and the concepts of spanning set and span. Major conjectures: (1) Students find two sets from their password generator (one which has all the vectors that allow for the generation of the numerical passwords and the other which contains the vectors that, after creating the linear combination, is obtained by the vector for each numeric password with them); (2) students identify common features between two sets connected to their password generator and the concepts of spanning set and span; (3) students distinguish between spanning set and span with the analogy table.  Referential activity. Students use the set notations along with the definitions of spanning set and span to find two sets linked to their password generator that in mathematical notation correspond to the concepts of spanning set and span, respectively. These two sets function as models-of with respect to their previous activity with passwords and with vectors.  Task 3: Conjecture what the rank of the matrix should be so that its rows generate R2. Major conjectures: (1) Students explore particular cases and identify some regularity; (2) students relate spanning set and span with other concepts; (3) students conjecture the number of vectors in a set needed to generate R2; (4) students determine that the number of vectors is not sufficient to indicate if a set generates the span of R2.  General activity. Students explore, reflect and make new conjectures about sets of vectors in Rn that correspond to spanning sets or spans and that do not refer to sets within the context of passwords. The two sets linked to the context of the referential level passwords function as models-for reasoning about properties of this type of set, but without hinting at the initial situation.  Task 4: Indicate if the set C = {(1,0,0,1),(0,1,0,0)} is a spanning set for span W = {(x,y,z,w)/x=w}. Major conjectures: Students to pose a solution: (1) they explore possible routes for resolution; (2) they find a spanning set or span (according to the resolution they decided); (3) they verify if the set C is a generator of W.  Formal activity. Students work with the conventional notations of spanning set and span as well as their properties in situations that differ from those presented in previous levels, allowing them to demonstrate an overall understanding of these concepts.  Table 2. Summary of the tasks and major conjectures of the learning trajectory of the HLT and how students are expected to manifest their activity levels in each of the tasks of HLT Task and its major conjectures of learning trajectory  Manifestation of activity levels  Task 1: create a password generator based on vectors. Major conjectures: (1) Students read information from the secure passwords; (2) students created a password generator by following the steps of the modelling cycle and using their previous knowledge of vectors and passwords.  Situational activity. Students begin the construction of the concepts of spanning set and span in the context of passwords. Students explore different ways of using vectors and linear combinations to propose a mathematical model that generates passwords.  Task 2: make an analogy table between their password generator and the concepts of spanning set and span. Major conjectures: (1) Students find two sets from their password generator (one which has all the vectors that allow for the generation of the numerical passwords and the other which contains the vectors that, after creating the linear combination, is obtained by the vector for each numeric password with them); (2) students identify common features between two sets connected to their password generator and the concepts of spanning set and span; (3) students distinguish between spanning set and span with the analogy table.  Referential activity. Students use the set notations along with the definitions of spanning set and span to find two sets linked to their password generator that in mathematical notation correspond to the concepts of spanning set and span, respectively. These two sets function as models-of with respect to their previous activity with passwords and with vectors.  Task 3: Conjecture what the rank of the matrix should be so that its rows generate R2. Major conjectures: (1) Students explore particular cases and identify some regularity; (2) students relate spanning set and span with other concepts; (3) students conjecture the number of vectors in a set needed to generate R2; (4) students determine that the number of vectors is not sufficient to indicate if a set generates the span of R2.  General activity. Students explore, reflect and make new conjectures about sets of vectors in Rn that correspond to spanning sets or spans and that do not refer to sets within the context of passwords. The two sets linked to the context of the referential level passwords function as models-for reasoning about properties of this type of set, but without hinting at the initial situation.  Task 4: Indicate if the set C = {(1,0,0,1),(0,1,0,0)} is a spanning set for span W = {(x,y,z,w)/x=w}. Major conjectures: Students to pose a solution: (1) they explore possible routes for resolution; (2) they find a spanning set or span (according to the resolution they decided); (3) they verify if the set C is a generator of W.  Formal activity. Students work with the conventional notations of spanning set and span as well as their properties in situations that differ from those presented in previous levels, allowing them to demonstrate an overall understanding of these concepts.  Task and its major conjectures of learning trajectory  Manifestation of activity levels  Task 1: create a password generator based on vectors. Major conjectures: (1) Students read information from the secure passwords; (2) students created a password generator by following the steps of the modelling cycle and using their previous knowledge of vectors and passwords.  Situational activity. Students begin the construction of the concepts of spanning set and span in the context of passwords. Students explore different ways of using vectors and linear combinations to propose a mathematical model that generates passwords.  Task 2: make an analogy table between their password generator and the concepts of spanning set and span. Major conjectures: (1) Students find two sets from their password generator (one which has all the vectors that allow for the generation of the numerical passwords and the other which contains the vectors that, after creating the linear combination, is obtained by the vector for each numeric password with them); (2) students identify common features between two sets connected to their password generator and the concepts of spanning set and span; (3) students distinguish between spanning set and span with the analogy table.  Referential activity. Students use the set notations along with the definitions of spanning set and span to find two sets linked to their password generator that in mathematical notation correspond to the concepts of spanning set and span, respectively. These two sets function as models-of with respect to their previous activity with passwords and with vectors.  Task 3: Conjecture what the rank of the matrix should be so that its rows generate R2. Major conjectures: (1) Students explore particular cases and identify some regularity; (2) students relate spanning set and span with other concepts; (3) students conjecture the number of vectors in a set needed to generate R2; (4) students determine that the number of vectors is not sufficient to indicate if a set generates the span of R2.  General activity. Students explore, reflect and make new conjectures about sets of vectors in Rn that correspond to spanning sets or spans and that do not refer to sets within the context of passwords. The two sets linked to the context of the referential level passwords function as models-for reasoning about properties of this type of set, but without hinting at the initial situation.  Task 4: Indicate if the set C = {(1,0,0,1),(0,1,0,0)} is a spanning set for span W = {(x,y,z,w)/x=w}. Major conjectures: Students to pose a solution: (1) they explore possible routes for resolution; (2) they find a spanning set or span (according to the resolution they decided); (3) they verify if the set C is a generator of W.  Formal activity. Students work with the conventional notations of spanning set and span as well as their properties in situations that differ from those presented in previous levels, allowing them to demonstrate an overall understanding of these concepts.  4. Results In this section, we describe how the reasoning used by the student groups progressed through each of the four levels of activity using the tasks of the HLT that they developed. In accordance with Rasmussen and Blumenfeld (2007) ‘these different levels offer a journey through students’ mathematical thinking without imposing a strict hierarchy’ (p. 201). 4.1 Situational activity With respect to the first of the four levels of activity, ‘situational activity involves students working toward mathematical goals in an experientially real setting’ (Zandieh & Rasmussen, 2010, p. 58). The context of generating secure passwords serves as an experientially real starting point for initiating the construction of spanning set and span. In their development of task 1, student groups showed they were at the level of situational activity because they explored different ways of using vectors and linear combinations towards the goal of creating a numerical password generator based on vectors. This is exemplified by the following dialogue of group A where it was observed that they made use of their previous conceptions of vectors and that the students S3 and S1 of this group proposed different ways of using the vectors to propose a solution: S3: First, what are we going to use? What operations are we going to do with the vectors? Are we going to add them up, multiply them or something like that? S1: A number that multiplies each vector and then, add the vectors. S3: How many vectors will we make? S4: Three. S2: Three vectors. S1: Three vectors. Student S3 started questioning how they would use the vectors to propose a solution for task 1. Student S3 asked to his companions ‘What operations are we going to do with the vectors? Are we going to add them up, multiply them or something like that?’. Student S1 suggested ‘A number that multiplies each vector and then, add the vectors’. That is, implicitly, student S1 proposes to make a linear combination. Then, student S3 asked about the number of vectors their model would contain and the other students in the group answered ‘three’ vectors. The ideas proposed by the students of group A led them to propose their mathematical model to generate the numerical passwords presented in Table 3. Table 3. Some of the written work that the groups A and B carried out to propose a solution for the task 1 Group A  Group B  Mathematical model      Codification      Example of how their password generator operates      Group A  Group B  Mathematical model      Codification      Example of how their password generator operates      Table 3. Some of the written work that the groups A and B carried out to propose a solution for the task 1 Group A  Group B  Mathematical model      Codification      Example of how their password generator operates      Group A  Group B  Mathematical model      Codification      Example of how their password generator operates      Table 3 presents some of the written work that the groups A and B carried out to propose a solution for the task 1, which they carried out following the steps of the modelling cycle (Blum & Leiss, 2007), but adapted to task 1 (Cárcamo et al., 2016). Table 3 shows that the mathematical model for generating numerical passwords for group A corresponded to a linear combination of vectors of R3 while that of group B was a linear combination of vectors of R4. To develop passwords encoded from numerical passwords, group A designed a mixed coding set, unlike group B, which created their own coding set. Group A created a two-line code: the first line of ‘code’ contains letters and symbols designated by them, whereas the second row ‘ASCii’ consists of characters taken from the ASCII code. The coding of group A operated as follows: those numerical password numbers that were in an even position were replaced by ‘code’ and those in an odd position were replaced by ‘ASCii’. On the other hand, group B designed their own code composed of numbers, letters and symbols that is divided into two rows: ‘code 1’ and ‘code 2.’ To transform a numeric password to a coded one, group B changed the number of the second position of the numeric password using a character from ‘code 1’ and in addition, it replaced both the first and last number of it using ‘code 2’. The groups showed how they operated their password generator by giving an example (Table 3). Group A gave specific values to the letters of its mathematical model and obtained as a numerical password the vector (3,5,4), corresponding to the numeric password 354. Consequently, they used their own code and had the encrypted password ETXZEOT. Group B followed a process similar to Group A and generated the numeric password 52302166, which was transformed into the encrypted password//430216\\. The solution given by the two groups for task 1 allowed for these students’ level of situational activity to be shown. Also, their solutions allowed for the observation that they made use of their previous conceptions of vectors, and used the steps of the mathematical modelling cycle to propose a mathematical model that generates numeric passwords. They will use this vector-based mathematical model to obtain two sets that are related to the context of the passwords, but also with the concepts of spanning set and span. In this way, they are given an initial familiarization with these concepts of Linear Algebra. 4.2 Referential activity After students established their mathematical models to create a numerical password generator based on vectors, the teacher introduced the definitions of generating set and generated space in relation to the work and ideas established by the students. The introduction of these definitions served to support students in the transition from situational activity (work done within the context of passwords) to referential activity (work done with sets within the context of passwords that are examples of spanning set and span). With respect to the four levels of activity, Task 2 led the students towards the referential activity that, according to Gravemeijer (1999), implies the use of descriptions, concepts and procedures that relate to the problem of the situational activity; in this case, of task 1. Task 2 asked the students to make an analogy table between their password generator that they created in task 1 and the concepts of spanning set and span. In this task, the students showed that they were at the reference level, due to the fact that they used set notations and definitions to find two sets linked to their password generator: one representing a spanning set and another describing a span. To find these two sets, the students used the mathematical model they had designed in task 1. In the following dialogue between students S1 and S3 of group A, we observe how their ideas for relating the span with a set from their password generator came about: S1: Okay, span. S3: It’s the formula. S1: Describes all the vectors that can be made. S3: Describes the operations to find the numerical vector that generates a password. S1: All the passwords that could exist. To respond to the relationship between the password generator and span, student S3 made a very general comment, noting that ‘it’s the formula’, in response to which student S1, said that the span ‘describes all the vectors that can be made’. This comment from student S1 is general because it does not specify what the purpose of those vectors is and does not mention the link with the passwords. Perhaps, for this reason, student S3 specified that the span ‘describes the operations to find the numerical vector that generates a password’. Student S3 mentioned that the span describes operations. He linked this concept to the context of passwords when he specified that these ‘operations’ served to find the numeric vector that generates a password. From the written response of group A (Table 4), it follows that when student S3 talked about operations, he was referring to those that have a linear combination (addition and multiplication). The answer was the set {(a,b,c) ε R3: a(1,0,1) + b(1,1,0) + c(0,1,1)}. However, student S1 disagreed with the last statement made by student S3, and he pointed out that the span has ‘all the passwords that could exist’. This statement made by student S1 allows it to be inferred that he is thinking of the span as the set that contains all the vectors that allow for numerical passwords to be generated. Table 4. Analogy tables of task 2 of groups A and B Group A  Mathematical name for this concept  Name given in your password generator  How it is written in mathematical language  Spanning set  Vectors that generate numeric passwords.  {(1,0,1),(1,1,0),(0,1,1)}  (a) Span  Describes the operations to find the numerical vector that generates a password  {(a,b,c)∈R3:a(1,0,1)+b(1,1,0)+c(0,1,1)}  Group B  Spanning set  Spanning set V allows to generate from a pattern different numerical passwords  V={(5,0,0,0),(0,−10,0,0),(0,0,−18,0),(0,0,0,3)}  (b) Span  Span V provides the characteristics of said pattern while creating our passwords.  V={(5a,−10b,−18c,3d)∈R4}  Group A  Mathematical name for this concept  Name given in your password generator  How it is written in mathematical language  Spanning set  Vectors that generate numeric passwords.  {(1,0,1),(1,1,0),(0,1,1)}  (a) Span  Describes the operations to find the numerical vector that generates a password  {(a,b,c)∈R3:a(1,0,1)+b(1,1,0)+c(0,1,1)}  Group B  Spanning set  Spanning set V allows to generate from a pattern different numerical passwords  V={(5,0,0,0),(0,−10,0,0),(0,0,−18,0),(0,0,0,3)}  (b) Span  Span V provides the characteristics of said pattern while creating our passwords.  V={(5a,−10b,−18c,3d)∈R4}  Table 4. Analogy tables of task 2 of groups A and B Group A  Mathematical name for this concept  Name given in your password generator  How it is written in mathematical language  Spanning set  Vectors that generate numeric passwords.  {(1,0,1),(1,1,0),(0,1,1)}  (a) Span  Describes the operations to find the numerical vector that generates a password  {(a,b,c)∈R3:a(1,0,1)+b(1,1,0)+c(0,1,1)}  Group B  Spanning set  Spanning set V allows to generate from a pattern different numerical passwords  V={(5,0,0,0),(0,−10,0,0),(0,0,−18,0),(0,0,0,3)}  (b) Span  Span V provides the characteristics of said pattern while creating our passwords.  V={(5a,−10b,−18c,3d)∈R4}  Group A  Mathematical name for this concept  Name given in your password generator  How it is written in mathematical language  Spanning set  Vectors that generate numeric passwords.  {(1,0,1),(1,1,0),(0,1,1)}  (a) Span  Describes the operations to find the numerical vector that generates a password  {(a,b,c)∈R3:a(1,0,1)+b(1,1,0)+c(0,1,1)}  Group B  Spanning set  Spanning set V allows to generate from a pattern different numerical passwords  V={(5,0,0,0),(0,−10,0,0),(0,0,−18,0),(0,0,0,3)}  (b) Span  Span V provides the characteristics of said pattern while creating our passwords.  V={(5a,−10b,−18c,3d)∈R4}  In Table 4 we present the analogy tables made by each group, in which it is shown that they correctly associated the sets in mathematical notation that arise from their mathematical model to generate passwords using the concepts of spanning set and span. However, we observed that group B designated the letter V for both sets. From this, it could be inferred that the students from this group considered that the two sets are equal, but they give a description of these in the context of their password generator that shows that they differentiate between them, and that the fact they assign the letter V to both set is due only to a notation error. On the other hand, in the second column of the analogy table that asks for the name or what corresponds in their password generator. Group B wrote that the spanning set ‘allows to generate from a pattern different numerical passwords’ while the span ‘provides the characteristics of said pattern while creating our passwords.’ We deduced that group B used the term ‘pattern’ to refer to the vectors of the spanning set, since it was these vectors that allowed the generation of numerical passwords. It was important for the students to make the analogy table and to link the real context with the mathematical one. From this form, they continued to expand their grasp of spanning set and span, given that they created a preliminary example of both concepts in analytical notation, and established a distinction between these concepts when they were required to relate them to sets derived from a real situation. In the referential activity, the two sets linked to the context of the passwords, which were also used as examples of spanning set and span, functioned for the students as a model-of their previous mathematical activity with the passwords and the vectors. In the next section, students are presented with a more challenging question that allows them to progress towards a general level of activity. In other words, the students begin to change the model they use: from thinking of two sets that are examples of spanning set and span to thinking in terms of properties of these types of sets, but without alluding to the scenario with the passwords. 4.3 General activity One of the goals of task 3 was to shift students away from situational and referential activity with respect to the context of the passwords, and towards a general level of activity with respect to the concepts. General activity involves exploring, reflecting and generalizing the material that appeared at the previous level, but without referencing the initial context (Zolkower & Bressan, 2012). The students showed they were at this level of activity when they explored, reflected and made conjectures about sets of vectors in Rn that corresponded to spanning sets or spans and that did not refer to the scenario in task 1. For example, for question (a) of task 3, students conjectured: what should be the rank of the matrix that has a set of R2 as vector rows such that this set generates R2? The groups, after calculating the rank of four matrices (M1, M2, M3, and M4), correctly conjectured that the rank of the matrix having a set of R2 as vector rows must be two in order for it to generate R2. This is shown in Table 5. Table 5. Written answer of groups A and B to question (a) of task 3 Group A  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: If the rank is 2, the set can generate R2.  A={(0,−3)}  M1=(0−3)  1  B={(5,0),(7,0)}  M2=(5070)  1  C={(1,0),(1,−1)}  M3=(1   01−1)  2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  2  Group B  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: The rank of the matrix must be equal to 2.  A={(0,−3)}  M1=(0−3)  RM1=1  B={(5,0),(7,0)}  M2=(5070)  RM2=1  C={(1,0),(1,−1)}  M3=(1   01−1)  RM3=2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  RM4=2  Group A  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: If the rank is 2, the set can generate R2.  A={(0,−3)}  M1=(0−3)  1  B={(5,0),(7,0)}  M2=(5070)  1  C={(1,0),(1,−1)}  M3=(1   01−1)  2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  2  Group B  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: The rank of the matrix must be equal to 2.  A={(0,−3)}  M1=(0−3)  RM1=1  B={(5,0),(7,0)}  M2=(5070)  RM2=1  C={(1,0),(1,−1)}  M3=(1   01−1)  RM3=2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  RM4=2  Table 5. Written answer of groups A and B to question (a) of task 3 Group A  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: If the rank is 2, the set can generate R2.  A={(0,−3)}  M1=(0−3)  1  B={(5,0),(7,0)}  M2=(5070)  1  C={(1,0),(1,−1)}  M3=(1   01−1)  2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  2  Group B  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: The rank of the matrix must be equal to 2.  A={(0,−3)}  M1=(0−3)  RM1=1  B={(5,0),(7,0)}  M2=(5070)  RM2=1  C={(1,0),(1,−1)}  M3=(1   01−1)  RM3=2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  RM4=2  Group A  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: If the rank is 2, the set can generate R2.  A={(0,−3)}  M1=(0−3)  1  B={(5,0),(7,0)}  M2=(5070)  1  C={(1,0),(1,−1)}  M3=(1   01−1)  2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  2  Group B  Sets  Matrix whose rows are the vectors of set  Rank of matrix  Conjecture: The rank of the matrix must be equal to 2.  A={(0,−3)}  M1=(0−3)  RM1=1  B={(5,0),(7,0)}  M2=(5070)  RM2=1  C={(1,0),(1,−1)}  M3=(1   01−1)  RM3=2  D={(−1,−4),(2,8),(0,−1)}  M4=(−1−4   2  8   0−1)  RM4=2  To answer question (a) of task 3, in group A, student S1 indicated that the rank of the matrix M1 ‘is 1’ and then, referring to the other matrices, noted that ‘this is also 1 (showing M2). Yes, this is two (indicating M3). One, one, two, two’. When student S1 mentions one, one, two, two, he alludes to the ranks of the matrices M1, M2, M3, and M4, respectively, for that is what appears in the written answer (Table 5). From this, the students of group A surmised that if the rank is 2, the set can generate R2. The fact that group A has put in its conjecture the word ‘may’ indicate to us that they left open the possibility for there to be a set for which the associated matrix has rank 2, but does not generate R2. The students from group A did not express any idea about the characteristics of the vectors that are contained in each matrix. However, student S3 from group B did comment on it, as he pointed out to his group that the vectors of set B = {(5,0),(7,0)} ‘form a linear combination between them’ and then also indicated that ‘they are not independent of each other.’ Therefore, the rank of the matrix associated with this set is one. The affirmations of student S3 from group B allowed for us to infer that he was able to identify when the vectors are linearly dependent, as was the case of the set B = {(5,0),(7,0)}. From the written response of both groups (Table 5), we infer that they noticed that a spanning set of R2 can have more than two vectors (such as set D), but that the minimum must be two vectors that are linearly independent (such as set C). That is, it is not enough that the set has two vectors (such as set B). From this it follows that the students observed that the number of vectors is not sufficient to indicate if a set generates R2, because apart from the fact that they are two vectors, they must be linearly independent. The formation of this conjecture was fundamental, because it made the students relate spanning set and span with the concepts of the rank of a matrix and linear independence. In addition, they identified a property that characterizes the sets that generate the span of R2, which indicates that the number of vectors is not sufficient to indicate if a set generates R2. At a general level, the two sets related to the context of the referential-level passwords functioned for students as models-for, to reason about properties of this type of set, but without referring to the scenario of task 1. 4.4 Formal activity Formal activity, as described by Gravemeijer (1999), involves reasoning with conventional symbolism, which is no longer dependent on the support of models-for mathematical activity. For their part, Zolkower and Bressan (2012) specify that formal activity involves students working with conventional procedures and notations disconnected from the situations that gave them their initial meaning. We consider that the development made by students in task 4 of the HLT gave indications that they may have reached the level of formal activity, because they worked implicitly with the concepts of spanning set and span in a situation different from those presented for the previous levels. In task 4, the students were asked questions different from those they had previously been asked in the tasks of the HLT. For example, they answered the question: indicate if C = {(1,0,0,1), (0,1,0,0)} is a spanning set for the span W = {(x,y,z,w)/x = w}. The students of the groups gave evidence that they were at the formal activity level when they answered this question (see answer in Table 6), because we have inferred that they used the definitions of spanning set and span to solve it, although this was not indicated in their process for finding the solution to this question. Table 6. Written answers of groups A and B to the question: indicate if C = {(1,0,0,1), (0,1,0,0)} is a spanning set for the span W = {(x,y,z,w)/x = w} Group A    Group B    Group A    Group B    Table 6. Written answers of groups A and B to the question: indicate if C = {(1,0,0,1), (0,1,0,0)} is a spanning set for the span W = {(x,y,z,w)/x = w} Group A    Group B    Group A    Group B    Both groups pointed out that the set C is not a spanning set for W (Table 6). In the case of group A, to reach this conclusion, they followed a process to determine a spanning set of W, as shown in the dialogue between students S1 and S2 of this group: S2: (looking to the set W) condition x is equal to w. S1: we are going to replace it. S2: you do the operation. S1: Look at this (S1 shows what he wrote to S2) … there w 1 0 0 1. I was left 0 1 … 0 0. No, it is not because of the z component. S2: the z nothing, it does not influence. S1: this (indicating the set {(1,0,0,1), (0,1,0,0)}) is not a spanning set of this (showing W). Look (showing them w(1,0,0,1) + y(0,1,0,0) + z(0,0,1,0)). Student S2 expressed (looking to the set W) that ‘condition x is equal to w’ and immediately, student S1 told his colleagues ‘we are going to replace it’. Student S1 referred to substitution, the condition that was indicated by student S2, in the generic vector of the span W, since that was what they did and wrote the linear combination w(1,0,0,1) + y(0,1,0,0) + z(0,0,1,0). On seeing it, student S1 stated that the set {(1,0,0,1),(0,1,0,0)} is not a spanning set for W because of ‘component z’ and, although student S2 pointed out that z does not influence, student S1 added that ‘this (indicating the set{(1,0,0,1), (0,1,0,0)}) is not a spanning set of this (showing W). Look (showing them w(1,0,0,1) + y(0,1,0,0) + z(0,0,1,0))’. Student S1 proposed to his colleagues that they compare the expression of the three vectors with the set that gave them two. From this, they concluded that it is not a spanning set of W. On the other hand, group B pointed out in its written answer (Table 6) that C is not a spanning set of W because it lacks the vector (0, 0, 1, 0). From this, it is inferred that these students carried out a process similar to that of group A. This is because the vector they mention is associated with the z component in the linear combination shown by students in group A. In the responses of both groups, we observe that they verify if the set C is generator of W and give a correct answer. This is similar to the description of the THA proposed for task 4 (Table 2). We consider that the students would have given evidence that they were at the formal activity level in this question in task 4 (indicate whether C = {(1,0,0,1), (0,1,0,0)} is a (x, y, z, w)/x = w}) if they had explicitly stated the use of spanning set or span during their attempt to solve the problem. For example, if they had indicated: ‘we are going to determine a spanning set for W to see if it is C’ or ‘for C to be a spanning set, any vector of W must be a linear combination of the vectors of C’. However, this did not occur. It should be noted that emergent models do not specify what happens when students do not give evidence of being at the formal level, but rather that they are in the transition from the general level to the formal level. This may be an indicator of the existence of a new level between the general and formal levels, or the existence of sub-levels within each level. 5. Discussion and conclusions The main contribution of this research is to show how the emergent models can be a good alternative for the design of a HLT that favors the construction of the students of the concepts of spanning set and span through an HLT for which the initial context is the creation of secure passwords. In this way, it is also intended to contribute to the field of application of emergent models at the university level, since few studies have been conducted in this area. In order to respond to the objective of the study, we reconstructed the reasoning groups of students used throughout each of the levels of activity. We observe that students transitioned from their model-of informal mathematical activity (using passwords and vectors) to a model-for formal mathematical reasoning (on the concepts of spanning set and span). Therefore, we can point out that the emergent models supported the students’ development of the concepts of spanning set and span. The level of situational activity implied that students would work towards mathematical goals in an experientially real environment, particularly in the context of creating secure passwords. In task 1, students determined how to create a password generator based on vectors. Each group developed a mathematical model that, in both cases, was a linear combination (Table 3) and that complied with the requested characteristics (create a password generator based on vectors). Consequently, the level of referential activity required students to work with set notations and with the definitions of spanning set and span in the password scenario. This was done to determine two sets that referred to the context of the passwords, but that also were examples of spanning set and span (Table 3). For the students, these two sets were models-of of what worked in the context of passwords and vectors. At the level of general activity, the two sets that were models-of at the referential activity level became models-for for reasoning about properties of this type of set of Rn, but without hinting at the initial situation. For example, Table 5 shows that students gave correct conjecture as to what the rank of the matrix should be such that the rows generate R2. Finally, students gave indications of being at the formal activity level, because, for example, in solving a question from task 4 (Table 6), they worked implicitly with the concepts of spanning set and span in a situation different from those presented at previous levels. The choice of the emergent model that will support students in the transition from the model-of to the model-for is considered to be essential. This information will allow the teacher to better guide the students in their process of constructing mathematical knowledge. In this study, the emergent models were the two sets that initially emerged as models-of of what worked in task 1, that is, with passwords and vectors. Gradually, however, these two sets changed in nature as students began to change their outlook on these two sets; from seeing the two sets as examples of spanning set and span to reasoning about properties of these types of sets. Finally, these two sets became models-for for reasoning about the properties of spanning set and span. These emergent models allowed students to transition from real experience (passwords and vectors) to a new mathematical reality (spanning set and span). We agree with the results obtained by Rasmussen and Blumenfeld (2007), in their research conducted on the differential equations course, in relation to which analytic expressions can support the entire transition from the model-of to the model-for. This occurred in our study with the construction of the concepts of spanning set and span. In our case, we observed that students began their work with vectors and linear combinations (situational activity), then continued with sets of vectors (referential activity and general activity) and finally with conventional mathematical procedures and notations (formal activity). With respect to the tasks applied in this study, Table 2 shows a summary of the HLT and how students were expected to manifest their transition by activity levels. The results provided evidence that students progressed through the different levels of activity by solving the task sequence of the HLT. This indicates to us that there was a close relationship between the HLT and the current trajectory of student learning. However, in order for the teacher to enhance the students’ reasoning in their process of solving tasks 3 and 4, we suggested that the teacher ask the students to justify the steps of their solution to a given question by, for example, the use of the definitions of spanning set or span or its properties, or indicating how these concepts were used in their proposed resolution. We recognize that the tasks and the order in which they are presented to students are critical in facilitating their transition from their informal knowledge to more formal mathematical reasoning. We consider that it is important for each task to offer the student the possibility of fully passing through the level of activity that is intended for this activity, otherwise it will need to be presented differently. In this study, we observed that the sequence of tasks of the HLT tended to lead students to transit through different levels of activity. This was not a coincidence, but rather a result of the two previous refinements of the HLT, that is to say, as indicated by Wawro et al. (2013), it ‘was made possible by the cyclical process of the design research’ (p. 911). On the other hand, we agree with Dawkins (2015) that the teacher should support the students in the basic processes that are involved in the levels of activity. However, we also consider it important that, prior to this, the teacher should do the following: choose an initial activity that is experientially real, conjecture about what the emergent model will be (the model guiding the transition between model-of and model-for) and develop key questions that promote the advancement of students between levels of activity. The results showed that the HLT based on the emergent models contributed to the students’ progression from their informal mathematical activity to a more formal mathematical form of reasoning regarding the concepts of spanning set and span. Therefore, this didactic proposal could be useful as a teaching tool to be applied in another classroom (previously adapted to its context), but also as a tool for teachers to use as an example in their consideration aimed at devising instructional sequences based on this heuristic design. We consider that a limitation of emergent models is that they do not detail possible sub-levels within each level which would be useful to help students progress within each level and bring them to the next level. Another limitation is that it does not detail what happens in the transitions between one level and another. For future research, the challenge will be to design HLTs based on the emergent models for other Linear Algebra concepts with the objective of determining if this approach also contributes to the construction of these. Andrea Cárcamo is a PhD candidate in the area of mathematics education and experimental sciences teaching at the Autonomous University of Barcelona. Professor at the Centre for Teaching Basic Science Engineering at the Austral University of Chile. Her research focuses on teaching Linear Algebra and mathematical modelling. Josep Fortuny is PhD in Mathematics. Professor at the Department of Teaching Mathematics and Experimental Sciences at the Autonomous University of Barcelona. His research focuses on learning and interaction. Claudio Fuentealba is a PhD candidate in the area of mathematics education and experimental sciences teaching at the Autonomous University of Barcelona. Professor at the Centre for Teaching Basic Science Engineering at the Austral University of Chile. Her research focuses on teaching Linear Algebra and Calculus. Acknowledgement We would like to express a gratitude to the participants of the seminar led by Dr. Asuman Oktaç for their suggested improvement about this article. References Blum W., Leiss D. ( 2007) How do students and teachers deal with modelling problems? 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( 2010) Defining as a mathematical activity: a framework for characterizing progress from informal to more formal ways of reasoning. J. Math. Behav ., 29, 57– 75. Google Scholar CrossRef Search ADS   Zolkower B., Bressan A. ( 2012) Educación matemática realista. Educación Matemática. Aportes a la Formación Docente desde Distintos Enfoques Teóricos  ( Pochulu M., Rodríguez M. eds). Argentina: UNGS–EDUVIM, pp. 175– 200. © The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com

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Teaching Mathematics and Its Applications: International Journal of the IMAOxford University Press

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