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Using an economics model for teaching linear algebra
Linear Algebra and its Applications 438 (2013) 1779–1792
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Using an economics model for teaching linear algebra María Trigueros, Edgar Possani ∗ Department of Mathematics, Instituto Tecnológico Autónomo de México, ITAM, Río Hondo No.1, Progreso Tizapán, 01080, D.F., Mexico City, Mexico
ARTICLE INFO
ABSTRACT
Article history: Received 30 November 2010 Accepted 8 April 2011 Available online 12 May 2011
We present an approach for teaching linear algebra using models. In particular, we are interested in analyzing the modeling process under an APOS perspective. We will present a short illustration of the analysis of an economics problem related to production in a set of industries. This problem elicits the use of the concepts of linear combination, linear independence, among other linear algebra concepts related to vector space. We describe cycles of students’ work on the problem, present an analysis of the learning trajectory with emphasis on the constructions they develop, and discuss the advantages of this approach in terms of students’ learning. © 2011 Elsevier Inc. All rights reserved.
Submitted by V. Mehrmann Keywords: Teaching Leontief Model Linear independence and dependence Linear algebra education Vector space Modeling
1. Introduction Linear algebra has been recognized as a difficult subject for students. Day and Kalman [1] point out that recently the mathematical community has lively debated and creatively discussed how to improve the teaching of linear algebra, and that an effort is needed to understand why students find this material difficult, and how to help them to learn it effectively. In the last decade several studies have been carried out to understand the main obstacles that students face when learning linear algebra and have presented some suggestions to overcome them [2–6]. For example, Carlson et al. [7] suggest the use of problems that go beyond simple exercises, especially if they come from other subject areas, to facilitate the learning of linear algebra concepts in different contexts. The research reported here is about a project similar to one described in an earlier paper in Linear Algebra and its Applications [8], which was by the present authors and two others. The premise for both projects is that students can learn a theoretical concept more effectively by beginning with an intriguing problem in which the need for that concept becomes apparent. In [8], a traffic flow problem was studied, in order to help students understand systems of linear equations. Here a supply problem is ∗ Corresponding author. E-mail address: [email protected] (E. Possani). 0024-3795/$ - see front matter © 2011 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2011.04.009
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presented to introduce the concepts of linear combination and linear (in)dependence. In both projects, the authors created specific teaching sequences to help students construct those concepts, and the materials that we developed and our analysis of students’ work were based on APOS theory [9] and Models and Modeling theory [10]. In this paper, we also used, for the first time, ideas from [11] to design a learning trajectory. We later used these theories to analyze students’ work. Particularly, the concepts of linear dependence and independence are new concepts that students have not encountered in previous courses and, without a doubt, are a fundamental stepping stone for linear algebra advanced courses. The research questions in the project discussed here were: (1) Is it possible to introduce students to important concepts such as linear combination, linear dependence, and linear independence through the use of models? (2) In particular, how to introduce linear (in) dependence through modeling? (3) What role do activities based on APOS theory play in the learning trajectory of students? (4) Which aspects of students’ mathematical knowledge can be retrieved and enhanced through the use of the above mentioned approach? One of the aims of this work is to study how the use of mathematical modeling together with specially designed activities, based on complementing mathematics education theories, contributes to students’ learning of abstract concepts. This study can be achieved through the analysis of the work developed by the students, and the evolution of their solving strategies. A second aim of this investigation is to test the modeling activity itself in terms of its potential for providing valuable insight for researchers and teachers on students’ understanding. Specifically, we want to see if it gives the teacher the opportunity to observe and analyze subtle aspects of each student’s mathematical development and provide a clearer view on the student’s process of reasoning, verification and justification, as opposed to just observing their failure or success at producing an expected answer. Since the construction of abstract concepts is known to be a difficult process, we consider that the use of modeling activities by themselves can provide the setting for students to use their previous knowledge and to confront new conceptual needs. These needs can then be addressed in the teaching process by introducing activities that would help the students make the necessary constructions to learn the abstract concepts of linear independence and linear dependence. In Section 2, we present a brief description of the theories that sustain this study. In Section 4, we describe students’ work and difficulties with the model, the learning trajectory analysis, and the cycles observed in students’ understanding process. We conclude by signaling some of the opportunities that this approach opens for the teaching of linear algebra as well as some of the difficulties involved when this approach is used in the classroom. 2. Previous considerations and theoretical concepts Studies [12,13] have shown that linear combination, linear dependence, and linear independence are closely connected abstract concepts, and that students face many difficulties when they are introduced to them. Moreover, other previous studies [14–16] have shown that a geometric introduction to these concepts is confusing to students, and difficult to relate to its algebraic representations. Furthermore, this geometric approach creates a conceptual image that is difficult to overcome when working with more general vector spaces. Understanding of these concepts is important not only because they are needed to learn more advanced linear algebra concepts, but also because they appear in other mathematical areas as well as in many applications. Many studies report that the use of models in the classroom increases students’ motivation, and suggest that students are able to develop important mathematical concepts when working with appropriately designed “real life” problems [10]. Hence, we decided to use models to introduce linear dependence and independence concepts and to analyze if this didactic strategy can help students to develop a schema for these concepts that results in a more effective learning experience. 2.1. APOS theory Action-Process-Object-Schema (APOS) theory [9] was built on Piaget’s work and constructivist ideas. It intends both to model the way students learn advanced mathematical topics in order to design
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teaching sequences, and to analyze the knowledge that students display when solving a specific activity at a particular moment in time. When using APOS theory to design activities, researchers take into consideration students’ previous knowledge. Students work collaboratively in groups discussing and responding to specific tasks contained in pre-designed activities which have particular aims based on the analysis of the way students construct the concepts of interest. Actions here consists of specific guided operations on familiar mathematical objects. Students work on activities involving these operations where they have the opportunity to reflect on their actions. This is the case, for example, when actions are needed in order to multiply two vectors by scalars and add them to obtain a new vector. The purpose of these activities is to guide the construction of new concepts. To enable the process conception, other tasks have as a goal to interiorize those actions into procedures that students can perform without external help, or to generalize actions performed on specific objects into processes that are valid for any object of the same kind. Reflection on how and why these processes work helps students abstract their main characteristics, take control over them, and become flexible in using them. An example of a process conception is when a student takes appropriate decisions to efficiently state if one vector is a linear combination of others. For example, instead of stating and solving a system to verify linear dependence, finding, by simple observation, the values that verify that a vector is a combination of others. To enable an object conception, there are also tasks which intend to make students reflect on a process and be aware of it as a totality so that they can apply new actions to it. When this happens APOS theory talks about an object having been constructed or encapsulated. For example, when students think of linear combinations as a whole and compare them in terms of their properties, they have constructed an object conception of a linear combination. To enable the development of a schema, different actions, processes, and objects need to be related if they are to be used together in the solution of problems or in the construction of new knowledge. Tasks are also designed to help students be aware of the relationships among actions processes and objects and also of the relationships that exist with other concepts. The theory refers to these collections as schemas. For example, when a student is able to use the concept of linear independence to find the rank of a matrix, he demonstrates that he has constructed a schema of linear independence that includes matrices as objects, in particular to de-encapsulate them to work with the rows (columns) of the matrices as vectors. Schemas evolve as new relations between new and previous action, process, and object conceptions and other schema are constructed and reconstructed. The application of APOS theory to describe particular constructions by students requires researchers to develop a genetic decomposition. It consists of a description of specific mental constructions a person may make in the process of understanding mathematical concepts and their relationships. A genetic decomposition for a mathematical concept or a topic is not unique; it is a general description of how such a concept may be constructed. Different researchers can develop different genetic decompositions of how students in general construct that particular concept. However, once one is proposed, it needs to be supported by research data from students in order to be used in the design of teaching materials. We now explain how we used this theory about learning and teaching abstract concepts in a coherent way to design a genetic decomposition of linear dependence and linear independence. This genetic decomposition is as follows (see Fig. 1). We considered that students should use three basic schemas they have previously constructed. One is an algebraic schema that allows students to perform basic algebraic operations, work with the concept of variable, and enable students to set, identify, and work with variables in a system of equations. The second schema is a vector schema that requires students to be familiar with vectors in R2 , and R3 , their graphical representations, and basic operations as sum and multiplication by a scalar. The third schema is a set schema that requires students to use the set notation and perform actions on sets, such as finding intersections or unions, and understanding the solution of a system of equations as a set. Actions are performed on vectors to form linear combinations of given vectors and to represent them geometrically when possible. These actions are generalized in a process that allows the consideration of all possible linear combinations of those vectors. This process is coordinated with the set schema
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Fig. 1. Genetic decomposition.
in order to consider the resulting set of vectors and its geometric representation. These processes can be encapsulated as the object linear combination. Next the action of finding a linear combination that results in a given vector, in any representation, is interiorized and coordinated with the set schema to consider a set of linear combinations that can represent a given vector, including zero. This process is encapsulated into the object “vector as linear combination”. Actions needed to form linear combinations and make them equal to unknown vectors, to find the corresponding system of equations and its solution set are interiorized in the process of considering all vectors that can be obtained from the linear combinations of all the elements in a set. This process is coordinated with the process of adding new vectors to the given set and comparing the resultant vector set with the previously obtained one, to interiorize linear dependence and linear independence of a set of vectors as processes. These processes are encapsulated into the “linearly independent set” object. The processes described beforehand can be coordinated into a schema that includes linear dependent, and independent sets of vectors (including vectors in spaces – different from Rn ) and which we refer to as linear dependence and independence schema. With this schema students are able to construct, based on new activities, other notions such as span and basis, that further contribute to the development of the linear dependence and independence schema. We used the above genetic decomposition to design a sequence of activities to guide students’ construction of the desired concepts. These activities together with ideas from the two theories explained below guided the teaching sequences in the classroom. 2.2. Models and Modeling The Models and Modeling approach is a useful theoretical framework for developing model-eliciting activities to help students develop ideas in a meaningful realistic context [10]. The modeling perspective focuses on the development of conceptual tools which are useful in decision making. Researchers working on this perspective have developed criteria that the problems to be posed to the students must satisfy in order to be successfully applied in the classroom if one intends to contribute to the learning process of students. The Models and Modeling perspective’s main idea consists in introducing realistic complex situations where students engage in mathematical thinking. Complex conceptual tools are produced by students in order to accomplish the intended goal. These tools are constructed during cycles of work and reflection and can be, in each cycle, self-evaluated by the students. Model eliciting activities provide a context where new mathematical ideas arise. Together with the genetic decomposition, tasks can be designed and hypothetical learning trajectories can be formulated [11]. The whole process can be used to analyze students’ work, and to introduce new concepts.
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2.3. Learning trajectories and integrating theories The hypothetical learning trajectory [11] is used as a tool to guide teachers’ decisions in the classroom, and to link APOS theory with Models and Modeling. The main idea of the proposed trajectory is to include a presentation of a situation where students can use their previous knowledge, and use this knowledge as a starting point to construct new concepts. Important steps in this trajectory include discussion of the situation to arrive to a mathematical model, decisions on when to introduce the data, and when to compare solutions. Activities designed using the genetic decomposition from APOS theory are used to develop new concepts. In a circular dynamic, students go from working on the problem to theoretical activities and back to the problem where they use their newly constructed concepts to better deal with the problem. Using ideas from these theories it is possible to design activities for the classroom where students face rich context problems to work on and which lead them to develop mathematical ideas which can be taken as a starting point in sessions where more controlled activities based on a genetic decomposition are introduced. When this is done, these last activities also respond to students’ conceptual needs which arose within the modeling process.
3. Problem at hand The context selected for this modeling experience was a production planning problem. Students were presented with the following description of the problem: There are three production plants A, B, and C belonging to the same company. They each manufacture a different product, a fraction of its production is used to fuel its manufacturing process and other fractions of its production are used by the other plants. Part of the production of each plant is used to satisfy an external demand. At the end of each week, each plant decides how much to produce after observing the external demand for its product. Initial question: 1. Could you design a method for the company to determine the amount of product each industry must produce to satisfy the internal and external demands? Students are organized in small groups to work on this initial question and the modeling of the problem. It is important to mention at this stage that there are two other questions that are posed as students progress in their modeling experience: Table 1 External (consumer’s) demand for 9 months. Units are millions of pesos. Industry A B C
Period 1 30 20 20
2 20 20 20
3 30 20 30
4 20 30 30
5 15 10 15
6 50 50 60
7 10 0 10
8 10 0 0
9 10 10 0
8 13.044 2.120 3.319
9 16.548 14.704 5.623
Table 2 Internal (groups of industries) demand for 9 months. Units are millions of pesos. Industry A B C
Period 1 53.515 36.967 40.470
2 40.470 34.847 37.150
3 57.202 39.687 53.422
4 47.661 50.150 52.408
5 28.601 19.843 26.711
6 104.86 89.836 105.83
7 16.732 4.838 16.271
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2. If we have a set of data of 9 periods of observed internal and external demand for the products, as shown in Tables 1 and 2 do you need all of the periods to estimate the parameters of the model; if not how would you choose them? 3. Can you use your conclusions to verify the internal demand levels in the 6th period? Can you find the internal demand levels of each product in the 10th period when the external demand is 75 for industry A, 70 for B, and 90 for C?
4. Modeling experience and working cycles Students in five undergraduate courses on linear algebra (mathematics, actuarial science, engineering, and economics majors at Instituto Tecnológico Autónomo de México), taught by four different teachers, were presented with the problem. Each teacher had on average 28 students in a group. Students had already taken one or two calculus courses, and had been previously introduced to vector geometry. They worked on the solution of the problem through six class periods of 2 h. In each period the session was broken so that students worked on small groups of three students for awhile and then in whole class discussion where they could present their progress on the problem and where other students and the teacher could ask and discuss questions. In each session, all the students’ work was collected and classroom discussion was recorded in video or audio. After each session, the teachers and the researchers had meetings where they analyzed and discussed students’ work and designed tasks and activities to guide the conceptual construction of students in the following session, and selected some to be given as homework. Work on these activities was also collected and analyzed by each researcher separately. Results of those analyses were always discussed and negotiated between them in order to arrive at a consensus that would validate the resulting analysis of the teaching experience. In the five groups of this study, we identified four general cycles that students go through while working with the modeling situation. A first cycle has to do with the exploration and analysis of the situation, where students develop hypotheses and draw diagrams to select the variables, their relationships, and to construct an initial model for the problem at hand. A second cycle has to do with the operational use of the variables, the manipulation of the sets of data to identify patterns in different sets, and the use of this data to estimate the parameters of the model. A third cycle was identified, where students compare and discuss their solutions, and a new language to describe differences arises. The fourth and final cycle deals with the use of model as a tool for prediction, and as a reference in new situations. Each cycle was characterized by the type of work that students were doing, and in each of them some specific difficulties were detected. Through this analysis the teachers and researchers were able to identify the constructions developed by each group of students, and proposed selected activities to help them formalize their ideas and thus construct the intended mathematical concepts. In what follows we briefly describe results obtained in each of the four cycles, and some of the activities proposed. 4.1. Situation analysis and diagram During the first phase students work in small groups with the problem description, making sense of the problem and exploring possible ways to answer the question posed. Most groups drew diagrams to represent the situation. Students got easily involved with the problem, and showed that they were motivated and enthusiastic about the activity. One frequent approach, as shown in Fig. 2(I), was to represent each industry by a circle and drawing lines to represent the demands posed on each of them by all of the industries and the external demand. There is a line drawn to itself to represent the amount of product for self-consumption. This diagram is a rudimentary schematic representation of the problem. A more useful approach is the one shown in Fig. 2(II) where the demands are grouped together in one arrow, and clearly marked between each industry. Note that the external demand is abstracted as one entity.
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Fig. 2. Student’s diagram for the industries setting.
Fig. 3. Algebraic approach to modeling.
Fig. 4. Use of diagram to set equations.
Other students tried writing formulas that verbalized the problem, but were not proper equations as exemplified in Fig. 3(I), where they write “consumption by industry A of product A, plus external demand, plus consumption by industry B of product A, plus consumption by industry C of product A, is the quantity needed of A”. Others started setting equations using concepts they had learnt in other courses, as shown in Fig. 3(II) where equations are set to estimate a total production (PT) in terms of a given net production (PB), an inventory level (Inv), and given external demand (x) for a specific period. Some students proposed models learned in their economics lectures having to do with supply-demand, and yet others recalled their high school linear regression models. These first approaches to the problem exemplify how, when faced with a modeling situation, students use previously constructed schemas to help them in the modeling process. After about 30 min of small group work a whole class discussion was conducted where different groups presented their approach to the problem and questions were posed both by other students and the teacher. After discussion of several approaches, it was concluded that the diagram representations of the type shown in Fig. 2(II) facilitated the selection of variables by naming the arrows and setting a system of equations to describe how the demands on each industry could be satisfied. After another period of small group work, some of the groups presented new suggestion to model the problem, as the one shown in Fig. 4. Here students named each arrow and tried to set equations to represent flows of productions between the industries and the external demand. We observed in all classrooms that students showed a tendency to propose linear models and many of them introduced a hypothesis about the production being equal to consumption, which was accepted by all of them later in whole class discussion. Also, at least one of the small groups was able to propose a model with a system of
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Fig. 5. System of equations for the problem.
equations to represent the problem in terms of variables that could be related clearly to the proposed situation. In some cases, they would put this in terms of a simple relationship that the demand for A is the sum of its demand by all industries and the external sector, and then immediately put in terms of a proportion (α, β, γ ) of the total production demanded by each industry, as shown in Fig. 5. It was also observed that they could respond to other students’ questions although the meaning of the parameters introduced was not clearly defined but was considered as “αi is the fraction of production of industry A that industry j needs for its production so that the external demand is satisfied". Note this is the Leontief’s [17] economic Input–Ouput Model. 1 It can be presented in the following form: x1
= a11 x1 + a12 x2 + · · · + a1n xn +e1
x2
.. .
= a21 x1 + a22 x2 + · · · + a2n xn +e2 .. .. . .
xn
= an1 x1 + an2 x2 + · · · + ann xn +en
(1)
where xj is the production level for each industry in the economy, aij is the number of units produced by industry i that is necessary to produce one unit by industry j, and ej is the external demand of product j (all referenced to the currency used). At the end of this first cycle, which lasted two hours, students decided to use this model to answer the two other proposed questions (see Section 3), and the model became the object of study. In most textbooks and traditional lectures the model is presented as in the above system (1), and not developed by students on their own. We have observed that students arrive at it without major difficulties. However, in our case, we are not interested in the economic ramifications of the model. Instead, we give the students a set of actual production vectors x’s in different periods in time (a collection of data vectors x = (x1 , . . . , xn )) and expect them to use the data to estimate the parameters aij , as explained in the following subsection (second cycle of modeling). 4.2. Data manipulation and parameter setting After the model had been agreed upon in whole class discussion, students were asked question 2: If we have a set of data of 9 periods of observed internal and external demand for the products, do you need all of the periods to estimate the parameters of the model, if not how would you choose them? Students are given either all 9 periods if they ask for it or a subset of them to work with in order to find parameters aij . The data used is shown in Tables 1 and 2. Most students asked for only three periods to estimate the parameters (aij ’s), since they have a system with three equations, and sometimes the teachers gave them a linear dependent, and sometimes a linear independent, set of data. Others asked for all the data. The first hurdle students encountered 1
Wassily Leontief won the Nobel Prize in Economics in 1973 for this development and its application to the US economy.
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Fig. 6. Incorrect use of data.
Fig. 7. Graphing data.
was using this data to find the parameters for the model. Some students got confused understanding what the parameters to estimate were. Even though they used a correct system of equations, it was not uncommon for several of them to think that the data given are the parameters aij ’s instead of the actual productions xi ’s as shown in Fig. 6. This approach resulted in students finding an inconsistent system of equations. Sometimes students forgot momentarily about the system and, when faced with all nine periods, opted for graphing them and finding repeated trends between internal and external demands as seen in Fig. 7. This was used by teachers during whole group discussion to point out the existence of relationships between both data tables. These students showed they had not internalized the model as a tool for answering the questions. Others used operations to find relationships between the data of the tables and were able to find a constant relation between the sum of the data of each table for the different periods. Teachers used this to point out to students that this was a confirmation of the linearity of the model. In this cycle students performed actions on the proposed model and on the given data either to understand the model or to find the necessary information to answer the questions. In fact when using the proposed model, students need to solve three different system of equations, one for each industry, taking into account three periods. Once they realized this, students were ready to solve the system of equations. Some did it by hand, and some of them chose to use Maple or the calculator to estimate the parameters. 4.3. Pattern recognition validation and concept development When reviewing and comparing students’ solutions for the parameter’s estimation, we observed three different types of results in the different small groups: (1) Those students that had a data set of three linear independent pairs of data-vectors found a unique solution and it was the same for all of them. An example of the calculations in Maple by students for the first row of parameters (a11 , a12 , a13 ) is shown in Fig. 8. (2) Those students that had a data set of three linear dependent pairs of data-vectors found: (2.a) An infinite number of solutions, as shown in their work in Fig. 9. (2.b) Incongruent solutions, as negative numbers or unexpected big numbers, when using Maple or a calculator. An example of this case can be seen in Fig. 10, where their Maple solution to the system gave negative values for the parameters.
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Fig. 8. Unique solution when solving the system.
Fig. 9. Infinite number of solutions.
Fig. 10. Incongruent solutions.
(3) Among those that had more than three pairs of data-vectors, some small groups got unique solutions, equal to that found by other groups, and other did not. During a long period of whole group discussion, students were asked to review these results and try to explain why they occurred. As there were several groups that had the same solution they all agreed on those being the correct estimate for the parameters. They then tried to find an explanation for the differences in solutions obtained by different groups. At the beginning some thought that there were errors in their calculations, or approximation errors in the calculations with the program or calculators. However, when they reviewed the solution by hand they found they had an infinite number of solutions. An interesting discussion started about the differences in the data used. Students began to relate their solutions to properties of systems of equations, considering the system as an object, and also to some vector concepts they knew. For example, when looking at all data some students found that period six could be obtained from the sum of periods three and four. One group of students related this to a previous discussion about planes in R3 where if a third vector is the sum of multiples of two other vectors in the plane, the resultant vector is still on the plane. Another group of students related it with previous work on systems of equations where one of the equations “disappears” during the solution process. In both cases students attributed this to the fact that some of the periods did not provide useful information for the estimation of the parameters. The expression “redundant information”, or a similar one, appeared as a way to talk about this fact. This kind of discussion appeared in all of the groups, and the teachers took advantage of students’ ways of reasoning to introduce new concepts. At this moment, activities that had been designed based on the genetic decomposition were introduced so students could formalize the definition of linear combination, linear dependence, and linear independence. Some other activities were introduced to explore properties of linear independent and linear dependent sets of vectors, as well as their geometric representation, when possible. An example of such activities is as follows:
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Activitiy No. 7. (a) Can the vector v = (3, 4, 1, 2) be expressed in a unique way as a linear combination of the set S1 = {(1, 2, 1, 0), (3, 0, 1, 2), (2, 1, 0, 1), (4, 0, 3, 5), (5, 3, 0, 3)} ? (b) Can the vector v = (3, 4, 1, 2) be expressed in a unique way as a linear combination of the set S2 = {(1, 2, 1, 0), (3, 0, 1, 2), (2, 1, 0, 1), (4, 0, 3, 5)} ? (c) Can the vector 0 = (0, 0, 0, 0) be expressed in a unique way as a linear combination of the set S1? (d) Can the vector 0 = (0, 0, 0, 0) be expressed in a unique way as a linear combination of the set S2? (e) State a sentence where you describe what the constants of the linear combination must satisfy if 0 can be expressed as a linear combination of a set of vectors in a unique way.
The purpose of this activity is to help students coordinate the process of linear combination with that of writing a system of equations and identifying the actions needed to verify if a given vector, including vector 0, can be expressed in a unique way as a linear combination of a set of vectors. This coordination promotes the construction of the linear dependence and linear independence processes. The given sets are similar (S2 = S1 − {(5, 3, 0, 3)}, S1 linear dependent and S2 linear independent) so students begin to reflect on the process of adding vectors to a given set and to consider the result of this action. In part (e), the intention is that students generalize their actions to a process where they describe the meaning of the definition of linear independence. Then students went back to the model and tried to find data-vectors that could be written as a linear combination of a given set of data-vectors from the table. Students concluded that when there is “redundant information” in their data, there is always a vector that can be represented as a linear combination of the other vectors in the set, and when there is no “redundant information" this is not possible. Students were then asked their intuition of why this happened and, again, they used the idea of “redundant information" to explain linear dependence or independence of vectors. This idea became a powerful tool to understand these abstract concepts by making them somehow concrete. 4.4. Validation and use as a prediction tool One characteristic of the Models and Modeling approach [10] is that students should be able to validate their models. In our case students proceeded to do this, without difficulties, using the calculated parameters and the 6th period of external demand to obtain the given internal demand as shown in the table. Teachers then proceeded to reinforce students’ concepts of linear dependence and independence thorough a brief discussion. This discussion served as an opportunity to use the concepts they had constructed from the model to introduce new abstract concepts such as span, generating set, basis and dimension through new activities. An example is given below: Activitiy No. 13. (a) Is the set {(3, −1, 3), (−5, 4, 9), (−2, 4, 18), (0, 4, 24), (7, −2, 9)} linearly dependent or linearly independent? Why? (b) Describe geometrically the space “generated” by the given set. (c) Is the set {(−5, 4, 9), (−2, 4, 18), (0, 4, 24), (7, −2, 9)} linearly dependent or linearly independent? Why? (d) Describe geometrically the space “generated” by the set given in part (c). (e) Compare your answers to part (a) and part (b). What can you say? (f) Is the set {(−5, 4, 9), (−2, 4, 18), (0, 4, 24)} linearly dependent or linearly independent? Why? (g) Describe geometrically the space “generated” by the set given in part (f). (h) Compare your answers to parts (b), (d), and (g). What can you say? (i) What is the minimum number of vectors needed to “generate” the same set?
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In these activities students do actions on vectors to decide if the given set is linearly dependent or independent. They also generalize these actions to describe the set of vectors that can be written as a linear combination of the vectors in the given sets, and coordinate this process with that of representing this new set geometrically (part (b), (d) and (g) represent the same plane in R3 ). Comparison of the different sets helps students to reflect on the process of adding or subtracting vectors from a given set, to coordinate it with the processes of span and to begin, in part (i), to reflect on what later on will be defined as basis of a vector space. It is important to underline that when discussing their work on these activities with the whole group, many times, students resorted to the model and to the idea of “redundant information" to explain spans of different sets of vectors, basis or dimension. Students then worked on the last question of the problem where they had to use the parameters together with their model to predict the internal demand when given a new external demand. All students were able to solve the problem. We observed that the Models and Modeling principles were satisfied at different stages in the students’ modeling process. Later in the course, students were introduced to matrices, their operations and properties. They worked on activities related to the model and used the concepts of linear dependence and independence to decide if the matrix of the system (I − A)x = e was or not invertible. They were able to generalize the need of linear independence for the existence of an inverse matrix and for the uniqueness of a solution of any given nxn system of equations. Furthermore, we observed that these students effortlessly related linear independence of the columns or rows of a squared matrix with its invertibility. In other classes where the model had not been used, students had difficulties with this result, and it did not come as naturally to them. A question related to fare wages of three different handymen that shared each others’ work was presented to students in an exam. Most of the students were able to identify this question as a specific case of the Leontief model, and were able to solve it correctly. This is evidence of the use of the model as a reasoning tool in new situations. An external researcher who is interested in students’ understanding of the concepts of basis, generating set and span conducted interviews with several groups of students, including ours, and students from other universities. Preliminary results (working Ph.D. thesis) showed that our students were able to answer correctly and explain better which sets of vectors generated a given space, and the differences between generating set and basis. This researcher found that students from our group consistently used the phrase “redundant information” when asked to explain their answers. In contrast students from other groups and universities had many difficulties with these questions.
5. Conclusions The analysis of the work done by students shows that this modeling situation was hard for them to interpret and to start working with. Although the expected model consists of a system of equations and they had experience with this topic, students found it difficult to select variables and interpret possible relations among them. Most of the groups of students tried to write a system of equations without considering explicitly the meaning of variables. It was the use of a diagram which prompted them to reconsider possible hypotheses and the role played by variables and parameters in the model. The description of the meaning of the parameters needed the guidance of the teachers in most groups. Once students agreed on the model to use, they were faced with a new difficulty related to the use of variables. There were students who were unable to interpret the parameters as unknowns in the system, while others were confused about how to use the data to find them. Students found it hard to realize they needed to consider three periods for each industry in order to calculate the parameters. Whole class discussion was used by the teachers to help students overcome these difficulties. Even though students had no previous experience with the use of data sets when modeling situations or in problem solving, they were able to examine them and to find trends in their behavior. Some students got distracted from the model when presented with the sets of data. However, the discussion of their findings was useful to discover relations among periods, and to use them to confirm the linearity of the model. In some cases the use of data allowed students to better understand the model. Students were motivated by the use of data and these data provided a means of verifying their model.
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This experience highlights the importance of using sets of “real” data in problem solving, since they contribute to students’ mathematical learning. The work on the solution of the system of equations opened the possibility of relating sets of data vectors with previously learnt concepts. Comparison of solutions and discussion about discrepancies found when choosing different sets of data-vectors provided the opportunity for students to develop a powerful idea: “redundant information”. Teachers used this as a link to the concepts of linear dependence and linear independence through activities based on the genetic decomposition of these concepts. Students were able to use the idea of “redundant information” to give a “concrete” meaning to linear dependence and independence properties of a set of vectors. The use of mathematics education theories proved helpful in the interpretation of students’ work and needs, as well as in the design of concept construction activities that are linked to the modeling process. It guided teachers’ decisions to help students throughout the whole process, and also served in developing some assessment instruments which helped teachers make sense of students’ work and students’ possible constructions. Teachers and researchers found this valuable in deciding how to help students overcome their difficulties and misconceptions. We can finally conclude that in spite of students’ difficulties in each cycle, they were highly motivated while working with the model. Students’ insights constituted a solid base for the introduction of the concepts of linear combination, linear dependence and independence. Our findings about their understanding of the existence of inverse matrix and uniqueness of solution for n × n systems of equations, and the interview results by the external researcher on the differences between a generating set and basis, make it clear that the use of this model, together with the learning trajectory, contributed to students’ learning of linear combination, linear dependence and linear independence concepts. It is also clear that students were able to apply this knowledge to the learning of new concepts. Results found in this experience will also be helpful in re-designing the learning trajectory in a way that can be better utilized when new groups work with this problem. Similar teaching strategies may prove as successful as this one in teaching new and complex linear algebra concepts to students through the use of rich contextual problems. Acknowledgments This project was funded by CONACyT Grant No. 62375, and also partially supported by Asociación Mexicana de Cultura A.C. This work was undertaken by the authors in collaboration with other researchers, in particular the authors thank Gustavo Preciado, Dolores Lozano, Javier Alfaro Pastor, María del Carmen López Laiseca, and Ivonne Twiggy Sandoval for their work on the experience presented here. Finally, the authors thank the external researcher for sharing her preliminary results, and the reviewers for their useful comments and suggestions. References [1] J.M. Day, D. Kalman, Teaching linear algebra: issues and resources, College Math. J. 32 (3) (2001) 162–168. [2] L. Dorier, A. Sierpinska, Research into the teaching and learning of linear algebra, in: Derek Holton (Ed.), The Teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publisher, 2001, pp. 255–273. (Printed in The Netherlands). [3] G. Gueudet, Rôle du géométrique dans l’einsegnement de l’algébre linéaire, RDM 24 (1) (2004) 81–114. [4] M. Trigueros, A. Oktaç, La théorie APOS et l’enseignement de l’algébre linéaire, Annales de Didactique et de Sciences Cognitives 10 (2005) 157–176. [5] D. Kú, M. Trigueros, A. Oktaç, Comprensión del concepto de base de un espacio vectorial desde el punto de vista de la teoría APOE, Educación Matemática. 20 (2) (2008) 65–89. [6] M. Parraguez, A. Oktaç, Construction of the vector space concept from the viewpoint of APOS, Linear Algebra Appl. 432 (8) (2010) 2112–2124. [7] D. Carlson, C.R. Johnson, D.C. Lay, A.D. Porter, A.E. Watkins, W. Watkins (Eds.), Resources for Teaching Linear Algebra, MAA Notes, vol. 42, 1997. [8] E. Possani, M. Trigueros, J.G. Preciado, M.D. Lozano, Use of models in the teaching of linear algebra, Linear Algebra Appl. 432 (8) (2009) 2125–2140. [9] E. Dubinsky, M. McDonald, A constructivist theory of learning in undergraduate mathematics education research, in: D. Holton (Ed.), The Teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publishers, 2001, pp. 273–280.
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[10] R. Lesh, L. English, Trends in the evolution of models and modeling perspectives on mathematical learning and problem solving, ZDM 37 (6) (2005) 487–489. [11] M.A. Simon, Reconstructing mathematics pedagogy from a constructivist perspective, J. Res. Math. Educ. 26 (2) (1995) 114–145. [12] M. Maracci, Combining different theoretical perspectives for analyzing students’ difficulties in vector spaces theory, ZDM 40 (2) (2008) 256–276. [13] S. Stewart, M. Thomas, Student learning of basis span and linear independence in linear algebra, Int. J. Math. Educ. Sci. Technol. 41 (2) (2010) 173–188. [14] H. Harel, Students’ understanding of proof: a historical analysis and implications for the teaching of geometry and linear algebra, Linear Algebra Appl. 302–303 (1999) 601–613. [15] F. Uhlig, The role of proof in comprehending and teaching elementary linear algebra, Educ. Stud. Math. 50 (3) (2002) 335–346. [16] M. Bogomolny, Raising students’ understanding: linear algebra, in: J.H. Woo, H.C. Lew, K.S. Park, D.Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, vol. 4, PME, Seoul, 2007, pp. 201–208. [17] W.W. Leontief, Input–output economics, Sci. Am. (1951) 15–21.
Contents lists available at SciVerse ScienceDirect
Linear Algebra and its Applications journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / l a a
Using an economics model for teaching linear algebra María Trigueros, Edgar Possani ∗ Department of Mathematics, Instituto Tecnológico Autónomo de México, ITAM, Río Hondo No.1, Progreso Tizapán, 01080, D.F., Mexico City, Mexico
ARTICLE INFO
ABSTRACT
Article history: Received 30 November 2010 Accepted 8 April 2011 Available online 12 May 2011
We present an approach for teaching linear algebra using models. In particular, we are interested in analyzing the modeling process under an APOS perspective. We will present a short illustration of the analysis of an economics problem related to production in a set of industries. This problem elicits the use of the concepts of linear combination, linear independence, among other linear algebra concepts related to vector space. We describe cycles of students’ work on the problem, present an analysis of the learning trajectory with emphasis on the constructions they develop, and discuss the advantages of this approach in terms of students’ learning. © 2011 Elsevier Inc. All rights reserved.
Submitted by V. Mehrmann Keywords: Teaching Leontief Model Linear independence and dependence Linear algebra education Vector space Modeling
1. Introduction Linear algebra has been recognized as a difficult subject for students. Day and Kalman [1] point out that recently the mathematical community has lively debated and creatively discussed how to improve the teaching of linear algebra, and that an effort is needed to understand why students find this material difficult, and how to help them to learn it effectively. In the last decade several studies have been carried out to understand the main obstacles that students face when learning linear algebra and have presented some suggestions to overcome them [2–6]. For example, Carlson et al. [7] suggest the use of problems that go beyond simple exercises, especially if they come from other subject areas, to facilitate the learning of linear algebra concepts in different contexts. The research reported here is about a project similar to one described in an earlier paper in Linear Algebra and its Applications [8], which was by the present authors and two others. The premise for both projects is that students can learn a theoretical concept more effectively by beginning with an intriguing problem in which the need for that concept becomes apparent. In [8], a traffic flow problem was studied, in order to help students understand systems of linear equations. Here a supply problem is ∗ Corresponding author. E-mail address: [email protected] (E. Possani). 0024-3795/$ - see front matter © 2011 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2011.04.009
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presented to introduce the concepts of linear combination and linear (in)dependence. In both projects, the authors created specific teaching sequences to help students construct those concepts, and the materials that we developed and our analysis of students’ work were based on APOS theory [9] and Models and Modeling theory [10]. In this paper, we also used, for the first time, ideas from [11] to design a learning trajectory. We later used these theories to analyze students’ work. Particularly, the concepts of linear dependence and independence are new concepts that students have not encountered in previous courses and, without a doubt, are a fundamental stepping stone for linear algebra advanced courses. The research questions in the project discussed here were: (1) Is it possible to introduce students to important concepts such as linear combination, linear dependence, and linear independence through the use of models? (2) In particular, how to introduce linear (in) dependence through modeling? (3) What role do activities based on APOS theory play in the learning trajectory of students? (4) Which aspects of students’ mathematical knowledge can be retrieved and enhanced through the use of the above mentioned approach? One of the aims of this work is to study how the use of mathematical modeling together with specially designed activities, based on complementing mathematics education theories, contributes to students’ learning of abstract concepts. This study can be achieved through the analysis of the work developed by the students, and the evolution of their solving strategies. A second aim of this investigation is to test the modeling activity itself in terms of its potential for providing valuable insight for researchers and teachers on students’ understanding. Specifically, we want to see if it gives the teacher the opportunity to observe and analyze subtle aspects of each student’s mathematical development and provide a clearer view on the student’s process of reasoning, verification and justification, as opposed to just observing their failure or success at producing an expected answer. Since the construction of abstract concepts is known to be a difficult process, we consider that the use of modeling activities by themselves can provide the setting for students to use their previous knowledge and to confront new conceptual needs. These needs can then be addressed in the teaching process by introducing activities that would help the students make the necessary constructions to learn the abstract concepts of linear independence and linear dependence. In Section 2, we present a brief description of the theories that sustain this study. In Section 4, we describe students’ work and difficulties with the model, the learning trajectory analysis, and the cycles observed in students’ understanding process. We conclude by signaling some of the opportunities that this approach opens for the teaching of linear algebra as well as some of the difficulties involved when this approach is used in the classroom. 2. Previous considerations and theoretical concepts Studies [12,13] have shown that linear combination, linear dependence, and linear independence are closely connected abstract concepts, and that students face many difficulties when they are introduced to them. Moreover, other previous studies [14–16] have shown that a geometric introduction to these concepts is confusing to students, and difficult to relate to its algebraic representations. Furthermore, this geometric approach creates a conceptual image that is difficult to overcome when working with more general vector spaces. Understanding of these concepts is important not only because they are needed to learn more advanced linear algebra concepts, but also because they appear in other mathematical areas as well as in many applications. Many studies report that the use of models in the classroom increases students’ motivation, and suggest that students are able to develop important mathematical concepts when working with appropriately designed “real life” problems [10]. Hence, we decided to use models to introduce linear dependence and independence concepts and to analyze if this didactic strategy can help students to develop a schema for these concepts that results in a more effective learning experience. 2.1. APOS theory Action-Process-Object-Schema (APOS) theory [9] was built on Piaget’s work and constructivist ideas. It intends both to model the way students learn advanced mathematical topics in order to design
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teaching sequences, and to analyze the knowledge that students display when solving a specific activity at a particular moment in time. When using APOS theory to design activities, researchers take into consideration students’ previous knowledge. Students work collaboratively in groups discussing and responding to specific tasks contained in pre-designed activities which have particular aims based on the analysis of the way students construct the concepts of interest. Actions here consists of specific guided operations on familiar mathematical objects. Students work on activities involving these operations where they have the opportunity to reflect on their actions. This is the case, for example, when actions are needed in order to multiply two vectors by scalars and add them to obtain a new vector. The purpose of these activities is to guide the construction of new concepts. To enable the process conception, other tasks have as a goal to interiorize those actions into procedures that students can perform without external help, or to generalize actions performed on specific objects into processes that are valid for any object of the same kind. Reflection on how and why these processes work helps students abstract their main characteristics, take control over them, and become flexible in using them. An example of a process conception is when a student takes appropriate decisions to efficiently state if one vector is a linear combination of others. For example, instead of stating and solving a system to verify linear dependence, finding, by simple observation, the values that verify that a vector is a combination of others. To enable an object conception, there are also tasks which intend to make students reflect on a process and be aware of it as a totality so that they can apply new actions to it. When this happens APOS theory talks about an object having been constructed or encapsulated. For example, when students think of linear combinations as a whole and compare them in terms of their properties, they have constructed an object conception of a linear combination. To enable the development of a schema, different actions, processes, and objects need to be related if they are to be used together in the solution of problems or in the construction of new knowledge. Tasks are also designed to help students be aware of the relationships among actions processes and objects and also of the relationships that exist with other concepts. The theory refers to these collections as schemas. For example, when a student is able to use the concept of linear independence to find the rank of a matrix, he demonstrates that he has constructed a schema of linear independence that includes matrices as objects, in particular to de-encapsulate them to work with the rows (columns) of the matrices as vectors. Schemas evolve as new relations between new and previous action, process, and object conceptions and other schema are constructed and reconstructed. The application of APOS theory to describe particular constructions by students requires researchers to develop a genetic decomposition. It consists of a description of specific mental constructions a person may make in the process of understanding mathematical concepts and their relationships. A genetic decomposition for a mathematical concept or a topic is not unique; it is a general description of how such a concept may be constructed. Different researchers can develop different genetic decompositions of how students in general construct that particular concept. However, once one is proposed, it needs to be supported by research data from students in order to be used in the design of teaching materials. We now explain how we used this theory about learning and teaching abstract concepts in a coherent way to design a genetic decomposition of linear dependence and linear independence. This genetic decomposition is as follows (see Fig. 1). We considered that students should use three basic schemas they have previously constructed. One is an algebraic schema that allows students to perform basic algebraic operations, work with the concept of variable, and enable students to set, identify, and work with variables in a system of equations. The second schema is a vector schema that requires students to be familiar with vectors in R2 , and R3 , their graphical representations, and basic operations as sum and multiplication by a scalar. The third schema is a set schema that requires students to use the set notation and perform actions on sets, such as finding intersections or unions, and understanding the solution of a system of equations as a set. Actions are performed on vectors to form linear combinations of given vectors and to represent them geometrically when possible. These actions are generalized in a process that allows the consideration of all possible linear combinations of those vectors. This process is coordinated with the set schema
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Fig. 1. Genetic decomposition.
in order to consider the resulting set of vectors and its geometric representation. These processes can be encapsulated as the object linear combination. Next the action of finding a linear combination that results in a given vector, in any representation, is interiorized and coordinated with the set schema to consider a set of linear combinations that can represent a given vector, including zero. This process is encapsulated into the object “vector as linear combination”. Actions needed to form linear combinations and make them equal to unknown vectors, to find the corresponding system of equations and its solution set are interiorized in the process of considering all vectors that can be obtained from the linear combinations of all the elements in a set. This process is coordinated with the process of adding new vectors to the given set and comparing the resultant vector set with the previously obtained one, to interiorize linear dependence and linear independence of a set of vectors as processes. These processes are encapsulated into the “linearly independent set” object. The processes described beforehand can be coordinated into a schema that includes linear dependent, and independent sets of vectors (including vectors in spaces – different from Rn ) and which we refer to as linear dependence and independence schema. With this schema students are able to construct, based on new activities, other notions such as span and basis, that further contribute to the development of the linear dependence and independence schema. We used the above genetic decomposition to design a sequence of activities to guide students’ construction of the desired concepts. These activities together with ideas from the two theories explained below guided the teaching sequences in the classroom. 2.2. Models and Modeling The Models and Modeling approach is a useful theoretical framework for developing model-eliciting activities to help students develop ideas in a meaningful realistic context [10]. The modeling perspective focuses on the development of conceptual tools which are useful in decision making. Researchers working on this perspective have developed criteria that the problems to be posed to the students must satisfy in order to be successfully applied in the classroom if one intends to contribute to the learning process of students. The Models and Modeling perspective’s main idea consists in introducing realistic complex situations where students engage in mathematical thinking. Complex conceptual tools are produced by students in order to accomplish the intended goal. These tools are constructed during cycles of work and reflection and can be, in each cycle, self-evaluated by the students. Model eliciting activities provide a context where new mathematical ideas arise. Together with the genetic decomposition, tasks can be designed and hypothetical learning trajectories can be formulated [11]. The whole process can be used to analyze students’ work, and to introduce new concepts.
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2.3. Learning trajectories and integrating theories The hypothetical learning trajectory [11] is used as a tool to guide teachers’ decisions in the classroom, and to link APOS theory with Models and Modeling. The main idea of the proposed trajectory is to include a presentation of a situation where students can use their previous knowledge, and use this knowledge as a starting point to construct new concepts. Important steps in this trajectory include discussion of the situation to arrive to a mathematical model, decisions on when to introduce the data, and when to compare solutions. Activities designed using the genetic decomposition from APOS theory are used to develop new concepts. In a circular dynamic, students go from working on the problem to theoretical activities and back to the problem where they use their newly constructed concepts to better deal with the problem. Using ideas from these theories it is possible to design activities for the classroom where students face rich context problems to work on and which lead them to develop mathematical ideas which can be taken as a starting point in sessions where more controlled activities based on a genetic decomposition are introduced. When this is done, these last activities also respond to students’ conceptual needs which arose within the modeling process.
3. Problem at hand The context selected for this modeling experience was a production planning problem. Students were presented with the following description of the problem: There are three production plants A, B, and C belonging to the same company. They each manufacture a different product, a fraction of its production is used to fuel its manufacturing process and other fractions of its production are used by the other plants. Part of the production of each plant is used to satisfy an external demand. At the end of each week, each plant decides how much to produce after observing the external demand for its product. Initial question: 1. Could you design a method for the company to determine the amount of product each industry must produce to satisfy the internal and external demands? Students are organized in small groups to work on this initial question and the modeling of the problem. It is important to mention at this stage that there are two other questions that are posed as students progress in their modeling experience: Table 1 External (consumer’s) demand for 9 months. Units are millions of pesos. Industry A B C
Period 1 30 20 20
2 20 20 20
3 30 20 30
4 20 30 30
5 15 10 15
6 50 50 60
7 10 0 10
8 10 0 0
9 10 10 0
8 13.044 2.120 3.319
9 16.548 14.704 5.623
Table 2 Internal (groups of industries) demand for 9 months. Units are millions of pesos. Industry A B C
Period 1 53.515 36.967 40.470
2 40.470 34.847 37.150
3 57.202 39.687 53.422
4 47.661 50.150 52.408
5 28.601 19.843 26.711
6 104.86 89.836 105.83
7 16.732 4.838 16.271
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2. If we have a set of data of 9 periods of observed internal and external demand for the products, as shown in Tables 1 and 2 do you need all of the periods to estimate the parameters of the model; if not how would you choose them? 3. Can you use your conclusions to verify the internal demand levels in the 6th period? Can you find the internal demand levels of each product in the 10th period when the external demand is 75 for industry A, 70 for B, and 90 for C?
4. Modeling experience and working cycles Students in five undergraduate courses on linear algebra (mathematics, actuarial science, engineering, and economics majors at Instituto Tecnológico Autónomo de México), taught by four different teachers, were presented with the problem. Each teacher had on average 28 students in a group. Students had already taken one or two calculus courses, and had been previously introduced to vector geometry. They worked on the solution of the problem through six class periods of 2 h. In each period the session was broken so that students worked on small groups of three students for awhile and then in whole class discussion where they could present their progress on the problem and where other students and the teacher could ask and discuss questions. In each session, all the students’ work was collected and classroom discussion was recorded in video or audio. After each session, the teachers and the researchers had meetings where they analyzed and discussed students’ work and designed tasks and activities to guide the conceptual construction of students in the following session, and selected some to be given as homework. Work on these activities was also collected and analyzed by each researcher separately. Results of those analyses were always discussed and negotiated between them in order to arrive at a consensus that would validate the resulting analysis of the teaching experience. In the five groups of this study, we identified four general cycles that students go through while working with the modeling situation. A first cycle has to do with the exploration and analysis of the situation, where students develop hypotheses and draw diagrams to select the variables, their relationships, and to construct an initial model for the problem at hand. A second cycle has to do with the operational use of the variables, the manipulation of the sets of data to identify patterns in different sets, and the use of this data to estimate the parameters of the model. A third cycle was identified, where students compare and discuss their solutions, and a new language to describe differences arises. The fourth and final cycle deals with the use of model as a tool for prediction, and as a reference in new situations. Each cycle was characterized by the type of work that students were doing, and in each of them some specific difficulties were detected. Through this analysis the teachers and researchers were able to identify the constructions developed by each group of students, and proposed selected activities to help them formalize their ideas and thus construct the intended mathematical concepts. In what follows we briefly describe results obtained in each of the four cycles, and some of the activities proposed. 4.1. Situation analysis and diagram During the first phase students work in small groups with the problem description, making sense of the problem and exploring possible ways to answer the question posed. Most groups drew diagrams to represent the situation. Students got easily involved with the problem, and showed that they were motivated and enthusiastic about the activity. One frequent approach, as shown in Fig. 2(I), was to represent each industry by a circle and drawing lines to represent the demands posed on each of them by all of the industries and the external demand. There is a line drawn to itself to represent the amount of product for self-consumption. This diagram is a rudimentary schematic representation of the problem. A more useful approach is the one shown in Fig. 2(II) where the demands are grouped together in one arrow, and clearly marked between each industry. Note that the external demand is abstracted as one entity.
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Fig. 2. Student’s diagram for the industries setting.
Fig. 3. Algebraic approach to modeling.
Fig. 4. Use of diagram to set equations.
Other students tried writing formulas that verbalized the problem, but were not proper equations as exemplified in Fig. 3(I), where they write “consumption by industry A of product A, plus external demand, plus consumption by industry B of product A, plus consumption by industry C of product A, is the quantity needed of A”. Others started setting equations using concepts they had learnt in other courses, as shown in Fig. 3(II) where equations are set to estimate a total production (PT) in terms of a given net production (PB), an inventory level (Inv), and given external demand (x) for a specific period. Some students proposed models learned in their economics lectures having to do with supply-demand, and yet others recalled their high school linear regression models. These first approaches to the problem exemplify how, when faced with a modeling situation, students use previously constructed schemas to help them in the modeling process. After about 30 min of small group work a whole class discussion was conducted where different groups presented their approach to the problem and questions were posed both by other students and the teacher. After discussion of several approaches, it was concluded that the diagram representations of the type shown in Fig. 2(II) facilitated the selection of variables by naming the arrows and setting a system of equations to describe how the demands on each industry could be satisfied. After another period of small group work, some of the groups presented new suggestion to model the problem, as the one shown in Fig. 4. Here students named each arrow and tried to set equations to represent flows of productions between the industries and the external demand. We observed in all classrooms that students showed a tendency to propose linear models and many of them introduced a hypothesis about the production being equal to consumption, which was accepted by all of them later in whole class discussion. Also, at least one of the small groups was able to propose a model with a system of
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Fig. 5. System of equations for the problem.
equations to represent the problem in terms of variables that could be related clearly to the proposed situation. In some cases, they would put this in terms of a simple relationship that the demand for A is the sum of its demand by all industries and the external sector, and then immediately put in terms of a proportion (α, β, γ ) of the total production demanded by each industry, as shown in Fig. 5. It was also observed that they could respond to other students’ questions although the meaning of the parameters introduced was not clearly defined but was considered as “αi is the fraction of production of industry A that industry j needs for its production so that the external demand is satisfied". Note this is the Leontief’s [17] economic Input–Ouput Model. 1 It can be presented in the following form: x1
= a11 x1 + a12 x2 + · · · + a1n xn +e1
x2
.. .
= a21 x1 + a22 x2 + · · · + a2n xn +e2 .. .. . .
xn
= an1 x1 + an2 x2 + · · · + ann xn +en
(1)
where xj is the production level for each industry in the economy, aij is the number of units produced by industry i that is necessary to produce one unit by industry j, and ej is the external demand of product j (all referenced to the currency used). At the end of this first cycle, which lasted two hours, students decided to use this model to answer the two other proposed questions (see Section 3), and the model became the object of study. In most textbooks and traditional lectures the model is presented as in the above system (1), and not developed by students on their own. We have observed that students arrive at it without major difficulties. However, in our case, we are not interested in the economic ramifications of the model. Instead, we give the students a set of actual production vectors x’s in different periods in time (a collection of data vectors x = (x1 , . . . , xn )) and expect them to use the data to estimate the parameters aij , as explained in the following subsection (second cycle of modeling). 4.2. Data manipulation and parameter setting After the model had been agreed upon in whole class discussion, students were asked question 2: If we have a set of data of 9 periods of observed internal and external demand for the products, do you need all of the periods to estimate the parameters of the model, if not how would you choose them? Students are given either all 9 periods if they ask for it or a subset of them to work with in order to find parameters aij . The data used is shown in Tables 1 and 2. Most students asked for only three periods to estimate the parameters (aij ’s), since they have a system with three equations, and sometimes the teachers gave them a linear dependent, and sometimes a linear independent, set of data. Others asked for all the data. The first hurdle students encountered 1
Wassily Leontief won the Nobel Prize in Economics in 1973 for this development and its application to the US economy.
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Fig. 6. Incorrect use of data.
Fig. 7. Graphing data.
was using this data to find the parameters for the model. Some students got confused understanding what the parameters to estimate were. Even though they used a correct system of equations, it was not uncommon for several of them to think that the data given are the parameters aij ’s instead of the actual productions xi ’s as shown in Fig. 6. This approach resulted in students finding an inconsistent system of equations. Sometimes students forgot momentarily about the system and, when faced with all nine periods, opted for graphing them and finding repeated trends between internal and external demands as seen in Fig. 7. This was used by teachers during whole group discussion to point out the existence of relationships between both data tables. These students showed they had not internalized the model as a tool for answering the questions. Others used operations to find relationships between the data of the tables and were able to find a constant relation between the sum of the data of each table for the different periods. Teachers used this to point out to students that this was a confirmation of the linearity of the model. In this cycle students performed actions on the proposed model and on the given data either to understand the model or to find the necessary information to answer the questions. In fact when using the proposed model, students need to solve three different system of equations, one for each industry, taking into account three periods. Once they realized this, students were ready to solve the system of equations. Some did it by hand, and some of them chose to use Maple or the calculator to estimate the parameters. 4.3. Pattern recognition validation and concept development When reviewing and comparing students’ solutions for the parameter’s estimation, we observed three different types of results in the different small groups: (1) Those students that had a data set of three linear independent pairs of data-vectors found a unique solution and it was the same for all of them. An example of the calculations in Maple by students for the first row of parameters (a11 , a12 , a13 ) is shown in Fig. 8. (2) Those students that had a data set of three linear dependent pairs of data-vectors found: (2.a) An infinite number of solutions, as shown in their work in Fig. 9. (2.b) Incongruent solutions, as negative numbers or unexpected big numbers, when using Maple or a calculator. An example of this case can be seen in Fig. 10, where their Maple solution to the system gave negative values for the parameters.
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Fig. 8. Unique solution when solving the system.
Fig. 9. Infinite number of solutions.
Fig. 10. Incongruent solutions.
(3) Among those that had more than three pairs of data-vectors, some small groups got unique solutions, equal to that found by other groups, and other did not. During a long period of whole group discussion, students were asked to review these results and try to explain why they occurred. As there were several groups that had the same solution they all agreed on those being the correct estimate for the parameters. They then tried to find an explanation for the differences in solutions obtained by different groups. At the beginning some thought that there were errors in their calculations, or approximation errors in the calculations with the program or calculators. However, when they reviewed the solution by hand they found they had an infinite number of solutions. An interesting discussion started about the differences in the data used. Students began to relate their solutions to properties of systems of equations, considering the system as an object, and also to some vector concepts they knew. For example, when looking at all data some students found that period six could be obtained from the sum of periods three and four. One group of students related this to a previous discussion about planes in R3 where if a third vector is the sum of multiples of two other vectors in the plane, the resultant vector is still on the plane. Another group of students related it with previous work on systems of equations where one of the equations “disappears” during the solution process. In both cases students attributed this to the fact that some of the periods did not provide useful information for the estimation of the parameters. The expression “redundant information”, or a similar one, appeared as a way to talk about this fact. This kind of discussion appeared in all of the groups, and the teachers took advantage of students’ ways of reasoning to introduce new concepts. At this moment, activities that had been designed based on the genetic decomposition were introduced so students could formalize the definition of linear combination, linear dependence, and linear independence. Some other activities were introduced to explore properties of linear independent and linear dependent sets of vectors, as well as their geometric representation, when possible. An example of such activities is as follows:
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Activitiy No. 7. (a) Can the vector v = (3, 4, 1, 2) be expressed in a unique way as a linear combination of the set S1 = {(1, 2, 1, 0), (3, 0, 1, 2), (2, 1, 0, 1), (4, 0, 3, 5), (5, 3, 0, 3)} ? (b) Can the vector v = (3, 4, 1, 2) be expressed in a unique way as a linear combination of the set S2 = {(1, 2, 1, 0), (3, 0, 1, 2), (2, 1, 0, 1), (4, 0, 3, 5)} ? (c) Can the vector 0 = (0, 0, 0, 0) be expressed in a unique way as a linear combination of the set S1? (d) Can the vector 0 = (0, 0, 0, 0) be expressed in a unique way as a linear combination of the set S2? (e) State a sentence where you describe what the constants of the linear combination must satisfy if 0 can be expressed as a linear combination of a set of vectors in a unique way.
The purpose of this activity is to help students coordinate the process of linear combination with that of writing a system of equations and identifying the actions needed to verify if a given vector, including vector 0, can be expressed in a unique way as a linear combination of a set of vectors. This coordination promotes the construction of the linear dependence and linear independence processes. The given sets are similar (S2 = S1 − {(5, 3, 0, 3)}, S1 linear dependent and S2 linear independent) so students begin to reflect on the process of adding vectors to a given set and to consider the result of this action. In part (e), the intention is that students generalize their actions to a process where they describe the meaning of the definition of linear independence. Then students went back to the model and tried to find data-vectors that could be written as a linear combination of a given set of data-vectors from the table. Students concluded that when there is “redundant information” in their data, there is always a vector that can be represented as a linear combination of the other vectors in the set, and when there is no “redundant information" this is not possible. Students were then asked their intuition of why this happened and, again, they used the idea of “redundant information" to explain linear dependence or independence of vectors. This idea became a powerful tool to understand these abstract concepts by making them somehow concrete. 4.4. Validation and use as a prediction tool One characteristic of the Models and Modeling approach [10] is that students should be able to validate their models. In our case students proceeded to do this, without difficulties, using the calculated parameters and the 6th period of external demand to obtain the given internal demand as shown in the table. Teachers then proceeded to reinforce students’ concepts of linear dependence and independence thorough a brief discussion. This discussion served as an opportunity to use the concepts they had constructed from the model to introduce new abstract concepts such as span, generating set, basis and dimension through new activities. An example is given below: Activitiy No. 13. (a) Is the set {(3, −1, 3), (−5, 4, 9), (−2, 4, 18), (0, 4, 24), (7, −2, 9)} linearly dependent or linearly independent? Why? (b) Describe geometrically the space “generated” by the given set. (c) Is the set {(−5, 4, 9), (−2, 4, 18), (0, 4, 24), (7, −2, 9)} linearly dependent or linearly independent? Why? (d) Describe geometrically the space “generated” by the set given in part (c). (e) Compare your answers to part (a) and part (b). What can you say? (f) Is the set {(−5, 4, 9), (−2, 4, 18), (0, 4, 24)} linearly dependent or linearly independent? Why? (g) Describe geometrically the space “generated” by the set given in part (f). (h) Compare your answers to parts (b), (d), and (g). What can you say? (i) What is the minimum number of vectors needed to “generate” the same set?
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In these activities students do actions on vectors to decide if the given set is linearly dependent or independent. They also generalize these actions to describe the set of vectors that can be written as a linear combination of the vectors in the given sets, and coordinate this process with that of representing this new set geometrically (part (b), (d) and (g) represent the same plane in R3 ). Comparison of the different sets helps students to reflect on the process of adding or subtracting vectors from a given set, to coordinate it with the processes of span and to begin, in part (i), to reflect on what later on will be defined as basis of a vector space. It is important to underline that when discussing their work on these activities with the whole group, many times, students resorted to the model and to the idea of “redundant information" to explain spans of different sets of vectors, basis or dimension. Students then worked on the last question of the problem where they had to use the parameters together with their model to predict the internal demand when given a new external demand. All students were able to solve the problem. We observed that the Models and Modeling principles were satisfied at different stages in the students’ modeling process. Later in the course, students were introduced to matrices, their operations and properties. They worked on activities related to the model and used the concepts of linear dependence and independence to decide if the matrix of the system (I − A)x = e was or not invertible. They were able to generalize the need of linear independence for the existence of an inverse matrix and for the uniqueness of a solution of any given nxn system of equations. Furthermore, we observed that these students effortlessly related linear independence of the columns or rows of a squared matrix with its invertibility. In other classes where the model had not been used, students had difficulties with this result, and it did not come as naturally to them. A question related to fare wages of three different handymen that shared each others’ work was presented to students in an exam. Most of the students were able to identify this question as a specific case of the Leontief model, and were able to solve it correctly. This is evidence of the use of the model as a reasoning tool in new situations. An external researcher who is interested in students’ understanding of the concepts of basis, generating set and span conducted interviews with several groups of students, including ours, and students from other universities. Preliminary results (working Ph.D. thesis) showed that our students were able to answer correctly and explain better which sets of vectors generated a given space, and the differences between generating set and basis. This researcher found that students from our group consistently used the phrase “redundant information” when asked to explain their answers. In contrast students from other groups and universities had many difficulties with these questions.
5. Conclusions The analysis of the work done by students shows that this modeling situation was hard for them to interpret and to start working with. Although the expected model consists of a system of equations and they had experience with this topic, students found it difficult to select variables and interpret possible relations among them. Most of the groups of students tried to write a system of equations without considering explicitly the meaning of variables. It was the use of a diagram which prompted them to reconsider possible hypotheses and the role played by variables and parameters in the model. The description of the meaning of the parameters needed the guidance of the teachers in most groups. Once students agreed on the model to use, they were faced with a new difficulty related to the use of variables. There were students who were unable to interpret the parameters as unknowns in the system, while others were confused about how to use the data to find them. Students found it hard to realize they needed to consider three periods for each industry in order to calculate the parameters. Whole class discussion was used by the teachers to help students overcome these difficulties. Even though students had no previous experience with the use of data sets when modeling situations or in problem solving, they were able to examine them and to find trends in their behavior. Some students got distracted from the model when presented with the sets of data. However, the discussion of their findings was useful to discover relations among periods, and to use them to confirm the linearity of the model. In some cases the use of data allowed students to better understand the model. Students were motivated by the use of data and these data provided a means of verifying their model.
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This experience highlights the importance of using sets of “real” data in problem solving, since they contribute to students’ mathematical learning. The work on the solution of the system of equations opened the possibility of relating sets of data vectors with previously learnt concepts. Comparison of solutions and discussion about discrepancies found when choosing different sets of data-vectors provided the opportunity for students to develop a powerful idea: “redundant information”. Teachers used this as a link to the concepts of linear dependence and linear independence through activities based on the genetic decomposition of these concepts. Students were able to use the idea of “redundant information” to give a “concrete” meaning to linear dependence and independence properties of a set of vectors. The use of mathematics education theories proved helpful in the interpretation of students’ work and needs, as well as in the design of concept construction activities that are linked to the modeling process. It guided teachers’ decisions to help students throughout the whole process, and also served in developing some assessment instruments which helped teachers make sense of students’ work and students’ possible constructions. Teachers and researchers found this valuable in deciding how to help students overcome their difficulties and misconceptions. We can finally conclude that in spite of students’ difficulties in each cycle, they were highly motivated while working with the model. Students’ insights constituted a solid base for the introduction of the concepts of linear combination, linear dependence and independence. Our findings about their understanding of the existence of inverse matrix and uniqueness of solution for n × n systems of equations, and the interview results by the external researcher on the differences between a generating set and basis, make it clear that the use of this model, together with the learning trajectory, contributed to students’ learning of linear combination, linear dependence and linear independence concepts. It is also clear that students were able to apply this knowledge to the learning of new concepts. Results found in this experience will also be helpful in re-designing the learning trajectory in a way that can be better utilized when new groups work with this problem. Similar teaching strategies may prove as successful as this one in teaching new and complex linear algebra concepts to students through the use of rich contextual problems. Acknowledgments This project was funded by CONACyT Grant No. 62375, and also partially supported by Asociación Mexicana de Cultura A.C. This work was undertaken by the authors in collaboration with other researchers, in particular the authors thank Gustavo Preciado, Dolores Lozano, Javier Alfaro Pastor, María del Carmen López Laiseca, and Ivonne Twiggy Sandoval for their work on the experience presented here. Finally, the authors thank the external researcher for sharing her preliminary results, and the reviewers for their useful comments and suggestions. References [1] J.M. Day, D. Kalman, Teaching linear algebra: issues and resources, College Math. J. 32 (3) (2001) 162–168. [2] L. Dorier, A. Sierpinska, Research into the teaching and learning of linear algebra, in: Derek Holton (Ed.), The Teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publisher, 2001, pp. 255–273. (Printed in The Netherlands). [3] G. Gueudet, Rôle du géométrique dans l’einsegnement de l’algébre linéaire, RDM 24 (1) (2004) 81–114. [4] M. Trigueros, A. Oktaç, La théorie APOS et l’enseignement de l’algébre linéaire, Annales de Didactique et de Sciences Cognitives 10 (2005) 157–176. [5] D. Kú, M. Trigueros, A. Oktaç, Comprensión del concepto de base de un espacio vectorial desde el punto de vista de la teoría APOE, Educación Matemática. 20 (2) (2008) 65–89. [6] M. Parraguez, A. Oktaç, Construction of the vector space concept from the viewpoint of APOS, Linear Algebra Appl. 432 (8) (2010) 2112–2124. [7] D. Carlson, C.R. Johnson, D.C. Lay, A.D. Porter, A.E. Watkins, W. Watkins (Eds.), Resources for Teaching Linear Algebra, MAA Notes, vol. 42, 1997. [8] E. Possani, M. Trigueros, J.G. Preciado, M.D. Lozano, Use of models in the teaching of linear algebra, Linear Algebra Appl. 432 (8) (2009) 2125–2140. [9] E. Dubinsky, M. McDonald, A constructivist theory of learning in undergraduate mathematics education research, in: D. Holton (Ed.), The Teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publishers, 2001, pp. 273–280.
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