Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 93
A commognitive approach for teaching functions: The discursive
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 94
requirement to avoid ambiguousness. A modern definition of a function does not restrict the nature of the dependency between the variables. There is also no restriction of the variables themselves. Both are arbitrary. The two properties are the result of the historical development of the function (Boyer, 1959; Kleiner, 1989). The same properties are also among the demands for the MKT by Even (1993).
A commognitive approach
A commognitive approach was used on several levels. First, the learning goal for functions is formulated within a commognitive framework. Second, the analysis is done with a commognitive approach.
Sfard regards communication and cognition as intertwined: “[t]hinking is an individualised version of (interpersonal) communication” (2008, p. 81). With the blending of communication and cognition, a new word is merged of the two nouns:
commognition. An implication is that narratives can be regarded as expressed thinking.
Sfard (2008) describes mathematics as a special discourse with four properties that makes it distinct: word use, visual mediators, narratives and routines. Visual mediators, visualisations used as mediators, are operated upon as an important part of communication. As participants in a discourse the mathematical activity consists of producing narratives that can be endorsed. According to Sfard (2008) and Lavie, Steiner and Sfard (2018) mathematical narratives are governed by metadiscursive rules split into three categories of routines, corresponding to their use: rituals, deeds, and explorations. An exploration will end with an endorsed narrative about change of mathematical objects containing the story of what, and why, the change appeared.
Before the learners end up with the full-fledge explorations, they will undergo a development through deeds and rituals. As interlocutors in the discourse, learners will meet mathematical objects. A mathematical object is defined as a signifier with a corresponding realisation tree (Sfard, 2008). A realisation tree is a hierarchical organisation of realisations (representations in other frameworks) of the signifier. An example is the signifier “quadratic function” where realisations can be the expression 𝑥", the corresponding graph or a table of values. All of them are perceptually accessible and called primary objects (p-objects). Sfard (2008) argues for an objectification as a discursive process where discursive objects (d-objects) are individualised. The participants must create narratives about p-objects. Simplifying is a driving force in the communication and a next step will be to signify p-objects by assigning a noun or pronoun. Thus, a concrete d-object is created. More advanced processes are involved when d-objects are put together and individualised: saming, encapsulation, and reification. Through saming, “the act of calling different things the same name” (Sfard, 2008, p. 170), a search for common attributes will end by assigning one signifier to many realisations; e.g., when “linear function” is used in communication both about linear polynomials on the form 𝑎𝑥 + 𝑏 and the graph as a straight line. When a noun, or pronoun, will signify a specific set of objects, the act is called encapsulation. Several objects are turned into a single entity. This act can be observed as a change from
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 95
narratives in plural being changed to singular when talking about a property of all the set members. Finally, reification happens by introducing a noun or pronoun for processes on objects to create narratives about relations between objects (Sfard, 2008).
An abstract d-object will substitute narratives about processes. In the discourse of functions a crucial reification will be that 𝑓(𝑥) signifies the function as a d-object and will not be realised as a process of “putting values into an expression and calculate a value”.
THE DESIGN EXPERIMENT
According to DR the implementation of the teaching is a design experiment, build upon a conjectured Local Instruction Theory (LIT) as a foundation for a Hypothetical Learning Trajectory (HLT). I will now describe how the design experiment took place, the participants, and the basis for the planning.
The students and data collection
The participants of the design experiment were twenty-six PST in their second year of teacher training. All had chosen mathematics as their primary subject and completed a mathematics course of thirty ECTS in their first year. Combined with the second year course of thirty ECTS, the PST shall be prepared for teaching mathematics to 11-16 year old pupils in Norwegian schools. None of them had any experience from teaching except for the practice during teacher training education.
The collected data consist of written material and recordings of the group- and plenary discussions. Both the participants and me wrote logs. As attendance was voluntary, the number of students who took part of the different lectures may vary.
Some of the narratives from students, who can typically exemplify the discourse, were chosen. These narratives will serve as examples to illustrate the changes of the whole discourse. These students are designated S1 to S5.
The planning and the implementation
A considerable constrain, caused by national, institutional, and collegial premises, was a time limit of four plenary lectures. The duration of each lecture was three hours.
Between each lecture the PST had to read and complete assignments. As a consequence of the limited lecture time two instructional videos, were the teacher’s monologue played the main part, were made. After watching the videos the students were required to write short summaries and to write down questions for the next lecture. The foundation for the teaching was that participation in an orchestrated discourse would contribute to a transition from processes, and p-objects, to a reification of the function object. Thus, a discursive practice, with all PST engaged as participants, had to be created. Then the role of more experienced participants, as the teacher, becomes important. All narratives, both orally and written, has to be carefully considered by the teachers. The use of “function” has to signify a reified object. Hence, the discourse must be guided in the same direction by allowing the participants to produce narratives that can be collectively corrected.
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 96
Digital tools are important mediating artefacts for the learners. In this design they include Dynamic Geometry Software (DGS), graph plotters and Computer Algebra System (CAS). The main digital tools were GeoGebra and Desmos. A conjecture was that these artifacts can contribute to a rich use of realisations of the function and provide visual mediators for the interlocutors.
RESULTS AND ANALYSIS
I will now present an overview of what happened during the teaching periode. First, the discursive change in the narratives of functions, then some episodes that argues for an individualisation of the function object, and last the role technology played.
The students’ initial definitions of a function
Before the experiment, the PST had to write down an answer to the question: What is a function? All writings were collected, and a plenary discussion took place. This served as a pre-test for the experiment. The narratives were analysed according to how they responded to the signifier “function”. S1 wrote: “I would say that the concept of a function is how to do calculations from a formula where you can use variables. The result will be a graph in a coordinate system.” In this narrative “function” can be analysed as a signifier for a process where the result of the process is a p-object, the graph. When S2 responded: “It is an expression that can be written with letters and numbers”, there is a reference to a p-object with a limited realisation tree. The same applies for the narrative of S3: “Something that shows a picture/line of values that changes during certain time periods (hours, days, years)”. The p-object is a graph and limited to a time unit, and “function” is not used as a signifier. These narratives are typical for sixteen, of the twenty students who took part in the pre-test. None of these respondents’ narratives made explicit requirements of univalence and arbitrariness, and none showed an object-driven use of “function” and a reified function object. The rest of the respondents produced vague, or incorrect, answers.
The students’ definitions of a function at the end
At the end of the last lecture a plenary discussion on how a teacher should perceive a function, was conducted. Before the discussion everyone had to write down their own description of a function. These written responses were analysed with the narratives given at the plenary discussion. Twenty-two students participated. The signifier
“function” was used as a noun describing a covariation, or coherence, between variables, in a majority of the respondents. Sixteen of the PST showed that clearly. The utterance of S1: “A function is a connection between various factor, often x and y. x depends on y. Each value of x gives one y”, is an example of that category. A total of seventeen narratives express explicit one-valuedness.
None of the narratives contain a direct statement about “function” as a process, but three can be interpreted as implicit utterances of a process. An example is S4: “A function is about relationship of values. Every argument value will affect the value of the function. It is exactly one dependent value”. The use of “is about” is imprecise, but
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 97
interviewed later the respondent corrected the narrative to: “It is the function that expresses the co-variation between the values”.
The written, and oral, narratives show a change in the discourse. The signifier
“function” had changed from signifying a process to signify a function object. The different narratives of S2 provides an example. In the first, S2 expressed a process. The last was: “In order to call something a function there has to be correspondence between the value you use as a variable and the value given by the function. For each input only one output”. This is a phrase-, or object, driven use of a d-object with explicit univalence. Thus, an example of the development of the majority of the students.
Arbitrariness was also examined, but is not as evident as the discursive change from processes to objects. It was more clearly drawn into the plenary discussion.
A change of the discourse
During the experiment, there was a change in the discourse. The narratives of functions transformed from being about processes and p-objects into the use of the function as an object, including the characteristics of univalence and arbitrariness. Of course, endorsed narratives cannot give an exact answer on how the object has been individualised – they may be phrase driven. As shown by others (Tall & Vinner, 1981;
Vinner & Dreyfus, 1989) there may be a distinction between definitions and by how they are actually treated. On the other hand, incorrect narratives would imply direct shortcomings in the MKT. The change in discourse and the proper narratives are indications of correctness of the hypothesis in the LIT and HLT about the facilitation of the discourse. At the end, most narratives can be categorised as explorations.
An objectification can also be observed from the use of signifiers. Both “function”, and various use of symbols like 𝑓(𝑥), gives rise to narratives where they are used to signify a function object with a noun. This is typical for an individualisation of the function object by the participants. The narratives also show a rich use of realisation trees.
HOW DID TECHNOLOGY PLAY A ROLE?
The result of the experiment is due to many factors and the commognitive approach regards learning as both situated and distributed. Thus, it is not easy, or even possible, to isolate each factor. Nevertheless, I will try to elaborate on the role and contribution of ICT without insisting on a direct isolated effect.
First, ICT was important as an organiser. A Learning Management System (LMS) was used as tool to structure the learning experiment. It was also used to assign, and collect, responses. The LMS offered the use of a diary and logbooks which proved important for the production of the narratives. These narratives sought to increase both intra- and interpersonal communication. As the narratives could be both peer reviewed, and read by me, they contributed to many productive plenary talks, comments, and served as a reflection tool for interpersonal communication. With the immediate access in the LMS, they also served as an important guide line for adjusting the discourse by the
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 98
teacher. Another contribution of technology was the use of instructional videos and the collection of the responses to the videos in the LMS.
Digital tools were also important mediating artefacts and supportive for an inquiry process. I will now outline some of these contributions. The challenges reported on conversions between realisations (cf. Duval, 2006) were considered for the experiment.
In the commognitive framework transitions can be considered as including realisations in a discourse and let these play a part in the objectification. The dynamic nature and the easy transitions of realisations offered by the tools constitute an important part of the narratives. Through saming, encapsulation and reification, realisations are crucial for the individualisation of the function object. The digital artefacts can supply realisations in form of objects which can serve as visual mediators. In the function discourse, examples of visual mediators are 𝑓(2) − 𝑓(3) and 𝑓(𝑔(𝑥)). Both should be regarded as objects and treated that way. I will provide some examples. At the first lecture, the PST were given a task with a graph of a third degree polynomial function in a coordinate system. The students should solve the equation 𝑓(𝑥) = 0 and find 𝑓(2) without access to the expression of the function. Only nine out of twenty-two student interpreted 𝑓(2) as the correct –2. Seven did not answer. In the same task, they were also given a function by 𝑔(𝑥) = 𝑥"+ 1 and supposed to find 𝑔(3). In the plenary discussion that followed they explained that as an easy task, but without an expression given, they were unsure about what to do. The students’ explanations can be taken as evidence of a lack of realisation of the signifier 𝑓(𝑥), and an incorporated routine for calculating function values. During the experiment the use of digital tools as artefacts for producing visual mediators sought to improve the students’ realisation tree.
Examples are conversions of physical situations into realisations, tracing graphs with coordinates visualised, and an extended use of symbols. In reflections about the initial task of this example, most of the students expressed astonishment about the fact that they were unable to solve the task.
In a modelling task with the draining of a water tank, the students had to find the rate of change in intervals. They could do that by calculating 0(121)30(14)
2314 . First, the students calculated every value, but by the tools they saw the use of simplifying and treating 𝑓(𝑡) as an object. There was also a change in the discourse as the students started to form narratives like “what is being drained between the third and the second minute”
as a replacement of 𝑓(3) − 𝑓(2). At the end, all participated in a discourse where symbols were replaced the function values. That supports the claim of objectification as well as the contribution of ICT to provide function values without the process of calculation as interference. In another context, the signifier ℎ(𝑡) was used as “all the heights” in a situation concerning the height of a tree over time. Another contribution for the objectification is the object nature of CAS commands. CAS syntax often demands a need to operate on the function object given by an expression. The operation is carried out on the function as an object, and the result can either be properties of the function or a new function. Tasks that emphasised both the result, and reflections about
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 99
operations on the function object, showed important for the reification. An example is the command Derivative(f) where the input is the declared function and the result is a new function object – the derivative. Some of the reflections about the nature of these commands were crucial for the individualisation of the function object. The same example also provided discussions about univalence: What if univalence was not required? With a CAS command as a visual mediator narratives like “it is obvious that the function cannot give multiple values” were typical. The arguments were found by the impossibility of ambiguous input and output of commands.
A modelling approach was used during parts of the experiment. With digital tools, the students had to collect data from real life situations to make a model. As they had to produce narratives about the process, several important aspects for the properties of the function came up. Arbitrariness could be discussed when an expression was found by regression: Did the expression really delineate the covariance correct? In some cases, the conclusions were that regression gave the best fit, but the covariance could not be expressed through an expression. The gap between the model found, and reality was used for discussions about the arbitrary property of a function.
Throughout the experiment, a main goal was to support how different discourses could be subsumed into one about functions. As ICT provided easy, and rapid, transitions between physical situations, graphs, expressions, and tables, narratives of the each realisation could be transformed into narratives of the function object. This helped the different discourses to be subsumed into one.
CONCLUSION AND DISCUSSION
An explicit change in the discourse could be observed and a conclusion is that technology provided an important contribution to the change. All PST had been taught functions before teacher education and mastered common process related to function, but important properties and a reified function object was missing. The discursive change supports a claim of an individualisation of the function object. Without the support of digital tools this change is hard to imagine within the limited time available.
Naturally, there are several adjustments I would have made. One is to make more videos for the students to see between lectures. That could have provided more time for discussions and work in groups. A more thorough examination of each individual would also be useful. That could have revealed a phrase driven use of narratives as an approach to provide what as intended by the teacher.
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