Reflections on what happens in a mathematics class and video recordings as resources

The ability to assess what happens in a classroom seems to be one of the basic skills that a mathematics teacher must have. It is therefore necessary to have conceptual frameworks and appropriate methodological tools ( ^{GODINO; BATANERO, 2009} ). For Seckel, a teacher expresses or demonstrates the competence of reflection on teaching practice, his own or someone else’s if “[...] he or she critically analyzes their pedagogical practice and that of other teachers in terms of its impact on student learning and proposes and provides a basis for changes to improve it” ( ²⁰¹⁶ , p. 40). This research seeks to define possible strategies that lead to such an assessment, so that the teacher can notice or identify situations of interest and explain them to make sense of alternative practices ( ^{MASON, 2002} , ²⁰¹⁷ ).

Lesson Study ( ^{CLARKE et al., 2006} ; ^{ISODA et al., 2007} ; ^{LEWIS; TSUCHIDA, 1998} ; ^{YOSHIDA, 1999} ) is a strategy or methodology used for the improvement of mathematics education which has attracted interest in the study of processes of initial training and professional development of in-service mathematics teachers.

Although Lesson Study can be used to create activities or good practices, Yoshida states that this is not his goal. His main concern is that teachers who are outside the context of study or outside of school communities who study lessons “[...] adopt without any reflection these practices in which there are gaps in their understanding, due to the lack of mathematical and pedagogical knowledge , and of skills necessary to observe and evaluate the learning of their students in the classroom” ( ²⁰¹² , p. 141). Although this author focuses on the case of teachers in the US, this concern is valid in a large part of the educational community.

The use of videos to analyze instructional processes is also a strategy currently used to evaluate the abilities and competencies of a teacher based on selected situations ( ^{BORKO et al., 2011} ; ^{KAISER et al., 2015} ).

Kaiser and other authors (2015) state that this strategy is important because it provides a broader picture of situations in the classroom and allows better assessment of the capacity and knowledge of the teachers who are evaluated (as compared, for example, to evaluation of situations described using paper and a pencil). These authors also warn that there could be an important limitation on this strategy when the classes recorded are artificially constructed (imposed situations) because the actions that actually occur in a real class are lost. There are already several theoretical frameworks that support the possible uses of videos of mathematics teachers depending on the purposes for which they are used ( ^{COLES, 2014} ).

Providing theoretical support to the benefit of video analysis goes beyond the objective of this article. In this research, the teacher analyzes his or her own mathematical instruction in the classroom, assessing their work and analyzing recorded videos, situations, decisions or behaviors that were made consciously or unconsciously.

Given the international trends mentioned previously, the use of video to study real situations in instruction has attracted increasing attention and some research provides specific results on teacher training and video recording ( ^{ROSAEN et al., 2008} ; ^{KLEINKNECHT; SCHNEIDER, 2013} ).

The DMKC model

The didactic analysis of classroom activities can be approached from different perspectives. In particular, the DMKC model ( ^{GODINO et al., 2017} ; BREDA; FONT; PINO-FAN, 2017) based on the Onto-semiotic Approach (OSA) ( ^{GODINO; BATANERO; FONT, 2007} ) offers theoretical tools to structure this analysis.

The DMKC model considers that one of the key skills of the mathematics teacher is the general competence of analysis and didactic intervention, which is broken down into five sub competences linked to the analysis of: a) global meanings, b) mathematical practices, c) didactic configurations, d) standards, and e) didactic suitability. This model is used in the present study, with an emphasis on the assessment of the didactic suitability sub competence. The didactic suitability of the OSA is viewed as a coherent articulation of six specific suitability criteria ( Table 1 ).

Each of these criteria can be broken down into components and indicators as a category, in order to make them operational. For this research, ^{Font’s (2015)} proposal is used. The criteria and components of didactic suitability of the aforementioned proposal are detailed below (due to space issues, the indicators are not detailed) ( Table 2 ).

The methodology used in this study is described in the next section.

Description of the participant

A 38-year-old Secondary Education math teacher with a Bachelor’s Degree in Mathematics Teaching from a state university in Costa Rica in 2008 was selected. By 2014, she had 14 years of teaching experience (she began working before graduating). In 2010 she began working as a teacher at the school where she was recorded. The selection of the participant was based on convenience ( ^{STAKE, 1995} ).

Protocol

A recording of a math class of a teacher was made as she taught the logarithmic function in a public secondary school in Costa Rica. In the first stage, the recording (1 hour, 5 minutes, and 40 seconds long) was examined as follows:

1. An expert analysis of the first four types of didactic analyses proposed in the didactic analysis model based on the OSA constructs was carried out following ^{Godino, Contreras and Font (2006)} and ^{Pochulu and Font (2011)} , to identify mathematical practices, objects and processes, teacher and student functions, didactic configurations, semiotic conflicts, patterns and norms.

2. With the information obtained from Point 1, an expert assessment of didactic suitability was carried out (the fifth type of didactic analysis of the didactic analysis model based on OSA constructs) to be used as a reference for the study of the teacher’s own evaluations (see Montiel and other authors, 2009, as a model of the study of valuations).

In a second stage, the teacher who taught the class was asked to analyze her activities to determine her explicit intentions and assess the situations that occurred. To do so, a workshop was held with the teacher in which the video was shown on two consecutive occasions; afterwards, she was asked to record in writing the elements of the recorded mathematical activity that she considered to be important to analyze (after the two continuous showings she was allowed to control the presentation of the video in additional showings using forward, delay, pause or stop). The activity lasted 3 hours and 30 minutes.

In a third stage, the teacher’s ratings were classified using the criteria and components of didactic suitability described in the section on the DMKC model ( ^{FONT, 2015} ).

Finally, in a fourth stage, the previously categorized information about the teacher’s observations (third stage) and the analyses carried out in the first two stages were contrasted to show which assessments coincided with the expert assessment of the criteria, components and indicators of suitability, and which were not identified or evaluated by the teacher; the content of the teacher’s evaluation was analyzed to determine its relevance in her justification.

Context and description of the class

The teacher worked with her students on the concept of the logarithmic function, which is presented in the fourth year of high school as specified in the Transition Plan between the previous and current Programs of Studies ( ^{COSTA RICA, 2012} ). The students were between 15 and 17 years old. She organized the class based on the specific suggestions contained in the Mathematics Studies Program of the Ministry of Public Education of Costa Rica (COSTA RICA, 2012). Table 1 describes the instructions of the Mathematics Studies Program.

Figures 1a, 1b, 1c, 1d, 1e, 1f, show the sequence of work used by the teacher in her class

In the first part ( Figure 1a ), the teacher reminds students of the problem raised in the previous class and writes the mathematical model that solves that problem. She defines the names of parameters of the model that she indicates will serve to resolve the problem that she has presented and asks students to form work groups. When starting work after the question, she tells them to use the trial and error method. In Figure 1b , students mention some approximations. A group of students indicated that the value was between t = 8 and t = 9. She states that it is obvious that they do not have tools to solve the exponential equation, so she is going to introduce them to the logarithmic function to solve the problem.

Source: Photographs taken from the video recording

Figures 1a, 1b, 1c, 1d, 1e, 1f Sequence of the notes on the whiteboard 

Figure 1c shows that the teacher wrote an incorrect definition of the logarithmic function, and the way to convert an exponential expression to a logarithmic expression. She explains the definition and introduces the procedure to convert them. She does this using a specific example (2³= 8 if and only if log₂8 = 3) and uses an associative memory technique based on a diagram whose shape is similar to that of a heart. Figure 1d shows how she returned to the previous problem and used the note explaining the conversion using the heart technique (a heart shape as a graphical mnemonic rule). The teacher asks them to indicate where to put the data in the model written on the board to perform the conversion of (53/50)^t= (5/3) <=> t = log_(53/50)(5/3). She then says that they should enter the information in their calculators.

In the penultimate part ( Figure 1e ) the teacher graphs a logarithmic function [f (x) = log₃x]; to do this, she writes a table on the whiteboard with the values of the variable x and ask them to complete the table using the calculator (the values of the dependent variable y). She deliberately includes values not included in the R⁺domain so that they can see what happens using the calculator. Next, the teacher talks about the positive values of the table she wrote on the board and asks them to add a 1 in the table (in the domain), because she forgot to do so (her intention was to talk about the intersection with the X axis later). She comments on the characteristics of the values of the dependent variable y (which have an increasing trend). She then draws a line on the Cartesian plane, mentions that the line does not touch the Y-axis, and reminds them of the meaning of an asymptote (the analogy she used was that there was an electrified fence and the line could not touch it).

She then tells the students that the intersection with the X-axis is (1,0) and asks them about the intersection with the X-axis of the graph of the exponential function. She discusses the injective and surjective nature of the function and indicates that the exponential function is the inverse function of the logarithmic function; next, she graphs g (x) = x and reminds students that the inverse functions, graphically, look like a reflection with respect to the y-axis, and asks them for the name of the reflected line. Since no one answers her, she tells them that it is the graph of the exponential function.

In Figure 1f , the teacher uses the property that a⁰= 1 and the heart technique to infer that log_a1 = 0. She then relates this to the value of the table that she had previously forgotten to write and says that this is always true. In the same way, she generalizes this for a1 = a if and only if log_a1 = a. Finally, she tells them that in the next class they will plot f (x) = log_(3/4)(x).

Elements of suitability present in the teacher’s evaluation of the instructional process

The components and indicators of didactic suitability criteria for the class taught by the teacher were analyzed in two ways. On the one hand, the elements that the teacher used when she made her assessment were discussed. On the other hand, the elements that she did not use in her analysis and that could be of interest for the analysis and assessment of her teaching (as indicated in the expert assessment) were also mentioned.

Components and indicators of epistemic suitability [ES]

[ES1] an erroneous definition of the logarithm. With respect to this component, the teacher indicated that possibly because she was nervous, she made an error when defining the logarithmic function with a base different from 0 and greater than 1. It should be noted that this error caused several problems, such as when a student tried to use the definition to justify the domain when graphing f (x) = log ₃ x . In addition, the teacher failed to notice that she made an error in the use of the concepts of approximation and exact values; she even argued that the value obtained using a calculator is the exact one, neglecting or devaluing the approximations that she made in the earlier part of the class. This caused several students to care more about rounding decimals than about what they were solving.

[ES2a] Regarding possible ambiguities, the teacher’s assessment noted that she tried to make a didactic transposition that was perhaps inadequate, when she stated that the class members would test t values using the calculator to find an approximation. [ES2b] She also noted that she has never used the symbol “does not exist” (∄) in her class and indicated that she did not remember that she needed it until that moment, and did not define it correctly. Another ambiguity that she detected is the analogy of an “electrified fence” when discussing the asymptote; she acknowledged that this analogy might be inadequate, although she indicated that she would use it again [ES2c] . The problem with this metaphorical analogy is that it did not convey everything that she wanted to, and every time she referred to the asymptote, she had to use further analogies or metaphors (like “approaching, but not touching”) to convey her idea. ( ^{FONT; BOLITE; ACEVEDO, 2010} , ^{FONT et al., 2010} ).

An ambiguity that the teacher did not identify explicitly was that which determined the choice of the values of the table and the graph that she drew, since it allowed, at least visually, a logarithmic adjustment, but also seemed to be linear; even a student noticed this (when the teacher asked a question, a student stated that the graph was a straight line). With reference to this problem, she only indicated in her written notes that “I should have used a more exact scale (I do not like to graph)” [ES2d] . Another element that the teacher did not mention is that, at various times, she talked about the model function that they were working with as if it were an exponential function, which is not at all rigorous. The correct way to discuss it is to consider it as a particular case of an equation with parameters.

[ES3a] Regarding the richness of processes, the teacher indicated that at certain times more effort might have been made, such as assisting students to understand the use of 6% or 0.06. She was aware that she explicitly instructed the students to solve the equation “using an algebraic strategy”, but noted that they responded by indicating what actions could be taken to simplify the equation to the form a = b ^x that she intended [ES3b]. Another aspect that she indicated that she did not take advantage of was the discussion that could have been had about converting an expression t = 8.77 years to a more common expression like years, months and days [ES3c] . The teacher evaluated as “great” the fact that they made the generalization that in the table “only the negatives” did not work [ES3d] .

There were several elements that she did not identify. The exposition of the problem was linked to a function model which initially suggested to the students the method they should use to find a solution (trial and error); this made the problem a very simple exercise and limited the richness of the process. It is important to point out that the introduction of the topic began with a model to be applied, when it might have been more useful to begin with a problem and finding a way to model it. In addition, the important processes of regulation were carried out completely by the teacher, without considering the contributions made by students during the solution of the problem. It finally stated all the properties and the students tested numbers, applied the algorithms and filled in values in the table.

Suggesting trial and error and later disqualifying it as an appropriate method not only reduced the continuity of the class, but could implicitly suggest to the students that using this approach is not a good way to start a problem (contradicting the structure of the class). It is sufficient to note that a student arrived at the same answer through trial and error as the teacher provided as an exact answer when she applied logarithms ( t = 8.77 ). Another aspect related to the richness of processes is that there were some indicators that the teacher did not evaluate, but which are of interest: there was a conversion of records (table - graph), there was reasoning about negative values within the table, the properties of logarithms were presented, and the connection between exponential logarithmic functions was discussed (confusingly). However, all these processes were carried out by the teacher.

[ES4] With respect to representativeness, the teacher indicated that the need to use algorithms to solve the problem was evident among students (used as a technique more than as a concept). She was able to try to solve some logarithms to give a clearer idea of what a logarithm is (perhaps through an arithmetic approach).

Although the concept of a logarithmic function was treated in a manner that was consistent with specific indications of the official curriculum, strictly following this approach sacrifices the opportunity to talk about other situations that could be addressed, such as those mentioned in the methodology of the Program of Study ( ^{COSTA RICA, 2012} ): 1) Using cuttings from magazines, newspapers and statistical data (p 421); 2) History of logarithms (p 426); 3) Sound intensity (p. 426); the Ritcher scale (p 426); 4) Newton’s Law of Cooling (p 426); 5) pH measurement (p 427); 6) Compound interest model (p. 427) (the one used by the teacher). Regarding representations, the teacher did present different ways of representing a mathematical object (tabular - symbolic and graphic).

Components and indicators of Cognitive Suitability [CS]

[CS1] previous knowledge. The teacher stated that the first thing she did was to briefly discuss what she explained in the previous class because she considers it to be “always convenient”. She stated that students understood the concepts of domain, rank and scope, which was incorrect, based on later evidence. Regarding curricular adaptation to individual differences, she did not carry out reinforcement or expansion activities (although this is of course influenced by her management of time versus the time available). Regarding student learning, there is not enough information to confirm that such learning took place, but it was observed in the video that the teacher works to solve students’ difficulties.

[CS3] learning. Regarding logarithms, the teacher stated that “it is not possible to deduce this knowledge. I think you can only show its usefulness.” It is possible that there is a relationship between what she thinks about the learning achieved with the fact that there has not been high epistemic suitability. This may be due to the specific indications of the Program of Studies, since they suggest taking the model as true and applying it to a situation that is suggested as real.

[CS4] cognitive demand. The only cognitive achievement the teacher believed that students obtained was being able to read a graph. It should be noted that it was the teacher herself who pointed out that this was prior knowledge.

Components and indicators of interactional suitability [IS]

[IS1] Teacher-student interaction. When the teacher wanted to show how to use the model in an algebraic manner, she repeatedly consulted the students about what they had observed when they worked in groups, which she considers to be valuable in facilitating the inclusion of students since she allowed them to discuss what they had worked on. This positive assessment is debatable, since student comments are based on her answers when they had asked about solutions (in some ways she had suggested the answer and there is no evidence that the students had constructed those concepts themselves).

The teacher indicated that she does not know if the way she ignored students who provided erroneous answers was incorrect, but indicated that she had decided to direct questions to the persons that she already knew had solved the problem (to obtain the answer she was waiting for) [IS1a] .

Despite the filming, the teacher stated out that her class “developed naturally” [IS1b] . In addition, she indicated that she had to “deal” with them several times, because the group is particularly undisciplined. She also added that she valued very positively the fact that students are trying to respond even “using an adequate vocabulary” [IS1c] . However, she indicated her awareness of the fact that although a large part of the class consisted of lectures, the students do the work that they are assigned [IS1d] .

Regarding her way of dealing with the students, she indicated that she considers it to be important that she addressed them in a respectful way and always motivated them to improve, although she added that she considers herself to be a teacher that is “completely dominant and controlling (I always demand attention )” [IS1e] .

With regard to the elements that she did not evaluate, it can be stated that her classroom management has lead to very limited discussions on the part of the students, and that she has ignored erroneous arguments (which can be interpreted as evidence of exclusion for those that have participated but have not arrived at the answer that the teacher expects). Another point is that the teacher almost always resolves from the start the difficulties or doubts that students have.

Although she did not consider the point as valuable, she has facilitated interaction among students by having them work in small groups. However, this did not promote autonomy because she provided them with the model to solve the problem. Finally, there is no evidence of having proposed a formative assessment of her class.

Components and indicators of Mediational Suitability [MS]

[MS1] With regard to resources, the teacher concluded that students did not know how to use calculators well, even though the use of calculators was a valuable resource, and that this negatively affected class fluidity. Even so, the teacher instructed students to use calculators as a black box ( ^{LINDSAY, 1975} ) to calculate logarithms, without understanding how the calculation is actually carried out. She also believed that her use of the whiteboard was inadequate (for example, when presenting the table and the graph), and suggested that perhaps she should have used software to present her ideas: “it would have allowed me more flexibility (in showing asymptotes, intersections, and the inverse)” [MS1a] .

As for classroom space, there are not too many students or other problems related to physical space. The teacher indicates that she effectively adapted her use of time to the planned activities [MS1b] .

Components and indicators of Affective or Emotional Suitability [AS]

[AS2] Attitudes. The teacher does not assess any of the needs, attitudes and emotions of the students, but considers that the fact that students are motivated, and explain their answers and show interest in verifying their work is very positive; regarding this, she stated that she believes that she has convinced the students of the importance of this topic [AS2] . Although the students do indeed work, this activity in itself does not seem to be especially motivating for the students; on the other hand, the teacher does promote students’ interaction and listens to their arguments (even when they are those which she herself previously suggested).

Components and indicators of Ecological Suitability [CIES]

[CIES1] The teacher indicated that she had to modify or adapt the model suggested in the Program of Studies. Her experience working with other groups was that, as formulated, the Program created an obstacle due to the amount of calculations that students had to perform, so she presented the problem with the time data in years. Perhaps the issue that should be considered is that the curricular guidelines were completely followed without a deeper analysis of what could be achieved by following these specific recommendations. With respect to intra- and interdisciplinary connections, the teacher noted that she connected the subject with the exponential function [CIES2] , but she had not realized that she left out connections with other disciplines.

Although she did not comment on it, the topic that was being presented seems to have social and labor utility, because it is useful content for this context, although perhaps not at this moment. There is no evidence of any curriculum innovation (since she followed all that is indicated in the curriculum), but the traditional way of teaching was modified in the sense that an effort was made to include student participation.

Although several factors have been presented that could be taken into account in redesign of her class, it should also be noted that the management of the class was carried out responsibly by the teacher, who chose appropriate options when taking into account the amount of time available to introduce this subject. This is consistent with the vision of ^{Huang and Li (2009)} discussed in the Introductory section of this document.

Some implications for the development of the teaching function and open questions

The results of the work carried out in this investigation offer tools that may be relevant for teachers who wish to evaluate their own teaching practices. In addition, the methodology and framework of this research could be used as a strategy for systematic self-assessment of what happens in the classroom. The results of this research agree with other investigations ( ^{BREDA; FONT; LIMA, 2015} ) which confirm that if the teacher manages to create a conceptual framework such as that of didactic suitability, with its components and indicators, through a rubric-type instrument that operationalizes it (such as the one of ^{FONT, 2015} ), then it could provide not only a strategy or protocol for self-assessment, but also for obtaining elements to guide the search for resources to improve knowledge in mathematics, pedagogy and mathematics didactics, as well as to increase the ability of teachers to identify and reflect on situations in mathematical instruction. It can also serve as support for training of teachers who wish to engage in self-assessment.

This type of self-assessment can be an excellent resource, but it is only a tool that must be evaluated through triangulation using evaluations by peers (as in the case of Lesson Study , for example), by supervisors or advisors, or by their own students, which can also provide valuable information about what is happening in the classroom.