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Prospective mathematics teachers’ interpretative knowledge: focus on the provided feedback
Jeannette Galleguillos, Miguel Ribeiro
To cite this version:
Jeannette Galleguillos, Miguel Ribeiro. Prospective mathematics teachers’ interpretative knowledge:
focus on the provided feedback. Eleventh Congress of the European Society for Research in Mathe- matics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. �hal-02422519�
Prospective mathematics teachers’ interpretative knowledge: focus on the provided feedback
Jeannette Galleguillos1 and Miguel Ribeiro2
1Universidad de Valparaíso, Chile; [email protected]
2State University of Campinas – UNICAMP, Brazil; [email protected]
In this work, we focus on the meaning prospective mathematics teachers assign to a set of students’
productions and the feedback provided to these students. Based on these findings, we discuss prospective teachers’ interpretative knowledge required for providing a fruitful feedback to students, having their reasoning as the starting point. Data collection (written responses) for this investigation took place at a workshop involving 19 prospective teachers from a Chilean university.
Thematic analysis of their responses revealed two distinct types of feedback, namely that grounded in the nature and focus of the interpretation provided, and that focusing on the content of prospective teachers’ knowledge.
Keywords: Mathematics teachers’ interpretative knowledge, error analyses, teachers’ feedback.
Introduction
Teachers’ knowledge is one of the core factors considered when developing highly demanding mathematical practices (Ball, Thames, & Phelps, 2008). Thus, in that sense, it is essential to focus on the content of such knowledge and, in particular, on the dimensions that would allow teachers to develop such mathematical practices, having the students’ own reasoning and understanding as a starting point for the mathematical discussions, as well as for determining the feedback to be provided to these students. By better understanding the provided feedback, the aim is to enrich (prospective) teachers’ knowledge and experience related to teaching practice. The knowledge grounding such type of practice is perceived as specialised.
In order to deepen and broaden our understanding on the content of teachers’ interpretative knowledge and promote its development, we have been developing tasks for teacher education with a specific focus of developing teachers’ knowledge (e.g., Jakobsen, Ribeiro, & Mellone, 2014;
Policastro, Mellone, Ribeiro, & Fiorentini 2018, submitted). We have been using tasks that probe into teachers’ interpretative knowledge in different teacher education contexts (initial, continuous, and complemental) as a prompt to orchestrate mathematical discussions (Bussi, 1996). The aim is to develop the solvers’ awareness (Mason, 2001) and specialised knowledge—in this case, the Mathematics Teachers’ Specialized Knowledge (Carrillo et al., 2018). Such specialised knowledge would permit teachers to give meaning to non-standard student reasoning (Mellone, Jakobsen, &
Ribeiro, 2015), as well as to better understand the causes of students’ errors (Tulis, 2013), thus treating them as real learning opportunities in Borasi’s (1994) sense.
In the scope of a broader project, the aim of the present study is discussing secondary prospective teachers’ Interpretative Knowledge (IK) based on the feedback they provide to students when analysing and commenting on productions students provided to a specific mathematical problem.
Theoretical Framework
Shulman (1986) emphasised the need for integration of pedagogical and disciplinary knowledge in teacher education. Other researchers contributed to the mathematics teacher development,
proposing to move away from a segregated and advanced training in “pure” mathematics to focus on the development of specialised knowledge that allows the teacher to teach mathematics.
Amongst several teachers’ knowledge conceptualisations that have emerged, in the scope of the work we have been developing, we assume the specialised nature and content of teachers’
knowledge in the sense of the Mathematics Teachers Specialized Knowledge (MTSK) conceptualisation (Carrillo et al., 2018). Teachers’ specialised knowledge grounds teachers’
interpretations of the students’ comments and productions (e.g., Di Martino, Mellone, Minichini, &
Ribeiro, 2016).
Interpretative Knowledge is perceived as the knowledge that allows teachers to give sense to pupils’
answers, in particular to those that contain errors, or to “non-standard” solutions, i.e., adequate answers that differ from those teachers would give or expect (Jakobsen et al., 2014). It sustains the teachers’ ability to support the development of pupils’ mathematical knowledge, starting from their own reasoning, even if students’ ideas are incomplete and/or non-standard. Moreover, IK is associated with the notion of discipline of observation and, in particular, with the idea of teachers working “on becoming more sensitive to notice opportunities in the moment, to be methodical without being mechanical” (Mason, 2001, p. 61). Such knowledge thus allows teachers to consider errors and non-standard reasoning as learning opportunities (Borasi, 1994; Mellone et al., 2015).
One way to enrich prospective teachers’ interpretive knowledge is to offer opportunities to face student productions and to provide and discuss feedback, as part of their teaching practice. The term feedback is defined as “information provided by an agent (e.g., teacher, peer, book, parent, self, experience) regarding aspects of one’s performance or understanding” (Hattie & Timperly, 2007, p.
81). The feedback can be constructive, i.e., one that includes hints, corrections, examples or explanations. Kulhavy and Stock (1989) termed this type of feedback as elaborative and defined it as any method that goes beyond indicating only the answer is correct or incorrect. Still, there is no consensus that providing elaborative feedback benefits learning more than simply indicating whether the answer is correct (Shute, 2008). Some authors even argue that complex and detailed information can counteract the effectiveness of feedback. Santos and Pinto (2010) studied the evolution of written feedback provided by a middle school mathematics teacher, reporting that her feedback evolved, developing plasticity, adjusting to the specific students or to the tasks, creating reflection moments by delivering elaborate clues, and encouraging the correction of the situation.
In the present study, we examined prospective mathematics teachers’ interpretative knowledge by analysing the feedback they provided to students. The quality of the feedback has been the greatest influence on student performance (Black & Wiliam, 1998), thus opportunities for interpreting students’ productions are perceived as chances for improving feedback and students’ learning.
Methods
In the present study, we focus on the answers given by 19 prospective mathematics teachers (PTs), who worked in small groups on interpreting students’ productions related to a math problem adopted from Cañadas, Castro, and Castro (2008). Most participating PTs were in the seventh semester of a 10-semester course and had already completed courses in mathematics, pedagogy, and mathematics education, and had some school practice experience.
The tasks were conceptualised following a particular design (e.g., Ribeiro, 2016; Ribeiro, Mellone,
& Jakobsen, 2013), whereby the first part focused on accessing and developing PTs’ MTSK, while the Interpretative Knowledge – Interpretative Task – and the provided feedback was examined in the second part. Here, we focus on this last part of the task, as the goal is to examine PTs’ IK when interpreting students’ productions and the feedback provided. In the first step of the task implementation, PTs were required to solve a problem that would later be explored with their students (Figure 1).
Complementarily, PTs were asked to refer to and reflect upon their own difficulties when solving the problem. In the next phase of the task, PTs were provided with three student productions, and were required to give meaning to each one of these productions, as well as provide what they consider to be a fruitful feedback to each student. The productions included in the task either contained errors, were incomplete, or involved a non-standard approach to problem-solving. Here, we discuss some productions included in the task, as shown in Figure 2(a)−2(c).
Figure 1: Tiles problem (Cañadas et al., 2008)
Figure 2(a): Camilo’s productions
Figure 2(b): Aracelli’s productions
Imagine you have white and grey square tiles of the same size. We make a row with the white tiles and then surround the white tiles with grey tiles, as the drawing shows:
1. How many grey tiles would you need if you had 1320 white tiles and wanted to surround them in the way you did in the drawing?
2. How many grey tiles are necessary for any number of white tiles?
white grey white grey
Through the proportions, you can determine the number of grey tiles needed:
, grey tiles.
(1) Grey tiles
(2) According to the relationship that occurs in the rule of three (5 is to 16, as x is to y).
Figure 2(c): Javiera’s productions
The data utilised in the analyses included PTs’ productions when solving the Interpretative Task, focusing on PTs’ perception of the problem difficulty and the feedback given to three students based on the solutions offered. The written feedback that the prospective teachers provided to the students’ answer was analysed categorizing the emergent feedback.
Analyses and discussions
Data was analysed in two phases. As noted earlier, all participating PTs were first asked to solve the problem in small groups and discuss any difficulties they encountered. In the next phase, feedback the PTs provided to Camilo and Aracelli based on their productions was discussed (these students utilised a similar strategy for solving the problem).
Difficulties in solving the problem
Eight of the nine groups of PTs solved the problem correctly, while the remaining group adopted the “rule of three” which yielded incorrect answer (Figure 3).
When this group examined the solutions provided by other PTs, they were able to identify their mistake (pertaining to task interpretation) and subsequently provided a correct answer.
Concerning the PTs’ difficulties, four out of nine groups stated that they had difficulty understanding how to represent the figure when the number of tiles increased (e.g., “We doubt how the white tiles were placed.”; “Yes, because we did not know in what way the tiles would be grouped.”). These results confirm that PTs had difficulties in understanding the statement of the problem, even though the manner in which the tiles should be grouped was explained explicitly in the problem statement “we make a row of white tiles” (see Figure 1).
Categorising the feedback
Most of the PT groups pointed out that both Camilo and Aracelli (Figure 2(a) and (b)) solved the problem by incorrectly applying the “rule of three” without giving further explanation of how they should have proceeded, while some groups provided indication on the way the problem should be addressed. When analysing the feedback provided to these two students based on their productions, four categories emerged in relation to Camilo’s and Aracelli’s solutions:
(i) Feedback on how to solve the problem: Guide on how the students should proceed to solve the problem, particularly stating that they should think inductively.
Figure 3: Mistake in understanding the problem
1320*2 = 2640
2640 + 6 = 2646 grey tiles
if x = white tiles, the grey tiles will be: 2x+6 because:
2x to cover above and below the white (tiles), +6 to expand on each side, so that it protrudes 1 on each side and to fill the sides.
(ii) Confusing feedback: When the feedback seems to be correct, but it can be confusing for the student.
(iii) Counterexample as feedback: An example is used to refute the error exposed.
(iv) Superficial feedback: The content of such feedback was insufficient (too broad or inconsistent) to allow the solver to understand its meaning.
(i) Feedback on how to solve the problem
Two PT groups focused their feedback on explaining to the student how he/she should proceed in order to solve the problem. They specifically indicated that the problem should be approached inductively, suggesting that the student should think what happens if there is one white tile before considering, what happens if there are two, three white tiles and so on. How many grey tiles would there be in each case? Then, they prompted the student to think about what happens in general terms. We observed that these PTs could correctly interpret the students’ mistakes, and thus indicate the correct approach to solving the problem.
(ii) Confusing feedback
One of the groups focused on explaining why the number of grey tiles is not proportional to that of the white tiles. They stated that “proportional thinking should not be used, since the number of grey tiles does not increase by the same amount as white tiles; thus, they are not proportional” (see Figure 4).
Figure 4: Confusing feedback
Perform the activity by proportion, that is, for every group of 5 white tiles, there are 16 grey tiles.
.
The way to approach the problem is erroneous, since proportional thinking should not be used, because the number of grey tiles does not increase by the same amount as the white tiles. Therefore, they are not proportional. So, yes, we can say that the tiles that are above and below the white tiles are in a 1:2 ratio, not counting the edges.
Then ask what happens with the number of tiles at the edges when the white tiles increases or decreases? Lead the student to make the connection between the number of upper and lower tiles plus the tiles at the edges before prompting him/her to generalise for any number of white tiles.
We posit that the last expression can lead the student to incorrectly conclude that, if the number of grey tiles increases/decreases by the same amount as that of the white tiles, those variables are proportional. Instead, feedback should focus on the fact that the number of tiles does not increase/decrease “by the same proportion.”
Second, they stated, “we can affirm that the tiles that are above and below the white tiles are in proportion 1:2, without counting the edges.” Thus, they explained when the figure would be proportional. Although this reasoning is true, it could be confusing to the student, because the problem is not one of proportionality. As a result, this was termed as confusing feedback. In sum, although the PTs in this group could interpret the students’ mistakes, when providing an elaborated feedback, they opted for a rather complex solution, which can confuse the students.
(iii) Counterexample as feedback
Two groups used a counterexample to respond to the student (Figure 5). In this excerpt, a group provided feedback suggesting that the student should draw a figure for the case of 6 white tiles with their 18 grey tiles (if proportional), while also applying the “rule of three” that would yield 19.2 grey tiles. The group interpreted this as a contradiction, along with the impossibility of having 0.2 tiles. However, it would have been more informative to argue that there is no multiplicative relation between white and grey tiles.
This group used a counterexample in their feedback to confirm that the number of grey tiles is not proportional to that of the white tiles. The use of a counterexample requires that the teachers have a comprehension of the mathematics knowledge involved in the problem (KoT), as this would allow them to provide an explanation for student’s errors. From the MTSK perspective, teachers are required to have the mathematics knowledge of the theme that allows approaching the problem in the correct way, but also ability to explain why other ways do not work. Explaining why something is incorrect requires interpreting the students’ reasoning and building feedback on these mistakes.
Figure 5: Counterexample as feedback (iv) Superficial feedback
Some of the feedback on Camilo’s and Aracelli’s productions was superficial, that is, too vague for a clear understanding by the student. For example, PTs stated: “I would ask what is proportion, so
(S)he used “simple rule of three” because (s)he assumed a direct proportion between the number of white and grey tiles.
I would tell him/her to try a case where you can draw a diagram. For example, we can assume that there are 6 white tiles. So, BB = 6, BN = 18 and try the simple rule of three to see if it works.
, . Clearly, it does not make sense to have 0.2 tiles.
that [students] can argue if it can actually be used in the exercise,” or “I would ask if the rule of three is used for all cases. . . . Analyse if there is a proportion to apply the rule of three.” The nature of such feedback would neither allow students to understand their mistakes nor to envisage how to proceed in order to solve the problem correctly.
Some final comments
Some prospective mathematics teachers experienced difficulties in understanding and interpreting the posed problem, which is in the space of problems (and content) they will need to work on with their students in the near future. This finding highlights the need for enhanced preparation that would ensure that prospective mathematics teachers can solve the problem independently in order to, at least, be able to perceive a correct typical answer (Ribeiro et al., 2013).
Data analysis further revealed two dimensions of feedback study participants provided to students:
the nature and focus of the interpretation provided, and the content of prospective teachers’
knowledge. In the feedback on how to solve the problem, the groups focused on explaining how students should proceed in order to obtain the correct solution, disregarding explaining why the original answer was incorrect. Feedback that was classified as confusing feedback and counterexample as feedback focused on explaining why there is a mistake. Such feedback is particularly linked with the elements of PTs’ space of solutions, usually referred to as having a single approach to guide to students (Jakobsen et al., 2014). Further, when the PTs attempted to provide elaborate feedback, they supported their explanations with erroneous arguments, or worded it in a way that can be confusing for the students. These feedback categories reveal some critical points that need to be addressed in teacher education in order to enrich and expand the superficial or confusing feedback, empowering a meaningfully interpretation that should have students’ reasoning, knowledge, and understanding as a starting point for the mathematical discussions to be developed.
Acknowledgements: This research is partially supported by the grant 2016/22557-5, São Paulo Research Foundation (FAPESP) and is a part of the activities comprising the project CONICYT PCI/Atracción de capital humano avanzado del extranjero, nº 80170101 (Chile).
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