KeywordsMathematics teachers’ specialized knowledge Interpretative approach Mathematics teacher education Approach to errors in teaching mathematics Standardized assessment
Mathematics teacher knowledge has a central role in the research on mathematics teacher education. On the one hand, researchers investigate connections between practice and teacher knowledge within different theoretical perspectives (Potari 2012). On the other hand, from a more theoretical point of view, the issue of identifying the scope and the nature of the mathematical knowledge needed for teaching is dominant in our field (Chapman 2013). Here we refer to a particular kind of mathematical knowledge, named Interpretative Knowledge (IK), which allows teachers to give meaning to students’ reasoning and productions, even when incorrect or nonstandard, whenever aiming at supporting the development of their mathematical knowledge having such reasoning as the core element of the mathematical work in practice. Considering its nature and specificities, a rationale and background will be presented, and afterward, the meaning of such IK will then be deepened.
Mathematics education lies at the intellectual crossroads of many different domains and often deals with and develops constructs originally emerged in other domains (Sierpinska et al. 1993). Concerning teacher knowledge, Shulman’s (1986) notion of different types of knowledge has strongly affected the research in mathematics education. Shulman develops a general framework for classifying both the domains and categories of teacher knowledge, regardless of the subject matter. In particular, from the seven defined categories, only three of them are related to the learning of the specific topics: (a) subject (or content) matter knowledge (SMK), (b) pedagogical content knowledge (PCK), and (c) curricular knowledge.
Several studies were developed – also within the field of mathematics education (“Pedagogical Content Knowledge in Mathematics Education”) – in order to (a) sharpen theorizations of PCK, (b) measure PCK, and (c) get insights for teacher education based on PCK development (Carrillo et al. 2018).
Teaching may require a specialized form of pure subject matter knowledge – “pure” because it is not mixed with knowledge of students or pedagogy and is thus distinct from the pedagogical content knowledge identified by Shulman and his colleagues and “specialized” because it is not needed or used in settings other than mathematics teaching. This uniqueness is what makes this content knowledge special. (Ball et al. 2008, p. 396)
Several program for preservice mathematics teacher development focus (i.e., some sense) on specialized content knowledge (“Models of Preservice Mathematics Teacher Education” and “Mathematics Teacher Education Organization, Curriculum, and Outcomes”). For this relevance, it appears crucial to analyze in more detail the construct of SCK, recognizing and describing its main components.
In their description of the knowledge for teaching, Ball et al. (2008) give a special attention to the management of errors. In particular, they differentiate between the kind of knowledge needed to diagnose incorrect strategies or to understand correct but nonstandard ones, from the kind of knowledge needed to develop didactical actions to prevent typical errors. The first kind of knowledge is considered within the SCK, while the latter is seen as included in the PCK (“Mathematical Knowledge for Teaching”).
On the other hand, the seminal work of Raffaella Borasi (1996) changed completely the approach to the errors in the field of mathematics education, highlighting the educational potential of mathematical errors and, consequently, defining new needs for teacher education. The reconceptualization of the role of the errors in mathematics education proposed by Borasi is developed on the basis of the etymological meaning “to get lost” of the Latin term “Errare.” Borasi promotes the metaphor of the error, also in educational setting, as getting lost and therefore as a chance to know new places, to foster curiosity and to build new knowledge. In Borasi’s view error is not something to be avoided but an opportunity for learning and inquiry, a resource for mathematics teaching: she offers several examples of reorganization of the mathematical teaching around the capitalization of errors (Borasi 1987). This capitalization of errors is grounded on their interpretation.
The belief that teachers’ capability of exploiting the potential of students’ mathematical errors is an essential aspect for the quality of mathematics teaching has highlighted the need to go beyond the marginal role that “errors” and their interpretation have in the first theorizations of MKT. Combining the idea of a specialized mathematical knowledge involved and required for teaching with the approach to errors and nonstandard reasoning as learning opportunities, Ribeiro et al. (2013) introduce the construct of Interpretative Knowledge. It refers to a deep and wide mathematical knowledge that enables teachers to support students in building their mathematical knowledge starting from their own reasoning and productions, no matter how not standard or incorrect they might be. IK completes the knowledge of typical errors or solution strategies, with the knowledge of possible source for typical or atypical error and the knowledge of possible use of errors in the sense that Borasi developed.
In particular, a high level of IK permits to understand unexpected students’ strategies or approaches, giving sense to students’ reasoning. IK also includes the ability of developing specific feedbacks based on the sense given to the students’ reasoning; therefore it should allow to exploit the potential of erroneous or unexpected strategies.
Also in the case of IK, two issues appear particularly relevant: to measure the level of IK of teachers and to create activities for the development of the level of IK in teacher education program. The two issues are clearly related because the discussion around activities for the measurement of teachers’ IK can be used for the development of new awareness about the specific error/strategy both in terms of specific mathematical knowledge and in terms of possible approach to erroneous or unexpected strategies.
As it is evident, the knowledge of the content plays a crucial (but not sufficient) role in the level of IK. Therefore the way to explore and develop IK has to be different with respect to the school level of the teachers, but some general and common guidelines can be traced.
In particular, Ribeiro et al. (2013) developed and experimented specific activities to measure the level of IK of preservice and in-service teachers. Since the argumentative nature of the IK, these activities are mainly based on open questions and structured into two phases. In the first phase, teachers have to solve a problem: in this way, they face with the difficulties of the problem, and their favorite strategy can emerge. In the second phase, teachers are asked to evaluate several and different strategies developed by real students and to propose specific feedbacks for each strategy.
For example, Jakobsen et al. (2014) proposed the following mathematical problem to future teachers (both primary and secondary levels): “If we divide five chocolate bars equally among six children, what amount of chocolate would each child get?”. As predictable, the most common strategy used to solve the problem was to divide each of the 5 chocolate bars into 6 equal pieces and distribute 5 of the 30 equal pieces obtained among the 6 children. Following this procedure, solvers typically conclude that each child will get five little pieces of chocolate corresponding to 5/6 of a chocolate bar.
In several experimentations, prospective teachers – in different countries and independently by the school level – consider this solution as erroneous. This position is essentially related to two causes not mutually exclusive. The first is a low level of mathematical knowledge: it appears evident as mathematical knowledge – in this case the rational numbers in the context of partitioning – plays a crucial role in the approach to unusual strategy such as Marianna’s one. The second cause is related to the belief that there is only one way of answering and reasoning on this problem. Often this latter aspect is related to the more general belief that in mathematics there is always a unique way to approach a problem: this belief is clearly a big obstacle to develop IK because it leads to consider incorrect all strategies different by the one recognized as the only one right. In this case, as it emerges by the results of the quoted researches, future teachers’ feedbacks are mainly focused on the judgment of (presumed) incorrectness and on sharing the favorite strategy.
In conclusion, a strong common content knowledge (CCK) is needed to develop a good level of IK, but it is not sufficient as showed by the results of researches that involve teachers with a strong mathematical background (Jakobsen et al. 2016). On the other hand, a teacher with a strong common content knowledge can have difficulties in considering correct and accepting strategies different by his own strategy. For this reason, the notion of IK is introduced as part of the SMK, in the intersection between the common content knowledge and the specialized content knowledge.
Main Implications for Teacher Education
In our research on IK in teacher education, the notion of IK was introduced together with a set of possible tasks used both for measuring IK and for developing it (Di Martino et al. 2016; Jakobsen et al. 2014; Ribeiro et al. 2013, 2016). In teacher education practice, a third phase is added to the two phases described in the previous section: prospective teachers are engaged in a collective discussion on the mathematical aspects involved in students’ productions in order to recognize the potentialities of exploring these productions in the mathematical activity, and, after a while (about 1 month), teachers are asked to write (a new) interpretation and reflection of the students’ productions.
One of the crucial parts in the design/choice of the task for the teachers is the selection of students’ productions to be interpreted. We need a certain number of unexpected strategies in order to challenge teachers’ IK.
From the teacher education point of view, the results of the research show that IK does not develop only with practice over time (Ribeiro et al. 2013): also in-service, sometimes experienced, teachers can reveal low level of IK. Therefore the development of IK should be an explicit focus of attention in teacher education programs.
IK is evidently crucial also in order to exploit the formative value of mathematical assessment, overcoming a simplistic approach based on a right/wrong judgment. Particularly interesting is the case of standardized assessments in mathematics whose importance has increased in the last years. Frequently the results of these assessments have repercussions on educational decisions (Kanes et al. 2014).
Recently, Di Martino and Baccaglini-Frank (2017) defined the informational and the developmental potential of standardized mathematical tests, seen, respectively, as the information that can actually be obtained by interpreting and analyzing students’ performance results on standardized tests and as the educational opportunities offered by a critical approach to standardized tests and by a re-elaboration of the informational potential. The authors show that teachers’ lack of IK can inhibit their possibilities to exploit these two kinds of potential of standardized tests.
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