RE-DEFINING HCK TO APPROACH TRANSITION

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1 RE-DEFINING HCK TO APPROACH TRANSITION Saínza Fernández, Lourdes Figueiras, Jordi Deulofeu, Mario Martínez Universitat Autònoma de Barcelona Abstract Mathematics learning is a continuous process in which students face some abrupt episodes involving many changes of different natures. This work is focused on one of those episodes, transition from primary to secondary school, and targets teachers and their mathematical knowledge. By including expert teachers views about transition and characterising the mathematical knowledge that they need to smooth transition processes, we aim to highlight their importance in the continuity in mathematics education. The concept of Mathematical Knowledge for Teaching (MKT) developed by Ball among others(ball, Thames & Phelps, 2008) and, within this framework, the construct Horizon Content Knowledge (HCK) emerge as our theoretical response to the knowledge for teaching mathematics in a continuous way, particularly relevant during transition to secondary school. The enrichment of the idea of HCK and its expression in the teaching practice intends to develop a theoretical tool to approach transition from teachers mathematical knowledge perspective. INTRODUCTION Mathematics learning is a continuous process in which students face some abrupt episodes which involve many changes of different nature that derive in a variety of alterations in their educational path. This work is focused on one of those episodes, transition from primary to secondary mathematics, a compulsory transition for students that involves many external changes and targets teachers mathematical knowledge. An extensive survey let us detect that previous research on transition from primary to secondary school has been mostly focused on its effects in students academic attainment (McGee, Ward, Gibbons and Harlow, 2003) whereas a focus on teachers and their perspectives has been mostly absent. Considering specifically transition in mathematics, the following questions arise immediately: which characteristics describe it? Looking at previous research foci of attention and conclusions the mathematical content emerges as a determinant factor: the step from arithmetic to algebra (Boulton-Lewis et al., 1997; Boulton-Lewis, Cooper, Atweh, Pillay & Wilss, 1998; Cooper et al., 1997; Flores, 2002; Gonzales & Ruiz Lopez, 2003), the learning of integer numbers (Gallardo, 2002; Pujol, 2006) or the development of the need of proofs in geometry (Berthelot & Salin, ; Sdrolias & Triandafillidis, 2008) and other fields appear as particular well-known problems embedded in the teaching and learning of mathematics that involve transition to

2 secondary school. Since teachers shape the access of students to this content, we consider their role in transition as crucial. In conclusion, albeit transitions are processes with a broad range of effects and consequences in the whole students educational experience, we believe that it is necessary to investigate the specificity of mathematics in transition and particularly, the role of the teacher and his/her professional knowledge during this process. More specifically, the objectives of our investigation are: O1. Find out mathematics teachers views about transition. O2.Characterise the mathematical knowledge that teachers need to smooth transition processes. With regard to O1, we believe that teachers opinions and practical everyday experience on transition must be taken into consideration in order to broaden our understanding of transition in mathematics, since they are the only professionals that experience the process of transition along with the students in the classroom. By involving the views of expert mathematics teachers we want to enlarge the multiple perspectives from which transition to secondary school can be considered and thus gain a better comprehension of it. Experts responses will constitute the justification of the appropriateness of the theoretical perspective adopted in O2. In order to accomplish O2 we require a solid theoretical frame that suits our purpose of focusing on the role of the teacher and, particularly, on the relationship of his or her mathematical knowledge and practice with primary to secondary transition. The concept of Mathematical Knowledge for Teaching (MKT) (Ball, Thames & Phelps, 2008; Hill, Rowan & Ball, 2005; Hill, Blunk, Charalambous, Lewis, Phelps, Sleep, & Ball, 2008) and its division in different domains and sub-domains appears as a suitable framework for this part of our investigation. Within this theory, the construct Horizon Content Knowledge (HCK) emerges as our theoretical response to the knowledge for teaching mathematics in a continuous way, particularly relevant during transition to secondary school. Hence, our research focus in this part of the investigation is the enrichment of the idea of HCK in order to use it as a theoretical tool to approach transition. EXPERTS VIEWS We aim to find out experts opinions about transition to secondary mathematics regarding the following questions: which elements affect in a determinant way the transition to secondary mathematics? And which of these refer to the teaching practice? Moreover, not only we want to collect judgements of individual experts but we also want to offer a scenario of group decision, where participants are capable of interact anonymously and reach a consensus. In the following, we will consider an expert a mathematics teacher that has taught in both levels and has shown a particular concern about transition issues.

3 In order to create an experts group discussion we use the classical Delphi method, which allows us to create a scenario of group decision where experts feel free to express their opinions and interact anonymously in a very efficient way, ensuring that the information obtained is of good quality and reliable (Plans and León, 2010). The versatility of the Delphi method seems particularly appropriate for our research, since it includes open and closed questions as well as qualitative and quantitative analysis. Figure 1 shows the steps included in a typical three round classical Delphi method (Skulmoski, Hartman and Krahn, 2007, p3) which guides the design of this part of our work. Figure 1: Classical three round Delphi method (Skulmoski, Hartman and Krahn, 2007, p3) Delphi study Our own teaching experience led to the problem of transition to secondary school and a previous survey detected a theoretical gap in transition research focused on teachers. Our uncertainty involves expert teachers opinions on the elements that affect transition and particularly those related to the teaching practice in both levels. In the study 15 experts answered our open questionnaire, which included three questions: What factors affect students mathematical learning during transition to secondary school? Which qualities should primary teachers have in order to smooth students transition to secondary mathematics? Which qualities should secondary mathematics teachers have in order to smooth students transition from primary school? Among the factors that affect student learning they identified methodological issues and the influence of teachers initial training as especially remarkable. For example, scarce use of manipulatives or group work in secondary school, and no balance between teachers pedagogical and mathematical knowledge. Experts pointed out that in order to manage transition, teachers must have a set of skills, attitudes and knowledge related to general education such as preparing primary students for a greater autonomy, valuing diversity, managing the interaction, using active and constructive methodologies or encouraging participation. We can infer from data that they refer to those skills from a theoretical framework that conceives a pure pedagogical knowledge. Experts also refer to a specific mathematical knowledge for teaching, which, in terms of Ball, Thames and

4 Phelps (2008) we call Mathematical Knowledge for Teaching. For example, they refer to manage with different degree of mathematical rigor, know the mathematics taught at both levels and have a global vision of the contents. Some kind of experience connecting the two levels appears essential, as the need to know about the previous and/or forthcoming stages. Following Ball et al. (op cit.), we find that experts refer to characteristics that describe the long-term perspective embedded in the idea of Horizon Content Knowledge. This way, experts responses become our way in and justify the second part of our investigation, in which the framework of the Horizon Content Knowledge appears crucial to approach teachers role in transition. MKT AND HCK In this part of the investigation we explore the concepts of MKT and HCK in order to clarify the latter s place in the diagram and introduce a different approach to the MKT s organisation above that concludes with the refinement and placement of HCK in this framework and thus, with the inclusion of the notion of continuity in MKT s theory. Mathematical Knowledge for Teaching overview Ball, Thames and Phelps (2008) distinguish two domains within the MKT, namely Pedagogical Content Knowledge and Subject Matter Knowledge (see Figure 2). These are not independent from each other, but it is their combination which defines the knowledge needed for teaching mathematics. Figure 2: Categories of Mathematical Knowledge for teaching (Ball, Thames and Phelps, 2008, p.403) Pedagogical content knowledge is subdivided attending three foci of attention within the teaching practice: students, methodology and curriculum. Knowledge of content and students (KCS) involves students expected difficulties, questions, motivations, etc. and teacher s preparation and responses to those. Knowledge of content and teaching (KCT) concerns methodology issues such as the design of the

5 sequence of a topic or the use of appropriate tasks, representations and examples, etc. Finally, knowledge of content and curriculum (KCC) is the curricular knowledge needed for teaching. This includes not only the knowledge of the mathematical topics that are included in a particular curriculum, but also the specific moments when they have to be taught and how they are developed in the educational path. Subject matter knowledge is sub-divided in three categories: common content knowledge (CCK), which is the mathematical knowledge that is common to other professions and specialised content knowledge (SCK), which is the specific mathematical knowledge needed for the teaching practice (Ball, Thames and Phelps, 2008). Before centring our attention on the last sub-domain, the HCK, we detect that KCS, KCT and SCK arise and are expressed only during the teaching practice in mathematics or in the observation of other s teaching practice, while the KCC and the CCK are not necessarily linked to the teaching practice. We highlight this observation by considering KCC and CCK as foundation knowledge and KCS, KCT and SCK as having an in-action nature. It is important to remark here that the word foundation denotes our idea of this theoretical knowledge being the basis and it is not related to the more complex construct of Foundation knowledge included in the Knowledge Quartet of Mathematical Knowledge in Teaching (MKiT) framework (Rowland, Huckstep and Thwaites, 2005). Figure 3 shows our interpretation of the categories of MKT from this approach, not considering yet HCK. Figure 3: Categories of Mathematical Knowledge for teaching without the HCK Horizon Content Knowledge Our main interest is focused on the Horizon Content Knowledge (HCK), provisionally included within subject matter knowledge. About HCK the authors say We are not sure whether this category is part of subject matter knowledge or whether it may run across the other categories. (Ball, Thames and Phelps, 2008, p.403) HCK refers to the general awareness of the previous and the forthcoming, and requires an overview of students mathematical education so that it can be applied to the mathematics taught in the classroom (Ball, Thames and Phelps, 2008). Teachers

6 consciousness of the past and the future within their subject is actually very closely related to continuity in mathematics education and thus, our view of HCK comprises this teacher s longitudinal perspective required for continuity. However, this longitudinal view that we understand as HCK encompasses a complex combination of pedagogical and mathematical knowledge, skills and experience that must be clarified in order to successfully approach transition issues in mathematics from this framework. Firstly, despite the fact that the HCK may be related to the knowledge of the curriculum (KCC), it is independent from the curriculum itself. HCK is not only an awareness of how mathematical topics are related over the span of mathematics included in the curriculum but also refers to the global knowledge of the evolution of the mathematical content and the relationship among its different areas needed for the teaching practice. This general knowledge does not depend on the curriculum context and it is different to the curriculum awareness that a teacher must have in order to teach the appropriate topics at a particular grade. In other words, a teacher could have good level on KCC but fail on approaching this knowledge from a longterm perspective. Secondly, HCK influences the KCS, the KCT and the SCK. For example, HCK must include the ability of the teacher to find out students previous mathematical ideas and to prepare them for the future. This ability involves KCS (knowledge of previous, current and future students difficulties, misconceptions or questions) and KCT (different ways students might have seen that represent the same idea or types of tasks that facilitate students learning in the future). Also, the SCK of a teacher for a particular grade depends on whether that teacher currently teaches in that level. If so, his/her SCK for that grade will be obviously greater than for the other grades. The inclusion of HCK in the framework implies the extension of the SCK to those topics that may really have an effect in what students are learning at the moment or to those future topics for which a teacher is setting up the basis. Thirdly, HCK has a different nature than the other sub-domains since it does not seem sensible to find the presence of HCK in a particular teaching situation if there is a previous absence of KCS, KCT or SCK. In fact, these are the required bases that allow the posterior gradual inclusion of HCK in the professional practice. From this perspective HCK is not another category in the diagram but it adds a more sophisticated (continuous) perspective to the teaching practice. For instance, we would not expect to observe a teacher recognising previous misconceptions in a particular topic, dealing with students difficulties or conveying a prospective mathematical view of the future if that teacher does not know the topic and its methodological issues at first. The previous ideas lead us to consider HCK, not as another sub-domain of MKT, but as a mathematical knowledge that actually shapes the MKT from a continuous mathematical education point of view since it must be present in every in-action category in order to attend transition. Figure 4 shows our interpretation of the

7 framework with the inclusion of the HCK. The idea of opening the previous diagram (Figure 3) and shaping it when including the HCK intends to indicate: a) the difference in the expression of each in-action category in the teaching practice with or without HCK; b) the connection with past and future mathematical levels, particularly important from a transition (or a continuous) point of view and c) the fact that HCK has a different nature than the rest of categories since it does not appear in the diagram but its presence modifies the teaching practice. Figure 4: HCK shapes MKT and outlines its nature Characterisation of HCK At this point, our consequent aim now is to characterise more specifically our idea of HCK. Hill et al (2008) highlight the need of clarifying how teachers knowledge affects classroom instruction by carrying out an investigation in which the relationship between teachers MKT and the quality of their practice is analysed. We follow this idea for the particular case of refining the construct of HCK. Since we do not consider HCK as a theoretical cluster itself but a type of knowledge that shapes the in-action knowledge needed for teaching and also because our purpose is to obtain a useful tool for future research on transition, we adopt a practical perspective from which the following question emerges immediately: how does HCK get expressed in teaching practice? In order to obtain an answer to this question, a series of non-participant observations were carried out. Mathematics lessons in the last year of primary school as well as the first year of secondary school were observed during three months. Participating schools were city centre comprehensive schools that were chosen in order to cover the different possibilities in transition: schools with transitional programs and links among primary and secondary teachers, schools with transitional programs but no communication among teachers and schools without transitional programs. The following HCK s characterisation is based on real situations derived from those classroom observations. Table 1 describes the three main competences or

8 characteristics of a teacher which help us to concrete and identify HCK in the teaching practice: strategist, coach and transformer. STRATEGIST COACH TRANSFORMER Preparation of activities from a continuity perspective. Identification, prevention and reorientation of misconceptions and difficulties from a continuity perspective. Adaptation of the classroom activity from students contributions and level. Table 1: HCK competences Once we have concluded this first investigation, future research questions based on the characterisation above particularly aim to develop a useful tool to identify the presence of HCK in the teaching practice. A questionnaire is being developed mostly based on real classroom situations, intending to detect the degree of HCK that a mathematics teacher holds. The question shown in Figure 5 exemplifies these ideas by targeting, in this case, the well documented misconception between area and perimeter. Figure 5: Example of a real classroom situation in which HCK may get expressed The variety of possible teacher s responses is an indicator of the degree of HCK in the teaching practice. For instance, if the teacher is aware of this common

9 misconception, one way he or she could proceed would be to return the mistake to the student or the classroom. The Coach competence could be visible in this particular situation by observing a series of teachers questions that target the origin of the student s confusion: if two rectangles have the same perimeter, do they also have the same area? With the help of the teacher (questions, counterexamples, etc) and/or the rest of the class, the responsibility of correcting the mistake would be returned to the student or the class. Moreover, with an eye on the future and in order to strength the basis for the future learning of more complex geometry concepts, the teacher could also extend these ideas to other 2D shapes or even 3D solids. The appropriateness and effectiveness of the activities suggested by the teacher could also exemplify the Strategist and Transformer competences. This way, we consider that posing questions like these and discussing them in teacher training programs could be a future way in to attend continuity in mathematics education. FINAL REMARKS This work intends to move towards a better understanding of continuity in mathematics education. HCK emerges as a noteworthy construct within MKT framework that allows the researcher to approach continuity questions in mathematical education by looking at teachers professional knowledge. The future systematic investigation of HCK s expression in the teaching practice and its consequences on transition shows a path to the potential inclusion of this mathematical knowledge as part of teacher training programs designed to smooth transition. REFERENCES Ball, D.L., & Hill, H. (2009). R and D: The curious - and crucial - case of mathematical knowledge for teaching. Phi Delta Kappan, 91(2), Ball, D.L., Thames, M.H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59 (5), Berthelot, R. & Salin, M.H ( ) L enseignement de la géométrie au début du collège. Comment concevoir le passage de la géométrie de constat à la géométrie déductive? Petit x, 56, Boulton-Lewis, G.; Cooper, T.; Atweh, B.; Pillay, H.; Wilss, L.; Mutch, S. (1997). Transition from arithmetic to algebra: A cognitive perspective. 21st Annual Conference of the International Group for Psychology of Mathematics Education (Vol. 2, ). Lahiti, Finland. Boulton-Lewis, G.; Cooper, T.; Atweh, B.; Pillay, H.; Wilss, L. (1998). Arithmetic, pre algebra and algebra: A model of transition. Mathematics Education Research Group of Australasia, Gold Coast.

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