Mental mathematics with mathematical objects other than numbers: The case of operation on functions

Journal of Mathematical Behavior 39 (2015) 156–176

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Mental mathematics with mathematical objects other than numbers: The case of operation on functions Jérôme Proulx Département de mathématiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre-Ville, Montreal, QC H3C 3P8, Canada

a r t i c l e

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Article history: Received 29 November 2014 Received in revised form 24 June 2015 Accepted 2 July 2015 Available online 21 July 2015 Keywords: Mental mathematics Graphical environment Operations on functions Enactivist theory of cognition

a b s t r a c t This article reports on a study, part of a larger research program, focused on issues of mental mathematics with mathematical objects other than numbers. The study is about operations on functions in a Cartesian graph environment with two groups of 30 high school students (grade-11). Grounded in aspects of the enactivist theory of cognition, the research aims at characterizing students’ emerging mathematical activity by analyzing the strategies they put forth in this mental mathematics environment. It illustrates how students pose their own problems when solving tasks and thus made emerge tailored strategies for solving the very problems that they posed. The data analysis highlights three specific approaches that students engaged with/in for solving tasks: algebraic/parametric, graphical/geometric, and numerical/graphical. This characterization offers understandings of how students have engaged in and succeeded in solving the various tasks, leading to a discussion of the generation of strategies for solving these tasks. Triggered by the nature of students’ engagements, the article closes with future research avenues and issues to investigate in mental mathematics. © 2015 Elsevier Inc. All rights reserved.

1. Introduction To argue for the relevance and importance of teaching mental mathematics with numbers, defined broadly as the solving of mathematical tasks without paper-and-pencil, Thompson (1999) in his literature review highlights that (a) most of everyday calculations in adult life are done mentally, (b) mental mathematics contributes to the development of ones’ number system or number sense, (c) mental mathematics deepens one’s problem-solving skills, and (d) mental mathematics improves one’s written calculations (see also Threlfall, 2002). These aspects stress the nonlocal character of doing mental mathematics with numbers, where the skills being developed extend to wider mathematical abilities and understandings. Indeed, diverse studies show significant effects of mental mathematics practices with numbers in classrooms: on students’ problem-solving skills (Butlen & Pézard, 1992; Butlen & Pézard, 2000; Leutzinger, Rathmell, & Urbatsch, 1986; Trafton, 1986), on the development of their number sense (Boule, 2008; Butlen & Pézard, 1992; Butlen & Pézard, 2000; Heirdsfield & Cooper, 2004; Leutzinger et al., 1986; Murphy, 2004), on their paper-and-pencil skills and standard algorithms (Butlen & Pézard, 1992; Butlen & Pézard, 2000) and on their estimation strategies and skills (Heirdsfield & Cooper, 2004; Schoen & Zweng, 1986). For Butlen and Pézard (1992), the practice of mental mathematics with numbers can enable students to develop new ways of doing mathematics and solving arithmetic problems that the traditional paper-and-pencil context rarely provides because it is often focused on techniques and

E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmathb.2015.07.001 0732-3123/© 2015 Elsevier Inc. All rights reserved.

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algorithms that are in themselves efficient and do not create a need to step outside them. These outcomes are also found in students’ own discourse, which explains their feeling that doing mental mathematics with numbers has helped them when they subsequently had to solve problems (Butlen & Pézard, 2000). Thus in addition to the often seen stimulating character of these practices for students’ learning (Butlen & Pézard, 1992; Carlow, 1986), what is convincing about the significance of doing mental mathematics is the overall agreement of research results. Studies conducted in USA (Reys & Nohda, 1994; Schoen & Zweng, 1986), France (Butlen & Pézard, 1992; Butlen & Pézard, 2000; Douady, 1994), Japan (Reys & Nohda, 1994), and UK (Murphy, 2004; Thompson, 2000, 2009; Threlfall, 2002, 2009) show abundant evidence of how the practice of mental mathematics with numbers enriches students’ learning and mathematical work (paper-and-pencil activity) about calculations and numbers. This being so, as Caney and Watson (2003) and Rezat (2011) explain, most if not all research studies on mental mathematics have focused on numbers/arithmetic. However, mathematics involve much more than numbers, and these mathematics are predominantly studied through paper-and-pencil activities. This rouses interest in knowing what doing mental mathematics with mathematical objects other than numbers might contribute to students’ mathematical reasoning and understanding, as well as in knowing the nature of the mathematical activity so generated (e.g., algebra, functions, trigonometry; mathematical objects significantly studied in the school curriculum). The study reported here is part of a larger research program that aims to probe mental mathematics with objects other than numbers: that is, solving mathematical tasks without paper-and-pencil or other computational (material) aids in algebra, geometry, statistics, functions, trigonometry, and so forth. In this article, I report on a study on functions conducted in two grade-11 (15–16 years old) high school classrooms, where students had to operate mentally on functions in a graphical environment. 2. What is meant by “mental mathematics”, with objects other than numbers Because most work on mental mathematics is on numbers (often referred to as mental arithmetic or mental calculations), no definition of mental mathematics that would encompass other mathematical objects appears in the literature. For Thompson (2009), mental calculations represent a subset of mental mathematics; however, he offers no definition of mental mathematics. This said, even if considered in terms of numbers and mental arithmetic, definitions about mental calculations can be adapted to other mathematical objects to help define mental mathematics. Building on Hazekamp (1986), who offers a definition that summarizes what is generally considered by mental calculations, mental mathematics is defined here as the solving of mathematical tasks through mental processes without paper-and-pencil or other computational (material) aids. Thus, this research program is situated in the existing research literature on mental mathematics (calculations/arithmetic) where it is the context of study, that is, the fact that there are no paper, pencils, or other material aids available, that defines what mental mathematics is. This focus on context is important, because it would be natural to argue that most of the mathematics produced (in classrooms, by mathematicians, by engineers in their workplace, etc.) has a “mental” dimension. Hence the intention in this work is not to distinguish what is mental and what is not, but mainly to study how students engage in strategies and the nature of their mathematical activity when they have to solve tasks without recourse to paper-and-pencil or other materials. Moreover, numerous dimensions are found in the literature about mental strategies with numbers (e.g. Boule, 2008; Butlen & Pézard, 1990, 1992, 2000, 2007; Kahane, 2003; Ministre Jeunesse Éducation Recherche [MJER], 2008), adaptable to other mathematical objects, which can help in developing a finer sense of what is meant by mental mathematics. One of these dimensions is about reasoned computations, implying the elaboration of personal strategies, often nonstandard and adapted to the problem, versus automatized computations, which imply access to an immediate result through the use of known facts or memorized procedures. An example of this could be, for area, between using the formula ((D × d)/2) to find the area of the rhombus versus cutting the figure into triangles to find or compare the area. A second set of dimensions is about approximate computations, based on estimation and approximation to gain an order of magnitude for the answer versus mentally applying an algorithm or a fact to obtain an exact answer. An example for trigonometry could be between using the fact that sin 30◦ = ½ versus establishing a visual order of magnitude that the opposite side of a 30◦ angle enters approximately twice in the hypotenuse. A third dimension is about rapid computations, which require quick execution to find the answer. Often criticized because it is perceived as a speed exercise detrimental to sense-making, it can also be seen as helping to develop new solving methods because it forces the solver, in trying to be economical, to abandon methods that may be slower (e.g. standard procedures) or less efficient for completing the task (e.g. one-on-one counting). In the case of algebra, an example could be the development of a global reading of an equation like x + (x/4) = (x/4) + 6 giving x = 6, avoiding numerous algebraic manipulations in order to isolate x (Bednarz & Janvier, 1992). These dimensions illustrate possible entries for solving mental mathematics tasks, helping to refine what mental mathematics can mean, but also valuable for reinvestment in data analysis, as a first orientation for giving meaning to the strategies developed by solvers for operations on functions in a graphical environment. For this study, graphical calculations are conceived in line with Tournès (2000) as “any process using drawings in a 2D environment (or 3D), with the help of any mechanical device, that aims to avoid, in totality or in part, the recourse to numerical calculations for solving the problem” (pp. 127–128, my translation). The same is true in the case of this mental mathematics environment, but here no recourse is allowed to any physical device or material aid to solve the task. It also means that the task is to be solved in relation to the graph itself (by any conceptual means available in the mental mathematics context).

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Fig. 1. An addition of function task, f(x) + g(x).

For example, a typical task would be to have two functions represented in the same Cartesian plane for which the students would have to add or subtract the two functions. Fig. 1 shows graphs that would be offered for solving f(x) + g(x). The purpose of the study is to scrutinize students’ mathematical activity generated in this mental mathematics context by paying attention to their mathematical strategies for solving tasks about operations on functions.1 The main intention here is to study mental mathematics contexts and what happens in them with regard to students’ mathematical activity. However, clearly the choice of functions as a topic is not random, as it is argued to be one of the most important topic in the curriculum (Dubinsky & Wilson, 2013) and one where, “although there has been some progress, the conceptual difficulties observed more than 50 years ago are, for the most part, still with us today” (p. 86). Without asserting that a contribution is made directly to these issues, in the final section I raise issues about learning and teaching functions in relation to mental mathematics. The following section outlines the theoretical perspective that grounds the conceptualization of the study about the nature of strategy-development processes in mental mathematics contexts by presenting aspects of the enactivist theory of cognition. Section 4 describes methodological aspects of the design of activities, data-collection and analysis procedures. Section 5 outlines this analysis, which illustrates and discusses strategies that students engaged in for solving tasks. The final Sections 6 and 7 revisit the data analysis and extends the discussion to issues to be explored further in subsequent research. 3. Theoretical grounding for the study: conceptualizing mathematical activity in mental mathematics Recent work on mental mathematics has begun to critique the notion that students choose from a previously developed toolbox of predetermined strategies to solve problems in mental mathematics. Threlfall (2002, 2009), for example, insists on the organic emergence and contingency of strategies in relation to the tasks and the solver (e.g., what he/she understands, prefers, knows, has experienced with these tasks, is confident with; see also Butlen & Pézard, 2000; Murphy, 2004; Plunkett, 1979; Rezat, 2011). This idea of emergence is also stated by Murphy (2004), who discusses Lave (1988) situated cognition work, which conceptualizes mental strategies as flexible emergent responses adapted and linked to a specific context and situation. As I argue in Proulx (2013b), aspects of the enactivist theory of cognition (e.g. Maturana & Varela, 1992; Maturana, 1987, 1988; Varela, Thompson, & Rosch, 1991; Varela, 1999), particularly Varela’s distinction between problem-posing and problem-solving, offer theoretical answers to questions about the emergence and characterization of mathematical strategies generated for solving tasks.2 Enactivism is an encompassing term given to a theory of cognition that views human knowledge and meaning-making as processes understood and theorized from a biological and evolutionary standpoint. By

1 It is worth noting that this study could be seen as situated in the large field of visualization studies (Presmeg, 2006; see also Arcavi, 2003), defined broadly as implicating mental-image reasoning of many kinds in teaching and learning mathematics. This study, however, emerged from an interest in mental mathematics/calculations, that is, in solving without paper-and-pencil or any material aids, and for investigating the nature of the mathematical activity engaged with in this context. This explains why it is approached here through issues of mental graphical calculations, where graphical representations are given up front in the tasks. Unlike other visualization studies, this study is not interested in probing the kinds of imagery that students used/created for solving tasks, on how useful images are for solving tasks, on outcomes of teaching through imaging or effective pedagogy, on what aspects of instruction might encourage solvers to use visualization, on how using images helps them to overcome their difficulties or to learn concepts, on the differences of images using between students, and so forth: all topics of deep interest in visualization studies. However, because of “the diffuse nature of continuing research on visualization” (Presmeg, 2006, p. 225), which at times incorporates issues of representations, gestures, metaphors, symbols, etc., in the learning of mathematics, it is possible for the reader to establish some links. I come back to this and discuss some of these issues in Section 7.2. 2 Note that Varela’s distinction does not refer to Brown and Walter’s work on mathematical problem-posing/solving.

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adopting a biological point of view on knowing, enactivism considers the organism as interacting with/in an environment. A biological perspective has often been adopted as a metaphor for thinking about knowledge and learning, for example, with ideas of adaptation and evolution.3 However, for Maturana and Varela, knowing is a biological phenomenon. Thus, a student is considered as an organism that evolves with/in its environment in an adapted fashion: his strategies or mathematical solutions are not necessarily optimal, but are functional (what Reid, 1996, and Zack & Reid, 2003; Zack & Reid, 2004, call “goodenough”), with/in a context, a problem, that is itself evolving under the influence of the solver. In this sense, mathematical strategies are not considered as existing a priori (or a posteriori) of the moment when they appear: they are the real-time product of interaction, of the meeting, of the solver and his/her environment, directly and continually influenced by both. At the core of this theorization, and grounded in Darwin (1867) theory of evolution, are concepts of structural coupling, structural determinism and, specifically relevant for our case here, problem-posing (for more details, see Proulx, 2013b). For Varela (e.g. 1996), problem-solving implies that problems are already in the world, lying “out there” waiting to be solved, independent of us as knowers. Varela explains that we specify the problems that we encounter through the meanings we make of the world in which we live, which leads us to recognize things in specific ways. We do not choose or take problems as if they were lying “out there,” objective and independent of our actions: we bring them forth. The most important ability of all living cognition is precisely, to a large extent, to pose the relevant questions that emerge at each moment of our life. They are not predefined but enacted, we bring them forth against a background, and the relevance criteria are oriented by our common sense, always in a contextualized fashion. (1996, p. 91, my translation) The problems that we encounter and the questions that we ask are thus as much a part of us as they are a part of our environment: they emerge from our interaction with it. We are not acting on preexisting situations: our constant interaction with the environment creates the possible situations for us to act upon. The problems that we solve, then, are implicitly relevant for us, because we allow these to be problems for us while the environment triggers them in us. Some issues of the environment that would trigger elements in some persons do not trigger the same elements in others. This is also a phenomenon to which we have been attuned through the work of Piaget on assimilation schemes and structures (see, e.g., La construction du reel chez l’enfant [Piaget, 1963] or Conversation libres avec Jean Piaget [Bringuier, 1977]). In mathematics education research, among others, Simon (2007) through Piaget and Glasersfeld’s work refers to this issue by using the expression “we see what we understand”.4 Hence it is claimed that reactions to a task do not reside inside either the solver or the task: they emerge from the solver’s interaction with the task, through posing this task. If one adheres to this perspective for (mental) mathematics teaching and learning, one cannot assume, as René de Cotret (1999) notes, that instructional properties are present in the (mental mathematics) tasks offered and that these will determine learners’ reactions. Butlen and Pézard (1992), Heirdsfield and Cooper (2004) and Rezat (2011) have indeed shown the occasional futility in mental mathematics of varying the type of problem or its didactic variables to encourage students to use specific strategies. Strategies emerge in the interaction of solver and task, influenced by the task but determined by the solver’s experiences in mental mathematics in solving similar and different problems: in his/her solving habits for similar or different tasks, in his/her successes in mathematics with specific approaches, in his/her understanding of the tasks, and so forth. In this perspective, the solver does not choose from a group of predetermined strategies to solve the task, but engages with the problem in a specific way and generates a strategy tailored to the task. This is a powerful distinction because it offers an explanation that issues addressed and explored in mental mathematics tasks are those that resonate with and emerge from students, although these are triggered by the task offered. Thus students transform the mathematical tasks for themselves, making them their own (which is often different from the designer’s intentions, as René de Cotret, 1999, says). By doing this, students generate a strategy tailored to the problem (they) posed. Mathematical strategies are thus seen as emergent, enacted at the moment of interaction with tasks (Davis, 1995; Thom, Namukasa, Ibrahim-Didi, & McGarvey, 2009), emerging from both and being “new” to some extent,5 dependent on or influenced by the task and its context, but determined by the learner in accordance with his/her own complex histories and situations (Davis, Sumara, & Kieren, 1996). As Threlfall (2002) explains: As a result of this interaction between noticing and knowledge each solution ‘method’ is in a sense unique to that case, and is invented in the context of the particular calculation – although clearly influenced by experience. It is not learned as a general approach and then applied to particular cases. The solution path taken may be interpreted later as being the result of a decision or choice, and be called a ‘strategy’, but the labels are misleading. The ‘strategy’ (in the holistic sense of the entire solution path) is not decided, it emerges. (p. 42)

3 See e.g. the work of Glasersfeld (1995), Morss (1990), Siegler (1996), or the review of Davis (2004). Even Piaget, trained as a biologist, explicitly acknowledged in the famously called Piaget-Chomsky debates (Piatelli-Palmarini, 1979) that his biological understandings of adaptation and evolution were used as metaphors to make sense of learning processes and knowledge construals. 4 A proposition that I have studied (see e.g. Proulx, 2008) in order to initiate discussion, inspired by enactivism, of how solvers’ emergent responses are dependant on the environment one is put in but are determined by one’s own structure, all happening in the interaction of solver and environment. 5 The expression “new” is in quotation marks because it does not mean that strategies are new in a sense that nothing similar has been attempted before. It is mainly to say that strategies are generated for the tasks faced, tailored to them, and thus reflect both the task and the solver.

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What enactivism offers, through the distinction between problem-solving and problem-posing in particular, is not a static framework to apply on data, but mostly a sensitivity to issues of emergence and adaptiveness of strategies in mental mathematics environments. It represents a dynamic perspective that offers a specific way of talking about strategies, avoiding ideas of possession (acquisition of, choice of, of “having” things, etc.) in favor of issues about emergence, flux, movement, interactions, relations, actions, and so forth. It is this perspective that orients this proposed research, offering ways of understanding (what happens in) the solving process in mental mathematics and how it helps in understanding (what happens in) this solving process. 4. Methodology, data collection and data analysis 4.1. Classroom activities and design The classroom activities included presenting various tasks for students to solve mentally. Tasks were designed in collaboration with the grade-11 (15–16 years old) mathematics teacher in whose classrooms the study took place. We organized five one-hour research meetings to collaborate in developing and designing the classroom activities and tasks, thus negotiating expertise as well as intentions or constraints. These meetings consisted of merging each other’s earlier work: my earlier work and analyses conducted with future teachers on operations on functions in a graphical environment (Proulx, 2010, 2012) and the teacher’s work and teaching plans on the topic accumulated in his more than 15 years of experience in the classroom. This allowed the material that we developed to be well grounded in didactical and conceptual analyses (Brousseau, 1998) based on diverse mathematics education literature on functions (e.g. Hitt, 1998a, 1998b, 2001; Janvier, 1978, 1983, 1987; Lovric, 2008; Markovits, Eylon, & Bruckheimer, 1986; see also Dubinsky & Wilson, 2013, concise but efficient review). It was also tailored to the teacher’s classroom context as we benefited from his professional knowledge and expertise (Barry, 2009; Bednarz, 2000, 2009). In addition, these meetings served also to clarify some boundaries and constraints that played a role in the classroom organization as well as the nature of the tasks. We thus had to play within the teachers’ boundaries (e.g. time constraints, program of studies) as well as he had to play within ours (e.g. organization as explained below in Section 4.2, having various students explain at length even less efficient strategies, not offering the “right” answer on the first occasion). These boundaries from both sides played a role in the design of the activities (see also Footnote a). 4.2. Classroom organization and examples of tasks The classroom activities took place in two grade-11 classrooms (with approximately 30 students each) and were conducted by the regular classroom teacher (on some occasions, I asked students for clarifications or held discussions with them). The activities followed a structure similar to that described by Douady (1994): (1) Graphical tasks are shown on the whiteboard where the teacher gives oral instructions for the task to be solved; (2) Students listen and then have 20 s to think about their solutions; (3) At the teacher’s signal, students have 10 s6 to write their answer (on a small sheet of paper that includes a Cartesian graph with y = x drawn for reference) and then raise it for the teacher to see; (4) The teacher asks various students (with or without adequate answers) to explain their answers to the class in detail (and in some cases to come to the front to illustrate on the board); (5) If answers are given orally, the teacher writes them on the board (and in many cases explains them again); (6) The answers are occasions for justifications, and other students are invited to question or intervene in some of the solutions if they are not fully convinced or do not understand what is offered; (7) The teacher also invites other students who may have solved differently (or who have thought of other ways of solving); (8) The various solutions are compared if possible and discussed by teacher and students with regard to their effectiveness, links, efficiency, advantages/inconveniences, possibilities for other problems, etc. Six sets of approximately 6–10 tasks were organized (summarized in Table 1). These grade-11 students had never before explicitly worked on operations on functions. The first set of tasks was used to introduce them to the ideas. In this case, both the graph (shown in Fig. 2) and the algebraic expression for linear functions were offered, as in the following for solving f(x) + g(x). In this set of tasks, various linear functions and constant functions were offered. For the second set, graphs of two functions (sometimes three) were given without their algebraic representation, and students were asked to solve them mentally. In these tasks, as in Fig. 1, the functions to add varied from a combination of constant functions with linear, quadratic, square root, constant, rational, and step-function. In the third set, on the same Cartesian plane, students were given the representation of one function and the result of an operation. In this case, students

6 This follows a structure that we have used in a number of studies (e.g. Osana & Proulx, 2013; Proulx, 2013a; Proulx & Osana, 2013), based on earlier studies on mental mathematics (re: the literature cited in the introduction) as well as based on current teaching practices on mental mathematics in classrooms (and advocated by the participating teacher).

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Table 1 Summary of the nature of tasks given to students.a Set of task

Nature of the tasks

Set of task 1

Addition and subtraction of two functions, where both the graph and the algebraic expression for linear functions are given.

Set of task 2

Graphs of two or three functions to add, without their algebraic representation. Functions varied from a combination of constant functions with linear, quadratic, square root, constant, rational, and step-function.

Set of task 3

One function f and the resulting function f + g are given. Students have to find the function g that had been added or subtracted to give the resulting function f + g. Functions were two linear functions, two square-root functions, or a combination of a constant function with a linear or square-root function.

Set of task 4

Graphs of two or three functions to subtract, without their algebraic representation. Functions were two linear functions that were spread in the Cartesian plane and did not pass through the origin. Only algebraic expressions of functions were given. Students have to compute operations like f + g, f − g, g + f, g − f, f + f, etc., and draw the resulting function. Functions were a combination of, e.g., f(x) = |x| or f(x) = [x] with g(x) = x or g(x) = x2 . Addition and subtraction of two (or three) functions that looked symmetrical in the graph, and that did not pass through the origin. Functions were a combination of two linear functions, two quadratics or two piece-wise functions.

Set of task 5

Set of task 6

a One reviewer wondered why trigonometric functions (e.g. sinx + x) were not used or why multiplication tasks were not designed (e.g. x |x|) in the study. As mentioned in Section 4.1, tasks were designed in collaboration with the teacher and thus some of his teaching constraints/intentions played a role, as well as ours. Questions of time were an issue, as the focus was placed on + and − for the main part, but also because (a) multiplication is not explicitly part of Quebec’s program of studies, hence the lesser need felt by the teacher to work on them, and (b) trigonometric functions (and others not worked on) were not felt as adapted by the teacher in his current teaching sequence.

Fig. 2. Example of a task with both graph and algebraic equation written.

had to find the function g that had been added or subtracted to give the resulting function, as in Fig. 3. In these, the functions f and the resulting function (f + g or f − g) could be two linear functions, two square-root functions, or a combination of a constant function with a linear or square-root function. The fourth set of tasks was similar to the second, but focused on subtractions like f(x) − g(x) and g(x) − f(x). However, in these, f and g were two linear functions that were spread in the Cartesian plane and did not pass through the origin. The fifth set of tasks was different from the others in that no graph, but only algebraic expressions of functions were given: algebraic

Fig. 3. Example of a task requiring to find the function g added to function f.

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Fig. 4. Example of a task focused on symmetry.

expressions that could not be directly computed as in the first set. For these, students had to compute operations like f + g, f − g, g + f, g − f, f + f, and draw the resulting function. The functions given were a combination of f(x) = |x| or f(x) = [x] with g(x) = x or g(x) = x2 . The sixth set of tasks focused on symmetry, where students were asked to add two (or three) functions that looked symmetrical in the graph. These functions were spread in the Cartesian plane (did not pass through the origin) and were a combination of linear functions, quadratics or piece-wise functions, as in Fig. 4. 4.3. Data collection and analysis As mentioned above, the study took place in the middle of the school-year (December)in two grade-11 classrooms of approximately 30 students each; it was the students’ first experience with operations on functions as they had not explicitly studied such operations. (They had, however, worked a great deal with functions during the school-year, with for example linear, quadratic, square-root, piece-wise, constant, and rational functions.) Two consecutive sessions of 75 min for each class were organized. Students’ responses (in the form of verbal explanations and notes made on the board) generated the data for the study (quotes from students were often directly noted, and are reported below in SMALL CAPITALS), which were recorded in note form by myself (PI) and a research-assistant (RA) to prepare for the subsequent analysis meetings where the notes were combined with other means, as explained below.7 The data analysis aims to characterize the nature of the mathematical activity generated in this mental mathematics context on operations on functions in a graphical environment. Grounded in Varela’s (1996) problem-posing mentioned in Section 3, the analysis pays attention to the aspects of the tasks students engage with: hence the strategies that emerged, which give rise to the tasks they posed in an attempt to scrutinize the nature of their mathematical activity in this mental mathematics context. Following Douady (1994), the goal is thus not to report on all the learning that took place, nor to discuss the long-term outcomes for student in other contexts, but to gain a better understanding, to make sense and characterize students’ mathematical activity when solving mental mathematics tasks (ways of engaging in tasks, of posing them, the strategies developed, etc.). Answers in both classrooms were similar if not identical, so no difference is made in the data analysis. This data analysis was carried out in two phases aiming to generate robust understandings about the nature of the mathematical activity that emerged during the classroom activities. The first phase consisted of on-the-spot three-person analysis meetings (PI, RA, and teacher) after each classroom session to discuss events that occurred during the sessions and analyze how students engaged with the tasks. These meetings offered a first level of analysis that included the teacher’s voice, which afforded interpretations of the events from a practitioner’s perspective, combined with observations (notes taken) from both PI and RA. This combination of observations, field notes, and impressions enabled discussions on specific aspects that stood out and deserved attention, and completed/bolstered the content of the notes taken in order to develop an understanding of them (and considering them in subsequent sessions). This first level of analysis revealed salient issues about the use of the graphs, the algebraic expressions and the numerical/pointwise data, which offered an initial orientation toward the kinds of problems students posed and solved. This three-pronged orientation was used to orient the subsequent data analysis. To investigate and understand better the nature of the strategies students engaged with, the second phase consisted of attending to the data in relation to issues of graphs, algebra, and numerical data highlighted in the first phase. To do this, we

7 Some reviewers raised important questions about gathering data in a classroom environment from students’ spoken explanations and its reliability for gaining insight into students’ thinking. This is indeed relevant, since links between the verbal reports and the “actual” doings of a subject have been questioned for a long time in psychology (see e.g. Nisbett & De Camp Wilson, 1977). Thus we could say that our interest was in their “verbal mathematics” and not in their “mental mathematics” because, as mentioned, mental mathematics defines the context of study, not the processes in which students engage (i.e., there is no mathematics without any “mental” anyway). Thus asking students to explain how they solve (orally and by coming to the board) was in order to collect strategies for solving problems in that context. The other point concerns the classroom and its relevance as a data-gathering site. We decided to go into actual classrooms in order to realize our exploratory study. The next steps of the study are taking interest in working with smaller groups (about 10) and also interviewing students one on one. This will give access to more students and more intimate access to the strategies in which they engage.

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referred to Desgagné (1998) notion of available constructs from the literature, here on representations of functions (e.g. Duval, 1988; Hitt, 1998a, 1998b; Janvier, 1987; Yerushalmy, 2006) combined with issues from the literature on mental mathematics. (Examples of specific constructs drawn on to enrich this analysis were the global and pointwise approach to functions of Even (1998), the concept of generic example from Mary (2003), the approximated and exact calculations from Kahane (2003).) These constructs refined our understanding of the strategies that students developed and the meaning that they gave to functions, which thus related to the kinds of problems that they posed and attempted to solve. Following this, we conducted repeated interpretative readings of the field notes about the strategies (oral explanations, notes made on the board) in order to gain a better understanding of and to strengthen the categories, which gave rise to characterizations (algebraic/parametric, graphical/geometric, graphical/numerical) as a way of making sense of the nature of students’ mathematical activity that emerged in this mental mathematics context. 5. Characterization of strategies The analyses of students’ strategies and ways of dealing with the tasks, of posing them, offer a way to conceive of the nature of students’ mathematical activity when they are solving mental mathematics tasks, illustrating its emergent, adapted, and creative nature. The analysis of students’ ways of dealing with the mental mathematics tasks also leads to an appreciation of the potential of doing mental mathematics for students’ learning of mathematical concepts, here on functions. The following subsections present and discuss the algebraic/parametric, graphical/geometric, and numerical/geometric approaches to the tasks. While reading and making sense of these approaches in which students engaged, it is important to keep in mind that these emerged in a mental mathematics context. This means that students had limited time to “get into” the tasks, pose them, and develop a strategy to solve them. Also, no paper-and-pencil were used to support the reasoning or draw lines, compare solutions, validate, and so on. This context is significant in helping to explain the specific aspects of these approaches. 5.1. Strategy 1. The algebraic/parametric approach Students have a strong tendency to think algebraically rather than visually. Moreover, this is so even if they are explicitly and forcefully pushed towards visual processing (Eisenberg & Dreyfus, 1991, p. 29) Even if tasks were proposed in a graphical context, many students were observed engaging in algebraic-related solving. Using Butlen and Pézard (2007) concept of automatisms in mental mathematics procedures, these algebraic engagements can be interpreted for the most part as automatized behavior when students were faced with functions, as if it were a procedure to follow. This seemed to enable students to generate information quickly and reinvest it for solving the task. This can be observed in the first set of tasks, where both graphical and algebraic representations were given for the functions (see Fig. 2). In these cases, students explained that they were aiming for the algebraic expressions of both functions (e.g., 4x − 7 and − 2x + 2), added these expressions mentally (4x with – 2x and −7 with +2) to obtain the resulting algebraic expression (e.g., 2x − 5) and then drawing the function on their Cartesian plane when asked to do so, alternating between algebraic and graphical representations (in their solving or explanations). Other than in the first set of tasks when the algebraic expressions were not available, students’ actions can be conceived of as bringing forth, as making emerge, what Duval (1988) calls significant units for reading the graphical representation of a linear function and offered an interpretation in relation to the algebraic expression (to go from one to the other). That is, students can be seen as paying attention to the parameters of the algebraic expression (the a and b of the linear function f(x) = ax + b) rapidly to make sense of the graphs and to add them. However, again, because the resulting function had to be expressed graphically afterward, answer and strategy were (fluently) explained algebraically by blending aspects of graphical information. For example, in the following addition task (Fig. 5), where neither function has an algebraic expression attached, many students explained that “both functions looked symmetrical, so the ‘a’ parameter of each line would cancel out, as well as the ‘b’ and thus give x = 0”.8 Similarly, in the second set of tasks, with Fig. 6, even if no algebraic expression was attached to the functions, students would explain that the a parameter of the function f does not change when added with a constant function that “DOES NOT HAVE AN ‘a’ PARAMETER, SO THE FUNCTION’S STEEPNESS STAYS THE SAME AND ONLY THE ‘b’ CHANGES” giving a function parallel to f with a y-intercept at “b” instead of at 0. 5.1.1. Discussion of Strategy 1 As is seen in these cases, students’ actions can be understood as making algebraic information emerge from the functions, from the graphs, in order to operate on them and develop solutions to the task. This strategy enabled them to solve many of the tasks easily, as well as to organize how to handle them (e.g., first take the y-intercept, then . . .). Thus even if the tasks were offered in a graphical context, as observers we saw students recurrently engaged with them algebraically, drawing out for themselves algebraic aspects of the functions (expressions, parameters, etc.). Paper-and-pencil contexts might also

8

Again, quotations in small capitals are taken from students’ words, translated from French to English.

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Fig. 5. Example of an addition task with two linear functions.

have led students to similar algebraic approaches to solving. But in a paper-and-pencil context, students often take time to work with specific examples and make “tests”, whereas here they could not do this and thus resorted to the parameters of the functions, talked about them in these terms, and worked with them on an abstract/generic level, talking about the effect not on a specific slope or on a specific y-intercept, but on a generic slope and a generic y-intercept. This makes the nature of the mathematical activity in which they engaged quite specific and focused on the general algebraic/parametric properties of the functions. However efficient, one can interpret this algebraic orientation toward solving tasks as problematic or even restrictive in the sense that it “kills” the graphical exercise that aims to have students thinking graphically in order to solve the tasks. This need not be so, as the adequacy of the strategies engaged in, directly adapted to the tasks, and their capacity for solving an important number of these tasks needs to be appreciated (especially in a short time frame given to them). This analysis of strategies offers ways to understand students’ actions in order to draw out, to pose, an algebraic context with which they were comfortable/confident and to solve the problem in this context. This said, interpreting students’ actions through an algebraic prominence/preference when working with functions is not new and has been amply documented elsewhere (e.g., Vinner’s, 1989, “algebraic bias”). In Knuth’s (2000) study, for example, more than ¾ of first-year algebra calculus students engaged in algebraic methods to solve tasks on functions even if a graphical approach seemed easier and more efficient. Knuth linked these results with curricular influences, where “instructional emphasis dominate[s] by a focus on algebraic representations and their manipulations” (p. 505; see also Slavit, 1998). Along this line, the curriculum in which the study took place (Quebec, Canada) focuses heavily on algebra, which accounts for more than 65% of the curricular objectives in the final years of high school (Bednarz, Maheux, & Proulx, 2012; Mary, 2003). Schmidt and Bednarz (1997) have also highlighted the higher social role attributed to algebra by future elementary and secondary schoolteachers (in Quebec) for solving problems, where algebra is seen as more “mathematical” or a “better” option than arithmetical or other approaches. This could be interpreted as influencing students’ fluency or propensity for algebra, for posing their problem, where they would be oriented toward and led to interpret the problem in specific ways because of their lived-history with/in mathematics and their mathematical experiences. In this case, what is interpreted as students’ automatized behavior in the face of functions, here in algebraic ways, is seen as leading them to engage in tasks algebraically. Thus even if in the task design (except for the first set of tasks) we thought that most students would enter the graph to solve them, this understanding of students’ actions leads one to see that there were potential algebraic pathways in these tasks. This is also what happened in Knuth’s (2000) study, where “the problems were designed to ‘force’ the use of a graphical-solution method, yet the students’ initial responses illustrated overwhelming

Fig. 6. Example of an addition task, with a linear and constant function.

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Fig. 7. Example of an addition task with a non-linear and a constant function.

reliance on algebraic-solutions methods” (p. 505). The same happened in Habre and Abboud’s (2006) study, in which they developed a course focused on graphical/visual representations: “Thus, even though the experimental sections, the objects of our study, were geared almost completely geometrically, students thinking remained algebraic” (p. 62, see also Slavit, 1998). Here, however, one can see this approach as even “more” than algebraic, as mentioned above, as it is discussing in terms of parameters at an abstract/generic level concerning the effect on these parameters and how it affects the function itself, and not in terms of x and y or of 3x + 7. This enables a focus on the significant units of analysis of Duval (1988), as mentioned above, but without having time to zero in on it (like estimating it numerically). Hence, “a” for slope was a slope of a, same for the y-intercept, that was affected by transformations, by operations. It is algebraic, but reasoned on the algebraic parameters (a for the slope, b for the y-intercept) and not on numerical parameters (like “3” for slope, etc.). Students can be seen to have transformed the tasks we thought we had designed: they posed them, making them their own, making them algebraic/parametric. We are again reminded of René de Cotret’s (1999) note that one cannot assume that instructional properties, in this case a graphical entry, are present in the tasks and that these will determine solvers’ reactions. Those reactions are triggered by, and emerge in, interaction between the solver and the task. Here one can say that these tasks triggered algebra in students, which led them to engage in an adapted fashion to solve the task that they posed and made emerge/developed a well-tailored algebraic strategy. 5.2. Strategy 2. The graphical/geometric approach Mental calculations represent a brilliant and new aspect of our teaching. The teacher and even the students constantly invent new ways of running without failing. This sort of exercise is sane for the spirit. (Alain, 1932/1967, p. 81) In various instances, when facing a function that was not linear (e.g. quadratic, square root, rational, hyperbolic), students’ actions can be analyzed as the generation of particular ways of working with already known concepts. This relates to mental strategies that Boule (2008) and Butlen and Pézard (1992) call reasoned computations, which imply the elaboration of personal solving strategies, often non-standard and contingent on the type of tasks worked on. The two concepts in this case were slope and parallelism, as explained below. 5.2.1. Example 1. (Constantly varying) slope When facing a nonlinear function, students explained that the functions with which they were dealing had a constantly varying slope. For example, with the addition of a quadratic and a constant function (Fig. 1), they explained that the slope of the quadratic function was not affected by the addition of a constant function, because a constant function “DID NOT HAVE A VARIATION” and thus the slope of the quadratic function “WILL CONTINUE TO VARY IN A CONSTANT WAY”.9 When students said constant, one can see this as a focus on the fact that its appearance, its style, is not affected. Thus the resulting function of their addition would have the “SAME SLOPE AS THE QUADRATIC FUNCTION” but would simply be “TRANSLATED DOWN” in the graph because the constant function was “NEGATIVE”. Although it is not clear what exactly students meant by this “CONSTANTLY VARYING” slope for nonlinear functions (especially, e.g., when they were dealing with f(x) = 1/x, as in Fig. 7, or the graphs shown in Figs. 3 or 4), they can be seen as offering a language for solving that enabled the solving of the tasks (and how to talk about it) and not worrying about the variation inherent in the function. As one student said about the square-root function: “ITS SLOPE IS LEFT UNTOUCHED WHEN I ADD THE CONSTANT FUNCTION, SINCE IT HAS NO VARIATION”. Interesting here too is the idea that the constant function has no slope, whereas

9 For students, in how they talked about the functions, functions like the quadratic one, or the rational one as in Fig. 7, all had a slope. This seemed to mean “variation”, but they used the expression “slope” for it. I return below to this issue in 5.2.3 in terms of a geometrical view of functions.

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it would normally be seen as having one, albeit 0. However, through the meaning they offer, the constant function did not have a slope. 5.2.2. Example 2. Parallelism (of curves) In cases where students faced more than one nonlinear function, the constantly varying slope strategy appeared insufficient for solving, as they were observed analyzing functions in terms of “parallelism”. For example, take the case shown in Fig. 3 where function g is to be found. In this case, some students expressed that “EACH FUNCTION WAS PARALLEL TO THE OTHER,” that g had to be a constant function “FOR THE CURVE TO BE TRANSLATED DOWN,” and that this constant function was “NEGATIVE FOR BRINGING THE CURVE LOWER” as a result of their addition. Here again, one can see that this idea about parallelism, which can be questioned mathematically, emerged as a way of making sense of the task without considering or going in detail about the fluctuation in image-length for each function. 5.2.3. Discussion of Strategy 2 When required to explain what happened to the graph of the resulting function, students often showed it with their hands on the board or with the pen, expressing a curving movement to illustrate what the function did (varying, being parallel, etc.). When solving with issues of constantly varying slopes, students did not seem to see the slope as a value that varies along the function because the function is nonlinear (Herbert & Pierce, 2009). Mainly, the analysis leads one to see students focusing on how the function itself behaves, its appearance or curvedness. To some extent, as a way of rapidly engaging with the tasks, one can see that these students were bringing forth a geometrical interpretation of slope as a property not of the function, but of the curve present on the graph, making the task, posing it, about graphical/geometric issues.10 Students’ reasoned calculation can then be conceived of as a way of talking about a geometric slope; something reminiscent of Zaslavsky, Hagit and Leron (2002) argument about slope being seen as a geometric concept versus slope being seen through the lens of analytical geometry. Students’ interchange of expressions taux de variation and pente (see Footnote 9) hints at this, as do expressions like “TRANSLATION OF THE FUNCTION”. Each case, analytical and geometrical, has different referents. The former is an analytical geometry concept at the bridge between algebra and geometry existing uniquely in the Cartesian plane with numbered coordinates and related to its algebraic expressions. The latter, the geometric slope, is a geometric concept, a curve, independent of the Cartesian plane and not necessarily understood in relation to its algebraic expression or its numbered position in the plane.11 Hence in a geometrical context, it is the constancy of the curve, its appearance, that is looked into and “LEFT UNTOUCHED” when added with a constant function.12 Thus through the lens of this constantly varying geometrical slope, students can be considered to have made emerge a (well-tailored and adapted) way of engaging with the nonlinearity of nonlinear functions, and of entering into the(ir) posed mental mathematics task just as they did for linear functions with their usual slope (particularly through algebraic expression). This geometrical view of slope was also observed when students solved tasks similar to those shown in Figs. 5 or 10, where one student explained that he “SIMPLY SAW IT, JUST ONE LINE REMAINING, SINCE THEY ARE BOTH THE INVERSE OF THE OTHER”. In this sense, students can be seen to have posed the task as a geometrical one, about solving in regard to the appearance of the curve, its geometrical aspects. Issues of parallelism can also be interpreted in geometrical terms as properties of the curves. For example, students had to solve a task like that shown in Fig. 1, which produced a “parallel” quadratic function as in Fig. 8. When students drew the f + g function on the board in front of the class, neither drawings of f nor f + g looked like the two functions reported in Fig. 8, as they were drawn “parallel” to each other more as in Fig. 9. The issue here is about the appearance of both curves. Those in Fig. 8 could be said to not look parallel even if they are the correct graphical representation, as opposed to those in Fig. 9; the issue is one of horizontal versus vertical equidistances.13

10 Note that students used the French expression pente and taux de variation interchangeably in their discourse. I use slope as a translation throughout because one reviewer noted that rate of change, which seemed to me an adequate translation of taux de variation in English, is almost exclusively used in calculus contexts. In French, taux de variation and pente are both used early in schooling to describe, e.g., linear functions; in fact in many documents taux de variation is preferred in the study of functions in schools. Thus using both interchangeably, even if taux de variation is the usual expression used in schools, can be thought of as an illustration of students’ geometrical view for solving because pente can be argued to be related to a geometrical view (hence less fitting in a study of analytical environments). 11 For a more extensive discussion on this issue, see Zaslavsky et al. (2002) as well as Stump (1999). 12 In contrast to earlier approaches for algebra, students were not observed as making use of the a parameter of the quadratic function (e.g., f(x) = ax2 ) to reason as above for linear functions; whereas students in Habre and Abboud (2006) did, however, in a pencil-and-paper context. This also oriented an interpretation of students’ actions as geometrical understandings of these nonlinear functions’ slope, not directly related to algebraic expressions/parameters. This same geometrical understanding can be seen in the idea that the constant function has “no variation”, where its geometric shape illustrates no variation, but its algebraic expression does, which is 0 [in f(x) = 0x + b; null variation and not “no variation”]. 13 This is reminiscent of Goldenberg (1988) illusion that one parabola is more “obtuse” that the other, which he explains as a perceptual phenomenon and not a consequence of lack of expertise with the concept. As well, one could relate this to issues of establishing the “smallest distance” between two points and of drawing a perpendicular along the (tangent of the) curve to determine that smallest distance; which leads to Fig. 9 and not to Fig. 8. In his article on visualization, Arcavi (2003) directly discusses this issue in relation to the presence or absence of the Cartesian graph to orient the attention from the notion of distance between lines toward “vertical displacement” from one line to the other. He argues that the system of reference in which the lines are considered (e.g. Cartesian plan, vertical lines, no reference system) plays an important role in how they can be assessed and managed, which could be related to what happened here when the parabolas were drawn by students (or visually mentally-imagined). Another link can be made with Presmeg’s

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Fig. 8. The quadratic function (f + g) obtained from the addition of f and g in Fig. 1.

Fig. 9. Reproduction of an example of the drawing of the “parallel” quadratic function.

However, students did not explicitly define what is meant by parallelism. For example, do curves have to maintain a certain distance between one another? If so, a distance in relation to what: horizontal, vertical, or other? And how can we be sure that this distance does not fade along the x-axis or y-axis? Does neither curve ever touch? If so, can asymptotes be considered parallels? Or simply, are both branches of the function in Fig. 7 considered parallels? How? Students’ action can be interpreted as a way of defining the meaning of parallelism of curves through its use, in its emergent use for solving tasks (see Ronda, 2009, on this issue). One can say that students saw the functions as geometrical entities, as wholes, that had properties and that were “parallel”, “inverse of the other”, “translated”, “constantly varying”, and so forth.14 This orientation can be seen as enabling them to

(1986) issue of “irrelevant details” or “false data” when visualizing a problem, where visual images constructed can be misleading and produce difficulties in how visualization can be used to solve the task. However, in this case it did not play a role, but it could have. I come back in Section 7.2 to other links between this study and visualization studies. 14 These two geometrical ways of handling functions (slope and parallelism) can be related to what has often been called the object perspective for discussing various mathematical topics (e.g. algebra, functions, induction, numbers), an orientation that can be seen to have its source in Piaget’s concept of reflective abstraction (see e.g. concerning induction, Dubinsky & Lewin, 1986; Dubinsky, 1986; Dubinsky, 1989). Discussing the object/process perspective specifically in relation to functions and how they can be handled and viewed, Moschkovich et al. (1993) explain this perspective as follows: “From the process perspective, a function is perceived of as linking x and y values: For each value of x, the function has a corresponding y value. From the object perspective, a function or relation and any of its representations are thought of as entities – for example, algebraically as members of parameterized classes, or in the plane as graphs that, in colloquial language, are thought of as being “picked up whole” and rotated or translated” (p. 71). This said, there exist various orientations to the process/object perspective, e.g. Dubinsky (Breidenbach, Dubinsky, Hawks, & Nichols, 1992; Dubinsky & Harel, 1992) on the mechanisms of encapsulation; (Sfard, 1991, 1992, Sfard & Linchevsky, 1994) on issues of operational/structural conception and reification. As well, this perspective can be linked to the work of the late and founding editor of the journal, Robert B. Davis, in relation to the notion of “naming a process” (see, e.g., Davis, 1973).

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Fig. 10. (a) Addition task (b): Point brought forth by students to solve the task.

engage with these functions in the graphs as entities.15 Through these generated geometrical ideas, students’ actions can be understood as making sense of the tasks and engaging with them rapidly, developing ways of solving that fit the task that they posed. Furthermore, this fit can be seen as functional as, for example, no other student stopped the discussion and asked for clarification about these ideas, and neither did their teacher. That is, no students explicitly expressed surprise about the use of these concepts, their meanings, or even the drawings of the curves. As Zack and Reid (2003), Zack and Reid (2004) would have it, these can be interpreted as “good enough” for solving the tasks. Thus apart from questions of mathematical adequacy, which could open debate, students’ engagement in the mental mathematics tasks can be understood as adequately adapted to the tasks that they solved/posed and enabled them to solve the(ir) posed task. 5.3. Strategy 3. The graphical/numerical approach A third strategy in which students were observed engaging was paying attention to specific numerical points in the graphs of functions. Through these numerical points, students’ actions can be conceived of as bringing forth what the Kahane report (2003) calls exact and approximated answers, which were combined to find the resulting function. Exact calculations are often seen as more “noble”, where one uses precise strategies to find exact solutions. Approximated calculations, on the other hand, mainly refer to obtaining an order of magnitude for the answer, for controlling the nature of that answer, and for having an indication of it. This was not necessarily the case here in the mental mathematics context, and through this students were seen to engage in two entries, as explained below. 5.3.1. Example 1. Using both exact and approximated cues to draw the resulting function The observations lead to understanding students’ actions as eliciting particular cues from specific points in the graph of both given functions, drawing them to determine in an exact or approximated fashion what the resulting function would look like for these points. In the example in Fig. 10(a), students had to find the function resulting from the addition of f and g. In this case, their actions can be seen as fixing their attention on specific points, namely: (1) where f crossed the x-axis

15 For Goldenberg (1988), it represents a language that reflects our natural way of thinking about functions, for example, when we say that the equation “ax2 + 5 (a > 0) generates an upward-opening parabola with its vertex 5 units up from the origin” (p. 159).

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(x-intercept); (2) where both f and g intersected; (3) where f and g crossed the y-axis (y-intercept); (4) where g crossed the x-axis (x-intercept) (see Fig. 10(b)). For case (1), the operation is an exact calculation as the addition of the image for f (which is of length 0) with that for g resulted in an image for f + g that is the same as that for g (it has the same image-length for g to which 0 was added). For case (2), the operation is an approximated calculation, as the addition of both image-length at f and g is the same, and thus the resulting image will be of double-length of where the intersection point is; but a precise location is impossible unless one knows the exact location of the intersecting point in terms of precise length. For case (3), the same approximated calculation applies as both image lengths are added. Then for case (4), an exact answer is obtained as in case (1). Seen in this way, students’ actions seem to mingle both exact and approximated calculations to find points for the resulting function. (This said, rarely would all four be brought forth in the same task by one student, but rather a varied combination of two or three.16 ) As observers, we also thought it interesting that for many students, even if the x-intercept gives an exact value, the point at the intersection of both lines and the y-intercept seemed to be favored. This may be because students are in the habit of finding the b parameter in the linear equation (y = mx + b, where “b” is the intersection of a line and the y-axis) as mentioned for Strategy 1, and the fact that in systems of equations the intersecting point is usually the point of interest that is studied. 5.3.2. Example 2. Obtaining exact points to situate the resulting function In other cases, students’ actions can be understood as evaluating the image of each function at a specific point (using the algebraic expression) and then computing them to obtain an indication of where the function would be for that point. For example, when facing in the fifth set of tasks, the addition of two algebraic expressions that were difficult to solve algebraically, e.g., f(x) = |x| and g(x) = x, the students explained splitting the function for each side of the y-axis and then evaluating points on each side, by which they would gain an indication of where it would be situated (in which quadrant) and what it would look like. So for x = 2, this led them to obtain f(2) = |2| = 2 and g(2) = 2, making f + g(2) = 2 + 2 = 4, and to position the resulting function in the first quadrant with a slope the double of y = x, then checking for a point on the left side of the y-axis, say x = – 5, and obtaining f(− 5) = |– 5| = 5 and g(– 5) = – 5, making f + g(– 5) = – 5 + 5 = 0. When their actions are understood as a mingling of both exact and approximated answers, students are seen to gain an indication of where the function would be, as values obtained for the left side of the y-axis were exact and along y = 0, and values on the right side of the y-axis were approximated for obtaining f + g(x) = 2x and drawing it. As well, students’ choice of points, or simply the separation of the Cartesian plane in two regions, can be seen to have enabled them to produce what is known as generic examples (Mary, 2003). The x = 2 does not need to be seen as a specific example, but can be conceived of as a point that represents the group of points on the right side of the y-axis. Similarly, x = – 5 can be seen not as a specific point, but as one that acts as representative of all points on the left side of the y-axis. In this view, each numerical point is then seen as offering a view of the appearance of the function on each side of the Cartesian plane: taking these points as representative of examples that generate a family of all other examples for each region. This interpretation of the generic character of the values chosen can also be seen when other students came to the board to explain their solution, often choosing other points almost at random, saying “IF I CHOOSE, LET’S SAY, x = 3, THEN. . .”. In fact, as other students came to the board for other tasks, they chose other xs to gain an idea of the appearance of the function on each side of the y-axis. 5.3.3. Discussion of Strategy 3 For both kinds of examples cited, the analysis conducted leads one to see students’ actions as ways of making points emerge to obtain indications of the appearance of the graph of the function as well as to obtain exact points to draw the function. In this view, it is through the interplay of both kinds of exact and approximated answers/points that the resulting functions were produced. This approach is quite different from that of a paper-and-pencil context, because more time on material would have led, e.g., to precise measurements or to placing points directly on the graph. Through this, students are seen not as in an algebraic context, but mainly in a blend of numerical and graphical contexts, where attention is paid to numbers or coordinates that have meaning in the graph. For example, when students explained referring to the x-intercept, it did not result in finding its meaning in the algebraic expression (see Moschkovich, 1999), but was interpreted as remaining in the graphical context where information was brought forth to compute the resulting function. The same goes for the yintercept, not treated as a parameter b, but mainly as a point on the graph (hence quite differently from Strategy 1). As well, when students explain splitting the Cartesian plane, this can lead to an understanding of the action as a way of engaging with generic examples. Attention is then paid to where a change in the graph occurs, that is, at the origin where for example the absolute value function changes orientation and so changes the kind of resulting point that can be produced. Here also, this reading can be interpreted as graphical, even geometric, as the focus is on the curve itself. Thus the problem seems to be posed in these terms.

16 One can wonder why students did not seem to be content with only two points to draw the line, as it could have been sufficient. The explanation may be that often students are told that finding three points reduces the risk of errors caused by wrong calculations for finding the position of their line. As well, students asked if the addition of two linear functions always gave a linear function. Whereas some students were strongly assertive that it did, and the teacher discussed this with them, some appeared to continue being sceptical. However, this is not an issue that I probe here, but one that could help explain this “more than two points” practice.

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This interpretation of the interrelation of exact and approximated answers is important. Although often seen in opposition, the Kahane report (2003) discusses at length the importance of interrelating both exact and approximated calculations for solving mathematical problems and how each influences the other. Thus as the following quote stresses, depending on the kind of problem one faces, both types of calculations can intervene. The relationship between exact calculations and approached calculations has to be thought about, not in opposite ways but as complementary, in relation to the problems to be solved (Kahane report, 2003, p. 14, my translation) And in particular for the Kahane report, mental mathematics is said to have an essential and privileged role for developing the link between both kinds of calculations. This complementarity is perceptible in the understanding offered about students’ solving. The Kahane report insists that both calculations are equally important for solving problems in mathematics. In this understanding of students’ ways of solving, approximated calculations were seen not only to be aimed at when exact calculations were impossible (although this could have been a possibility), but to enable the solving of the problem itself and for finding an entry into the problem. No students asserted that exact calculations were “more precise” or “better” than the approximated ones; in fact students were seen more frequently bringing forth the approximated “less precise” way of finding points of the function. These examples illustrate why it is said that exact and approximated calculations are interpreted as realized in complementarity, and not in opposition, in students’ solving of tasks. This analysis offers an enrichment of our understanding of how both calculations can complement each other, and especially in mental mathematics when issues of time are a factor. The undistinguished nature of both kinds of calculations in their interplay between exact answers and orders of magnitude can be understood as participating in the control that students are seen to exert over their solutions. The solutions can thus be seen doubly robust: oriented by an order of magnitude and confirmed by exact answers, and vice versa.17 6. Additional discussions on students’ generated strategies: “cues”, prompts and affordances Common to all these strategies is what in mental mathematics is called a global reading of the situation, where time constrains/forces the development of a larger view of the situation on which to operate (in contrast to focusing on isolated parts, because there is not enough time to pay attention to all parts needed to solve the task). Hence the focus is not on one tiny aspect of the function, but on its general shape; not on one algebraic value for slope, but on the parameter representing the slope; and not on x = 2, but on a generic example that produces the entire family of x > 0. It is then a provoked necessity to gain a larger view of the situation in order to engage with it in full in a time-constrained situation, in its entirety, and to create a strategy for this entirety that enables one to enter and solve the posed task. This understanding of students’ actions in regard to the strategies generated contributes to illustrating Threlfall (2002, 2009), assertions about students’ development in action of ways of solving, as emergent adapted responses to the task. These analyses of strategies lead one to see these as enacted on-the-spot, as emergent reactions tailored for the tasks, determined by students’ mathematical histories (their past, preferences, habits, etc.), and influenced by tasks (one does not react in the same way to two square-root functions as one does to two linear intersecting functions). These analyses invite us to see students’ actions and mathematical activity through a specific lens. Hence by their entry into the tasks, students are seen to pose the tasks offered to them: posing their own problems, making them algebraic, geometric, procedural, etc. This posing of tasks is seen to generate strategies for finding the resulting functions. In fact, as Simmt (2000) explains, it is not tasks that are given to students, but rather prompts that are taken up by students, who then create tasks with them. Prompts become tasks when students engage with them, when as Varela (1996) would say, they pose problems. Hence students make the “wording” or the “prompt” a multiplication task, a ratio task, a function task, an algebra task, and so forth. Through this understanding of the data gathered, an algebraic reading of functions produced an algebraic approach; a graphical reading produced a graphical approach; a numerical/pointwise reading produced a numerical pointwise approach. Depending on the task and on who they were as solvers, various strategies emerged. This leads one to see that students posed the problems that they could pose, from the prompts they were given, and solved their own problems that they had made emerge. One aspect that stands out in this explanation of the data is the generation of cues for the tasks, or anchors with which students worked. Through this/their posing, students can be seen as generating aspects for working within the tasks, making them emerge. For example, even if they were given only in a graphical context, students can be seen eliciting algebraic aspects about these tasks (whether algebraic expressions or parameters). As Varela (1996) would explain, this algebra was not there in itself, waiting: students brought it out and made it present. The same view can be imposed on the interpretation of the various points on which students were seen to focus through their approximated/exact calculations. They are seen to have made these points emerge as possible cues with which to engage, as significant points in the graph, for advancing toward the resulting function. Obviously one can see these points in the graph (especially after they have been pointed out), but they are not there explicitly, giving information on the function (e.g., not all tasks caused similar cues emerge, as other cues

17 In addition, except for hinting where the complementarity between both sorts of calculations has played a role historically and where it could play one in our current era, no concrete examples are given in the Kahane report (2003) to illustrate what this interplay looks like in action. Thus understanding these data under that lens can be seen to offer illustrations of this.

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were elicited for other tasks). Nonetheless, each prompt is always designed following specific intentions in specific ways, as is the case here, which can play a role in how solvers pose problems. In sum, each prompt can be seen to have what Gibson (1986) refers to as affordances, a concept that has recently received attention in mathematics education research (see Brown, 2014; Gresalfi, 2013): The affordances of the environment are what it offers the animal, what it provides or furnishes [. . .] I mean by it something that refers to both the environment and the animal in a way that no existing term does. It implies the complementarity of the animal and the environment [. . .]. If a terrestrial surface is nearly horizontal (instead of slanted), nearly flat (instead of convex or concave), and sufficiently extended (relative to the size of the animal) and if its substance is rigid (relative to the weight of the animal), then the surface affords support [. . .]. Note that the four properties listed – horizontal, flat, extended, and rigid – would be physical properties of a surface if they were measured with the scales and standard units used in physics. As an affordance of support for a species of animal, however, they have to be measured relative to the animal. They are unique for that animal. They are not just abstract physical properties. (Gibson, 1986, p. 127, emphasis added) These affordances are relative to the interaction of the students and the prompts, as affordances for those students interacting with these prompts: they do not exist in themselves, they are not inherent properties of the prompts, but are evoked in interaction with the prompt when one poses the task and makes them emerge. This process of problem-posing and problem-solving contributes to understandings of issues of the emergence of strategies in a mental mathematics context (I also discuss these ideas in Proulx, 2013b). More than a theoretical argument, understanding data in these terms offers illustrations of this process that unfolds in the doing, through solving the tasks. By generating affordances, students are seen to offer illustrations of how they posed the problem (it became a task about those affordances) and how they solved it (regarding those specific affordances). It can thus be seen as a double-emergent phenomenon, from the posing to the solving. And this posing/solving is also nested in the tasks themselves where, as Davis (1995) explains, mathematical strategies are inseparable from the solver and from the task itself, emerging from both. Cues are not interpreted as generated out-of-the-blue: it was algebra drawn out of the linear functions; it was a geometrical slope from the nonlinear functions; it was points that could be pointed to in the graph; and so forth. Along these lines, in this mental mathematics context, students are conceived of as posing the(ir) problems in order to have a problem to solve, for finding a way to “get into” the problem. As I have mentioned elsewhere about students solving algebraic equations mentally (Proulx, 2013a), by posing the problem, students are seen to generate a context in which to solve it: here an algebra context, a graphical context, a point-by-point context. The task then becomes about this. The objective is to find an answer, but it is first and foremost to find a way in, to generate cues to grasp for solving the task. These generated strategies, these cues enacted on-the-spot in meeting with the tasks in the context of mental mathematics, characterize the understanding developed in this study about the nature of the mathematical activity students engage in. This analysis offers a way to make sense of students’ interactions with the mental mathematics tasks and the kind of mathematical activity triggered in them through this mental mathematics context.

7. Issues to probe in further research 7.1. On functions studies Students’ strategies and ways of engaging raise questions about their developing understanding of functions. In effect, in addition to the fact that their actions were not interpreted as illustrating significant difficulties in solving the mental mathematics tasks presented to them, one aspect that stands out is their apparent ease in navigating through various representations. Through their explanations, some of which are quoted above, students were seen as fluent in linking algebraic (symbolic expression), numerical (coordinate points in x or/and y) and graphical (Cartesian plane) aspects of the tasks, going back-and-forth between each while solving, as well as explaining this to the whole class. This seems to contrast with what has been shown in other studies, where students are frequently reported as experiencing many kinds of difficulties when linking graphs of functions with other representations (Duval, 1988; Goldenberg, 1988; Hitt, 1998a, 1998b; Knuth, 2000; Leinhart, Zaslavsky, & Stein, 1990; Mitchelmore & Cavanagh, 2000; Moschkovich, Schoenfeld, & Arcavi, 1993; Moschkovich, 1999; Slavit, 1998; see also Dubinsky & Wilson, 2013, review). This raises questions about the outcomes/contributions that doing mental mathematics can be seen to have on students’ fluency with representations.18 In addition, through their solving, many students’ actions were understood as engaging flexibly in what are known as global and pointwise approaches to functions (Even, 1998; see also Bell & Janvier, 1981). As Even explains, a pointwise

18 It is to note that other researchers, in vizualisation studies for example, have also raised issues about a growing level of comfort with graphical representations in the context of functions (see e.g. Stylianou’s work [2001, Stylianou & Silver, 2004] with undergraduate advanced mathematics students). However, one main difference in this mental mathematics context, among others, is the absence of paper-and-pencil to produce the imagery and work on it as we see solvers do in Stylianou’s work. I return to this issue in Section 7.2.

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Fig. 11. Illustration of points brought forth by students to solve Fig. 4 task.

approach means dealing with discrete points of the function, whereas one can consider and deal with the function globally by paying attention to its behavior. Her study shows how students “stuck” in one or the other way of dealing with functions experience significant difficulties when they engage in various tasks about functions. Navigating through both approaches is thus seen as the most powerful approach for solving (see also Moschkovich et al., 1993, on this issue). Students’ actions in this study can be understood not only as being able to switch depending on the problem for pointwise or global approaches when needed (e.g., in the approach/exact calculations and the use of slope), but also as engaging at times in a combination of both for the same task. For example, in the addition task in Fig. 4, some students offered a strategy that can be seen as attending to points and global aspects: working with points like those from 1 to 5 (see Fig. 11) and at the same time considering the symmetry of other parts of the function for finding the resulting function, a line with double the value of image-lengths of points 1 and 5. This rouses interest in investigating the outcome of doing mental mathematics on the development of this fluency between pointwise and global approaches to functions, and about the effects of these mental mathematics activities on students’ understandings of functions (e.g., students “concept image” [Vinner, 1983] of functions, their “standard” written work on operations on functions, their fluency between functions’ representations, etc.). We know from the literature that work on mental mathematics with numbers can have significant effect on broader aspects of students’ understandings like number sense, written work related to algorithms, and problem-solving skills to name a few. Thus further studies are needed to inquire into what mental mathematics activities can contribute to students’ understandings of functions (and also for other mathematical objects). 7.2. On visualization studies As mentioned in Footnote 1, even if the visualization literature did not directly inform this study, some findings can be seen to parallel some of those found in visualization research (especially as a number of studies on visualization refer to the study of functions). In relation to the definition of mental mathematics offered in Section 2, any manipulation/calculation/computing with objects that are mental can be seen as examples of mental mathematics, provided the manipulation is done without the use of paper-and-pencil or any material aid. Hence by being performed without the use of paper-and-pencil, such operations on functions can also be seen as being performed through visualization. Referring to Presmeg (1986, p. 42), the ability “to perceive clear mental pictures” was indeed needed to operate on these functions mentally, especially as in the Set of Tasks 5 only algebraic expressions were given, implying mental-image reasoning to solve the problems. Hence connections can be established between this study and some other work in visualization studies, particularly in relation to the enacted capacities of students to visualize and operate on their visualizations to solve the problems. I raise some of these issues below. One aspect raised earlier concerned the level of comfort that students demonstrated in managing visual representations for operating on functions. However, one main difference, as mentioned, is that the representations were built into the tasks or imposed in their solving, and thus students were confronted with them immediately. Hence issues of preference of using visual representations, or of reluctance, distrust, avoidance (to use expressions often referred to in visualization studies, see e.g. Eisenberg & Dreyfus, 1991; Stylianou, 2001, Stylianou & Silver, 2004; Vinner, 1989) are not much at issue here. An issue to probe further could then be related to aspects of the availability of visual representations. Does giving the task in a visual form, as was the case here, play a role in how visual representations are managed, referred to, and so forth by solvers? What if some of the tasks given were given only orally? For example, if students were asked orally to operate on, to add, f(x) = x and g(x) = x2 , how would they go about solving? Would their ways of visualizing the functions differ from those gathered here? Students’ observed comfort with the visual representations and operating visually on them suggests issues, at least for work on mental mathematics, about the possible differences between asking a question orally and asking it in writing (and even concerning the various ways that it can be offered in writing). How does this affect students’ solution processes? How

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are visual representations referred to in the solution processes of each? What kinds of visual representations are referred to or created when they are not given first-hand in the tasks? It is significant to understand more deeply students’ ease and comfort with visual representations (as well as which ones), because in this context graphs were directly available. This is to be probed further in future research for understanding how visualizing the task, or simply giving a visual task, plays a role in its solving. As well, taking Arcavi’s (2003) definition of visualization as the ability, the process and the product of creation, interpretation, use of and reflection upon pictures, images, diagrams, in our minds, on paper or with technological tools, with the purpose of depicting and communicating information, thinking about and developing previously unknown ideas and advancing understandings (p. 217) then what has been asked of students in this study is clearly an exercise in visualization. In this sense, Strategy 2, with issues of constantly varying slope and parallelism of curves, fits well with the idea of previously unknown ideas and advancing understandings. However, the “unknown” aspects need to be thought about in relation to students themselves. Even if an external experienced solver could assert having already known these strategies (e.g. issues of resorting to parameters or exact and approximate points), as mentioned in Section 6 these are to be thought of in terms of innovative ways of posing and solving the problems at hand (and thus were not pre-decided as tools to be reused). In this sense, these strategies contributed to advancing students’ understanding. One can ask how these “new understandings” and “ways of solving” can affect and be taken up in students’ future work when they subsequently undertake paper-and-pencil tasks. Linking with issues of the presence of paper-and-pencil to produce the imagery and being able to work on it afterwards, as we see solvers do in Stylianou’s work (2001; Stylianou & Silver, 2004) as mentioned in Footnote 18, one can wonder in what ways can these visualizations of operations on functions in a graphical environment play a role in students’ subsequent solving? And, perhaps most interestingly, how can these ways of solving be pushed further when possibilities exist to produce and work on the imagery in paper-and-pencil context? These are some of the issues that outline possible links between the study of mental mathematics and studies of visualization, and that need further explorations. There would be more links, obviously, especially because visualization (even of numbers, of symbols, etc.) is everywhere in mathematics; just as one could argue that most of the mathematics done has a mental dimension. Hence the issue should not be about what is visual and what is not, but about exploring and investigating the links between these various lines of research and enriching each one through studying connected avenues. Raising these questions is but one attempt toward this.

8. Concluding remarks The strategies and meanings expressed in the activities underline the potential of this mental mathematics activity for engaging students in solving processes and generating strategies, especially because the analysis leads to see that almost all students were successful in solving tasks in the sense of engaging with the task, of developing ways of getting in, of generating/creating adapted strategies, of playing with ideas and representations, and so forth. However, showing fluency with functions and being successful is one thing, but the goal of the research is not this. Of interest is what student did in this mental mathematics context, how they engaged in (their) tasks, the graphs, without any paper-and-pencil, and the mathematical potential that these strategies evoke. From a teaching point of view, many mathematical issues could have been reinvested afterwards by a teacher in his/her classroom: transition from algebraic to graphic to numerical representations; reflections on the nature of slope outside a linear context; engaging with multiplication/division of functions through their pointwise approach, on the global reading of functions in order to handle parameters of functions [a,h,k], a specific teaching unit in the Quebec curriculum; and even on the parallelism of curves, to name but a few. Although mental mathematics and graphical environments can be seen as unusual contexts for working on operation on functions, the analysis leads one to see students coping, engaging with (their) tasks, and developing efficient solving strategies. In that sense, even if it was their first explicit experience with operations on functions, students’ actions can be understood as ways of solving by generating strategies adapted to the(ir) tasks. In the literature about mathematics teaching with numbers, one frequent recommendation is to work on concepts through mental mathematics first before working on them on paper (Butlen, 2007). With the analysis developed through this study about students’ mathematical activity, one is tempted to believe that this could also become a possibility for mathematical objects other than numbers. Clearly, it shows some potential. However, much more research is needed before such a route can be suggested, supporting the intention to continue to study mental mathematics with other objects in order to understand its potential and its possibilities regarding mathematics teaching and learning.

Acknowledgements This study is part of a research program funded by the Social Sciences and Humanities Research Council of Canada (SSHRC Grants #430-2012-0578 and #435-2014-1376) and the Fonds québécois de la recherche sur la société et la culture (FQRSC Grant #164724).

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