A Tale of Two Researchers: Commonalities, Complementarities, and Contrasts in an Examination of Mental Computation and Relational Thinking

This paper describes a research collaboration between an educational psychologist and a mathematics education researcher, namely a didacticien des mathématiques. Our joint project aimed to explore the mental computation strategies of preservice teachers in an elementary mathematics methods course and to investigate the relationship between mental computation and relational thinking. The primary objective of the paper, however, is to go beyond the data and their interpretation. We describe the commonalities, complementarities, and points of contrast that emerged between us as researchers who hail from different disciplines, but who have the same overarching interests in mathematical thinking. In particular, we untangle issues we encountered during our collaboration related to our research questions, methodologies, and epistemological stances. We detail the ways in which we navigated these issues in the context of the research and describe what we learned about our own disciplinary perspectives and each other’s. We conclude by discussing what our story offers as a means of reflecting on our individual fields and potential interactions between them.


The Beginning of the Collaboration and Initial Emerging Issues
The research team consisted of the two authors of this paper and their two doctoral research assistants.In this section, the backgrounds and research foci of the two authors are described, which is then followed by a description of the genesis of the collaborative project.

The Research Team
The first author (henceforth referred to as HPO) is an educational psychologist whose research objectives are to understand the characteristics of instruction that promote (a) the appropriate use and interpretations of mathematical representations, and (b) the development of conceptual and procedural knowledge in instructional contexts.Her research is influenced by cognitive psychology, but her focus is on the application of cognitive theories to domain-specific accounts of how children learn mathematics in educational settings.
Consistent with the historical traditions of educational psychology (Derry, 1992;Mayer, 1992Mayer, , 1993)), HPO most often collects experimental data using methodologies that employ random assignment to conditions, sample sizes that meet minimum statistical requirements, several outcomes measures, and quantitative analyses.This said, HPO's research is not always experimental.Depending on her purposes, she entertains questions that are best answered through descriptions about how students work through particular tasks or how they interpret specific mathematical representations.These descriptions serve to provide teachers with insight on children's thinking in particular contexts, assist to explain or illustrate quantitative results, or act as a springboard for future experimental research.
The second author (JP) works in a mathematics department and teaches future teachers of mathematics.As a didacticien des mathématiques, he focuses heavily on mathematics as a discipline and its specificity in the teaching and learning of mathematics.His research interests center on problem-solving processes and the mathematics that emerges in mental computation environments.His work addresses how mathematical and epistemological issues bear on mathematical cognition: Studies of cognition for JP are linked to epistemological questions about the nature of mathematics and its development (as much for mathematics as a discipline as for the person engaging in the mathematics; see Proulx, 2017a).Therefore, the data JP collects are often in the form of classroom observations of students' activity, because they offer unique opportunities -being in realtime and in action -to investigate and conceptualize the nature of problem solving.

The Beginning
The impetus for the collaboration came from one incident in a mathematics methods course HPO taught in an undergraduate elementary teacher training program in Canada ii .Her goal for the students was to expand their repertoire of mental computation strategies.In one class, HPO was explaining the compensation strategy by using 500 × 4 as an example.She demonstrated the strategy "one factor gives to the other" (Parker & Baldridge, 2004) and wrote 500 × 4 = 1000 × 2 on the board.At this point, a student raised her hand and said, "but I don't understand: 500 × 4 is not equal to 1000." It became apparent to HPO that this student had difficulty grasping the meaning of the equal sign, at least in this specific context.HPO hypothesized that the student held what has been called an "operator" view of the symbol (Kieran, 1981;Seo & Ginsburg, 2003;Sherman & Bisanz, 2009) -a signal to compute rather than a representation of a relation between 500 × 4 and 1000 × 2. In this specific case, HPO entertained the possibility Osana & Proulx 61 that this student's apparent misconception of the meaning of the equal sign impeded her ability to reason relationally about the equation.As a result of this classroom exchange, HPO wondered about the possible relationship -correlational or causal -between mental computation and relational thinking.She was particularly drawn to explaining such a relationship and whether it could be leveraged to support the development of relational thinking in learners of mathematics.With this in mind, HPO contacted JP to mount a collaborative research project around this problem.

Perspectives and Objectives
Through discussion, it soon became apparent that HPO and JP approached the problem in different ways.In this section, these perspectives are described, as are the objectives formulated by the authors as a result of their different approaches.

A View From Educational Psychology
The conceptualization of relational thinking offered by Carpenter, Levi, Franke, and Zeringue (2005) and Jacobs, Franke, Carpenter, Levi, and Battey (2007) served as an apt theoretical frame for HPO's hypothesis.
Jacobs et al. defined relational thinking as a form of reasoning that entails looking at numbers and expressions holistically and noticing relations among them.Within this perspective, relational thinking is viewed as a form of algebraic reasoning that provides a conceptual foundation for the formal study of algebra (Carpenter et al., 2005).To illustrate, consider again the equation 500 × 4 = 1000 × 2. One way to determine the truth value of this equation is by computing the values on either side of the equal sign and then comparing the two amountse.g., 500 × 4 is 2000 and 1000 × 2 is 2000 -concluding that the equation is true.Such a strategy could be considered an example of "computational thinking" (Stephens, 2006) and can be completed strictly procedurally, with little understanding of the meaning of the quantities and operations involved.In contrast, thinking about the equation relationally entails reflecting on the numerical relationship between both sides of the equal sign.For example, seeing that 1000 is twice 500, it would need to be multiplied by a number that is half of 4 (on the left side) to make the equation true, invoking compensation as a way to make the amounts on each side of the equation the same.
The illustrations of computational and relational thinking in the above example are both dependent on a view of the equal sign as "the same as."According to Jacobs et al. (2007), there is more to relational thinking than a conceptual grasp of the symbol, however.While the equal sign features centrally in many descriptions of relational thinking (Freiman & Lee, 2004;Knuth, Stephens, McNeil, & Alibali, 2006;McNeil, 2014;Molina, Castro, & Mason, 2008), it is also contingent on a conceptual knowledge of numbers, operations, and number properties, all foundational to algebraic reasoning.In this example, the connection between relational thinking and algebraic reasoning is apparent because the same conceptual understandings undergird the procedure for solving for x in 500x = 1000 × 2.An expanded conceptualization of relational thinking and the theoretical overlaps with mental computation is offered in the Appendix.

A View From Didactique des mathématiques
The generation of a hunch about the explanation of a psychological phenomenon -in this case, one that is closely tied to teaching practice -was HPO's entry into the research.JP had a different entry, however: As a didacticien des mathématiques, educational problems of this sort, or the search for causal links, are seldom addressed.When HPO introduced the research idea to JP, he first engaged in problematizing the links between Commonalities and Contrasts: Educational Psychology and Mathematics Education 62 500 × 4 = 1000 × 2 and 500x = 1000 × 2. For him, more was to be said about the mathematics at stake.
Although he agreed on the surface about the connection between the two equations, in his conceptualization, arithmetic and algebra do not share the same roots and the transition from one to the other is complex because they engage in different mathematical processes (Bednarz & Janvier, 1996).In short, an algebraic equation requires looking for all the values of x that would make the equation true, and the solver has to accept -that is, operate with -unknown values in similar ways as with known values (i.e., givens).This results, for example, in isolating the x on the left side of the equation by dividing both sides by 500 (Filloy & Rojano, 1989), or considering equivalent equations (see Arcavi, 1994), where one operates on the complete equation to find ones that are easier to solve (500x = 1000 × 2 becomes 1000x = 1000 × 2 × 2 = 4000 leading to x = 4).If, on the other hand, one wishes to make the connection between 500 × 4 = 1000 × 2 and 500x = 1000 × 2, this leads to viewing the x in 500x = 1000 × 2 not as algebraic, but rather as a place-holder, as commonly seen in the elementary mathematics curriculum (e.g., ◻ + 6 = 9, what is ◻?).The place-holder conception requires an entry where operations are arithmetically "undone," in terms of unwinding (Hewitt, 1996;Nathan & Koedinger, 2000) to arrive at a value for x: "I undo the operations made on x: x was multiplied by 500, so 1000 × 2 is to be divided by 500 too."As Filloy and Rojano (1989) explained, when using this entry, "it is not necessary to operate on or with the unknown" (p.20), as it becomes a series of arithmetical operations performed on numbers, avoiding the usage of algebraic reasoning altogether.
Such distinctions in the approaches to "the problem" highlighted other differences about the contexts in which each researcher's objectives were grounded iii .For HPO, the means with which to explore whether the preservice teachers indeed learned -i.e., improved in their relational thinking -was to examine the cognitive processes that characterized Carpenter et al.'s (2005) theoretical framework.Central questions for HPO were: Did the preservice teachers engage in computational strategies to determine the truth value of mathematical equations or were they able to think about the relationships among the quantities in the equations?Can we find evidence to support the claim that the mental computation intervention facilitated their relational thinking?If the research generates such evidence, what can be said of the theoretical overlap between mental computation and relational thinking?Such questions make evident that HPO was driven by the search for evidence that would make recommendations for teaching practice credible.
In contrast, HPO's experience with the preservice teacher would not have led JP to question the links between relational thinking and mental computation.Rather, some of the questions he might have generated are: "What sorts of 'equal signing' is happening in a mental computation session?"; "What mathematical meanings are invoked during the mental computation sessions?";"What sorts of mathematical understandings of the equal sign are the students developing and interacting with?"; and "What sorts of obstacles (epistemological, didactical, cultural, historical;Brousseau, 1983Brousseau, , 1989) ) are the students facing during the mental computation sessions?".These questions focus directly on, and emerge from, the nature of the mathematics itself and of students' mathematical understanding.

A Problem-Solving Perspective
JP was intrigued by HPO's proposal, but his interests in it were motivated by a continued need for investigating and conceptualizing problem-solving processes in mental computation environments.Most studies concerned with analyzing students' mental computation strategies are (implicitly) based on the assumption that has often been termed the "toolbox metaphor" (or the "selection-then-execution" hypothesis).The toolbox metaphor implies that when solving a problem, students choose a strategy (i.e., a tool) from a group of predetermined Osana & Proulx 63 others.This section focuses on the reasons for each decision related to data and design, and highlights issues marking significant disciplinary differences.

Design
HPO and JP needed to make decisions about how the intervention would be delivered and the tasks that would be used in the unit.In this section, the mental mathematics unit is described, followed by a discussion of key decision points related to tasks and instructional procedures.

Course Context and the Mental Mathematics Unit
The participants were 33 preservice teachers in the first of three required mathematics methods courses in an elementary teacher training program at a large, urban university in Canada.Six 45-minute class sessions were devoted to mental computation activities.In three of the sessions, students were asked to solve a number of arithmetic tasks mentally (e.g., 741 -75; 184 ÷ 8).HPO read each task out loud, and students were given 20 seconds to arrive at an answer.They were not permitted to use any tools (e.g., paper and pencil, fingers) during the solving activity.When time was called, the instructor called on the students to describe their strategies, and at times invited them to the board.
During whole-class conversations about strategies, HPO's focus was to help students understand the underlying principles that justified each strategy (e.g., the meaning of multiplication, commutativity, distributive property) and to see the connections between different strategies for the same task.While HPO focused on such principles for instructional purposes (i.e., this was a portion of the course's curriculum and reflected in the assigned readings), she was also conscious of ensuring that the unit indeed fostered appropriate mental computation strategies for research purposes.For JP, this aspect of the unit was significant because the explanations and interactions that occurred in the session were, for him, the object of inquiry.
In the other three sessions in the unit, students worked individually on practice problems (e.g., exercises) delivered on worksheets.In these practice sessions, the students were required to solve tasks mentally and describe in writing the strategies they used.The worksheets were graded and returned to students with feedback.

Choice of Tasks
The tasks used in the unit were not chosen randomly.As instructor of the course, HPO aimed for tasks that were likely to generate specific strategies from the course readings (Parker & Baldridge, 2004).For her, this was required to investigate the theoretically hypothesized transfer to relational thinking.For JP, the chosen tasks needed to be grounded in mathematics, developed through and oriented by a conceptual analysis (a particular research tool from the French didactique scholars, e.g., Brousseau, 1998).In short, a conceptual analysis aims to draw out, among other things, the key understandings and main obstacles, difficulties, and common errors that could theoretically occur in relation to a specific mathematical idea or group of ideas, in this case numbers and the four arithmetical operations (see Proulx, 2017b, for an example of a conceptual analysis in the domain of statistical average).
As a way of illustrating the use of the conceptual analysis in the present context, consider the task 184 ÷ 8.This task was chosen by JP in particular because a simple decomposition of 184 into

Instructional Procedures
There were a number of methodological decisions that needed to be made regarding the delivery of the mental computation unit.First, it was important to HPO to pinpoint the elements of the instruction that might be responsible for the differences in relational thinking after the unit was completed.To her, this was particularly critical because there was no treatment integrity built into the study design.As such, HPO had initially intended to "directly teach" specific strategies to her students by labeling them and explicitly describing how and when to apply them.For example, for a computation involving ÷ 5, a useful strategy would be divide by 10 and double.
By directly teaching the strategies, HPO felt she could ensure that the instruction targeted mental computation -and as much as possible nothing else -so that the data would bear on its relationship to relational thinking.
In contrast, explicit instruction (or "telling") of specific strategies ran against the research goals of JP.To meet JP's objectives, the students needed to engage in whatever methods were meaningful to them (arising in situ; Lave, 1988;Varela et al., 1991;Varela, 1996), and to avoid having students merely pleasing the instructor by using what was shown beforehand (something akin to what didacticiens des mathématiques call the didactic contract; Brousseau, 1998;Sarrazy, 1995).The tension between control (from a design perspective) on one hand, and the investigation of authentic, emergent mathematical practice on the other, was a significant issue that needed to be resolved.HPO accepted to allow students to use strategies that were meaningful rather than reproduce "pre-fabricated" ones.She conceded that such reproductions might actually decrease the likelihood that mental computation actually took place, which would ultimately compromise her ability to draw conclusions about its connection to relational thinking.
Furthermore, when asked to solve mental computation tasks, children have been known to perform standard algorithms mechanically (Heirdsfield & Cooper, 2004;Sowder, 1992;Varol & Farran, 2007).Knowing this, HPO wanted to prevent the preservice teachers from doing the same and ensure that they engaged in mental computation that was driven by flexible mental transformations of quantities and mathematical expressions.As such, she specifically instructed the students not to use standard algorithms.JP accepted to limit the students' use of the standard algorithm, even if it could still be considered a mental strategy.He thus proposed a 20second time limit for each mental computation task; his suggestion was informed by previous work, showing that reasonable time limits tend to encourage strategies that are idiosyncratic and personally meaningful (Butlen & Peizard, 1990).This was to eliminate exposure to non-canonical equations as a way to account for any potential differences in relational thinking after the unit; JP found this acceptable because it did not compromise the nature of his observational data.

Data Sources
The data came from two sources, chosen to address each researcher's specific objectives.A paper-and-pencil measure of relational thinking was administered before and after the unit, and observational field notes were collected during the unit.

Measuring Relational Thinking
Establishing a causal link between mental computation and relational thinking was the ideal objective for HPO.
To establish cause, however, a researcher would need to collect evidence under specific design conditions.
HPO's ultimate objective was to design a study that would eliminate as many alternate explanations for the results as possible iv .This was important for HPO so that she could claim, and hence explain v , that it was in fact the focus on mental computation, and not something pre-existing or co-occurring, that was responsible for any improvement in the students' relational thinking.A "controlled randomized trial" would have served this purpose, but was not possible given the constraints present at the time of the research: There was only one section of the methods course available and not enough time could be devoted to the delivery of a control intervention to half the students.Thus, given such constraints, HPO strived to reduce the number of alternative explanations for the results rather than eliminate them entirely.
To gather data that could coordinate with the corresponding theory (Kuhn, 2010;Shavelson & Towne, 2002) about the theoretical relationship between mental computation and relational thinking, HPO and JP created the Relational Thinking Test (RTT), a paper-and-pencil instrument to assess the preservice teachers' relational thinking.JP did not require a measure of relational thinking to meet his research objectives, but he understood that this measure was required for HPO's part of the research and thus accepted to co-construct the measure.
The RTT, shown in Table 1, consisted of six arithmetical expressions involving whole numbers.Each item required the student to "indicate whether the following number sentence is true or false" and to justify his or her response (see Carpenter et al., 2005).The items on the RTT were constructed to measure the students' ability to transfer what they learned from the mental computation unit to contexts requiring relational thinking.For example, consider item 2 on the posttest.

Osana & Proulx 67
To think about this item relationally, one may invoke (implicitly or explicitly) the distributive property: Four groups of 156 would be the same as three groups of 156 and an additional group of 156.Recognizing the resulting inequality by adding a group of 157 to the three groups of 156 would obviate the need to compute the amounts on both sides of the equal sign -the hallmark of relational thinking according to Carpenter et al. (2005).The psychological notion of transfer (e.g., Barnett & Ceci, 2002;Mestre, 2006) is important here.In HPO's sessions during the instructional unit, her goal was on helping students see how principles of arithmetic justified their mental computation strategies, but outside the context of judging the truth value of mathematical equations.Switching the context to one that would require relational thinking (at least as explicated by Carpenter et al., 2005) by transferring the concepts learned during instruction is a common approach used by educational psychologists to measure understanding.

Observations of Mental Computation Strategies
To investigate mathematics problem-solving processes in action and the mathematics students enact in their strategies, JP required data that captured how students solve mental computation tasks.Thus, observational data of students in the throes of mathematical activity were gathered.JP and the two research assistants attended the class sessions during which students solved and then explained their strategies to their peers.
Data were taken in note form, where all three observers took detailed field notes of the strategies described orally by the students and the explanations that ensued.One of the assistants also took pictures of the students' work that remained on the blackboard after they had explained their strategies.The combination of the three sets of field notes, the photographs, and the on-the-spot short discussions among the team members after the sessions made a rich corpus of data vi .
place, but may in fact serve as evidence of an explanation or justification of some underlying mathematical reasoning (relational or not).The rubric was thus expanded to include distinct sub codes for two instantiations of relational thinking -namely, a sub code [RT -No Computation] for instances of the "no computation" benchmark proposed by Carpenter et al., and a sub code [RT -Justification] to recognize cases where students used computation to support or explain a form of relational thinking.

Mental Computation Strategies
The contextual background against which the data were considered, grounded in issues of emergence and situativity, welcomed any idiosyncratic or even locally-functioning strategy offered by the preservice teachers during the unit.Embracing Plunkett's (1979) notion that mental strategies are fleeting, variable, flexible, active, and also local, tailored, and connected to the task at hand, the data analysis aimed to characterize the mathematical activity generated in this mental computation context, untangling the steps taken as students solved tasks and scrutinizing the nature of their problem-solving processes.Following Douady (1994), the intention was not to report on the learning that took place, nor to discuss the long-term effects of such problemsolving processes in other mathematical situations.Rather, JP aimed to gain finer understanding and conceptualization of these processes, both in terms of their functionality for the tasks at hand and the meanings created by the students.
The observational data were analyzed in a two-phase process.The first phase consisted of brief team meetings (JP, HPO, and the research assistants) after each instructional session to review and summarize the events that occurred during the session, focusing on students' strategies.The primary purpose of these meetings was to supplement the field notes with additional observations and insights.The team meetings also offered a first level of analysis that included the teacher educator's voice (HPO), which afforded interpretations of the events from a practitioner's perspective, and observations from other members of the research team on specific aspects of the session that merited attention.These discussions sharpened the team's understanding of solving processes, which were in turn used to interpret the observations in subsequent sessions.The second phase of the analysis consisted of repeated interpretive readings by JP of all field notes, after which the analyses were presented to HPO and the assistants.The resulting discussions centered on the relevance and nature of the analysis, which were strengthened through joint conceptualizations and interpretations.

Results and Findings
The following reports on both the results and the findings vii of the research.Each data set is presented separately (i.e., data on relational thinking and observational data on mental computation strategies) as well as the interpretations of each researcher's analyses.

Results: Assessing Improvement in Relational Thinking
Given that HPO was primarily interested in determining the increase in relational thinking from pretest to posttest, all justifications on the RTT that were coded C (i.e., computational thinking) and OT (i.e., other) were collapsed into one non-relational category (NR).This resulted in two possible codes for each item on the RTT: proportion of RT codes assigned in total on the RTT at both time points.A conservative threshold of at least 5 out of 6 RT codes was used -that is, students who were assigned an RT code on at least 5 of 6 RTT items on the pre-and posttest were considered relational thinkers.The others were considered non-relational thinkers.
The proportion of relational thinkers at pretest was 48.5% (16 of 33 participants), whereas the proportion of relational thinkers at posttest was 78.8% (26 of 33 participants).A McNemar test found that the change in proportions of students becoming relational thinkers was significant, χ 2 (1, N = 33) = 6.25, p = .012.
Table 2 illustrates the nature of the observed change for two students, Brittany and Claude (pseudonyms), from before to after the mental computation unit.Their responses serve as exemplars of the types of computational and relational thinking described in the literature (Carpenter et al., 2005;Empson et al., 2011;Stephens, 2006).
Before the unit, both Brittany and Claude were engaged in computational thinking by calculating the amounts on each side of the equal sign and then comparing them.After the unit, both students examined the expressions in their totalities, taking into account the quantities on both sides of the equal sign in relation to the operations involved.Brittany's work illustrates an application of the distributive property to reason about the quantities involved.Seeing that 156 × 4 would be the same as (156 × 3) + 156 allowed her to conclude that it could not be equivalent to (156 × 3) + 156 + 1. Claude's work illustrates intuitive knowledge of the inverse element for addition that for every real number a, a + (-a) = 0.The implications of these results for practice are important, in HPO's view.As Carpenter et al. (2005) pointed out, the fundamental properties of arithmetic, which explain almost all procedural transformations in the school curriculum, are hidden behind the "notational elegance" of the standard algorithm.Too many students enter high school with procedural knowledge (i.e., knowing how to perform symbolic transformations) that is divorced from the conceptual rationales that give meaning to mathematical activity.Should mental computation be related to relational thinking, and especially if a causal relation is found in future research, one may conclude that engaging students in activities that increase their strategy flexibility, such as mental computation, lays the groundwork for genuine understanding in mathematics.
Commonalities and Contrasts: Educational Psychology and Mathematics Education 70

Findings: Observations and Conceptualizations of Mental Strategies
JP's intention is seldom to report on all the strategies themselves.Rather, he uses strategies generated by individuals as vehicles for extending and contributing to existing conceptualizations on strategy processes in mental computation.In similar ways to psychology, the intention of many didacticiens des mathématiques is theory building, but it is first and foremost to offer conceptualizations of, and fruitful distinctions about, mathematical thinking processes (Proulx, 2015).Therefore, the findings that follow are not results per se, but are orientations, ways of drawing out distinctions about problem solving in mental computation.The discussion below addresses such distinctions in the context of one specific strategy described orally by a preservice teacher, Amy (pseudonym).Amy's case is also well suited for illustrating issues about strategy generation and their emergence in relation to the task at hand.c.To do so, I do 5+1 is 6, 2+4 is 6, and I have 600, so 666.Focusing on the nature of the strategy as emergent, but contingent on the task itself, one can argue that going for 700 is not a step that came out of the blue, as it is related to the fact that 75 can be easily subtracted from 700 compared to, for example, 67, which would be arguably more difficult.Attempting to take 67 from 700 would have likely produced a different strategy altogether.Again, the remaining steps taken by Amy could be seen as directly related to the task, or more precisely said, as a function of the numbers (and in relation to what the student can see as possibilities with those numbers).Furthermore, each step of Amy's strategy can be seen as a "new" task to solve, thus necessitating additional "ways in" that allow her to continue reasoning Osana & Proulx 71 through it.Specifically, step (a) in Amy's strategy is a way to "enter" the task, the problem, and the outcome of this step (having to compute 700 -75) places her in front of another "problem to solve."Again, she must find a way to continue and thus opts for finding a money context ($7 minus 3 quarters) to carry out the computation.
Again, the outcome leads Amy to another sort of "obstacle" to overcome, which is to find a way to compute 625 + 41, leading to a strategy of splitting ones, tens, and hundreds to add like units.Amy's strategy shows that solving 625 + 41 mentally is not "obvious" to everyone, but is instead dependent on the unique characteristics of the solver, including her understandings and ways of doing mathematics.
Of interest is the fact that this specific strategy was generated for this task.Amy opted for splitting 741 into 700 and 41, and then subtracted 75 from the 700.Although fitting, this is likely not a strategy Amy would have used if the work had been carried out with pencil and paper.In pencil-and-paper contexts, students often resort to the standard algorithm, working their way through the task in the way they are accustomed, mostly independent of the task given (e.g., its numbers).Indeed, algorithms are intentionally created and used to solve an entire class of problems regardless of the specific quantities involved, and their power is that they "work" every time (Bass, 2003).
In contrast, students' first intention in a mental computation context is to find a way to "enter" the problem.That is, when the task 741-75 was given, Amy's first action (or first "reaction with knowledge," to use Threlfall's, 2002, expression) was to find a way to solve it, to "find a way in."It is in this sense that the mental computation strategies generated by the solver are contingent on the task given, tailored to it, generated through it, and thus also emergent, developed on the spot when the task is received.As Threlfall (2002) explained, "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.[…The strategy…] is not decided, it emerges" (p.42).
Mental strategies are then conceived as unique, new, generated in the act of solving, and creatively produced in a problem-solving context viii , a conceptualization epistemologically aligned with the nature of mathematicians' practice, defined as a creative and productive activity (e.g., Burton, 2004;Livingston, 2015;Lockhart, 2009;Papert, 1972).This alignment represents an illustration of how, for didacticiens des mathématiques, findings become epistemologically laden (Proulx, 2017a).

Final Remarks
Have you heard the one about the educational psychologist and the didacticien des mathématiques?An educational psychologist meets a didacticien des mathématiques and says, "we need to control the variables."The didacticien replies "hmm, but that's not easy since algebraic variables imply co-variation in a functional context"!
In this paper, we described a collaboration between an educational psychologist and a didacticien des mathématiques who investigated the mathematical activity of elementary preservice teachers.In our description, we deviated from the format traditionally used to report empirical results to instead highlight disciplinary commonalities as well as contrasts in our collaboration.Table 3 highlights a number of themes that captured these contrasts.
Commonalities and Contrasts: Educational Psychology and Mathematics Education research.Some 45 years ago, Piaget maintained that psychological methods, at least those that existed at the time, are not by themselves useful when they are "parachuted" (our term) into a complex setting, such as a classroom: There is a necessity to constitute a special study of didactics that is both supported by psychology and distinct from it […].Adapting to a classroom is really different than doing psychology with students of the same age.It is absolutely excluded that one can directly draw didactics lessons from psychology.
(Piaget, in Morf, 1971, pp. 4-6, our translation) Echoing Piaget's thoughts are those of Schoenfeld (1999), who argued that practically useful and theoretically enriching educational research should entail more than the traditional "research leads to development" perspective (see also Stokes, 1997).Instead, Schoenfeld proposed a model for educational research that places psychology and the research conducted in real-world settings (e.g., classrooms) in a dialectical arrangement.That is, researchers from both domains can work together so that the efforts of one informs the efforts of the other in an iterative, and even recursive, fashion, but at the same time, in such a way that can further the disciplinary objectives of both fields.In our specific case, we propose that the respective fields of educational psychology and didactique des mathématiques can be informed, and indeed enriched, by teasing apart and maintaining explicit the points of contrast in their respective questions, objectives, techniques, and interpretations.
Our experience confirmed that in collaboration, the work of educational psychologists and didacticiens des mathématiques cannot, and should not, be reductionist -that is, it should not be the objective to produce one single view on the study of students' mathematics.In this regard, we concur with Battista (2010), who delivered the compelling argument that the study of mathematics thinking and learning is best informed by research from a multitude of disciplines.A singular approach is unwelcome, as it would dampen the diversity of perspectives, thereby preventing opportunities for the fields to grow and enrich one another.

Conclusion
The collaboration we lived and outlined in this paper illustrates that it succeeded not by one "applying" the ideas of the other, nor by imposing on the other alternate ways of doing.Rather, its success can be conceived as an augmented sensitivity toward the need to agree and accept the other's perspective.In a very real sense, our interactions served to make the familiar unfamiliar; through our common goal of studying and supporting mathematical thinking, each of us had our ways of addressing that objective in ways that were to us routine, even comfortable.The collaboration pushed us out of our respective spheres of familiarity and forced us to recognize and confront our differences, rendering them explicit objects of discussion (and even of analysis for this paper!).From this emerged a critical examination of our own perspectives and sensitivities to those of the other.While we are not claiming that we produced a synergy that can serve as a model for all other collaborations between our two fields -a "how-to" guide for how our fields should collaborate -the experience worked for us in part because of our fundamental belief that learning from each other allowed us to more strongly justify, and at times question, our own assumptions and approaches.This appears to us as a central rationale for collaboration, whether at the individual or broader disciplinary level.
Osana & Proulx 75 mental strategy, a child must then execute the calculations mentally, which can occur in the context of relational thinking, but is not a defining feature.

Operational view of equal sign
The misconception that the equal sign indicates a signal to "do something," such as "add all the numbers" and "the answer comes next" (Carpenter et al., 2003;Falkner, Levi, & Carpenter, 1999;McNeil, 2014;Rittle-Johnson et al., 2011;Seo & Ginsburg, 2003).

Relational view of the equal sign
The understanding that the equal sign represents an equivalence relation and means "the same as" (McNeil & Alibali, 2005;Sherman & Bisanz, 2009).Understanding the equal sign can also involve the notion of "substitution" (Jones et al., 2012).
Relational thinking A form of reasoning that entails looking at numbers and expressions holistically and noticing relations among them (Carpenter et al., 2005;Empson et al., 2011;Jacobs et al., 2007).
Computational thinking A focus on computation when analyzing open-number or true-false number sentences rather than on relations between the expressions on both sides of the equal sign (Stephens, 2006).

Finally,
HPO based the instruction on existing research in educational psychology revealing that the types of equations to which children are exposed can impact their conceptions of the equal sign, even with minimal teacher involvement(McNeil, 2008;Osana & Sokol, 2014).During the mental computation unit, she made certain that the equations written on the board were always in "canonical," or standard, form -that is, those that Commonalities and Contrasts: Educational Psychology and Mathematics Education 66 have one number to the right of the equal sign and the operations only to the left of it (e.g., a + b = c; a × b = c).
(a) RT (relational thinking), which included both RT sub codes, and (b) NR (non-relational thinking).Each participant was then assigned either an overall RT or NR code at pretest and posttest depending on the Osana & Proulx 69

For
the task 741-75, Amy's work is shown in Figure 1.She explained her mental computation strategy orally and at the board in the following way: a. 741 -75 is like 700 -75 + 41. b. 700 -75 is like having 7 dollars and subtracting 3 quarters.I am left with $6.25.And, 6.25 is six-twentyfive, so I add 41 to 625.

Figure 1 .
Figure 1.Amy's presentation of her strategy on the board.
traditionally used in elementary and middle-school mathematics classrooms, where there are no operations on the right side of the equal sign.Examples include: a + b = c; a + b -c = d.
(e.g., a = a), have operations only on the right side of the equation (e.g., a = b + c), or operations on both sides of the equation (e.g., a + b = c + d).Open-number sentence An equation with a missing number.Examples include __ + b = c + d; a = b + __; a + __ = c + d.True-false number sentence Two expressions separated by an equal sign.The two expressions may or may not be equal, which would render the "equation" false.Examples include: 28 + 14 = 27 + 13 (false); (124 × 3) + 124 = 124 Proulx  65same time permits a number of different entry points for mental computation.For example, the versatility of 184 ÷ 8 is related to the iterative nature of 8 as a divisor -that is, 184 can be divided repeatedly by all the factors of 8 (i.e., 8 can be thought of as dividing by 2 three times), yielding 184 ÷ 2 = 92, 92 ÷ 2 = 46, and finally 46 ÷ 2 = 23.In addition, 184 ÷ 8 is amenable to a number of different applications of the distributive property, namely that "If number a can be decomposed into parts and if each part is divisible by the number b, then the entire number a is divisible by the number b."For example, 184 can be decomposed in a variety of ways such that 180 and 4 would not permit a straightforward mental computation strategy (i.e., 180 ÷ 8 and 4 ÷ 8 do not yield whole numbers) and at the Osana &

Table 1
Items on the Relational Thinking Measure at Pretest and Posttest

Table 2
Illustrative Examples of Responses on Selected Items From the RTT Before and After the Mental Computation Unit

Table A1
Definitions of Key Theoretical Terms in Relational Thinking