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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.

This paper describes a collaborative effort between two researchers from different fields with common interests in the nature of mathematical thinking. One of us, an educational psychologist, and the other, a mathematics educator (more precisely, from the field of ^{i}), together explored the nature of mathematical thinking in preservice elementary teachers in a three-week mental computation unit. Our goal for this paper, however, is to go beyond a standard research report. We aim to describe our interactions during the various phases of the project, highlight contrasting and complementary issues that emerged, and illustrate how the collaboration shaped each other’s approaches to the object under investigation.

We view this paper as a case study of what others have portrayed as differences between disciplines investigating the nature of mathematical thinking and development. In a recent issue in this journal,

Gauvrit argued that psychologists have a different focus; they often aim to uncover general processes in mathematical thinking, such as the role of working memory and arithmetic fluency, or to examine specific learning disabilities (e.g., dyscalculia). His summary yields an assessment each field has of the other: Psychologists underestimate the complexity of mathematics and

Gauvrit also underscored methodological differences. Whereas many psychologists conduct experimental studies to maximize internal validity,

As researchers, we could affiliate ourselves with these caricatures of psychology and mathematics education/

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 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 (

The second author (JP) works in a mathematics department and teaches future teachers of mathematics. As a

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” (

It became apparent to HPO that this student had difficulty grasping the meaning of the equal sign, at least in this specific context. HPO

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.

The conceptualization of relational thinking offered by

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

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

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

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;

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 strategies they have acquired or created in the past (i.e., the toolbox). The subsequent data analysis often becomes centered on outlining and cataloging the various strategies used, and, in turn, instruction becomes a matter of explicitly teaching these strategies so that students use them in appropriate ways.

Recently, however, the toolbox metaphor has been questioned by a number of researchers, who illustrate that there is more to solving mental computation tasks than the “simple” re-using of already predetermined strategies.

JP’s work is aligned with this perspective and draws on the enactivist theory of cognition to study mental computation solving processes (see, e.g.,

These initial discussions led to two specific local objectives for the research. HPO’s intention was to collect evidence that might demonstrate a relationship between mental computation and relational thinking, which could provide evidence of learning. JP’s objective was to capture in action and investigate solving processes generated in the context of mental computation tasks so that their emergent mathematical nature could be further investigated. Through this collaboration, both HPO and JP followed and addressed each other’s research approaches, techniques, and objectives, thereby generating an understanding, or curiosity, about how the differences in their disciplinary perspectives could enhance cross-fertilization of ideas and deepen their research.

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.

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.

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 (

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 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 same 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

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

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

Furthermore, when asked to solve mental computation tasks, children have been known to perform standard algorithms mechanically (

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 (

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.

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 ^{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 (

Item | Pretest | Posttest |
---|---|---|

1 | 347 + 108 = 108 + 347 | 228 + 319 = 319 + 228 |

2 | 82 × 3 = 82 × 2 + 83 | 156 × 4 = 156 × 3 + 157 |

3 | 10000 ÷ 4 = 5000 ÷ 2 | 5000 ÷ 4 = 2500 ÷ 2 |

4 | 117 + 56 – 117 = 58 | 4508 + 124 – 4508 = 124 |

5 | 25000 – 7899 + 1 + 7899 = 25001 | 24780 – 893 + 1 + 893 = 24781 |

6 | 8607 × 56 = (60 - 4) × 8607 | 1350 × 48 = (50 - 2) × 1350 |

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. 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

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}.

This section offers descriptions of how the data were treated, coded, and analyzed. These descriptions serve to explain and justify how conclusions were drawn from the data collected on the project.

HPO developed a rubric based on the frameworks of relational thinking outlined by

Discussing the rubric with JP, however, highlighted yet another disciplinary difference, this time with respect to analyzing data on mathematical thinking. Whereas HPO was (initially) bound by

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

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.

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.

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: (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 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,

Student | Before Unit |
After Unit |
||
---|---|---|---|---|

Item | Response | Item | Response | |

Brittany | True or False: |
Student computed amounts on each side of the equal sign and compared numbers (246 ≠ 247). | True or False: |
With no visible computations, the student wrote: “ |

Claude | True or False: |
Student computed amounts on each side of the equal sign and then wrote: “ |
True or False: |
With no visible computations, the student wrote: “ |

The implications of these results for practice are important, in HPO’s view. As

JP’s intention is seldom to report on all the strategies

For the task 741–75, Amy’s work is shown in

741 – 75 is like 700 – 75 + 41.

700 – 75 is like having 7 dollars and subtracting 3 quarters. I am left with $6.25. And, 6.25 is six-twenty-five, so I add 41 to 625.

To do so, I do 5+1 is 6, 2+4 is 6, and I have 600, so 666.

Amy’s presentation of her strategy on the board.

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 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

In contrast, students’

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.,

Have you heard the one about the educational psychologist and the

In this paper, we described a collaboration between an educational psychologist and a

Themes | HPO’s perspective |
JP’s perspective |
---|---|---|

Research Questions | ||

General research interests and objectives | Develop evidence to make credible recommendations for teaching practice | Make sense of the nature of the mathematics generated by students, solvers, and so on, through their activity |

Local research interests and objectives | Explore the link between mental computation and relational thinking; |
Investigate problem-solving processes in mental computation environments; |

Methods | ||

Design of intervention tasks | Aligned with specific strategies hypothesized to foster relational thinking | Grounded in mathematics, designed through a conceptual analysis about numbers and operations |

Mathematical content – equal sign | For algebraic preparation | No direct link between arithmetic and algebra; Engages in different mathematical domains (place holder vs. unknown/variable) |

Role of mental computation unit | Intervention | Observational data |

Instructional constraints | No standard algorithm and no presentation of the equal sign; |
To unleash solvers’ natural strategies; |

Data | Measure: written test administered before and after the intervention | Observational field notes of solving actions; |

Nature of the evidence | Aggregated data | Idiosyncratic strategies, from one person locally solving |

Rubric development | Aligned with theoretical frame of relational thinking | Aligned with the mathematical historical development in the discipline in relation to sense making and symbolizing |

Reporting | Results; |
Findings; |

Epistemology | ||

Views on knowledge | Positivist/empiricist | Postmodern |

Epistemological goals | Generalize results on links between mental computation and relational thinking | Generate ideas about mathematical solving processes |

Disciplinary differences that surfaced within our collaboration included research questions, methodological choices, and epistemology. First, our research questions underlined significant discrepancies in the way we each entered the research project, reminiscent of

If we mark sciences off from one another […] by their respective “domains,” even these domains have to be identified not by the types of objects with which they deal, but rather

Second, as

In somewhat analogous, but perhaps more nuanced ways, Berch’s epistemological dichotomy featured in our collaboration as well. More precisely, HPO assumed a positivist/empiricist orientation to the research through the coordination of theory and data, and JP a postmodern one, focused on emergence, connectedness, and situativity of events. The collaboration provoked confrontation in our disparate epistemological perspectives. Such tensions were salient as we grappled with issues on how the data would grant us understandings about mathematics, mental computation, and relational thinking, and the ability to explain them — that is, to generate scientific knowledge for our fields of study. Ultimately, for HPO, her objectives required data that would demonstrate at least a correlation between two constructs that were empirically observed: a necessary, but not sufficient, condition for establishing a causal connection. For JP, on the other hand, the intention was mainly to generate new ideas and new distinctions that would push forward or trigger the field’s knowledge about the themes under study.

Regardless of our seemingly discordant research questions, methodologies, and epistemological paradigms, our interactions permitted

With this in mind, then, we do not concur with Berch’s rather gloomy outlook on the apparent disciplinary rift between mathematics education and cognitive psychology – or educational psychology in our case. More concretely, we argue that there are more productive responses to what he calls a “developmental disconnection syndrome” between the two fields than a facile application of psychology to practices in 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

Echoing Piaget’s thoughts are those of

Our experience confirmed that in collaboration, the work of educational psychologists and

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

The core defining feature of relational thinking depends on an understanding of the equal sign (“=”;

Even with a “sameness” conception (

Students who think relationally engage in transformations that are justified, often implicitly, by properties of whole number operations. When asked to think relationally about 99 × 3, for example, students can, and often do (e.g.,

The transformations that occur in a mental computation are often achieved through implicit or explicit knowledge of number properties (

Key terms and their definitions from this literature are presented in

Term | Definition |
---|---|

Canonical equation | An equation, traditionally used in elementary and middle-school mathematics classrooms, where there are no operations on the right side of the equal sign. Examples include: |

Non-canonical equation | An equation, rarely used in elementary and middle-school mathematics classrooms ( |

Open-number sentence | An equation with a missing number. Examples include __ + |

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 × 4 (true) ( |

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” ( |

Relational view of the equal sign | The understanding that the equal sign represents an equivalence relation and means “the same as” ( |

Relational thinking | A form of reasoning that entails looking at numbers and expressions holistically and noticing relations among them ( |

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 ( |

The research was supported by the Social Sciences and Humanities Research Council of Canada (410-2011-1310; 430-2012-0578; 435-2014-1376) and the Fonds québécois de la recherche sur la société et la culture (FQRSC, 164724).

We would like to thank Emmanuelle Adrien and Deborah Nadeau for their diligent assistance in collecting and coding the data on this project. We are also grateful to Jean-François Maheux for producing the drawing in the figure.

In French, the term “

HPO teaches in an elementary teacher education program housed in a child development unit at a large university in Canada. The program is not housed in a department of curriculum and instruction, as is traditionally the case in US institutions.

Other examples of this sort, where

Ways to maximize what is called the internal validity of a study have been covered extensively elsewhere (e.g.,

Here is another disciplinary difference, where explanations for JP follow mathematical definitions, where “an explanation is the not the quest for underlying causes, but the

One additional discussion between HPO and JP that occurred concerned the nature of the verbal explanations as data and its reliability for gaining insight into students’ thinking. Indeed, links between the verbal reports and the “actual” doings of a subject have been questioned for a long time in psychology (see e.g.,

The expression “findings” is used instead of “results” to describe the

Note that these strategies are not “new” in the sense that nothing similar has been attempted before in mathematics or even by Amy herself, but that they are generated

The authors have declared that no competing interests exist.

Both authors contributed equally to writing this paper and to the research reported.