The aim of this dissertation was to investigate how 5th grade students might be taught about instantaneous speed. To answer this question a design research approach was followed, which was executed in a series of three design experiments, and was preceded by a small number of one-on-one teaching experiments. All of which was reported upon in the preceding chapters. However, instead of giving a continuous narrative detailing this design research, this dissertation focused on four separate studies that emerged from the design research project. In this chapter, a summary of these studies is given, followed by a discussion of the over-all findings.

Summary

Introduction

The aims and scope of this thesis are elucidated in Chapter 1. In answer to the call for a new STEM education in primary school for the 21st century (Gravemeijer, 2013; Gravemeijer & Eerde, 2009; Keulen, 2009; Léna, 2006; Millar & Osborne, 1998)—innovative STEM education should be inquiry-based, ICT-rich, close to students’ world-view, and able to integrate in the primary school curriculum—, a design research project was started to answer the research question How to teach instantaneous speed in 5th grade? The topic of instantaneous speed was chosen because the interpretation, representation, and manipulation of dynamic phenomena are becoming key activities in the information society and is firmly rooted in the realm of STEM and key to a better understanding of dynamic phenomena is the concept of instantaneous rate of change. In the context of primary school the term “speed” is preferred because students of this age will not have developed the level of abstraction associated with the use of “rate of change” in the literature.

Design research is well-suited to explore how to teach topics in earlier grade levels than usual (Kelly, 2013), and instantaneous speed is conventionally taught first in a calculus course, which is not part of the primary school curriculum. Design research is a reaction to a perceived gap between educational practice and research (Eerde, 2013; Reeves, McKenney, & Herrington, 2010; The Design-Based Research Collective, 2003), which is to be bridged by exploring theoretical issues regarding students’ learning processes through studying students’ actual learning processes as they happen in a real classroom instead of in a laboratory setting (Collins, Joseph, & Bielaczyc, 2004). It is an interventionist process of iterative refinement that takes real-world classroom practice into account to create some instructional artifact and a theory on how that artifact works (Bakker & van Eerde, 2013; Barab & Squire, 2004; Cobb, 2003; Plomp, 2013; Reimann, 2011; The Design-Based Research Collective, 2003). It further aims at exploring theoretical issues that go beyond the scope of a particular design research project (Barab & Squire, 2004).

In this thesis, the approach to design research outlined in Gravemeijer & Cobb (2013) was applied to develop a local instruction theory (LIT) on teaching instantaneous speed in grade five. A design research project starts by formulating the learning goals, the instructional starting points, and an initial LIT, based on the literature and any other source that might contribute to the researcher’s understanding of students’ prior conceptions (Chapter 2). These other sources, such as text books, teacher guides, prior experiences, and basically everything deemed useful by the researchers, feature strongly in the starting-up phase because design research is often applied in areas without much relevant literature (Gravemeijer & Cobb, 2013).

Once formulated, the LIT is elaborated, adapted, and refined in multiple design experiments (Cobb, 2003; Gravemeijer & Cobb, 2013) (Chapter 4). A design experiment consists of three phases:

1. The LIT is (re)formulated and, based on that LIT and any other sources deemed relevant by the researchers, an instructional design is developed.

2. That design is tried and adapted in a teaching experiment consisting of a sequence of micro design-cycles of (re)developing, testing, and evaluating instructional activities and materials.
   

3. During the retrospective analysis (Chapter 3), an analysis of what happened during the teaching experiments informs the refinement of the design, and an analysis why this happened informs the refinement of the LIT.

This refined LIT is the starting point of the next design experiment. This process yields both a working prototype and a theory on how that prototype works. In line with the theory-driven nature of design research the focus in this thesis is on the LIT rather than on the prototype. The proposed LIT is presented in Chapter 5.

The design research project consisted of four partial projects: a starting-up phase followed by three subsequent design experiments. As design research is a process of iterative refinement, experiences gained and results found in earlier partial projects are reflected and apparent in the 3rd design experiment. As a result, this thesis focuses on this last design experiment and its substantive relationships with the preceding partial projects. This thesis is not set up as a monograph offering a continuous narrative detailing the design research project. Instead a format is chosen that consists of four separate studies that did emerge from the design research project:

• In the starting-up phase, 5th grade students’ level of covariational reasoning was investigated by conducting a one-on-one teaching experiment with nine 5th grade students and analyzing their behavior using the covariation framework (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002).

• The generation of new theory in design research through abductive reasoning was explored in the context of the retrospective analysis of the third design experiment.

• In an effort to contribute to the codification of educational design, the researchers’ own learning process was discussed as an example of how educational-design knowledge and practices might be documented.

• Finally, the proposed LIT was formulated in terms of students’ key learning moments that emerged from the data collected during the whole design research project.

These studies are discussed next. Furthermore, to get a complete overview of the design research project, after discussing the starting-up phase, a short overview of the three design experiments is included as well.

Starting up the design research project

The starting-up phase is described in Chapter 2. To start up the design research, two avenues were taken to explore 5th graders’ prior conceptions of speed, a literature review and a small-scale study, both of which are discussed next. Finally, the first ideas regarding a LIT on instantaneous speed are formulated.

Literature review on students’ conceptions of speed and teaching calculus-like topics early

In both the literature on primary school students’ conceptions of speed and the primary school curriculum, speed is treated as average speed and interpreted as a ratio of distance and time. In general, primary school students have problems with relating distance and time to speed (Groves & Doig, 2003), and 5th grade students have trouble developing an understanding of speed as a rate (Thompson, 1994). Stroup (2002) attributes these problems to conventional approaches to teaching speed that favor ratio-based understanding of speed over students’ intuitive understanding. Instead, he argues for developing students’ qualitative understanding of speed. This “qualitative calculus” approach deviates further from conventional approaches by starting with exploring non-linear situations of change (Stroup, 2002). In most other studies on primary school students’ conceptions of speed instantaneous characteristics have not been explored. Because instantaneous speed is usually taught first in a calculus course, the literature review was extended to teaching calculus-like topics early on in the mathematics curriculum. Studies on teaching calculus-like topics in primary school (Boyd & Rubin, 1996; Ebersbach & Wilkening, 2007; Galen & Gravemeijer, 2010; Kaput & Schorr, 2007; Nemirovsky, 1993; Nemirovsky, Tierney, & Wright, 1998; Noble, Nemirovsky, Wright, & Tierney, 2001; Stroup, 2002; Thompson, 1994) share two characteristics: computer simulations and Cartesian graphs.

Research suggest that young students are able to explore speed mathematically using computer simulations, which are a natural fit to explore dynamic phenomena. Computer technology enables inquiry-based learning approaches because it allows students to explore more authentic, realistic, and complex problems (Ainley, Pratt, & Nardi, 2001; Chang, 2012). However, because they can function as black boxes, Gravemeijer, Cobb, Bowers, & Whitenack (2000) argue that computer simulation only induce adequate learning processes when embedded in a suitable instructional sequence. Furthermore, if students are sufficiently supported in using graphs, they appear to be able to express their understanding through graphs, even when they lack graphing experience (Leinhardt, Zaslavsky, & Stein, 1990). More specifically, social-cultural approaches to graphing (Roth & McGinn, 1997) and (guided) reinvention approaches (diSessa, 1991) have been proposed.

Although the results of the literature review did not offer sufficient information to formulate an initial LIT for teaching instantaneous speed, it did offer enough basis for developing and performing a small-scale study to explore 5th graders’ understanding of speed in situations with two co-varying quantities.

A small-scale study to explore 5th graders’ understanding of speed

To explore 5th graders’ understanding of speed in a situation with two co-varying quantities in the context of filling glassware, a short instructional sequence and computer simulation was developed. The students were asked to perform similar activities with a highball glass, a cocktail glass, and an Erlenmeyer flask:

1. turn the glass into a measuring cup by dragging hash marks to the right place on the glass

2. evaluate the solution

3. draw a graph of filling that glass

4. and evaluate the graph by comparing it with the computer-drawn graph.

Eight one-on-one teaching experiments (Steffe & Thompson, 2000) with above average performing 5th graders were performed. The videos of the experiments were transcribed and, together with the screen-captured computer sessions, analyzed using the covariation framework (Carlson et al., 2002). Students’ utterances were coded for one of five developmental levels of covariational reasoning defined in the covariation framework. These levels ranged from a basic understanding of co-varying quantities to an understanding of a dynamic situation in terms of instantaneous speed. It was found that the students came to reason at levels two and three: they were quite capable in estimating the relative amount of change at certain points or intervals. The students almost never talked about (instantaneous) speed, however, and when they did it had to be classified as pseudo-behavior (Vinner, 1997).

However, it was also concluded that the covariation framework did not offer the grain size nor the focus on instantaneous aspects of change to be a good instrument to study 5th grade students’ conceptions of instantaneous speed. A serious limitation of the usefulness of the covariation framework in primary school appears to be in the limited vocabulary and graphing skills of the primary school students. Conversely, the investigation with the framework was valuable in that it pointed to the importance of graphing skills and the fact that 5th grade students’ skills were insufficient.

Starting points for an initial LIT

Nevertheless, the experiences during the one-on-one teaching experiments and the findings of the literature review together allowed for the formulation of starting points for the initial LIT. This initial LIT aimed at expanding on Stroup (2002)’s qualitative calculus approach by supporting 5th graders in developing both a qualitative and a quantitative understanding of speed in the context of filling glassware, based on the following line of reasoning:

Assuming that students’ intuitive notion of instantaneous speed is close to a historical notion of instantaneous velocity that did not build on the complex process of taking the limit of average speeds, students might come to equate the instantaneous rising speed in the cocktail glass with the constant rising speed in an (imaginary) highball glass. After exploring the relation between a glass’ shape and the rising speed, students may realize that the instantaneous rising speed in the cocktail glass at a given height is equal to the constant rising speed in an highball glass that has the same width as the cocktail glass has at that height. This would eventually enable students to quantify the instantaneous rising speed in a point by computing the constant rising speed of the corresponding highball glass. Building on the correspondence between a highball glass’s graph and the steepness in a point of a curve, the students may construe the tangent line in a point of a curve as an indicator for the instantaneous speed in that point. Given students’ familiarity with constant speed and graphing linear situations, students then may quantify the instantaneous speed by computing the rise over run of the tangent line.

From the one-on-one teaching experiments it was learned that conceiving filling a highball glass as a linear process was self-evident for the students. This was also the case for the linearity of the graph. The cocktail glass revealed that they easily fell for the linearity illusion, which they would overcome, however, when they saw the simulation. Important was that the students showed to understand the mechanism; they realized that the rising water level in the cocktail glass slows down, because the glass gets wider. Graphing proved ambivalent, they could not draw the correct graph, but they nevertheless seemed to appreciate the curve—even though they could not explain it.

Overview of the design experiments

Design experiment 1

The first design experiment revolved around an instructional sequence that started with students making measuring cups from a highball glass, a cocktail glass, and an Erlenmeyer flask in a computer simulation. The second lesson was dedicated to exploring speed in the highball glass through drawing its graph and computing its speed. Only then, in the third lesson, would the attention shift to the problem of instantaneous speed. It was expected that, once the students had realized that the speed in a cocktail glass and a highball glass would be the same when the widths are the same, would come to see that they could determine the instantaneous speed at a specific point in the cocktail glass by computing the constant speed of a (computer-drawn) virtual highball glass with the same width as the cocktail glass at that point. Next, by directly linking the virtual highball glass with its graph as a tangent line on the cocktail glass’ curve, the students were expected to come to accept the tangent line as a tool to measure instantaneous speed in a graph. It was anticipated that this would enable them to reason about speed in other contexts as well.

Through Design Experiment 1, it became even more clear that the power of the context of filling glassware lies in the fact that it offers the students a powerful theory to reason about the covariation process on the basis of their understanding of the relationship between a glass’ width and its speed. It was further learned that the students had no problem with answering the question, “When is the speed in the cocktail glass equal to the speed in the highball glass?” This answer presupposes that one thinks of speed in a point. The students could handle the tangent-line-tool, but their level of understanding was doubted. Further, they still did not manage to come up with a continuous graph; it is believed that starting with the highball glass might have put them on the wrong track.

Design experiment 2

In design experiment 2, to overcome the problem of a discrete learning environment, it was decided to remove the measuring cup activities. Furthermore, the students’ learning process was to revolve around the non-linear situation of filling the cocktail glass. To increase conceptual discussion about speed a modeling-based learning (MBL) approach was selected to allow students to express their understanding more explicitly while presenting, discussing, and evaluating their models in class.

The instructional sequence started by asking the students to model filling a cocktail glass, and to improve their models after exploring the situation in a computer simulation. They were not expected to start using Cartesian graphs, but once the graph was introduced by the teacher, it was expected that they would see its value and be able to use it to predict the water level height at any moment and to connect its shape to their image of filling glassware. They would extend their understanding of the relationship between a glass’ width and speed to include the steepness of the curve. After a brief exploration of computing constant speed in the highball glass, the students would be guided towards construing the highball glass as a tool for measuring instantaneous speed by exploring when the speed is the same in both glasses. The students were expected to come to see that the cocktail glass’ curve is as steep as the highball glass’ straight line in the point they have the same width. Finally, moving to the context of toy-car racing, the students were expected to explain a race given a graph, indicate where the car went fastest or slowest, and use the tangent line to quantify its speed.

The (small scale) Design Experiment 2 taught the value of MBL; the modeling activity did indeed help to make the students’ thinking visible and topic of discussion. Again the students’ lack of understanding of, and fluency with, measures of speed revealed itself. Again there was no success in developing continuous graphs, which could suggest that maybe students of this age are chunky thinkers (Castillo-Garsow, 2012). Nevertheless, the students construed the tangent line to the cocktail glass’ curve parallel to the graph of the highball glass as an indicator of the speed in a given point.

Design experiment 3

Design Experiment 3 was a turning point because the students in one of the two classrooms showed that they were able to invent a continuous graph by themselves. The catalyst proved to be a critical reflection on the shape of the segmented-line graph, while using their understanding of the relationship between a glass’ width and speed, and thus of the character of the covariation. Thus, they were not chunky thinkers; they were continuous thinkers (Castillo-Garsow, 2012) who have difficulty with graphing. In addition, it showed that there is another road to the continuous graph, which was followed by the students in the other classroom where they did not have the benefit of a segmented-line graph. These students came to understand the continuous graph via shrinking the intervals of a bar chart. Next the students construed the tangent line as an indicator of the speed at a given point, by drawing a line parallel to the linear graph of the highball glass. It was further learned that the students could build on those ideas to come to grips with the shape of the graphs of a cooling process by assuming that the tangent line in a given point depends on the difference between the actual temperature and the final temperature.

However, even though the students had a tool to measure instantaneous speeds, they did not yet develop a sound understanding of how to quantify speed. They were hampered by the fact that they did not have a sound basis for calculating speeds; they clearly needed more experience with quantifying constant speeds in a variety of ways (with different units), and relating those with the corresponding graphs.

The generation of new explanatory conjectures in design research

Chapter 3 focuses on one of the key aspects of design research: the generation of new theory during the retrospective analysis. It is shown how a process of abductive reasoning during the retrospective analysis of the third design experiment led to the generation of the conjecture that primary school students come to the classroom with a continuous conception of speed and only switch to discrete reasoning because of a lack of means for visualizing continuous change. This, in turn, led to the realization that average rate of change is a hindrance rather than a necessity in teaching instantaneous rate of change in primary school.

Abductive reasoning is triggered by an unexpected event (Fann, 1970) during the teaching experiments. These surprising facts are treated as an indication of a misalignment between the researcher’s understanding of (anticipated) students’ learning processes and their actual learning processes. Abductive reasoning tries to resolve this conflict by generating new explanatory conjectures. Subsequently, the data collection is re-examined to determine the extent to which these conjectures are supported or have to be rejected. In Chapter 3 it is illuminated how abduction plays a specific role in design research. To that end, the teaching experiment leading up to the unexpected event is described in detail, followed by a two-step retrospective analysis.

Sketching the unexpected event and the context leading up to it

The third design experiment started by reformulating the LIT based on the findings of the previous design experiments. Based on this LIT a four-lesson instructional sequence was developed around this LIT, which was tested in two gifted 4th-6th grade classrooms taught by the same teacher. During the second lesson, a surprising difference between what happened in the two classrooms resulted in process of abductive reasoning in the retrospective analysis.

In the first lesson, the students modeled filling a cocktail glass four times to allow for an iterative improvement. During these activities, although many of the first models were quite realistic depictions of the situation, the discrete snapshots model became taken-as-shared in both classrooms. At the start of the second lesson the students were asked to create a minimalist model based on an earlier model. Most of these minimalist models were discrete representations, but in each classroom there was one graph-like model with continuous characteristics. At this point, what happened in both classrooms deviated: in classroom 1 (C1) this continuous graph-like model was discussed in class, in classroom 2 (C2) it was not.

C2 followed the anticipated learning trajectory. Under the guidance of the teacher, the students condensed the discrete minimal models into a bar graph first. The bars came to signify water heights in the glass at specific moments in time. Next, these bars were connected by arrows, signifying the change between the bars. Finally, after introducing the computer-drawn curve, the students were expected to come to see the curve as signifying both the changing value and the speed. In C1, however, the students’ actual learning trajectory was different. While discussing the student-drawn graph-like minimalist model, which was a segmented straight-line graph, the students argued that the straight line segment had to be a curve: they invented the curve by themselves. This unexpected event led to a process of abductive reasoning during the retrospective analysis.

Retrospective analysis

The retrospective analysis followed a two-step method based on Glaser and Strauss’s (1967) comparative method, in particular, the elaboration of Cobb & Whitenack (1996) on this method was used:

1. conjectures about What happened? were formulated and tested against the data. This process resulted in six conjectures.

2. conjectures about Why did this happen? are formulated and tested against the data, which resulted in the generation of the following new explanatory conjecture: The students come to the classroom with a continuous conception of speed. They only switch to discrete reasoning because of a lack of means for visualizing continuous change.

This retrospective analysis showed that students’ reasoning was grounded in continuous reasoning, while discrete reasoning functioned as a tool to get a handle on continuous processes. Furthermore, students easily reasoned about constantly changing speed, which implies a conception of instantaneous speed. This observation triggered a process of abduction at the design research level, which generated the question: “Do the students ever use average speed?” With one exception, it never showed that the students were thinking of average speed. This stands in sharp contrast with the common practice of starting instruction on speed by introducing average speed. Starting with average speed is problematic: it promotes discrete thinking and could be the source of Castillo-Garsow (2012) problematic chunky thinkers. It would make more sense to explore constant speed, which subsequently can be connected to the students’ notion of instantaneous speed.

Generalizability and the generating new ideas in design research

Mark that the generalization of these findings are possibly limited by the uniqueness of the classroom situation—it was tried in two mixed gifted 4th-6th grade classrooms, after all—and the fact that the line graph was introduced by students in C1 seemed more a lucky accident than a controlled act on behalf of the teacher. On the other hand, once the line graph was presented, the teacher did recognize its didactical value and was able to guide the discussion successfully to have students deepen their understanding. In a sense, the teacher was the perfect match: He was one of few primary school teachers who had followed a calculus course in high school. It cannot be expected from the average teacher to have as much insight and experience with calculus-related topics and graphs: these topics are simply not part of the primary school curriculum.

Admittedly abduction does not offer the same rigor as deduction and induction. However, the primary goal of design research is to find out how things work, not to establish for a fact how things are. By being explicit about the abductive argument underlying the development of the LIT, special attention is paid to the justifications common to design research, such as ecological validity, trackability (Smaling, 1990), process oriented causality (Maxwell, 2004) and consilience (Gould, 2011). With respect to abductive reasoning, the unexpected events that trigger it are an indication that more attention should be paid to letting the object speak (Smaling, 1992) which might reveal a clear misalignment between researchers’ prior understanding of students’ learning processes and the actual learning processes. To resolve this distortion, the researchers have to re-examine their prior conceptions while considering students’ perspective more strongly and formulate new explanatory conjectures. Subsequently, if these new conjectures can be grounded in the data collected, and are added to the LIT, the LIT itself becomes more objective as it does do better justice to the students’ learning process.

Design research as an augmented form of educational design

Design research builds on educational design to create both a product and a theory detailing how that product works. Due to the theory-driven nature of design research, that product often does not evolve beyond a prototype (Burkhardt & Schoenfeld, 2003), which limits the utility of the theory. Schoenfeld (2009) recommends project teams that balance expertise in design and research, but that might not be a realistic option for many design research projects. This suggests that researchers have to acquire practical design skills, but as the educational design community lacks both an institutionalized form of schooling and professional literature (Schunn, 2008), this is not straightforward. To overcome this problem, codification of design practices has been proposed (Schoenfeld, 2009), to which design research can play a role.

To develop theory implies a commitment to strengthen the credibility of the theory by justifying its claims and a commitment to allow other researchers to assess the trustworthiness of the process leading up to those claims. The latter can be satisfied by enabling outsiders to retrace the process by which those claims are produced (Smaling, 1990), which means to give a detailed account of the design research process and the researcher’s own learning process embedded in it. It is argued that this method of reporting on the learning process of the researchers can function as a paradigm for the way educational designers might want to document their practices and knowledge. To offer an example, the researcher’s own learning process is elaborated in Chapter 4 by tracking the development of the instructional sequence from the starting-up phase through the three subsequent design experiments. Both the starting-up phase and the three design experiments have been summarized in a previous section.

Empirically-grounded theory and educational design

The development of the prototypical instructional sequence is illustrated in terms of the researcher’s own learning process, which encompassed both the design decisions and the rationale for those decisions. They offer a framework of reference on the basis of which teachers may adapt the instructional sequence to their own classroom and they offer support for instruction design and further theory development. Such a framework of reference may take the form of a LIT that offers a rationale for the prototypical instructional sequence that is developed alongside the LIT. In this manner, design research offers a different kind of support for teachers than most textbooks do.

Apart from offering an example of documenting instructional design decisions and practices, Chapter 4 also elaborates on what makes design research credible, even though it does not follow the classical research method of an (quasi-)experimental design. These theoretical findings can be substantiated by the virtual repeatability of one’s research by other researchers (Smaling, 1990). The goal of design research is to generate a theory on how the intervention works. This kind of research employs a process-oriented perspective on causality, stating that, in principle, causal claims could be based on a single case. When aiming at LITs, a single case is the classroom as a whole. To justify that there is a causal relation two methods are used in design research: validating existing conjectures and generating new explanatory conjectures.

Conjectures that are confirmed by the students’ actual learning process remain part of the LIT and are tried and refined again in the next design experiment. As a result, conjectures are confirmed or rejected in multiple different situations, offering a form of triangulation that adds to the understanding of students’ learning processes in terms of these conjectures. Those observations enable to develop some theories about the mechanisms that were at play here. In addition to this, new explanatory conjectures are generated through abductive reasoning which could be tested on the available data. However, only claims about the students who participated in the experiments can be made and the claims have to be grounded carefully in the observational data to make sure that the conclusions are valid for the majority of the students in the teaching experiments. It was judged that this was predominantly the case in the classrooms where the experiments were carried out. In addition, one usually also carries out a retrospective analysis, which results limit themselves to statements about the actual design experiments. However, a design experiment can be treated as a paradigm case. The goal then is to come to understand (the role of) the specific characteristics of the investigated learning ecology in order to develop theoretical tools that make it possible to come to grips with the same phenomenon in other learning ecologies. The LIT offers a theory of how the intervention works, which teachers and instructional designers can adjust and adapt.

To develop theory implies a commitment to the credibility by justifying the theoretical claims and a commitment to allow other researchers to assess the trustworthiness of the process leading up to those claims. The latter commitment can be satisfied by trackability (Smaling, 1990), which means giving a detailed account of the design research process and the researchers’ own learning process embedded in it. This will encourage design researchers to pay attention to practical issues of design and will allow design researchers to create theories that are more practical applicable. Design research can also be taken as a paradigm that may show educational designers the value of documenting design decisions and anticipated students’ learning processes. Design research can be seen as an augmented form of educational design, which offers educational designers indications on how to handle the issue of documenting their practices and knowledge.

A proposed LIT on teaching instantaneous speed in 5th grade

By capitalizing on the results of the various design experiments a LIT on teaching instantaneous speed in 5tg grade is proposed in Chapter 5. This chapter starts by summarizing the theoretical background and detailing the methodology used in this research project: design research. More specifically, the theoretical underpinnings of the LIT are elaborated in terms of the three instructional design heuristics of Realistic Mathematics Education: guided (re)invention, didactical phenomenology, and emergent modeling. The proposed LIT is based on the patterns in students’ learning processes that were identified in the data of the various design experiments. This allowed for a triangulation on two levels. At the level of a single design experiment a multitude of data is collected and used to validate and generate conjectures, and these conjectures are validated or refuted in multiple design experiments, strengthening their empirical basis.

Reporting on the results, basically Cobb et. al.‘s (in press) recommendation for an argumentative grammar for design experiments was followed, which requires the justification of the theoretical findings of a design experiment by a) showing that the students’ learning process is due to their participation in the design experiment, b) describing that learning process, and c) enumerating the necessary means of support for that learning process to occur. The first requirement is self-evident as 5th graders are not taught on this topic. The other two requirements are split into a documentation of the key learning processes, and a separate description of the envisioned learning process and the means of support that learning process.

Students’ key learning moments

The patterns that emerged with respect to students’ key learning moments are put in context by a description of students’ instructional starting points. These starting points were the same in each teaching experiment: the students had limited graphing experience, they had trouble computing speeds, and they were thinking in terms of instantaneous speed from the start. There is no indication that the situation will be much different in other classrooms. Given these starting points, several key learning moments were identified in the data. Notably, the students were familiar with linearity, but broke through the linearity illusion (Bock, Dooren, Janssens, & Verschaffel, 2002) easily when seeing the cocktail glass fill up. They understood the relationship between a glass width and its speed, allowing them to realize when the speed in the cocktail glass and highball glass is the same. Despite the students’ limited graphing experience, once the curve was introduced—in one classroom the students even invented it themselves—they accepted it as a better model than the discrete snapshots models they had created earlier. They were able to construe the tangent line as an indicator of the speed in a given point by combining the cocktail glass’ graph and the graph of a highball glass.

Proposed LIT

Based on these key learning moments, the proposed LIT is formulated in terms of the postulated students’ learning processes and the potential means of support necessary for those learning processes to emerge. The students’ learning processes may be summarized as follows:

Given a cocktail glass, students are given the task to make a drawing of how the water height changes when the glass fills up. After observing it fill up, they notice that the water level rises slower and slower, and they realize this is the result of the glass’ increasing width. This realization allows the students to form valid expectations about the process of filling glassware and they come to depict it both as a discrete bar chart as well as a continuous graph. It is expected that the students link the curve of the continuous graph with the continuous change of the speed of the rising water: at every moment that speed is different. From this perspective, students come to interpret the speed of rising as an instantaneous speed.

That conception is deepened both qualitatively and quantitatively by exploring two avenues of thought. First, by comparing the speed in the cocktail glass with the constant speed in a cylindrical highball glass in order to answer the question when the water rises with the same speed in both glasses. The constant speed of an imaginary highball glass becomes a measure for the instantaneous speed in the cocktail glass. Second, building on that understanding, trying to measure speed in a graph by interpreting the straight line graph of the highball glass as a tangent line on the curve of the cocktail glass. Throughout this process, the representations of the speed in the highball glass act as an emergent model of measuring instantaneous speed. Finally, students’ understanding of speed in terms of graphs and tangent-line can be translated to other contexts as well.

For this learning process to emerge, three potential necessary means of support are identified:

1. The context of filling glassware is well-suited to explore covariation because it is very familiar to students and it visually connects constant speed (represented by a highball glass) to instantaneous speed (in a cocktail glass). In a concrete setting, picturing the highball glass together with the cocktail glass offers a very powerful image that allows students to construe a measure for instantaneous speed based on constant speed. In other contexts, this is not straightforward.

2. Computer simulations combined with graphs supported the students in developing both a discrete and continuous understanding of graphs despite students’ limited graphing experience. Furthermore, the highball-glass-tool allowed the students to put their invention of the highball glass as measure for instantaneous speed to the test, making it more tangible. Finally, the tangent-line-tool allowed students to quantitatively explore filling glassware as well as other contexts. It allowed an inquiry-based learning approach.

3. The modeling-based learning approach not only gave the teacher indirect access to the students’ mental models, it also allowed him to better support students in constructing a deeper understanding of graphs in relation to speed. In one classroom it even enabled students to reinvent the curve.

Characterizing the proposed LIT and its application

Unique to the presented approach to teaching instantaneous speed is that it circumvent the troublesome limit concept while supporting students to coming to understand an quantify instantaneous speed. Students are supported in expanding their intuitive understanding of instantaneous speed by having them compare constant speed with instantaneous speed in a computer simulation. This means that compared to conventional approaches, average speed plays a minimal role. Furthermore, the tangibility of the relationship between the constant speed in a highball glass and the instantaneous speed in a cocktail glass allows students to develop a quantitative measure of instantaneous speed that relies on their understanding of constant speed. For, despite their limited experience with graphs, students were able to translate this understanding to graphs, accepting the curve as a fitting model for their intuitive understanding of instantaneous speed. By linking the highball glass’ graph to the tangent line at the cocktail glass’ graph, a curve, the students came to accept the tangent line as a quantitative measure of instantaneous speed. From there, the students were able to shift their understanding to the context of warming and cooling as well, suggesting they developed a more general understanding of instantaneous speed and graphs.

Although the proposed LIT is considered a potential viable theory, it clearly cannot easily be used in regular classrooms. Apart from the fact that the corresponding instructional sequence has to be elaborated to cater for the students’ limited graphing abilities and limited understanding of quantifying constant speed, there are issues of teacher professional development. For others to adapt it to their own situation, the proposed LIT should be transferable (Smaling, 2003). This means that the findings should both be plausible for others and that they should be able to ascertain the potential applicability to their situation. Some aspects of the proposed LIT are problematic in this regard, implying potential avenues for further development and research. Therefore, a collaboration between researchers, educational designers, and teachers is envisioned to further explore teaching instantaneous speed in grade five.

The proposed LIT and the aims of the research project

After summarizing the various studies this thesis is comprised of, the focus now shifts towards discussing the findings of the design research project. The proposed LIT for teaching instantaneous speed in 5th grade is discussed in light of the aims put forth in the Introduction of this thesis. This research project was started to answer a call for innovative STEM education in primary school (Galen & Gravemeijer, 2010; Gravemeijer, 2009; Léna, 2006; Millar & Osborne, 1998) suitable for the information age, which should be inquiry-based, close to students’ world view, ICT-rich, and integrated into the primary school curriculum. These characteristics and their implications gave rise to four themes characterizing the proposed LIT and its potential application:

• an innovative approach to teaching instantaneous speed,

• MBL and teachers’ professional development,

• computer-enhanced learning in the information society,

• and bridging the gap between students’ world-view and STEM.

Before discussing these four themes, it is noted and emphasized that the proposed LIT is not intended as a ready-made product but as a potentially viable theory for other researchers, educational designers, and maybe teachers to apply and adapt to their situation. This aspect of the proposed LIT is intertwined throughout this discussion and will be elaborated further when discussing design research in the next section.

An innovative approach to teaching instantaneous speed

This research can be placed in a long tradition of calculus reform (Tall, Smith, & Piez, 2008). In particular, it can be placed among initiatives to teach calculus-like topics in primary school (Boyd & Rubin, 1996; Ebersbach & Wilkening, 2007; Galen & Gravemeijer, 2010; Nemirovsky, 1993; Nemirovsky et al., 1998; Noble et al., 2001; Stroup, 2002; Thompson, 1994). According to Tall (2010), calculus reform at large can be characterized by adapting and improving conventional approaches to calculus with the use of ICT (Tall, 2010). Building on Stroup’s (2002) qualitative calculus, the proposed LIT deviates significantly from conventional approaches to rate of change in that it tries to support students in developing a non-ratio based understanding of rate. More so, and this makes the proposed LIT truly an innovative approach to learning instantaneous speed, it circumvents the problematic limit concept (Tall, 1993, 1997, 2009) all together while still enables students to quantify instantaneous speed.

The proposed LIT tries to support students in expanding their intuitive understanding of instantaneous speed by having them compare constant speed with instantaneous speed in a computer simulation. In the context of filling glassware, the tangibility of the relationship between the constant speed in a highball glass and the instantaneous speed in a cocktail glass allows students to develop a quantitative measure of instantaneous speed that relies on their understanding of constant speed. As a result, average speed takes a less prominent place in the proposed LIT than in conventional approaches to instantaneous speed, which is fortunate because students of this age do have a limited understanding of average speed although it is part of their curriculum.

Moreover, although the proposed LIT is for teaching instantaneous speed in grade five, it might be interesting to explore the usability of its innovative approach to instantaneous speed in higher grade levels as well. A deeper understanding of instantaneous speed might provide students with a strong conceptual basis for coming to understand the limit concept. It might make students more susceptible to the ideas explored in a conventional calculus course.

Modeling-based learning and teachers’ professional development

The characteristic of inquiry-based learning—students’ learning is supported by involving them in real-life situations with an emphasis on questioning, hypothesizing, and experimenting (Léna, 2006; Osborne & Dillon, 2008; Rocard et al., 2007)—is covered in the proposed LIT by means of using MBL. MBL is a form of inquiry-based learning built around the idea that modeling is a core activity of science and developing knowledge. From that perspective, MBL seems a natural basis for STEM education. Because it is impossible to know ones mental model, the only way to get any access to the students’ mental models is having them express their understanding (Coll, France, & Taylor, 2005; J. Gilbert & Boulter, 1998). An expressed model can be presented, discussed, and evaluated in class, allowing students to refine their mental models, all the while giving the teacher (and researchers) indirect access to their thinking. Although tangible (scale) models seem to be more common in primary education, with “model” is emphatically also meant more abstract models, such as a flowchart, a Cartesian graph, or a simulation (J. Gilbert, 2004; S. Gilbert, 1991).

Key to MBL are suitable classroom social norms that support students to freely express their opinions, ask questions, indicate their doubts or disagreements, and explore alternatives. The teacher plays an important role in creating and maintaining such a supportive learning environment, which is a matter of concern. First of all, Lehrer & Schauble (2010) warns that modeling is not straightforward. It is a skill that students and teachers alike have to develop through modeling. However, because MBL is different from conventional approaches to instruction in primary education, teachers will need support to start with MBL. Secondly, because primary school teachers do not have much expertise teaching STEM and have a poor understanding of STEM (Léna, 2006), modeling as an activity might not be well-understood. In either case, however, we think that the proposed LIT could function as a starting point to introduce primary school teachers to MBL, for example when embedded in a professional development effort.

Computer-enhanced learning in the information society

In establishing and maintaining a suitable classroom culture for MBL there is a role for ICT. In particular, flexible and interactive computer simulations enable students to explore phenomena they normally do not have access to (Chang, 2012), therefore enabling students to solve more meaningful, complex, and realistic problems (Ainley, Nardi, & Pratt, 2000). Indeed, although students could explore the context of filling glassware using concrete glasses and water, by offering them a computer simulation of filling glassware, they were able to repeatedly explore filling various glasses without creating a wet mess. The computer simulations offered the students a save environment to explore the situation in detail, formulate and test hypotheses, and discuss, evaluate, and critique their ideas by using the computer simulation in their arguments.

However, what made the computer simulations a truly necessary means of support in the proposed LIT for teaching instantaneous speed was its support for students to construct a dual understanding of graphs and for them to discover and use tools to quantify instantaneous speed. The computer simulation could draw both a line graph and a bar graph. This emphasized and supported the dual understanding of graphs the students were to construct: both as a way to reason about individual data points and a way to reason about the whole continuous process of filling a glass. Moreover, by increasing the number of bars shown, slowly the shape of the curve comes to the fore, offering the students a powerful image to support them in accepting the curve as a good fit to describe their understanding of the situation. It creates an affordance for shuttling back and forth between a continuous and a discrete image of change.

Similarly, once the students construed the highball glass as a measure for instantaneous speed, the highball-glass-tool not only allowed them to put their invention to the test, it also made their theory tangible. The tangent-line-tool allowed students not only to quantitatively explore filling glassware through graphs, but other contexts as well. More interestingly, however, by moving the mouse cursor over the curve while using the tangent-line-tool, the changing speed is aptly visualized, strengthening students’ understanding of the relationship between speed and steepness of a curve. Furthermore, this simulation of the changing angle of the tangent line over time offers a potential avenue to support students in developing an understanding of instantaneous speed as a rate. Furthermore, it makes for a tool that could potentially be used to explore all kinds of topics in primary school where change or growth plays a significant role.

It is noted, however, that the use of these computer tools in the proposed LIT is not an end in itself but a means to support students’ learning processes while exploring instantaneous speed. In a sense, this use of computer tools is a reflection of how ICT has come to dominate our interactions with dynamic phenomena in our society. As using computer technology has become an integral part of doing STEM, it should also be an integral part of STEM education. As more and more dynamic phenomena are being monitored, and more (real-time) data is becoming available, the potential for meaningful and realistic exploration of these phenomena by students and teachers is enormous. Key to unlocking this potential is realizing that computer technology is an inseparable part of learning and teaching STEM in the information society.

Bridging the gap between the students’ world-view and STEM

The proposed LIT is close to students’ world-view. Obviously, they are very familiar with the context of filling glassware: they have filled glasses and bottles all of their lives. At the same time, although they do have a conception of instantaneous speed in this context from the start, they probably have never thought much of it. This makes filling glassware an ideal context to explore students’ covariational reasoning (Carlson et al., 2002, 2002; Castillo-Garsow, Johnson, & Moore, n.d.; Gravemeijer, 1984-1988; Johnson, 2012; McCoy, Barger, Barnett, & Combs, 2012; Swan, 1985; Thompson, Byerley, & Hatfield, 2013). More so, the context of filling glassware is intrinsically tied to the proposed LIT because it connects constant speed to instantaneous speed in a tangible and visible way that is difficult to realize in the motion context that is commonly used to explore (average) speed in primary school.

On the other hand, due to its simplicity, filling glassware is not a very inspiring context. During the classroom teaching experiments it was observed that some students started getting bored with exploring filling glassware over and over. This context is intended as a starting point for an exploration of a wide range of topics. Any practical adaptation of the proposed LIT should aspire to explore various dynamic phenomena, which binds into the characteristic of integration of new STEM education in the primary school curriculum. There are a number of topics in both the curriculum and current events where rate of change, speed, or growth play a significant role, such as biological growth processes, historical demography, financial crises, physical education, weather, traffic, and so on. In a MBL setting, students can explore these topics in more detail by using graphs and the tangent line tool, which could result in a richer experience. Ultimately, exploring authentic and meaningful situations with tools from STEM might help close the gap between the students’ world-view and the reality propagated by the scientific world-view (Osborne & Dillon, 2008). After discussing doing design research in the next section, a proposal is made for a possible fruitful way to adapt the proposed LIT that takes into account these characteristics of innovative STEM education.

Reflections on design research

This thesis is as much about doing design research as it is about the development of the proposed LIT for teaching instantaneous speed in grade five. Design research is still an evolving research methodology. Despite the growing body of literature on design research (see for instance Plomp & Nieveen (2013)), there is no text book or manual that delineates how to do a design research project as outlined in Gravemeijer & Cobb (2013). Without any pretension to create a practical manual, this thesis illustrates getting started with design research (Chapter 2), the iterative nature of the design research (Chapter 4), and the place of generating new theory in design research (Chapter 3). In the following, these three characteristics are discussed, followed by the limitations of design research and this design research project in particular.

Doing design research

Design research can be characterized as an iterative process of refining a LIT in multiple design experiments (Gravemeijer & Cobb, 2013). Key to this process is the initial LIT. Remember that design research is often used to explore learning that is new somehow, of which not much is known (Kelly, 2013). It is unlikely that a topical literature review will amount to a thorough understanding of the problem at hand: other sources have to be tapped, such as the researcher’s prior experiences or a preliminary study. In this respect, design research entails a creative process in which design choices can have long-reaching consequences.

For example, the small-scale study to explore 5th graders’ understanding of speed in a situation with two co-varying quantities (Chapter 2) shaped the initial LIT and the design of the learning trajectory in the first design experiment. In the one-on-one teaching experiments of the starting-up phase, the students seemed to react well to the measuring-cup activities, while having more trouble with the graphing activities. Without much thought, the measuring-cup activities were adopted to introduce the students to the context of filling glassware in the first lesson of the learning trajectory in design experiment 1. At the same time, the graphing activities were pushed to the next lessons and were scaffolded by the graphing-component in the computer simulation. Unfortunately, the resulting learning environment promoted discrete thinking, which interfered with students’ development of continuous conceptions of speed.

One could wonder if this problem of a discrete learning environment reinforcing discrete thinking could have been prevented. But should it? The design research would have been significantly different: Because discrete thinking was reinforced in the beginning, it became a topic of concern, which directed the retrospective analysis towards inspecting the dichotomy between discrete and continuous images of change and the discovery of the literature on “smooth” and “chunky” thinkers (Castillo-Garsow, 2012; Castillo-Garsow et al., n.d.). Subsequently, discrete and continuous thinking became a central theme in the design research project (Chapter 4). This binds in with the secondary aim of design research to develop theory on more encompassing issues than just the LIT. In this regard, the central theme can be seen as such a more encompassing theoretical issue.

As an aside, it is interesting to note that this dichotomy can also be traced back to the development of the concept of instantaneous rate of change, from a qualitative approach to change in classical antiquity to the quantitative differential calculus approach in the early modern period (Boyer, 1959; Dijksterhuis, 1950). A salient detail in this regard is the fact that Heytesbury’s (1335) intuitive notion of instantaneous velocity (Clagett, 1959)—this was the inspiration of using the highball glass’ constant speed as a measure for the instantaneous speed in the cocktail glass in the LIT—was formulated in the beginning of this development towards the differential calculus. It might be a fitting start for students to learn about instantaneous speed and offer them a strong conceptual basis for coming to understand the concepts of the differential calculus.

Here the power of iterative development comes to the fore. By refining the LIT in multiple design experiments, this central theme could be tracked across different instructional settings. It adds to the justification of the conjectures generated around the central theme through triangulation. In particular, it contributed to the ecological validity by exploring how similar students in different classrooms struggled to align discrete and continuous images of change. It seems therefore plausible that students in other classrooms will have similar struggles and by elaborating these struggles, potential users of the proposed LIT can recognize these struggles from their own experiences even if they are unfamiliar with teaching instantaneous speed in 5th grade. Smaling (2003) speaks of communicative generalizability in this regard, but beyond that, concerns of generalizability of the findings of the proposed LIT have to be raised. These limitations are discussed in the next section.

Justification of conjectures does not only stem from repeated validation in multiple teaching experiments, but also from the systematical way they are generated during the retrospective analysis by means of abduction. The use of abduction is distinctive to design research and it differentiates design research from experimental research, which builds on induction and deduction. In Chapter 3 it is illuminated how abductive reasoning was applied in the retrospective analysis of the third design experiment to generate the conjecture that students come to the classroom with a continuous conception of speed and only switch to discrete reasoning because of a lack of means for visualizing continuous change. Writing Chapter 3 and explicating the abductive reasoning process was quite enlightening to get a better understanding of how new conjectures are actually generated and why these conjectures are credible. There is a clear relation between an surprising fact that triggers abductive reasoning and the proposed explanation for that fact, the generated conjecture. However, the new claim is justified by it being grounded in the data collection. Here comes a second level of triangulation in play: during the teaching experiment a multitude of data sources is collected, all of which can contribute to support the validation of the newly generated conjecture.

With respect to the iterative nature of design research it is worth noting that the development of the LIT and prototypical instructional sequence is mostly a continuous process. Most conjectures in the LIT are carried over from one design experiment to the next. Similarly, the various computer simulations developed in this thesis share many characteristics. There is a clear line of development from the first computer simulation in the starting-up phase to those in the last design experiment. Doing design research means continuously building on previous experiences, materials, and ideas; design research is emphatically not a sequence of isolated design experiments. It is an iterative and cumulative process

Limitations

Due to the nature of design research there are concerns about the generalizability of the findings of design research. In particular, the classroom teaching experiments took place in gifted (and mixed) classrooms, the instructional sequences that were tried out were very short, and over-all it was tried to set up the learning environment most conducive to gathering data for research. Furthermore, because the proposed LIT is not intended as a ready-made product, there are issues with the application of these findings as the proposed LIT has to be adapted by others to their situation. A potential practical adaptation needs further support. These limitations of design research are discussed in terms of the design research presented in this thesis.

The conjectures of the proposed LIT were generated and validated throughout the design research project based mainly on transcripts of video captured whole-class discussions and collected student products. The extent to which these data faithfully represent the reasoning of all students largely depends on the established classroom-culture during the teaching experiments. It appeared that in each design experiment there was a classroom-culture wherein students felt safe to actively participate, freely express themselves, and ask questions. Some students were more strongly present than others, however, who adopted a more cautious attitude towards participation in the whole-class discussions. On the other hand, the aim of this design research was to explore innovative ways of teaching instantaneous speed in 5th grade. This resulted in a potential viable learning trajectory, which is elaborated in the proposed LIT, that delineates a way to support 5th grade students in developing a notion of instantaneous speed that could lead to a more quantitative understanding of instantaneous speed.

At the same time, however, especially because most students participating in the teaching experiments were above average performing students or gifted students, it is likely that teaching instantaneous speed in 5th grade will take more time and effort than suggested by what happened during the teaching experiments. This binds into the issue of generalization of the findings. In the end, the findings are based on the small number of students and classrooms that participated in the teaching experiments. These findings can only be generalized by means of “communicative generalization” (Smaling, 2003), which means that:

‘it is not so much the researcher but the reader (or potential user) of the research report who figures out to what degree the research results and conclusions are generalizable to people, situations, cases, et cetera, that are relevant to him.’ (Smaling, 2003, p. 59)

It is up to the researcher to best support the potential user to transfer the findings, if at all, to their situation (Smaling, 2003). In this sense, the proposed LIT acts as a theory on how instantaneous speed can be taught in 5th grade as a starting point for potential users to build on when they want to teach instantaneous speed early in the mathematics curriculum, develop educational materials for that purpose, or do further research on this topic. As a potential limitation of the transferability of the findings to real-world classrooms, it is noted that to create and maintain a classroom-culture that is conducive to learning instantaneous speed as proposed in the LIT, the teacher plays a key role. However, applying a MBL approach is not straightforward; teachers, in particular those with little affinity with and exposure to STEM, will need further support to successfully apply MBL in their own classrooms.

Furthermore, it is emphasized that parts of the proposed LIT have to be developed further, in particular students’ command and understanding of the quantification of constant speed. In this regard, an approach that starts with exploring discrete movements, such as used by Doorman (2005) and Galen & Gravemeijer (2010), might offer an interesting perspective. Simultaneously, students’ experience and skill with graphs has to be developed, for which a reinvention approach (diSessa, 1991; Galen, Gravemeijer, Mulken, & Quant, 2012) seems a fruitful candidate. Finally, during the design experiments, the generalization of students’ use of the tangent line has only briefly been explored. In particular, generalization of students’ understanding of quantifying instantaneous speed using the tangent line has to be expanded to other contexts as well.

Adapting the proposed LIT: A proposal for further research

Having discussed the proposed LIT and the process of doing design research that led to that proposed LIT, it has become abundantly clear that the proposed LIT is not ready-made. And neither is the instructional sequence from the third design experiment that is instantiated by it. Yes, it is to be qualified as a potentially viable theory for other researchers, educational designers, and teacher to adapt to their situation, but is it actually useful? To answer this question affirmative, a proposal for a viable adaptation is made.

Given the average primary school teacher’s limited affinity with and knowledge of (teaching) STEM (Léna, 2006), adapting the proposed LIT by the average teacher on his own is unlikely, and maybe even undesirable. Indeed, because most primary school teachers will not be familiar with calculus (Kaput & Roschelle, 1998) and instantaneous speed is not part of the primary curriculum, it is likely that the average teacher does not possess the necessary understanding of issues related to (teaching) instantaneous speed. This suggests that any adaptation of the proposed LIT requires sufficient and suitable support for the teachers implementing it in their classrooms. This suggests an effort to contribute to teachers’ professional development, as was suggested before in subsection 6.2.2.

The proposed LIT has some issues regarding the instructional starting points, in particular the students’ lack of graphing experience (Leinhardt et al., 1990) and limited command of average and constant speed. To successfully adapt the proposed LIT means to pay extensive attention to overcome these issues and support students in deepening their understanding of graphs and speed side by side. Furthermore, because this thesis is small in scope, the aim put forward in the Introduction to integrate new STEM education in the primary school curriculum has not been treated at all. This suggests a significantly longer learning trajectory than developed in this thesis.

Taking these issues into account, a long-term design research project is needed that aims at exploring the integration of instantaneous speed in the primary school curriculum. Following the suggestion of Schoenfeld (2009) to perform design research in balanced project teams, ideally researchers, experienced educational designers, and teachers collaborate. The teachers will need (initial) support for learning content-knowledge about instantaneous speed, applying modeling-based learning, including initiating and maintaining suitable classroom social norms, and developing PCK about teaching instantaneous speed in primary school. Gravemeijer & Eerde (2009) speak about “dual design research” in this regard, where the teacher has to learn about (teaching) a topic while supporting students in learning that topic (Gravemeijer & Eerde, 2009). Throughout the design research project, however, they will construct PCK about teaching instantaneous speed, gain expertise with inquiry-based learning approaches and in particular MBL, and deepen their own understanding of instantaneous speed. During this professional development, they become invaluable resources for the project team.

For example, they can help the educational designers in selecting and elaborating suitable topics from the primary school curriculum where change, growth, or speed can play an important role. These topics are ideal candidates for exploration in the long-term learning trajectory on instantaneous speed, taking care of both integrating STEM in the curriculum and exploring topics that are close to the students’ world view. Furthermore, as experts on their students’ instructional starting points and capabilities, the teachers play an important role in intertwining the learning trajectory on instantaneous speed with the curriculum regarding speed and graphs.

Beyond the practical adaptation of the proposed LIT, in the proposed design research project there is ample room for researchers to study different aspects of learning in real-world classrooms, teaching, and professional development. In doing a long-term design research with a large project team that includes teachers, the promise of design research to bridge the gap between research and practice becomes more credible. Ultimately, the proposed design research project to explore teaching instantaneous speed will result in a better understanding of how to teach instantaneous speed in grade five.

If nothing else, even when instantaneous speed is not taught—it is not part of the primary school curriculum, after all—, the findings presented in this thesis do have practical implications for upper primary education:

1. Delay teaching of average speed. For one thing, because most problem situations that primary school students will encounter at school are linear in nature, average speed does not have much meaning beyond constant speed. Instead, focus on deepening students’ qualitative and quantitative understanding of constant speed and ensure they develop a flexible command of different units for constant speed.

2. Support primary school students in developing a good understanding of Cartesian graphs by having them reinvent graphs. The students should be able to express and discuss their understanding of dynamic phenomena by means of representing and interpreting their ideas with graphs.

Bibliography