# Summary

## Exploring Instantaneous Speed in Grade Five

### Huub de Beer

### May 11, 2016

Chapter 1 introduces the topic and set-up of this thesis: In answer to a call for new STEM education in primary school (Gravemeijer, 2013; Gravemeijer & Eerde, 2009; Keulen, 2009; Léna, 2006; Millar & Osborne, 1998), a design research project was started to answer the question *How to teach instantaneous speed in grade five?* 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 it is firmly rooted in the realm of STEM. Key to a better understanding of dynamic phenomena is the concept of instantaneous speed, which is conventionally taught first in a calculus course in upper secondary education or college. A design research approach was chosen, as design research is well-suited to explore how to teach topics in earlier grade levels than usual (Kelly, 2013).

Design research is an interventionist process of iterative refinement that takes real-world classroom practice into account when creating 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). 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. In this approach, a design research project starts by formulating an initial LIT. This LIT is then elaborated, adapted, and refined in multiple design experiments (Cobb, 2003; Gravemeijer & Cobb, 2013). A design experiment consists of three phases: developing an educational design based on the LIT, trying out and adjusting that design in multiple teaching experiments, and performing a retrospective analysis on the data collected during the teaching experiments to refine the LIT, which is then used in the next design experiment. The retrospective analysis is typically based on the constant comparison approach of Glaser & Strauss (1967). In this project the elaboration of Cobb & Whitenack (1996) was used. In line with the theory-driven nature of design research the focus in this thesis is on the development of the LIT rather than on the development of the educational design. 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, which are summarized 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

In Chapter 2 it is described how the design research project started by exploring 5th graders’ prior conceptions of speed through a literature review that was followed by a preliminary study. 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. To overcome students’ problems with speed (Groves & Doig, 2003; Thompson, 1994), Stroup (2002) proposes a “qualitative calculus” approach, which builds on students’ qualitative understanding of speed instead of following conventional approaches that favor ratio-based understandings. Because instantaneous characteristics of speed have not been explored in this literature, this review was extended to cover teaching calculus-like topics early in the mathematics curriculum as well. In this research it was found that young students are able to explore speed mathematically using computer simulations and graphs (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) when these are embedded in a suitable instructional sequence (Gravemeijer, Cobb, Bowers, & Whitenack, 2000), even provided that they lack graphing experience (Leinhardt, Zaslavsky, & Stein, 1990).

Based on this literature review, a small-scale study was performed to to explore 5th graders’ understanding of speed in the context of filling glassware. To that end, a short instructional sequence and a computer simulation were developed and subsequently tried in eight one-on-one teaching experiments (Steffe & Thompson, 2000) in which the students were asked to turn a highball glass, a cocktail glass, and an Erlenmeyer flask into a measuring cup, draw a graph of filling the glass, and evaluate their work using the computer simulation. The data collected during these experiments were analyzed using Carlson et. al.’s covariation framework (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002): Students’ utterances were coded for one of five developmental levels of covariational reasoning, which 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 were quite capable in estimating the relative amount of change at certain points or intervals, but they almost never talked about speed. It was also concluded that, due to the students’ limited vocabulary and graphing skills, the covariation framework did not offer the clarity on instantaneous aspects of change to be a good instrument to study 5th grade students’ conceptions of instantaneous speed.

Nevertheless, the findings of the literature review and the experiences during the teaching experiments allowed for the formulation of an initial LIT that 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. After exploring the relation between a cocktail 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. 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. Next, building on the correspondence between a highball glass’s graph and the steepness in a point of a curve, the students might construe the tangent line in a point of a curve as an indicator for the instantaneous speed. Given students’ familiarity with constant speed and graphing linear situations, students then could quantify the instantaneous speed by computing the rise over run of the tangent line.

## Overview of the three design experiments

After the starting-up phase of the design research project, three design experiments were performed. The procedure of analysis used during each of the three design experiments was a two-step retrospective analysis modeled after Glaser and Strauss’s (1967) comparative method. In particular, the elaboration of Cobb & Whitenack (1996) on this method was used. After formulating conjectures about *What happened?* during the teaching experiments and testing these conjectures against the data collected, a second round of analysis was carried out by formulating conjectures about *Why did it happen?*. These conjectures were also tested against the data collection. The data was collected during the following three design experiments:

The first design experiment revolved around an instructional sequence that started with students making measuring cups from various glasses, followed by a lesson on exploring quantification of speed in the highball glass. Only then, in the third lesson, the attention shifted to instantaneous speed. Following the initial LIT, it was expected that students 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. 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. Although the students could handle the tangent-line-tool, their level of understanding was doubted. Furthermore, they did not manage to discover a continuous graph; it was conjectured that starting with the linear situation of the highball glass might have put them on the wrong track.

In the small-scale design experiment that followed, it was decided to remove the measuring cup activities and have the students’ learning process 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, which helped to make the students’ thinking visible and a topic of discussion. During the teaching experiments the students were repeatedly asked to model filling a cocktail glass, and to improve their models after exploring the situation in a computer simulation. It was expected that, once the Cartesian graph was introduced by the teacher, they would 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. 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. During the teaching experiments, the students did construe 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. However, there was no success with developing continuous graphs, which seemed to suggest that maybe students of this age are chunky thinkers (Castillo-Garsow, 2012). Furthermore, the students’ lack of understanding of, and fluency with, measures of speed revealed itself.

Design Experiment 3 was a turning point. The students in one of the two classes that the experiment was tried in, 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. This meant that 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 the students came to understand the continuous graph via shrinking the intervals of a bar chart. Furthermore, the students could build on those ideas to come to grips with the shape of the graph 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 appeared to be hampered by the fact that they did not have a sound basis for calculating speeds.

## The generation of new explanatory conjectures in design research

Chapter 3 focuses on the generation of new theory during the retrospective analysis, highlighting the role of abductive reasoning (Fann, 1970)—which can be characterized as noticing that a certain observation needs an explanation and seeking to find the simplest and most likely explanation of that observation. It is shown how abductive reasoning led to the generation of the conjecture during the retrospective analysis of the third design experiment 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. It is illuminated how abduction plays a specific role in design research by describing both the teaching experiment leading up to the unexpected event and the two-step retrospective analysis that followed.

In design experiment 3, a four-lesson instructional sequence was developed and tested in two gifted 4th-6th grade classrooms (C1 and C2) taught by the same teacher. In the first lesson, the students modeled filling a cocktail glass four times, during which 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, most of which were discrete representations. However, in each classroom, there was one graph-like model with continuous characteristics. In C1, while discussing the student-drawn graph-like minimalist model, the students argued that the straight line segment had to be a curve: they invented the curve by themselves. In C2, the graph-like model was not discussed and the students’ learning trajectory followed the one anticipated in the LIT. This unexpected difference led to a process of abductive reasoning during the retrospective analysis.

The retrospective analysis was based on a two-step method based on Glaser and Strauss’s (1967) comparative method: after testing conjectures about what happened, conjectures about why that did happen were formulated and tested against the data collected during the teaching experiments. It showed that the students’ reasoning was grounded in continuous reasoning, while discrete reasoning functioned as a tool to get a handle on continuous processes. Furthermore, the 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 led to the conjecture that starting with average speed is problematic: it promotes discrete thinking and could be the source of problematic chunky thinkers (Castillo-Garsow, 2012). It would make more sense to explore constant speed, which subsequently can be connected to the students’ notion of instantaneous speed.

However, the generalization of these findings were possibly limited by the uniqueness of the classroom situation. Furthermore, 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).

## Design research as an augmented form of educational design

Design research builds on educational design to develop both a product and a theory detailing how and why that product works. To develop theory implies a commitment to strengthen the credibility of the theory by substantiating 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. In Chapter 4, 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 by tracking the development of the instructional sequence from the starting-up phase through the three subsequent design experiments in terms of a framework encompassing both the design decisions and the rationale for those decisions. Such a framework of reference may take the form of a LIT that offers a rationale for the 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. The theoretical findings can be substantiated by the virtual repeatability or trackability of one’s research by other researchers (Smaling, 1990). The goal of design research is to generate a theory on how the intervention works. To develop theory, 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, offering a form of triangulation that adds to the understanding of students’ learning processes in terms of these conjectures. New explanatory conjectures are generated through abductive reasoning, but only claims about the students who participated in the experiments can be made. They 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.

The goal is to come to understand 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. Design research can be taken as a paradigm that may show educational designers the value of documenting design decisions and anticipated learning processes. In this way, 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 5th 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 was collected and used to validate and generate conjectures, and these conjectures were validated or refuted in multiple design experiments.

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.

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, which allowed them to realize when the speed in the cocktail glass and highball glass would be 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.

Based on these key learning moments, the proposed LIT is formulated in Chapter [ch5] in terms of a postulated students’ learning processes and the potential means of support necessary for those learning processes to emerge. The students’ potential learning processes may be summarized as follows:

Given a cocktail glass, students are given the task of making a drawing of how the water height changes when the glass fills up. After observing the glass 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—based on their intuitive notion of instantaneous speed.

This 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 and answering the question when the water rises with the same speed in both glasses. Then the constant speed of an imaginary highball glass can become 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: the model of water height become a model for 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: the context of filling glassware, computer simulations combined with graphs, and the modeling-based learning (MBL) approach to learning. The context of filling glassware visually connects constant speed, which students already know, to instantaneous speed; picturing the highball glass with the cocktail glass offers a very powerful image enabling students to invent the highball glass as a tool to measure instantaneous speed. Combining interactive simulations with graphs enabled an inquiry-based learning approach where students and teacher could safely explore speed and graphing. Furthermore, offering both discrete and continuous representations allowed for an affordance for shuttling back and forth between continuous and discrete reasoning. By using an MBL approach the class-based discussions about speed became more conceptual in nature while giving the teacher indirect access to the students’ mental models. This allowed the teacher to better support students in constructing a deeper understanding of graphs in relation to speed.

Unique to the presented approach to teaching instantaneous speed is that it circumvents the troublesome limit concept while supporting students to come to understand and quantify instantaneous speed. This means that compared to conventional approaches, average speed plays a minimal role.

Although the proposed LIT is considered a potential viable theory, it clearly cannot easily be used in regular classrooms due to issues of teacher professional development and the fact that the corresponding instructional sequence has to be elaborated to cater for the students’ limited graphing abilities and their limited understanding of quantifying constant speed. For others to adapt the LIT 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.

## Discussion

In Chapter 6, the findings of the design research project are discussed in light of the aims put forth in the Introduction, by reflecting on doing design research, and by proposing a viable adaptation of the LIT.

### Characterizing the proposed LIT

The proposed LIT and its potential application can be characterized by four themes:

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 and by circumventing the problematic limit concept (Tall, 1993, 2009), which makes it a truly innovative approach to learning instantaneous speed.

MBL is a form of inquiry-based learning built around the idea that modeling is a core activity of science and seems a natural basis for STEM education. 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: teachers will need support to start with MBL, but as 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 establishing and maintaining a suitable classroom culture for MBL there is also a role for ICT: 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). During the teaching experiments, 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 an understanding of graphs and for them to discover and use tools to quantify instantaneous speed; it creates an affordance for shuttling back and forth between a continuous and a discrete image of change.

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.

### Reflections on design research

This thesis is as much about doing design research as it is about the development of the proposed LIT. Design research is still an evolving research methodology. Despite the growing body of literature on design research (see for instance Plomp & Nieveen (2013) and Prediger, Gravemeijer, & Confrey (2015)), there is no text book or manual that delineates how to do a design research project as outlined in Gravemeijer & Cobb (2013). 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).

However, due to the nature of design research there are concerns about the generalizability of the findings. Moreover, 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. 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. Some students were more strongly present than others, which raises the question to which extend the data faithfully represents the reasoning of all students. Furthermore, the findings can only be generalized by means of “communicative generalization” (Smaling, 2003), which means that 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.

### Adapting the proposed LIT: A proposal for further research

The proposed LIT is not ready-made. And neither is the instructional sequence from the third design experiment that is instantiated by it. However, a long-term design research project is proposed that aims at exploring the integration of instantaneous speed in the primary school curriculum, ideally with collaboration of researchers, experienced educational designers, and teachers. The teachers will need (initial) support for learning content-knowledge about instantaneous speed, applying modeling-based learning, including establishing and maintaining suitable classroom social norms, and developing PCK about teaching instantaneous speed in primary school. During this professional development, they become invaluable resources for the project team to 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 might be 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 such a design research project there would be 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. Even when instantaneous speed is not taught in primary school, it is not part of the curriculum after all, the findings presented in this thesis do have practical implications for upper primary education as well. Delay teaching of average speed, but focus on deepening students’ qualitative and quantitative understanding of constant speed and ensure they develop a flexible command of different units for constant speed. And support primary school students in developing a good understanding of Cartesian graphs by having them reinvent graphs to allow them to express and discuss their understanding of dynamic phenomena through graphs.

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