Videogame Construction by Engineering Students for Understanding Modelling Processes: the Case of Simulating Water Behaviour

We present some results of an ongoing research project where university engineering students were asked to construct videogames involving the use of physical systems models. The objective is to help them identify and understand the elements and concepts involved in the modelling process. That is, we use game design as a constructionist approach for promoting a modelling activity and the learning of the elements involved. In this paper, we focus on the case studies of two students, in their last year of studies, who built a videogame where they had to model liquid water behaviour while working within the restrictions of the game engine. By analysing students' written work and group discussions, we observed that students, through this videogame-building task, were able to deepen and refine how they conceive the process of mathematical modelling, in a fun and engaging way in which they were receptive and open to experimentation, and learned from other students, as well as from making mistakes.

Introduction, Background and Theoretical Framework
Learning modelling is an important activity in the life of an engineering student, because it is the first step to develop the design of any device, system, product or process. It requires students to apply mathematical skills in order to build appropriate representations and models.
In this paper, we present some results from an ongoing project where university engineering students are asked to build videogames in order for them to engage in modelling processes and understand the concepts and elements involved. Some of the videogames require simulating physical systems; in this case, we focus on a videogame that required modelling liquid water behaviour.
The idea of asking students to build a videogame for engaging in mathematical modelling tasks, is based on the constructionist paradigm. Constructionism (Papert and Harel, 1991) considers that learning is facilitated when the learner engages in the active construction and sharing of external objects; it emphasises the importance of providing learners with opportunities and a context to build hardware or software artefacts that engage them and are meaningful to them. Thus, in our quest to help students conceive and apply mathematical modelling processes, we thought of engaging our engineering students in a construction activity -that of programming a videogame -that would require modelling. Since the early days of computers in education, game design and programming have been proposed as a methodology for engaging learners, as in the case of the work of Harel (1990). In particular, the works of Kafai and colleagues (e.g. Kafai, 1995;Kafai, 2006), present some of the foundations and structure of this methodology, and their relationship with the constructionist paradigm, even though those works are with younger learners than the ones in our study. One of the key ideas is that: Game design provided a situation that naturally combined issues of practice and theory, and reflection on those relationships and game design provided opportunities for discussion, reflection, and collaboration within a meaningful context. (Kafai et al., 1998, p.180) In our work, the programming activities of the game design were conceived within a learning microworld, which we considered a suitable context for engaging in the activity of "building a meaningful product." Hoyles and Noss (1987) defined a microworld as composed of four components: the student component (concerned with the existing understandings and partial conceptions that the learner brings to a learning situation); the technical component (constituted by the software or programming language and a set of tools which provides the representational system for understanding a mathematical structure or a conceptual field); the pedagogical component (the didactical materials and interventions that take place during the computer-based activity); and the contextual component (the social setting of the activities).
Since we want students to go through, explore and think about modelling processes, the programming activities of the videogame -which are part of the pedagogical component of the microworld -were conceived as a set of what Lesh and Doerr (2003) have named model-eliciting activities (MEAs) which they define as problem-solving activities that are: so called because the products that students produce go beyond short answers to narrowly specified questions -which involve sharable, manipulatable, modifiable, and reusable conceptual tools (e.g., models) for constructing, describing, explaining, manipulating, predicting, or controlling mathematically significant systems. (Lesh and Doerr, 2003, p. 3) MEAs involve six principles of design (Lesh et al., 2000), (Hamilton, 2008): Reality, model construction, model documentation, self evaluation, model generalization, simple prototype. We took into consideration those six principles to integrate the modelling process, in this case of water behaviour, as part of a sequence of activities (see next section) that lead to the programming of the videogame.

Research Aims and Methodology
In our activities, we seek, through the videogame design and programming, for students to construct and reconstruct how they conceive and put into practice the modelling processes. The activities were carried out as part of an optional course that we created, for engineering students, at the National Polytechnic Institute in Mexico City, Mexico.
We began with an the initial questionnaire, where students had to express and reflect on what it means, in general terms, to model and the general processes involved in it, simulation and videogame construction. For example, we asked students to define, in general terms, modelling, what a model was, what a simulation was among other questions. In particular, the main question was: What steps are needed to build a model? Their written answers gave us, as researchers, and insight into their conceptions, with regard to the modelling process, before and after the activities and allowed us to observe if and how students transformed their conceptions through the activities.
During the activities, participants were organized into teams of two, within a learning microworld, as defined above, that involved the following components, and which describe our methodology.

The Technical Component
Within the technical component, we have the game engine. In the examples given in this paper, we used Game Maker Studio (Yoyo Games, 2014) and its physics engine Box2D (Catto, 2014), through which students built a videogame that used the mathematical model of water to simulate this element in a virtual environment governed by the basic physical laws of the real world. These technological environments are meant to be rebuildable and modifiable products.

The Student Component
The students involved in the case presented in this paper, were 12 engineering university students of the National Polytechnic Institute, in their 8th semester of studies. They voluntarily signed up for this optional course, because they are interested in engaging in videogame development after college. It is important to mention that these students had previously taken at least one course in classical modelling and simulation.

The Pedagogical and Contextual Components
The pedagogical component includes a sequence of three activities designed taking into account the six principles, mentioned above, of MEAs (Lesh et al., 2000), (Hamilton et al., 2008) (see Table 1) to encourage students to construct and reconstruct concepts about mathematical modelling of physical systems.
After the reflection in the initial questionnaire, Activity 1 involved working on to the idea and collaborative design of the videogame. Activity 2 required each student to individually propose a mathematical model, and Activity 3 was again collaborative work to solve the actual modelling problem and incorporate a modified (adapted and simplified) version of the mathematical model into the videogame. Thus, in terms of the context, the activities (1 & 3) combined collaborative work in pairs, but also individual work (Activity 2 -also the written questionnaires where answered individually). In this way students discussed and worked on their proposed solutions about the modelling of the water behaviour, developing a genuine engagement in the creation of the conceptual design of the videogame. We now describe each of these activities: In this activity, each team of students had to develop an idea for their videogame through a brief "story board" of up to 4 images, and adding a brief description of the gameplay. We required that the gameplay used water as the main element to solve puzzles and complete levels. This is a task meant to genuinely engage students in creating a meaningful product. Activity 2.

•
This was an individual activity, where students had to build a model of water behaviour, that later could be used in the programming of the videogame designed in activity 1. The creation of this model requires students to have, and use, previous knowledge of algebra, vector calculus, differential calculus and fluid mechanics (statics and dynamics of fluids). The task is presented, not as a traditional problem statement, but as an open problem, as it happens in real life. Activity 3.

•
Here students had to collaboratively program the game using the models that they created (i.e. discussing the best way to apply one of them, or defining together a new one), into the game engine (in this case, Game Maker Studio), thus rendering those models more concrete and meaningful. This requires going through a stage of refining and simplifying the model, proposing appropriate restrictions and relationships (e.g. of scale, metric system, degree of freedom, etc.); as well as defining the characteristics of the computational objects that will simulate the particles of water, giving them attributes (shape, colour, and size) and some physical properties (gravity, density and potential and kinetic energy) represented by algebraic mathematical equations.
In Table 1, we illustrate how the design principles of the MEA's apply to each of the activities.

Sample Results
We now illustrate the unfoldment of each of the above activities, through a case study of two students -Felipe and Javier -who worked as a team.

The Initial Questionnaire
In the Table 2, we give the replies that these students gave to the question: What steps are needed to build a model? Table 2 Question / answers in the initial questionnaire

Student Question / Answer
What steps are needed to build a model?

Felipe
Movements, variables, and (degrees of freedom) Javier Analyse the phenomenon, represent it mathematically, analyse its behaviour, implement an action based on the above.
We observe that Felipe only mentions some aspects to take into account and doesn't give actual steps needed to build a model; Javier has a more clear and structured (although perhaps incomplete) idea. Previously, as mentioned above, the students had stud- Table 1 Relationship of the six principles of MEAs and the activities carried out by students MEA design principle Related activities

Model documentation
In the questionnaires, students already have to describe, for a general and abstract case, the steps of building a model. In Activity 1, the story board is a first documentation of the model. Then, in Activity 2, students have to describe the mathematisation of the water model. And a final documentation is found in the actual program, in the game engine, of the videogame. Reality The problem of designing (Activity 1) and programming (Activity 3) a model of water behaviour is realistic problem. Furthermore, constructing a real videogame is something that genuinely engages students in the task of mathematical modelling.

Model construction
Students build a first version of the mathematical model (Activity 2) and they make a refinement and simplification of this model when having to adapt it for, first, constructing a simulation in the game engine (Activity 3), and then implementing it in the actual videogame. Thus, students need to have clear understanding of the relationships necessary to program the simulation and, later, the videogame. Simple prototype When students set appropriate relationships and constraints for establishing a model with simplified equations (Activity 3), they can also provide simple and structured steps to generate a set of tests, such as the simulation of the model, in order to test its effectiveness.

Model generalization
As described in the previous point, in Activity 3, students need to establish appropriate restrictions and relationships that lead to a general model that can be used again in other situations and with other technological tools.

Self evaluation
During the stage of refining and simplifying the model (Activity 3), students need to evaluate their models and through discussions take decisions for finding an implementable version of the model into the game engine. Also, through the post questionnaire, a reflection is carried out about what was involved in the modelling process and how the entire process of the videogame creation, transformed and enriched their concepts from their initial conception (as given in the initial questionnaire).
ied courses on modelling, so they know how to build models in practice; but, as seen here, they have difficulties in expressing the steps needed to build a model.

Activity 1
We recall that in this activity, students constructed a brief "story board" (see Fig. 1) to describe the gameplay. The main idea of Felipe and Javier's game is to present the player and two containers (one empty and one filled with water). These containers can appear anywhere in the playing area. The player must take the water of the filled container, and fill the empty one, aided by gravity or other resources that may appear in the play area.
Felipe and Javier, as did all the other students, showed great interest in this activity, because they want to engage in videogame design when they graduate.

Activity 2
As explained above, in this activity students had to construct a mathematical model of the water behaviour that included realistic physics, for implementing it in the previously designed game. Each student built a mathematical model of fluid mechanics, addressing the water model either as a molecular model (Felipe's case) or as a continuous model (Javier's case) -see Table 3.

Activity 3
Here the two students shared their mathematical models and, in discussing them, noted that it would not be possible to simulate them for validation in the game engine (Game Maker Studio), concluding that they must simplify them.

Simplifying the Model
For constructing an appropriately simple model, students had to take into account certain constraints and relationships of the physical engine of GameMaker Studio, such as the metric system, scales, and that instead of the model being applied to a real 3D environment, it had to work in a virtual 2D world. This implied adapting the formulas. More specifically: Because the game took place in a 2D environment, students had to be aware that • the bodies were constrained in their movement to three degrees of freedom: i.e. two translational coordinates and one rotational coordinate. The students observed that the physics engine (inside the game engine) uses as • measuring system the metric one, but limits objects to have a size equivalent to between 0.1 m and 50 m; however it also sets that pixels should not be in a oneto-one relationship with meters (i.e. one must not set the equivalence of 1:1 meter to pixel), and radians are used as angles. Thus, the scaling ratio used by Felipe and Javier, inside the virtual physical world, was 10:1 (i.e. 10 pixels equal 1 m). This  relationship was expressed through the code: physics_world_create(1/10). Using this relationship, the students created virtual objects inside the physical world: e.g., water was represented as a set of circular-shaped particles with a radius equal to 4 pixels (i.e. equivalent to 0.4 m). Another important relationship in the simulation was to establish the value of • the constant of gravity acceleration (g). For this, the students decided that the constant, which is equal to 9.8 m/s 2 , should be rounded to 10 m/s 2 , due to the scaling factor that they needed to use; this meant that the acceleration would be of 100 pixels per second, every 60 steps, in the direction 270°. This was expressed through the following segment of code: physics_world_gravity(0, 10); and with a room_speed of 60. The value of g affects every object in the virtual world differently, depending on the properties of that object: friction coefficient, linear damping, angular damping, and density.
To set the value of density for each water particle simulated in the virtual world, • students had to take into account that the density used was not a volumetric density one (3D), but a surface density (2D): that is, ρ = 1000 Kg/m 2 (with square meters, instead of cubed as in ρ = 1000 Kg/m 3 ). This is expressed in code as: physics_fixture_set_density (Water_particle, 1000) where Water_particle is the name of the object representing a particle of water and 1000 is the value of water density.
Having established these relationships to the density, students established oth-• er relationships, such as the friction coefficient between the water particles. In GameMaker Studio, friction is defined as the force that resists movement or slippage of a material over another, as can happen when two materials collide in the virtual physical world. The friction value must be between zero and one. In this case, because they were modelling water which is a fluid, the students tried to define this coefficient as a coefficient of viscosity expressed in Kg/m * s, giving it a value of 0.01, considering that the viscosity value between water particles is very close to 0. This is expressed as:

physics_fixture_set_friction(Water_particle, 0.01)
Another property that students had to define, is the restitution coefficient that indi-• cates the amount of kinetic energy that remains after a collision between particles or objects; that is, this coeffcient refers to the level of "bounce" between particles. The restitution value is between zero and one. The water in the real world has a coefficient of restitution of 0.98 and in the virtual world, Felipe and Javier assigned it the value of 0.9, which is expressed as: Finally, the students had to establish relationships for the linear and angular damp-• ing, which refers to the forces that oppose the linear and rotational (angular) motion of a particle (like air friction). Air friction dampens water motion. A simplified representation of air friction is the formula Af = -K . v, where K is the damping coefficient dependant on the shape of the body, and v is the velocity of each particle, dependent on its interactions in the virtual world. Damping forces are normally expressed with values between zero and infinity, but the game engine uses a simpler method, giving shapes that have less air resistance (i.e. round ones) values close to zero and rectangular shapes that have more resistance, values near 1. Thus, in this case, since water particles are round, students assigned the value of 0.1 for the damping coefficient. This is expressed in code as: Having defined the relationships, constraints and properties, as shown above, students were able to develop a water model simulation.

The Simulation
Felipe and Javier concluded that it was more feasible to build a model of the water behaviour, based on a simplified molecular model -where each particle of water is affected by the following factors: specific gravity (γ = ρ . g), density (ρ = m / V), kinetic energy (E c = 1/2 . m . v 2 ) and potential energy (E p = m . g . h) -instead of building a continuous model (as in the macroscopic point of view), because of the restrictions of the game engine.
They then conducted tests to validate the model, including simulations to show the Archimedes' principle which relates to the upward buoyant force that is exerted on a body immersed in a fluid (Fig. 2) as well as other tests (e.g. for Pascal's principle, which says that any change in pressure in any point of a resting enclosed fluid, is transmitted undiminished throughout the fluid). With the validated model, it was relatively easy for the students to then program the game they had designed. A screenshot of the videogame can be seen in Fig. 1.

The Post-Questionnaire
The post-questionnaire focused on the question: What steps are needed to build a model? requiring to draw a block diagram for the steps given. Table 4 shows Felipe's and Javier's answers. We can observe the evolution of the conception of the modelling process as compared to the answers in the initial questionnaire. However, in Felipe's case, from his diagram, we see that he doesn't conceive the modelling process as an iterative activity; in contrast, Javier's block diagram (without taking into account if his representation is correct or not) includes feedback loops in the internal processes.

Reflections and Discussions on the Process
After the activities, all the students engaged in a group discussion of the process of modelling they had carried out. In Activity 2, approximately half of the students had defined water behaviour models using a molecular approach, and the other half a continuous approach (as in Table 3). But all models used differential calculus (partial derivatives) and vector calculus. However, in order to implement these models to the game engine in Activity 3, students had to adapt and simplify them using only simple algebra. This meant that students needed a clear understanding of the variables and components of the mathematical model, in order to identify the most important elements and relationships. Thus, in the discussion, students reflected deeply about the process of modelling in engineering work: on how, when implementing in practice mathematical models, these models have to be adapted; but how their previous teachings had given them an incorrect conception of what is involved in a modelling process. They reflected on how they understood now a modelling process as an iterative and perfectible activity, replacing the traditional concept of "problem statement" requiring a fixed, final answer. Thus, through the MEAs and the experience of building the videogame, students transformed and enriched the way they conceive a modelling process.

Final Remarks
The activities faced students with a real problem (the design and programming of a videogame) that is significant in its sociocultural context. Through the programming of the videogame, students engaged in producing a working model that was meaningful to them and gained a deeper understanding of all the elements involved in the modelling process.
As described above, the mathematical models of water behaviour were first stated using advanced mathematics. But, what happens when the equations cannot be simulated with the technological tool being used (in this case, the game engine)? This is the question that gives meaning to the proposed activities: through the restrictions imposed by the game engine, we lead the students to adapt the model to the context. Thus, students not only had to produce mathematical representations of the physical phenomena (water behaviour), but also understand these in the context and dynamics of the game they were building, establishing relationships between appropriate variables and restrictions (imposed by the technological restrictions of the game engine), and physical formulas in order to adapt their mathematical model. Specifically, they had to establish relationships between the computational objects (in this case water particles) -which also had to be attributed values such as form, colour and size-, and their physic-mathematical properties (gravity, density and potential and kinetic energies). Students were surprised by the great challenge that this was (they had assumed it would be much easier), especially in terms of working with scales and finding the relationships, but learned from their mistakes and thoroughly enjoyed it as well as working and learning from their peers. This constructivist process also helped them appreciate how -unlike traditional problem statements in class that look for fixed answers -in the real world, modelling processes are cyclic (iterative) and perfectible, and often collaborative.
We continue work in our project with other engineering students, designing and building video games in other topics (e.g. robotics, digital systems), as well as working with other game engines.
A. Pretelín-Ricárdez received his M.Sc. in Physics Education at Mexico's National Polytechnic Institute in 2010 and is currently pursuing a PhD degree in Mathematics Education at Centre for Research and Advanced Studies (Cinvestav-IPN), Mexico. He is also a lecturer at the National Polytechnic Institute, UPIITA-IPN, Mexico. His research interest focuses on serious games, videogame-based learning, and virtual and constructionist technological environments in education.
A.I. Sacristán, PhD has been a researcher since 1989, at the Department of Mathematics Education of the Centre for Research and Advanced Studies (Cinvestav-IPN) in Mexico City. Her main area of research is the teaching and learning of mathematics through digital infrastructures, as well as the professional development of teachers in that area. In particular, she is an advocate of computer programming in mathematical education. Other interests include advanced mathematical thinking and the articulation of representations. She has been part of many international committees and been visiting professor and collaborator with institutions in several countries, including the UK, France and Canada. Homepage: http://www.matedu.cinvestav.mx/~asacristan


Socrates in the schools: Gains at three-year follow-up

Three recent research reports by Topping and Trickey (2007a; 2007b), by Fair and colleagues (2015), and by Gorard, Siddiqui and Huat See (2015) have produced data that support the conclusion that a Philosophy for Children (P4C) program of onehour-per-week structured discussions has a marked positive impact on students. This article presents data from a follow up study done three years after the completion of the study reported in Fair et al. (2015). The data show that the positive gains in scores on the Cognitive Abilities Test (CogAT in the USA and CAT in the UK) were still present and had not faded after three years. Given the strength of these confirmations of the positive durable impact of the P4C program of structured discussions and given the relatively low cost of implementing the P4C program, it is recommended that it become a standard part of the school curriculum.

Introduction
After successfully replicating the results of an earlier Scottish study of the effect of a P4C program of one-hour-per-week philosophy discussions on students' scores on the Cognitive Abilities Test (CogAT in the USA, CAT in the UK), our research group in Texas extended the replication by doing a three-year follow-up study to determine if the relative improvements in the CogAT/CAT scores for the experimental group over the control group that were visible in the original 2010-2011 study were still detectable in the spring of 2014.

The original studies done in Scotland by Topping and Trickey
In 2009, when we formed our Philosophy for Children (P4C) interest group, we looked for hard evidence for the educational impact of P4C programs. When we found Topping and Trickey's study of the effects of a P4C program on students in Clackmannanshire, Scotland (Topping & Trickey 2007a), our Philosophy for Children (P4C) study group decided to replicate their study in a middle school in Texas. We had several reasons for our choice. First, Topping and Trickey used the Cognitive Abilities Test (CogAT in the US, CAT in the UK) as their tool to measure the impact of the P4C program, and thus they used an instrument that was wellregarded and widely-studied. From the publisher's website here is a description of the CogAT/CAT: Reasoning abilities have substantial correlations with learning and problem solving, both in and out of school. CogAT's measurement of three different content domains ensures that educators receive a balanced view of the child. take extensive and expensive staff preparation. All that was done was to take the participating teachers, those who had been randomly selected, through one day on training with the materials. The training involved an introductory overview of P4C, and then working through three lessons in the book in the same was that they would in turn work with their students.
The point is simply that an educational intervention that is costly in terms of instructional time, materials, or teacher training is less likely to be adopted as a standard component of the school curriculum even if it has a beneficial effect on students. We were looking for a program that school leaders would be more likely to adopt once its educational effectiveness was demonstrated, and, with that aim in mind, less costly is better.
In the Scottish study, Dr Keith Topping of the University of Dundee supervised the work of a doctoral student, Steven Trickey, who was at that time working as a psychologist for the Clackmannanshire Educational Authority where the study took place. In addition, Paul Cleghorn served as a Head Teacher in the school system and, along with Stephanie Baudet, had put together a set of materials for reflection in a series of books entitled Thinking through philosophy. This particular study used Thinking through philosophy: Book 4 and that book contained an extensive description of how to engage students to form a community of inquiry in the classroom. Each philosophy discussion session was structured as a process with a series of seven stages, beginning with a focusing exercise to create a relaxed, meditative frame of mind and followed by a brief link with the previous week's discussion. Then came the stimulus for discussion-usually reading a story from Thinking through philosophy. Next the students worked in pairs discussing the story and reflecting on some open-ended questions suggested by the story. This was followed by dialogue in larger groups where the teacher has encouraged the students to form a community of inquiry by: Finally, the teacher brings closure by encouraging the students to reflect on the discussion and how their thinking might have progressed, and by providing a 'thought for the week' that highlights an idea to serve as a basis for 'homework' to be reflected on in order to relate the idea to situations outside the original stimulus (Topping & Trickey 2007b; for more detail see Cleghorn & Baudet 2002). Topping and Trickey (2007a) reported achieving substantial pretest to posttest gains in CogAT/CAT scores for their experimental group versus their control group, and our P4C interest group decided those results were worth serious attention. But then Topping and Trickey did what few others had done, namely, they conducted a follow-up study two years later in 2005. This was important because of the tendency of the impact of many educational interventions to fade over time (Cascio & Staiger 2012). In their follow-up study Topping and Trickey (2007b) were clearly able to document the persistence and durability of the effects on the students' cognitive abilities of the P4C intervention, and that was very encouraging, given the correlation of CogAT/CAT scores with academic achievement as documented by Deary et al. (2007) and others. The results of the original study and the two-year follow-up can be seen in Figure 1 provided to us by Dr Trickey.

I. Our initial study in 2010-2011
In this section we provide a brief description of our 2010-2011 initial study in order to provide background information for our report of our follow-up study three years later. If a more complete description of the initial study is desired, please see the article which reports on it (Fair et al. 2015). During the school year of 2010-2011, we attempted to replicate the Topping and Trickey (2007a) study by looking at the impact of the same P4C program on seventh graders (12-13 years old) in a Texas middle school. We structured the situation to emulate, as closely as we could, the intervention that Topping and Trickey employed in their study. Thus we used the same materials from Thinking through philosophy: Book 4 as the main source to prompt the weekly discussions. In addition, we spent one day before the start of school with the teachers to train them by having them role-play as students. We followed the same 7-stage process for each discussion. Prior to the start of the school term in late August of 2010, the teachers were trained by having them role-play as students. Dr Trickey was able to consult with us and take part in the teacher-training session. The participating seventh grade teachers were Language Arts teachers who were randomly selected on the morning of the training. Those seventh grade teachers who were not selected were strongly advised not to incorporate any of the P4C program into their classes in order to maintain the integrity of the control group. We structured the student sessions to emulate, as closely as we could, the intervention that Topping and Trickey employed in their study. We used the same materials from Thinking through philosophy: Book 4 as the main source to prompt the weekly discussions. We followed the same 7-stage process for each philosophical discussion.
The result was an experimental group containing 186 seventh grade students and a control group containing 79 seventh grade students (for further details, see the report by Fair et al. 2015).

Analyses of the quantitative data
To According to Cohen (1988), the effect size for the difference was moderate to large.
The results of our replication study are displayed in Figure 2. As one can see-and the statistical analysis bears this out-the 22-26 weeks of onehour-per-week Philosophy discussions within the seventh grade experimental group had a significant impact on CogAT/CAT scores (Fair et al. 2015). This meant that our results for the seventh graders replicated the results that Topping and Trickey (2007a) had obtained. Additionally, our results showed that 22-26 weeks was a sufficient period of time to produce the results, and that length of time could be easily accommodated in one school year in the USA. In contrast, the original Topping and Trickey study concerned the effects of a 58-week program of weekly discussions-somewhat over twice as many weeks as in our study. That led us to wonder whether our program, one involving substantially fewer weeks of discussions, would have a similarly durable impact, or whether the results would be more susceptible to fading over time.

II. The three-year follow-up study
It happened that we were able to conduct our own follow-up study. The question was whether a follow-up at the three-year mark, instead of the two year mark for the Topping and Trickey study, might display a noticeable 'fading' of the results. Here is what we found.

Analyses of the quantitative data
Analyses of data were conducted to determine whether a statistically significant difference was present between the pretest scores as seventh grade students and posttest scores of experimental group and control group participants three years later as tenth grade students. The reader will note that the means are slightly adjusted as plotted on the previous figure for the first and second round of testing for seventh grade students. This is due to changes in the number of participants in the final round of testing for comparison of all three rounds of testing.
Parametric dependent samples t-tests were conducted to determine differences between pretest and posttest scores among the two groups.  Cohen (1988) the effect size for the difference was moderate to large. In reference to pretest and posttest scores for tenth grade students, a greater difference in posttest scores was present for participants in the experimental group than for participants in the control group, which is consistent with the results noted between the two groups in 2010-2011.
Presented in Table 1 are the descriptive statistics for all three rounds of pretest and posttest scores and the results of both studies are presented in Figure 3.

III. Discussion
In the article describing the results of their follow-up study, Topping and Trickey write: This study provides evidence that gains in cognitive ability which opened up over the intervention period in primary school can be maintained over the subsequent 2 years into secondary school in the same measure, even when pupils have not had any further experience of collaborative inquiry in the secondary school. (2007b, pp. 793-794) Our study, now including the three-year follow-up report, strengthens and extends their work. First, our study strengthens their results by showing that the program of collaborative philosophical discussions that they used had the same powerful effect on students who were not only half a world away from Scotland, but who were more ethnically diverse. The particular Texas school district enrollment in 2010 was comprised of 46% white, 26.9% African American and 25.8% Hispanic students (Fair et al. 2015). Also the school district is not in a wealthy community since in 2009 the median annual income was US $33,359 in contrast to the state median income of US $49,433. Indeed, at the time of the study 60.2% of the students in the district were classified as 'economically disadvantaged' (Fair et al. 2015 showing that the presence of the originators of the program was not essential for the program to have an effect. Finally, as we report now, the fact that we could detect a significant difference between the experimental and control groups of students after three years, not just two, gives greater credence to the likelihood that the intellectual effects produced by the one-hour-per-week philosophy discussions are as durable as one could wish.

IV. Conclusion
Historically, the Philosophy for Children movement was inspired by thinkers such Foundation in the UK. Here are two key conclusions from the report: 1. There is evidence that P4C had a positive impact on Key Stage 2 attainment. Overall, pupils using the approach made approximately two additional months progress in reading and math.
2. Results suggest that P4C had its biggest positive impact on Key Stage 2 results among disadvantaged students (those eligible for free school meals). (Gorard et al. 2015, p. 3).
Given the results of these studies, the powerful impact of the process of one-hourper-week philosophy discussions is clear, and, since the particular program that structures this process of discussion is not costly, we are led to recommend that the program be incorporated into the school curriculum as a standard component.