Investigating how students collaborate to generate physics problems through structured tasks

Student groups discussed and made decisions regarding the learning goals for their problem, and expectations of what the targeted students should learn. The learning goals of the problem consisted of enabling secondary school students to utilize their physics knowledge (e.g., “The idea is that they would exercise with the problem we give them”). The school level of the targeted secondary students emerged in multiple opportunities in favor (e.g., “If we think on the students’ age, they should know how to do such operations.”) or against (e.g., “Maybe that is too much for a kid in 10th grade. Because that is something they would do in 11th grade.”) the difficulty of the problem in terms of mathematics representations and concepts. In Group 2, this discussion concentrated on the reasons for using circular motion as the key concept, its importance for them (university students) and the targeted secondary students. From the example below, it is possible to perceive the intention of making sense of the activity and the overall objective of learning concepts and principles of circular motion:

Student A: What is the goal of this (activity)? we mean, of teaching this?

Student B: What thing?

Student A: All this equations and concepts.

Student B: For us or for students who would be solving this?

Student A: Well, for both.

Student B: It is assumed that almost every movement is circular, as there is rare to find truly straight movements. These do not exist. You will see that this (movement) has no angles, but in reality it has.

Student A: It would be like explaining to them (high school students) how these (movements) work.

The discussion about the learning goal emerged from the explicit objective of the task, yet was not necessarily supported by the argument made by student B in regards to the ever-present circular motion, which attempted to highlight its importance by transferring and extending the real nature of motion upon a combination of circular displacements. The latter argument is interesting, as students explicitly attempted to make sense of the content to be learned for the targeted students, yet this idea received no follow up from other team members.

Group 2 linked the goal of the task to what science teachers would do in the face of a similar task. The following segment shows this brief moment of reflection, where the group attempted to convey the appropriateness of their problem in coherence with the learning objectives:

Student C: We have to be clear with the goal, which consists of teaching and learning kinematics of circular motion to 12th grade students. So, it is like a teacher preparing to teach circular motion. That way, each element of circular motion could be linked to different contexts from daily life, or just one. It has to be didactic for them (high school students) to understand.

Student D: Let us do problems similar to the ones our instructor has used.

Student C: That is it, didactic.

This segment provided evidence that, in finding the learning goal, students mirrored what experts [secondary school teachers] would do in contextualizing the content, and projected their own expectations into what a physics problem should look like. Finally, in Group 3, we observed a deeper reflection of the learning goal, where a student highlighted the importance of the real-life context for learning:

So, if you include a difficult exercise, but they do not know how to solve it through equations, they would remember that they tackle a problem involving a laundry machine where there was a circular motion and were able to calculate the speed. Consequently, and lastly, they would understand and know how to calculate angular and tangential speed for a laundry machine, and they would imagine the same type of motion but on different problems.

According to this quote, the context would mediate the difficulty of the problem in the case that the targeted students were incapable of using the needed equations, as they would ultimately associate the context of the laundry machine with circular motion. Through this link, the student argued that learners would draw similarities in the use of equations for the purpose of calculating quantities across different scenarios. This, in essence, is the notion of transferring knowledge, or in other words, the use of information from a well-known to an unknown situation.

During this process, teams identified, posed, and decided on the physics concepts to use in the problem. In addition, they looked at how these concepts aligned for the generated problem to be well-structured, and consistent with task requirements. Defining procedures was in direct connection with the learning goal, as teams engaged in the former process to meet the expectations previously defined. When addressing this theme, groups attended to and emphasized different sets of elements, such as the algebraic steps through manipulation of equations, concepts and the combination of quantities for an appropriate problem structure. Figure 2 would allow readers to identify the set of concepts used by each group on their respective problems. For instance, Groups 1 and 2 from section 1 decided to use the angular version of speed, distance, acceleration as data to determine the magnitudes defined as questions. In contrast, Groups 3 and 4 from section 2 selected the linear version of speed, distance and acceleration as initial conditions that would allow solvers to determine the number of revolutions completed at the end of motion, the final distance covered, speed, and other questions.

We observed two different sets of strategies for addressing the early stages of this process and making decisions: Equation-driven and Concept-driven. The first approach (Equation-driven) was observed in teams that primarily focused on the mathematical dimension of the problem for decision making, whereas concept-driven strategies emphasized the conceptual dimension of the situation to then reflect on mathematical representations. Group 1 mostly utilized the Equation-driven approach when defining concepts and procedures. In doing this, they proposed to include the equations in the problem so that students could easily ‘plug and chug’ and find the solution (e.g., “I supposed we need to include the equations. That way they only need to replace what’s there.”). Consequently, this process was guided by the (implicit) idea that a problem is constructed in the same way that one may solve it, which refers to following very structured set of steps (e.g., “So the problem must be in order. First you calculate one (value), which then allows you to find other.”). The example illustrates the algorithmic nature of developing a problem for Group 1. Similarly, Group 4 engaged in such a strategy for defining concepts and procedures, yet transitioned towards a Concept-driven description of the phenomena after establishing the situation to be used:

It will start from rest, and that way we could calculate the movement of the barrel. So then, we would tell them that the barrel is accelerating constantly and that it needs certain time in seconds to hit the target. Because after some seconds the barrel will be there, at its final position. That is, it will impact the target, so then they could begin their calculations for different things, like angle and everything.

This quote from a member of Group 4 showed a simple physics analysis of the situation used (i.e., a barrel is thrown to a person), as the student analyzed the position and evolution of the object through time, and enabled further understanding of the procedures the targeted students are expected to go through for solving the problem.

Even though deciding on concepts and procedures utilizing Concept-driven approach is not absent from the attention to equations for decision making, the subtle difference is that the equations emerged after deciding on the concepts first. For instance, a student in Group 2 stated: “So let’s create a situation where we combine angular speed, acceleration and everything else, like a situation that involves circular distance.” This approach helped the group to create a problem based on the relationship between concepts rather than on the exclusive use of equations. In Group 3, this was insinuated by a member arguing against the equation-driven approach in the description of the physics regarding the situation selected:

More than the equations, it would be better to say the there is a force acting over there, whereas there is another force but in that direction… We need to be more specific. For instance, say that there is a force acting to the inside, and another to the outside.

Even though the use of forces is beyond what is expected for the target students to know, this qualitative description provided a conceptual framework for the group to decide on the physics for the problem.

Groups posed ideas and decided on the contextualization (i.e., place, subjects, actions etc.) and wording of the problem. This process required groups to invest considerable amount of time (21,4% of the data, see Table 2), which is not surprising taking into account the need to select a daily situation where circular motion is observed. Groups experienced some conflict to find the right contextual elements to use. For instance, Group 1 started by only focusing on objects with wheels, yet members attempted to achieve some degree of novelty by pushing the conversation towards situations beyond wheels (i.e., “Is there one without wheels? I cannot think of anything.”). Group 4 wanted to create something fun for students to be motivated with (e.g., “Let us do something cool, like a wooden spinning top.”). Other participants suggested ideas like a fisherman moving his fishing rod and describing a circular motion, or an ant walking on the inner wall of a bottle. Here, originality was controlled by their level of confidence in exploring such situations through their physics knowledge, (e.g., “We do not need to complicate ourselves with that.“) and ended up using more familiar situations.

As seeing in Figure 2, Group 1 selected the wheels of a car moving and covering a trip between places; Group 2 decided on the use of a person trapped in a barrel in motion; Group 3 decided on the motion of a bicycle; and Group 4 utilized references from a problem created by their instructor to design a situation where Donkey Kong moves a barrel.

The wording of the problems illustrated the use of technical language, and became an opportunity for participants to challenge their own conceptual understandings. A simple example emerged from Group 1, where a student asked the following when deciding on the right wording of the problem: “If I want to say that the car wants to move from A to B, is that displacement? “ Because displacement is defined as a vector, then to properly use it, the group needs to incorporate a direction. There is no evidence in the data when this correction was made, but the problem is worded using scalars rather than vectors, and framed the phenomena as “a car wants to move”.

Another example of negotiating the wording with a correct use of physics concepts was observed in group 4, when students discussed the conditions under which Donkey Kong would make the barrels move:

Student P: For this, he tosses the barrel from rest?

Student O: You cannot toss a barrel from rest. It releases the barrel then. He let the barrel go.

Student P: Ah, okay.

Student O: Or better, he tosses it with an initial speed, is that okay?

This segment showed students’ understandings of motion in connection with an appropriate use of language to convey the idea that releasing and tossing the barrel implies different physical conditions. Here, a body that begins its motion from rest must be release to accelerate due to the presence of an external force (e.g., gravity), and will therefore gain speed. Differently, tossing implies an interaction (i.e., force) that boosted energy and therefore speed to the object that was moving. Both ideas showed comprehension of motion and the implications of forces for the motion defined in the problem.

During this process, groups engaged in discussions and decision making regarding relative values and magnitudes, as well as measurement units for the physics concepts (e.g., 10 km/h, 20 s, 2.5 km) to be introduced into the problem’s description. This process is important as it brings a sense of ‘reality’ to the physics of the phenomena and situation under design. This process enabled groups to identify and select appropriate values to link with the physics concepts used as data. The validity of these magnitudes was tested through the calculation of their unique responses. For instance, Group 2 discussed the appropriateness of a high angular acceleration for the barrel that yield to 1,400 rpm (revolutions per minute), and decided to ‘Lower the values’. Similarly, in Group 1, we observed the following interaction for deciding on the acceleration of the car:

Student A: What do we say the acceleration is?

Student B: 20.

Student A: 20 what?

Student B: Meters by square second.

Student A: Is that too much?

Student B: I know it is a lot. Do you want the car to get to its destination fast or slow?

Student A: I want it to get there at a normal speed.

This dialogue reflects the intention to utilize magnitudes that resemble real life situations. Later on in this process, the same group tested the problem with an acceleration of 10 m/s, and obtained a final speed of 200 m/s, an unrealistic result for a common car moving in the city.

This process relates to algebraic steps that the group went through to obtain physical quantities needed to solve and test the appropriateness of their designed problems. This included suggesting strategies to determine a physics quantity (e.g., “Here we will use a proportionality rule. If one revolution is 2$\pi$, then x revolutions will be…”); and/or requesting advice on how to proceed in order to get the right value (e.g., “How do we transform this to radians? Does someone know how to?”). Most of the evidence found here emerged when students wanted to achieve either of the latter two goals.

To contextualize the use of algebra in this context, it is important to remember that kinematics problems rely on three fundamental physical quantities: position [$\vec{r}(t)$], velocity [$\vec{v}(t)$] and acceleration [$\vec{a}(t)$], all functions of time $t$. Even though these concepts are defined as vectors, in this context students utilized these mathematical representations to determine scalar quantities, or the magnitudes of the vectors at any given time. In circular motion, these concepts are written in an angular form: angular position [$\theta(t)$], angular speed [$\omega(t)$] and angular acceleration [$\alpha(t)$]. The link between these linear and angular magnitudes comes from $s=\theta R$ (i.e., arc) , $v=\omega R$ and $a=\alpha R$. Consequently, in order to test their problems, students manipulated some or all of the latter mathematical relations. For instance, students in Group 4 had the following argument to determine the angular position of distance covered by the barrel:

Student L: And how would I get the angle?

Student M: With the (angular) acceleration that is obtained from the equation. With the angular speed. Because you have the angular speed at 3 s, which is 10 $\pi$, so 10 $\pi$ is equal to (angular) acceleration plus the initial angular speed. So then you clear and get the (angular) acceleration.

Here, student M suggested the use of angular speed [$\omega(t)$] at time 3 s, to determine the value of angular acceleration by isolating this value from the equation, because all other elements were given. Once this was done, the argument, although not explicitly mentioned, oriented to the use of this value of angular acceleration into the equation for angular position, and calculate $\theta$ at 3 s. Once again, this was possible because all other elements in the equation are defined.

Another interesting example was observed in Group 1, as they used the equations $\vec{r}(t)=\vec{r}_{0}+\vec{v}_{0}t+1/2\vec{a}t^{2}$ and $\vec{v}(t)=\vec{v}_{0}+\vec{a}t$ to determine the time that it will take the car to reach its destination. The following interaction depicts the set of algebraic steps suggested for one member to achieve this goal:

Student A: So we have that the initial position is zero, and the initial speed is zero. And we have that (in the equation), only the acceleration for the square time will remain.

Student B: And then?

Student A: That will give you 10 m/s (magnitude of the acceleration) times the square time, and with that you can obtain the time. So, it will be the square root of something, and then we will use only the positive root.

This interaction shows an appropriate use of the equation $\vec{r}(t)=\vec{r}_{0}+\vec{v}_{0}t+1/2\vec{a}t^{2}$ to obtain the time, given that all the elements but the time are known (final distance is given in the heading of the problem and is equal to 2 km). Because mathematical manipulation may be perceived as a rather individual exercise, it is not surprising that the audio recording only captured brief descriptions of the strategies to be implemented to find numerical values, and the request for advice on how to calculate them.

This theme consisted on groups addressing the qualitative description of the physics regarding the circular motion in the context under consideration for the problem. This process enabled access to students’ conceptualization of the physics phenomena in question and the ways in which they would explain such situations. The sample size of examples that illustrate this process is rather small and does not provide evidence on the most difficult concepts. Consequently, the observed frequency (see Table 2) reflects the disparity between algebraic-versus-qualitative strategies students displayed. Moreover, qualitative descriptions emerged from the data when students tried to make sense of the situations and physical objects considered for the problem.

There is evidence that students attempted to explain revolutions, tangential velocity, and inertia, the last being used to determine what would happen if someone were to fall from a fast-spinning carousel. The concept of a revolution was conceptualized through the perimeter of a wheel: “Suppose that first there is a point that moves along the perimeter until here. That will be one revolution”. This description is very simple and does not really reflect deep understanding of a physics concept that one were to associate with motion.

A more interesting example was provided by Group 4 in an attempt to understand the relationship between angular and linear speed in the context of a wheel moving. First, angular speed is defined as the change of angular position per unit of time (i.e., $\omega=\Delta\theta/\Delta t$), and may be difficult to understand because it does not imply distance units, such as meters or kilometers. Secondly, an object spinning will measure the same angular speed at any distance from the center of rotation (i.e., radius) at a particular time. However, the linear or tangential speed will increase according to the distance from the center of rotation as shown by the equation ($v=\omega R$). Next, the discussion unfolded as follow:

Student E: If this is supposed to be in the same wheel, then why? If you advance five meters, you will complete the same number of revolutions.

Student G: Yes, you are right. So, how many…

Student F: The angular speed will change at different points of the wheel.

As one would notice, the claim made by student F suggested a misconception regarding the nature of angular speed, because this quantity remains constant regardless of the distance from the center of rotating object. Consistent with the definition of angular speed, the comment made by student E would have made more sense if instead of using 5 meters as the distance covered, he would have used angular measurements to highlight the distinction with linear speed.

The last example that illustrates the nature of qualitative descriptions used by students emerged in Group 3 when discussing the nature of acceleration and forces on circular motion:

Student S: There is also velocity, this velocity that goes to the middle. This is the one that enables… This was related to forces if I remember. The topic of the two forces pointing out to one side. Now I remember, centripetal and centrifugal force.

Student T: Centrifugal force was like…

Student U: It is the one that points inside.

Student T: No.

Student S: Centripetal force points to the inside.

Student T: Okay, centripetal force points to the inside. But centrifugal points to the outside. And those two forces would make that… “There were like equals and…

Student U: Both forces allow the circular motion.

Student T: But this centrifugal force was something like hypothetical, or something that was not real…

According to the interaction, students attempted to make sense of the physical interactions that enable circular motion: in this instance, centripetal force. The ideas related to centripetal force are correct: it is an interaction that is directed to the center of the circumference described by the motion, and it is responsible for the circular motion. However, centrifugal force, as corrected by student T at the end of the interaction, is not a force but rather the effect of inertia, defined as resistance to change the state of motion and is often referred to as a “fictitious force”. Finally, this interaction provides some insights on students’ understandings, but again fall short to give substantial evidence to assess whether students are actually understanding the underlying physics of circular motion beyond the use of equations.