Exploring Students' Understanding of Linear and Quadratic Relationships in a Projectile Motion Context

This study demonstrates that a digital teaching experiment involving projectile motion can foster middle school students' covariational reasoning and deepen their understanding of linear and quadratic relationships, particularly when task design emphasizes comparing these relationships and interpreting graphs as representations of covarying quantities.

The Gist

Imagine you are watching a movie of a ball being thrown into the air. You see it go up, slow down, stop at the very top, and then come crashing back down. Now, imagine trying to draw a picture of that exact moment-to-moment movement on a piece of paper.

For many students, this is like trying to translate a foreign language they haven't studied yet. They see the ball moving and they see the graph, but they don't realize the graph is the story of the ball's journey. They often think the graph is just a picture of the path the ball took (like a map), rather than a chart showing how two things (height and time) are changing together.

This paper is about a study that tried to teach two middle school students how to read that "story" using a special digital game. Here is the breakdown in simple terms:

The Problem: The "Map" vs. The "Story"

Most students look at a graph of a ball's flight and think, "Oh, that's just a curve." They treat the graph like a map of the hill the ball rolled over.

• The Reality: A graph isn't a map; it's a dance partner. It shows how two dancers (Height and Time) move in sync. As Time moves forward, Height goes up, then down. The graph is the record of that dance.

The Experiment: A Digital Playground

The researchers set up a digital game called "How Accurate Is Your Aim?" for two students, Fania and Bianca.

• The Twist: Usually, math graphs put Time on the bottom (horizontal) and Height on the side (vertical). This game flipped it! Time was on the side, and Height was on the bottom. It was like looking at a movie screen sideways.
• The Goal: This "weird" view forced the students to stop looking at the graph as a picture of a hill and start thinking about the numbers changing.

The Journey: From "Gross" to "Smooth"

The researchers watched how the students' thinking evolved, using a framework that measures how well they understand two things changing together.

1. The "Gross" Coordination (The Big Picture)
At first, Fania and Bianca looked at the graph and said, "It looks like a half-oval, then a straight line."

• What they got: They understood the ball went up, came down, and then stopped.
• What they missed: They saw the shape, but they didn't fully grasp how the numbers changed together. They were looking at the "big picture" without counting the steps.

2. The "Chunky" Coordination (The Step-by-Step)
The teacher then drew a straight line on a whiteboard and asked, "Why is the ball's graph curved, but this one is straight?"

• The "Aha!" Moment: The students realized that a straight line means the ball is moving at a constant speed (like a car on cruise control). Every second, it goes the exact same distance.
• The Breakthrough: They looked back at the curved ball graph and realized: "Ah! The ball isn't moving at a constant speed. In the first second, it flies high. In the next second, it doesn't go as high. In the next, even less."
• The Analogy: They started thinking of the graph as a video with a pause button. They could pause every second and see that the "height gain" was shrinking. This is called Chunky Continuous Covariation—understanding change in distinct, measurable chunks.

3. The "Smooth" Coordination (The Flow)
Finally, when looking at the part of the graph where the ball hit the ground and stopped, the students realized something profound.

• The Insight: The line went straight up. The students said, "The ball stopped moving, but time kept going."
• The Analogy: Imagine a runner who stops at the finish line but the race clock keeps ticking. The runner's position (height) is frozen, but the time (the clock) is still flowing. The students understood that the two variables were still linked, even when one stopped changing. This is Smooth Continuous Covariation—seeing the flow of time and height as a continuous stream, not just a series of stops.

Why This Matters

The study found that comparing the curved ball graph with a straight-line graph was the key.

• It's like teaching someone to taste a complex spice by first giving them plain water. Once they taste the water (the straight line/constant speed), the spice (the curved line/changing speed) makes much more sense.

The Takeaway

By using technology that links the moving ball to the graph in real-time, and by flipping the axes to confuse their usual habits, the researchers helped the students stop seeing graphs as "pictures" and start seeing them as "stories of change."

In short: The students learned that math isn't just about drawing shapes; it's about understanding the rhythm of how things change together. They moved from seeing a static drawing to hearing the music of the motion.

Technical Summary

1. Problem Statement

Students frequently struggle to develop a deep understanding of linear and quadratic relationships, particularly in connecting algebraic forms, graphical representations, and real-world contexts. A common misconception is the spatial interpretation of graphs, where students view a graph as a literal picture of a physical trajectory (e.g., seeing a parabolic graph as the physical path of a ball) rather than as a representation of how two quantities covary (change together).

While covariational reasoning (the ability to mentally envision two quantities changing simultaneously) is identified as a critical mechanism for understanding functions, there is a need for empirical evidence on how specific instructional designs can foster this reasoning to help students distinguish between linear (constant rate of change) and quadratic (varying rate of change) relationships in dynamic contexts like projectile motion.

2. Methodology

The study employed a teaching experiment design, a qualitative research approach focused on characterizing student activity and conceptual development.

• Participants: Two ninth-grade female students (pseudonyms: Fania and Bianca) from a public middle school in Yogyakarta, Indonesia. They were selected for diverse academic abilities and strong verbal communication skills.
• Context: The study took place in a digital learning environment using Desmos Classroom (Amplify). The researcher acted as a teacher-researcher (TR) facilitating the session.
• Task Design: The core intervention was the "Height-Time Relationship Task," part of a sequence titled "How Accurate Is Your Aim?" Key design features included:
  • Real-world Context: Projectile motion (launching a ball).
  • Non-Canonical Graphing: The axes were reversed compared to standard conventions. Height was plotted on the x-axis, and Time on the y-axis. This was intended to disrupt spatial associations and force quantitative reasoning.
  • Technological Affordances: The task synchronized a video animation of the ball's motion with dynamic graphical representations (moving line segments and a tracing point).
  • Comparative Prompting: The final screen required students to interpret a curved graph (quadratic) and compare it against a linear graph drawn by the researcher on a whiteboard.
• Data Collection & Analysis:
  • Data included screen recordings, webcam footage of facial expressions, and audio/video of gestures.
  • Analysis focused on the third screen of the task.
  • Theoretical Framework: Data was coded using Thompson and Carlson's (2017) Covariational Reasoning Framework, which ranges from "No coordination" to "Smooth continuous covariation."
  • Focus: The analysis traced the evolution of the students' reasoning, specifically looking for evidence of coordinating values, chunky continuous variation, and smooth continuous variation.

3. Key Results

The study identified a developmental shift in the students' reasoning as they engaged with the task:

• Initial Interpretation (Gross Coordination):

  • Initially, students described the graph as a "trace" or "record" of points.
  • They correctly interpreted the non-canonical graph (Height on x, Time on y) quantitatively. For example, they identified the upward-left curve as the ball descending (height decreasing while time increases).
  • At this stage, they demonstrated Gross Coordination of Values, recognizing the general trend (one goes up, the other goes down) but without explicitly pairing discrete values or calculating rates.

• Shift via Comparison (Chunky Continuous Covariation):

  • When prompted to compare the curved graph with the researcher's linear graph, the students' reasoning advanced.
  • They identified that the linear graph represented a constant rate of change ("every second it's the same").
  • They contrasted this with the curved graph, noting that for equal time intervals, the change in height was not constant. Fania described how the ball moves "quickly" in the first second and "slows down" in subsequent seconds.
  • This indicated a shift to Chunky Continuous Covariation, where students coordinated discrete chunks of time with varying changes in height, understanding the concept of a changing rate.

• Advanced Reasoning (Smooth Continuous Covariation):

  • When interpreting the vertical line segment of the graph (representing the ball stopped on the ground while time continues), Bianca explained that the height remained unchanged while time flowed continuously.
  • This demonstrated Smooth Continuous Covariation, the highest level in the framework, where the student envisions variables varying simultaneously and continuously without discrete "chunks."

4. Key Contributions

• Pedagogical Strategy: The study provides empirical evidence that comparing linear and quadratic relationships is an effective strategy for fostering sophisticated covariational reasoning. The contrast helped students articulate the difference between constant and varying rates of change.
• Task Design Validation: It validates the use of non-canonical graphing tasks (reversed axes) to prevent spatial misinterpretations. By forcing students to reason quantitatively rather than visually, the task successfully elicited covariational reasoning.
• Role of Technology: The study highlights how synchronized digital animations and dynamic graphs help students construct the "quantities" involved in a phenomenon, bridging the gap between physical motion and abstract representation.
• Theoretical Application: It successfully applies the Thompson and Carlson framework to middle school students, showing that even younger learners can access "chunky" and "smooth" continuous reasoning when supported by appropriate scaffolding.

5. Significance

This research addresses a critical gap in mathematics education: helping students move beyond static, spatial graph interpretations to dynamic, functional reasoning.

• For Instruction: It suggests that teachers should explicitly encourage students to compare different functional relationships (linear vs. quadratic) to deepen their understanding of rate of change.
• For Curriculum: It supports the integration of technology that links physical phenomena directly to graphical representations, particularly using non-standard orientations to challenge student assumptions.
• For Future Research: The study lays the groundwork for investigating how covariational reasoning can be extended to parametric representations and the concept of the "rate of change of the rate of change" (the second derivative), which is fundamental to understanding quadratic functions fully.

Limitations: The study is limited by its small sample size (two students) and a focus on only one specific screen of a larger task sequence, which limits the generalizability of the findings to broader populations.