Students Learn Quadratic Functions Better When Formulas Come After Patterns of Change

Quadratic functions are often the first genuinely nonlinear mathematics that students meet, and a new study suggests that the order in which representations are introduced may matter more than the representations themselves. In a design research study published in Discover Education, researchers from Universitas Pendidikan Indonesia and Universitas Riau worked with 160 tenth-grade students at two public senior high schools in Riau Province, Indonesia, under the Kurikulum Merdeka curriculum. Their goal was to help students understand quadratic functions not as formulas to memorize but as patterns of change, and to trace how students move between contextual situations, numerical tables, symbolic expressions, and graphs.

The starting point was sobering. Before any new instruction, 80 students completed an initial mathematical representation test with five written items scored on a 0-to-4 rubric across three indicators: visual, symbolic, and verbal. Combined achievement across the three indicators was only 34.53 percent. The symbolic indicator was weakest at 17.66 percent, followed by the verbal indicator at 40.63 percent and the visual indicator at 45.31 percent. In other words, translating a situation, a graph, or a table into an algebraic expression was where students struggled most, even though they could often recognize a parabola when they saw one.

Classroom observations reinforced the test results. Quadratic functions were typically introduced through the general form f(x) = ax² + bx + c and its algebraic characteristics, such as maximum or minimum values and intersections with the coordinate axes. Instruction emphasized symbol manipulation over meaning. On the symbolic items, none of the 80 students completed the graph-based item in full. Although 13 students, 16.25 percent, correctly derived the quadratic formula, none went on to determine the corresponding pattern of function values that the item also required. On the contextual item, which asked students to model a ball’s trajectory and decide whether it would pass through a goal, only 7.5 percent answered correctly.

Semi-structured interviews with 12 purposively selected students revealed how students framed the concept. Three described a function purely as a computational tool, with one student saying, “A function is just a formula, Ma’am, used for calculation.” Two recalled a memorized set-based definition without understanding its meaning. The remaining seven, the largest group, identified a quadratic function primarily by the presence of the x² term or the shape of a parabola rather than by any relationship between changing quantities. “If there is x², then it is a quadratic function,” one student explained. Graphical work showed a related pattern: 58 of the 80 students, 72.5 percent, produced graphs that were not parabolic, and 29 students, 36.25 percent, connected plotted points with straight-line segments, treating quadratic situations with strategies characteristic of linear graphs.

Perhaps most striking, none of the twelve interviewed students identified a pattern of change as a defining feature of a quadratic function, regardless of their written performance. Four of them had already derived a correct quadratic formula on the graph-based item yet still could not identify the underlying pattern when asked directly. Other students judged a quadratic data set to be linear “because the values kept increasing,” or said the numbers seemed random. This suggests that pattern recognition functions as a distinct competency rather than a natural extension of symbolic proficiency, a dissociation that became the specific starting point for the instructional design.

The learning sequence integrated three tools, each matched to a distinct representational purpose. A physical walking activity opened instruction because it grounded the idea of changing quantities in an embodied experience: students walked first at a constant pace and then at an accelerating pace, producing an intuitive contrast between constant and non-constant change before any numerical or symbolic representation appeared. The PhET Moving Man simulation followed immediately, reproducing the same motion context under controlled conditions with position = 0, velocity = 0, and acceleration = 2, allowing students to generate stable position data that the walking activity alone could not consistently provide. GeoGebra came last, after students had constructed a symbolic generalization, so they could manipulate sliders for the coefficients a, b, and c and observe how each shapes the resulting graph.

The teaching experiment involved a separate group of 80 students, 40 per school, organized into ten heterogeneous groups of four per class. Each cycle ran as a single 135-minute session covering a contextual walking activity, the PhET simulation, a table-completion task, a GeoGebra exploration, and practice items. Cycle 1 exposed two obstacles. In the walking activity, conducted without a structured measurement procedure, 8 of 10 groups produced data too irregular to discern a consistent pattern of first or second differences. In the PhET activity, only 4 of 10 groups independently derived the correct symbolic generalization, and 3 of 10 groups required teacher assistance to identify the constant second difference. Four groups were also confused by position values that were not whole numbers.

The researchers revised the design in response. Students were instructed to walk in stages and stop briefly to measure distance at each one-second interval, and a generalization row represented by x was added to the existing table to help students move from particular values to general symbolic forms. Two brief prerequisite reviews were also added: decimal rounding before the simulation, and number classification before the GeoGebra slider activity. In Cycle 2, only 3 of 10 groups produced walking data too irregular to show a trend, only 1 of 10 groups required assistance identifying the constant second difference, and 8 of 10 groups derived the correct symbolic generalization independently. All 10 groups concluded that the coefficients a, b, and c were real numbers, compared with 8 of 10 in Cycle 1.

One group’s worksheet illustrates the mechanism. Using the simulation, the group recorded position values of 0, 1.085, 4, and 8.752, then rounded them to 0, 1, 4, and 9. From this sequence they computed first differences of 1, 3, and 5, identified a constant second difference of 2, correctly predicted the values 16 and 25 for the following two seconds, and concluded that f(x) = x². This episode shows that second differences did not function merely as a computational step; they provided an intermediate representation through which students could connect numerical growth with symbolic structure. Overall, 7 of 10 groups in Cycle 2 reached representational coordination, consistently linking the numerical pattern, its symbolic generalization, and the underlying motion context without teacher support, compared with 4 of 10 in Cycle 1. Mean quiz scores rose from 5.05 out of 8 (63.13 percent) to 6.73 out of 8 (84.13 percent), with correct answers increasing from 25 percent to 55 percent of students.

The researchers caution that the two cycles occurred in different classes at different schools, so the improvement cannot be attributed solely to the design revisions, and the study examined only the first meeting of a longer sequence. Still, the findings yield three concrete design principles: begin with contrastive situations that allow students to compare linear and quadratic change; use second differences as a representational bridge between numerical patterns and symbolic generalization; and formalize the general form f(x) = ax² + bx + c, with a ≠ 0, only after students have recognized the structure of change in values. The lesson for teachers is that learning quadratic functions should not begin directly from formulas or graphing procedures, but from activities that help students construct meaning through connected representations, with the formula emerging as a symbolic generalization of a pattern students have already seen.

Subject of Research: Supporting students’ understanding of quadratic functions as patterns of change through representational transitions

Article Title: Supporting students’ understanding of quadratic functions as patterns of change through representational transitions

Article References: Siregar, H. M., Turmudi, T., Nurjanah, N., Darhim, D., & Nurlaelah, E. (2026). Supporting students’ understanding of quadratic functions as patterns of change through representational transitions. Discover Education, 5(1), Article 1177. https://doi.org/10.1007/s44217-026-02298-9

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02298-9

Keywords: mathematics education, quadratic functions, representational transitions, design research, patterns of change, PhET simulation, GeoGebra, Indonesia, Supporting, students, understanding, quadratic