9th grade students’ understanding and strategies when solving $x(t)$ problems in 1D kinematics and $y(x)$ problems in mathematics

We design, validate, and administer a 24-item test to study student understanding of linear functions in 1D kinematics [$x(t)$] and mathematics [$y(x)$] in the 9th grade. The items assess identification and comparison of initial position and velocity in 1D kinematics and of the $y$ intercept and slope in mathematics using a graph or an algebraic formula. Results show that students’ performance on most mathematics items is significantly better than on their isomorphic kinematic counterparts, but also that most of the easiest as well as the most difficult items are kinematics items. Students achieve the highest accuracies on graphical questions in which they must compare two positive slopes, and they achieve the lowest accuracies on questions in which they must determine or compare a negative slope. We find that students have more difficulties with the $y$ intercept in mathematics and with the slope in kinematics. Furthermore, questions in symbolic representation result in far lower accuracies compared to questions in graphical representation, particularly when the $y$ intercept or the slope has to be determined instead of compared. We also analyze the results qualitatively by categorizing the students’ strategies and errors. We find frequent confusion between the $x$ intercept and the $y$ intercept in mathematics, but far less in kinematics. Negative velocities in kinematics are by far the largest pitfall, whereas negative slope in mathematics is rarely an issue. The results also show a significant frequency of interval or point confusions in kinematics but very little in mathematics. We reaffirm the occurrence of the interval or point confusion in questions with graphs and discuss three different cases of interval or point confusions in questions with algebraic expressions: numerical, algebraic, and unit based. Our results indicate a weak link between kinematics and mathematics and we suggest that closer integration between these two contexts during education could benefit student understanding of linear functions and linear phenomena in kinematics.

Figure 1

Test structure: two signs for the slope (positive and negative); two concepts (slope and $y$ intercept); three question types using two tasks (compare or determine) and two representations (graph and formula); two contexts [kinematics (K), mathematics (M)]. The items marked in gray require an explanation.

Figure 2

Three kinematics (K) items and the three isomorphic mathematics (M) examples illustrating the different factors in the test design.

Figure 3

GEE results for the term context * concept. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. Highly significant differences are found between the contexts for each of the concepts.

Figure 4

GEE results for the term context * slope sign. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. A highly significant difference is found between the contexts for negative slopes.

Figure 5

GEE results for the term context * QType. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. Highly significant differences are found between the contexts for compare given a graph and for determine given a formula question types.

Figure 6

GEE results for the slope in the term context * QType * concept. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. Highly significant differences are found between the contexts for determine given a graph and for determine given a formula question types.

Figure 7

GEE results for the slope in the term context * concept * slope sign. The $p$ value of the comparisons is indicated by * for $p<0.05$, ** for $p<0.01$, *** for $p<0.001$. Highly significant differences are found for the slope between the contexts for both slope signs.

Figure 8

Example of case 1 with question K10 in which the equation $x=4+8t$ was provided: a numerical interval or point confusion and additionally the $\mathrm{\Delta }$ in the formula for $v$ is written but not taken into account.

Figure 9

Example of case 2 with question K12 in which the equation $x=3–12t$ was provided: an algebraic interval or point confusion in which the equation of motion is substituted in the formula for velocity. Additionally, the isomorphic question M12 from the same student is shown, in which strategy S1 is used: the location of $-6$ is underlined in the question and the explanation translates as “the slope is located before the $x$ in the form $f(x)=mx+q$.”

Figure 10

Example of case 2 with question K10 in which equation $x=4+8t$ was provided: an algebraic interval or point confusion in which the equation of motion is used for $x$ and a manipulated version for $t$. The comparable question M12 from the same student is solved using strategy S1 by identifying the slope through its location in the equation, made clear by comparing with a standard form $f(x)=mx+q$. The written explanation translates as: “the slope is -6.”

Figure 11

Example of case 3 with question K10 in which equation $x=4+8t$ was provided: a unit-based interval or point confusion in which the equation of motion and a term in the equation are thought to be expressed in units of meters and units of seconds, respectively.

Figure 12

Example of case 3 with question K10 in which equation $x=4+8t$ was provided: a unit-based interval or point confusion in which the coefficients from the equation were each linked with a unit and then the appropriate ratio was taken to calculate the velocity.

Figure 13

Example of isomorphic questions K8 and M8 solved by the same student. In both cases strategy S7 is used, but in kinematics the minus sign has not been taken into account in contrast to M8. Translated wording in K8 and M8 are “The speed or velocity of the car is 0.5 meters per second.” and “The slope is -3.”

Figure 14

Example of isomorphic questions K6 and M6 (with positive slopes) solved by the same student. In K6, the interval or point confusion occurs resulting in an incorrect answer, but in M6 a triangle is drawn (S7) with a horizontal step size of $+1$ resulting in a correct answer. The text in the K6 translates as “speed or velocity train.”

Figure 15

Example of isomorphic questions K8 and M8 (with negative slopes) solved by the same student who answered the questions in Fig. 14. In K8, the interval or point confusion occurs and the minus sign is omitted. In M8 a triangle with horizontal step size of $+1$ is drawn and the minus sign is included. In K8, the text translates as “velocity: distance/time.”