How numbers influence students when solving linear equations

Figures & data

Table 1. Overview of the ten equations in the test.

Typeof coefficients	Additive structure	Multiplicative structure	Additive and multiplicative structure
Natural numbers	$352=63+\mathit{x}$	$42=3\cdot \mathit{x}$ $819=39\cdot \mathit{x}$	$37=7\cdot \mathit{x}+2$
Numbers with decimals	$2.01=0.434+\mathit{x}$	$0.12=0.4\cdot \mathit{x}$ $0.657=0.045\cdot \mathit{x}$*	$2.5=0.8\cdot \mathit{x}+2.1$
Negative numbers		$-24=6\cdot \mathit{x}$
Symbolic		$\mathit{a}=\mathit{b}\cdot \mathit{x}$

*The equation was added to the test after the first class had taken it.

Table 2. Comparison of solution rates for different equations.

Structure	Equation 1 $\mathit{a},\mathit{b},\mathit{c}\in {\mathbb{Z}}^{+}$	Correct (%) (95% CI)	Equation 2 $\mathit{a},\mathit{b},\mathit{c}\in \mathbb{Q}$	Correct (%) (95% CI)	p * value
$\mathit{a}=\mathit{b}\cdot \mathit{x}$	$819=39\cdot \mathit{x}$	89.1 (83.2-95.0)	$0.12=0.4\cdot \mathit{x}$	69.1 (60.4–77.8)	<0.001
			$0.657=0.045\cdot \mathit{x}$	78.7 (71.0–86.4)	0.006
			$-24=6\cdot \mathit{x}$	74.5 (66.4–82.7)	<0.001
$\mathit{a}=\mathit{b}+\mathit{x}$	$352=63+\mathit{x}$	90.9 (85.5–96.3)	$2.01=0.434+\mathit{x}$	83.6 (76.7–90.6)	0.011
$\mathit{a}=\mathit{b}\cdot \mathit{x}+\mathbf{c}$	$37=7\cdot \mathit{x}+2$	89.1 (83.2–95.0)	$2.5=0.8\cdot \mathit{x}+2.1$	52.7 (43.4–62.1)	<0.001

*p values concerning the hypotheses H₀: p₁= p₂ and the alternative hypothesis H₁: p₁>p₂ where p₁ is the probability for a student to solve Equation 1 but not the corresponding Equation 2, and p₂ is the probability of a student to solve Equation 2 but not Equation 1. McNemar test was used, binominal distribution and exact significance (1-sided).

Figure 1. Categorization of the wrong answer, 43 students in total.

* 3 of the 8 students (concerning $-24=6\cdot \mathit{x}$) use division to solve some other multiplicative equation on the test. The same accounts for 3 of the 7 students (for $0.12=0.4\cdot \mathit{x}$), and 4 of the 10 students (for $0.657=0.045\cdot \mathit{x}$).

Figure 2. One student’s answers from the same occasion (Glenn).

Table 3. Outcome space of the conceptions.

The linear equation is experienced as …	Meaning discerned in the structure	Aspect in foreground
A. … a relation between operations and elements	Syntactic meaning	Structure
B. … a relation between numbers	Syntactic meaning	Numbers and structure
C. … a relation that must represent a situation	Semantic meaning	Numbers and structure
D. … numbers to compute	Searches for a meaning	Operation
E. … an unfamiliar equation	Searches for a meaning	Numbers