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<p>One of the benchmarks of algebraic thinking is that of being able to operate on the unknown in equations where there is an occurrence of the unknown on each side of the equal sign. Research has provided ample evidence of the difficulties that students encounter in learning to do this. However, there has been relatively little research devoted specifically to equations where the two occurrences of the unknown are on the same side. More recently, it has come to light that such equations, that is, equations of the type <i>ax</i> ± <i>b</i> ± <i>cx</i> = <i>d</i>, are more challenging than has previously been suspected and warrant further investigation. In our large-scale data study, we explored the solutions produced by 37,293 Danish students, aged 11 to 15, in an online environment for three different types of equations (<i>ax</i> ± <i>b</i> ± <i>cx</i> = <i>d</i>, <i>ax</i> ± <i>b</i> = <i>cx</i> ± <i>d</i>, <i>ax</i> ± <i>b</i> = <i>c</i>). Analysis of the most popular incorrect solutions for the equations with both occurrences of the unknown on the same side allowed us to infer the use of a solving strategy in which the unknowns were ignored and “solutions” were calculated by balancing all the visible numbers on the left side with those on the right side (i.e., <i>a</i> ± <i>b</i> ± <i>c</i> compared with <i>d</i>, to yield the “solution”, d&#xa0;–&#xa0;(<i>a</i> ± <i>b</i> ± <i>c)</i>)—a strategy that we have named the Ignoring-and-Balancing strategy. This strategy was especially prominent when the leading coefficient “a” was absent, thereby pointing to the influence of form in students’ solving strategies. Additional analyses of the students’ solutions for the other two types of equations provided evidence of the same strategy, but at a weaker rate. These results underscore the need for more focused instructional attention to the meaning that students attribute to the unknowns, especially in the equation&#xa0;type <i>ax</i> ± <i>b</i> ± <i>cx</i> = <i>d</i> and its related form <i>x</i> + <i>b</i> + <i>cx</i> = <i>d</i> with the unadorned leading unknown.</p>

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The surprising obstacle of two unknowns on the same side of the equal sign in solving equations of the type ax ± b ± cx = d and the phenomenon of ignoring the unknown

  • 期刊论文

  • 发表日期：

    2025年11月19日

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  • Morten Elkjær,
  • Carolyn Kieran

摘要

One of the benchmarks of algebraic thinking is that of being able to operate on the unknown in equations where there is an occurrence of the unknown on each side of the equal sign. Research has provided ample evidence of the difficulties that students encounter in learning to do this. However, there has been relatively little research devoted specifically to equations where the two occurrences of the unknown are on the same side. More recently, it has come to light that such equations, that is, equations of the type ax ± b ± cx = d, are more challenging than has previously been suspected and warrant further investigation. In our large-scale data study, we explored the solutions produced by 37,293 Danish students, aged 11 to 15, in an online environment for three different types of equations (ax ± b ± cx = d, ax ± b = cx ± d, ax ± b = c). Analysis of the most popular incorrect solutions for the equations with both occurrences of the unknown on the same side allowed us to infer the use of a solving strategy in which the unknowns were ignored and “solutions” were calculated by balancing all the visible numbers on the left side with those on the right side (i.e., a ± b ± c compared with d, to yield the “solution”, d – (a ± b ± c))—a strategy that we have named the Ignoring-and-Balancing strategy. This strategy was especially prominent when the leading coefficient “a” was absent, thereby pointing to the influence of form in students’ solving strategies. Additional analyses of the students’ solutions for the other two types of equations provided evidence of the same strategy, but at a weaker rate. These results underscore the need for more focused instructional attention to the meaning that students attribute to the unknowns, especially in the equation type ax ± b ± cx = d and its related form x + b + cx = d with the unadorned leading unknown.