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This article was submitted to the journal Frontiers in Human Neuroscience.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Algebra typically represents the students’ first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students’ ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16–17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15–16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students’ transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.

United States National Council of Teachers of Mathematics defines algebra as “a way of thinking and a set of concepts and skills that enable students to generalize, model, and analyze mathematical situations” (

In line with arguments suggesting that understanding algebra concepts may be difficult for children in primary schools, research has shown that students indeed often face difficulties in moving from the arithmetic to the algebraic form of reasoning (

A more overarching evaluation of students’ success and difficulties in acquiring fundamental algebra concepts is introduced by large international surveys, such as PISA (Program for International Student Assessment) and TIMSS (Trends in International Mathematics and Science Study) that give insights into the quality and efficiency of school systems across many countries. The findings of PISA testing conducted in 2012 with a particular focus on mathematics indicate that students in the highest-performing countries are “more frequently exposed to formal mathematics than students in most of the other PISA-participating countries and economies” (

However, when discussing the acquisition of basic algebra concepts, it is important to highlight that these represent a broad part of school mathematics. As was mentioned earlier, at its fundamental level, algebra includes solving simple algebraic equations that were the focus of the present study. These equations were chosen because equation rearrangement represents a very important skill required for problem solving in many school subjects. Within different teaching frameworks, it is often assumed that, once students learn to solve a simple equations such as, e.g., they can solve such equations for any unknown. This would mean that they are able to solve equivalent simple equations containing both numbers, letters or other symbols. However, physics and chemistry teachers know that students struggle with equations rearrangements, especially for “all-symbol” equations.

Motivated by these differences, as well as the practical relevance of this topic, the present study was aimed at investigating the developmental trajectory of students’ ability to solve simple algebraic equations. Based on the

The participants in the present study included 331 students from five primary and four secondary state schools in Zagreb. With respect to primary school students, all state primary schools in Croatia have the same curriculum, so their students have comparable experiences with algebra education. With respect to secondary schools, we tested students from two gymnasiums (general education and foreign language type schools) and two technical secondary schools. These schools were chosen to represent the average secondary school population in Zagreb mostly preparing for university studies. Specifically, graduates from the two gymnasiums included in the present study typically continue their education at university, typically studying non-mathematics or science related majors. In comparison, graduates from the tested technical schools often continue their education majoring in technical fields. Students from gymnasiums that specialize in natural sciences and mathematics were not included in this study.

The participants in the present study included students from the seventh grade of primary school (age 13–14 years) to the second grade of secondary school (age 16–17 years). Hence, our sample included the students of four age groups, i.e., different school grades: the 7th and 8th grade of primary, and the 1st and 2nd grade of secondary school. Given that in Croatian schools, equation rearrangement is taught at the end of the sixth grade of primary school roughly corresponding to the students’ age of 12–13, all our participants were taught how to solve the task used in the study at least one year prior to this measurement. The number of tested female and male students in each grade is shown in

Number of students according to their age and gender.

7th grade | 8th grade | 1st grade | 2nd grade | |
---|---|---|---|---|

Male | 36 | 41 | 62 | 54 |

Female | 36 | 39 | 33 | 30 |

Total | 72 | 80 | 95 | 84 |

The study was approved by the Ethics Committee of the Ministry of Science, Education and Sports, as well as by the schools’ headmasters. Each student’s parents gave an informed written consent before the child took part in the experiment.

Raven’s Progressive Matrices were used to assess general cognitive ability (

A computerized test of equation rearrangement was prepared using E-Prime (Psychology Software Tools Inc., Pittsburgh, PA, USA). In each trial, simple equations consisting of three elements (numbers or letters) were presented in the centre of the visual field. The presented numbers and letters were black, displayed in 24 pt size Ariel font on the white background. Participants’ task was to make

Three types of equations were used in the study:

A equations: x ⋅ a = b,

B equations:

C equations:

The offered answers were of the following types: x ⋅ a = b,

The participants were tested during two school periods (45 min long). During one school period, Raven’s Progressive Matrices and d2 Test of Attention were administered to students in their classrooms. On the same or on another day, students solved the computerized test of equation rearrangement and completed a post-measurement questionnaire in the computer lab.

Before administering the equation rearrangement test, participants were familiarized with the task. They were instructed to respond as quickly as possible by pressing one of the two mouse buttons with their index and middle fingers, corresponding to correct and incorrect answers, respectively. Prior to experimentation, the participants performed a training block consisting of 6 equations equivalent to those used in subsequent experimental trials.

During both practice and experimental trials, each equation was presented until the participant responded, up to a maximum of 30 s. If the participant did not respond within 30 s, the equation disappeared from the screen and another 30 s were available to give an answer. However, these late responses (<0.1% of all trials) were not included in the analysis. After each response, the next equation was presented after a delay of 1 s. Reaction times (RTs) were measured automatically by the computer from the stimulus onset to the participant’s response. No feedback was given to the participants.

During the experiment, the participants were presented with the three previously described types of equations, which were randomized across four blocks. Each block consisted of 15 equations of each equation type, amounting to an overall of 45 presented equations per block. Two blocks contained equations with numbers, while the other two blocks contained equations with letters (symbols). Equations in the first and third blocks contained numbers while those in the second and fourth blocks consisted of letters. The participants could take a break between blocks if needed.

After having finished the computerized test, the participants completed a questionnaire designed for assessing their strategies during equation solving. While responding to these questionnaires, the participants described how they solved each equation type and ranked them by difficulty. In addition, they indicated whether their response depended on the type of the offered answers, and whether they changed their problem solving strategies during the time course of the experiment.

For each participant and each condition, reaction time and accuracy were evaluated. Only correct responses were included in the analysis of RTs. Inverse efficiency was also calculated as the ratio of reaction time and accuracy (

To determine the effects of age, gender, level of abstraction, repetitions and equation type, a two-way repeated measures analysis of variance (ANOVA) on accuracy and RTs was conducted. Repeated-measures

To evaluate participants’ strategies in equation solving, we analyzed their answers in the administered

Two-way ANOVAs with factors Age (7th vs. 8th vs. 1st vs. 2nd grade) and Gender (Male vs. Female) were conducted to compare the mean accuracy and RTs. The obtained results for accuracy indicated a statistically significant main effect of Age [

A corresponding comparison for RTs revealed a statistically significant main effect of Age [

To test the differences between participants’ accuracy and RTs in solving equations with numbers and letters across different age, we used the two-way mixed-design ANOVAs with between-subjects factor Age (7th vs. 8th vs. 1st vs. 2nd grade) and within-subjects factor Abstraction level (numbers vs. letters). With respect to accuracy, the statistically significant main effects of Age [

For the RTs, results revealed a statistically significant main effect of both factors, Age [

We have used two-way mixed-design ANOVAs with between-subjects factor Age (7th vs. 8th vs. 1st vs. 2nd grade) and within-subjects factor Equation type (A vs. B vs. C equation) to test the differences between participants’ accuracy and RTs for different types of equations across different age. For the accuracy, a significant main effects of both Age [

Corresponding results for the RTs again revealed significant main effects of both Age [

Two-way mixed-design ANOVAs with between-subjects factor Age (7th vs. 8th vs. 1st vs. 2nd grade) and within-subjects factor Block (first vs. second block) were used for testing the differences between participants’ accuracy and RTs across time course of the experiment. The results showed a statistically significant main effect of both factors, Age [

For the RTs, results indicated corresponding significant main effects of both factors, Age [

The relation between the students’ equation solving efficacy and their cognitive abilities was addressed by calculating the partial correlation coefficient between participants’ inverse efficacy and their scores on Raven’s Progressive Matrices score, while controlling for the age effects. The obtained results indicate a statistically significant correlation between equation solving efficacy and cognitive abilities [

Evaluation of participants’ answers in the questionnaires confirmed that they used different strategies for solving equations with letters. We categorized their answers and divided them into two groups – concrete strategies and rule-based strategies. The most frequently used concrete strategy (37% of all participants) was inserting numbers instead of letters. 11% of participants used a “triangle” memory technique and 4% used a “biggest on the top” strategy that is based on a belief that products and numerators are “big.” For the equation

The most common rule-based strategy (38% of all participants) was a standard application of multiplication/division operations on the equation. 11% of participants reported correctly moving letters to the other side of the equation and often indicated the operation with arrows. The most frequently used incorrect strategy (6%) was to “move letters other than

Participants ranked different equation types by difficulty in the questionnaires. 28 participants reported that all equation types are equally difficult. Three participants thought that equations with multiplications (A type) are easier than equations with division (B and C). Eight participants did not provide an answer to this question.

The results obtained in the present study suggest that the tested students were overall rather successful in equation rearrangement, with accuracy levels amounting to an average of 85%. Although this may seem quite high, if the true-false nature of test items is taken into account this becomes a less satisfactory result, especially for all-symbol equations which were correctly solved by 82% of the participants. However, our data indicate that students become more efficient, i.e., more accurate and faster, in higher school grades.

With respect to gender differences, the girls in our sample were on average more accurate in equation rearrangement than boys, while no significant differences in their speed were revealed. This finding is in disagreement with a common belief that boys are better in mathematics than girls which is based on reports that boys outperform girls on standardized tests like SAT (e.g.,

It is important to emphasize that students’ success in solving simple algebra equations differed across different types of equations. Specifically, within the present study we compared equivalent equation formats that contained either symbols or numbers. As expected, the obtained results indicate that the younger participants were more accurate and faster in solving equations with numbers than with letters although these were equivalent. This indicates that younger students still struggle with more abstract equations. In contrast, students in the 2nd grade (age 16–17 years) had a comparable level of accuracy and RTs for equations with numbers and letters. This indicates that they reached an adequate level of formal reasoning (

Next, we compared participants’ efficacy in solving three different types of equations. The lowest accuracy and the longest RTs obtained for the C equations (

In addition to exploring age, gender and equation type effects, within the present study we also explored practice effects across all equation types. Our participants became faster and more accurate in equation rearrangement during the time course of the measurement. This finding is in agreement with a previous report indicating how children become faster during a 5-day practice in algebra equation solving (

Furthermore, our results showed that the participants with higher cognitive abilities were more efficient in equation solving. This is in line with the previous longitudinal testing which indicated that students with higher IQ scores tended to demonstrate higher cognitive levels and made faster progress through algebra levels than students with lower IQ scores (

Half of the participants used concrete strategies for equation rearrangement and the most frequently used concrete strategy was inserting numbers into equations. When using this strategy, the students think of an equivalent equation with numbers, solve it and then apply the solving algorithm on the equation with symbols. For example, for the A equation (

A considerable number of participants (11%) used the “triangle” method often taught by physics teachers to “simplify” equation rearrangement for their students. Within this strategy, a triangle is divided into three parts. Two quantities that are multiplied together are written side-by-side at the bottom of the triangle. The remaining quantity (their product) is written at the top. For

The “biggest on the top” strategy also has origin in concrete way of thinking. As few participants reported, they always considered product in multiplication equations and numerator in division equations as the biggest object that helped them in the rearrangement. For example, in the A equations (

More than half of the participants (56%) were reasoning more abstractly while solving at least one equation type, i.e., they were applying rules. During the testing, few participants made a transition from concrete substitution of letters by numbers to the recognition of patterns and rules. The most frequently used rule-based strategy was multiplication and division of equation with the “letter next to

As in our previous study (Susac et al., under revision), some participants reported moving letters to the other side of equation. This corroborates findings showing that spatial reasoning is closely related to the number sense (as in the case of mental number line; e.g.,

Some participants reported strategies based on some types of rules that they developed by themselves. By repeated exposure to equation rearrangement, they recognized some patterns from which they derived some general rules. Although participants’ rules were not always correct, they possibly represent a step in developing more consistent and correct solving strategies. A number of participants recognized that they do not have to perform two steps of multiplication and division for

Overall, the obtained results suggest that the proportion of concrete strategy usage decreases at the same time as the proportion of rule-based strategies increases with the age of participants. This progression is gradual and it probably continues after the 2nd grade of secondary school (age 16–17 years). Our data confirm that the development of algebraic thinking is a process which unfolds over a long time. Consequently, we can conclude that children at the age 14–15 are in transition from concrete to abstract strategies in algebra that is in agreement with previous research (

To determine the difficulty of different types of equations with letters, we evaluated inverse efficiency across the age groups of our participants. In all age groups, participants were the least efficient in solving the C equations, which suggests that these are the most difficult equation types. This finding was expected, because the C equations are usually solved in two steps while only one step is needed for the A and B equations. Not all participants performed two operations in solving

Inverse efficiency measures indicated that the A equations were of similar difficulty as the B equations. The B equations,

It is interesting to note that a large majority of participants reported that the B equations are more difficult than the A equations although this is not supported by the obtained results. Probably their self reports were again influenced by the fact that division is perceived as more difficult than multiplication. However, in judging equation difficulty, participants failed to take into account the fact that correctly solving these equations also includes these operations. Still, the increased number of participants who ranked B equations as easiest among older students suggests that some older participants (from secondary school) became aware of the patterns in the task. In addition, it seems that metacognitive skills improve with age as secondary school students, on average, ranked equation difficulty more accurately than younger participants. This finding concurs the previous reports on the importance of metacognitive activities for success in problem solving in mathematics (

The goal of the present study was to investigate the development of students’ abstract reasoning skills on a simple equation rearrangement task. Although all our participants learned equation rearrangement in mathematics at least one year prior to our testing, and were required to solve simple equations in mathematics and science problems, they still had difficulties with some equation types. However, accuracy and speed of equation rearrangement increased with the participants’ age. Younger participants were more accurate and faster in solving equations with numbers than with letters, suggesting that they are still concrete thinkers. The difference in the efficacy of solving equations with numbers and letters disappeared for participants from the 2nd grade of secondary school (age 16–17 years), indicating their ability to think more abstractly, at least on our task. The transition from concrete to formal reasoning was also reflected in strategies that the participants used for solving equation with letters. Younger participants from the primary school (age 13–15 years) mostly employed concrete strategies such as inserting numbers, while secondary school participants (age 15–17 years) mainly used rule-based strategies.

Our results indicate that the transition from concrete to abstract reasoning represents quite a long process, even for simple algebraic task used in this study. Teachers and educational policy makers should be aware that it is not enough to learn about equation rearrangement in mathematics once. It should not be presumed that students master this skill quickly and that they can easily apply it in other context such as problem solving in physics. On the contrary, teachers should use every opportunity to encourage students to use formal reasoning – both pattern recognition and effective application of mathematical rules and known procedures.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This research was supported by a grant from the University of Zagreb Development Fund (198002) to Ana Susac.