This study, using a quantitative approach, examined Spanish and South African pre-service teachers’ responses to translating word problems based on direct proportionality into equations. The participants were 79 South African and 211 Spanish prospective primary school teachers who were in their second year of a Bachelor of Education degree. The study’s general objective was to compare the students’ proficiency in expressing direct proportionality word problems as equations, with a particular focus on the extent of the reversal error among the students’ responses. Furthermore, the study sought to test the explanatory power of word order matching and the static comparison as causes of the reversal error in the two contexts. The study found that South African students had a higher proportion of correct responses across all the items. While nearly all the errors made by Spanish students were reversals, the South African group barely committed reversal errors. However, a subgroup of the South African students made errors consisting of equations that do not make sense in the situation, suggesting that they had poor foundational knowledge of the multiplicative comparison relation and did not understand the functioning of the algebraic language. The study also found that the word order matching strategy has some explanatory power for the reversal error in both contexts. However, the static comparison strategy offers explanatory power only in the Spanish context, suggesting that there may be a difference in curriculum and instructional approaches in the middle and secondary years of schooling, which is when equations are taught.

Over the past decades there has been much research about the learning of early algebra and algebra researchers have documented many misconceptions of students (e.g. Bush & Karp,

The reversal error has been identified at different levels of study, among high school students in their initial stage of formal algebra learning (MacGregor & Stacey,

The present study looks into the cognitive processes that underly the reversal error, by exploring the extent of the reversal error and the explanatory power of the theoretical models suggested by previous research. The participants are prospective primary school teachers from two countries (South Africa and Spain) which represent two different educative contexts in terms of educational opportunities and resources. We take a comparative approach with the two groups (South African and Spanish prospective primary school teachers). This is accomplished by means of two versions of a questionnaire in which specific changes were made to adapt each one to the idiomatic features but keeping the same core structure in both countries. By considering these two different contexts and using these specifically designed questionnaires, this study looks at the incidence of the reversal error across the two samples. Therefore, as each sample is taken from a different context, the study can identify whether the reversal error is equally likely to occur in the two groups, which would suggest that the difficulty is embedded within the learning of algebra in general. In addition, if the strategies used by students differ across the groups, then this would suggest that there may be differences in curriculum approaches in the early learning of algebra, which could be explored in further studies. Hence, we can gain insights into how the educative context may affect some sources that seem to prompt the reversal error, while there may be other sources that seem to be inherent to the learning of algebra.

Much research in mathematics education has focused on systematic misconceptions that commonly occur with students from different countries and students who speak other languages. As part of the learning process of any concept, students develop misconceptions as they grapple with the concept at hand. A misconception is often revealed when a teacher is alerted by a consistent error made by a student in response to a particular situation. Instructors can be alerted by a consistent error made by students, which points to a misconception. Barmby (

Some students have misconceptions about the meaning of the equal sign, which seem to persist up to tertiary level studies (Booth,

Many studies have investigated the influence of factors such as variable symbol choice, sentence structure, semantic cues, context familiarity, among others, on students’ tendency to commit the reversal error (e.g. Cohen & Kanim,

Basically, studies that assess the explanatory power of different students’ strategies that lead them to commit a reversal error have consisted of modifying some variable of the research instrument and then observing whether the change prompted different results in the students’ responses. In this study we investigate the explanatory power of the word order matching and static comparison strategies as well as their contextual dependence or independence.

The word order matching strategy involves a ‘literal, direct mapping of the words of English into the symbols of algebra’ (Clement et al.,

Students who opt for the reversal error strategy because they have been misled by the translation cues have some algorithmic understanding of equations; however, their algebraic skills may not have included a consideration of the modelling component (Martin & Bassok,

Research has also identified that some students may display the reversal error even in statements without syntactic obstruction (Cohen & Kanim,

With the static comparison strategy, the student conceives the equation as representing a correspondence or ratio between groups instead of an algebraic equivalence between quantities. In the previous student and professor example, using the equation 6

the expression 6S is used to indicate the larger group and P to indicate the smaller group. The letter S is not understood as a variable that represents the number of students but rather is treated like a label or unit attached to the number 6. The equals sign expresses a comparison or association, not a precise equivalence. (p. 288)

Regarding the assessment of the explanative power of the static comparison model, most studies (e.g. Clement,

Note that each model involves different cognitive processes. Indeed, within the static comparison the student understands the correct ratio between quantities but falls short when trying to translate the relationship into an equation. In contrast, word order matching can be applied without comprehending the multiplicative relationship between quantities, but by incorrectly trying to replicate the verbal syntax in algebraic language.

The purpose of this study was to consider the explanatory power of certain theoretical models as causes of the reversal error using items set within two languages: Spanish and English. This study was guided by the following research objectives with respect to the samples of Spanish and South African prospective primary school teachers:

To examine the proficiency of the students from the two contexts in being able to express direct proportionality word problems into equations.

To test the explanatory power of word order matching and the static comparison as causes of the reversal error within the two groups.

In order to assess the explicative power and the context’s (in)dependence of the word order matching and static comparison as models for the reversal errors, we followed a quantitative approach with two samples. First, within a repeated measures design in each sample, we conduct hypothesis testing so as to assess in each context the student’s proficiency when translating verbal proportional statements into equations and the explicative power of each model for the reversal error. Afterwards, as both experiments were analogous, we follow Ellis (

The participants were made up of two groups of prospective primary school teachers from universities in Spain and South Africa based on convenience sampling since the institutions were accessible to the researchers. However, the sampling criteria were taken as the educational backgrounds, admission requirements, field of study and future job qualification which were equivalent in both countries. Both groups of students were in their second year of a Bachelor of Education (B.Ed.) degree with a primary school specialisation. The sample from South Africa consisted of 79 students most of whom (95%) spoke English as a second language. Two-thirds of the group had attended non-fee-paying schools showing that they come from very poor socio-economic backgrounds. Researchers have found that students from non-fee-paying schools in South Africa generally perform less well in mathematics national and international exams when compared to their peers from more affluent schools (Reddy et al.,

The sample from Spain consisted of 211 students at a public university in their second year of study towards a B.Ed. degree with a primary school specialisation. Regarding their socio-economic background, they mostly belong to the low-middle and middle class and attended non-fee-paying schools. The analysis of the international assessment tests conducted in Spain also conclude that students belonging to high socio-economic backgrounds obtain better academic results than their peers from poorer backgrounds (MEFP,

Concerning the instrument, we adapted the questionnaire designed and already applied in quantitative research about the reversal error by Soneira et al. (

In the Spanish university, we used the original questionnaire of Soneira et al. (

Some examples of word problem statements used in Spain and South Africa.

Magnitude type: Syntactic obstruction | Language | Item |
---|---|---|

Yes | Spanish | |

Direct translation to English | According to the engineer, [it] is nine times higher the acceleration of the motorbike than that of the bicycle. | |

Modified and used | According to the engineer, the speed is nine times as high in the car as it is in the bike. | |

No | Spanish | |

Direct translation to English | According to the engineer, the acceleration of the scooter is four times greater than that of the skateboard. | |

Modified and used | According to the engineer, the speed of the scooter is eight times as great as that of the tricycle. | |

Yes | Spanish | |

Direct translation to English | At this university, [it] is six times greater the community of students than that of professors. | |

Modified and used | At this university, there are six times as many students as professors. | |

No | Spanish | |

Direct translation to English | At this hospital, the community of patients is five times greater than that of doctors. | |

Modified | At this hospital, the community of patients is eight times as large as that of doctors. | |

Yes | Spanish | |

Direct translation to English | In this lemonade drink, [it] is four times greater the volume of water that that of lemon juice. | |

Modified and used | In this drink, there is four times as much the volume of water as that of lime juice. | |

No | Spanish | |

Direct translation to English | In this cocktail, the volume of tonic is five times greater than that of gin. | |

Modified and used | In this cocktail, the volume of tonic is five times as much as that of gin. |

Data collection took place across two countries, first at the university in Spain and thereafter at the university in South Africa. At both sites, the university learning management system was used as a medium to administer the 12 word problem statements in random order that formed the data collection instrument. Each item was presented with a list of operators and numbers and with instructions to click and drag into position to create the equation (

Example of a word problem appearing on the learning management system.

When coding the students’ responses, we defined three variables: ‘Cor’ (correct), ‘Rev’ (reversal error) and ‘Other’ (other error) to code the response of each student to each problem. The variable Cor takes the value 1 if the response is correct and 0 otherwise. The variable Rev equals 1 if the answer corresponds to a reversal error and 0 otherwise. The variable Other takes the value 1 whenever the answer is incorrect, but the error is not a reversal, otherwise it equals 0. In the case of the students and professors problem, the responses

Regarding the statistical tests we use, note that some choices depend on the fulfilment of certain assumptions, which in turn relies on the sample sizes and the results of other statistical tests. For the sake of clarity and to avoid repetition, in this section we describe the general procedure and in the results section we provide the statistical values that were used to check if the assumptions were fulfilled.

In order to assess the explanative power of the word order matching, our assumption is that students who use the word order matching strategy are likely to respond correctly to items that are expressed without syntactic obstruction and for items expressed with syntactic obstruction, these students’ guiding strategy would prompt them to produce a reversal error. We take the factor ‘syntactic obstruction’ (our instrument is made up of six items with syntactic obstruction and six without syntactic obstruction), which has two levels: items expressed with syntactic obstruction and those that are not. Given the sample sizes (79 and 211), we opted to use the paired t-test which is robust for sample sizes greater than 30 (Pallant,

In order to assess the explanative power of the static comparison, as explained in the conceptual framework section, our assumption is that this strategy is more or less likely to be applied depending on the type of magnitude, which will be reflected in the rate of reversal errors. Hence, we take the factor ‘type of magnitude’. Specifically, this factor has three levels (intensive continuous, extensive continuous, extensive discrete) which are ordered from lowest to highest according to how likely it is that the statement triggers the static comparison strategy. Thus, we use a one-way within-subject factorial design. The sphericity assumption was checked by means of the Mauchly’s test. If the sample distributions meet the sphericity condition, the analysis of variance test is used, otherwise we follow the protocol suggested by Pardo and San Martin (

In all tests, we take a

The ethical committees granted ethical clearance at the universities for this study. The researchers complied with all prescribed ethical measures, such as getting informed consent by the participants and keeping their anonymity. Moreover, all participants were volunteers.

We first present the descriptive statistics as a general overview of the results, before going into more specific detail about the results of the statistical tests.

Regarding the factor of syntactic obstruction, for each student and each of the variables Cor, Rev and Other, we computed the mean in statements with syntactic obstruction, and the mean in statements without syntactic obstruction (

Differences depending on the syntactic obstruction. Descriptive statistics.

Response: Syntactic obstruction | South Africa ( |
Spain ( |
||
---|---|---|---|---|

Mean | Standard deviation | Mean | Standard deviation | |

Yes | 0.6962 | 0.27575 | 0.5940 | 0.36404 |

No | 0.7489 | 0.32344 | 0.6517 | 0.41136 |

Yes | 0.1013 | 0.14719 | 0.3926 | 0.35789 |

No | 0.0422 | 0.09038 | 0.3397 | 0.40462 |

Yes | 0.2025 | 0.25413 | 0.0134 | 0.06865 |

No | 0.2089 | 0.32846 | 0.0087 | 0.06549 |

Cor, Correct; Rev, Reversal error; Other, Other error.

The results for the variable Cor in

Concerning the type of magnitude, we computed three scores for each student and variable. Each one was the mean of the scores for the items involving discrete magnitudes, extensive continuous magnitudes and intensive continuous magnitudes (

Descriptive statistics by country and magnitude type.

Response: Magnitude type | South Africa ( |
Spain ( |
||
---|---|---|---|---|

Mean | Standard deviation | Mean | Standard deviation | |

Intensive continuous | 0.6677 | 0.31704 | 0.6600 | 0.41641 |

Extensive continuous | 0.7278 | 0.33539 | 0.6268 | 0.41717 |

Discrete | 0.7722 | 0.32073 | 0.5818 | 0.35792 |

Intensive continuous | 0.0601 | 0.13406 | 0.3258 | 0.40846 |

Extensive continuous | 0.0728 | 0.14518 | 0.3637 | 0.41487 |

Discrete | 0.0823 | 0.15873 | 0.4088 | 0.35225 |

Intensive continuous | 0.2722 | 0.31315 | 0.0142 | 0.07582 |

Extensive continuous | 0.1994 | 0.32369 | 0.0095 | 0.06835 |

Discrete | 0.1456 | 0.31165 | 0.0095 | 0.06385 |

Cor, Correct; Rev, Reversal error; Other, Other error.

Thus, the results depending on the type of magnitude (

Taken together, the descriptive results (

We report next (

Contrasts for differences depending on the syntactic obstruction.

Variable | South Africa ( |
Spain ( |
||||||||
---|---|---|---|---|---|---|---|---|---|---|

t(78) | 95%CI |
95%CI |
||||||||

LL | UL | LL | UL | |||||||

Cor | ™1.76 | 0.082 | ™0.112 | 0.007 | ™ | ™4.165 | 0.001 |
™0.085 | ™0.030 | 0.287 |

Rev | 3.37 | 0.001 |
0.024 | 0.094 | 0.379 | 3.774 | 0.001 |
0.026 | 0.081 | 0.260 |

Other | ™0.281 | 0.780 | ™0.051 | 0.039 | ™ | 1.226 | 0.221 | ™0.003 | 0.012 | - |

CI, confidence interval; LL, lower limit; UL, upper limit.

We obtain (

With respect to the South African context, the fact that there are not significant differences depending on the syntactic obstruction neither for the variable Cor nor the variable Other would mean that, among those students who have problems with the algebraic modelling process that we identified above, the syntactic obstruction is not the main source of the error. Otherwise, there would be differences depending on the syntactic obstruction. Furthermore, these students did not follow, or their command of the algebraic language is so poor that they were not able to follow, the word order matching strategy.

We report next the results of the hypothesis testing about the differences depending on the type of magnitude. For the sake of clarity, this is done for the variables Rev, Cor and Other in each context, separately. Concerning the variable Rev, in the South African context we cannot assume sphericity (

Differences depending on the type of magnitude for the variable Rev in the Spanish context.

Difference variables for Rev | Mean | Standard deviation | 95%CI |
Cohen’s d | |||
---|---|---|---|---|---|---|---|

LL | UL | ||||||

Extensive continuous –Intensive continuous | 0.03791 | 0.21069 | 2.614 | 0.030 |
0.00932 | 0.06651 | 0.17993 |

Extensive discrete – Intensive continuous | 0.08294 | 0.24566 | 4.904 | 0.001 |
0.04960 | 0.11628 | 0.33762 |

Extensive discrete – Extensive continuous | 0.04502 | 0.25422 | −2.573 | 0.022 |
0.01052 | 0.07952 | 0.17709 |

CI, confidence interval; LL, lower limit; UL, upper limit.

Therefore, in respect of the explanatory power of the static comparison as a model for the reversal error, firstly, our results point out that when taking together statements with and without syntactic obstruction, the rate of reversal errors among South African students is very low regardless of the type of magnitude (

Regarding the variable Cor, in the South African group, by Mauchly’s test, sphericity can be assumed (

Differences depending on the type of magnitude for the variable Cor in the Spanish context.

Difference variables for Cor | Mean | Standard deviation | 95%CI |
Cohen’s d | |||
---|---|---|---|---|---|---|---|

LL | UL | ||||||

Intensive continuous – Extensive continuous | −0.03318 | 0.19691 | −2.447 | 0.015 |
−0.05990 | −0.00645 | 0.16850 |

Intensive continuous – Extensive discrete | −0.07820 | 0.24113 | −4.711 | 0.001 | −0.11092 | −0.04548 | 0.32431 |

Extensive continuous – Extensive discrete | −0.04502 | 0.25069 | −2.609 | 0.04 |
−0.07904 | −0.01100 | 0.17958 |

CI, confidence interval; LL, lower limit; UL, upper limit.

Therefore, in relation to the second objective, in the light of

Note also that in the South African sample, reversal errors were barely made, and the highest rate of correct answers was obtained for discrete magnitudes and the lowest for intensive magnitudes. With Spanish students, the opposite happened (

Differences depending on the type of magnitude for the variable Other in the South African context.

Difference variables for Other | Mean | Standard deviation | 95% CI |
Cohen’s d | |||
---|---|---|---|---|---|---|---|

LL | UL | ||||||

Intensive continuous – Extensive continuous | 0.07278 | 0.29730 | 2.176 | 0.066 | 0.00619 | 0.13938 | 0.2448 |

Intensive continuous – Extensive discrete | 0.12658 | 0.27114 | 4.150 | 0.001 |
0.06585 | 0.18731 | 0.4668 |

Extensive continuous – Extensive discrete | 0.05380 | 0.22882 | 2.090 | 0.040 |
0.00255 | 0.10505 | 0.2351 |

CI, confidence interval; LL, lower limit; UL, upper limit.

Thus, although most errors in the South African context are not reversals, the type of magnitude still has an effect on the incidence of the errors. In addition, the performance by the South African students regarding these other kinds of errors is better with the extensive discrete magnitudes and worse with the intensive continuous ones. Comparisons of quantities with extensive discrete magnitudes are less abstract, while those for intensive continuous ones are the most abstract, with the extensive continuous in the intermediate position. This means that for the South African students of our sample, the more abstract the magnitudes involved were, the more difficult they found it to express the word problem into an equation. This did not happen in the Spanish sample, which contributes to the answer to the first research objective.

Overall, our results point out that the South African pre-service teachers in our sample outperform the ones in the Spanish sample when expressing proportionality word problems in terms of equations. Indeed, although the results are based on descriptive statistics (

Firstly, our results point out that the errors when expressing proportionality word problems in terms of equations differ between the South African and Spanish pre-service teacher participants. Specifically, in the Spanish case nearly all the errors were reversals, while in the South African case there were other errors with a higher incidence. Furthermore, even restricting our attention to the reversal error, its sources are different depending on the country. In the Spanish case the word order matching and the syntactic comparison would both have some explanatory power for the reversal error, which is in line with other results about Spain (González-Calero et al.,

The results of our study contribute to improve our understanding of translation of word problems involving multiplicative comparisons into equations in two ways. Firstly, we have shown that the South African students also have the tendency to keep the structure or the ordering of meaningful components of the statements in natural language when changing the register of representation, even if this leads them to make errors. Note that in this sample 90% of the students spoke isiZulu as a first language; this therefore shows that isiZulu-speaking students also display the tendency of the word order strategy. This tendency has been identified in studies with students who speak English (e.g. Cohen & Kanin,

Therefore, our results in the South African context point out that the tendency to apply the word order matching does not depend on the context but it is common for different contexts. On the one hand, this is in line with the theoretical ideas of Duval (

Secondly, our study points out that the static comparison may be context dependent, diminishing its explanatory power as a model for the reversal error despite being one of the classic models. Indeed, we did not find evidence of the static comparison being a tendency among South African pre-service teachers; therefore, it should be context dependent. This highlights the importance of word order matching as a source of the reversal error because it is shared by groups from different populations that differ regarding other strategies when structuring equations, specifically the static comparison.

It is also noteworthy that, in contrast to the students of the Spanish sample, the students of the South African sample committed the highest rate of errors with intensive continuous magnitudes, and the lowest with extensive discrete magnitudes. This suggests that the more abstract the magnitude, the harder they find it to mathematise the situation. Even more, when dealing with discrete magnitudes they perform clearly better than Spanish participants, but the difference fades with the items that have intensive magnitudes. This is another difference between contexts. It could be due to differences in curriculum or instructional approaches at school level, although more research about the matter is needed.

Moreover, success in comparisons of quantities with extensive continuous magnitudes, which are easy to visualise and to reduce to discrete magnitudes by means of taking units of measurement, placed between the other two types. This could mean that some students spontaneously apply a process of discretisation whenever the magnitude’s unit of measurement allows a corporeal interpretation. For example, in the cocktail problem, it is possible to think in terms of five tablespoonfuls of tonic for each tablespoonful of gin. But, if we do so, the situation and the subsequent reasonings would be the same as with discrete magnitudes. Indeed, the new magnitudes would be the number of tablespoonfuls of tonic and gin. Note that this process cannot be carried out with intensive magnitudes. In this case, the quantity of substance does not correspond to the quantity of magnitude. For example, the density of the sauce can be greater than that of the water, but the quantity of water greater than that of the sauce. Thus, it would be more difficult to mentally corporealise the unit of measurement, which in turn would make discretisation more difficult. Soneira et al. (

Regarding curriculum and instructional implications, our results indicate that the teaching interventions to be conducted in order to remediate the reversal error should be different depending on the country. Specifically, Spanish educators should take into account both the word order matching and the static comparison strategies. On the one hand, when first introducing the algebraic language in Spanish schools, the study suggests that teachers need to highlight the meaning of the equal sign as an algebraic equivalence instead of correspondence, and the fact that letters represent quantities instead of being labels for objects. This would be so because these aspects have been proposed as the reasoning underlying the use of the static comparison strategy (Clement,

On the other hand, the answers to problems with statements presented with syntactic obstruction point out that the cause of most errors made by the South African students was not word order matching. The errors they made, equations that did not make sense in the problem situation, suggest that their mental scheme to express multiplicative relationships by means of equations is poorly developed. Thus, although South African pre-service teachers’ performance was, overall, acceptable or, at least, better than that of the Spanish ones, there was a small subgroup of them that seem to lack the rudiments of algebraic language. This suggests that when students are introduced to the concept of equation during the intermediate and senior phase schooling, it should be done in a manner that allows them to develop a conceptual understanding and to make meaning of the notion of equation, instead of proving rules for manipulation of equations.

In terms of the study’s limitations, it would have been preferable to have a larger sample size, particularly for the South African context. Given that we did not find evidence of the static comparison being a tendency in the South African context and that it means a key difference with respect to others (e.g. Spain, United States), it may be convenient to increase the sample. Furthermore, it would be convenient to conduct similar experiments in the South African context, both with pre-service teachers and at other educative levels. In addition, since we have detected that South African students barely commit reversal errors, but other types of errors, and that the source of the former seems not to be static comparison, further studies could be interesting. In particular, qualitative studies would be helpful to provide deeper insights into the present quantitative one. These would shed light on how students think when translating proportionality word problems into equations. It could also be of interest to identify whether the students’ specialisation phase affects the extent to which the word order matching strategy usually arises to prevent the reversal error. Along similar lines, it could be interesting to identify whether South African students’ specialisation phase is related to the extent to which they face difficulties with intensive continuous magnitudes.

The authors declare that they have no financial or personal relationships that may have inappropriately influenced the writing of this article.

This article was developed collaboratively by the three authors. C.S. led the conceptualisation of the article, led the experimental design, carried out the analysis of the data, and drew up the results and discussion. S.B. led in the introduction, literature review, collaborated on the analysis of the South African data, and collaborated in preparing the final version of the discussion. R.G. developed the tool, cleaned and coded the data, led the methodology, technical aspects related to standard setting, and prepared the manuscript to comply with publication standards.

This work was supported by Spanish Government through the project PGC2018-096463-B-I00 as well as the National Research Foundation (NRF) grant number UID 118377.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The views, opinions, findings and conclusions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors.