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The use of the algebraic method for solving word problems is a challenging topic for secondary school students. Students' difficulties are usually associated with extracting the problem's network of relationships between quantities and with formalizing these relationships into algebraic language in a problem model. Both sources can coexist and interact; thus, it is usually not possible to determine which source of difficulty is more relevant. In addition, there are specific errors, such as the error by multiple referents for the unknown, which are directly linked to the wording of the problem text, and in which the same two sources of error coexist. In this work, we present the results of an experiment conducted with 255 secondary school students assessing the effect of two common difficulties on the accuracy of problem models and on the rate of multiple referents for the unknown. The first difficulty is the use of algebraic language in the construction of the problem model; the second is the use of the same expression to designate different quantities within the problem text. We used a 2 × 2 between-between design, with one factor related to the symbolic language (algebraic or arithmetic) in which the problem model is constructed, and the other factor related to the actual language features of the text problem. Our results indicate that overall, the main source of difficulty for students is the use of algebraic language to formalize a problem model, representing a large effect size.

The reversal error is a prevalent phenomenon that consists of reversing the relationship between two variables when writing equations from comparison in verbal language. A study with 169 Basque/Spanish bilingual pre-service teachers examined whether this error is mainly due to the students' tendency to employ a direct-translation strategy based on the syntax of statements. By using Basque and Spanish the factor of syntax was studied since in Basque, unlike Spanish (or English), the natural way of expressing comparisons prevents students from using direct-translations strategies. Results show a significantly lower number of reversal errors in sentences written in Basque compared with those in Spanish. This finding confirms the importance of syntax statement in the translation from words into equations and suggests that teachers should have awareness of this fact and should design lesson plans devoted to work on word problems accordingly.

We present two empirical studies with 241 and 211 pre-service teachers that evaluate the explanatory power of word order matching and static comparison as models for the reversal error. We used tasks consisting of generating an algebraic equation representing a comparison given in a verbal statement. We introduce the types of magnitude involved in the statement as variables of analysis, something that was not previously tackled in previous works. Our results show that there are no statistical differences in the production of reversal errors depending on the information included in the name used to designate the variable, and that there are statistical differences depending on the syntactic configuration as well as the type of magnitude involved in the statement. The interpretation of these results indicates that both word order matching and static comparison have some potential as explanatory models for the reversal error, and that neither one of them, alone, is enough to completely explain the phenomenon.

The term intensive scaffolding refers to any set of conceptual scaffolding strategies that always allow the user to find the solution to a problem. Despite the many benefits of scaffolding, some negative effects have also been reported. These are mainly related to the possibility that a student solves the problems without actually engaging in their content. In this paper, we have used an intelligent tutoring system (ITS) to analyze the effect that intensive scaffolding has on the learning of algebraic word problem solving (AWPS). Two different versions of the ITS, which differ in the amount of scaffolding that they provide, have been created. These two versions were used by two groups of students in Secondary Education, in a quasi-experimental study. The comparison of pre- and posttest scores shows a significant increase of the competence in AWPS in the group that used the ITS with intensive scaffolding.

An empirical study has been carried out to evaluate the potential of word order matching and static comparison as explanatory models of reversal error. Data was collected from 214 undergraduate students who translated a set of additive and multiplicative comparisons expressed in Spanish into algebraic language. In these multiplicative comparisons we used a format that can be translated from Spanish word-for-word as "n times more than" (increasing comparison) and "n times less than" (decreasing comparison) instead of "n times as many", which is usual in other studies. Data analysis shows a significantly lower incidence of reversal error in the decreasing comparisons compared to the increasing ones. Additionally, no significant differences were found between additive and multiplicative comparisons. These results cannot be explained by the static comparison model.

Designers of interactive learning environments with a focus on word problem solving usually have to compromise between the amount of resolution paths that a user is allowed to follow and the quality of the feedback provided. We have built an intelligent tutoring system (ITS) that is able to both track the user's actions and provide adequate supervision during the resolution. This is done without imposing any restriction on the resolution paths that are allowed. Instead, the system attempts to enforce metacognitive learning by requiring an appropriate definition of quantities before they are used. The program (a) supports both the arithmetical and algebraic way of solving problems; (b) allows the recurrence to one or more equations when solving a problem in an algebraic way; (c) determines the validity of the expressions when they are introduced; and (d) admits the incorporation of new problems without the need of being reprogrammed. In this paper, we explain the design foundations, which are mainly based on (a) a reflection of the steps that a student should follow to solve a problem in an algebraic way, and (b) the use of a domain specific notation to represent both the problem structure and the current state of the resolution process. In particular, hypergraphs are introduced as an adequate way to support tracking in both the arithmetical and algebraic case. Moreover, we offer an extensive experimental evaluation which highlights the potential of the ITS as a learning tool. (Contains 18 figures and 2 tables.)

The use of the algebraic method for solving word problems is a challenging topic for secondary school students. Students' difficulties are usually associated with extracting the problem's network of relationships between quantities and with formalizing these relationships into algebraic language in a problem model. Both sources can coexist and interact; thus, it is usually not possible to determine which source of difficulty is more relevant. In addition, there are specific errors, such as the error by multiple referents for the unknown, which are directly linked to the wording of the problem text, and in which the same two sources of error coexist. In this work, we present the results of an experiment conducted with 255 secondary school students assessing the effect of two common difficulties on the accuracy of problem models and on the rate of multiple referents for the unknown. The first difficulty is the use of algebraic language in the construction of the problem model; the second is the use of the same expression to designate different quantities within the problem text. We used a 2 × 2 between-between design, with one factor related to the symbolic language (algebraic or arithmetic) in which the problem model is constructed, and the other factor related to the actual language features of the text problem. Our results indicate that overall, the main source of difficulty for students is the use of algebraic language to formalize a problem model, representing a large effect size. [ABSTRACT FROM AUTHOR]

The reversal error is a prevalent phenomenon that consists of reversing the relationship between two variables when writing equations from comparison in verbal language. A study with 169 Basque/Spanish bilingual pre-service teachers examined whether this error is mainly due to the students' tendency to employ a direct-translation strategy based on the syntax of statements. By using Basque and Spanish the factor of syntax was studied since in Basque, unlike Spanish (or English), the natural way of expressing comparisons prevents students from using direct-translations strategies. Results show a significantly lower number of reversal errors in sentences written in Basque compared with those in Spanish. This finding confirms the importance of syntax statement in the translation from words into equations and suggests that teachers should have awareness of this fact and should design lesson plans devoted to work on word problems accordingly. [ABSTRACT FROM AUTHOR]

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Abstract: One of the most challenging steps in learning algebra is the translation of word problems into symbolic notation. This paper describes an Intelligent Tutoring System (ITS) that focuses on this stage of the problem solving process. On the one hand, a domain specific inference engine and a knowledge representation mechanism are proposed. These are based on a description language based on hypergraphs, and the idea of using conceptual schemes to represent the student’s knowledge. As a result, the system is able to simultaneously: (a) represent all potential algebraic solutions to a given word problem; (b) keep track of the student’s actions; (c) univocally determine the current state of the resolution process; (d) build a student model; and (e) provide adaptive automatic remediation. On the other hand, the Graphical User Interface (GUI) has been designed to force the student to follow the sequence of steps described in the Cartesian method. The ITS has been evaluated in an educational environment. The results show significant gains for the experimental group and hence support the use of the ITS in practice. [Copyright &y& Elsevier]