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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 aim of this pre-experimental study is to evaluate the acquisition level of counting skills of a 3-year-old classroom made up of 14 children through a specific instructional design. To this end, an instructional proposal to improve these mathematical skills was designed. Before and after the intervention, we measured the students' level regarding counting skills through an evaluation of their counting abilities. The results indicate that the designed intervention increased the acquisition level of skills related to counting principles, constituting an effective instrument to enhance counting skills for 3-year-old children. In particular, after the intervention children improved significantly in skills related to the one-to-one correspondence principle and the order-irrelevance principle, both showing a large effect size in their observed differences. The cardinality principle, stable-order principle and abstraction principle also showed gains, but the differences were found to be statistically nonsignificant. Finally, the role of the age of the participants was also analyzed in relation to their acquired counting skills, indicating that children in the older age range improved their counting skills more than children in the younger group.

The COVID-19 pandemic led to the lockdown of schools in many countries, forcing teachers and students to carry out educational activities remotely. In the case of mathematics, developing remote instruction based on both synchronous and asynchronous technological solutions has proven to be an extremely complex challenge. Specifically, this was the case in topics such as word problem solving, as this domain requires intensive supervision and feedback from the teacher. In this piece of research, we present an evaluation of how technology is employed in the teaching of mathematics, with particular relevance to learning during the pandemic. For that purpose, we conducted a systematic review, revealing the almost complete absence of experiments in which the use of technology is not mediated by the teacher. These results reflect a pessimistic vision within the field of mathematics education about the possibilities of learning when the student uses technology autonomously. Bringing good outcomes out of a bad situation, the pandemic crisis may represent a turning point from which to start directing the research gaze towards technological environments such as those mediated by artificial intelligence. As an example, we provide a study illustrating to what extent intelligent tutoring systems can be cost-effective compared to one-to-one human tutoring and mathematic learning-oriented solutions for intensive supervision in the teaching of word problem solving, especially appropriate for remote settings. Despite the potential of these technologies, the experience also showed that student socioeconomic level was a determining factor in the participation rate with an intelligent tutoring system, regardless of whether or not the administration guaranteed students' access to technological resources during the COVID-19 situation.

In this article, we analyze the potential of conversational frameworks to support the adaptation of existing tutoring systems to a natural language form of interaction. We have based our research on a pilot study, in which the open-source machine learning framework Rasa has been used to build a conversational agent that interacts with an existing intelligent tutoring system (ITS) called hypergraph-based problem solver (HBPS). This agent has been seamlessly integrated into the ITS to replace the previously available button-based user interface and allow the user to interact with HBPS in natural language. Once appropriately trained, the conversational agent was capable of identifying the intention of a given user utterance and extracting the relevant entities related to the message content, with average weighted F1-scores of 0.965 and 0.989 for intents and entities, respectively. Methodological guidelines are provided to generate a realistic training set that enables the creation of the required natural language understanding model and also evaluate the resulting system. These guidelines can be easily exported to other ITS and contexts to provide an enhanced interaction based on natural language processing methods.

This study analyses the potential of a learning analytics (LA) based formative assessment to construct personalised teaching sequences in Mathematics for 5th-grade primary school students. A total of 127 students from Spanish public schools participated in the study. The quasi-experimental study was conducted over the course of six sessions, in which both control and experimental groups participated in a teaching sequence based on mathematical problems. In each session, both groups used audience response systems to record their responses to mathematical tasks about fractions. After each session, students from the control group were given generic homework on fractions--the same activities for all the participants--while students from the experimental group were given a personalised set of activities. The provision of personalised homework was based on the students' errors detected from the use of the LA-based formative assessment. After the intervention, the results indicate a higher student level of understanding of the concept of fractions in the experimental group compared to the control group. Related to motivational dimensions, results indicated that instruction using audience response systems has a positive effect compared to regular mathematics classes.

Problem solving is often regarded as one of the most essential cognitive functions in our daily lives, and, for that reason, educational theorists have long stressed the need for its development. As cognitive flexibility is a fundamental characteristic necessary throughout the problem-solving process, the purpose of this study is to analyse students' problem-solving performance after following intra-task flexibility-based training mediated by an intelligent tutoring system. With this aim, 110 fifth and sixth grade students took part in a quasi-experimental study that included six forty-five-minute flexibility-based training sessions preceded and followed by test sessions in order to evaluate eventual problem-solving proficiency improvements. The findings show that intra-task flexibility training enhances students' ability to solve arithmetic word problems, especially when flexibility-related activities are completed sequentially in the same session and not in different sessions. Furthermore, the proposed instruction is particularly helpful in improving girls' problem-solving competence, which can minimize eventual gender gaps and strengthen their STEM vocation.

Patterning, as a component of early mathematic knowledge, is a common activity carried out at elementary levels in which children are not equally successful. This study aimed to measure different variables affecting performance on patterning tasks in early childhood. For this purpose, the success of Pre-K (N = 33), K (N = 31) and first-grade (N = 33) children when solving 14 repeating-pattern tasks, which varied in complexity, was analysed. The results revealed no significant differences between core-2 and core-4 length patterns, and greater success with patterns involving size. The study also introduces distractors, as a novel factor, in the patterning activities related to the presence of contradictory or surplus data. The impact of each factor and the relations between complexity and difficulty on the patterning tasks are discussed in order to contribute to the design of teaching itineraries. [Spanish language review by Mercè Rius.]

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.

When translating word problems into equations, a common error consists of using the same letter to refer to two different quantities. This error, called multiple referents for the unknown, can be connected to the existence of deictic or indexical expressions within the problem statement. This paper aims to analyse the influence of indexical expressions on the incidence of the multiple referents for the unknown error. Results from a quantitative study with 117 Spanish secondary students show a significantly higher number of such type of error when the problem statement contains indexical expressions. The analysis of students' performances in subsequent interviews indicates that this error may be due to the fragmented reading of statements.

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.

This paper presents an intelligent tutoring system (ITS) for the learning of arithmetical problem solving. This is based on an analysis of (a) the cognitive processes that take place during problem solving; and (b) the usual tasks performed by a human when supervising a student in a one-to-one tutoring situation. The ITS is able to identify the solving strategy that the student is following and offer adaptive feedback that takes into account both the problem's constraints and the decisions previously made by the user. An observational study shows the ITS's accuracy at emulating expert human supervision, and a randomized experiment reveals that the ITS significantly improves students' learning in arithmetical problem solving.

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.)

This study analyses the potential of a learning analytics (LA) based formative assessment to construct personalised teaching sequences in Mathematics for 5th‐grade primary school students. A total of 127 students from Spanish public schools participated in the study. The quasi‐experimental study was conducted over the course of six sessions, in which both control and experimental groups participated in a teaching sequence based on mathematical problems. In each session, both groups used audience response systems to record their responses to mathematical tasks about fractions. After each session, students from the control group were given generic homework on fractions—the same activities for all the participants—while students from the experimental group were given a personalised set of activities. The provision of personalised homework was based on the students' errors detected from the use of the LA‐based formative assessment. After the intervention, the results indicate a higher student level of understanding of the concept of fractions in the experimental group compared to the control group. Related to motivational dimensions, results indicated that instruction using audience response systems has a positive effect compared to regular mathematics classes. Practitioner notesWhat is already known about this topic Developing an understanding of fractions is one of the most challenging concepts in elementary mathematics and a solid predictor of future achievements in mathematics.Learning analytics (LA) has the potential to provide quality, functional data for assessing and supporting learners' difficulties.Audience response systems (ARS) are one of the most practical ways to collect data for LA in classroom environments.There is a scarcity of field research implementations on LA mediated by ARS in real contexts of elementary school classrooms.What this paper adds Empirical evidence about how LA‐based formative assessments can enable personalised homework to support student understanding of fractions.Personalised homework based on an LA‐based formative assessment improves the students' comprehension of fractions.Using ARS for the teaching of fractions has a positive effect in terms of student motivation.Implications for practice and/or policy Teachers should be given LA/ARS tools that allow them to quickly provide students with personalised mathematical instruction.Researchers should continue exploring these potentially beneficial educational implementations in other areas. [ABSTRACT FROM AUTHOR]