The previous body of research literature has reported several separate cognitive processes relevant in solving mathematics wps. Therefore, it is of the essence to seek for effective intervention and instruction for students in need for support in learning.

This article reports the outcome of an intervention targeted at mathematics word problem (wp) skills.

This study included three data collection points: (1) Premeasurements, (2) post-measurements and (3) follow-up measurements. Pre-measurements were performed in August, post-measurements immediately after the intervention period in October and follow-up measurements in December.

A programme, which included face-to-face support in mathematics wp strategies with the think-aloud protocol, was applied. The participants were 28 Finnish third-graders (14 training group students and 14 control students). Their mathematics wp skills were tested three times (pre-, post- and follow-up assessments). The groups were matched by gender, family type and the mathematics wp pre-measurement score level. The groups differed neither by literacy skills (i.e. technical reading, reading comprehension) nor by task orientation at baseline.

Some acceleration of mathematics wp skills among the training group students was found but the growth dramatically declined as soon as the face-to-face support stopped. The results further showed improvement in the efficacy of correct answers or attempted mathematics wp items among training group students.

The results suggested that training consisting of face-to-face support is crucial for accelerating mathematics wp strategies among students struggling with mathematics. Repeated, cyclic periods of support are suggested for sustained effect.

Arithmetic word problems (wps) constitute an important part of mathematics in elementary school. They integrate formal school mathematics and the real world, and require the learners to apply previously learned skills (Verschaffel, De Corte & Lasure

The previous body of research literature has reported several separate cognitive processes relevant in solving mathematics wps. Based on those studies, it may be argued that in addition to basic arithmetics skills (Schoppek & Tulis

In elementary school, most of the mathematics wps are presented in written form (e.g. Fuchs, Fuchs & Compton

As stated before, the mathematics wp-solving process integrates many essential knowledge-processing phases, some of which are parallel and others are concurrent. Previous intervention research seeking amelioration of mathematics wp-solving skills has emphasised training separate cognitive strategies (Swanson, Lussier & Orosco

The present study considers the think-aloud method as one cognitive strategy, which is suitable to certain contexts (Ericsson & Simon

Using think-aloud protocols as a tool for supporting learning is easier if the teaching context allows individual face-to-face contact. A comprehensive report by Gersten et al. (

Think-aloud skills are connected with good reading comprehension skills (Ghaith

As we now understand that the mathematics wps represent a cognitive ‘hub’ combining a large set of different cognitive processes such as reading fluency, reading comprehension, procedural skills and the ability to understand the context of each problem, the think-aloud protocol was chosen as one tool of the intervention, as language and verbalisation of thoughts are in the core of the approach. We utilised a think-aloud protocol by Ericsson and Simon (

The early theoretical background of the think-aloud method could be related to Wundt’s (

Along with teaching problem-solving strategies, another important feature of the think-aloud intervention is to teach efficacy in solving mathematics wps: it includes the ability to recognise items that are solvable items one finds too hard and items a student may try to solve, even though they seem hard. In the present study, the mathematics wp efficacy is determined to serve as the efficacy consisting of the input–output ratio between correctly solved items and attempted items. The larger the ratio is, the higher the efficacy in solving the mathematics wps. For example, if a student succeeds in solving three items correctly, but has attempted to solve 10 items, the input–output ratio is 0.30. If a student succeeds in solving 10 items and has attempted to solve 10 items, the ratio is 1.00. The number of attempted mathematics wps was considered important, because students with difficulties in mathematics (Passolunghi

The study questions and hypotheses were as follows:

To what extent do the effects of the intervention on mathematics wp skills differ between the training group students (receiving overall mathematics wp strategy instruction and face-to-face think-aloud strategy instruction) and control group students (only receiving an overall mathematics wp strategy instruction). The effects of intervention were calculated as the amount of correct answers to the wp tasks. We hypothesised that the intervention combining strategy instruction and the think-aloud protocol would accelerate math wp performance among the training group students (Rittle-Johnson

To what extent does the intervention improve math wp-solving efficacy? Mathematics wp efficacy was defined as the input–output ratio of correct answers and attempts to solve wp tasks. In addition, the efficacy was inspected over the three measurement time points (pre-, post- and follow-up measurements). An attempt was calculated, if the student had marked down calculations and an answer to the item, regardless of whether the answer was correct or not. We hypothesised that the intervention would improve the mathematics wp efficacy, as it is a combined measure of skills, metacognition and self-regulation (Montague 1998; Thorndsen

A total of 148 Finnish third-graders were invited to participate in the study at the beginning of the school year, in August, including 76 boys and 72 girls (_{age} = 8.72, SD [standard deviation] = 0.47). The Finnish school year starts in the middle of August and ends in late May or at the beginning of June. The summer vacation lasts from early June to around the 15th of August. Compulsory schooling in Finland lasts for 9 years. It starts from Grade 1 the year the child turns 7 years old. Finnish schools are becoming more multicultural. However, the present data consisted solely of students who speak Finnish as their primary home language.

Informed consent for participation in the study was obtained from 136 participants’ parents. The educational level of parents is usually relatively good in Finland. In the present data, 15.5% of the fathers had a university degree or a degree from a polytechnic university, 69.0% had a vocational school or vocational institute degree and again 15.5% of the fathers had only a compulsory school diploma. Regarding the mothers, 33.8% had a university degree or degree from a polytechnic university and 58.0% had a vocational school or vocational institute degree, while 5.4% of the mothers had only a compulsory school diploma.

This study included three data collection points: (1) Pre-measurements, (2) post-measurements and (3) follow-up measurements. Pre-measurements were performed in August, post-measurements immediately after the intervention period in October and follow-up measurements in December.

The training group inclusion criteria were (1) teacher referral and (2) each participant who had regularly received Tier 2 support in mathematics during earlier years (_{age} = 8.75, SD = 0.60). Tier 2 support for learning in the Finnish Response-To-Intervention-like framework is called ‘intensified support’ (for further details, see Fuchs & Fuchs

The control group students (_{age} = 8.77, SD = 0.46) were selected from the remaining 122 students. The gender, family type and baseline mathematics wp score were matched as closely as possible, by pairwise matching of each training group participant and a control group participant. The matched pairs were otherwise identical in terms of these criteria with one major exception: the control group students were not among those that the teachers had referred as in need of support in mathematics. Additionally, in one training group student–control group student pair, family type was ‘unmarried spouse and children’ for one and ‘other’ for the other student (other criteria practically identical; baseline mathematics wp score between-group difference

Demographic information and baseline measures.

Scale | TrG ( |
CG ( |
||
---|---|---|---|---|

% | % | |||

Comprehensive school | 1 | 7.1 | 1 | 7.1 |

Basic vocational school | 5 | 35.7 | 3 | 21.4 |

Vocational institute | 1 | 7.1 | 1 | 7.1 |

Higher education | 3 | 21.4 | 4 | 28.6 |

University degree | 4 | 28.6 | 5 | 35.7 |

Comprehensive school | 2 | 15.4 | 2 | 14.3 |

Basic vocational school | 2 | 15.4 | 8 | 57.1 |

Vocational institute | 4 | 30.8 | 2 | 14.3 |

Higher education | 2 | 15.4 | - | - |

University degree | 3 | 23.1 | 2 | 14.3 |

Spouse and children | 11 | 78.6 | 11 | 78.6 |

Unmarried spouse and children | 2 | 14.3 | 1 | 7.1 |

Single parent or other | 1 | 7.1 | 2 | 14.3 |

TrG, training group; CG, control group.

The students’ mathematics wp skills were measured using MATTE (Matematiikan sanallisten tehtävien ja laskutaidon arviointi [Evaluation of the student’s mathematical problem-solving and arithmetic skills]) (Kajamies et al.

Mathematical wp-solving was assessed by a parallel set (set A for pre-measurement, B for post-measurement and C for follow-up measurement) at each measurement time point. Each set consisted of 15 one- and multi-step wps (e.g.

Technical reading was used as one of the control measures in the study. It was assessed using the word recognition subtest of the ALLU reading test (Lindeman

Text comprehension was another control measure in the present study. It was measured using a subtest of the standardised primary school reading test (Lindeman

As the third control measure, the participants’ motivation towards mathematics tasks (Salonen et al.

This study applied a design with (1) mathematics wp strategy instruction to all the students in participating classrooms (including training group students, control group students and other students within the classrooms) and in addition to that (2) face-to-face think-aloud strategy instruction for the training group students, provided by a trained teacher. The intensity of mathematics wp strategy instruction was three times per week and the duration was 15 min. This way, exposure to the intervention (strategy instruction) was 3 × 15 min per week. Exposure to the intervention (strategy instruction + face-to-face think-aloud instruction) was 3 × 45 min per week for the training group. The overall duration for the intervention was 6 weeks. We included a 2-week ‘resting’ gap in the middle, so the actual intervention was given in 2 × 2-week sets. During the 2 week gap, all the students participated in their regular school days. In summation, training group students received a total of 540 min of (intensive) support during the intervention, whereas the control group students (only participating in the overall strategy instruction) received a total of 180 min of support during the intervention period.

Each intervention session started with an overall strategy instruction (duration 15 min) in solving mathematics wp tasks during mathematics lessons (see also

After the overall mathematics wp strategy instruction, the training group students started the face-to-face work with the trained teacher (duration 20 min). Those students within the control group, as well as other students in the classrooms participating in this study, started to calculate the MATTE (Kajamies et al.

In the present study, we applied a set of pre-determined questions, ‘a protocol’ (see Ericsson & Simon

After each session, the students were asked to rate the MATTE tasks using a form that comes with the material. Additionally, the training group students were interviewed to close the session and to get feedback on the tasks. The interviews were recorded and used to improve the instruction. For example, if a student suggested spending more time on example mathematics wps, this was individually taken into account immediately during the next session.

To answer the first research aim the effects of the intervention were inspected by the amount of correct answers to the wp tasks, repeated measures of analysis of variance (ANOVA) were calculated. There was no need for using literacy skills (i.e. technical reading and reading comprehension) or task orientation as control variables, as there were no statistically significant differences between groups in these measures (

Next, to answer the second research aim on the mathematics wp-solving efficacy, the input–output ratio of attempts and correct answers to solve wp tasks was inspected over the three measurement time points (pre-, post- and follow-up measurements). Again, repeated measures of ANOVAs with paired-samples

This article followed all ethical standards for research without direct contact with human or animal subjects.

Firstly, differences between the groups were calculated with repeated measures ANOVAs. See

Means and standard deviations at different measurement time points by group.

Scale | TrG ( |
CG ( |
Wilcoxon TrG/CG | |||
---|---|---|---|---|---|---|

Technical reading (baseline) | 10.23 | 2.62 | 9.29 | 2.43 | 0.97 | - |

Reading comprehension (baseline) | 27.40 | 7.53 | 29.50 | 12.47 | 0.53 | - |

Task orientation (baseline) | 15.23 | 3.30 | 14.14 | 3.00 | 0.90 | - |

Mathematics wp pre-measurement |
0.17 | 0.50 | 0.17 | 1.33 | 0.80 | 0.590/0.547 |

Mathematics wp post-measurement | 0.71 | 0.94 | 0.07 | 1.28 | 0.37 | - |

Mathematics wp follow-up measurement | 0.37 | 0.73 | 0.37 | 1.12 | 2.04 |
- |

Mathematics wp efficacy pre-measurement | 0.21 | 0.55 | 0.21 | 1.20 | 1.13 | 0.046 |

Mathematics wp efficacy post-measurement | 0.16 | 0.89 | 0.15 | 1.11 | 0.79 | - |

Mathematics wp efficacy follow-up measurement | 0.25 | 0.50 | 0.25 | 1.06 | 0.37 | - |

TrG, training group; CG, control group,

,

, All standardised values negative.

, Calculated from

However, an important trend suggesting actually quite opposite changes in mathematics wp scores may be detected in

Mathematics word problem performance between the training group and control group. (Estimated marginal means of a maths word problem.)

Next, group differences in the efficacy, measured as the input–output ratio of the correctly calculated items and attempted items, were inspected. The results showed again that group and measurement time in mathematics wp performance did not have a statistically significant interaction effect (

Overall, the results of the present study suggested that even though the interaction between the measurement time and group did not prove this particular think-aloud intervention as statistically significant, it is safe to say that applying a systematic face-to-face think-aloud strategy instruction could be used as one tool to accelerate and improve the mathematics wp strategy skills among students struggling with mathematics. Additionally, the results showed that the efficacy to solve mathematics wps grew significantly higher among the training group students: the efficacy in mathematics wps as an input–output ratio measured as correct answers or attempted items was ameliorated during the intervention period. These findings may contribute to the field of special education and mathematics intervention taking into account how the support should be provided for different learners.

Firstly, the extent to which the number of correct solutions in mathematics wps increased over pre-, post- and follow-up measurements was examined. It has been suggested that think-aloud questions are very task-specific (Ostad & Sorensen

However, it needs to be noted that mathematics wps are essential in the curriculums for third-graders and they are present in textbooks in almost all lessons. Therefore, the student would continue to be exposed to the mathematics wps even after the intervention was over. The differences between the training group and control group grew over time which suggested that the regular classroom instruction was not sufficient for the training group students. The dramatic decline of training group students’ mathematics wp-solving skills could be explained firstly by the fact that we utilised parallel versions of the same test, but not exactly the same items. Secondly, Fuchs et al. (

Secondly, the extent to which the input–output ratio of correct answers or attempted mathematics wps is increased over pre-, post- and follow-up measurements was examined. This was an important aim to be looked at, as students struggling with mathematics tend to learn to avoid even attempting to solve the tasks they find difficult (Aunola et al.

There are at least three limitations to be considered when attempting to generalise the results presented here. Firstly, the sample was very small in the present study. However, intervention studies are hard work, and therefore, small samples are not unusual within the educational intervention research paradigm. Secondly, this study did not aim to identify students with mathematics difficulties in a diagnostic sense. This means that no strict criteria for selecting students to be randomised in one of the intervention conditions were used. Instead, we asked the teachers to indicate the students who would benefit from the intensive intervention and who also have been engaged in part-time special education. Thirdly, we aimed to simulate an authentic situation in classrooms in terms of diversity in mathematics skills by selecting a little over 10% of the students as participants for the intervention (by teacher referral), as opposed to 25% used many times in a similar research design.

The present study adds to the previous literature by providing a set of results for a small-scale intervention study, utilising a face-to-face think-aloud strategy instruction as an added feature to general mathematics wp strategy instruction. Our research outlined that the individual face-to-face support provided was very intensive, as a trained interventionist spent all the intervention sessions solely instructing the target students. This way, we simulated one type of an extreme condition of co-teaching. There is an ongoing discussion on the effects of inclusive co-teaching versus small-group instruction outside the classroom (see Fuchs et al.

The authors have declared that no competing interests exist.

All authors equally contributed to the writing of the manuscript.

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Data sharing is not applicable to this article as no new data were created or analysed in this study.

The views and opinions 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.

A sample lesson plan of an intervention lesson (45 min).