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Learning fractions poses a challenge for many elementary school students, including applying fraction knowledge in novel contexts. For instance, there are substantial individual differences in students’ tendency of spontaneous focusing on quantitative relations (SFOR), which is related to the development of rational number knowledge. In this study, 4th grade students (N = 129) took part in a quasi-experimental study comparing an intervention condition (n = 71) aimed at improving students’ multiplicative relational reasoning and fraction knowledge with a control condition (n = 58) of business as usual fraction instruction. Five lessons of intervention activities were designed to promote students ability to recognize and describe multiplicative relations in their everyday surroundings. There was an overall positive effect on the students’ mathematical knowledge. Students who participated in the intervention improved their ability to recognize and describe multiplicative relations embedded in pictures representing everyday situations. There were no significant differences in the development of fraction knowledge despite replacing five traditional fraction lessons. These findings provide further evidence that researchers and educators should continue to pay attention to issues surrounding students’ spontaneous mathematical focusing tendencies.

Understanding mathematics is essential for understanding the world; as stated by Pythagoras and Galilei, “the world everywhere around us is written in the language of mathematics”. Yet, according to recent research on spontaneous mathematical focusing tendencies, children and adults alike differ in how much attention they focus on mathematical features in the world around them (

Solid knowledge of fractions is essential not only for higher mathematics learning, such as algebra (

Many primary school students struggle to understand fraction magnitudes (

A distinction can be made between part-whole interpretations of fractions and understanding fraction magnitudes (

The overuse of natural number features when learning and reasoning about rational numbers is referred to as the natural number bias. The natural number bias refers to the tendency to erroneously apply knowledge relevant to natural numbers even when it is not applicable to the rational number task at hand (

Despite these difficulties with formal fraction instruction, studies show that even very young children appear to have a basic understanding of ratios and an ability to reason multiplicatively on tasks (

Opportunities to notice and use fractions or multiplicative relations do not happen only in explicitly mathematical situations in the classroom. Rather, the opportunities to describe mathematical relations in the world are limitless if one directs students’ attention to the mathematical aspects all around (

Evidence suggests that the amount and intensity of attention paid to quantitative or multiplicative relations in situations that are not explicitly mathematical, referred to as SFOR tendency, substantially differs between children (

Importantly, it is possible to enhance an individual’s spontaneous mathematical focusing tendencies and get them to pay more attention to the mathematical aspects of their surroundings. For instance,

To increase students’ ability to use multiplicative relations in their everyday lives, intervention activities should make multiplicative relations explicit targets of focusing. This can be achieved by teaching students how to recognize and describe these multiplicative relations using explicit multiplicative phrasing, such as “twice as many” (

Secondly, alongside recognizing the mathematical features around them, it is necessary for students’ to be able to describe them using exact mathematical language. Therefore, the present intervention aimed to improve students’ mathematical language by providing explicit terms for describing everyday multiplicative relations. Knowledge of mathematical language may have a direct impact on mathematical development (

Increasing students’ tendency to make multiplicative relations explicit targets of focusing in their everyday lives allows for connecting the representational and analytic meanings of multiplicative relations and fractions (

The considerable body of evidence regarding primary school students’ difficulties in dealing with fractions encourages further research on how to support students’ learning (

The aim of the present study was to test how intervention activities aimed at improving students’ focusing on multiplicative relations impact their mathematical knowledge, including multiplicative relations and fractions. We carried out a pre-registered quasi-experimental classroom study comparing students who participated in the intervention activities with those who had their normal fraction instruction. We examined students’ mathematical knowledge at three measurement points: before, after, and three months after the intervention. We measured the students’ ability to describe multiplicative relations when their attention was guided toward quantitative aspects, as well as their fraction representation and size knowledge. The overall effectiveness of the intervention on the three knowledge measures was examined based on a pre-registered analysis plan (

Students (ages 10 to 11 years) from three schools and seven 4^{th} grade classrooms (

Permission to take part in the study was received from teachers, parents, students, and heads of the schools. The ethical guidelines of the University of Turku on research integrity were followed.

Both groups followed the same pretest–intervention–posttest–delayed posttest schedule. The tests lasted 45 minutes and were conducted during mathematics lessons. After the pretest, both intervention groups went through five consecutive 45-minute mathematics lessons, followed by the posttest on the next mathematics lesson. No other mathematics lessons took place during the intervention period. The classroom teacher taught the lessons, and the tests were run by a trained researcher. Pretest, intervention, and posttest were completed over a period of two weeks, and there were no other mathematics lessons during that time. The delayed posttest was completed three months after the intervention. Between the posttest and delayed posttest, the classrooms did not have any lessons specifically targeted at fractions.

The experimental group teachers (

The control group teachers (^{th} grade mathematics book normally. No specific instructions or restrictions were placed on the teaching of fractions during the lessons, nor did they have any professional development sessions.

Experimental group lessons were developed from previous interventions aimed at promoting spontaneous mathematical focusing tendencies (

The aim of the Lesson 1 was to introduce the students to the idea that multiplicative relations and fractions can be found in many different situations in their everyday lives. It provided guidance on how to recognize and basic vocabulary on how to describe these relations when they are found. The teacher was encouraged to organize the whole lesson so that it would elicit the most individual thinking and sharing of thoughts for the class (e.g., the think-pair-share method). First, the students were asked to think and share examples of daily situations in which mathematical thinking and multiplicative relations are relevant and applicable.

The teacher then presented a PowerPoint presentation, including examples of how to use multiplicative relations when describing pictures. The pictures were used to elicit discussions between the students and the teacher about how to describe multiplicative relations in everyday situations. For example, the students could describe

In Lessons 2 and 3, students did two different scavenger hunt-type activities similar to those used in previous intervention on SFOR tendency (

For the second scavenger hunt lesson, the teacher formed another scavenger hunt by taking a selection of students’ answers from the first hunt. Teachers then deleted one of the inputs that the students found on the first hunt to make a new set of statements (e.g., “____ is two times as tall as chair”). Students then went on and tried to solve these new scavenger hunt tasks.

The last two lessons used the format of “which one does not belong” tasks. During the lessons, the students rehearsed finding and describing multiplicative relations from pictures. First, the students were presented with a series of picture sets. In each set, three of the pictures contained an instance of the same multiplicative relation (e.g., one-half), and one did not. In the example in

During the last lesson, the students created picture sets in groups of two to four for other students to solve. The teacher had pedagogical autonomy on how to organize the creation process (i.e., whether to make digital sets of pictures or use materials found from the classroom). However, all teachers elected to organize the activity so that the students’ formed the picture set by using physical materials found around the classroom (e.g., pencils, books, etc.). The students then went around the classroom and tried to solve other groups’ tasks.

The 45-minute tests took place during the mathematics lessons and were taken using paper and pencil. The tasks presented were in the same order for all students. The tasks measured three different aspects of mathematical knowledge: multiplicative relations, fraction representations, and fraction size.

At all three time points, a transformation task was used to measure the students’ ability to recognize and describe multiplicative relations. This task was based on previous measures of guided focusing on quantitative relations (

In the writing task, students were given one point for each multiplicative relation they described per set of objects (e.g., three times more, half the amount, a third less). The relation did not have to be mathematically correct but needed to be a specific multiplicative relation, for example, saying that the items tripled when they actually doubled was still correct. In the drawing task, one point was given to each set of objects with the correctly drawn amount. There was a maximum of three points per item and 12 points overall.

Additional picture description tasks were included in the posttest and delayed posttest. The task measured the students’ ability to recognize and describe multiplicative relations from pictures. Students were shown photos of real-life situations and asked to describe mathematical relations as precisely as possible (e.g., “What mathematical relations do you see in the picture? Describe all the mathematical relations you find from the picture as precisely as possible”). On the posttest, only the car picture description item was included (

Students were given one point for each described multiplicative relation or fraction (e.g., “two out of six cars are red,” “half of the cars are on the other side”, “4/8 of the eggs are blue”). The relation did not have to be mathematically correct but should have been a specific multiplicative relation. For example, saying that one-third of the eggs on Item 2 are yellow, when the correct amount is one-fourth, would still get a point. There was no theoretical maximum number of points for the task. The highest score obtained by a student in a single item was 8 points.

The picture description tasks had an acceptable reliability (pretest α = .74; posttest α = .64, delayed posttest α = .85). As stated in the preregistration, the use of non-specific responses such as “less” and “more” were separately coded for the transformation writing tasks and picture description tasks. However, this measure was not included in the main analysis due to poor reliability (see Appendix A in the

Fraction representation knowledge was measured by two tasks at each measurement point.

The number sets task (

The coloring task was used to measure students’ fraction representation knowledge by asking them to color a specified proportion of eight figures, including four geometrical shapes and four pictorial representations (Appendix C and Appendix D,

Fraction size knowledge was measured with three tasks at each measurement point.

In the missing value task, the students were presented with two fractions, with one fraction missing either the numerator or denominator. The students were to fill in the missing numerator or denominator that would make the two fractions equal. The task contained five items: A)

In the fraction comparison task (

In the number line estimation task, students estimated the location of a given fraction on a 14.5 cm long number line with endpoints of 0 and 1 (e.g., “Below is a number line between 0 and 1. Place a fraction

The number line estimation task had a good reliability (pretest α = .91; posttest α = .91; delayed posttest α = .87).

The data were analyzed based on pre-registered analysis using IBM SPSS 25. The graphical figures were created using JASP version 0.14. To create separate composite measures of fraction representations and fraction size knowledge, the students’ scores were standardized for each task separately at each time point. The average of the standardized component task scores—in the case of fraction representations: number sets and coloring tasks, and, in the case of fraction size: missing value, comparison, and number line estimation—were then calculated. This process assigned equal weight to the different tasks used in the measure, no matter the scales. Lastly, an overall measure of mathematical knowledge was created by averaging the standardized knowledge aspect scores: multiplicative relations, fraction representations and fraction size (for Pearson correlation coefficients of the knowledge aspects see Appendix E,

Descriptive statistics of all tasks are reported in Appendix F,

Overall score |
Multiplicative relations |
Fraction representations |
Fraction Size |
|||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Pre | Post | Delayed | Pre | Post | Delayed | Pre | Post | Delayed | Pre | Post | Delayed | |

Experimental ( |
0.05 (0.80) | 0.14 (0.82) | 0.16 (0.85) | .056 (1.03) | .281 (1.02) | .271 (1.21) | .009 (0.95) | .066 (0.90) | .080 (0.88) | .091 (0.79) | .082 (0.81) | .131 (0.75) |

Control ( |
-0.05 (0.65) | -0.15 (0.74) | -0.16 (0.66) | -.057 (0.98) | -.286 (0.90) | -.276 (0.63) | -.009 (0.79) | -.067 (0.87) | -.081 (0.89) | -.093 (.0.68) | -.083 (0.79) | -.133 (0.81) |

Cronbach’s alpha | .81 | .82 | .87 | .74 | .68 | .87 | .83 | .86 | .85 | .71 | .68 | .66 |

ICC | 0.07 | 0.16 | 0.22 | 0.03 | 0.16 | 0.15 | 0.11 | 0.17 | 0.21 | 0.07 | 0.07 | 0.19 |

To examine the effects of the intervention on students’ knowledge, a repeated measures ANOVA was run with overall scores as the dependent variable, measurement point (pre-, post-, and delayed post-test) as a within-subject variable, and group membership (experimental, control) as a between-subject variable. There was a statistically significant interaction effect of measurement point by group,

Subsequent repeated measures ANOVAs were run with measurement points (pretest, posttest, and delayed posttest) as a within-subject variable and group membership (experimental, control) as a between-subject variable on each knowledge measure separately to investigate the effects of the intervention on different aspects of mathematical knowledge. There was a statistically significant interaction of measurement point by group for multiplicative relation knowledge,

There were no statistically significant interaction effects of measurement point by group for fraction representation knowledge,

In our pre-registered analysis, we planned to examine specific effects of the intervention on students’ natural number bias if the overall intervention effects on their size knowledge was statistically significant. Despite there being no statistically significant intervention effect on fraction size, there was some differentiation in students size knowledge from pretest to delayed posttest (see

Task / Group | Adjusted Mean ( |
95% CI |
Sig | |||
---|---|---|---|---|---|---|

Lower | Upper | |||||

Multiplicative relations | ||||||

Experimental | 9.50 (.83) | 7.86 | 11.13 | 10.44 | .002 | .09 |

Control | 5.65 (.86) | 3.96 | 7.35 | |||

Coloring | ||||||

Experimental | 4.48 (.21) | 4.06 | 4.90 | 0.005 | .942 | .00 |

Control | 4.46 (.22) | 4.03 | 4.89 | |||

Number sets | ||||||

Experimental | 9.14 (.48) | 8.18 | 10.10 | 2.79 | .089 | .02 |

Control | 7.98 (.50) | 6.98 | 8.97 | |||

Missing Value | ||||||

Experimental | 1.68 (.24) | 1.21 | 2.16 | 0.04 | .847 | .00 |

Control | 1.62 (.25) | 1.13 | 2.11 | |||

Fraction comparison | ||||||

Experimental | 1.71 (0.23) | 1.26 | 2.16 | 0.07 | .798 | .001 |

Control | 1.63 (0.24) | 1.16 | 2.10 | |||

Number Line Estimation PAE^{a} |
||||||

Experimental | 19.0 (2.00) | 15.10 | 22.90 | 7.63 | .007 | .06 |

Control | 26.8 (2.00) | 22.80 | 30.90 |

^{a}Number line estimation average PAE (%) scores (less is more accurate answer). PAE mean represents the average distance from correct mark per item.

In order to confirm these exploratory results were not dependent on the nested nature of the data within classrooms and differences in instruction after the intervention, two competing linear mixed models were run with delayed post-test estimation as the dependent variable, experimental condition as the grouping variable, and pre-test estimation as a covariate. In the first model, classroom effects were not included in the model. In the second, random intercepts and slopes for each classroom were estimated. Results show that the model fit decreased with the inclusion of random classroom effect (BIC = 307) when compared with the model without random classroom effects (BIC = 302) and we therefore did not follow-up these results with further analysis of classroom effects.

The present study aimed to explore how an intervention designed to promote the ability to recognize and describe multiplicative relations embedded in everyday life affects primary school students’ mathematical knowledge. The effects of the intervention on student’s ability to recognize and describe multiplicative relations, fraction representation knowledge, and fraction size knowledge were measured. The intervention was effective in promoting the students’ overall mathematical knowledge. This appeared to be concentrated on the students’ ability to recognize and describe multiplicative relations. There were no statistically significant effects of the intervention on students’ fraction knowledge when compared with traditional fraction instruction. However exploratory analysis revealed a potential positive effect on the experimental group students’ knowledge on the number line estimation task in the delayed posttest. The intervention replaced, rather than supplemented, five mathematics lessons on fractions, yet there was no difference in fraction knowledge between the two groups. Thus, the added value of the intervention on students’ multiplicative relational knowledge suggests an overall positive outcome.

The present study is a second iteration of an intervention program with classroom activities based on enhancing students’ focusing on multiplicative relations. Akin to the previous study by

The teaching and learning of fractions is an integral part of the primary school curriculum, yet students struggle with learning fractions (

Integrating guided focusing activities with normal fraction instruction could expand the ways in which fractions are taught and help students gain a deeper understanding of fractions and their representations. According to

One major factor contributing to the results may be that the lessons succeeded in supporting the students’ explicit mathematical language (

The present study has limitations that have to be acknowledged. The quasi-experimental design limits the conclusions that can be drawn about the learning outcomes. A larger randomized sample would lead to more clarity regarding the effects of the intervention; for example, examining whether there are differing effects of the intervention depending on students’ prior knowledge. As well, the sample size of the study does not allow for reliable analysis of the multilevel nature of the data. Thus, the presented results might be attributable to differences between schools or class level teaching between posttest and delayed posttest, especially within the exploratory analysis in which the posttest was not regarded. However, the reported intra-class correlations are within acceptable range and the pre-registration of the analysis suggest that the presented results should be fairly robust.

The multifaceted approach of the intervention activities makes it hard to determine which of the activities contributed towards the results. Hence, it is not possible to determine the causal mechanism that led to the development of mathematical knowledge based on the intervention. It may be that increased skills in how to recognize and describe multiplicative relations around them led to the students in the experimental group to gain more self-initiated practice with multiplicative relations (

The study places a great deal of emphasis on the teachers involved in the study. The study was relatively naturalistic, as the teachers carried out the lessons in both groups. This brings real-world limitations that would not apply in laboratory settings where teaching could be standardized. On the other hand, having the teachers carry out the intervention activities has the potential to show whether carrying out the activities and understanding the pedagogical idea of the intervention are feasible for teachers. Importantly, the results imply that teachers were able to adopt new pedagogical approaches and tools into their teaching, even with minimal introduction. The teachers voluntarily agreed to participate in the study. Therefore, the teachers may differ from the average teacher in their motivation and pedagogical approach toward mathematics instruction. Expanding this intervention to a larger group of teachers may not be straightforward, and the influences of teacher characteristics on the outcomes of the interventions should be examined in future studies.

The results of the present study support the existing empirical evidence of the effectiveness of interventions aimed at enhancing spontaneous mathematical focusing tendencies on students’ learning (

This work was supported by the Academy of Finland Strategic Research Council [grant number 336068] awarded to the second author and by Academy of Finland [grant number 331080] awarded to the last author.

We cannot publish the original data of the study due to the restrictions negotiated with the project associated with funding of the study. We do not have permission from participants and their guardians to publish data due to requirements of their school administration in the city that the study took place in.

The Supplementary Materials contain the following items (for access see

The preregistration protocol for the study

Six Online Appendices with additional images and tables

The authors have declared that no competing interests exist.

The authors have no additional (i.e., non-financial) support to report.