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Recent studies have tracked eye movements to assess the cognitive processes involved in fraction comparison. This study advances that work by assessing eye movements during the more complex task of fraction addition. Adults mentally solved fraction addition problems that were presented on a computer screen. The study included four types of problems. The two fractions in each problem had either like denominators (e.g., 3/7 + 2/7), or unlike denominators exhibiting one of the following relationships: one denominator was a multiple of the other denominator (e.g., 2/3 + 1/9), both denominators were prime numbers (e.g., 2/7 + 3/5), or both denominators had a common divisor larger than one (e.g., 5/6 + 3/8). Self-reports, accuracy, and response times confirmed that participants adapted their strategy use according to problem type. We analysed the number of eye fixations on each fraction component, as well as the number of saccades (rapid eye movements) between fixations on components. We found that participants predominantly processed the fraction components separately rather than processing the overall fraction magnitudes. Alternating between the two denominators appeared to be the dominant process, although in problems with common denominators alternating between numerators was dominant. Participants rarely used diagonal saccades in any of the problems, which would indicate cross-multiplication. Our findings suggest that adults adapt their cognitive processes of fraction addition according to problem type. We discuss the implications of our findings for numerical cognition and mathematics education, as well as the limitations of our current understanding of eye movement patterns.

Proficiency with fractions is important for everyday life as well as for further mathematical learning (

Common fractions^{i} are composed of two natural number components, the numerator and the denominator (e.g., 3/8 has a numerator of 3 and a denominator of 8). Thus, from a cognitive perspective, it is possible to process a fraction either

Given students’ apparent difficulty processing fractions holistically, research has investigated whether our cognitive system is capable of processing fractions in a holistic manner, and if so, to what extent people make use of holistic processing of fractions. Many studies address these questions by exploring how educated adults and academic mathematicians solve number comparison problems (

A limitation of the research described so far is that it relies on accuracy and response time measures, which are distal measures of cognitive processes since they do not capture potential differences in

During visually presented cognitive tasks, eye movements are assumed to correspond to mental operations (

Recent studies used eye tracking to investigate strategy use and the cognitive processes involved in fraction comparison (

Recent eye tracking research also explored whether individual fraction components are more or less difficult to process when solving fraction comparison problems.

The results from studies of fraction comparison reviewed above (that holistic and componential fraction processing depends on whether fractions have common components) may not be directly transferable to fraction addition. The cognitive processes involved in fraction addition might differ substantially from those in fraction comparison because addition problems require the use of different strategies than fraction comparison problems. For example, it is not evident that people use holistic processing in fraction addition. While holistic processing can be beneficial to estimate or to double-check the result of an addition problem,^{ii} it is not absolutely necessary. In fact, the standard addition algorithm taught in schools relies only on manipulating the fraction components and does not involve holistic processing. This standard strategy is generally valid for adding any two fractions (e.g., 5/6 + 3/4). It includes three steps:

Step 1: Find the least common denominator^{iii} of the two fractions (e.g., the least common denominator of 6 and 4 is 12).

Step 2: Multiply the numerators and the denominators of both fractions so the fractions will have like denominators (e.g., 5/6 = (5 x 2)/(6 x 2) = 10/12 and 3/4 = (3 x 3)/(4 x 3) = 9/12).

Step 3: Add the new numerators and use the least common denominator as the denominator (e.g., 10/12 + 9/12 = 19/12).

In the following, we refer to fraction addition problems that require all of these steps, and that are not among the special cases described below, as

Not every fraction addition problem requires these three steps. There are at least three special cases of fraction addition problems. The following three types of problems have specific affordances, in the sense that one or more of the steps described above are especially easy to carry out or can be skipped entirely.

If one of the two fraction denominators is a multiple of the other denominator (e.g., 5/6 + 7/12; hereafter referred to as a

If the two denominators are prime numbers^{iv} (that are different from one another, e.g., 2/3 + 5/7; hereafter referred to as a

If the two fractions have like denominators (e.g., 7/8 + 3/8; hereafter referred to as a

This study assesses eye movement patterns in different types of fraction addition problems. The aims of this study are to investigate how strongly adults rely on componential and holistic processing when solving fraction addition problems; whether adults adapt their strategies according to problem type; how the cognitive processes differ between problem type; and how adults distribute their attention over the two fraction components.

Our study extends previous research in at least five ways. First, while previous research largely focuses on fraction comparison, we study more complex fraction addition problems, thus providing a broader insight into the cognitive mechanisms of fraction processing in the context of fraction arithmetic. To our knowledge, no studies to date have assessed eye movements in fraction addition. Second, we not only analyse the number of eye fixations on the fraction components but also the saccades (rapid eye movements) between the fixations on these components. Arguably, saccades are a better measure of cognitive processes than fixations alone because they track which fraction components participants fixate on in succession. As in previous research (

We used the four types of addition problems (Standard, MultiDenom, PrimeDenom, LikeDenom) described above, for which the most efficient strategies include different processes (see section entitled Strategies and Cognitive Processes in Fraction Addition). In addition to eye movements, we also analysed accuracies and response times. We expected that both accuracy and response times would differ significantly between problem types, with mean accuracy decreasing (Hypothesis 1a) and response times increasing (Hypothesis 1b) in the following order: LikeDenom, MultiDenom, PrimeDenom, Standard. We expected that accuracies and response times would reflect the increasing difficulty level of these problem types, a result of the increasing cognitive effort required to reach a solution.

In regard to eye movement patterns, the first parameter we analysed was the number of fixations on numerators and denominators. We hypothesized that problem type would interact with the relative number of fixations on numerators and denominators (Hypothesis 2). In LikeDenom problems, we expected the number of fixations on numerators to be higher than those on denominators because in these problems it is sufficient to add the numerators without performing any operations on the denominators. In contrast, we expected that there would be more fixations on denominators than on numerators in problems of all other types because the denominators determine the most efficient strategy for solving the problem (see above). Moreover, participants would pay more attention to denominators because these problems require finding a common denominator, which is presumably a demanding process. In MultiDenom and PrimeDenom problems, we expected that the difference between fixations on numerators and denominators would be less pronounced than in Standard problems because of the reduced effort to find a common denominator.

The saccades between the fraction components were of primary interest. We used an approach similar to the one employed by

As a general indicator of difficulty level, we expected to find differences between problem types in the total number of the saccades described above, similar to accuracy and response times. The total numbers of saccades should increase moving from LikeDenom to MultiDenom to PrimeDenom to Standard problems (Hypothesis 3). Since previous research demonstrated that people use a variety of saccades in solving fraction comparison problems (

The participants in this study included 28 adults (9 females and 19 males). Their mean age was 24.3 years (

We constructed 40 fraction addition problems of four different types. In LikeDenom problems (10 problems), the two fractions had the same denominator. In MultiDenom problems (10), the denominator of one fraction was a multiple of the denominator of the other fraction. In PrimeDenom problems (10), the denominators were unequal prime numbers. Standard problems (10) did not belong to any of the before-mentioned types. In these problems, the denominators had a common divisor larger than one.

To reduce processes unrelated to addition, all fractions were presented in their simplified form. We only included proper fractions (i.e., all fraction values were smaller than 1). The numerators of all fractions were one-digit numbers, and the denominators were all one- or two-digit numbers smaller than 20, not including ten. The results of the addition problems were never equal to one. The problems were designed such that several problem features were comparable between problem types. Among these features were the average magnitudes of the numerators (problem type averages ranging from 4.25 to 4.60), of the denominators (8.20–11.05), of the two fractions (0.45–0.67), and of the results of the additions (0.90–1.34). In half of the problems, the first addend of the addition problem was larger, and in the other half, the second addend was larger. For a complete list of the addition problems used in this study, see the

The participants worked on the problems individually in a quiet room at the university. They sat in front of a 22-inch computer screen connected to a remote SMI eye tracker with a sampling rate of 500 Hz. Before the experiment began, participants performed a five-point calibration and solved a practice problem. The participants were instructed that they would see fraction addition problems on the screen and that they should try to verbally provide the correct answer as quickly and as accurately as possible. If they felt unable to solve a problem, they should say “continue.” The experimenter recorded the participants’ responses. As in the study conducted by

After the eye tracking session, we asked participants to record on a worksheet the strategies they had used to solve fraction addition problems of each type. The worksheet illustrated one sample problem of each of the four types that participants had worked on during the eye tracking experiment. We wanted to determine whether participants actually made use of the strategies we expected to be most efficient for the four different problem types.

Although eye movements are the main focus of this study, we first analysed accuracy and response times. We used analyses of variance (ANOVA), a priori contrasts, and post hoc ^{2}_{p}) for ANOVAs and Cohen’s

^{v}. For both PrimeDenom and Standard problems, the large majority of participants reported using the standard algorithm. While we expected this strategy in response to Standard problems, it is surprising that only two participants reported using cross-multiplication in PrimeDenom problems. This may suggest that the majority of participants were not aware that in PrimeDenom problems, the denominators are the numbers with which one has to multiply the other fraction. On the other hand, cross-multiplication is just a special way of finding common denominators and participants may not explicitly report using it even if they did.

Problem Type / Strategy | Frequency | Percent |
---|---|---|

LikeDenom | ||

Check denominators, then add numerators | 26 | 93 |

Just add numerators | 1 | 4 |

Just add fractions | 1 | 4 |

MultiDenom | ||

Find common denominator, multiply one fraction, add numerators | 17 | 61 |

Find common denominator, multiply both fractions, add numerators | 6 | 21 |

Multiply one fraction, then add numerators | 5 | 18 |

PrimeDenom | ||

Find common denominator, multiply both fractions, add numerators | 24 | 86 |

Find common denominator | 2 | 7 |

Find common denominator, add cross-product | 1 | 4 |

Add cross-product | 1 | 4 |

Standard | ||

Find common denominator, multiply both fractions, add numerators | 22 | 79 |

Find common denominator, multiply both fractions | 3 | 11 |

Find common denominator, add numerators | 1 | 4 |

Find common denominator | 1 | 4 |

Multiply both fractions, add numerators, find common denominator | 1 | 4 |

More generally, it is important to note that participants’ responses may not be completely reliable because some participants did not report all the steps necessary to solve these addition problems. For example, in response to MultiDenom problems, five participants did not report finding a common denominator. Rather, the first step these participants reported was multiplying one fraction, a step for which finding a common denominator is, however, a prerequisite. We assume that participants did not explicitly mention finding a common denominator because they assumed that this step was included in the first step they described (multiplying one fraction). Likewise, the strategy “find common denominator” in response to Standard problems (as reported by one participant) is clearly incomplete.

It is also noteworthy that none of our participants mentioned using holistic reasoning about the fraction magnitudes, which may have been a way to check whether or not their result is reasonable. However, we assume that even if participants used holistic reasoning, they might not have considered such reasoning part of their strategy and thus did not report it.

As a whole, however, these reports confirm our assumption that the problems we presented encouraged different processes and that the large majority of our participants were aware of the different affordances of these problem types.

^{2}_{p} = .69. These differences were ordered as expected, with accuracy being highest for LikeDenom problems, followed by problems of types MultiDenom, PrimeDenom, and Standard. All pairwise comparisons between problem types were significant (all ^{2}_{p} = .84, with response times increasing in the same order as the accuracies decreased (from LikeDenom to MultiDenom to PrimeDenom to Standard). All pairwise comparisons between problem types were highly significant (all

Problem Type | Accuracy (%) |
Response Time (ms) |
||
---|---|---|---|---|

LikeDenom | 96.1 | 5.7 | 4262 | 724 |

MultiDenom | 91.1 | 11.0 | 8683 | 1958 |

PrimeDenom | 70.4 | 23.2 | 23712 | 7916 |

Standard | 63.2 | 25.5 | 31564 | 10143 |

All | 80.2 | 22.7 | 17055 | 12829 |

In the analyses of eye movement data reported in the remainder of this section, we excluded four participants due to low calibration quality. This exclusion reduced the sample size to 24.

As an indicator of the relative importance of the numerators and denominators, we analysed the average number of fixations on numerators and denominators per problem for each problem type. ^{2}_{p} = .82, with the lowest number of fixations appearing in LikeDenom problems, followed by MultiDenom, PrimeDenom, and Standard problems. There was also a significant effect of ^{2}_{p} = .81, indicating that there were generally more fixations on denominators than numerators. Notably, there was also a significant interaction between ^{2}_{p} = .79, indicating that the difference in the number of fixations between numerators and denominators depended on problem type. Post-hoc

Problem Type | Number of Fixations |
|||
---|---|---|---|---|

Numerators |
Denominators |
|||

LikeDenom | 2.78 | 0.77 | 2.08 | 0.51 |

MultiDenom | 4.84 | 1.70 | 5.56 | 1.32 |

PrimeDenom | 10.30 | 4.94 | 13.74 | 4.12 |

Standard | 10.69 | 5.53 | 22.86 | 8.72 |

All | 7.15 | 5.10 | 11.06 | 9.38 |

We also analysed the number of saccades between each of the four AOIs. First, it is notable that the total number of saccades varied substantially between the four problem types, ranging from only seven saccades in LikeDenom problems to 30 saccades in Standard problems. As expected, the number of saccades for the other two problems types fell in between, with an increase from LikeDenom to MultiDenom, PrimeDenom, and Standard problems (see ^{2}_{p} = .68, as were the pairwise comparisons (all

Number of Saccades |
||
---|---|---|

Problem Type | ||

LikeDenom | 6.78 | 1.80 |

MultiDenom | 13.04 | 4.99 |

PrimeDenom | 23.84 | 12.03 |

Standard | 29.31 | 15.49 |

All | 18.24 | 13.38 |

To analyse the proportions of the different types of saccades within each problem type, we first calculated per problem the percentage of each saccade type relative to the total number of relevant saccades on that problem. We averaged this number across the ten problems of each type for each participant and then across participants. ^{vi}

The figure demonstrates that all types of saccades occurred in problems of any type. As expected (Hypothesis 4a), the saccades between numerators played the dominant role in LikeDenom problems, with their proportion being highest among the different saccades of interest. Particularly, they were more frequent than saccades between denominators, ^{2}_{p} = .75; Standard: ^{2}_{p} = .86; with ^{2}_{p} = .14. Post hoc comparisons showed that saccades of this type were more frequent in PrimeDenom problems than in LikeDenom problems,

It is also notable that for all problem types, holistic saccades (i.e., those between the numerator and the denominator of each fraction) were relatively common, regardless of problem type. These saccade types occurred nearly as often in all problem types, except for LikeDenom problems, where saccades between the numerator and the denominator of the right fraction occurred more often than those between the numerator and denominator of the left fraction.

The aim of this study was to explore the cognitive processes involved in fraction addition by adults, thereby extending previous research on fraction comparison (

The addition problems in our study differed with respect to the strategies that were thought to be most efficient for solving them. Participants’ self-reports largely confirmed that they adapted their strategies to the affordances of the different problem types. Response times and accuracy also reflected the different affordances of problems of different types. Participants were particularly correct and fast in adding fractions with like denominators, where the only required process was to add the numerators. For other addition problems, difficulty is presumably determined by the cognitive effort required in the different solution steps, namely finding a common denominator, multiplying the fraction components, and adding the numerators. In line with our analysis of these steps (see section entitled Strategies and Cognitive Processes in Fraction Addition), accuracy decreased and response times increased from MultiDenom problems to PrimeDenom problems to Standard problems. Overall eye movement parameters such as the numbers of fixations and the total numbers of saccades per problem also increased from one problem type to another in the same order, supporting our basic assumption that eye movement patterns correspond closely to the problem solvers’ cognitive processes. We can conclude that participants adapted their strategies to the affordances of the problems, which is in line with findings from fraction comparison studies (

As different strategies should require different cognitive processes, the results discussed in the previous section already suggest that different processes are involved in solving fraction addition problems of different types. Our analysis of the proportions of saccade types within each problem type allowed for a more detailed analysis of these processes. We had expected that, within each problem type, the majority of saccades would be componential, because fraction addition does not necessarily require holistic processing of the fraction magnitudes. The results show that for fractions with like denominators, the (componential) saccades between numerators account for the majority of saccades within this problem type. For all other problem types, the most frequently occurring saccades are (componential) saccades between the denominators. Although holistic saccades occur in all problems types, these saccades (counted separately for the two fractions within a problem) never represented the majority of saccades. These results offer further support for the assumption that adults increase their reliance on holistic processing of fractions only when this is required for solving the problem, presumably because holistic processing is more demanding than componential processing. For example, in the study by

Although holistic saccades for the left or the right fraction did not represent the majority of saccades for any problem type, the proportions of holistic saccades were remarkably high, given that the addition problems did not require holistic processing. In fact, the holistic saccades for the left and the right fraction

Somewhat unexpectedly, diagonal saccades did not play a major role in fraction addition problems, not even in those with prime denominators. Although cross-multiplication would be an efficient strategy in these problems, the participants in our study did not seem to make extensive use of this strategy. This finding corresponds to our participants’ self-reports, in which only two participants reported using cross-multiplication. It is also in line with the study by

We have assumed that fraction denominators play a key role in fraction addition problems because strategy choice depends largely on the relation between the denominators and because finding a common denominator—which was required in three fourths of the problems in our study—is a relatively demanding step (see section entitled Strategies and Cognitive Processes in Fraction Addition). Thus, we expected that processing the denominators would be more demanding than processing the numerators in all problem types except for LikeDenom.

The number of fixations and the saccades analyses are in line with our predictions. In LikeDenom problems, people fixate less often on the denominators than on the numerators, and they switch less often between denominators than between numerators presumably because it is sufficient to add the numerators to find the result. In contrast, in all other problem types, the denominators require more attention than the numerators, as indicated by both fixations and saccades. While previous studies on fraction comparison were inconclusive about whether denominators are generally more difficult to process than numerators (

It is well documented that school students who struggle with fraction problems rely heavily on componential processing and hardly employ holistic processing when working with fractions (see section entitled Cognitive Processing of Fractions). The specific cognitive processes of individual students are, however, less well understood. In further research, the eye movement patterns of educated adults identified in our study may help interpret eye movement patterns of students who process fractions ineffectively. For example, if a student solves a Standard fraction addition problem by purely relying on componential addition (i.e., adding the numerators and denominators separately), we would expect—relative to the pattern of our adult sample—larger proportions of saccades between numerators, and lower proportions of saccades between denominators. Moreover, the absence of holistic processing should be expressed by lower proportions of holistic saccades.

Assessing

A major limitation of our study is that eye movements may not be a perfectly valid measure of cognitive processes and strategy use in fraction addition, although previous research suggests that eye movements are a valid measure of strategy use in other mathematical problems such as number line estimation (

Another limitation of our study is that our interpretations of eye movement patterns are tentative to some extent. The reason is that although these interpretations are based on theoretical considerations and empirical evidence from previous fraction comparison studies, our knowledge about how to interpret eye movements is still limited. For example, it is not completely clear if all saccades between the numerator and the denominator of a fraction actually indicate holistic fraction processing. As discussed earlier (see section entitled Cognitive Processes Depending on Problem Type), some of these numerator-denominator saccades may indicate reading processes. Moreover, these saccades may also indicate additive rather than multiplicative comparisons between numerators and denominators. This means that participants may reason about the difference rather than the quotient of a fractions’ numerator and denominator, and thus may not process (holistic) fraction magnitudes. Further research is certainly necessary to better understand the relationship between specific cognitive processes and eye movement patterns.

Research suggests that adults compare fractions in both componential and holistic ways and that their strategies depend on the type of problem they encounter. Our study supports and extends these findings. Educated adults solve fraction addition problems mainly by focusing on and switching between the most informative fraction components. Whether these components are the numerators or the denominators depends on the specific type of addition problem at hand. In regard to methods, combining different measures such as self-reports, accuracy rates, response times, and eye movements is a promising way to gain insight into the cognitive processes of solving fraction problems.

Like Denominators (LikeDenom) | Multiple Denominators (MultiDenom) | Prime Denominators (PrimeDenom) | Standard |
---|---|---|---|

3/5 + 4/5 | 3/20 + 4/5 | 3/11 + 4/5 | 5/6 + 7/8 |

2/7 + 1/7 | 2/7 + 1/14 | 2/7 + 1/11 | 7/12 + 4/9 |

5/8 + 7/8 | 7/18 + 4/9 | 3/5 + 6/7 | 1/16 + 7/12 |

7/9 + 4/9 | 1/6 + 7/12 | 2/3 + 8/13 | 4/9 + 8/15 |

3/7 + 6/7 | 9/16 + 3/8 | 4/5 + 8/11 | 5/18 + 1/12 |

1/12 + 7/12 | 4/5 + 8/15 | 1/2 + 8/17 | 3/14 + 1/6 |

2/13 + 8/13 | 5/18 + 1/3 | 6/19 + 1/2 | 5/8 + 3/14 |

9/16 + 3/16 | 7/8 + 1/2 | 2/3 + 9/13 | 7/18 + 3/4 |

4/15 + 8/15 | 2/3 + 8/15 | 9/17 + 1/3 | 5/6 + 3/16 |

5/18 + 1/18 | 3/4 + 5/12 | 5/7 + 2/3 | 1/4 + 9/14 |

Note that we only consider positive fractions here.

For example, one might reject that 2/3 + 3/4 = 5/7, a result emerging from componential addition, without actually calculating the sum of the fractions, by arguing that adding two fractions larger than a half should result in a number larger than one, which 5/7 is not.

It is actually not necessary to use the

The strategy described for problems with prime denominators does not necessarily require prime denominators. This strategy is actually valid for any fraction addition problem. However, in problems with denominators that are not prime numbers, multiplying the denominators can result in large numbers, so that it is often more reasonable to find the least common denominator (as described in Standard problems), rather than multiplying the denominators.

Note that these participants’ responses imply that they used a common denominator that was larger than both denominators of the two given fractions, rather than the larger of the two denominators. For example, participants may have multiplied both denominators, although they did not explicitly mention doing so.

We ran a cluster analysis to identify potential subgroups of participants that might differ in their saccade patterns (and thus in their cognitive processes). This analysis showed, however, that we can consider our sample as a homogeneous group in terms of saccade patterns.

The authors have no funding to report.

The authors have declared that no competing interests exist.

The authors have no support to report.