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Many studies have used fraction magnitude comparison tasks to assess people’s abilities to quickly assess fraction magnitudes. However, since there are multiple ways to compare fractions, it is not clear whether people actually reason about the holistic magnitudes of the fractions in this task and whether they use multiple strategies in a flexible and adaptive way. We asked 72 adults to solve challenging fraction comparisons (e.g., 31/71 vs. 13/23) on a computer. In some of these comparisons, using benchmarks (i.e., reference numbers such as 1/2) was potentially beneficial. After each trial, participants provided verbal reports of their strategies. We found that participants used a large variety of strategies. The majority of strategies were holistic and relied on fraction magnitudes, and most of these strategies were based on benchmarks. Participants sometimes used gap comparison (i.e., comparing the differences between each fraction’s numerator and denominator), a heuristic that is not always valid and that does not rely on fraction magnitudes. Participants used strategies flexibly: they used many different strategies, they used highly efficient strategies most often, and they adapted their strategy use to features of the items. However, participants sometimes used gap comparison on items for which it did not yield the correct response, and this lack of adaptivity partly explained the “natural number bias” observed in this study.

How do people decide which of two fractions is larger? Although younger children often fail to compare even simple fractions accurately (

Understanding strategy use in fraction comparison is important for many reasons. First, flexible and adaptive strategy use is an important facet of mathematical competency (

Studies of educated adults reveal that most can make fraction comparisons quickly and accurately (e.g.,

Studies of educated adults also often report pronounced “distance effects” in fraction comparisons—that is, strong associations between the numerical distance between the two fractions and participants’ average accuracy and/or response times. Accuracy tends to be higher, and response times lower, with increasing numerical distance. Such distance effects suggest the processing of holistic (overall) magnitude representations, similar to those recruited for whole number magnitude comparisons (

One limitation of previous research is that almost all studies aggregated data across participants (e.g.,

Various facets of strategy use have been identified and characterized in previous work (

A second facet of strategy use is

A third facet of strategy use is

Two additional, related facets of strategy use are

Assessing all facets of strategy use simultaneously is a methodological challenge. For example, assessing strategy efficiency requires recording accuracy and response times, but these measures alone do not allow for straightforward conclusions about specific strategies. Accordingly, previous studies have investigated fraction comparison strategies using a variety of methods, including identifying error patterns across sets of items (

One approach that allows for individual-level analyses of strategy use is collecting verbal reports on a trial-by-trial basis. However, previous studies using verbal reports have not considered challenging fraction comparisons. For example,

One study that used a trial-by-trial analysis did include a few difficult comparisons.

Analyzing strategy use in fraction comparison may contribute to better understanding of the

The performance difference between congruent and incongruent items has been attributed to people’s use of simple heuristics (e.g., “larger numerator/denominator makes larger fraction” or “smaller numerator/denominator makes larger fraction” heuristics;

The aim of the present study was to assess adults’ strategy use on challenging fraction comparisons. We considered the facets of strategy use described earlier. Considering participants’

Regarding

Regarding

Finally, we were interested in whether we could evoke shifts in participants’ strategy use by highlighting the usefulness of a holistic strategy. To this end, we encouraged some participants to use benchmark strategies. If people can easily shift towards more frequent use of holistic strategies, it would suggest that strategy use is malleable, which would have implications for educational practice.

Participants were 72 undergraduates (32 male, 37 female, 1 non-binary, 2 not reported;

We used the same set of 56 fraction comparison items as in a previous study that did not involve strategy reports (

The item set was split into two subsets of 28 items each, such that each subset contained equally many items of each type, as described above. We added three filler items to each subset, and these fillers were not included in the analysis. Filler items were easy fraction pairs (with same numerators or with very large differences between fractions) that did not fall into any of the categories described above. These items were added to maintain participants’ motivation and attention.^{1}

Performance on items that followed filler items was comparable to performance on other items in speed, accuracy, and strategy use.

Thus, each participant was presented with a total of 31 items, consisting of one subset of 28 items and three filler items. Preliminary analyses revealed no differences in performance across the two subsets, so they were collapsed for analysis.Data were collected in small group sessions, with each participant working individually at a computer. Items were presented using E-Prime software. Participants wore headphones with microphones. They were presented with two fractions at a time and asked to choose the greater fraction as quickly and accurately as possible by pressing the corresponding left (“f”) or right (“j”) key within 15 seconds. After participants pressed the key, the fraction pair remained visible on the screen, and the question “How did you solve this problem?” appeared above the fraction pair. Participants then had another 20 seconds to respond by speaking into their microphones. After the given time limit, the next item appeared automatically.

There were two practice items before the experiment started. Accuracy feedback was provided for practice items but not for test items. Test items were presented in random order, and the position of the larger fraction (left or right) was counterbalanced across trials.

Participants in the tip condition saw an additional screen after the general instructions. The text on the screen suggested that it could be helpful to think of numbers such as 1/2, 1/4, or 3/4 that could be used as benchmarks to compare fractions. The example “5/8 vs. 3/7” was provided to illustrate the benchmark strategy. The text said that one could think that 5/8 is larger than 1/2, and that 3/7 is smaller than 1/2, so that 5/8 must be the larger fraction. No further explanation was provided.

We first provide an overview of accuracy and response time data (previously reported in

Participants’ verbal responses (

Category / Strategy | Description | Sample Response |
---|---|---|

Holistic | ||

Benchmark | Compare one or both fractions to another number (e.g., to ½) | [11/26 vs. 34/59] The right fraction is a little bit more than half while the left is less than half. |

Multiplicative reasoning | Reason about the multiplicative relationship between numerator and denominator of each fraction | [6/19 vs. 19/96] 6 goes into 19 a little more than 3 times whereas 19 goes into 96 probably 6 or so times, maybe 5, so 6 over 19 is bigger. |

Division/Conversion | Perform a division and/or convert one or both fractions to a decimal or percentage | [22/31 vs. 37/64] If you divide 22 by 31 it is a larger number than when you divide 37 by 64, by a great amount. |

Componential | ||

Gap comparison* | Compare the differences (gaps) between numerator and denominator of each fraction | [68/83 vs. 49/52] 49 is closer to 52 than 68 is to 83. |

Multiplication | Multiply one fraction to get similar numerators or denominators | [17/31 vs. 11/16] 11 over 16 can be 22 over 32 and 22 is much greater than 17, so the second one is greater. |

Component comparison* | Compare fraction components without calculation | [22/49 vs. 13/41] 22 over 49 because the numerator is bigger. |

Component approximation | Note that the numerators or the denominators are nearly equal, so compare the other two components | [17/91 vs. 13/41] 17 and 13 are close but 91 is way bigger than 41, so 13 over 41 is clearly bigger than 17 over 91. |

Multiplicative comparison | Compare the multiplicative relation between the two numerators with that between the two denominators | [32/57 vs. 11/27] The top number is like tripled while the bottom number only doubled so the one on the left is gonna be bigger. |

Other | ||

Unintelligible or unclear | Response is unintelligible or unclear | – |

Missing | No response or response unrelated to task | – |

Statement/Knowing | State the response without reasoning or by simply knowing | [13/43 vs. 6/37] 13 out of 43 is larger than 6 out of 37. |

Guessing/Intuition | Guess or refer to intuition | Just guessing. |

Don’t know | State that one does not know the answer | I don’t know. |

Verbal recordings from 18 randomly-selected participants (

Participants sometimes reported two strategies on the same trial. In most cases, one strategy was clearly the primary strategy, and another was mentioned only briefly (e.g., as a possible approach). In these cases, we coded only the primary strategy. Otherwise (in 17% of all trials), we coded both strategies.^{2}

Note that for these reasons the total number of strategies in the following analyses is larger than the number of verbal responses coded.

Because our primary interest was the distinction between holistic and componential strategies, we considered as holistic all strategies in which people relied on overall fraction magnitudes. Importantly, categorizing a strategy as holistic did not imply that people needed to

Average accuracy (Acc;

Source | Accuracy |
Response Time |
||||
---|---|---|---|---|---|---|

Wald χ^{2} |
Wald χ^{2} |
|||||

Congruency | ||||||

Item Type | ||||||

Tip | 0.15 | 1 | .695 | 0.00 | 1 | .969 |

Congruency x Item Type | 2.89 | 2 | .236 | 0.23 | 2 | .892 |

Item Type x Tip | 5.46 | 2 | .065 | 1.76 | 2 | .416 |

Congruency x Tip | 0.29 | 1 | .592 | |||

Congruency x Item Type x Tip | 1.03 | 2 | .598 | 0.80 | 2 | .669 |

There was a main effect of

One participant was excluded from the strategy analyses due to technical problems with the recording. As reported above, the effect of the tip about benchmark strategies on accuracy and response times was generally small, and the main effect of tip was not significant for either accuracy or response times. This was also the case for strategy use. Participants who received the tip used holistic strategies only slightly more often (52%) than participants who did not receive the tip (49%), and this difference was not significant, χ^{2}(1) = 2.46, ^{3}

We ran all analyses reported in the following separately for participants in the tip and no-tip group. All differences between the two groups were small and did not change the overall conclusions. For separate analyses, see

Participants used many different strategies (see

Category / Strategy | Percent Use Within Category | Percent Use Overall |
---|---|---|

Holistic | ||

Benchmark | 82 | 42 |

Multiplicative reasoning | 15 | 7 |

Division/Conversion | 3 | 2 |

Componential | ||

Gap comparison | 43 | 16 |

Multiplication | 23 | 9 |

Component comparison | 14 | 5 |

Component approximation | 11 | 4 |

Multiplicative comparison | 4 | 2 |

Other | 5 | 2 |

Other | ||

Unintelligible or unclear | 40 | 5 |

Missing | 30 | 4 |

Statement/Knowing | 13 | 2 |

Guessing/Intuition | 10 | 1 |

Don’t know | 6 | 1 |

The large majority of holistic strategies (82%) involved benchmarks. Benchmarks were either used as numbers straddled by the given fractions (“one fraction larger, the other smaller”; 19%) or, more often, as reference numbers to which one or both fractions were compared (“fraction close to benchmark”; 81%). The most prominent benchmarks were 1/2 (used in 42% of benchmark strategies), 1 (29%), 1/3 (10%) and 1/4 (4%). Several other numbers (including 0, 3/4 and 2/3) were infrequently used as benchmarks, as well. Holistic strategies that did not include benchmarks involved either multiplicative reasoning about the ratio between numerator and denominator or performing a division, often to convert fractions to an exact decimal or percentage. These strategies were relatively infrequent.

Among componential strategies, the most frequent was gap comparison, which was the second-most-frequent strategy overall. Other componential comparisons were multiplication strategies, in which fraction components were multiplied to get common (or similar) numerators or denominators. It is notable that component comparison (i.e., simple comparison of numerators or denominators) was used infrequently (5% overall).

Although mean accuracy was high overall, holistic strategies led to correct responses slightly more often (^{2}(1) = 4.47, ^{2}(1) = 1.92,

To evaluate how efficiently (i.e., quickly and accurately) different strategies were used, we calculated the Inverse Efficiency Score (IES), which combines accuracy and response times into one measure (

Category / Strategy | Accuracy (%) | Response Time (ms) | IES (ms) |
---|---|---|---|

Holistic | |||

Benchmark | 90 | 4695 | 52.2 |

Multiplicative reasoning | 90 | 5374 | 59.7 |

Division/Conversion | 87 | 5409 | 62.2 |

Componential | |||

Gap comparison | 85 | 4122 | 48.5 |

Multiplication | 87 | 6155 | 70.7 |

Component comparison | 77 | 4845 | 62.9 |

Component approximation | 98 | 4072 | 41.6 |

Multiplicative comparison | 95 | 4870 | 51.3 |

Across the 28 comparisons, participants used, on average, five of the strategies described in ^{4}

Note that this analysis did not include strategies that were coded as “other“.

No participant rigidly used a single strategy. Thus, participants used strategies flexibly. The number of different strategies participants used was not correlated with their overall accuracy,Participants used strategies adaptively in the sense that they frequently used highly efficient strategies—primarily benchmarking and gap comparison (ranked fourth and second, respectively, in

We also evaluated whether strategy use varied depending on item type. First, we considered the three item types (straddle, in-between, and close-to-0-or-1). Recall that participants were most accurate and fastest (i.e., most efficient) on close-to-0-or-1 items, followed by straddle items, and then in-between items (see section entitled Accuracy and Response Times). A major difference between the three item types was that participants used holistic strategies much more frequently than componential strategies for in-between, 60% vs. 27%; χ^{2}(1) = 82.51, ^{5}

Non-parametric chi-square test of the distributions of holistic vs. componential strategies.

, and straddle items, 53% vs. 34%; χ^{2}(1) = 44.15,

^{2}(1) = 16.70,

^{2}(2) = 47.85,

^{2}(2) = 50.31,

Finally, we were interested in whether strategy use could partially explain the congruency effect. We had hypothesized that this could be the case if gap comparison was used more often or more efficiently for incongruent than congruent items. ^{2}(1) = 13.58, ^{2}(1) = 0.82,

Strategy | Congruent |
Incongruent |
IES Difference |
||
---|---|---|---|---|---|

Percent Use | IES | Percent Use | IES | ||

Benchmarking | 40 | 56.7 | 43 | 48.0 | 8.7 |

Gap comparison | 14 | 67.9 | 18 | 37.4 | 30.6 |

Total | 57.4 | 45.6 | 11.8 |

Adults can solve complex fraction comparisons quickly and accurately. On average, participants needed fewer than 5 seconds per item and solved 86% of these challenging comparisons correctly, in line with earlier research on challenging fraction comparisons in educated adults (

Participants used holistic strategies, which rely on fraction magnitudes, far more frequently than componential strategies, which do not rely on fraction magnitudes. Thus, in line with earlier research with other methods (

Our study allowed us to characterize the specific strategies that adults used to reason about fraction magnitudes. The most prominent holistic strategy involved using familiar numbers (especially 1 and ½) as benchmarks, in line with earlier research on simpler fraction comparisons (e.g.,

Gap comparison was the second most frequent strategy overall, and the most frequent componential strategy. Although this strategy is not always valid, participants used it relatively frequently, as in previous studies of simpler fraction comparisons (e.g.,

We did not find a large effect of the tip that we gave some participants about the usefulness of benchmark strategies. This could mean that strategy use is not easily malleable. However,

Overall, participants displayed a reverse natural number bias. Our results suggest that this reverse bias may be associated with gap comparison more so than with simple comparisons between numerators or denominators, which people commonly employ when comparing fractions with common components (

What other factors could contribute to the reverse bias pattern? One possibility is subtle processing mechanisms that are not accessible to verbal strategy reports (e.g., intuitive processes). Although some studies have identified intuitive processes, such as automatic activation of natural number magnitudes, as contributing to the

One limitation of this work concerns the validity of retrospective self-reports of strategy use. Our experimental procedure was in line with accepted criteria for valid self-reports (

A second limitation is that this work does not provide definitive evidence about the sequential and temporal processes involved in fraction comparison. For example, people might start out by considering differences (rather than ratios) between numerators and denominators, and if these differences provide enough information for them to feel confident in their magnitude comparisons, they may base their responses on these differences (i.e., using the gap comparison strategy). If, however, gap comparison is inconclusive, or if other item features encourage holistic reasoning, they may then engage in more effortful holistic reasoning. Some adults, knowing that gap comparison is not a universally applicable heuristic, may bypass such reasoning altogether, and engage immediately in holistic reasoning. Further work is needed to delineate the processes involved in selecting and applying comparison strategies.

In sum, this study revealed that educated adults often rely on fraction magnitudes when comparing challenging fraction pairs. Thus, the fraction comparison task with challenging items is suitable for assessing fraction magnitude processing. The present work highlights both flexibility and adaptivity in adults’ strategies for fraction comparison, and it demonstrates that strategy choices contribute to the occurrence of the reverse natural number bias.

Congruent | Incongruent |
---|---|

Straddle | |

6/37 vs. 13/43 |
17/91 vs. 13/41 |

11/26 vs. 34/59 |
31/71 vs. 13/23 |

13/19 vs. 25/31 |
26/37 vs. 19/22 |

In-Between | |

11/37 vs. 42/95 |
19/63 vs. 16/35 |

21/37 vs. 41/59 |
17/31 vs. 11/16 |

2/21 vs. 21/97 |
10/99 vs. 9/43 |

21/26 vs. 59/62 |
68/83 vs. 49/52 |

The first author is an Associate Editor of the

This work was supported by a Feodor Lynen Research Fellowship granted to Andreas Obersteiner by the Alexander von Humboldt Foundation, Germany.

This study has been carried out in accordance with ethical and scientific standards. The study was approved by the Institutional Review Board at the University of Wisconsin–Madison, United States.

The Supplementary Materials contain one pdf file with separate analyses of strategy data for participants in the two experimental groups “With Tip” and “Without Tip”. Specifically, the file includes separate analyses of

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