^{a}

^{a}

^{a}

^{a}

^{b}

^{*}

^{a}

^{c}

Recent research suggests that bounded number line tasks, often used to measure number sense, measure proportion estimation instead of pure number estimation. The latter is thought to be measured in recently developed unbounded number line tasks. Children with dyscalculia use less mature strategies on unbounded number lines than typically developing children. In this qualitative study, we explored strategy use in bounded and unbounded number lines in adults with (N = 8) and without dyscalculia (N = 8). Our aim was to gain more detailed insights into strategy use. Differences in accuracy and strategy use between individuals with and without dyscalculia on both number lines may enhance our understanding of the underlying deficits in individuals with dyscalculia. We combined eye-tracking and Cued Retrospective Reporting (CRR) to identify strategies on a detailed level. Strategy use and performance were highly similar in adults with and without dyscalculia on both number lines, which implies that adults with dyscalculia may have partly overcome their deficits in number sense. New strategies and additional steps and tools used to solve number lines were identified, such as the use of the previous target number. We provide gaze patterns and descriptions of strategies that give important first insights into new strategies. These newly defined strategies give a more in-depth view on how individuals approach a number lines task, and these should be taken into account when studying number estimations, especially when using the unbounded number line.

The initial sample consisted of eight participants (four adults with dyscalculia and four TD adults). This small group enabled us to thoroughly explore strategy use in a qualitative manner. To validate our findings, we tested an additional group of eight participants. Since we found highly similar results between these two groups, in the remainder of this paper we will treat the samples as one sample. The final sample consisted of eight adults with dyscalculia (dyscalculia group) and eight TD adults (TD group). The dyscalculia group included participants with educational levels ranging from vocational education (two participants) to (pre-)university (six participants). The group consisted mainly of females (seven, one male), and the average age was 22.85 years (_{age}

All participants gave informed consent before participating in the study. Participants were tested individually in a quiet room. This room was depleted of distracting stimuli. Light conditions were held stable and direct light on the tracker or in the eyes was prevented. The experiment consisted of two tasks and the Cued Retrospective Reporting (CRR). After calibration, an unbounded NLT was administered, followed by a bounded NLT. Both NLTs (i.e., unbounded and bounded) started with a practice trial, which was excluded from the analyses. After both tasks were carried out, the CRR was administered. In total, the experiment took about 50 minutes per participant. During the tasks one of the researchers was present to answer the participants’ questions and to administer the CCR.

The unbounded NLT consisted of two subtasks, consisting of 15 trials each. Trials were randomized within each subtask and presented to the participants in a fixed order. A trial consisted of a presentation of a number line and a number, depicted below the line, that had to be estimated. All number lines were 757 pixels (20 cm) long and the numerical length ranged from 0 to 40 in the first subtask (

Examples of the 0-40 unbounded NLT (a) and the 0-400 unbounded NLT (b).

The bounded NLT was similar to the unbounded NLT, with the exception that the number lines were 816 pixels long and both begin-point and end-point were labeled. The first subtask ranged from 0-100 and the second from 0-1000. The target numbers were selected in a similar way as the unbounded number line. The selected numbers were 3, 12, 17, 25, 29, 37, 44, 51, 58, 63, 69, 76, 84, 88, and 99 for the 0-100 line and 36, 104, 153, 277, 308, 385, 422, 542, 594, 684, 723, 804, 880, 919, and 996 for the 0-1000 line.

After finishing the NLTs, eye movements and mouse clicks of some trials were replayed to the participants. After replaying the trial, the participants were asked to explain what strategy they used to position the target number (“can you explain what you did during this trial?”). Follow-up open-ended questions were asked when participant responses were unclear (e.g. “what do you mean by this?”) and/or to evoke more comprehensive reactions (e.g. “is there anything you would like to add?”). To be sure participants knew how to interpret the replays, participants were first shown an introductory replay and were explained how eye movements and mouse clicks were displayed. Subsequently, ten trials from both subtasks of the bounded NLT were replayed, and five trials from both subtasks of the unbounded NLT, since less variation in strategy use was expected in the unbounded NLT. For one participant with dyscalculia, no eye movements were recorded for the bounded number line tasks, due to equipment failure. As such, CRR data were available for eight participants with dyscalculia in the unbounded NLT and for seven participants in the bounded NLT. The CRR was audio-taped and transcribed afterwards.

During the NLTs, eye movement data were collected with a Tobii T60 Eye Tracker (60 Hz, manufactured: Dec 2008), containing a 17 inch screen (resolution: 1280 x 1024). A nine-point calibration was performed, and accuracy of calibration was assessed based upon visual inspection of length of error vectors and possible missing calibration points. Eye movements were analyzed using

To examine strategy use during both the bounded and the unbounded NLTs in adults with dyscalculia versus TD adults, we first analyzed the predefined strategies on the bounded NLTs based on the strategies defined by

First, we defined areas of interest (AOIs), corresponding with predefined reference point combinations within the proportion based strategy: begin-point, mid-point, and end-point. The size of the AOIs was defined with a margin of 5% of the length of each number line on each side of the particular point. We retrieved the number of fixations within each AOI for each trial. To evaluate which reference point combinations were used in each trial (i.e., (1) begin, (2) begin-mid, (3) mid, (4) mid-end, (5) end, (6) begin-end, or (7) all, based on

Next to general differences in the use of proportion based strategies, we determined the functionality of reference point combinations for both groups. We distinguished target number intervals: (1) 0-25 and 0-250, (2) 26-50 and 251-500, (3) 51-75 and 501-750, and (4) 76-100 and 751-1000. We computed the mean percentage of trials in which each reference point (combination) was used, separately for each interval, participant, and group. We defined functionality for each interval the following way: begin and begin-mid as functional for the first interval, begin-mid and mid for the second interval, mid and mid-end for the third interval, mid-end and end for the fourth interval, and other combinations as dysfunctional.

Due to equipment failure, for one participant in the dyscalculia group eye-tracking data were unavailable. As a result, this participant’s use of predefined strategies and functionality could not be calculated. When computing group percentages of the use of predefined strategies and reference point combinations and their functionality, missing values were excluded.

For the in-depth analysis of strategies we qualitatively explored strategy use in both the bounded and the unbounded NLTs. Utilizing an inductive thematic analysis (

To explore differences in accuracy of number estimation between bounded and unbounded NLTs and between TD adults and adults with dyscalculia, we defined accuracy as ‘the percentage of absolute error (PAE)’. To obtain the PAE for each trial we carried out several steps. First, we retrieved the horizontal pixel of the mouse click for each stimulus. Second, we computed absolute deviation scores by subtracting the horizontal coordinate of the mouse click from the horizontal coordinate of the correct location of the target number. In order to correct for different number ranges of the number lines in the tasks, we computed the deviations in pixels, rather than in numbers. Lastly, PAE was computed by means of next formula: (absolute deviation/length of line [in pixels]) * 100. These separate PAE scores were used to compute mean PAE for each participant, group, and subtask separately.

The use of reference points looked similar between the dyscalculia and TD group (see

^{a}Data on strategy use by Participant 3 from the dyscalculia group were missing, due to equipment failure.

Moreover, the percentage of trials that could not be specified as one of the reference point combinations was similar for both groups, and concerned around one-sixth (15%) of all trials. Results regarding the functionality of the use of reference points in both groups are presented in

Percentages of trials in which each proportion based strategy was used, seperated per target number interval.

The thematic analysis of the CRR on the bounded NLT resulted in several themes and subthemes. We found two substantive main themes: strategies, including the already established use of reference points as well as new strategies, and tools or additional steps to help with determining the target position. In

Next to strategies, six subthemes were interpreted as additional tools and steps.

Themes | Examples | Number of participants who used the strategy |
Percentage of trials in which the strategy was used |
||
---|---|---|---|---|---|

TD |
Dyscalculia |
TD |
Dyscalculia |
||

Newly defined strategies | |||||

Direct estimation | Just on intuition, that should be here. | 8 | 5 | 15.72 | 14.38 |

Proportion | First I looked at the half, but on the other side: so 25, divided in two. Then I thought: it has to be a little bit back. | 8 | 8 | 58.49 | 56.25 |

Taking steps | I think I started at 100 (…). I think: [I used] steps of 100 till 400. | 6 | 6 | 30.82 | 29.38 |

Use of previous number | This is a little bit further than the previous number. | 4 | 3 | 4.40 | 8.75 |

Additional tools and steps | |||||

Orientation on the line | I first looked where the line ended, [and I saw it was] at 100. Then I looked at the [target] number. | 7 | 7 | 10.69 | 9.38 |

Check | I often check whether I have the right number [in mind]. | 4 | 5 | 10.06 | 5.63 |

Estimation of small units^{a} |
I looked at 50, and then [moved] a little bit back. | 8 | 8 | 55.35 | 41.88 |

Rounding off | The 88, rounded off to 90 | 7 | 6 | 16.35 | 16.25 |

Use of cursor | I think I (...) tried to move the cursor to 900. (...) Actually, I used it to create a new [vertical] bar on the line. | 3 | 2 | 2.52 | 3.13 |

Decision based on multiple strategies | I think I started at 100, where I ended the previous one. I think [I used] steps of 100 till 400. Then I checked whether this [position] matched with [the place] where I think the 500 should be. And then 400 and a little bit further. | 3 | 2 | 3.77 | 2.50 |

Remaining theme | |||||

No strategy defined | I don’t know either. [No further comments] | 1 | 4 | 0.63 | 2.50 |

^{a}Units refers to small estimations in general (e.g. 15 in 115), not only to estimating the last unit in the decimal system (e.g. 5 in 115).

Representative examples of strategies used during the bounded NLT. On the left, gaze patterns are displayed, in which dots represent eye fixations and lines represent saccades (eye movements). On the right, heat maps are displayed, in which the color indicates the duration of fixations at a specific location. Green indicates short fixations, yellow indicates fixations of medium duration, and red indicates long fixations.

Subsequently, we explored differences between the dyscalculia and TD group in the use of these newly defined strategies, and additional steps and tools. For this purpose, the percentage of trials in which the strategies and additional tools or steps were used was calculated. The results (see

A similar pattern of strategies was found in the TD group in which

For the unbounded NLT we found main themes similar to those found in the bounded line for the dyscalculia and TD group (i.e., independent strategies, additional steps and tools, and being not able to define the strategy). In addition to the already known

The use of each strategy was calculated, as shown in

Regarding the tools, participants in the TD group seemed to be less likely to use

Themes | Examples | Number of participants who used the strategy |
Percentage of trials in which the strategy was used |
||
---|---|---|---|---|---|

TD Group | Dyscalculia Group | TD Group | Dyscalculia Group | ||

Previously established strategies | |||||

Direct estimation | I first looked at the unit, and then I determined where it approximately has to be (...). | 1 | 4 | 1.23 | 24.10 |

Dead-reckoning | I first took a step of five, and then I added this step of five again, and once again, and after that [I moved] a little bit back | 4 | 5 | 20.99 | 16.87 |

Newly defined strategies | |||||

Explicit use of unit | I used this unit to draw it repeatedly in my head, until I arrived at 14. | 7 | 8 | 39.51 | 39.76 |

Use of previous number | Yes. I still remembered that the previous number was 26, so I thought: this number has to be right after it. I tried to determine this, how many parts of one fitted behind this. | 7 | 8 | 25.93 | 31.33 |

Additional tools and steps | |||||

Orientation on the line | At the start I constantly looked at the begin-point, and then to the end, and back again, because I constantly tried to estimate what the length of the line is. | 3 | 5 | 11.11 | 9.64 |

Cursor as reference point | I repeatedly moved the cursor a bit further. I used the cursor as a measuring tape. | 1 | 4 | 2.47 | 8.43 |

Estimation of small units | … I took three steps of 10 and then again ten and after that a little bit back [target number 38]. | 5 | 7 | 14.81 | 19.28 |

Rounding off | I rounded off 38 to 40. | 0 | 2 | 0.00 | 3.61 |

Decision based onmultiple strategies | I thought I have to place it a little bit before the previous one. (…) then I thought I have to place it at one-third (…), because I thought it would end at 300. | 2 | 4 | 4.94 | 6.02 |

Looking back at targetnumber or unit | I often look back to the number, otherwise I forget which number it is | 3 | 3 | 4.94 | 4.82 |

Proportion^{a} |
So I thought one-third is approximately 100. 154… If the [end of the] line is a little more than 300, than 154 is a little bit before half [of the line]. | 1 | 2 | 9.88 | 4.82 |

Remaining theme | |||||

No strategy defined | 10. [no further comments] | 0 | 3 | 0.00 | 3.61 |

^{a}Before using ‘proportion’ the participant determined the length of the line with the use of dead-reckoning, which helped the participant to form reference points.

Gaze patterns examples of strategies used during the unbounded NLT. On the left, gaze patterns are displayed, in which dots represent eye fixations and lines represent utterances (eye movements). On the right, heat maps are displayed, in which the color indicates the duration of fixations at a specific location. Green indicates short fixations, yellow indicates fixations of medium duration, and red indicates long fixations

The accuracy on both the unbounded and bounded NLTs is displayed in _{unbounded}_{bounded}

Percentage absolute error per group and per NLT. Error bars display standard deviations in positive and negative direction.

The aim of our study was to explore strategy use and accuracy in both bounded and unbounded NLTs in adults with dyscalculia and typically developing adults. A combination of eye-tracking recordings and CRR enabled us to explore strategies more in-depth and as such establish additional strategies that were not captured in previous research. In addition, our study is the first to provide insight into the unbounded NLT in individuals with dyscalculia. We will first discuss the results with regards to strategy use, followed by the results on accuracy in both tasks.

Predefined strategy use on the bounded NLT was examined first based on the use of reference point combinations, i.e. proportion based strategies, as defined in previous research (

With regards to the functionality of the use of reference points, results showed mainly similarities between the individuals with and without dyscalculia. One difference between the groups on the bounded NLT was the tendency of the dyscalculia group to include a reference point below (i.e., smaller than) the target number in their estimations, whereas the TD group more often relied on a reference point larger than the target number only. An explanation for this difference could be that counting down requires more working memory capacity than counting up (

In the unbounded NLT, the predefined strategies

In-depth exploration of the strategies on the unbounded NLT revealed two additional strategies next to

With regards to accuracy, based on previous research, we expected that adults with dyscalculia would perform less well than adults without dyscalculia, at least on the bounded NLT (

Taken together, our results suggest that successful estimations on number lines up to 1,000 are reached in adults with dyscalculia with the use of similar strategies as typically developing adults. Previous research has shown that mathematical difficulties in individuals with dyscalculia do persist into adulthood (

Concerning possible limitations, one could argue that posing guiding questions during CRR evokes certain answers, by directing the attention of the participant, which could have influenced the in-depth analysis of strategy use (but not strategy use and performance itself since it was administered after the task). However, it has recently been suggested that combining specific questions with CRR and eye-tracking does have added value in comparison to retrospective think aloud methods without eye-tracking, because the questions are based on the eyemovement data observed during replay (

A second remark that could be made is the small sample used in this study. However, our main aim was to explore additional strategies used in bounded and unbounded number lines and see whether these strategies were used by both individuals with and without dyscalculia. As such, very detailed examination of the eye movements had to take place. Since identification of strategies that were not defined beforehand based on the eye movements was the main aim, we specifically choose an exploratory instead of experimental approach. The additional strategies that were discovered using this method serve as a basis for future quantitative research.

To conclude, our findings give some first insights into strategy use and performance of adults with dyscalculia on unbounded number lines. The results suggest highly similar strategy use and performance in adults with and without dyscalculia. Herewith, neither our hypothesis that adults with dyscalculia have difficulties with more complex mensuration skills (i.e., proportion based judgement in bounded NLTs), nor our hypothesis that adults with dyscalculia have difficulties with low level mensuration skills was confirmed. We suggest that adults with dyscalculia may be mainly impaired in fast access to numerical knowledge instead of lacking number knowledge per se. However, this suggestion requires further research including a broader range of tasks and a larger sample.

Most importantly, this study was the first to combine the use of eye-tracking with CRR to capture strategy use in number line estimation. We revealed multiple additional strategies, steps and tools as compared to earlier research on both the bounded and unbounded NLT. Combining results from quantitative and qualitative analyses also led to more specific insights in what should be considered functional strategy use. Alternative strategies in the bounded NLT were revealed, showing that the lack of use of the begin-, mid-, and end-points could still indicate the use of functional reference points when, for example,

Dutch guidelines for diagnosing dyscalculia are based on criteria of

Criterion of severity.

The level of mathematics is substantially below that of peers, and interferes with every day life.

Criterion of discrepancy.

The proficiency in mathematical skills is substantially below the level expected based on the cognitive development of the individual.

Criterion of didactic resistance.

Improvement in math skills based on individualized instruction is limited.

We thank Margot van Wermeskerken, Willemijn Schot, and Tamara van Gog for their helpful comments on an earlier version of this manuscript, and Jos Jaspers for his help with the eye-tracking apparatus. We also want to show our gratitude towards Diedre Batjes who greatly assisted us in data collection.

The authors have no funding to report.

The authors have declared that no competing interests exist.