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Park and Brannon (2013, https://doi.org/10.1177/0956797613482944) found that practicing non-symbolic approximate arithmetic increased performance on an objective numeracy task, specifically symbolic arithmetic. Manipulating objective numeracy would be useful for many researchers, particularly those who wish to investigate causal effects of objective numeracy on performance. Objective numeracy has been linked to performance in multiple areas, such as judgment-and-decision-making (JDM) competence, but most existing studies are correlational. Here, we expanded upon Park and Brannon’s method to experimentally manipulate objective numeracy and we investigated whether numeracy’s link with JDM performance was causal. Experimental participants drawn from a diverse internet sample trained on approximate-arithmetic tasks whereas active control participants trained on a spatial working-memory task. Numeracy training followed a 2 × 2 design: Experimental participants quickly estimated the sum of OR difference between presented numeric stimuli, using symbolic numbers (i.e., Arabic numbers) OR non-symbolic numeric stimuli (i.e., dot arrays). We partially replicated Park and Brannon’s findings: The numeracy training improved objective-numeracy performance more than control training, but this improvement was evidenced by performance on the Objective Numeracy Scale, not the symbolic arithmetic task. Subsequently, we found that experimental participants also perceived risks more consistently than active control participants, and this risk-consistency benefit was mediated by objective numeracy. These results provide the first known experimental evidence of a causal link between objective numeracy and the consistency of risk judgments.

If more numerate individuals make more use of numbers when making decisions

Recently,

This finding is important for multiple reasons. First, the method is a relatively easy-to-implement way to manipulate numeracy and, thus, is beneficial for researchers seeking to test causal effects of numeracy on various outcomes (e.g., judgment and decision making, JDM, performance). Second, the inexact, non-symbolic arithmetic practice transferred to a

Numeracy is not a single construct (

These numeracy components are linked. The ANS, for example, provides a “feel” for the quantities referred to by symbolic numbers, such that symbolic number mapping reflects ANS-acuity and the connection between symbolic numbers and ANS magnitudes (

In the present study, we used a version of

Like

Participants in this study were recruited online via Amazon Mechanical Turk (MTurk) over a 2-week period, in small cohorts of no more than 50 individuals. We initially recruited 935 individuals via MTurk who began the pretest. We excluded 66 participants who responded from outside of the United States, and an additional 18 who did not identify as native English speakers, leaving an initial sample

Participants were paid $2.00 to complete the pretest, $3.00 to complete the posttest, and $3.00 for each training session in which they participated (up to 6 possible), for a maximum total possible reimbursement of $23.00 in the training conditions and $5.00 in the non-intervention control condition. Participants were paid promptly, typically within 24 hours of each session, to encourage retention.

At pretest, participants first completed measures of subjective and objective numeracy. Next, they provided background information, answering questions about their demographics, “growth mindset” (

Day / Activities |
---|

Session 1 |

1. Pretest (15–20 min): |

Numeracy measures: |

SNS |

Arithmetic task |

Other questions (e.g., background information and task compliance) |

2. Assignment to condition |

3. Training (if applicable) (20–30 minutes) |

Sessions 2–5 |

1. Training (if applicable) (20–30 minutes, 24–72 hours after the prior session) |

Session 6 |

1. Training (if applicable) (20–30 minutes, 24–72 hours after the prior session) |

2. Posttest (20–30 minutes): |

JDM tasks: |

Bets task |

Consistency in risk perceptions |

Framing |

Numeracy measures: |

SNS |

Arithmetic task |

ONS |

SMap task |

Other questions (e.g., task compliance) |

Training participants were linked to the posttest immediately after their last training session. Non-intervention controls were sent this link when half their recruitment cohort completed training.

We measured participants’ subjective numeracy at pretest and posttest using the Subjective Numeracy Scale (SNS,

Participants completed three JDM measures at posttest-only: a Consistency-in-Risk-Perception task (see

Participants were randomly assigned to one of six possible conditions: one of the five training conditions (detailed below) or a non-intervention control. Participants assigned to the non-intervention control were invited to participate in the posttest in 1–2 weeks. Participants assigned to the training conditions were invited to participate in six training sessions and the posttest, all to be completed within 1–2 weeks. Participants in all training conditions (including the active memory-training control) were told that they would practice “skills related to math ability” to equalize demand characteristics between intervention conditions. Spatial and math skills have long been linked (

Each of the six training sessions typically took 20–30 minutes to complete. Participants were able to start each subsequent training session at any time within a 24- to 72-hour window after they had started the previous session. A reminder email with a link to the training webpage was sent after 24 hours had passed. If participants did not begin the next session within the 24–72 hour window, they were excluded from the study.

In the four numeracy intervention conditions, participants practiced approximate arithmetic following a 2 (addition, subtraction) × 2 (symbolic, non-symbolic) between-subjects design. In particular, they practiced either approximate addition OR approximate subtraction using either symbolic OR non-symbolic numeric stimuli. Otherwise, we followed the training procedures described by

In the addition conditions, values would fly in from the left and right of the screen and hide behind a grey square. In subtraction conditions, a value would fly in from the left, and a second would fly out from the right. Participants estimated the sum or difference between these values. Participants then compared their estimates to a third value, either by saying if their estimate was greater or less than this value (comparison trials) or by choosing the correct value from a pair where the comparison-value acted as the foil (match trials, see

The ratio between the actual sum or difference and this comparison-value became smaller (more difficult) when participants responded correctly, and larger (less difficult) when participants made mistakes. The fifth training condition was an active control following

After excluding individuals outside the USA and non-native English speakers, our eligible recruitment sample size was 851, with 138–144 participants assigned to each group. Of these participants, 48.3% (

Condition | Assigned | Other Exclusion | Final |
|||
---|---|---|---|---|---|---|

Declined | Training Incomplete | Posttest Incomplete | ||||

Memory | 144 | 29 | 51 | 0 | 2 | 62 |

Symbolic Addition | 143 | 32 | 42 | 0 | 2 | 67 |

Non-symbolic Addition | 139 | 35 | 34 | 1 | 5 | 64 |

Symbolic Subtraction | 138 | 28 | 40 | 0 | 3 | 67 |

Non-symbolic Subtraction | 144 | 26 | 48 | 0 | 9 | 61 |

Non-intervention | 143 | 4 | N/A | 43 | 6 | 90 |

Active memory training control retention (43.1%) did not differ from retention across numeracy training interventions, 45.9%; X^{2}(1, ^{2}(1,

We used G*Power version 3.1.9.2 (2014) to determine the sensitivity of our final sample size of 321. The sensitivity analysis showed that with an

Attribution in randomly assigned studies rarely changes results (_{10} = 0.070) or pretest Arithmetic (BF_{10} = 0.061). Pretest and posttest mean scores are available in

Condition | Pretest SNS | Posttest SNS | Pretest Arithmetic | Posttest Arithmetic | Posttest ONS | SMap^{a} |
---|---|---|---|---|---|---|

Possible range of scores | 1–6 | 1–6 | 0–96 | 0–96 | 0–7 | 0–193^{a} |

Memory ( |
4.63 (0.10) | 4.69 (0.12) | 40.31 (2.04) | 40.94 (2.17) | 3.44 (0.24) | 23.83^{a} (1.59) |

Symbolic Addition ( |
4.58 (0.09) | 4.71 (0.09) | 37.94 (1.80) | 39.06 (1.82) | 3.82 (0.20) | 23.12 (1.42) |

Non-symbolic Addition ( |
4.68 (0.11) | 4.83 (0.11) | 36.83 (1.94) | 38.06 (2.12) | 3.73 (0.21) | 25.11^{a} (1.51) |

Symbolic Subtraction ( |
4.88 (0.09) | 4.93 (0.10) | 42.54 (1.92) | 43.07 (2.06) | 4.04 (0.19) | 23.58 (1.46) |

Non-symbolic Subtraction ( |
4.65 (0.11) | 4.74 (0.11) | 39.64 (2.09) | 39.80 (2.06) | 3.87 (0.18) | 25.15^{a} (1.46) |

Non-Intervention Control ( |
4.67 (0.07) | 4.77 (0.08) | 37.91 (1.55) | 35.83 (1.51) | 3.57 (0.18) | 28.54^{a} (3.07) |

^{a}SMap scores reflect the mean absolute numeric distance from correct (ADC) of the participants’ placement of numbers on the 0–1,000 number line, excluding the placement of “71.” We excluded four participants whose resulting ADCs were more than 5

Participants generally followed the same pattern of improvement across the training sessions as seen by

Our participants’ arithmetic performance was substantially different than that seen by

To account for pretest individual differences and these multiple measures, we used SPSS version 25 to conduct a Generalized Estimating Equation (GEE) analysis of posttest objective numeracy with the two subscales—Arithmetic and ONS—treated as repeated measures. This analysis allowed us to simultaneously test whether numeracy training improved objective numeracy versus the active memory-training control and if it had differential effects on the two subscales, while controlling for individual differences in pretest objective numeracy. For ease of comparison, ONS scores and pretest and posttest Arithmetic scores were transformed into proportion correct; pretest Arithmetic scores were entered as a covariate. We used maximum likelihood estimation with a normal probability distribution, identity link function, and independent correlation matrix to examine the two-way interaction of objective numeracy subscale (posttest Arithmetic vs. ONS) and training (numeracy training vs. memory training).

We only partially replicated

We confirmed that posttest Arithmetic scores were not influenced by training condition via a Bayesian ANCOVA that included training condition as a fixed factor and pretest Arithmetic scores as a covariate. It was conducted with JASP 0.14.1 (_{10} = 0.153), the model including only numeracy training condition (BF_{10} < 0.001), and the null model (BF_{10} < 0.001) were all less likely. (Note: We report all BF_{10}s in reference to the best model, whose BF_{10} is fixed at 1). Additional analyses also found no benefits of training on the SNS and SMap measures (see

In order to determine whether this arithmetic benefit occurred both for participants receiving non-symbolic training (the training that most closely replicated that used by

We also ran a GEE analysis which found no significant differences among numeracy-training conditions (see _{10} = 0.146), the model including only numeracy training condition (BF_{10} < 0.001), and the null model (BF_{10} < 0.001) were all less likely.

Although detectable, the present effect size is substantially smaller than the medium-sized effect (

We found that Non-symbolic training was particularly likely to yield higher posttest numeracy scores for low subjective-numeracy participants. A GEE analysis found that, as hypothesized in H1, the interaction of symbolic versus non-symbolic condition and pretest SNS was significant, Wald χ²(1) = 5.42,

Participants responded to three JDM tasks: bets, framing, and risk-perception consistency. However, the anticipated, pre-requisite effects of objective numeracy in the between-subject framing and bets tasks did not replicate. Hence, only the within-subject risk-perception task could be used to test H2. We discuss this decision further in the

A GEE analyses showed that, consistent with H2, inconsistencies in risk judgments were less likely among numeracy-training participants (proportion-based analysis:

We were interested in determining whether the effect of numeracy versus memory training on risk-inconsistency errors was mediated by the numeracy-training effect on ONS scores. (We restricted our analysis to ONS scores because numeracy training did not influence the Arithmetic scores, and thus were not a possible mediator). Thus, we ran a simultaneous regression mediational analysis, details of which are available in the

Results supported our hypothesis. As expected based on prior analyses, numeracy training predicted higher ONS scores, both proportion- and binary-based analyses:

Although we found that non-symbolic training was particularly beneficial for individuals with lower subjective numeracy, overall, non-symbolic and symbolic training were equally beneficial. Specifically, objective-numeracy outcomes did not differ, on average, between these conditions. This finding surprised us. On the face of it, one might think that training focused on symbolic numbers would be more beneficial to other symbolic tasks than would non-symbolic training.

The current results did not support one of

The results also do not support the idea that non-symbolic training improved symbolic arithmetic by improving subjective numeracy: No subjective numeracy differences were detected between the numeracy-training and control conditions (see

Participants who had lower SNS scores (e.g., who rated themselves as worse with numbers and preferring to use them less) derived directionally more benefit from non-symbolic training, whereas those with higher SNS scores instead performed marginally better when they received training with traditional symbolic numbers. It is important to emphasize that this effect was detected while controlling for pretest objective numeracy, indicating that it was the participants’

Numeracy training yielded more consistent risk perceptions, and this benefit was mediated by post-intervention condition differences in objective numeracy (controlling for pre-intervention arithmetic scores). These results indicate that the benefits of numeracy training can extend beyond mathematical paradigms to improved judgments. In addition, training need not be rooted in traditional symbolic calculation. Specifically, approximate-arithmetic training can yield these benefits, using either symbolic or non-symbolic numbers.

The precise mechanism for how numeracy causes these improvements remains unclear (see

Our results diverge from

First, our online training may have been less effective than the in-lab training conducted by

It may be that numeracy versus memory training encouraged abstract versus concrete processing, instead of or in addition to the anticipated effect on objective numeracy. Abstract processing is thought to improve performance on word problems (like those in the Objective-Numeracy Scale, but NOT the Arithmetic task) because thinking abstractly encourages people to ignore superfluous details in the narrative component of the problem in order to focus on the key numeric information necessary to solve the problem (

Finally, the different results could reflect a difference in our samples. For example, Park and Brannon’s participants were primarily college students and may have been more accustomed to taking symbolic math tests than participants in our more diverse internet sample. The fact that

We concluded that differences existed in the consistency of Risk judgments between the numeracy and memory training groups that were presumably due to our manipulation of objective numeracy. Our ability to interpret the results was aided by the fact that the intervention improved objective numeracy scores and risk judgments in the absence of any posttest training-condition differences in subjective numeracy or symbolic number mapping. One might otherwise suspect that participants made more numerically consistent risk judgments simply because numeracy training increased their confidence in their own numeric ability and/or their understanding of values represented by symbolic numbers. Nevertheless, it remains possible that numeracy training primed numeric processing more generally and made people more likely to make use of their numeric skills (

The finding that, relative to memory training, numeracy training improved objective-numeracy scores specifically—and not subjective numeracy or symbolic-number mapping—is itself intriguing. Higher objective numeracy has been related to both subjective numeracy and symbolic number mapping in past studies (e.g.,

Although our effects were smaller than those found by

We thank Martin Tusler for his assistance. We thank William Chaplin for his help with Bayesian analyses. We thank Melissa Peckins for her help with power analyses.

Funding was provided by grants through the National Science Foundation (SES‐1155924, SES‐1558230, and SES-2017651) to E. Peters. The content of this paper is solely the responsibility of the authors and does not necessarily represent the official views of the funding agency.

For this article, a dataset is freely available (

Supplementary methods, results, and discussion, along with data files, and SPSS analysis scripts are available via the OSF repository:

The authors have declared that no competing interests exist.

Dana Chesney and Ellen Peters were at The Ohio State University while this research was conducted.

Some of these findings were presented at the 2016 annual conference for the Society for Judgment and Decision Making.