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Being able to perform computational estimations efficiently is an important skill. Furthermore, computational estimation experiments are used to study general principles in strategy development. Rounding strategies are common in computational estimation. However, little is known about whether and when children use a mixed-rounding strategy (i.e., both rounding up and down in one estimation) and how demanding this is in comparison to only rounding-down or only rounding-up. Therefore, we systematically varied the size of unit digits (i.e., the rightmost digit in a whole number) in 72 addition problems. These estimation problems were presented to fourth graders. Most children preferred to use mixed-rounding on mixed-unit problems and therefore adjusted their strategy choice to the individual unit digits in a calculation. Additionally, the sum of units barely influenced children’s strategy choice. On mixed-rounding calculations, the proportion of best strategy use was comparable to that of rounding-up and the latencies to produce an estimate with mixed-rounding were between those for rounding-down and rounding-up. Therefore, the mixed-rounding strategy was in the difficulty range of the two more frequently studied rounding strategies; it was also the preferred strategy for mixed-unit problems by children who adapted their estimation strategies. Based on these findings we argue that research into strategy development with estimation tasks should also include mixed-rounding to improve ecological validity.

Being able to perform computational estimations correctly and efficiently is an important skill in everyday life and in learning mathematics. The ability allows children to make approximate calculations without the need for a calculator or writing in real-world situations and to check the reasonableness of complex calculations found through other means. Additionally, it may help children to “develop a better understanding of place value, mathematical operations, and general number sense” (

In good computational estimators, three key processes or broad categories of strategies were identified for the approximation step: reformulation, translation, and compensation (

A variety of strategies for computational estimation has been documented. Most studies indicate that children and adults typically use multiple strategies and that most trials are solved with some sort of rounding strategy (e.g.,

Children typically use multiple strategies, and they typically adapt their strategy choice to features of the estimation problem. Rounding strategies can be divided into three types: rounding-down (i.e., all operands are rounded down), rounding-up (i.e., all operands are rounded up), and mixed-rounding (i.e., some operands are rounded down and the others are rounded up). Children favour the rounding-down strategy when rounding-down provides a close estimate and they prefer rounding-up for those problems, for which the rounding-up result is closer to the exact calculation (e.g.,

So far, the adaptive use of a mixed-rounding strategy in children has rarely been studied systematically. In several studies participants’ choice of strategies was restricted to rounding either both operands down or rounding both operands up (e.g.,

Studying mixed-rounding use in school-aged children in detail is important for at least two reasons.

In a choice-design, fourth graders completed 72 computational estimation problems for two-digit additions by rounding-down, rounding-up or mixed-rounding. Unit sizes were systematically varied, resulting in equal numbers of otherwise comparable small-unit problems, large-unit problems and mixed-unit problems. The present study focussed on three research questions related to the use of mixed-rounding.

First, do children preferably use mixed-rounding on mixed-unit problems (RQ1)? We expected that they would because children adaptively favoured rounding-down for small-unit and rounding up for large-unit problems (e.g.,

Second, do children consider the unit sums when choosing a rounding strategy (RQ2)? To give close estimations in a design that includes mixed rounding, the children can consider unit sums when deciding a strategy for some small-unit or large-unit problems. We expected that unit sums would have little impact in the fourth graders’ strategy use. This was because compensation, which requires that the units of different operands are taken into account, has been reported to be rare in similar age groups (

Third, is mixed-rounding a particularly complex estimation strategy for children or not (RQ3)? We had no clear expectation because mixed-rounding could be the most complex of the three rounding strategies as it combines the two others or could be less complex than the rounding-up strategy as one of the operands is rounded down. To answer this question, we analysed how often mixed-rounding was chosen as best strategy in comparison to rounding-down and rounding-up and analysed estimation latencies for the three rounding strategies, while controlling for problem size to take account of this robust effect of task difficulty (e.g.,

Eighty-eight children were tested (46 males; age in months: ^{th} grade. They were recruited from eleven classes in seven primary schools in urban and suburban areas in the state of Hesse (Germany). The study was approved by the local ethics committee. Parents provided their written informed consent, and children gave their verbal consent.

Computational estimation problems were drawn from three main categories: 24 small-unit problems with unit digits of both operands smaller than 5; 24 mixed-unit problems with one of the unit digits smaller and the other larger than 5; and 24 large-unit problems with unit digits of both operands larger than 5.

Each of the three categories were subdivided into three subcategories according to the sums of the unit digits for two reasons. First, unit sums are decisive for identifying the rounding strategy that leads to the closest estimate. Second, in previous research, unit sums determined for mixed-unit problems whether a problem was classified as heterogeneous small problem vs. heterogeneous large problem with rounding-down vs. rounding-up as (second) best strategy (see

Main Category | Subcategory |
Closest Estimate | Classification | ||
---|---|---|---|---|---|

define by size |
Name | defined by |
Example | by rounding | in previous |

Small-unit |
small1 | 3 or 4 | _1 + _2 | down | homogeneous small |

small2 | 5 | _3 + _2 | down / mixed | homogeneous small | |

small3 | 6 or 7 | _2 + _4 | mixed | homogeneous small | |

Mixed-unit |
mixed1 | 7, 8 or 9 | _7 + _1 | mixed | heterogeneous small |

mixed2 | 10 | _2 + _8 | mixed | - (not included) | |

mixed3 | 11, 12 or 13 | _9 + _3 | mixed | heterogeneous large | |

Large-unit |
large1 | 13 or 14 | _6 + _8 | mixed | homogeneous large |

large2 | 15 | _8 + _7 | mixed / up | homogeneous large | |

large3 | 16 or 17 | _7 + _9 | up | homogeneous large |

The small-unit problems were subdivided into

The mixed-unit problems were subdivided into

The subcategories for large-unit problems were as following:

To avoid systematic errors in the composition of the operands, for half of the problems in each subcategory the unit digit of the first operand was larger than the unit digit of the second operand. The pool of unit digit pairs (e.g., _1 + _4, _7 + _2) was combined with the pool of expected additions after rounding (e.g., 30 + 60, 80 + 50). For each subcategory, 50% of additions were without carry (estimates of 50-100) and 50% with carry (estimates of 110-170), and in 50% the first operand was larger than the second one. Additionally, the sums of the exact calculations and the estimates when using the best rounding strategy were matched as closely as possible for the subcategories. Further constraints comparable to previous research (

All participants completed a computational estimation task of addition problems divided into two sets. Children were tested in groups of up to 5 children. The estimation task was presented on laptops and implemented in E-Prime. Instructions were presented verbally over headphones and additional instructions were given if children did not respond or made errors during the practice trials. Two experimenters were present for further questions and instructions.

Children were asked to give an approximate answer for two-digit addition problems without calculating the exact sum. Using the example of 28 + 41, different possibilities to get to an estimate were introduced: rounding down both operands to 20 and 40 and giving 60 as an answer (rounding-down strategy); rounding up both operands to 30 and 50 with 80 as an answer (rounding-up strategy); and rounding the first operand up to 30 and the second one down to 40 with 70 as an estimate (mixed-rounding strategy). Participants were told they could decide how they estimate the result, but they should do it in a way that yields estimates close to the exact sum while being fast at the same time. Children responded by typing their answers. Therefore, at the start of the study before introducing the estimation task, children were familiarized with the number pad. Children were not instructed to indicate which strategy they were using, but strategy selection was inferred from their estimation results, e.g., for the trial 22 + 57, the rounding-down strategy was inferred when 70 was the response, the mixed-rounding strategy for a response of 80, and the rounding-up strategy was inferred when 90 was a response (for further details, see

Two test sets were given on different days. For each of the two sets, children received practice trials with adaptive feedback. After instruction and practice, no participant displayed any apparent difficulties with the task. Estimation problems were presented in black (Font: Arial, Font size: 150) on an otherwise white screen, until the participant had typed in their answer. No time-limit for responding was imposed. Within each set of the estimation task, children completed two task blocks of 18 trials split by a short break: therefore, in total 72 estimation trials.

Data of the estimation experiment had a cross-classified structure with trials nested in items and in participants. To examine effects of problem features and strategy use while taking random effects of participants and random effects of items into account, data were analysed with cross-classified multilevel models or (generalized) linear mixed models ((G)LMM). The advantages of (G)LMMs compared to ANOVAs include that logistic GLMMs avoid known problems that occur when proportion data are analysed with ANOVAs. In addition, in (G)LMMs predictors at the item level (main category or problem size) and at the trial level (the strategy used on a given item by a certain participant) can be analysed jointly (see

Data inspection showed that some children used the same rounding strategy on (almost) all computational estimation problems within the same task block (the children received four task blocks, each of 18 trials), while other children switched between different rounding approaches. To analyse this systematically, we computed the proportion of trials solved with the most common strategy of this block. The resulting histogram (see

In most blocks where an inflexible approach was used (102 out of 114), rounding both operands down was the dominant strategy, and mixed-rounding was very rare (2 out of 114 inflexible blocks).

The following analyses on flexible strategy selection excluded task blocks with an inflexible approach. Including all children would have been problematic because two subpopulations with distinct approaches and distinct central tendencies of responding were found. The results of the combined analyses would be representative neither of the flexible nor the inflexible approach (see also

To address the first and second research question about when children used mixed rounding, the effects of the main categories (RQ 1) and the subcategories of unit sums (RQ 2) on children’s strategy selection were investigated. Data were analysed with cross-classified GLMMs with random intercepts for items and for participants. Three logistic models were set up: a central one for mixed-rounding, as well as, for completeness, one for rounding-down and one for rounding-up. For a visual understanding of the data see

To examine the second research question, we tested whether subcategories defined by unit sums influence strategy choice. Within small-unit and large-unit problems,

Model Parameters | Strategy Chosen |
||||||||
---|---|---|---|---|---|---|---|---|---|

Rounding-Down | Mixed-Rounding | Rounding-Up | |||||||

Fixed Part | β | 95% CI | β | 95% CI | β | 95% CI | |||

Intercept | 3.20 | [2.65, 3.77] | < .001 | -3.34 | [-3.91, -2.83] | < .001 | -4.49 | [-5.27, -3.76] | < .001 |

Mixed (vs. small) | -5.70 | [-6.43, -4.99] | < .001 | 5.02 | [4.41, 5.70] | < .001 | 1.23 | [0.41, 2.08] | < .001 |

Large (vs. small) | -6.68 | [-7.49, -5.93] | < .001 | 1.19 | [0.59, 1.85] | < .001 | 6.45 | [5.73, 7.24] | < .001 |

Small2 (vs. small1) | -0.62 | [-1.27, 0.02] | .06 | 0.90 | [0.28, 1.55] | .006 | -0.58 | [-1.66, 0.47] | .28 |

Small3 (vs. small1) | -0.60 | [-1.26, 0.05] | .07 | 0.67 | [0.03, 1.33] | .05 | 0.29 | [-0.58, 1.20] | .52 |

Mixed1 (vs. mixed2) | 0.19 | [-0.48, 0.83] | .57 | -0.07 | [-0.54, 0.38] | .76 | -0.12 | [-0.76, 0.52] | .71 |

Mixed3 (vs. mixed2) | -0.34 | [-0.98, 0.31] | .30 | 0.33 | [-0.13, 0.79] | .15 | -0.29 | [-0.93, 0.36] | .37 |

Large1 (vs. large3) | 0.28 | [-0.48, 1.07] | .47 | 0.21 | [-0.27, 0.67] | .38 | -0.26 | [-0.71, 0.19] | .24 |

Large2 (vs. large3) | -0.14 | [-0.95, 0.66] | .74 | 0.10 | [-0.38, 0.56] | .68 | 0.04 | [-0.42, 0.49] | .84 |

Random Part | 95% CI | Δ |
95% CI | Δ |
95% CI | Δ |
|||

Item intercept | 0.15 | [0.01, 0.36] | 12.30 | 0.05 | [0.001, 0.17] | 3.10 | 0.03 | [0.001, 0.13] | -0.90 |

Participant intercept | 1.00 | [0.62, 1.55] | 192.30 | 0.19 | [0.08, 0.34] | 35.00 | 1.99 | [1.26, 3.03] | 314.00 |

To address the third question of whether mixed rounding is a particularly complex rounding strategy, we first examined how often children used mixed-rounding when it was the best strategy compared to rounding-down and rounding up when they were the best strategies. Following

Children with a flexible approach to strategy selection chose the best strategy on many trials. At the same time, there were clear inter-individual differences in how often children chose the best strategy (range: 22%-100%). How well they adapted their strategy choices was taken into account by the random participant intercept of the GLMM (for full model results see _{estimate=140} = 81% vs. PP_{estimate=70} = 86%). As expected based on previous studies with a restricted design, the rounding-up strategy as best strategy on large-unit problems (PP = 79%) was used slightly less often in comparison to rounding-down as best strategy on small-unit problems (PP = 89%; large-to-small contrast: β = -1.11, 95% CI [-1.46, -0.76],

To explore further the third question about the complexity of the mixed-rounding strategy, we additionally analysed how fast children were at solving estimation problems. Estimation latencies were timed between the item appearing on screen and children completing their responses. Therefore, estimation latencies include the time it takes to encode the problem, to select a strategy for the problem, to execute the estimation strategy including adding the rounded numbers and to type in the estimate. If children solved tasks with an inflexible approach not adapting the estimation strategy to the problems, but using one dominant strategy, problems were solved on average in 3.50 s, β = 3.50, 95% CI [3.14, 3.86].

All further analyses included only trials from blocks approached in a flexible manner. Please note that LMMs allow the modelling of effects at the trial level, which can involve the effects of how students responded to particular items, for example that on a given trial a child choses mixed-rounding. To examine estimation latencies in a design with strategy choice (and not strategy execution speeds in a no-choice condition; like

Children needed on average about 5 to 6 s to come up with estimates but clear differences between participants and items were present and modelled in the random effect variances. The problem size, defined as the size of the best estimate to an item, clearly had the expected effect: estimation latency increased by 0.16 s, β = 0.16, 95% CI [0.12, 0.20],

The present study is the first to report detailed insights into children’s mixed-rounding strategy use in computational estimation. The main results in relation to the research questions were, (1) that fourth graders when not sticking to one dominant strategy clearly preferred mixed-rounding for mixed-unit problems, (2) that unit sums had little impact on strategy selection and (3) that best strategy use and estimation latencies for mixed-rounding were in a similar range to those that occurred for rounding-down and rounding-up. About 75% of children adopted a flexible approach with a variety of strategies rather than a single strategy, which is in line with previous research (for a review see

Fourth graders with a flexible approach switched systematically between strategies (see

While the size of the individual unit digits being above or below five had a strong effect on children’s strategy choice, subcategories of problems distinguishing between different unit-sums had no or little impact on strategy choice. Therefore, the answer to the second research question is that fourth graders typically did not consider the sum of both unit digits in their strategy choice. They did not apply prior or post compensation to reduce the rounding distortion of their rounding approach. This is in line with previous research because compensation has been reported as uncommon among fifth graders (

Consistent with previous research that has used a restricted strategy set (e.g.,

Still, in the present study about 25% of children focused on one dominant strategy which was roughly comparable to the proportion reported by

Being the first study of its kind, there are uncertainties about whether these findings generalizes to other ages. But given the clear pattern of results that were consistent with less detailed results on mixed-rounding in undergraduates (

The work was partially funded by a grant awarded to Sebastian Poloczek by the Goethe University as part of the program Junior Researchers in Focus.

We are grateful to William J. Browne for his valuable comments on our thoughts on computing credible intervals for predicted probabilities. We also thank David J. Messer, the editor John N. Towse and the anonymous reviewers for their feedback helping to improve the manuscript.

For this article, a dataset is freely available (

The Supplementary Materials contain the following items (for access see

Research data

Supplementary text including additional result tables

Code for analyses

The authors have declared that no competing interests exist.

A previous version of this article was included in Svenja Hammerstein’s publication-based doctoral dissertation thesis (