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The author has declared that no competing interests exist.

Conceived and designed the experiments: RWP. Performed the experiments: RWP. Analyzed the data: RWP. Contributed reagents/materials/analysis tools: RWP. Wrote the paper: RWP.

Natural number arithmetic is a simple, powerful and important symbolic system. Despite intense focus on learning in cognitive development and educational research many adults have weak knowledge of the system. In current study participants learn arithmetic principles via an implicit learning paradigm. Participants learn not by solving arithmetic equations, but through viewing and evaluating example equations, similar to the implicit learning of artificial grammars. We expand this to the symbolic arithmetic system. Specifically we find that exposure to principle-inconsistent examples facilitates the acquisition of arithmetic principle knowledge if the equations are presented to the learning in a temporally proximate fashion. The results expand on research of the implicit learning of regularities and suggest that contrasting cases, show to facilitate explicit arithmetic learning, is also relevant to implicit learning of arithmetic.

In many learning domains behavioral researchers have sought to characterize the knowledge that separates the novices from the experts, and to describe how that knowledge is acquired. Learners have been shown to use principles in a variety of domains. Principles are defined as regularities or general rules within a particular learning domain. Research of learners principle knowledge includes areas such as counting

Though the current research focuses on arithmetic principles, findings regarding principle learning in other domains (e.g., language acquisition, counting) are informative. Research on implicit learning of artificial grammars is particularly relevant as researchers often examine participants' learning of regularities

These findings suggest that it may be effective to investigate the learning of arithmetic principles through the manipulation of the types of example arithmetic equations to which learners are exposed. It is certainly the case that arithmetic learners see many examples of arithmetic equations inside and outside of formal education settings. Thus determining whether arithmetic principles may be learned in this way would be particularly useful. The current study addresses two factors, exposure to principle-inconsistent examples and the temporal proximity of said examples.

Learning of principles in artificial grammar and language studies generally involves learning by positive examples only

Principle learning may be facilitated by input that highlights the regularity repeatedly within a small amount of time. Prior work suggests that in order to learn arithmetic principles, the learner needs to have the relevant regularity highlighted repeatedly within a short time span

A secondary goal this study is to characterize the correlates of arithmetic principle knowledge acquisition. Knowledge of arithmetic principles may correlate with learning processing of arithmetic equations and their performance in arithmetic.

Prior research has attributed changes in learners' knowledge to changes in their

Consider learners' encoding of simple arithmetic equations. The learner may note characteristics of the equation such as the values of the operands, the operation, the order of the operands, specific characteristics of the operands such as evenness or oddness, and so forth. The learner's encoding of the equation involves noting and prioritizing a set of characteristics. Some characteristics may be deemed very important while others are virtually ignored.

What the learner attends to can have consequences for performance, for example the relative spacing of digits can influence performance on arithmetic tasks

Past research suggests that principle knowledge contributes to expertise in a domain

The current study focuses on the

In the case of the relationship to operands principle, the characteristic of the arithmetic equation that may be crucial to encode is the relative magnitude of the operands and the result. Learners whose encoding prioritizes the relative magnitude of the operands and result will demonstrate principle knowledge. A learner who does not prioritize the relative magnitudes of the operands and result is unlikely to show knowledge of relationship to operands.

The research procedures described below were completed in accordance with approval from the Institutional Review Board at the University of Wisconsin – Madison. Written consent was obtained from all participants prior to testing.

Adult participants (

Experimental stimuli were presented either on paper or via computer, depending on the task. Computer presentations used the PsyScope 1.2.5 interactive graphic system for experimental design and control

The experiment used a multifaceted approach to characterizing principle knowledge

In this type of assessment the learner is given the opportunity to differentiate between examples that violate a principle and those that do not. Participants may indicate that equations that violate a principle are “worse” or “more wrong” than equations that do not. The evaluation task was based on a task used in prior research

Sets of equations were constructed in pairs. Each pair contained the same operands for each of the eight equations but varied in the answers. Answers were controlled for so that the average deviation from the correct answer was the same across both sets, and so that one set contained one principle principle-inconsistent, while the other contained no principle-inconsistent equations. Since the average deviation from the correct answer was controlled for participants could not differentiate equation sets based solely on answer plausibility. Participants were inferred to have knowledge of the principle if they rated principle-inconsistent sets lower than principle-consistent sets.

Participants viewed solved arithmetic equations (80 total) presented serially on a computer monitor, and were asked to judge whether each equation was correct (e.g., “54÷6 = 9”) or incorrect (e.g., “15÷5 = 2”). Participants were asked to respond as quickly and accurately as they could. Both speed and accuracy were recorded. The task can be performed by calculating the answer to each equation and comparing it to the presented answer. Participants likely used this strategy on many trials. However, a subset of trials can be solved via a “short cut”. Any equation that violates a principle can be verified as incorrect without calculation. For example, 225÷15 = 345 can be quickly dismissed by noting that 345>225 (in contrast, try 225÷15 = 13). Equations were selected to control for deviation from the correct answer. We took as an indication of participants' principle knowledge if they used significantly less time to reject violation equations as compared to principle-inconsistent equations.

Though both the equation evaluation task and verification task were designed to assess principle knowledge they use different types of evidence. The verification task is based on the application of procedures while the evaluation task is based on evaluation of examples. These two types of knowledge assessments do not necessarily yield similar results

Participants' experience was manipulated using a training task similar to the methods in typical artificial grammar learning experiments

The training task involved serial presentation of arithmetic equations on a computer screen. Participants were instructed that they were to view a display of equations solved by two students and to decide which student understood arithmetic better. Each equation was presented for 2000 ms. Equations' solutions were marked by color as correct (green) or incorrect (red). There were a total of 120 equations split between the two students. Equations were presented in blocks of 30 alternating by student.

Equation sets included a mix of correct, incorrect principle-consistent and incorrect principle-inconsistent equations. The proportion and placement of the principle-inconsistent equations depended on the condition to which the participant was assigned. The exact set of equations viewed by participants varied by condition. There were two factors manipulated in a 2×2 factorial design: number of principle-inconsistent equations (high, low) and temporal proximity of principle-inconsistent equations (blocked, interleaved). Thus the four conditions were as follows: high-blocked( n = 22), high-interleaved (n = 23), low-blocked (n = 21), and low-interleaved (n = 21).

In the high principle-inconsistent conditions there were 12 principle-inconsistent, 10 correct and 8 principle-consistent equations (40% principle-inconsistent) for each block. In the low there were 6 principle-inconsistent, 10 correct and 14 principle-consistent equations (20% principle-inconsistent) for each block. In both conditions violation equations are blocked such that they all appear in succession in the middle of the presentation. For high and low interleaved conditions the proportion of equations types is the same as previously stated. However, for interleaved conditions violation equations never appeared in succession, instead a principle-inconsistent or correct equation always followed.

In addition to these conditions, there was also group of participants (n = 32) who viewed no principle-inconsistent equations at all. In this condition, all of the equations in the training stimuli were consistent with the arithmetic principle; this included 10 correct and 20 incorrect principle-consistent equations. Thus, the full design was a 2×2 factorial design, with one additional control condition.

To address the possibility that improvements in equation encoding lead to principle knowledge, a task was devised to characterize learners' equation encoding. Assessments of equation encoding have been used in other research that investigates arithmetic knowledge

Participants viewed equations presented briefly (990 ms) on a computer screen. After each equation the participant saw a series of four letters. The participant was then asked to answer a question about the letters (e.g.,

Prior research suggests that arithmetic principle knowledge may correlate with word problem skills

To compare principle knowledge before and after the training task, a “learning” score was calculated for each participant. This was done by comparing the difference in ratings of principle-consistent and principle-inconsistent sets in the evaluation task pre and post training. For example, if a participant rated principle-consistent equations higher than principle-inconsistent equations by 0.05 on the pre-training evaluation task and by 0.70 on the post-training evaluation task, that participant's learning score would be 0.65. The data analysis compares learning scores across conditions.

Recall that participants viewed input that varied along two factors, number of principle-inconsistent equations (high vs. low), and proximity (blocked, interleaved). we conducted a 2 (number of principle-inconsistent equations: high or low) x 2 (proximity: blocked or interleaved) ANOVA with learning scores as the dependent measure. The data are presented in ^{2} = 0.046. Participants in the blocked conditions had higher learning scores (M = .269) than participants in the interleaved conditions (M = −0.052). Number of principle-inconsistent equations did not significantly affect improvement scores, F(1, 83) = .919, p = 0.34. Participants in the high-number conditions did not have significantly different learning scores from participants in the low-number conditions (Ms = 0.032 and 0.186, respectively). The interaction of the two factors was also not significant, F(1, 83) = 1.04, p = 0.31.

To examine whether principle-inconsistent equations facilitated learning more than principle-consistent examples, we performed a planned comparison between the blocked principle-inconsistent group and the bocked principle-consistent group. This comparison was marginally significant, t(73) = 1.89, p = .061,

We also examined whether number of principle-inconsistent equations affected learning within the blocked conditions. The difference in learning scores between the high-blocked and low-blocked conditions was not significant, t(41) = 1.26, p = 0.21.

The verification task was also used as a measure of principle knowledge. For this task there were two analyses done, reaction time and accuracy. For each dependent measure, reaction time and accuracy, we conducted a 2 (number of principle-inconsistent equations: high or low) x 2 (proximity: blocked or interleaved) ANOVA. For both analyses there were no significant effects of amount, proximity or their interaction; reaction times, number F (1, 83) = 1.35, p = .24, proximity F (1, 83) = 0.36, p = 0.54, number by proximity interaction F (1, 83) = 1.98, p = 0.16; accuracy, number F (1, 83) = 2.38, p = .12, proximity F (1, 83) = 0.20, p = 0.65, number by proximity interaction F ( 1, 83) = 1.72, p = 0.19.

The most relevant comparison is between learners' post-training principle knowledge and their encoding scores. Encoding scores were calculated for each of the four types of questions in the encoding task: relationship, identity, parity and operation. Scores were based on the number of trials answered correctly.

Based on participants' post-training principle knowledge scores, participants were divided into three equal size groups, with high, medium and low principle knowledge. For these groups, mean scores on the post-training evaluation task were 1.14, 0.32, and −0.29, respectively. For each of the four encoding question types, we ran a one-way ANOVA and conducted post hoc comparisons.

For both relationship and identity encoding scores, the overall ANOVA was significant, F(1, 116) = 3.66, p = .03, η^{2} = 0.059 and F(1, 116) = 5.94, p = .003, η^{2} = 0.093, respectively. For both relationship and identity encoding scores, participants in the low principle knowledge group performed significantly more poorly than participants in the medium or high knowledge groups. For relationship encoding scores, the low knowledge group (M = 0.71) scored significantly lower than the medium group (M = 0.81, p = .023,

Neither parity nor operation encoding scores yielded significant results, F(1, 116) = 0.384, p = 0.68 and F(1, 116) = 0.087, p = 0.91. Thus, there were no significant differences between the principle knowledge groups in encoding the parity of the numbers or the operations used in the equations. Note that the parity of the numbers is not relevant to the relationship to operands principle. Further, regardless of which operation is involved, the relative magnitudes of the operands and result are crucial to encode; thus, variations in encoding operation as a function of principle knowledge were also not expected.

These results of this task suggest that learners with greater arithmetic principle knowledge encode specifically the relative magnitudes and the identities of the numbers in arithmetic equations better than learners with less principle knowledge.

Because the training was in division only, we examined participants' performance on the division items on the word problem task. We conducted a 2 (number of principle-inconsistent) x 2 (proximity) ANOVA with division sub-scores as the dependent measure. Neither factor was significant, nor was the interaction, proximity F(1, 83) = 2.39, p = 0.12, number of principle-inconsistent equations F(1,83) = 0.37, p = 0.54, number by proximity interaction F (1, 83) = .34, p = .55. Participants in the blocked conditions were correct on 78% of the division items while participants in the interleaved condition were correct on 71% of the items.

The current results demonstrate that increases in arithmetic principle knowledge are related to exposure to temporally proximal principle-inconsistent equations in addition to principle consistent equations. This result is unexpected based on much of the implicit learning literature, which generally focuses on learning via principle-consistent examples only. This suggests that the learning of arithmetic principles may occur via a different mechanism. Since exposure that includes multiple types of examples seems to work best, it is possible that it is the action of contrasting between the types that facilitate learning of principles. Previous work in arithmetic learning suggest that contrasting cases is beneficial to mathematical conceptual and procedural knowledge

The results of this study also suggest that learners with relatively low principle knowledge were also poor at encoding the relative magnitudes of the numbers in the equations and the identity of the numbers in the equations. Although accurate equation encoding may be required for knowledge of the principle, it does not seem to be the case that accurate encoding automatically leads to principle knowledge. This is suggested by the fact that high principle knowledge participants did not encode the problems significantly more accurately than the medium principle knowledge group. Consistent with prior

This research investigated how learners acquire knowledge of arithmetic principles though structured input. We addressed both the possible mechanism of change in encoding and the types of experience that may facilitate that change. The results of this study suggest that equation encoding is indeed related to arithmetic principle knowledge. We initially hypothesized a direct relationship between encoding of relative magnitudes and knowledge of relationship to operands. The results suggest that this relationship may not be as straightforward as originally hypothesized; there may be other mediating factors.

Finally, the results suggest that structuring the types of examples so as to give the learner an opportunity for contrast may facilitate learning of arithmetic principles; specifically, a mix of correct, principle-consistent and principle-inconsistent equations seems beneficial. This result may have some application to formal educational settings. While it is certainly the case than explicit instruction can play a role in formal education, learners have much more experience and opportunity for implicit leaning than explicit formal instruction. Prior research in contrasting cases

The author would like to acknowledge Martha Alibali for insights into the design of this work.