^{*}

Edited by: Linda Bol, Old Dominion University, USA

Reviewed by: Douglas J. Hacker, University of Utah, USA; Peggy Chen, CUNY, USA; Jennifer R. Morrison, Johns Hopkins University, USA

*Correspondence: Noelle M. Crooks, Department of Psychology, University of Wisconsin - Madison, 1202 W. Johnson St., Madison, WI 53706, USA e-mail:

This article was submitted to Educational Psychology, a section of the journal Frontiers in Psychology.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This study investigated whether activating elements of prior knowledge can influence how problem solvers encode and solve simple mathematical equivalence problems (e.g., 3 + 4 + 5 = 3 + __). Past work has shown that such problems are difficult for elementary school students (McNeil and Alibali,

A crucial step in solving problems is noticing features that are relevant for their solution. For example, in solving an arithmetic problem, it is necessary to note whether the operator symbol is a plus, minus, times, or division sign. Misencoding the operator would lead one to enact an incorrect strategy and obtain an incorrect solution. Thus, proper encoding of relevant problem features can mean the difference between solving a problem correctly and getting it wrong. Past research in child development and in adult cognition has highlighted the importance of accurate encoding in a number of domains, including probability (Dean and Mollaison,

What factors guide solvers' problem encoding? One important factor is prior knowledge. Learners' prior knowledge may provide them with information about what sorts of problem features are important. In mathematical equations, for example, numbers and operation symbols are important features, whereas other features (e.g., the font in which a problem is printed) can be safely ignored. Evidence from a range of problem domains shows that learners with greater prior knowledge encode key problem features more accurately than learners with less prior knowledge (e.g., Chi et al.,

A second factor that influences learners' encoding of novel problems is the amount of perceptual overlap between new problems and previously encountered problems (e.g., McNeil and Alibali,

A third factor that affects problem encoding is the strategy that the solver plans to enact (e.g., Alibali et al.,

We suggest that, when presented with a novel problem, solvers activate elements of their existing knowledge, and this activated knowledge guides their problem encoding. For example, when presented with a problem, solvers may activate knowledge of possible problem-solving strategies, because of their past experience with problems of that type. They may also activate knowledge of related problems, due to perceptual overlap of the given problem with related problems. We propose that the specific knowledge that solvers activate should influence their encoding of novel problems, and we test this idea in the present experiment. If the activated knowledge is relevant to the given problem, solvers should accurately encode and solve the given problem. If, on the other hand, the activated knowledge does not align with the given problem, it may be detrimental for encoding and solving.

One domain in which the relationship between prior knowledge and current behavior may be particularly important is mathematical problem solving. Mathematics is a cumulative subject, in which new information builds on concepts and procedures that have been encountered previously. Although drawing on prior knowledge is often beneficial, there are some situations in which activating prior knowledge may actually hinder new learning and problem solving.

One area of mathematics in which previous experience has been posited to have a substantial negative effect is in equation solving. Elementary and middle school students often fail to correctly solve equations that have operations on both sides of the equal sign (e.g., 3 + 4 + 5 = 3 + __), called

Why do children perform poorly on equivalence problems? One possibility is that the instructional materials and techniques used in elementary education in the United States may lead children to misconstrue the equal sign as an operator (meaning “find the total”), rather than as expressing an equivalence relation. Both elementary and middle school textbooks in the United States typically present the equal sign in a very narrow set of contexts, and many textbooks do not provide explicit instruction about the symbol and its use (e.g., McNeil et al.,

Children's experience during elementary school appears to teach children three patterns—perceptual, conceptual, and procedural—that may hinder their abilities to solve more complex equations (e.g., McNeil et al.,

First, children learn the ^{1}

Second, children learn the

Third, children learn the

Although these patterns are pervasive in American educational settings and curricular materials, they are not universal. Children in countries where the equal sign is presented in more varied contexts, both in textbooks and in classrooms (e.g., China), perform significantly better than American students on equivalence problems, with sixth graders solving up to 98% of problems correctly (Li et al.,

In addition to correlational evidence for the relationship between certain types of experience in the mathematics classroom and trouble solving equivalence problems, research with undergraduate participants supports a causal link between activation of the operational patterns and such difficulties (McNeil and Alibali,

Reactivating knowledge of the operational patterns can also affect how undergraduate participants

Even simple practice solving traditional arithmetic problems (i.e., those with the equal sign and answer blank at the end of the problem) has been shown to negatively affect undergraduates' ability to solve equivalence problems. McNeil et al. (

The studies reviewed above suggest a possible causal link between the activation of the operational patterns and difficulty solving equivalence problems. One question that remains open, however, is exactly

One possibility is that these operational patterns hinder children's performance by affecting what they encode about mathematics problems. If children fail to notice relevant features of a problem's structure (e.g., the location of the equal sign), they are unlikely to correctly solve the problem. Equivalence problems are perceptually very similar to traditional arithmetic problems, with which elementary school children are often highly skilled (e.g., 3 + 4 + 5 = 3 + __ vs. 3 + 4 + 5 + 3 = __). This perceptual similarity, combined with the operational patterns children have learned, may activate children's arithmetic knowledge, leading them to focus on features that are relevant for solving a traditional arithmetic problem (e.g., the numbers) as opposed to those that are relevant for solving a more complex equation (e.g., the location of the equal sign).

Indeed, when children are asked to reconstruct equations after a brief viewing, they make significantly more errors on blank-final equivalence problems (e.g., 3 + 4 + 5 = 3 + __), which are perceptually highly similar to traditional arithmetic problems, than on non-blank-final equivalence problems (e.g., 3 + 4 + 5 = __ + 5), which are visually less similar (McNeil and Alibali,

Activating solvers' knowledge of basic arithmetic may also activate a problem-solving strategy such as “perform all given operations on all given numbers”, which is commonly used in arithmetic. When applied to an addition equivalence problem, this strategy is implemented as the

In sum, there is growing evidence that activation of the operational patterns typical of early arithmetic experience can negatively affect equivalence problem solving. There is also correlational evidence showing that children who have difficulty solving equivalence problems tend to have difficulty encoding them. To date, however, there is no evidence for a causal link between activation of the operational patterns and poor encoding of equations.

The purpose of the current study was to test whether activation of the operational patterns hinders problem solving by changing what solvers encode about the problems. Although accurate encoding is a prerequisite for problem solving, there are other ways in which students could err when solving an equation. For example, experience with one problem solving procedure (e.g.,

In the present study, we utilized the knowledge activation paradigm developed by McNeil and Alibali (

Participants in this study were randomly assigned to either a knowledge activation condition or a control condition, and were then asked both (a) to reconstruct equivalence problems after viewing them briefly (to assess encoding) and (b) to solve equivalence problems. We hypothesized that activation of the operational patterns would affect not only participants' solutions to the equivalence problems, as shown in past work, but also their encoding of the problems. Moreover, we hypothesized that inaccurate encoding would mediate the effect of knowledge activation on problem solving. Finally, we expected that activation of the operational patterns would lead participants to encode and solve the equations as if they were traditional arithmetic problems.

Participants were 181 undergraduate students recruited from the Introduction to Psychology class participant pool at a large Midwestern university. Because we were interested in reactivating the operational patterns that are typical in American educational settings, we limited the sample to participants who were educated in the US (

For the problem-solving task, a subset of participants failed to follow instructions and was therefore excluded from analyses of that task. Specifically, these participants continued to perform the previous task (reconstruction) even after they received instructions to switch to the next task (problem solving). Thus, for problem-reconstruction analyses, the sample consisted of the entire 115 participants and, for the problem-solving analyses, the sample consisted of 100 participants. The number of excluded participants did not differ by condition.

The current work was approved by and conducted in accordance with the human subjects guidelines of the University of Wisconsin-Madison Social and Behavioral Science Institutional Review Board (IRB). Informed consent was obtained from all participants and a debriefing was provided at the conclusion of the experiment. Participants received one point of extra credit in their Introduction to Psychology class in exchange for their participation.

Participants completed tasks, adapted from McNeil and Alibali (

Participants saw a target stimulus at the top of the screen followed by five sample stimuli. They were asked to indicate whether or not each of the sample stimuli matched the target, and to respond by pressing a button on the keyboard. In the knowledge activation version of this task, stimuli were equations presented in the “operations = answer” format (e.g., 365 + 694 = __), so as to activate the perceptual pattern. In the control version of the task, stimuli were letter strings (e.g., XxCxcX).

Participants were shown a target word at the top of the screen followed by five sample words. They were asked to indicate whether each of the sample words matched the target, and to respond via button press. In the knowledge activation version of the task, the words (i.e., total, add, sum, and plus) were selected to activate the concept of “the total.” In the control version of the task, the words were neutral, non-mathematical words (e.g., apple).

Participants saw a target stimulus at the top of the screen followed by five sample stimulus pairs. Participants were asked to indicate whether each of the sample pairs would combine to make the target. In the knowledge activation version of the task, the targets were numbers (e.g., 17) and the sample pairs were sets of numbers (e.g., 8 and 9), so as to activate the strategy of performing all of the given operations on all of the given numbers. In the control version, the targets were colors (e.g., pink) and the sample pairs were sets of colors (e.g., red and white).

To assess their abilities to accurately encode equivalence problems, participants completed a problem reconstruction task. Participants viewed four equivalence problems, one at a time, on a computer screen. Problems were presented for 1.5 s each and after each problem, participants were asked to write the equation exactly as they saw it on a paper answer sheet. Reconstruction tasks have been used extensively in the literature to assess encoding, both in the domain of mathematical equivalence (e.g., Rittle-Johnson and Alibali,

To assess their abilities to correctly solve equivalence problems, participants completed a problem-solving task. Participants viewed four equivalence problems, one at a time, on a computer screen. Problems were shown for 1.5 s each and participants were then asked to write the number that should go in the blank on a paper answer sheet.

Participants were randomly assigned to either the knowledge activation condition or the control condition. The samples in the two conditions did not differ in terms of gender, age, or percentile SAT/ACT scores.

In the knowledge activation condition, participants performed one block each of the three activation tasks. In the control condition, participants performed one block each of the three control tasks. Tasks were presented in a fixed order (perceptual, conceptual, and procedural) that did not vary by condition, following the procedure used by McNeil and Alibali (

Following the knowledge activation or control tasks, participants completed an assessment comprised of the problem reconstruction task and the problem-solving task. Because we expected that effects of knowledge activation might dissipate quickly, half of the participants completed the problem reconstruction task first and half completed the problem-solving task first. The order of problems within each assessment task was fixed (see Table

Problem reconstruction | 3 + 4 + 9 = 3 + __ |

Problem reconstruction | 5 + 8 + 7 = 5 + __ |

Problem reconstruction | 9 + 6 + 3 = 9 + __ |

Problem reconstruction | 6 + 7 + 3 = 6 + __ |

Problem solving | 4 + 3 + 6 = 4 + __ |

Problem solving | 3 + 9 + 5 = 3 + __ |

Problem solving | 7 + 5 + 4 = 7 + __ |

Problem solving | 8 + 4 + 6 = 8 + __ |

Problem solutions were coded as either correct or incorrect. The problem-solving test included 4 problems, so participants could receive total scores from 0–4. Additionally, incorrect solutions were further coded in terms of the specific strategy used, as inferred from the solution, based on a scheme developed by Perry et al. (

Add all | 17 | 37 |

Add to equal | 12 | 5 |

Idiosyncratic | 2 | 43 |

No response | NA | 15 |

Participants' reconstructions were coded using the system developed by McNeil and Alibali (

3 + 4 + 5 + 5 = __ 3 + 4 + 5 + 5 | Add all | Structure changed to standard addition problem, equal sign may be present or not | 50 |

3 + 4 + 5 = __ 3 + 4 + 5 | Add to equal | Structure changed such that the right side is omitted, equal sign may be present or not | 9 |

3 + 4 = __ | Add two | Structure is changed such that only the first two addends appear | 7 |

3 + 4 + 5 = 5 | No right plus, No blank | Reconstruction lacks a right plus sign and a blank | 15 |

Varied | Idiosyncratic | Reconstruction does not fall into another category | 19 |

We predicted that participants whose knowledge of the operational patterns was activated would encode equivalence problems less accurately than control participants. We tested this prediction using a Two-Way (condition x order) between-subjects ANOVA, with percentile SAT/ACT math score as a covariate. As predicted, participants in the knowledge activation condition correctly reconstructed fewer problems (_{(1, 110)} = 11.96, ^{2}_{p} = 0.098 (see Figure _{(1.110)} = 11.19, ^{2}_{p} = 0.09. No other types of reconstruction errors were affected by condition. There was also a significant effect of task order, _{(1, 110)} = 4.00, ^{2}_{p} = 0.04; participants who completed the encoding task first correctly reconstructed fewer equations (_{(1, 110)} = 11.43, ^{2}_{p} = 0.094. Overall, participants with higher scores tended to reconstruct more problems accurately.

We also predicted that participants in the knowledge activation condition would be worse at solving equivalence problems than control participants. We tested this prediction using a Two-Way (condition x order) between-subjects ANOVA, with percentile SAT/ACT math score as a covariate. As predicted, participants in the knowledge activation condition solved significantly fewer problems correctly (_{(1, 95)} = 6.25, ^{2}_{p} = 0.062 (see Figure _{(1, 95)} = 7.54, ^{2}_{p} = 0.07. No other incorrect strategy types were affected by condition. There was also a significant effect of task order, _{(1, 95)} = 23.91, ^{2}_{p} = 0.20; participants who completed the problem solving task first solved significantly fewer problems correctly (_{(1, 95)} = 10.16, ^{2}_{p} = 0.097. Overall, participants with higher SAT/ACT math scores tended to solve more problems correctly.

We next sought to test whether the relationship between knowledge activation and problem solving performance could be explained by decreased accuracy in problem encoding. To do so, we conducted a mediation analysis (Baron and Kenny,

In this experiment, we tested a hypothesized causal link between activation of the operational patterns and inaccurate encoding of mathematical equivalence problems. No previous studies have provided causal evidence for this connection. To obtain such evidence, we examined the effects of activating knowledge of the three operational patterns on undergraduate students' encoding of mathematical equivalence problems. We also examined effects on problem solving.

As predicted, activating knowledge of the operational patterns led participants to incorrectly reconstruct the equations, suggesting that being induced to think about the patterns affected what they noticed and encoded. Not only were participants in the knowledge activation condition more likely to make reconstruction errors, they were also more likely to encode the equivalence problems as if they were traditional arithmetic problems. These findings resemble the patterns found in children, in which knowledge of the operational patterns correlates with difficulties reconstructing equivalence problems (McNeil and Alibali,

The current findings also replicated previous work demonstrating that experience with the operational patterns hinders equivalence problem solving (McNeil and Alibali,

Our mediation analysis supports the claim that poor encoding is one mechanism by which exposure to the operational patterns negatively affects problem solving. Specifically, we found that the relation between condition and problem-solving performance was partially mediated by performance on the problem reconstruction task. Thus, knowledge activation affected problem solving, at least in part, by affecting encoding. However, it is worth emphasizing that the mediation we observed was partial, rather than complete; thus, there are likely other mechanisms at play as well.

In general, participants who performed better on standardized tests of mathematics were significantly better both at encoding and solving equivalence problems, regardless of condition. This finding was not surprising, as previous work with middle-school students has demonstrated that students who perform better on standardized tests of mathematics are more likely to have a relational understanding of the equal sign (Knuth et al.,

We also found that the effect of knowledge activation was fleeting—it was always strongest in the first task following the activation tasks. Thus, the order of the tasks affected participants' encoding and problem solving. Because the activation phase of the current study lasted only a few minutes, it is not surprising that the performance decrements began to fade quickly. It is also possible that the effects of knowledge activation might last different lengths of time for participants with different levels of mathematical knowledge (McNeil et al.,

It is worth noting that the present study utilized control tasks that were non-mathematical in nature (i.e., strings of letters for the perceptual control, concrete words for the conceptual control, and combining colors for the procedural control). In light of other work (e.g., McNeil et al.,

The current study suggests that activation of the operational arithmetic patterns may hinder problem solving by negatively affecting problem encoding. However, in the present study we could not assess the unique effects of each of the operational patterns on encoding. Prior knowledge could influence encoding in a number of ways, and it may be that the different operational patterns affect what participants notice in different ways. For example, the perceptual pattern of the “operations = answer” format may affect encoding because of the high level of perceptual similarity between traditional arithmetic problems and equivalence problems (e.g., McNeil and Alibali,

In considering the implications of this research, it is important to bear in mind that we studied adult participants in order to shed light on the mechanisms underlying a phenomenon typically found in children. We chose to study adults because we believe it would be unethical to purposely expose child participants to information that we expect to have negative effects on their learning. Work with adults can provide corroborating evidence for our hypotheses without potential harm to young students.

The current findings help build the case that knowledge of the operational patterns

We suggest that curricular activities and homework assignments that emphasize the operational patterns may lead children to activate knowledge of those patterns. Thus, it seems reasonable to argue that, in children's day-to-day mathematics experience, they often activate the operational patterns, and those patterns affect their encoding and solving of equations.

The present findings also highlight the importance of problem encoding in correctly solving equations. Thus, our data underscore previous work suggesting that guiding students to notice the appropriate features of mathematical inscriptions might foster their understanding (e.g., Lobato et al.,

Our findings also suggest that it may be wise to teach children arithmetic in ways that do not entrench the operational patterns. Recent work by McNeil et al. (

In sum, the current work adds to a growing body of literature emphasizing the effects of prior knowledge on later mathematical performance. In this study, we found that activating adults' knowledge of the three operational patterns common in arithmetic led to difficulties in their encoding and solving of equations. Moreover, poor encoding partially mediated the relationship between activation of the operational patterns and problem-solving performance, suggesting a mechanism by which knowledge activation might affect success in solving equivalence problems. Specifically, prior knowledge of the operational patterns affects equivalence problem solving by guiding solvers to encode the problems inaccurately. These findings help build the case that experiences with the operational patterns are a potential source of children's persistent difficulties with mathematical equivalence.

Noelle M. Crooks designed and conducted the experiment, coded and analyzed the data, and wrote the manuscript. Martha W. Alibali assisted in experimental design, analysis, and writing.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant #R305B090009 to the University of Wisconsin-Madison. The opinions expressed are those of the authors and do not represent views of the institute or the U.S. Department of Education. The authors thank Marc Hava, Lacaya Johnson, and Vinoadharen Nair Das for their help with data collection and coding. We also thank Adam Riggall for technical support.

Target: 375 + 695 = ___ Sample: 365 + 694 = ___

Target: XxxxX Sample: XcccX

Target: Total Sample: Sum

Target: Apple Sample: Banana

Target: 16 Sample: 8 and 7

Target: Orange Sample: Yellow and blue

^{1}Note that, while each of the patterns includes perceptual, conceptual, and procedural elements, their names reflect the primary components of each pattern.