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Algebraic thinking and strategy flexibility are essential to advanced mathematical thinking. Early algebra instruction uses ‘missing-operand’ problems (e.g., x – 7 = 2) solvable via two typical strategies: 1) direct retrieval of arithmetic facts (e.g., 9 – 7 = 2) and 2) performance of the inverse operation (e.g., 2 + 7 = 9). The current study investigated the strategies people choose when solving these problems, and whether some people are more flexible in their choices than others. U.S. undergraduates (n = 59) solved missing-operand problems and made speeded verifications of arithmetic sentences corresponding to the direct- and inverse-matched facts. To ‘decode’ their strategy as direct or inverse, each participant’s response times (RTs) for missing-operand problems were regressed on their RTs for the corresponding direct and inverse facts. Our findings replicated the problem size effect for the arithmetic verification task and extended this effect to missing-operand (i.e., one-step) algebra problems, suggesting that the two tasks draw on common representations and processes in the addition (but not subtraction) context. We found individual differences in strategy choice and flexibility such that participants varied both in terms of fluency for retrieving the direct fact and sensitivity to the potential benefit of switching to the inverse fact, which was validated by self-report. We did not find a predicted relation between strategy flexibility and standardized mathematical achievement. These findings inform our understanding of the cognitive processes involved in strategy flexibility in algebra and establish an RT-decoding paradigm for future examination of individual differences in students’ learning of early algebra concepts.

Success in algebra depends in large part on

Early algebra instruction formalizes ‘open number sentences’ from arithmetic (e.g., [ ] – 7 = 2;

The direct strategy involves bottom-up pattern-matching from the algebraic equation to its arithmetic fact, while the inverse strategy requires an additional top-down symbol-manipulation step before retrieval (i.e., transformation or re-representation;

Essential to a deep understanding of mathematics is the ability to conceptually understand and choose appropriately between multiple strategies (

Much of the research on strategy flexibility in algebra has primarily used more complex problems involving multiple steps and multiple variables. However, success in solving simple algebra problems is foundational for and predictive of learning to solve more complex problems (

The current study tested a new approach to ‘decoding’ strategy flexibility in simple algebra problems (i.e., one-step missing-operand problems) using participants’ response times. It also investigated individual differences in use of the direct vs. inverse strategy for solving missing-operand problems, as well as the potential association between strategy flexibility and mathematical achievement. As an antecedent step, it investigated temporal profiles of the retrieval of arithmetic facts during the solving of missing-operand problems. Accordingly, our hypotheses derived from the arithmetic problem solving literature.

Previous research has revealed reasoners’ cognitive processes in arithmetic (

Reliance on direct retrieval to solve arithmetic problems is a common strategy even for young children (

Earlier research has shown that young children struggle with open number sentences (e.g., [ ] – 3 = 5;

Finally, research has demonstrated that when individuals switch between strategies across problems, they may incur a ‘switch cost’ (

These findings provide an arithmetic basis on which to investigate strategy flexibility in simple algebraic problem solving. Given the importance of simple algebra problems and the foundational nature of arithmetic to algebra, we sought to assess whether individual participants’ arithmetic and algebraic problem-solving patterns are related for single-step missing-operand problems, and if so, whether an individual’s unique patterns could be used to ‘decode’ their strategies for missing-operand problems. The current study does this through addressing five research questions.

The first and second research questions concern whether arithmetic and algebraic problem solving utilize common mental representations and processes. After ensuring replication of the problem size effect on the arithmetic verification task, the first research question extends the problem size effect to one-step algebra problems: Is there a problem size effect on the missing-operand task? The second research question connects arithmetic and algebraic problem solving through an individual differences analysis: Is an individual’s problem size effect on the arithmetic verification task positively correlated with their problem size effect on the missing-operand task? Given the centrality of the problem size effect to the representations involved in arithmetic thinking, extending the problem size effect to one-step algebra problems and finding correlations between the effect in arithmetic and algebra tasks would suggest that participants may repurpose arithmetic facts for algebraic problem solving.

The third research question addresses the arithmetic-algebra relation at the problem level to measure individuals’ flexibility in algebraic problem solving. Recall that there are two primary ways to solve missing-operand problems: by retrieving the direct fact or by transforming and retrieving the inverse fact. We predict that people will differ in strategy use as a function of their fluency for the direct fact and their ability to flexibly switch to the inverse retrieval when it is beneficial. To assess individual variation in flexibility, we formalize a

The fourth research question concerns the validity of this switch benefit operationalization. We assessed this question by comparing individuals’ ‘switch benefit’ estimates from the regression model to their explicit strategy self-reports collected at the end of the study.

The fifth research question concerns the relation between strategy selection, flexibility, and mathematical achievement. If flexibility is important for mathematical thinking more generally, then is it the case that sensitivity to switch benefit when solving missing-operand problems is associated with higher mathematical achievement more generally (as measured by standardized ACT and SAT scores)? We predicted that sensitivity to switch benefit would be positively associated with mathematical achievement.

To address these research questions, we created an experimental paradigm to ‘decode’ participants’ strategy choice flexibility from their response times (RTs). In this paradigm, participants make speeded judgements in the missing-operand task of interest (e.g.,

We recruited an initial sample of 62 of an intended 80 undergraduate students at a university in the Midwest U.S., before the onset of COVID-19 brought an end to face-to-face data collection. Because the experimental paradigm is new, we could not rely on the literature to estimate the number of participants and trials. We were instead guided by a pilot study in our lab of 64 participants experiencing 224 trials across the two tasks, which detected notable individual variation between-participants. Although we were unable to increase the sample size of the current study for external reasons beyond our control, we almost doubled the number of trials across the two tasks, to 432. (This is the number that they could complete in approximately 40 minutes before beginning to experience fatigue.) Given that most of our analyses are

Participant ages ranged from 18 to 23 years old (

The experiment was implemented using PsychoPy (

Participants were given one-step algebra problems and asked to identify the unknown (e.g.,

Participants were shown arithmetic sentences, which they had to judge as ‘true’ or ‘false’ by pressing the corresponding key as quickly and accurately as possible. The stimuli consisted of 129 true trials such as 9 – 7 = 2 and 127 false trials such as 9 – 7 = 4, which appeared once in one of two blocks.^{1}

The intended design was 128 true and 128 false trials, but due to an error in the stimuli file discovered in analysis, the true trial 2 + 9 = 11 was presented a second time instead of its corresponding false trial, 2 + 9 = 13. For data analysis, we used the RT of the intended true trial.

Trial order was randomly shuffled for each participant with a break inserted halfway. The true trials were derived from the 88 missing-operand problems that participants also solved, e.g., for^{2}

There were two necessary deviations. For false subtraction trials with a correct difference of 1 or 2 (e.g., 4 – 3), the results were always increased by 2 to prevent the introduction of negative results. For false addition trials with 2 as an operand (e.g., 2 + 7), the results were always increased by 2 (e.g., to 11) to prevent the sum from being equal to the second operand, in which case the statement could trivially be seen to be false by using surface features alone (e.g., 2 + 7 = 7).

Participants’ scores on the math section of the ACT or SAT tests were obtained with consent from university records. The ACT and SAT are comprehensive standardized tests used in U.S. university admissions and are used as estimates of mathematical achievement. The mathematics subtests of these exams include topics such as early algebra (linear equations and systems-of-equations), polynomial equations, probability and statistics, geometry, and trigonometry.

For each task, participants first completed eight practice trials with feedback. The experimental trials that followed were divided into two blocks with a break halfway. The order of the two tasks was counterbalanced across participants. After finishing both tasks, participants completed a strategy questionnaire about how they solved the missing-operand problems and what percentage of the time they utilized each strategy type (i.e., direct vs. inverse). Lastly, they completed a demographics form.

The analysis script, data files, codebooks, and survey are provided as

Before answering our first research question, we assessed whether we replicated the problem size effect for the true trials in the arithmetic verification task. Each participant’s mean RT for each problem size and operation type (addition and subtraction) are visualized in ^{3}

For this and subsequent correlations, Kendall’s

The first research question was whether the problem size effect extended to the missing-operand problems (_{10} = 5.5 × 10^{13}). For the subtraction problems, participants were almost evenly split between positive and negative correlation coefficients (individual _{01} = 6.211, indicating the observed correlations are 6 times more likely under a null model. Thus, the problem size effect was extended from arithmetic problems to missing-operand problems, but only for the addition context. This provides initial evidence for some overlap in the mental representations and processes recruited for arithmetic, specifically addition, and simple algebra. Notably, we observe a dip in the mean RT for addition facts with a problem size of 10 (

The second research question concerned whether overlapping mental representations and processes are recruited for arithmetic and algebra. We addressed this question using an individual differences approach. Specifically, individuals varied in the strength of their problem size effect in both arithmetic and simple algebra contexts (^{2} = .23. A Bayesian linear regression indicated this association was more likely under the model where problem size effects are correlated across task than uncorrelated (_{10} = 191.1). This is consistent with the recruitment of the same mental representations and processes for the two problem classes. For the subtraction context, there was no such association, ^{2} = .04. Here, the Bayes factor was _{01} = 1.34, indicating the observed correlations are slightly more likely under a null model, i.e., weak support for the absence of this correlation.

The third research question asks whether participants are fluent and/or flexible across problems in their strategy selection during the missing-operand task. Can we ‘decode’ whether participants are sensitive to the benefit of switching to the inverse strategy when solving missing-operand problems? For this analysis, multiple linear regressions were run for each participant individually, using all cases where the missing-operand problem and the direct and inverse arithmetic facts were all three answered correctly and within the time cutoffs.

We operationalized the ‘switch benefit’ for missing-operand problems as the ratio of an individual’s speed in retrieving the direct vs. inverse arithmetic facts during the arithmetic verification task:

This switch benefit ratio reflects the degree to which switching from direct retrieval to the inverse strategy would

Thus, we regressed each participant’s RTs for missing-operand problems on their switch benefit term, and also on their RT for the directly matched arithmetic fact. The direct match RT operationalizes the participant’s retrieval fluency. We also included block number (1 or 2) as a predictor to control for practice effects. The full equation is:

Given the full regression model, the switch benefit ratio predictor thus measures strategy flexibility

The regression analyses indirectly estimated the switch benefit for each participant from their RTs. The fourth research question sought converging evidence for these analyses. Do the resulting estimates correspond to participants’ self-reports about their strategy usage? We evaluated this by correlating the

The fifth research question asked whether strategy flexibility on missing-operand problems was associated with higher mathematical achievement more generally as measured by college entrance exam scores. We obtained standardized test scores for 58 of the 59 participants, 50 for the ACT college entrance exam and 8 for the SAT; the latter were converted to equivalent ACT scores using a concordance table.^{4}

We used the 2018 ACT/SAT Concordance Tables published by ACT, Inc. We judged 2018 to be the year that most participants in our sample likely took the ACT, i.e., the Spring of their junior year and the Fall of their senior year in high school. The tables are available for download:

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The current study investigated the strategies that undergraduate students use when solving simple algebra (missing-operand) problems. We developed an RT-decoding paradigm to infer whether a person uses bottom-up pattern-matching to retrieve a

More specifically, we examined five research questions. First, we extended the problem size effect from arithmetic verification to missing-operand problems, but only for addition problems; participants did not consistently show this effect for subtraction problems. Second, we conducted an individual differences analysis of whether participants’ problem size effect for missing-operand problems predicted their problem size effect for arithmetic problems. We found a significant relationship for addition, suggesting that missing-operand problems in addition recruit similar mental representations and processes to arithmetic verification. There was no such relationship for subtraction. Third, we ‘decoded’ individual differences in missing-operand RTs through individual-level regression models, predicting missing-operand performance by an individual’s RTs for both the direct arithmetic fact and the ‘switch benefit’ ratio of direct vs. inverse facts. We found an overall trend where higher direct fact fluency was associated with higher sensitivity to the switch benefit, suggesting higher strategy flexibility. Fourth, we validated the RT-based switch benefit measures by participants’ self-reported use of the inverse strategy. Fifth, we failed to find a predicted relationship between our ‘decoded’ strategy flexibility variable (i.e., switch benefit) and standardized mathematics scores.

The predicted extension of the problem size effect to missing-operand problems (our first research question) and the correlation between problem size effects across arithmetic problems and missing-operand problems (our second research question) were only found for addition problems. The mathematical cognition literature offers multiple possible explanations for this asymmetry. It may be that subtraction facts are less fluent for direct recall compared to addition facts, perhaps due to less practice (

Of primary theoretical importance are the individual-level regression models that implicitly ‘decode’ participants’ strategies during the missing-operand task. Prior work has shown individual differences in students’ (i.e., children’s) strategy flexibility (

These ‘decoding’ results were validated by explicit self-report. While self-report of strategies can be informative (

Future studies should investigate performance characteristics of the direct and inverse strategies for missing-operand problems to further assess the validity of this study’s paradigm. One way to do so is by varying the instructions given to participants about which strategy to use when solving problems or whether they can choose (i.e. choice/no-choice); another is by varying problem characteristics that are more or less conducive to certain strategies. Such work has already improved our understanding of strategy choice in arithmetic problems (

This study built on prior work emphasizing the importance of strategy flexibility to student success in algebra (

For this article, a data set is freely available (

The Supplementary Materials contain the following items (for access see

Experiment script and stimuli

Data files

Codebooks

Analysis script

Survey

The authors have no funding to report.

The authors have declared that no competing interests exist.

The authors thank Kori Terkelsen, Em Hayward, and Corissa Rohloff for their assistance running participants.