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This study explores student flexibility in mathematics by examining the relationship between accuracy and strategy use for solving arithmetic and algebra problems. Core to procedural flexibility is the ability to select and accurately execute the most appropriate strategy for a given problem. Yet the relationship between strategy selection and accurate execution is nuanced and poorly understood. In this paper, this relationship was examined in the context of an assessment where students were asked to complete the same problem twice using different approaches. In particular, we explored (a) the extent to which students were more accurate when selecting standard or better-than-standard strategies, (b) whether this accuracy-strategy use relationship differed depending on whether the student solved a problem for the first time or the second time, and (c) the extent to which students were more accurate when solving algebraic versus arithmetic problems. Our results indicate significant associations between accuracy and all of these aspects— we found differences in accuracy based on strategy, problem type, and a significant interaction effect between strategy and assessment part. These findings have important implications both for researchers investigating procedural flexibility as well as secondary mathematics educators who seek to promote this capacity among their students.

Promoting the development of students’ problem-solving abilities, particularly procedural flexibility, is one important aim in mathematics educational practice and policy (

Among the many dimensions along which problem-solving strategies can differ are two that are of particular interest in this study: strategy appropriateness and strategy accuracy. Strategy appropriateness lies at the core of procedural flexibility (

Among other possible strategies, some are arguably better than the standard algorithm, where better (or “situationally appropriate”;

But is there a general relationship between strategy appropriateness and strategy accuracy? In other words, which strategies tend to be more accurately implemented by students, those that are standard algorithms or those that are more situationally appropriate? On the one hand, an argument can be made that a standard approach tends to be the strategy most related to accuracy. Standard approaches are by definition broadly applicable to a wide range of problems. As a result, such algorithms can be automatically executed without a great deal of attention to the specific structural features of a problem. Such routine execution of standard algorithms can be efficient and reduce the likelihood of error. Prior research has suggested a “freed resources” mechanism behind this potential benefit of standard algorithms, where highly routinized strategies enable students to focus on the relationships and operations in a problem with greater facility (

But on the other hand, one might hypothesize that strategies that are more appropriate or better than the standard algorithm would result in greater accuracy. Situationally appropriate strategies take advantage of structural features of problems and tend to have fewer operations, and this may reduce the opportunity and likelihood for errors. When students opt for a strategy that departs from a highly routinized, standard approach, they may engage more carefully in on-the-spot encoding of problem features (

As an alternative to either of these hypotheses, it may instead be the case that a general relationship does not exist between strategy appropriateness and strategy accuracy – but rather, that this relationship is an interaction related to the structural features of a given problem. In particular, for problems where it is relatively straightforward to identify an alternative, better strategy (such as in the specific linear equation and integer addition examples previously shown), perhaps the use of the more situationally appropriate strategy leads to greater accuracy, for the reasons noted above. As another example, consider the fraction addition problem 18/36 + 21/42 (

An additional consideration with regard to the relationship between strategy appropriateness and strategy accuracy concerns the particulars of how these constructs are assessed. Specifically, in prior studies of procedural flexibility (e.g.,

Finally, it is possible that any relationships that exist between strategy appropriateness and strategy accuracy vary across mathematical domains. Procedural flexibility has been frequently studied in linear equation solving (e.g.,

With respect to mathematical domain differences in the relationship between strategy appropriateness and accuracy, of particular interest here is whether such differences exist between arithmetic problems and algebraic problems. Students employ myriad strategies for computing in arithmetic, including both formal algorithms and informal strategies. With respect to the latter, the richness and variety of children’s informal strategies for solving arithmetic problems has been well-documented in the literature (e.g.,

In sum, there is considerable potential nuance in the relationship between strategy appropriateness and strategy accuracy within the context of procedural flexibility. Although there has been an increase in interest in and research on procedural flexibility over the past decade, little is known about this relationship.

Increasing our knowledge base about the relationship between strategy appropriateness and strategy accuracy is important to the field for the following reasons. First, scholars, educators, and policymakers have focused on increasing students’ ability to employ multiple strategies flexibly in problem solving; however, flexibility itself absent the dimension of accuracy in problem solving seems a less desirable outcome. Arguments in favor of flexibility as an instructional goal would be substantially advanced if there were clearer links between flexibility and accuracy, including relationships between strategy appropriateness and accuracy. Attending to the dimension of accuracy in efforts to advance flexibility in practice would better align with broader goals for improving students’ mathematical proficiency and conceptual understanding.

Second, understanding more about the relationship between strategy appropriateness and accuracy also relates directly to debates in mathematics education about the role of standard algorithms in the curriculum (e.g.,

The relationship between strategy appropriateness and strategy accuracy is the focus of the present study. We explore this relationship in the context of algebra equation solving problems and in arithmetic problems, as well as in a task that prompted students to solve problems in different ways. While prior literature has demonstrated the variability in individual students’ strategy choices on the same items and across occasions (e.g.,

A convenience sample of 450 high school students from 19 math classes in a single large high school in the Southeastern region of the United States participated in this study. We reduced the sample to the 449 students who completed the assessment. An additional 36 students who were missing demographic data were also excluded, leaving

Student Demographics | % |
---|---|

Gender | |

Girls | 47.2% |

Boys | 52.5% |

Non-Binary | < 1.0% |

Grade in School | |

9th | 53.3% |

10th | 26.2% |

11th | 13.6% |

12th | 7.0% |

Self-reported Math Grade | |

A | 41.2% |

B | 32.7% |

C | 17.7% |

D | 6.5% |

F | 1.9% |

Age | |

13 | 0.2% |

14 | 27.6% |

15 | 39.5% |

16 | 19.9% |

17 | 9.9% |

18 | 2.9% |

Participants completed a two-part assessment. In Part 1, students were prompted to complete five problems (see

Accuracy | Problem | Standard Approach | Better-than-Standard Approach |
---|---|---|---|

1 | Simplify: |
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2 | Solve: |
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3 | Solve: |
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4 | Simplify: |
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5 | Simplify: |

Problems 2 and 3 represent algebra problems, and Problems 1, 4 and 5 represent arithmetic problems. Our measure of the algebra domain consisted of both replications of Items 2 and 3 across the two assessment parts, totaling 4 items (α = .60). Similarly, our measure of the arithmetic domain consisted of both replications of Items 1, 4 and 5 across the two assessment parts, totaling 6 items (α = .60).

We coded student-problems for both accuracy and type of strategy; of particular interest here is the distinction between standard and better-than-standard strategies. Two coders independently coded all strategies for strategy type and accuracy and subsequently resolved all disagreements. We elaborate on these types of strategies below and provide student examples in

The standard approach for Problem 1 involved putting all the fractions in the same form over the common denominator of 9 to get 15/9, 5/9, 3/9, and 4/9, adding the numerators to get 27/9, and finally simplifying to get 3 (see

Student-problems were coded as better-than-standard if their approach demonstrated more elegance and innovation than the standard approach, based on similar determinations in prior studies (e.g.,

Because our research questions were concerned with differences in accuracy between standard and better-than-standard approaches, we restricted the analysis sample to student-problems that used either of these two types of strategies across both Parts 1 and 2. Given the well-established relationship between worse-than-standard strategies and inaccuracy, we were only interested in examining problem solving accuracy between standard or better strategies (e.g.,

Part 1 | Part 2 | Total (%) |
---|---|---|

Below Standard | Below Standard | 296 (14.3) |

Below Standard | Standard | 47 (2.3) |

Below Standard | Above Standard | 53 (2.6) |

Standard | Below Standard | 520 (25.2) |

271 (13.1) | ||

487 (23.6) | ||

Above Standard | Below Standard | 107 (5.2) |

160 (7.7) | ||

124 (6.0) | ||

2,065 |

To begin investigating differences in accuracy between standard and better-than-standard approaches (RQ1), we present descriptive statistics for the accuracy rate by strategy use in student-problems. To begin to answer our second research question (R2) concerning how the relationship between strategy selection and accuracy might differ depending on assessment part, we present the accuracy rate for each strategy within Parts 1 and 2 of the assessment separately. We also show patterns of strategy use and accuracy across the problems in the first and second attempt to contextualize student-problems in the two-part assessment task. We present similar descriptives for the accuracy rate by problem type to begin answering our third research question (RQ3) concerning differences in problem solving success on algebra problems compared to arithmetic ones.

We then fit a multi-level logistic regression model, with the dichotomous outcome variable corresponding to whether students’ answer to each problem was correct or incorrect (see

where

To answer our research questions about how accuracy may relate to strategy selection (RQ1) and problem type (RQ3), we included three dichotomous level one variables in the model: the strategy employed (standard or better-than-standard), assessment part (Part 1 or Part 2), and problem type (arithmetic or algebra). To fully answer RQ2, we modeled the interaction between the strategy used and assessment part to determine if the effect of strategy on accuracy depended upon whether the student was on Part 1 or Part 2 of the exam, in which students were asked to re-solve the same problem a second time using a different strategy from the one employed in Part 1. Finally, we conducted post-hoc tests to test for differences in the likelihood of an accurate response. All computing was completed using Stata version 17.0.

Accuracy | Strategy Type |
Problem Type |
Assessment Part |
|||||||
---|---|---|---|---|---|---|---|---|---|---|

Analysis Sample |
Part 1 Only |
Part 2 Only |
Analysis Sample |
Analysis Sample |
||||||

Stand. | Above-Stand. | Stand. | Above-Stand. | Stand. | Above-Stand. | Arithmetic Problems | Algebra Problems | Part 1 | Part 2 | |

Incorrect | 230 (19.3%) | 231 (25.8%) | 115 (15.2%) | 86 (30.3%) | 115 (26.7%) | 145 (23.7%) | 275 (24.9%) | 186 (19.0%) | 201 (19.3%) | 260 (25.0%) |

Correct | 959 (80.7%) | 664 (74.2%) | 643 (84.8%) | 198 (69.7%) | 316 (73.3%) | 466 (76.3%) | 831 (75.1%) | 792 (81.0%) | 841 (80.7%) | 782 (75.1%) |

Total | 1,189 | 895 | 758 | 284 | 431 | 611 | 1,106 | 978 | 1,042 | 1,042 |

However, we observed notable differences in accuracy by assessment part. In Part 1, the accuracy rate for student-problems that used standard approaches was 84.8%, compared to 69.7% for better-than-standard approaches. But in Part 2, the difference between the accuracy obtained between both approaches reverses, with standard and better-than-standard strategies demonstrating accuracy rates of 73.3% and 76.3%, respectively.

With regard to differences in accuracy by assessment part alone, we found that 80.7% of student-problems in Part 1 were completed correctly, compared to 75.1% in Part 2. Further, when it came to problem type, 75.1% of arithmetic student-problems were completed correctly, compared to 81.0% of algebraic ones. Students appeared to be more successful on algebra problems compared to arithmetic ones.

Strategy Across Parts 1 and 2 |
Accuracy Across Parts 1 and 2 |
Total | ||||
---|---|---|---|---|---|---|

Part 1 | Part 2 | Both Correct | Part 1 Correct Only | Part 2 Correct Only | Both Incorrect | |

Standard | Standard | 166 (61.3%) | 40 (14.8%) | 21 (7.7%) | 44 (16.2%) | |

Standard | Above Standard | 364 (74.7%) | 73 (15.0%) | 27 (5.5%) | 23 (4.7%) | |

Above Standard | Above Standard | 56 (45.2%) | 10 (8.1%) | 19 (15.3%) | 39 (31.5%) | |

Above Standard | Standard | 117 (73.1%) | 15 (9.4%) | 12 (7.5%) | 16 (10.0%) | |

703 | 138 | 79 | 122 |

The results from our mixed-effects logistic regression analysis elaborate on these findings. We begin with our third research question concerning differences in accuracy by problem type, which factors into our discussion of the findings for our first two research questions. We found further support for the finding that responses on algebra problems were more accurate than responses for arithmetic problems, even when controlling for assessment part and strategy type. Algebra problems had an estimated 0.34-logit greater likelihood of a correct response compared to arithmetic problems,

Predictor | 95% CI |
|||
---|---|---|---|---|

(Intercept) | 1.89*** | 0.20 | 1.49 | 2.29 |

Algebra | 0.34* | 0.13 | .08 | .60 |

Above-Standard Approach | -0.90*** | 0.20 | -1.30 | -.51 |

Part 2 of Assessment | -0.70*** | 0.17 | -1.03 | -.36 |

Above-Standard Approach*Part 2 | 0.89** | 0.26 | .38 | 1.40 |

*

With regard to our first research question concerning differences in accuracy by strategy type and our second research question investigating how this relationship may differ by assessment part, we found further evidence in support of the interaction between strategy use and whether a student was completing a problem for the first or second time. In Part 1 of the assessment, the standard approach was related to a greater likelihood of an accurate response compared to the above-standard approach,

*

However, this relationship between strategy use and accuracy changes in Part 2, as evidenced by the significant interaction term between strategy and part,

Examining the effectiveness of each strategy type across the two attempts, the standard approach was significantly more likely to yield a correct response in Part 1 compared to Part 2, but this is not the case for the above-standard approach, which demonstrated equal chances of success in both Parts 1 and 2. The standard approach had an estimated 10% greater chance of success in Part 1 (87%) compared to Part 2 (77%) for arithmetic problems and an estimated 8% greater chance of success in Part 1 (90%) compared to Part 2 (82%) for algebra problems,

In the present study, we investigated differences in the accuracy achieved with standard and better-than-standard strategies and the extent to which accuracy differed by whether a student was solving for the first or second time and whether the problem was arithmetic or algebraic. We found that there is a relationship between strategy use and accuracy, and that the relationship depends on whether students are doing a problem for the first or second time as well as on the problem domain. The standard approach was related to greater success in problem solving only when a student was solving a problem for the first time; when asked to solve the same problems a second time using a different strategy, the standard approach was no more accurate than better-than-standard approaches. Further, examining the patterns of strategy use across the problems in the first and second attempt showed a majority of student-problems beginning with the standard approach in Part 1 followed by an above-standard approach in Part 2, with the proportion answered correctly both times being 74.7%. Students’ flexible strategy use appears to show the dominance of the standard approach as the primary strategy of choice. A small proportion of student-questions began with the above-standard approach as the primary choice of strategy in Part 1 followed by a standard strategy in Part 2, with the proportion answered correctly both times comparable at 73.1%.

For students’ primary strategy selection in Part 1, the standard approach may be associated with a higher accuracy rate than the better-than-standard strategy for reasons that may seem intuitive: this approach is the more common, reliable, and routine way of solving a problem. The success of the standard approach here may be attributable to the “freed resources account” for student attention on problems, which posits that highly routinized and well-practiced strategies require fewer cognitive resources from students, “freeing up” their ability attend to the key features and relationships in the problem (

However, when students were asked to go beyond their primary strategy of choice, as our assessment prompted them to do in Part 2, both strategy types were equally related to accuracy. To add to this, the standard approach was significantly more successful in Part 1 compared to Part 2, but better-than-standard approaches were equally successfully across the two exam parts. It could be the case that the application of above-standard approaches is a robust indicator of greater flexibility and conceptual understanding, regardless of whether this type of approach is a student’s primary or secondary strategy choice. Recognizing the structural features in a problem is related to flexibility and conceptual knowledge (e.g.,

Our results point to the value in cultivating flexibility in problem solving for learners. The vast majority of students in our sample who correctly solved a problem twice used some combination of standard and above-standard approaches across the two parts (see

While our findings more generally indicated that the success of certain strategies depends on whether a student is solving a problem for the first or second time, we observed notable differences in this relationship depending on the problem domain. We found significant differences in accuracy across algebraic and arithmetic student-problems, even controlling for assessment part and strategy type. It is possible this difference in accuracy is due to structural differences between the two problem types. The multi-step equations found in algebra domains may reduce the cognitive burdens of problem solving, given their predictable and less varied nature compared to arithmetic problems, with common patterns of distributing coefficients, combining like terms, and isolating variables. It seems reasonable for students to have common templates for solving, given the common format and structural features found in such equations.

It could also be the case that equations that are “flexible-eligible” (

A limitation of the present study is the use of student-problems as the unit of analysis. Future studies could look across assessment questions for the same student and examine each student’s strategy selection and accuracy conditionally on how the student solved a problem for the first time, including those that used a below-standard strategy. Given that we excluded 2,046 student-problems (or 1,023 student-questions) that showed a below-standard strategy in at least one part of the assessment, our study does not generalize to flexible problem-solving situations in which a student uses a standard or better strategy in combination with a below-standard strategy; our results can only speak to the relationship between flexible strategy use and accuracy for students using standard or better strategies. We recommend future work that examines the relationship between below-standard approaches, procedural flexibility, and conceptual understanding.

Future work investigating flexible strategy use might also adapt the choice/no-choice method, which compares problem solving latencies or times-to-solution between a method of choice and a prescribed method within individual students (

In addition, further work could be done to explore more of the variation in strategy selection and accuracy within each problem type. For example, the two algebra equations we used, while typical in secondary mathematics and algebra curricula, do not capture the full array of algebraic problem features students encounter. Similarly, the arithmetic problems in our assessment are primarily concerned with fraction and integer operations. Our results may have been influenced more by the specifics of these problem features rather than arithmetic problems more generally, limiting the generalizability of our findings to these problem domains. Future studies exploring the relationship between accuracy and strategy choice in different problem domains may wish to better account for general characteristics of problems in the mathematical domains of interest as well as to increase the number of items and thus reliability. Similar to problem domain, future studies may wish to investigate the relationship between strategy selection and accuracy on word problems in mathematics. Prior work has shown that even expert mathematicians struggle to apply simple arithmetic procedures to word problems when the problem presents semantic content that is incongruent with the arithmetic solving procedure (

Another limitation of the current study is the limited age and grade range of students in the sample (86.9% of students in the sample were between the ages of 14 and 16 years old, and 79.5% were in grades 9 or 10), precluding us from examining the effect of student age, grade, and by proxy grade-based curriculum, on problem-solving success. We recommend a thorough examination of situational variables related to the learner, taking into account student characteristics such as age, grade, math placement, and math performance as they relate to procedural flexibility. Related to situational variables, our sample comes from one specific high school, limiting the generalizability of our findings— future studies on the relationship between flexible strategy use and accuracy may wish to vary the school site and examine contextual factors related to the phenomenon. Finally, more qualitative research on students’ encoding of flexible-eligible problems and how a student decides which strategy to apply is needed to better understand students’ rationale for which strategies they employ. For example, secondary school students in Spain tend to prefer standard algorithms and approaches (

Our findings have implications for mathematics educators seeking to promote procedural flexibility in their classrooms. There is potential value in using this kind of task (in which students are prompted to re-solve a previously completed problem) for two reasons. First, our results are consistent with prior calls for the inclusion of this type of task, both as a student learning task as well as an assessment task (e.g.,

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The authors have declared that no competing interests exist.

The authors have no additional (i.e., non-financial) support to report.