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Central elements of adaptive expertise in arithmetic problem solving are flexibility, using multiple strategies, and adaptivity, selecting the optimal strategy. Research shows that the strategies children actually use do not fully reflect the strategies they know: there is hidden potential. In the current study a sample of 147 third graders from the Netherlands completed a comprehensive assessment of adaptive expertise in the domain of multidigit subtraction, designed to measure, first, the strategies students know and use to solve subtraction problems (potential and practical flexibility). Second, it measured to what extent students know which strategy is optimal and to what extent they use the optimal strategy (potential and practical adaptivity). Findings for flexibility showed that most students consistently used the same strategy across all problems: practical flexibility was low. When prompted, students knew more strategies than they used spontaneously, suggesting hidden potential in flexibility. Findings for adaptivity showed that students hardly ever spontaneously used the task-specific strategy that is efficient for specific problems since it has the fewest and easiest steps. However, almost half of the students could select this strategy from a set of given strategies at least once. Furthermore, an innovative, personalized version of the choice/no-choice method showed that the task-specific strategy was usually not the optimal strategy (fastest strategy leading to a correct answer) for individual students. Finally, students used the strategy with which they performed best more often than the other strategies, but there is hidden potential for the adaptive use of task-specific strategies.

There are various ways to solve mathematics problems such as 84 – 67 = ?, for instance adding-on from 67, or sequentially subtracting 60 and 7 (

Importantly, flexibility research shows that what children do on a task, their actual strategy use, does not fully reflect what they know (

There is near consensus on the strategies that can be used to solve multi-digit subtraction problems, see

Strategy | Steps in solving 84 – 67 = |
---|---|

Jump | 84 – 60 = 24; 24 – 7 = 17 |

Split | 80 – 60 = 20; 4 – 7 = 3 short; 20 – 3 = 17 |

Compensation | 84 – 70 = 14; 14 + 3 = 17 |

Indirect Addition | 67 + 3 = 70; 70 + 14 = 84; 3 + 14 = 17 |

Jump and split strategies are universal strategies that can be used on all types of problems (

Compensation and indirect addition are task-specific strategies (also called ‘varying’ strategies), since they are efficient on specific problems (

Adaptive expertise concerns the ability to apply meaningfully learned procedures flexibly and creatively. It contrasts with routine expertise: simply being able to complete school mathematics exercises quickly and accurately without understanding (

Regarding flexibility and task-based adaptivity, research usually shows rather disappointing results: students tend to quite consistently use the same strategy across problems and efficient task-specific strategies, also called shortcut strategies, are rarely used (

Therefore, we propose a comprehensive conceptual model of adaptive expertise with six dimensions.

To investigate which strategy is optimal for an individual, the choice/no-choice methodology has been developed (

A major drawback, however, is that it has to be decided

There are several student factors that have been found to relate to aspects of adaptive expertise in arithmetic. One is students’ mathematics ability. It is a common assumption that the acquisition of strategy flexibility is more difficult for lower achieving students (

Besides mathematical knowledge, domain-general cognitive resources are likely to be related to students’ adaptive expertise. Recent meta-analyses showed that working memory, the ability to simultaneously store and process information (

Most of these studies in which the relation between student factors and students’ strategy use was investigated focused on practical flexibility and practical task-based adaptivity only, and findings were at times inconsistent. The current study therefore aims to broaden the scope and the insights on the relation between student factors and students’ adaptive expertise, by investigating the role of mathematics achievement level, adaptive number knowledge, working memory, and gender in the practical and potential components of the three constructs of adaptive expertise.

The current study aimed to not only chart the practical adaptive expertise that third graders show in the domain of multi-digit subtraction, but also to investigate their (hidden) potential. For each of the three constructs of adaptive expertise – flexibility, task-based adaptivity, and individual-based adaptivity – two measures were created: the first addressing the practical component (what students show) and the second addressing the potential component (what students know). The study was guided by four research questions:

To what extent do students use different strategies (practical flexibility) and to what extent do they know different strategies (potential flexibility)?

To what extent do students use strategies that are optimal for the task (practical task-based adaptivity), and to what extent do they know which strategy is optimal for the task (potential task-based adaptivity)?

To what extent do students use strategies that are optimal for themselves (practical individual-based adaptivity), and to what extent do they know which strategy is optimal for themselves (potential individual-based adaptivity)?

How are the six constructs of the adaptive expertise framework related to students’ mathematics achievement level, adaptive number knowledge, working memory capacity, and gender?

Participants were 147 third graders (49.7% boys) from 12 primary schools from the Netherlands. The research protocol was approved by the Institute’s IRB [number ECPW2019-242] and only children with written parental consent and individual consent participated. As an indicator of mathematics achievement level, we collected the students’ most recent score on the standardized mathematics subtest of CITO’s student monitoring system, which was usually the end of grade 2 assessment (

The arithmetic task consisted of four tasks, administered in two sessions. Tasks 1-3 were administered in classroom setting in Session 1 in maximally 60 minutes, Task 4 was administered individually in Session 2 in maximally 20 minutes.

All tasks consisted of subtraction problems up to 100. We distinguished four problem types, based on the number characteristics that fitted one of the four main strategies jump, split, compensation and indirect addition. Jump problems had a large difference (> 40) and involved crossing tens, e.g., 63 – 17. Split problems had a large difference but did not require crossing tens, e.g., 84 – 32. Compensation problems also had a large difference, and the units of the minuend were 8 or 9, e.g., 68 – 19. Finally, indirect addition problems had a small difference (< 10) and involved crossing the tens, e.g., 82 – 76.

Task 1 and 2 focused on flexibility and included all four problem types. Tasks 3 and 4 focused on adaptivity and therefore included only problems suited for the task-specific strategies compensation and indirect addition.

Construct | Task | Description |
---|---|---|

Practical flexibility | 1a | The number of different strategies used to solve eight subtraction problems (one attempt) |

Potential flexibility | 1, 2 | The number of different strategies used to solve eight subtraction problems (three attempts), plus the number of additional strategies completed correctly in Task 2 |

Practical task-based adaptivity | 1a | The number of task-specific strategies used on the four problems suited for task-specific strategies |

Potential task-based adaptivity | 3 | The number of correctly selected task-specific strategies from four strategy options |

Practical individual-based adaptivity | 1a, 4a | The number of optimal strategies used in Task 1a, based on individual speed and accuracy in Task 4a |

Potential individual-based adaptivity | 4 | The number of correctly identified optimal strategies in Task 4b, based on individual speed and accuracy in Task 4a |

Task 1 consisted of eight subtraction problems (two per strategy) with three calculation boxes per problem. Students were instructed to solve these problems with the strategy they preferred. They had to write the solution steps in the upper calculation box (Task 1a). After they had completed all eight problems, they had to solve the same eight problems with different strategies than in their first attempt, for which they could use the two additional calculation boxes (Task 1b).

Strategies were coded into one of seven categories: jump, split, compensation, indirect addition, split-jump (combination of split and jump strategy; e.g., 84 – 67 via 80 – 60 = 20; 24 – 7 = 17), simplifying (changing both minuend and subtrahend to simplify the problem; e.g., 84 – 67 via adding 3 to both operands and solving 87 – 70 = 17), or indirect subtraction (computing how much has to be subtracted from the minuend to reach the subtrahend; e.g., 84 – 67 via 84 – 10 = 74; 74 – 7 = 67, so the answer is 10 + 7 = 17). For a random selection of 20% of the students (

Task 2 consisted of four problems (one per strategy) that were presented with a picture of a child solving that problem with the strategy most suited for the problem (see example in

In Task 3 students were offered four problems (two for each of the task-specific strategies compensation and indirect addition). Each problem was solved by four different children each using one of the four main strategies. Students were asked to select the optimal strategy for the problem (

In Task 4 students solves the two compensation problems and two indirect addition problems from Task 1 again. Task 4a was a personalized version of the no-choice conditions of the choice/no-choice design (

Strategy instruction cards and verbal instructions were used to instruct students which strategy they had to use. For each solution attempt, the experimenter recorded the solution time and scored the answer as correct or incorrect. Those speed and accuracy data were used to determine which strategy was optimal, i.e., the fastest strategy leading to a correct answer. This information was used to determine whether the strategy used in Task 1a was the optimal one for that particular student on that particular problem (practical individual-based adaptivity). Finally, in Task 4b students were shown their two no-choice solutions from Task 4a and had to select the strategy they thought was optimal for them. This captured whether they knew which of the two strategies they completed was optimal (potential individual-based adaptivity).

To measure students’ adaptive number knowledge we used the Arithmetic Production Task (

The forward and backward digit-span tasks from the WISC-III (

In the first attempt to solve the eight problems of Task 1, 60% of the solutions involved the jump strategy, 21% involved the split strategy, and 9% involved the split-jump strategy. The other strategies were used very rarely. This did not change much when students solved the same problems for a second and third time.

Number of different strategies | Attempt 1 | Attempts 1-3 combined |
---|---|---|

0 | 5 (3%) | 3 (2%) |

1 | 118 (80%) | 94 (64%) |

2 | 18 (12%) | 39 (27%) |

3 | 6 (4%) | 10 (7%) |

4 | 0 (0%) | 1 (1%) |

1.17 (0.541) | 1.40 (0.679) |

Construct | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | Pr Flex | 1.17 | 0.541 | 147 | ||||||||

2 | Pot Flex | 1.82 | 0.897 | 147 | .57** | |||||||

3 | Pr Ad (t) | 0.07 | 0.372 | 147 | .45** | .25** | ||||||

4 | Pot Ad (t) | 0.71 | 0.878 | 147 | .09 | -.01 | -.08 | |||||

5 | Pr Ad (i) | 0.79 | 2.070 | 117 | -.06 | -.13 | -.04 | -.03 | ||||

6 | Pot Ad (i) | 0.53 | 2.182 | 119 | .07 | .14 | .09 | -.12 | .26** | |||

7 | ML | 3.77 | 1.242 | 145 | .21* | .40** | .16 | .00 | .03 | .22* | ||

8 | ANK | 12.83 | 4.863 | 143 | .11 | .26** | .06 | .02 | -.15 | -.02 | .51** | |

9 | WM | 4.05 | 1.391 | 142 | .11 | .20* | -.01 | -.01 | -.16 | .03 | .21* | .27** |

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Potential flexibility was conceptualized as the number of strategies students know, and comprised two parts. The first part was the number of different strategies used across all three attempts in Task 1. The second part was based on Task 2, where students had to complete the four main strategies. For each of these strategies, one point was given if a strategy that was not part of the student’s strategy repertoire across all three attempts in Task 1 was completed correctly in Task 2. There were 13 students with hidden potential for the jump strategy, 13 for the split strategy, 24 for the compensation strategy and 12 for the indirect addition strategy. On average students knew 0.42 more strategies than they used in Task 1. A composite measure of

The number of times the two indirect addition problems and the two compensation problems were solved with their respective task-specific strategies in Task 1a was very low: 140 students (95.2%) never used the task-specific strategy, the other seven students used it on one to three problems. The measure of

In Task 3, 79 students (53.7%) never selected the task-specific strategy as the optimal one across the four problems, and the remaining 68 students made one to three adaptive strategy selections. The measure of

To determine the optimal strategy for each student on each problem, we compared the accuracy and speed of the task-specific and the other strategy in Task 4a to determine the fastest strategy leading to a correct answer, in a similar way as in previous studies (

The indirect addition strategy was the optimal strategy in 19% of the indirect addition problems, and the compensation strategy in 39% of compensation problems. The other, non-task-specific strategy that students themselves had used in Task 1a, was optimal in 60% of the indirect addition problems and in 39% of the compensation problems.

For each student we counted the number of times the optimal strategy as determined from Task 4a was the strategy used in Task 1a. Eleven students (7.5%) never used their optimal strategy, and 106 students (72.1%) used it one to four times. The remaining 30 students had a missing value because they did not use the intended strategies in the no-choice conditions, or they did not use one of the no-choice strategies in Task 1a. On average, students used their optimal strategy on 2.13 problems and used the non-optimal strategy (i.e., the incorrect strategy, or the slower strategy when both strategies yielded the correct answer) on 1.33 problems, a significant difference,

For each student we counted the number of times the optimal strategy as determined from Task 4a was the strategy students selected as they thought was optimal in Task 4b. Fourteen students (9.5%) never selected their optimal strategy, 116 students (78.9%) selected it one to four times. The remaining 17 students had a missing value because they had two or more missing values on the optimal strategy determination. On average, students selected their optimal strategy on 2.01 problems and selected the non-optimal strategy on 1.48 problems, a significant difference,

Correlation analyses were used to address the relation between the six adaptive expertise constructs on the one hand and students’ mathematics achievement level, adaptive number knowledge, and working memory on the other (see

The current study aimed to provide a comprehensive investigation of third graders’ adaptive expertise in the domain of multidigit subtraction, focusing on three main constructs – flexibility, task-based adaptivity, and individual-based adaptivity – and distinguishing between a practical component (what students show) and a potential component (what students know) for each construct.

Students’ level of practical flexibility and practical task-based adaptivity was rather low, which is in line with previous research (

To address individual-based adaptivity a personalized version of the choice/no-choice method was used (

Taken together, these findings imply on the one hand that the task-specific strategy, although it has the fewest and the easiest steps, is usually not optimal for an individual student. On the other hand, there is a discrepancy between the very low frequency of spontaneous use of the task-specific strategies (1-4%) and the percentage of trials where the task-specific strategy is optimal for an individual student (19-39%). This discrepancy suggests hidden potential for the adaptive use of task-specific strategies, particularly for the compensation strategy. However, although students on average knew which strategy was optimal for them, their potential individual-based adaptivity did not differ from their practical individual-based adaptivity, so there was no indication of hidden potential in individual-based adaptivity.

Finally, we addressed the relation between mathematics achievement level, adaptive number knowledge, working memory, and gender on the one hand, and the six components of adaptive expertise on the other. Mathematics achievement level was positively related to both practical and potential flexibility, as well as to potential individual-based adaptivity. This is in line with earlier findings that students with higher mathematics level show more strategy variability (

Unexpectedly, all other measures of adaptivity were unrelated to mathematics achievement, adaptive number knowledge, or working memory. For practical task-based adaptivity the absence of significant relations can probably be explained by a floor effect, given the low frequency with which the task-specific shortcut strategies were used spontaneously. For individual-based adaptivity it is important to note that the current study was the first to compare the task-specific strategy to students’ own preferential strategy. That is, if students are competent in their preferential strategy, such as the jump or split strategy, but not in the task-specific strategy, they can score high on individual-based adaptivity by consistently using their own strategy. This could possibly explain that for this type of adaptivity, mathematical knowledge and domain-general cognitive capacities do not play a prominent role.

Finally, there were no gender differences in the six components of adaptive expertise. Previous studies show inconsistent findings. Studies that have found gender differences show that girls tend to rely more on standard strategies and less often use shortcut strategies than boys (

Although there is hidden potential in the components of adaptive expertise, even when students would fully exploit this potential, their level of flexibility and adaptivity would still be rather low in an absolute sense and also compared to other studies capturing potential flexibility. A salient question is therefore how to raise the level of adaptive expertise. Some instructional factors seem to have a positive impact on children’s flexibility and adaptivity: stimulating children to invent, reflect, and discuss strategies, compare worked-out examples of different strategies presented side-by-side, and teachers asking open questions (

Looking at the types of strategies, previous research has already shown that primary school students can apply the indirect addition strategy efficiently and adaptively after a brief instruction (

Several methodological issues and limitations are important to discuss. First, the measure of potential flexibility comprised two components: prompting students to use a different strategy on the same problem (as was done in previous studies) and prompting students to complete a particular strategy, where the first step and a general verbal description of the strategy were given. This second approach has not been used before, to our knowledge. It is arguably different than the first approach, since part of the solution strategy is given away. It could be that students produced these alternative strategies for the very first time given this prompt. But even if this were so, this would mean that a single hint could lead students to use a strategy they did not used before, and thus that this strategy knowledge is very much at the surface and as such does give insights into their potential flexibility. Future attempts to measure potential flexibility in mathematics could add a similar measure to their instruments to investigate this issue further.

Second, by focusing on the number domain up to 100, the structural features of the problems intended to elicit the task-specific shortcut strategies compensation and indirect addition might not have been very salient. That is, in subtraction op to 1000 more solution steps have to be taken, making the benefits of rounding the subtrahend in for instance 743 – 399 or adding on in for instance 602 – 596 more salient. Future research could extend the domain to subtraction up to 1000 to gain more insight into the balance between benefits of task-based shortcut strategies and students’ individual strategy performance with these strategies in students’ strategy choices.

Third, a drawback of the implemented choice/no-design is that students solved identical problems in two different tasks (free choice and no-choice conditions), which could lead to retesting effects. However, because the two sessions were at least one week apart and subtraction problems are a frequent topic in mathematics textbooks (

Fourth, another weakness related to the choice/no-choice design is that it is necessary that strategies are identifiable on a problem-by-problem basis (

Finally, the study has a cognitive-psychological perspective on adaptive expertise and how to foster it, but affective factors and the socio-cultural context are also important (

The current study showed that students’ strategy flexibility and task-based adaptivity is rather low. However, including individual-based adaptivity (i.e., which strategy is optimal for an individual student) as well as students’ potential in flexibility and adaptivity presents a more nuanced, and also more positive, picture of students’ adaptive expertise. Students might not use task-specific strategies because these strategies do not work well for them. Furthermore, many students know more than they show regarding flexibility and task-based adaptivity.

Marian Hickendorff is one of the Guest Editors of this JNC Special Issue but played no editorial role in this particular article or intervened in any form in the peer review process.

The research was funded by a personal grant from the Dutch Research Council (NWO) for the principal researcher (Hickendorff): 016.Veni.195.166/6812. These funding sources did not have a role in the research design, execution, analysis, interpretation and reporting.

The author is indebted to the undergraduate students and research assistants who assisted in data collection and coding.