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The present study aims to examine inter-individual differences in adaptive number knowledge in secondary school students. Adaptive number knowledge is defined as a well-connected network of knowledge of numerical characteristics and arithmetic relations. Substantial and relevant qualitative differences in the strategies and expression of adaptive number knowledge have been found in primary school students still in the process of learning arithmetic. We present a study involving 879 seventh-grade students that examines the structure of individual differences in adaptive number knowledge with students who have completed one year of algebra instruction. Results of a latent profile analysis reveal a model that is similar than was previously found in primary school students. As well, arithmetic fluency and the development of arithmetic fluency are strong predictors of adaptive number knowledge latent profile membership. These results suggest that adaptive number knowledge may be characteristic of high-level performance extending into secondary school, even after formal instruction with arithmetic concludes.

Adaptivity with arithmetic refers to the ability to solve a particular problem in the most efficient and appropriate way in a particular situation (

Another of these individual characteristics, procedural flexibility had been described as the ability to switch between different problem solving strategies (e.g.

Adaptive problem solving strategies with whole-number arithmetic may be reliant on students’ understanding of the numerical characteristics of the numbers presented in the problem and a rich understanding of the arithmetic relations between these numbers (

Adaptive number knowledge has so far been explored most extensively in the form of arithmetic sentence production tasks (see

Given that there are a number of possible global approaches to the arithmetic sentence production tasks, previous studies have aimed to examine the nature of inter-individual differences in adaptive number knowledge. These person-centered approaches have revealed substantial, and relevant, qualitative differences in the strategies and expression of adaptive number knowledge on the arithmetic sentence production task in primary school students (

Due to the data-driven approach of the previous analysis, it is possible that these results are idiosyncratic and would not appear in other samples or different age students. In particular, students’ adaptive number knowledge may change as they learn more advanced mathematical content. More advanced skills and robust conceptual understanding of arithmetic that comes from learning algebra may change how some students respond on the arithmetic sentence production task. In order to examine this, the present study will investigate the structure of individual differences in adaptive number knowledge in a sample of lower secondary school students with at least one year of experience learning algebra.

Although a good deal is known about how adaptive number knowledge is related to other arithmetic knowledge and skills, little is known about the precursors of adaptive number knowledge. In particular, we aim in the present study to examine the relation between a number of relevant domain specific and general knowledge and skills and adaptive number knowledge.

Previous studies have revealed a relation between adaptive number knowledge and other arithmetic knowledge and skills. Among sixth graders, arithmetic fluency and arithmetic conceptual knowledge were both found to predict adaptive number knowledge, though the students’ grades in mathematics class did not (

Arithmetic skill does not merely develop through explicit practice with arithmetic procedures during early, explicit instruction on these topics (

Adaptive number knowledge has been found to be uniquely related pre-algebra knowledge, even after taking into account arithmetic fluency and conceptual knowledge (

Adaptive number knowledge has proved to be a unique and well-founded feature of arithmetic knowledge and skills (

Similarly, the arithmetic sentence production task may require students to rely on their working memory (

In general, more evidence is needed to better situate adaptive number knowledge in the framework of adaptivity with arithmetic problem solving. Examining the nature and predictors of individual differences in adaptive number knowledge will better support our understanding of how adaptive number knowledge fits into the development of arithmetic skills. To achieve this end, the present study asks the following questions:

1. Do profiles of secondary students’ adaptive number knowledge follow similar patterns as those profiles that were previously found among late primary school students?

Previously, both quantitative and qualitative differences in adaptive number knowledge were found among fourth to sixth graders (

2. Do arithmetic fluency, the development of arithmetic fluency, general mathematical achievement, non-verbal intelligence, and working memory predict membership in profiles of adaptive number knowledge?

Previously, arithmetic skills and knowledge have been found to be related to adaptive number knowledge (e.g.

The present study is part of a broader longitudinal study (

Participants’ ages ranged from 12 to 15 years (

Students’ data were collected during normal school days. All tests and questionnaires were group-administered by trained testers (there were always two trained research assistants present for each test situations). Students’ non-verbal intelligence was tested in the Fall semester of Grade 6 and working memory in the Spring semester of Grade 6. Arithmetic fluency was tested both in the Fall semester of Grade 6 and in the Spring semester of Grade 7. Students’ math performance and adaptive number knowledge were tested in the Spring semester of Grade 7.

The Raven Standard Progressive Matrices (

The Counting Span task was conducted using the touchscreen interface of an Android tablet (10.1 inches) with OpenSesame Runtime for Android (version 2.8.3) (

Arithmetic Fluency was assessed with the Basic Arithmetic Test (

Students’ mathematical performance was assessed using a test for basic mathematical skills (

Adaptive number knowledge was assessed using the arithmetic sentence production task (extensive details and analysis of this task can be found in

Item | Given | Target | Type | Mean Correct | Max Correct |
---|---|---|---|---|---|

1 | 2, 4, 8, 12, 32 | = 16 | Dense | 5.42 | 16 |

2 | 1, 2, 3, 5, 30 | = 59 | Sparse | 1.98 | 8 |

3 | 2, 4, 6, 16, 24 | = 12 | Dense | 5.85 | 16 |

4 | 2, 3, 6, 10, 18 | = 38 | Sparse | 2.99 | 14 |

Two types of items are included in the arithmetic sentence production task (see

For the LPA modeling, the participants’ responses were separately coded as either

All analysis were pre-registered prior any analysis or detailed examination of the data; pre-registration protocol can be found at [masked for blind review]. In order to examine the profiles of adaptive number knowledge, LPA was used to model students’ performance on the arithmetic sentence production task using four indicators of the number of correct simple responses on dense items (Dense Simple), complex responses on dense items (Dense Complex), simple responses on sparse items (Sparse Simple), and complex responses on sparse items (Sparse Complex). The normal procedure of modelling an LPA in which a one-class solution is estimated and the number of classes in the model is increased one class at a time. As well, since we aimed to examine if the LPA model used to describe 4^{th} to 6^{th} graders’ performance on the arithmetic sentence production task could be also found in the present sample, we also constructed a confirmatory LPA model of this model. This was estimated using relative constraints for each of the four indicators using the findings from the previously identified model, in which there were five classes that aligned as: Basic, Simple, Complex, Strategic, and High. Since these four values capture all the solutions students provided, total score is not included as an indicator in the LPA model. The following constraints were placed on the confirmatory model: Dense Simple: Basic < Simple = Complex = Strategic = High; Dense Complex: Basic = Simple = Strategic < Complex < High; Sparse Simple: Basic = Complex = Strategic < High < Simple; Sparse Complex: Basic = Simple < Complex = Strategic < High.

LPA modelling was conducted with Mplus version 7.4 (

A three-step approach will be taken to identify the relation between adaptive number knowledge profiles and the cognitive and mathematical covariates, this allows for the determination of the latent classes without influence from the covariates. The three-step approach examines the relation between external variables and profile membership only after the LPA model has been identified (as a

Variable | Skewness ( |
Kurtosis ( |
Range | ||
---|---|---|---|---|---|

Dense Simple | 10.21 | 3.92 | 0.41 (.09) | 0.48 (.17) | 0 − 29 |

Dense Complex | 1.06 | 1.45 | 1.80 (.09) | 3.71 (.17) | 0 − 9 |

Sparse Simple | 1.90 | 1.76 | 1.05 (.09) | 1.15 (.17) | 0 − 10 |

Sparse Complex | 3.08 | 2.20 | 1.08 (.09) | 2.37 (.17) | 0 − 15 |

Arithmetic Fluency (Grade 7) | 15.46 | 3.54 | -0.54 (.09) | 0.97 (.17) | 0 − 26 |

Arithmetic Change | 1.85 | 2.60 | -0.02 (.09) | 0.29 (.18) | -7 − 11 |

Math Achievement | 16.97 | 5.64 | -0.28 (.09) | -0.01 (.17) | 0 − 33 |

Non-verbal Intelligence | 22.63 | 3.67 | -1.18 (.08) | 2.31 (.17) | 5 − 29 |

Working Memory | 6.97 | 3.51 | 0.28 (.09) | -0.32 (.17) | 0 − 19 |

Number of Classes | BIC | aBIC | Entropy | BLRT | LMR |
---|---|---|---|---|---|

3 | 8628 | 8571 | .91 | .0000 | .02 |

4 | 8515 | 8442 | .86 | .0000 | .003 |

5 confirmatory | 8467 | 8407 | .77 | .500 | NA |

6 | 8384 | 8279 | .86 | .0000 | .60 |

The mean values for each indicator by latent profile in the five class solution can be found in

Profile | Dense Simple | Dense Complex | Sparse Simple | Sparse Complex | Total Correct | Percent of Sample |
---|---|---|---|---|---|---|

Basic | 8.45 | 0.37 | 1.33 | 2.17 | 12.32 | 58.8 |

Simple | 13.99 | 0.64 | 4.94 | 2.32 | 21.89 | 10.2 |

Complex | 11.32 | 2.61 | 1.96 | 4.37 | 20.26 | 19.6 |

Strategic | 14.65 | 0.56 | 1.71 | 5.99 | 22.91 | 7.4 |

High | 12.68 | 5.63 | 2.46 | 6.53 | 27.30 | 4.0 |

In general, the five class solution showed a large amount of similarity with the previous LPA of adaptive number knowledge. ^{th} graders, the Basic, Simple, and High classes appear larger and the Strategic class smaller, while the Complex class remains relatively large. In all, it appears that even though the confirmatory model was not the most appropriate in this sample, the results of the previous LPA of adaptive number knowledge were fairly well replicated.

Standardized mean values for each latent profile for the present study involving 7^{th} graders on the left and the study by ^{th} to 6^{th} graders on right. Error bars represent +/- 1 standard error.

The three-step approach provided an examination of the differences between the adaptive number knowledge profiles on a variety of cognitive and mathematical measures. All covariates were added simultaneously and revealed that there were differences across the profiles in arithmetic skills, the development of arithmetic skills, and general mathematical achievement.

Multinomial logistic regression coefficients based on the three-step procedure. Basic class is the reference class and all values represent profile derivations from Basic class values (set at 0). Error bars indicate +/- 2 standard error of mean difference between Basic class and each class (and cannot be used to compare other classes with each other, e.g. Strategic versus High).

In particular, two distinct patterns emerged with regard to arithmetic knowledge. First, the Basic profile had lower arithmetic fluency than the Complex, Strategic, and High profiles. Second, the Basic profile had a smaller changes in arithmetic than the Strategic and High profiles, and the Simple profile had smaller changes in arithmetic than the Complex, Strategic, and High profiles. As well the Basic profile had lower mathematical achievement than the Simple, Complex, and High profiles. There were no statistically significant differences between the profiles in non-verbal intelligence nor working memory (except for a small difference between the Complex and Basic profiles for working memory). In all, one striking finding is that for these five predictors there were no significant differences between the three groups with the strong adaptive number knowledge, namely, the Complex, Strategic, and High profiles.

Adaptive number knowledge has been identified as a feature of arithmetic development and described as an underlying feature of adaptivity with arithmetic problem solving alongside procedural flexibility (

The profile structures of adaptive number knowledge in seventh graders in the present study were remarkably similar as was found previously in fourth to sixth graders (

In both the present study and in the previous study, the Basic profile was the most prominent, with the majority of students in both samples (50% among 4-6 graders and 59% among 7 graders in this study). While this profile is labeled as Basic, the mean scores suggest some success in coming up with correct solutions, even the most advanced type of solutions, complex solutions on sparse items. This result suggests that adaptive number knowledge may be most differentiated at the top-levels of arithmetic performance. That the Basic performance level improves between fourth and seventh grade is not surprising. What is surprising is that more students would be placed in this category among older students. This suggests a widening gap in the development of adaptive number knowledge across grade levels, with fewer students exceeding their peers in exceptional ways. Increasing disaffect towards math in lower secondary school might be an issue at play, especially given the open-ended nature of the task (e.g. find as many solutions as you can). Longitudinal investigation into the development of adaptive number knowledge would be beneficial for better answering these questions.

Despite the large(r) number of students in the Basic profile in the present study, there still appears to be substantial and meaningful differentiation in adaptive number knowledge with these secondary school students at the upper levels of this knowledge. In particular the Simple, Complex, Strategic, and High profiles all have ostensibly high levels of success on the arithmetic sentence production task in terms of overall scores. But, they do so in with different solution patterns. In particular, there seems to be a difference in the response patterns for the Complex and Strategic profiles on the Dense items. In general, students did not use complex solutions on the dense items; presumably this is because it is not necessary to do so, given the large number of both simple and more complex arithmetic relations between the given and target numbers. While the Strategic profile appeared to vary their strategies depending on the item type, the Complex profile was exceptional in their use of complex solutions on the dense items (outside of the High profile).

Potentially, these patterns for producing answers on the task are a conscious strategy, though this also may be a reflection of the open nature of the task. Both groups were just as likely to produce complex solutions on the Sparse items, however. The Strategic profile even had a higher number of complex solutions on these items than the Complex profile. This suggests that the Strategic profile may more reflect a group of students with a higher general flexibility in solution strategies, who is able to adapt their solution strategy to fit the problem type. More details on the actual solutions provided by these groups may shed light on the nature of these differences and provide more insight into how adaptive number knowledge may be promoted in the classroom.

The present study provides more evidence that adaptive number knowledge is a distinct component of arithmetic skills and knowledge. Arithmetic fluency and change in arithmetic fluency were found to predict profile membership even after taking into account general mathematical achievement and domain general cognitive abilities. However, there appeared to be substantial individual differences in adaptive number knowledge that were not explained by these mathematical and other cognitive skills and knowledge.

Crucially, the present study was able to include a measure of the change in arithmetic fluency from the beginning of Grade 6 until the end of Grade 7, when arithmetic procedures are not any longer explicitly taught in the classroom. Improvements in arithmetic fluency during this period of non-explicit instruction may be representative of increased automatization of basic numerical facts (

However, it appears that performance on the arithmetic sentence production task is not related to general non-verbal intelligence, after taking into account more specific mathematical skills. Nor does working memory seem to distinguish between profiles of adaptive number knowledge. It is possible that the effects of these domain general abilities are already embedded in individual differences in arithmetic and mathematical skills and knowledge and therefore do not provide any unique explanation for differences in adaptive number knowledge (

The notion of adaptive number knowledge as high-level knowledge is supported by the lack of differences between the Complex, Strategic, and High profiles for any of the covariates included in the study. Thus, adaptive number knowledge may be the main cause for individual differences in performance on the arithmetic sentence production task at these upper levels. Previously, in the sample of fourth to sixth graders these same three top-level profiles were found to significantly differ in arithmetic fluency (

The present study is the first to present evidence that students’ development of arithmetic fluency positively predicted adaptive number knowledge. The question that arises, given that this was not a causal relation, was whether this change led to better adaptive number knowledge or whether better adaptive number knowledge led to more improved fluency. A third possibility is that both of these directional relations are true, with an positive iterative loop being the cause of this relation (e.g.

In the present study one of the biggest clarifying correlations is the differences between the Simple profile and more advanced profiles in arithmetic fluency and change in arithmetic fluency. While there were no differences between the Simple profile and the more advanced profiles in their arithmetic fluency in Grade 6, the Simple profile had significantly less development of arithmetic fluency over the next two school years than the three advanced groups. This suggests that the Simple group do not seem to draw out the key aspects of arithmetic implicitly from more advanced mathematical topics such as algebra, which would support their development of arithmetic fluency in Grades 6 and 7.

There are a number of limitations to the present study which should be addressed in future studies on adaptive number knowledge. The first being that the present study would be better suited to include more diverse measures of adaptive number knowledge. So far, the arithmetic sentence production task is the only task that has been used in assessing adaptive number knowledge, and it is not clear that this is a more generalizable type of knowledge. More measures which tap into assessing students’ knowledge of numerical characteristics and arithmetic relations in novel tasks would be valuable for determining more exactly the nature of adaptive number knowledge. Relatedly, the present study only included arithmetic measures of procedural fluency. Future studies should aim to assess how conceptual knowledge, procedural flexibility, and general adaptivity with arithmetic problem solving are related to adaptive number knowledge. Including a more comprehensive battery of arithmetic measures would better situate adaptive number knowledge in this domain.

Despite these limitations, the present study provides a strong step forward in deepening our understanding of the nature of adaptive number knowledge and its role in the development of arithmetic. As more and more mathematical curricula turn their focus to adaptable skills and knowledge that fall under the guise of adaptive expertise, finding foundational skills and knowledge that would support such goals is crucial (

This study forms part of the STAIRWAY-From Primary School to Secondary School Study (Ahonen & Kiuru, 2013-2017). We would like to thank all the participants, teachers, and assistants without whom this research could not have been carried out.

The authors have declared that no competing interests exist.