Shortcut Strategies Research Articles

How Do Multiplicative and Relational Thinking Contribute to Multidigit Division?

Multiplicative thinking (MT), relational thinking (RT), and multidigit division are highly important for facilitating the transition from arithmetic to algebra in the middle grades. The present study analyzed the extent to which the contribution of MT to multidigit division was mediated by 472 fourth- and fifth-grade students’ RT and how their RT supported the use of multiple and clever shortcut strategies for multidigit division. The MT measurement instrument mapped the progression from additive to multiplicative thinking in two proficiency levels, and the RT measurement instrument operationalized the progression from understanding and using relationships between numbers to understanding and using relationships between terms in four proficiency levels. RT was a complete mediator of the impact of MT on multidigit division and had a particularly strong influence on the use of 10 × N patterns and advanced strategies, including shortcut strategies.

Grade six students’ multidigit arithmetic strategy adaptivity and flexibility: evaluating a novel tri-phase assessment tool

In this paper we present a novel adaptation of a tri-phase assessment tool, originally devised to investigate students’ linear equations-related strategy flexibility, to evaluate Danish grade-six students’ multidigit arithmetic-related strategy adaptivity and flexibility. Participants, 731 students, median age 12 years and drawn from 20 demographically different schools, completed a series of positive integer addition, subtraction and multiplication tasks designed to elicit shortcut strategies. First, students solved each task by means of their preferred strategy, with those using a shortcut strategy being defined as showing adaptivity for that task. Second, students solved the same tasks by means of whatever alternative strategies they could devise; those that offered at least two strategies were defined as showing flexibility for that task. Third, for each task, students were asked to indicate which of their strategies they believed was optimal. Statistical analyses, based on the total number of tasks on which students demonstrated adaptivity or flexibility, found students to be more flexible than adaptive, with 55% of students showing moderate or high levels of flexibility but only 21% for adaptivity. Both strategy adaptivity and flexibility, which differed according to operation, were implicated in students’ solution accuracy, with adaptivity being a stronger predictor than flexibility.

Danish third, sixth and eighth grade students’ strategy adaptivity, strategy flexibility and accuracy when solving multidigit arithmetic tasks

In this paper, the multidigit arithmetic-related strategy adaptivity, strategy flexibility and solution accuracy of Danish compulsory school students is examined. Participants, 749 grade three, 731 grade six and 818 grade eight, were drawn from twenty demographically different schools. Drawing on a tri-phase assessment tool, each student completed a series of tasks designed to elicit shortcut strategies. First, students solved each task by means of their preferred strategy; those using shortcut strategies were construed as adaptive for that task. Second, students solved the same tasks by means of whatever alternative strategies they had available; those offering at least two strategies were construed as flexible for that task. Third, for each task, students were asked to indicate which of their strategies they believed was optimal. Across all grades, students were more flexible than adaptive. Overall, sixth graders exhibited higher levels of flexibility than third graders and marginally lower levels than eighth graders. Sixth graders exhibited higher levels of adaptivity than those in either grade three or grade eight. Students’ accuracy, which improved with maturation, was influenced positively by both adaptivity and flexibility, with flexibility having the greatest influence in grade three and adaptivity in grade six. The findings raise further questions concerning, inter alia, culture’s influence on students’ strategy choices and the interaction of adaptivity, flexibility and maturity on accuracy.

The Effect of Heuristics on the Reasoning of the Pre-Service Science Teachers on the Topic of Melting and Boiling Point.

The purpose of this study was to explore the effects of the heuristics on the reasoning processes of pre-service science teachers on the topic of melting and boiling point using the ten heuristic model proposed by Talanquer. In this phenomenographic study carried out in the spring semester of the 2018-2019 academic year, interviews were conducted with 30 teacher candidates enrolled in the Science Teaching Program of Firat University Faculty of Education. Participants were asked to answer three different questions during the interviews. These questions were about the ranking of some compounds according to their melting or boiling points. Six different answer patterns for each question were obtained from the answers. The findings of this study showed that students generally used shortcut strategies instead of analytical/scientific reasoning, as all ten heuristics affected participants' reasoning. This study also revealed that although not included in the model proposed by Talanquer, periodic trends heuristic also influenced participants' reasoning about the melting and boiling point.

The Role of Heuristics in the Reasoning Process of Pre-Service Science Teachers on the “Chemical Structure – Acidity/Basicity Relationship” Topic

The purpose of this research is to examine the effects of 10 heuristics proposed by Talanquer on the reasoning processes of science teacher candidates on the "chemical structure - acidity-basicity relationship" topic. In this phenomenographic research, interviews were conducted with 30 prospective teachers enrolled in the Science Education Program, Education Faculty, Firat University in the spring semester of the 2018-2019 academic year. In the first stage of the two-stage interview, the participants were asked to rank some chemical compounds according to their increasing acidity strength, while in the second stage, they were asked to rank some chemical compounds according to their increasing basicity strength. In the interviews, participants were also asked to explain in detail the reasons for their ranking. From the answers given by the participants to the questions, six different answer patterns were obtained for acidity strength, while five different answer patterns were obtained for basicity strength. It was determined that all ten heuristics affect the reasoning of the participants, and because of the effects of heuristics, students generally use shortcut strategies instead of scientific reasoning. In addition, this study revealed that although it was not included in the model proposed by Talanquer, periodic trends heuristic also affected the reasoning of the participants on the "chemical structure - acidity/basicity relationship".

Adiabatic theorem revisited: The unexpectedly good performance of adiabatic passage

Adiabatic passage employs a slowly varying time-dependent Hamiltonian to control the evolution of a quantum system along the Hamiltonian eigenstates. For processes of finite duration, the exact time evolving state may deviate from the adiabatic eigenstate at intermediate times, but in numerous applications it is observed that this deviation reaches a maximum and then decreases significantly towards the end of the process. We provide a straightforward theoretical explanation for this welcome but often unappreciated fact. Our analysis emphasizes a separate adiabaticity criterion for high fidelity state-to-state transfer and it points to new effective shortcut strategies for near adiabatic dynamics.

Analysis of Pre-Service Science Teachers’ Heuristic Reasoning Processes about Hydrogen Bonding

The purpose of this research was to determine the heuristics used by pre-service science teachers in understanding the details of hydrogen bonding. The reasoning processes demonstrated were evaluated based on ten heuristic models suggested by Talanquer (2014). Phenomenographic assessment of the 30 participants indicated that all ten heuristics were utilized to make interpretations about hydrogen bonding. It was found that most students used short-cut strategies rather than efficient analytical reasoning processes. A total of 12 answer patterns were determined based on the answers of the participants. The percentage of students who gave the correct answer was low. The frequency sequencing of participants' heuristics demonstrated in this study was fluency, associative activation, recognition, one-reason decision making, attribute substitution, overconfidence, surface similarity, generalization, rigidity, and affect.

Interleaved Learning in Elementary School Mathematics: Effects on the Flexible and Adaptive Use of Subtraction Strategies.

Empirical findings show that students are often not capable of using number-based strategies and the standard written algorithm flexibly and adaptively to solve multi-digit subtraction problems. Previous studies have pointed out that students predominantly use the standard written algorithm after its introduction, regardless of task characteristics. Interleaved practice seems to be a promising approach to foster the flexible and adaptive use of strategies. In comparison to the usual blocked approach, in which strategies are introduced and practiced successively, they are presented intermixed in interleaved learning. Thus, the students have to choose an appropriate strategy on the basis of every task itself, and this leads to drawing comparisons between the different strategies. Previous research has shown inconsistent results regarding the effectivity of interleaving mathematical tasks. However, according to the attentional bias framework, interleaved practice seems to be a promising approach for teaching subtraction strategies to enhance the students’ flexibility and adaptivity. In this study, 236 German third graders were randomly assigned to either an interleaved or blocked condition. In the interleaved condition the comparison processes were supported by prompting the students to compare the strategies (between-comparison), while the students of the blocked approach were encouraged to reflect the adaptivity of a specific strategy for specific subtraction tasks (within-comparison). Both groups were taught to use different number-based strategies (i.e., shortcut strategies and decomposition strategies) and the standard written algorithm for solving three-digit subtraction problems spanning a teaching unit of 14 lessons. The results show that the students of the interleaved condition used the shortcut strategies more frequently than those of the blocked condition, while the students of the interleaved condition applied the decomposition strategies as well as the standard written algorithm less frequently. Furthermore, the students of the interleaved condition had a higher competence in the adaptive use of the shortcut strategies and the standard written algorithm. A subsequent cluster analysis revealed four groups differing in their degree of adaptivity. Being part of clusters with a comparatively high level of adaptivity was positively related to the prior arithmetical achievement and, even more so, to the interleaved teaching approach.

On the importance of avoiding shortcuts in applying cognitive models to hierarchical data

Psychological experiments often yield data that are hierarchically structured. A number of popular shortcut strategies in cognitive modeling do not properly accommodate this structure and can result in biased conclusions. To gauge the severity of these biases, we conducted a simulation study for a two-group experiment. We first considered a modeling strategy that ignores the hierarchical data structure. In line with theoretical results, our simulations showed that Bayesian and frequentist methods that rely on this strategy are biased towards the null hypothesis. Secondly, we considered a modeling strategy that takes a two-step approach by first obtaining participant-level estimates from a hierarchical cognitive model and subsequently using these estimates in a follow-up statistical test. Methods that rely on this strategy are biased towards the alternative hypothesis. Only hierarchical models of the multilevel data lead to correct conclusions. Our results are particularly relevant for the use of hierarchical Bayesian parameter estimates in cognitive modeling.

Dutch sixth graders\u2019 use of shortcut strategies in solving multidigit arithmetic problems

Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth graders (12-year-olds) use shortcut strategies in solving multidigit addition, subtraction, multiplication, and division problems, to what extent student factors and task instructions affected this frequency of shortcut strategy use, and to what extent the strategies differed in performance. A sample of 648 sixth graders from 23 Dutch primary schools completed a paper-and-pencil task of 12 multidigit arithmetic problems, designed to elicit specific shortcut strategies such as compensation. Based on the students’ written work, strategies were classified into whether a shortcut strategy was used or not. Results showed that the frequency of shortcut strategies ranged between 6 and 21% across problem types, and that boys and high mathematics achievers were more inclined to use shortcut strategies. An explicit instruction to look for a shortcut strategy increased the frequency of these strategies in the addition and multiplication problems, but not in the subtraction and division problems. Finally, the use of shortcut strategies did not yield higher performance than using standard strategies. All in all, spontaneous as well as stimulated use of shortcut strategies by Dutch sixth graders was not very common.

Memory and learning for visual signals in time and space.

Vision is often characterized as a spatial sense, but what does that characterization imply about the relative ease of processing visual information distributed over time rather than over space? Three experiments addressed this question, using stimuli comprising random luminances. For some stimuli, individual items were presented sequentially, at 8 Hz; for other stimuli, individual items were presented simultaneously, as horizontal spatial arrays. For temporal sequences, subjects judged whether each of the last four luminances matched the corresponding luminance in the first four; for spatial arrays, they judged whether each of the right-hand four luminances matched the corresponding left-hand luminance. Overall, performance was far better with spatial presentations, even when the entire spatial array was presented for just tens of milliseconds. Experiment2 demonstrated that there was no gain in performance from combining spatial and temporal information within a single stimulus. In a final experiment, particular spatial arrays or temporal sequences were made to recur intermittently, interspersed among, non-recurring stimuli. Performance improved steadily as particular stimulus exemplars recurred, with spatial and temporal stimuli being learned at equivalent rates. Logistic regression identified several shortcut strategies that subjects may have exploited while performing our task.

Spontaneous Usage of Different Shortcuts Based on the Commutativity Principle

Based on research on expertise a person can be said to possess integrated conceptual knowledge when she/he is able to spontaneously identify task relevant information in order to solve a problem efficiently. Despite the lack of instruction or explicit cueing, the person should be able to recognize which shortcut strategy can be applied – even when the task context differs from the one in which procedural knowledge about the shortcut was originally acquired. For mental arithmetic, first signs of such adaptive flexibility should develop already in primary school. The current study introduces a paper-and-pencil-based as well as an eyetracking-based approach to unobtrusively measure how students spot and apply (known) shortcut options in mental arithmetic. We investigated the development and the relation of the spontaneous use of two strategies derived from the mathematical concept of commutativity. Children from grade 2 to grade 7 and university students solved three-addends addition problems, which are rarely used in class. Some problems allowed the use of either of two commutativity-based shortcut strategies. Results suggest that from grade three onwards both of the shortcuts were used spontaneously and application of one shortcut correlated positively with application of the other. Rate of spontaneous usage was substantial but smaller than in an instructed variant. Eyetracking data suggested similar fixation patterns for spontaneous an instructed shortcut application. The data are consistent with the development of an integrated concept of the mathematical principle so that it can be spontaneously applied in different contexts and strategies.

Decision-making in the aisles: informing, overwhelming or nudging supermarket shoppers?

Recent findings suggest that while shopping people apply `fast and frugal' heuristics: short-cut strategies where they ignore most product information and instead focus on a few key cues. But rather than supporting this practice, mobile phone shopping apps and recommender systems overwhelm shoppers with information. This paper examines the amount and structure of product information that is most appropriate for supermarket shoppers, finding that in supermarkets, people rapidly make decisions based on one or two product factors for routine purchases, often trading-off between price and health. For one-off purchases, shoppers can be influenced by reading customer star ratings and reviews on a mobile phone app. In order to inform decision-making or nudge shoppers in supermarkets, we propose using embedded technologies that provide appropriate feedback and make key information salient. We conclude that rather than overwhelming shoppers, future shopping technology design needs to focus on information frugality and simplicity.

Engaged at the Extremes

Although academic centers rely on assessments from medical trainees regarding the effectiveness of their faculty as teachers, little is known about how trainees conceptualize and approach their role as assessors of their clinical supervisors. In 2010, using a constructivist grounded theory approach, five focus group interviews were conducted with 19 residents from an internal medicine residency program. A constant comparative analysis of emergent themes was conducted. Residents viewed clinical teaching assessment (CTA) as a time-consuming task with little reward. They reported struggling throughout the academic year to meet their CTA obligations and described several shortcut strategies they used to reduce their burden. Rather than conceptualizing their assessments as a conduit for both formative and summative feedback, residents perceived CTA as useful for the surveillance of clinical supervisors at the extremes of the spectrum of teaching effectiveness. They put the most effort, including the crafting of written comments, into the CTAs of these outliers. Trainees desired greater transparency in the CTA process and were skeptical regarding the anonymity and perceived validity of their faculty appraisals. Individual and system-based factors conspire to influence postgraduate medical trainees' motivation for generating high-quality appraisals of clinical teaching. Academic centers need to address these factors if they want to maximize the usefulness of these assessments.

Intercepting moving targets on foot: Can people learn to anticipate multiple trajectories?

Previous work (Fajen & Warren, 2004; submitted) modeled human interception of moving targets based on first-order information about the target's bearing (constant bearing strategy). The model has accurately predicted paths to constant velocity and accelerating targets and to targets moving on a curved trajectory. Owens & Warren (VSS 2005) found that with repeated presentation of the same trajectory, most participants learn heuristic strategies to anticipate target motion and take “shortcuts” to accelerating and curving targets, although there was considerable variation. Here we investigate the number and variety of target trajectories that participants can learn to anticipate by varying trajectory characteristics within blocks of trials. Participants walked in the VENLab, a 12m x 12m virtual environment with a head-mounted display (60 deg H x 40 deg V) and a sonic/inertial tracking system (latency 50–70 ms). One block of 20 trials consisted of two randomized target trajectories that curved in the same direction, but had different radii and speeds. Another block of 20 trials contained two mirror-image trajectories with identical radii and speed. Presentation order of these blocks was counterbalanced. The final block of 40 trials presented four novel trajectories with different radii and speeds. Most participants developed shortcut strategies in all three conditions, although more learned them for the same-side targets than for the mirror-image targets. The trajectories in the last block of trials were the most quickly and consistently learned. A second experiment investigates how the addition of a color cue to target trajectory affects the acquisition of anticipatory strategies.

The relationship between adults’ conceptual understanding of inversion and associativity.

Children's understanding of the mathematical concepts of inversion and associativity are positively related, as measured by the use of conceptually based shortcut strategies on 3-term inversion problems (i.e., a + b - b, d x e / e) and associativity problems (i.e., a + b - c, d x e / f; Robinson & Dubé, 2009; Robinson & Ninowski, 2003). Individuals who use the inversion shortcut (e.g., 3) are more likely to use the associativity strategy (e.g., 3 x 12 / 4. 12 / 4 = 3, 3 x 3 = 9), which is almost never used by an individual who does not also use the inversion shortcut (Robinson & Dubé, 2009). One possible reason for this relationship is that directing attention to the right-most operation during problem solving may be required to prime the conceptually based shortcut strategies for both problem types. This study investigated the relationship between adults' understanding of inversion and associativity. Adults (N = 42) solved inversion and associativity problems in 1 of 2 conditions. The participants were either presented with the left-most operation and then the whole problem or presented with the right-most operation and then the whole problem. A positive relationship between the use of the conceptually based strategies was found, and it was strikingly similar to the relationship found in childhood. There was evidence that the presentation of the right-most operation first primed the inversion shortcut.

Acquisition and use of shortcut strategies by traditionally schooled children

This study aimed at analysing traditionally taught children’s acquisition and use of shortcut strategies in the number domain 20–100. One-hundred-ninety-five second, third, and fourth graders of different mathematical achievement levels participated in the study. They were administered two tasks, both consisting of a series of two-digit additions and subtractions that maximally elicit the use of the compensation $$\left( {45 + 29 = \_;45 + 30 - 1 = 75 - 1 = 74} \right)$$ and indirect addition strategy ( $$71 - 68 = \_;\,68 + 2 = 70,\,70 + 1 = 71$$ , so the answer is 2 + 1 or 3). In the first task, children were instructed to solve all items as accurately and as fast as possible with their preferred strategy. The second task was to generate at least two different strategies for each item. Results demonstrated that children of all grades and all achievement levels hardly applied the compensation and indirect addition strategy in the first task. Children’s strategy reports in the second task revealed that younger and lower achieving children did not apply these strategies because they did not (yet) discover these strategies. By contrast, older and higher achieving children appeared to have acquired these strategies by themselves. Results are interpreted in relation to cognitive psychological and socio-cultural perspectives on children’s mathematics learning.

The Use of Multi-Criteria Decision Analysis Weight Elicitation Techniques in Patients with Mild Cognitive Impairment

To test the applicability of multi-criteria decision analysis preference elicitation techniques in cognitively impaired individuals. A convenience sample of 16 cognitively impaired subjects and 12 healthy controls was asked to participate in a small pilot study. The subjects determined the relative importance of four decision criteria using five different weight elicitation techniques, namely simple multi-attribute rating technique, simple multi-attribute rating technique using swing weights, Kepner-Tregoe weighting, the analytical hierarchical process, and conjoint analysis. Conjoint analysis was judged to be the easiest method for weight elicitation in the control group (Z = 10.00; p = 0.04), while no significant differences in difficulty rating between methods was found in cognitively impaired subjects. Conjoint analysis elicitates weights and rankings significantly different from other methods. Subjectively, cognitively impaired subjects were positive about the use of the weight elicitation techniques. However, it seems the use of swing weights can result in the employment of shortcut strategies. The results of this pilot study suggest that individuals with mild cognitive impairment are willing and able to use multi-criteria elicitation methods to determine criteria weights in a decision context, although no preference for a method was found. The same methodologic and practical issues can be identified in cognitively impaired individuals as in healthy controls and the choice of method is mostly determined by the decision context.

Mental calculation in an autistic savant: A case study

We present a detailed investigation of mental calculation in RH, an autistic savant. In Experiment 1 RH performed faster and more accurately than control subjects in mentally solving multi-digit multiplication problems. However, we found no evidence that RH employed short-cut strategies to solve these problems. His response times were consistent with application of a left-to-right computational procedure employed by expert calculators. He also responded extremely quickly to 2-digit squares, suggesting that he may have memorized the answers. In Experiment 2 we taught RH a new computational algorithm for converting temperatures from Celsius to Fahrenheit. RH was able to learn the new algorithm, but required much assistance and made only slight gains with practice. RH did not recognize an easy solution strategy based on inverting steps in the conversion algorithm. We suggest that the inflexible nature of RH's performance may reflect his autistic tendencies coupled with limited intelligence.

Block pivoting and shortcut strategies for detecting copositivity

In recently proposed quadratic optimization algorithms, copositivity detection procedures are frequently employed which deliver a feasible direction yielding a negative value of the considered quadratic form, if the answer is negative. To improve the computational performance of this routine, here (1) recursive characterizations of copositivity are presented which enable efficient reduction of the dimension of the problem using block pivoting techniques, and (2) shortcut strategies are described which are connected with diagonalization.