Possible number systems

Number systems—such as the natural numbers, integers, rationals, reals, or complex numbers—play a foundational role in mathematics, but these systems can present difficulties for students. In the studies reported here, we probed the boundaries of people’s concept of a number system by asking them whether “number lines” of varying shapes qualify as possible number systems. In Experiment 1, participants rated each of a set of number lines as a possible number system, where the number lines differed in their structures (a single straight line, a step-shaped line, a double line, or two branching structures) and in their boundedness (unbounded, bounded below, bounded above, bounded above and below, or circular). Participants also rated each of a group of mathematical properties (e.g., associativity) for its importance to number systems. Relational properties, such as associativity, predicted whether participants believed that particular forms were number systems, as did the forms’ ability to support arithmetic operations, such as addition. In Experiment 2, we asked participants to produce properties that were important for number systems. Relational, operation, and use-based properties from this set again predicted ratings of whether the number lines were possible number systems. In Experiment 3, we found similar results when the number lines indicated the positions of the individual numbers. The results suggest that people believe that number systems should be well-behaved with respect to basic arithmetic operations, and that they reject systems for which these operations produce ambiguous answers. People care much less about whether the systems have particular numbers (e.g., 0) or sets of numbers (e.g., the positives).     

Arithmetic knowledge from the spontaneous attention to relations

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A ? B = C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants’ Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants’ APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non‐symbolic number does not generalize to attention to relations with symbolic arithmetic.     

The role of task rules and stimulus–response mappings in the task switching paradigm

Switch costs occur whenever participants are asked to switch between two or more task sets. In a typical task switching experiment, participants have to switch between two task sets composed of up to four different stimuli per task set. These 2 (task sets) x 4 (stimuli) contain only 8 different stimulus-response (S-R) mappings, and the question is why participants base their task performance on task sets instead of S-R mappings. The current experiments compared task performance based on task rules with performance based on single stimulus-response mappings. Participants were led to learn eight different S-R mappings with or without fore-knowledge about two underlying task sets. Without task set information no difference between shifts and repetitions occurred, whereas introducing task sets at the beginning led to significant switch costs. Most importantly, introducing task sets in the middle of the experiment also resulted in significant switch costs. Furthermore, introducing task rules at the beginning of the experiment lead to slower RTs when simple stimuli (Experiment 1) had to be processed. This detrimental effect disappeared with more complex stimuli (Experiment 2). Results will be discussed with respect to cognitive control.     

Pseudoneglect for the bisection of mental number lines

Patients with unilateral neglect of the left side bisect physical lines to the right whereas individuals with an intact brain bisect lines slightly to the left (pseudoneglect). Similarly, for mental number lines, which are arranged in a left-to-right ascending sequence, neglect patients bisect to the right. This study determined whether individuals with an intact brain show pseudoneglect for mental number lines. In Experiment 1, participants were presented with visual number triplets (e.g., 16, 36, 55) and determined whether the numerical distance was greater on the left or right side of the inner number. Despite changing the spatial configuration of the stimuli, or their temporal order, the numerical length on the left was consistently overestimated. The fact that the bias was unaffected by physical stimulus changes demonstrates that the bias is based on a mental representation. The leftward bias was also observed for sets of negative numbers (Experiment 2)—demonstrating not only that the number line extends into negative space but also that the bias is not the result of an arithmetic distortion caused by logarithmic scaling. The leftward bias could be caused by a rounding-down effect. Using numbers that were prone to large or small rounding-down errors, Experiment 3 showed no effect of rounding down. The task demands were changed in Experiment 4 so that participants determined whether the inner number was the true arithmetic centre or not. Participants mistook inner numbers shifted to the left to be the true numerical centre—reflecting leftward overestimation. The task was applied to 3 patients with right parietal damage with severe, moderate, or no spatial neglect (Experiment 5). A rightward bias was observed, which depended on the severity of neglect symptoms. Together, the data demonstrate a reliable and robust leftward bias for mental number line bisection, which reverses in clinical neglect. The bias mirrors pseudoneglect for physical lines and most likely reflects an expansion of the space occupied by lower numbers on the left side of the line and a contraction of space for higher numbers located on the right.     

Pseudoneglect for the bisection of mental number lines

Patients with unilateral neglect of the left side bisect physical lines to the right whereas individuals with an intact brain bisect lines slightly to the left (pseudoneglect). Similarly, for mental number lines, which are arranged in a left-to-right ascending sequence, neglect patients bisect to the right. This study determined whether individuals with an intact brain show pseudoneglect for mental number lines. In Experiment 1, participants were presented with visual number triplets (e.g., 16, 36, 55) and determined whether the numerical distance was greater on the left or right side of the inner number. Despite changing the spatial configuration of the stimuli, or their temporal order, the numerical length on the left was consistently overestimated. The fact that the bias was unaffected by physical stimulus changes demonstrates that the bias is based on a mental representation. The leftward bias was also observed for sets of negative numbers (Experiment 2)--demonstrating not only that the number line extends into negative space but also that the bias is not the result of an arithmetic distortion caused by logarithmic scaling. The leftward bias could be caused by a rounding-down effect. Using numbers that were prone to large or small rounding-down errors, Experiment 3 showed no effect of rounding down. The task demands were changed in Experiment 4 so that participants determined whether the inner number was the true arithmetic centre or not. Participants mistook inner numbers shifted to the left to be the true numerical centre--reflecting leftward overestimation. The task was applied to 3 patients with right parietal damage with severe, moderate, or no spatial neglect (Experiment 5). A rightward bias was observed, which depended on the severity of neglect symptoms. Together, the data demonstrate a reliable and robust leftward bias for mental number line bisection, which reverses in clinical neglect. The bias mirrors pseudoneglect for physical lines and most likely reflects an expansion of the space occupied by lower numbers on the left side of the line and a contraction of space for higher numbers located on the right.     

Children's Acquisition of Arithmetic Principles: The Role of Experience

The current study investigated how young learners' experiences with arithmetic equations can lead to learning of an arithmetic principle. The focus was elementary school children's acquisition of the Relation to Operands principle for subtraction (i.e., for natural numbers, the difference must be less than the minuend). In Experiment 1, children who viewed incorrect, principle-consistent equations and those who viewed a mix of incorrect, principle-consistent and principle-violation equations both showed gains in principle knowledge. However, children who viewed only principle-consistent equations did not. We hypothesized that improvements were due in part to improved encoding of relative magnitudes. In Experiment 2, children who practiced comparing numerical magnitudes increased their knowledge of the principle. Thus, experience that highlights the encoding of relative magnitude facilitates principle learning. This work shows that exposure to certain types of arithmetic equations can facilitate the learning of arithmetic principles, a fundamental aspect of early mathematical development.     

The representations of the arithmetic operations include functional relationships

Current theories of mathematical problem solving propose that people select a mathematical operation as the solution to a problem on the basis of a structure mapping between their problem representation and the representation of the mathematical operations. The structure-mapping hypothesis requires that the problem and the mathematical representations contain analogous relations. Past research has demonstrated that the problem representation consists of functional relationships, or principles. The present study tested whether people represent analogous principles for each arithmetic operation (i.e., addition, subtraction, multiplication, and division). For each operation, college (Experiments 1 and 2) and 8th grade (Experiment 2) participants were asked to rate the degree to which a set of completed problems was a good attempt at the operation. The pattern of presented answers either violated one of four principles or did not violate any principles. The distance of the presented answers from the correct answers was independently manipulated. Consistent with the hypothesis that people represent the principles, (1) violations of the principles were rated as poorer attempts at the operation, (2) operations that are learned first (e.g., addition) had more extensive principle representations than did operations learned later (multiplication), and (3) principles that are more frequently in evidence developed more quickly. In Experiment 3, college participants rated the degree to which statements were indicative of each operation. The statements were either consistent or inconsistent with one of two principles. The participants' ratings showed that operations with longer developmental histories had strong principle representations. The implications for a structure-mapping approach to mathematical problem solving are discussed.     

Access to information in working memory: exploring the focus of attention

Participants memorized briefly presented sets of digits, a subset of which had to be accessed as input for arithmetic tasks (the active set), whereas another subset had to be remembered independently of the concurrent task (the passive set). Latencies for arithmetic operations were a function of the setsize of active but not passive sets. Object-switch costs were observed when successive operations were applied to different digits within an active set. Participants took 2 s to encode a passive set so that it did not affect processing latencies (Experiment 2). The results support a model distinguishing 3 states of representations in working memory: the activated part of long-term memory, a capacity limited region of direct access, and a focus of attention.     

Number line estimation and complex mental calculation: Is there a shared cognitive process driving the two tasks?

It is widely accepted that different number-related tasks, including solving simple addition and subtraction, may induce attentional shifts on the so-called mental number line, which represents larger numbers on the right and smaller numbers on the left. Recently, it has been shown that different number-related tasks also employ spatial attention shifts along with general cognitive processes. Here we investigated for the first time whether number line estimation and complex mental arithmetic recruit a common mechanism in healthy adults. Participants’ performance in two-digit mental additions and subtractions using visual stimuli was compared with their performance in a mental bisection task using auditory numerical intervals. Results showed significant correlations between participants’ performance in number line bisection and that in two-digit mental arithmetic operations, especially in additions, providing a first proof of a shared cognitive mechanism (or multiple shared cognitive mechanisms) between auditory number bisection and complex mental calculation.     

Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.     

The use of heuristics in intuitive mathematical judgment

Anecdotal evidence points to the use of beauty as an indication of truth in mathematical problem solving. In the two experiments of the present study, we examined the use of heuristics and tested the assumption that participants use symmetry as a cue for correctness in an arithmetic verification task. We manipulated the symmetry of sets of dot pattern addition equations. Speeded decisions about the correctness of these equations led to higher endorsements for both correct and incorrect equations when the addend and sum dot patterns were symmetrical. Therefore, this effect is not due to the fact that symmetry facilitates calculation or estimation. We found systematic evidence for the use of heuristics in solving mathematical tasks, and we discuss how these findings relate to a processing-fluency account of intuition in mathematical judgment.     

Belief bias in the perception of sample size adequacy

In two experiments participants were instructed to set aside their own, complete knowledge of a statistical population parameter and to take the perspective of an agent whose knowledge was limited to a random sample. Participants rated the appropriateness of the agent's conclusion about the adequacy of the sample size (which, objectively, was more than adequate). They also rated the agent's intelligence. Whereas previous work suggests that unbelievable statistical conclusions impact reasoning by provoking critical thought which enhances the detection of research flaws, the present studies presented participants an unflawed scenario designed to assess effects of believability on bias. The results included the finding that participants’ complete knowledge did indeed bias their perceptions not only of the adequacy of the sample size, but also of the rationality of the agent drawing the conclusion from the sample. The findings were interpreted in the context of research on belief bias, social attribution, and Theory of Mind.     

Contingency learning with evaluative stimuli: testing the generality of contingency learning in a performance paradigm

In two experiments, we tested the generality of the learning effects in the recently-introduced color-word contingency learning paradigm. Participants made speeded evaluative judgments to valenced target words. Each of a set of distracting nonwords was presented most often with either positive or negative target words. We observed that participants responded faster on trials that respected these contingencies than on trials that contradicted the contingencies. The contingencies also produced changes in liking: in a subsequent explicit evaluative rating task, participants rated positively-conditioned nonwords more positively than negatively-conditioned nonwords. Interestingly, contingency effects in the performance task correlated with this explicit rating effect in both experiments. In Experiment 2, all effects reported were independent of subjective and objective contingency awareness (which was completely lacking), even when awareness was measured at the item level. Our results reveal that learning in this type of performance task extends to nonword-valence contingencies and to responses different from those emitted during the performance task. We discuss the implications of these findings for theories about the processes that underlie contingency learning in performance tasks and for research on evaluative conditioning.     

Cue combination in human spatial navigation

This project investigated the ways in which visual cues and bodily cues from self-motion are combined in spatial navigation. Participants completed a homing task in an immersive virtual environment. In Experiments 1A and 1B, the reliability of visual cues and self-motion cues was manipulated independently and within-participants. Results showed that participants weighted visual cues and self-motion cues based on their relative reliability and integrated these two cue types optimally or near-optimally according to Bayesian principles under most conditions. In Experiment 2, the stability of visual cues was manipulated across trials. Results indicated that cue instability affected cue weights indirectly by influencing cue reliability. Experiment 3 was designed to mislead participants about cue reliability by providing distorted feedback on the accuracy of their performance. Participants received feedback that their performance with visual cues was better and that their performance with self-motion cues was worse than it actually was or received the inverse feedback. Positive feedback on the accuracy of performance with a given cue improved the relative precision of performance with that cue. Bayesian principles still held for the most part. Experiment 4 examined the relations among the variability of performance, rated confidence in performance, cue weights, and spatial abilities. Participants took part in the homing task over two days and rated confidence in their performance after every trial. Cue relative confidence and cue relative reliability had unique contributions to observed cue weights. The variability of performance was less stable than rated confidence over time. Participants with higher mental rotation scores performed relatively better with self-motion cues than visual cues. Across all four experiments, consistent correlations were found between observed weights assigned to cues and relative reliability of cues, demonstrating that the cue-weighting process followed Bayesian principles. Results also pointed to the important role of subjective evaluation of performance in the cue-weighting process and led to a new conceptualization of cue reliability in human spatial navigation.     

Working memory in event‐ and time‐based prospective memory tasks: Effects of secondary demand and age

An experiment is reported examining the role of working memory in two laboratory‐based prospective memory (PM) tasks. Participants viewed a film for a later recognition memory task while simultaneously monitoring auditorially presented arithmetic problems for incorrect solutions. The arithmetic verification task was either low demand or high demand. In addition, participants were required either to indicate whenever an animal appeared in the film (event‐based PM task), or whenever 3 min had elapsed (time‐based PM task). PM performance was higher when the arithmetic task was low demand than when it was high demand. Young participants were more successful in both PM tasks than older participants, but only under high demand. Age did not interact with PM task type overall, and the young participants were faster overall in both types of PM task. Taken together, the results indicate that working memory plays an important role in PM tasks.     

The Impact of Implicit Self-Theory on Rated Creativity in Third- and Sixth-Graders

A total of 114 3rd- and 6th-graders from a suburban elementary school participated in a study examining the influence of implicit self-theories on rated creativity in the domains of art and literature in a quasi-experiment. Creativity was measured via the Consensual Assessment Technique. Participants were divided into two groups and received different sets of instructions emphasizing either an incremental or entity-implicit theory of creativity, before completing a drawing task and a writing task. Implicit theories of creativity of participants were measured before and after the instructions were given. The results indicated that participants in the incremental group showed increased incremental views following the manipulation; participants in the entity group showed no significant change. The writing (but not the artwork) from the incremental group was rated as more creative than those from the entity group. This effect was stronger for third graders than for sixth graders, suggesting that implicit theory interventions are more effective with younger children. Implications are discussed.     

Eye fixation behaviour in the number bisection task: Evidence for temporal specificity

Together with magnitude representations, knowledge about multiplicativity and parity contributes to numerical problem solving. In the present study, we used eye tracking to document how and when multiplicativity and parity are recruited in the number bisection task. Fourteen healthy adults evaluated whether the central number of a triplet (e.g., 21_24_27) corresponds to the arithmetic integer mean of the interval defined by the two outer numbers. We observed multiplicativity to specifically affect gaze duration on numbers, indicating that the information of multiplicative relatedness is activated at early processing stages. In contrast, parity only affected total reading time, suggesting involvement in later processing stages. We conclude that different representational features of numbers are available and integrated at different processing stages within the same task and outline a processing model for these temporal dynamics of numerical cognition.     

Preschoolers prior formal mathematics education engage numerical magnitude representation rather than counting principles in symbolic ±1 arithmetic: Evidence from the operational momentum effect

In numerical cognition research, the operational momentum (OM) phenomenon (tendency to overestimate the results of addition and/or binding addition to the right side and underestimating subtraction and/or binding it to the left side) can help illuminate the most basic representations and processes of mental arithmetic and their development. This study is the first to demonstrate OM in symbolic arithmetic in preschoolers. It was modeled on Haman and Lipowska's (2021) non-symbolic arithmetic task, using Arabic numerals instead of visual sets. Seventy-seven children (4–7 years old) who know Arabic numerals and counting principles (CP), but without prior school math education, solved addition and subtraction problems presented as videos with one as the second operand. In principle, such problems may be difficult when involving a non-symbolic approximate number processing system, whereas in symbolic format they can be solved based solely on the successor/predecessor functions and knowledge of numerical orders, without reference to representation of numerical magnitudes. Nevertheless, participants made systematic errors, in particular, overestimating results of addition in line with the typical OM tendency. Moreover, subtraction and addition induced longer response times when primed with left- and right-directed movement, respectively, which corresponds to the reversed spatial form of OM. These results largely replicate those of non-symbolic task and show that children at early stages of mastering symbolic arithmetic may rely on numerical magnitude processing and spatial-numerical associations rather than newly-mastered CP and the concept of an exact number.     

An eye for relations: eye-tracking indicates long-term negative effects of operational thinking on understanding of math equivalence

Prior knowledge in the domain of mathematics can sometimes interfere with learning and performance in that domain. One of the best examples of this phenomenon is in students’ difficulties solving equations with operations on both sides of the equal sign. Elementary school children in the U.S. typically acquire incorrect, operational schemata rather than correct, relational schemata for interpreting equations. Researchers have argued that these operational schemata are never unlearned and can continue to affect performance for years to come, even after relational schemata are learned. In the present study, we investigated whether and how operational schemata negatively affect undergraduates’ performance on equations. We monitored the eye movements of 64 undergraduate students while they solved a set of equations that are typically used to assess children’s adherence to operational schemata (e.g., 3 + 4 + 5 = 3 + __). Participants did not perform at ceiling on these equations, particularly when under time pressure. Converging evidence from performance and eye movements showed that operational schemata are sometimes activated instead of relational schemata. Eye movement patterns reflective of the activation of relational schemata were specifically lacking when participants solved equations by adding up all the numbers or adding the numbers before the equal sign, but not when they used other types of incorrect strategies. These findings demonstrate that the negative effects of acquiring operational schemata extend far beyond elementary school.     

Preserved spatial memory over brief intervals in older adults

Two studies compared young and older adults' memory for location information after brief intervals. Experiment 1 found that accuracy of intentional spatial memory for individual locations was similar in young and older participants for set sizes of 3 and 6. Both groups also encoded individual locations in relation to the larger configuration of locations. Experiment 2 showed that like young adults, older adults' latency to respond to a test probe in a letter working memory task was negatively influenced by spatial information that was irrelevant to the task. This interference effect indicated preserved incidental memory for spatial information in older adults. Together, these data suggest that initial encoding of spatial information for relatively small numbers of items is largely preserved in healthy older adults and that representations of spatial information persist over short intervals.