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Grammatical units for the algebraic notation considered in this paper, and the predicted order in which the elements of the units are scanned.
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The present study investigates how experienced users of mathematics parse algebraic expressions. The main issues examined are the order in which the symbols in an expression are scanned and the duration of fixation. Two experiments tracked the order in which the symbols of an expression were scanned. The results were analysed using Markov Chain mod...
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Context 1
... algebraic notation considered in this study is the most commonly used subset of modern algebraic notation. It consists of the seven types of grammatical units illustrated in Figure 2, where each grey square can be replaced by any syntactically well-formed subexpression. For example, I'/I can become 8x 3 ' 1 x where the first grey box corresponds to the subexpression 8x 3 , and the second grey box corresponds to 1 x : The right side of Figure 2 contains our hypothesised order (given by arrows) in which the components of a grammatical unit are processed, based on a left-to-right, top-to-bottom processing direction. ...
Context 2
... consists of the seven types of grammatical units illustrated in Figure 2, where each grey square can be replaced by any syntactically well-formed subexpression. For example, I'/I can become 8x 3 ' 1 x where the first grey box corresponds to the subexpression 8x 3 , and the second grey box corresponds to 1 x : The right side of Figure 2 contains our hypothesised order (given by arrows) in which the components of a grammatical unit are processed, based on a left-to-right, top-to-bottom processing direction. (Note that the first component to be examined, indicated by the longer arrow, can be approached from any direction.) ...
Context 3
... the proportion of trials for which the initial symbol was one of the first three symbols fixated was influenced by the different expression groups, F 1 (7, 154) 0/3.97, p B/.05; F 2 (7, 56) 0/5.32, p B/.05. A further break- down of these results indicated that the initial symbol is one of the first three fixations in a significantly lower proportion of trials if the expression begins with a fraction (groups 1, 2, and 4), than if it does not (a .06 difference), F 1 (1, 22) 0/9.73, pB/.05; ...
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... In the present study, there was no evidence that the participants paid little attention to the task, as the total time devoted to read the example was high (about 6 times the time needed to read it once). In addition, students' mean reading speed for text segments (122 word/min) was clearly lower than the reading speed in educated adults for normal texts, 200-400 word/min (Rayner et al., 2016), and the value for algebra (129 symbol/min) was also slower than the one (250 symbols/min) obtained by Andrà et al. (2015) in a study of students' fixation times reading algebraic expressions, and by Jansen et al. (2007) with experts. Regardless, more studies are needed to reliably determine the normal reading speeds in similar monitoring tasks. ...
This study explores the process itself of comprehension monitoring of worked-out examples in mathematics. A ‘reversal error’ was embedded in a worked-out example of algebraic nature. Ninety-four engineers in a master’s degree program to become secondary teachers of technology were asked to judge the comprehensibility of the statement and the resolution provided, and to report in writing any incoherence, inconsistency, or error they might detect. The participants’ mental processes throughout the task were operationalized through behavioural variables based on a psychological mechanism proposed for inconsistency detection. The behavioural variables focused on the monitoring of important mathematical processes, the algebraic translation, and the interpretation of the numerical solution of the worked example. The software ‘Read and Answer’ was used to record online data on each participant’s behaviour while monitoring the example, as well as his/her written partial and final reports (the task products). An individual short interview was conducted to increase the reliability of the study. Data from each participant were first analysed. Secondly, data from all the participants were considered together in statistical analyses aimed at relating behavioural variables to task products. Four student monitoring profiles were identified corresponding to different combinations of detection/overlooking the embedded algebraic inconsistency, and detection/overlooking the subsequent inconsistency in the result: ‘competent monitoring’, ‘delayed monitoring’, ‘blocked monitoring’, and ‘poor monitoring’ students. Implications for teaching are discussed.
... To our knowledge, no baseline measure exists of symbol insertion order when writing fractions by hand, so any comparisons of digital models to handwriting must assume a natural sequence. A left-to-right, top-tobottom order has been suggested when handwriting mathematical expressions in general [4], and it has been shown that mathematical expressions are read in a similar scanning order [3,6]; does the insertion of symbols during the writing of fractions follow this same sequence? Here we present Markov transition probability matrices, generated from transcription tasks which took place during an eye-tracking experiment designed to evaluate the performance of software applications for digitally typesetting mathematics [17]. ...
What-you-see-is-what-you-get word processing editors that support the typesetting of mathematical content allow for direct manipulation of 2-dimensional mathematical notation within a 1-dimensional environment. The models used to represent mathematical structure generally fall into two categories: (1) free-form, in which symbols are placed and moved freely in the workspace and (2) structure-based, where 2-dimensional structures are inserted and then populated with other structures or symbols. To understand the consequences of these models for writing mathematical expressions we examined the sequential order of symbol insertion when transcribing fractions, and compare this to the natural order employed when handwritten. Markov transition matrices were created from recordings of fraction transcriptions to quantify the probability of inserting a symbol immediately following insertion of other symbols. The three writing techniques yielded different dominant transition patterns and also differed in their ability to tolerate differences from the dominant pattern. This study likely represents the first empirical assessment of symbol insertion order probabilities when handwriting fractions, while also identifying differences in insertion order during use of different types of digital editors for typesetting mathematical content. These results have important implications for the usability of mathematical typesetting software applications and should be considered in the future design of such programs.KeywordsMathematical softwareSoftware user interfacesFractionsMathematics education
... Kim and colleagues [37] found that Markov models are well suited for paradigms that contain moving stimuli. Jansen et al. [38] reported which reading strategies are used by experts who read algebraic expressions. Moreover, Markov models have been used to predict human eye fixations [39] and to highlight differences in scan path patterns among subjects [40]. ...
Autism spectrum disorder (ASD) is a neurodevelopmental condition in which visual attention and visual search strategies are altered. Eye-tracking paradigms have been used to detect these changes. In our study, 18 toddlers with ASD and 18 toddlers with typical development (TD; age range 12–36 months) underwent an eye-tracking paradigm where a face was shown together with a series of objects. Eye gaze was coded according to three areas of interest (AOIs) indicating where the toddlers’ gaze was directed: ‘Face’, ‘Object’, and ‘No-stimulus fixation’. The fixation sequence for the ASD and TD groups was modelled with a Markov chain model, obtaining transition probabilities between AOIs. Our results indicate that the transition between AOIs could differentiate between toddlers with ASD or TD, highlighting different visual exploration patterns between the groups. The sequence of exploration is strictly conditioned based on previous fixations, among which ‘No-stimulus fixation’ has a critical role in differentiating the two groups. Furthermore, our analyses underline difficulties of individuals with ASD to engage in stimulus exploration. These results could improve clinical and interventional practice by considering this dimension among the evaluation process.
... For algebra units Jansen et al. (2007) computed mean values of 240 ms for fixation times when experts read algebraic expressions. Andrà et al., (2015) obtained mean values of 190-250 ms for fixation times in a sample of students. ...
... Andrà et al., (2015) obtained a mean value of about 240 ms/symbol for students' average fixation times reading algebraic expressions. This mean value is very similar to the mean value obtained if experts' data obtained by Jansen et al. (2007) is applied to units R1-R5 of the experimental problem used in the present study. However, in the present experiment participants' mean reading speed was clearly slower, 536 ms/symbol (SD = 335), or 112 algebra symbols per minute. ...
This study aimed at testing an extension of a theoretical model for the metacognitive monitoring mechanism implied in the detection of inconsistencies when the information provided includes abstract symbols in addition to plain text. Ninety-four postgraduates of STEM specialities were asked to read a worked-out algebra-problem example and to report any incoherence, inconsistency, or error detected in the statement or in the solving procedure. A set of model-inspired indexes was defined to describe participants’ behaviour along the task. The Read & Answer software was used to record online individual processing data and participants’ reports. Results supported model predictions. Indexes correctly predicted participants’ outcomes in the task with high accuracy. Specific students’ behaviours could be associated to observed task outcomes with sufficient reliability within the limitations of the study. In addition, algebra processing was compared with plain text processing.
... In [13], Hayes et al. presented a novel method to process sequential eye-tracking data by using successor representation on Raven's advanced progressive matrices (APM). They pointed out the limitations of other methods such as transition probability matrices and Markov models [14,35] and gained unprecedented accuracy to predict APM scores. Mavrikis and Chen et, al. discussed the relationship between eye-movement patterns and GPA [4,26]. ...
The difficulty level of learning tasks is a concern that often needs to be considered in the teaching process. Teachers usually dynamically adjust the difficulty of exercises according to the prior knowledge and abilities of students to achieve better teaching results. In e-learning, because there is no teacher involvement, it often happens that the difficulty of the tasks is beyond the ability of the students. In attempts to solve this problem, several researchers investigated the problem-solving process by using eye-tracking data. However, although most e-learning exercises use the form of filling in blanks and choosing questions, in previous works, research focused on building cognitive models from eye-tracking data collected from flexible problem forms, which may lead to impractical results. In this paper, we build models to predict the difficulty level of spatial visualization problems from eye-tracking data collected from multiple-choice questions. We use eye-tracking and machine learning to investigate (1) the difference of eye movement among questions from different difficulty levels and (2) the possibility of predicting the difficulty level of problems from eye-tracking data. Our models resulted in an average accuracy of 87.60% on eye-tracking data of questions that the classifier has seen before and an average of 72.87% on questions that the classifier has not yet seen. The results confirmed that eye movement, especially fixation duration, contains essential information on the difficulty of the questions and it is sufficient to build machine-learning-based models to predict difficulty level.
... With this 'rigged-up' visual system that has resulted from perceptual learning, experts are able to more efficiently parse the syntax of complex expressions. It has been demonstrated that adults are sensitive to arithmetic syntax, as memory tasks show that fragments of arithmetic equations are more easily recalled when they are syntactically valid ("2 + 5") rather than invalid ("x 5 +") (Jansen, Marriott & Yelland, 2007). Perceptual learning, therefore, has crafted perceptual routines that can enforce procedural knowledge of an abstract nature. ...
Research into symbolic processing of algebra suggests that expertise is characterised by a reliance on perceptual routines to carry out tasks of an abstract nature. While symbolic mathematics is fundamentally an abstract exercise, evidence suggests that long-term exposure to algebraic and arithmetic symbols ‘rig-up’ the visual system to perceptually group together higher-order sub-expressions, and selectively attend to these sub-expressions like objects (object-based attention). The perception of object-hood for these sub-expressions confers attentional benefits, insofar as shifting attention within an object is more fluent than between objects. Syntax-knowers, who understand the order of operations, have demonstrated an ability to attend to multiplicative sub-expressions in a manner consistent with how people attend to singular objects. Moreover, this attentional benefit is not observed in syntax non-knowers. This implies that knowledge of the abstract-relations of operation hierarchy forms the basis of perceptual grouping. However, said effects have not been found in arithmetic expressions. This is noteworthy, as a difference in notational flexibility for algebra may be driving this effect of perceptual grouping. Therefore, the current study examined whether syntax-knowers demonstrated such attentional benefits for arithmetic expressions as has been observed in algebra. Forty-three undergraduates (who were syntax-knowers) completed a task that required them to identify numbers that were either clustered around a multiplication symbol (within a perceptual group) or an addition sign (between perceptual groups). If attention to sub- expressions were sensitive to operation hierarchy, syntax-knowers should be able to react to stimuli faster in the perceptual group, compared to outside the perceptual group. Results showed that syntax-knowers did not experience any attentional benefits for stimuli located within the perceptual group, compared to outside it. Future studies should adopt certain paradigm-based changes (proposed in Discussion) before any conclusive statements regarding
attentional benefits in arithmetic expressions are made.
... Behavioral studies have indicated that processing of A-expressions is likely based on the syntactic structures 4,5 . For example, participants moved their eyes along with syntactic structures of sentences 6 . ...
Abstract The presence of a shared neural system for the syntactic processing in language and arithmetic is controversial. Recent behavioral studies reported a cross-domain structural priming between language and arithmetic. Using functional magnetic resonance imaging, we examined whether the neural activation reflects the structural interaction between language and arithmetic. We prepared sentences and arithmetic expressions (A-expressions) with same and different syntactic structures and presented structurally congruent/incongruent pairs consecutively. By directly comparing activations in the congruent and incongruent conditions, we observed significant repetition suppression effect in the regions including the bilateral inferior frontal gyrus, i.e., neural activation with an A-expression decreased after a sentence with the same syntactic structure (and vice versa). The results strongly support the idea that arithmetic and language share the neural basis for processing syntactic structures.
... Despite differences between texts with and without mathematical notation, there are also similarities in how they are read. A study analysing reading patterns reveals that pauses are taken in the same way and parsing (identifying parts and relations between parts) are done in similar ways when reading mathematical notation in algebraic expressions as when reading natural language (Jansen, Marriott, & Yelland, 2007). ...
When students solve mathematics tasks, the tasks are commonly given as written text, usually consisting of natural language, mathematical notation and different types of images. This is one reason why reading and interpreting such texts are important parts of being mathematically proficient, at least within the school context. The ability utilized when dealing with aspects of mathematical text is denoted in this thesis as a mathematical reading ability; this ability is useful when reading mathematical language, for example, in task text. There is, however, a lack of knowledge of what characterizes this mathematical language, what students need to learn regarding the mathematical language, and exactly which mathematical language that tests should preferably assess. Therefore, the purpose of this thesis is to contribute to the knowledge of aspects of difficulty related to textual features in mathematics tasks. In particular, one aim is to distinguish between a difficulty that has to do with a mathematical ability and another that has not. Different types of text analyses are utilized to capture textural features that might be demanding for the students when reading and solving mathematics tasks. Aspects regarding vocabulary are investigated both in a literature review and in a study where corpora are used to analyse word commonness. Other textual analyses focus on textual features that concern mathematical notation and images, besides natural language. Statistical methods are used to analyse potential relations between the textual features of interest and both task difficulty and task demand on reading ability. The results from the research review are sparse regarding difficult vocabulary, since few of the reviewed studies analyses word aspects separately. Several of the analysed textual features are related to aspects of difficulty. The results show that tasks with more words that are uncommon both in a mathematical context and in an everyday context, may favour students with good reading ability rather than students with good mathematical ability. Another textual feature that is likely to be demanding for students, is if the task texts contains many meaning relations, for example, when several words refer to the same or similar object. These results have implications for the school practice both regarding textual features that are important from an educational perspective and regarding the construction of tests. The research does also contribute to an understanding of what characterizes a mathematical language
... Apparently, syntactic structure influences encoding and subsequent recall of algebraic expressions. Several computational models assume that human processing of algebraic expressions begins with, and is subsequently guided by, syntactic parsing (Anderson, 2005(Anderson, , 2009Jansen, Marriott, & Yelland, 2007). ...
... Syntactic parsing, by contrast, depends only on syntactic structure, not on content, and should therefore be insensitive to the number values involved in an expression. This insensitivity to number values is implicit in the computational models of algebra processing mentioned earlier 1 (Anderson, 2005(Anderson, , 2009Jansen et al., 2007) and explicit in a recent model of arithmetic processing . Maruyama et al. (2012) proposed that such processing begins with a syntactic stage, in which syntactic structure is determined based on the positions and identities of bracket and operator symbols. ...
The idea that cognitive development involves a shift towards abstraction has a long history in psychology. One incarnation of this idea holds that development in the domain of mathematics involves a shift from non-formal mechanisms to formal rules and axioms. Contrary to this view, the present study provides evidence that reliance on non-formal mechanisms may actually increase with age. Participants – Dutch primary school children – evaluated three-term arithmetic expressions in which violation of formally correct order of evaluation led to errors, termed foil errors. Participants solved the problems as part of their regular mathematics practice through an online study platform, and data were collected from over 50,000 children representing approximately 10% of all primary schools in the Netherlands, suggesting that the results have high external validity. Foil errors were more common for problems in which formally lower-priority sub-expressions were spaced close together, and also for problems in which such sub-expressions were relatively easy to calculate. We interpret these effects as resulting from reliance on two non-formal mechanisms, perceptual grouping and opportunistic selection, to determine order of evaluation. Critically, these effects reliably increased with participants' grade level, suggesting that these mechanisms are not phased out but actually become more important over development, even when they cause systematic violations of formal rules. This conclusion presents a challenge for the shift towards abstraction view as a description of cognitive development in arithmetic. Implications of this result for educational practice are discussed.
... d on the ease with which subsequent evaluation may be performed under different possible parsings. This conclusion offers a challenge to theoretical models in which perceived structure depends only on syntactically-relevant featuresi.e. order and position of operators and parenthesesand not on specific number values (Anderson, 2005(Anderson, , 2009A. R. Jansen et al., 2007;Maruyama et al., 2012). Additionally, it offers a challenge to any serial model of processing in which perception of structure precedes and influences evaluation, but is not influenced by it (e.g., . ...
Most existing models of reasoning in arithmetic and algebra assume that formal syntactic structure guides human processing of symbolic expressions. The present experiment investigated the possibility that formally extraneous features also exert systematic influences on perceived structure. Participants – Dutch primary school children – evaluated three-term arithmetic expressions in which physical spacing between symbols and specific number values were manipulated to encourage perception of sub-expressions either consistent or inconsistent with formally correct order of evaluation. Participants solved the problems as part of their regular mathematics practice through an online study platform, and data were collected from over 50,000 children representing approximately 10% of all primary schools in the Netherlands, suggesting that the results have high external validity. Analysis revealed systematic effects of both symbol spacing and number values on perceived structure. First, adjacent symbols were more likely to be evaluated as part of a single group if they were spaced closely together. Second, sub-expressions that were easy to evaluate due to the specific number values involved were more likely to be prioritized for evaluation. Both effects led to systematic violations of formal rules regarding order of evaluation. The findings indicate a need to incorporate perceptual and procedural constraints, as well as formal constraints, into theoretical accounts of human perception of structure in symbolic expressions. Furthermore, the strength of the aforementioned effects increased with participants’ grade level, suggesting that they do not result from transitional strategies that are phased out during development, but may instead reflect processes intrinsic to mature competence in mathematics. Implications of this result for educational practice are discussed.
