Analysis of Students’ Misconceptions and Error Patterns in Mathematics: The Case of Fractions

This study analyzed three fifth grade students’ misconceptions and error patterns when working with equivalence, addition and subtraction of fractions. The findings revealed that students used both conceptual and procedural knowledge to solve the problems. They used pictures, gave examples, and made connections to other mathematical concepts and to daily life topics. Error patterns found include using addition and subtraction of numerators and denominators, and finding the greatest common factor.

Convolutional Codes for Network-Error Correction

In this work, we introduce convolutional codes for network-error correction in the context of coherent network coding. We give a construction of convolutional codes that correct a given set of error patterns, as long as consecutive errors are separated by a certain interval. We also give some bounds on the field size and the number of errors that can get corrected in a certain interval. Compared to previous network error correction schemes, using convolutional codes is seen to have advantages in field size and decoding technique. Some examples are discussed which illustrate the several possible situations that arise in this context.

Network Error Correction for Unit-Delay, Memory-Free Networks Using Convolutional Codes

A single source network is said to be memory-free if all of the internal nodes (those except the source and the sinks) do not employ memory but merely send linear combinations of the symbols received at their incoming edges on their outgoing edges. In this work, we introduce network-error correction for single source, acyclic, unit-delay, memory-free networks with coherent network coding for multicast. A convolutional code is designed at the source based on the network code in order to correct network- errors that correspond to any of a given set of error patterns, as long as consecutive errors are separated by a certain interval which depends on the convolutional code selected. Bounds on this interval and the field size required for constructing the convolutional code with the required free distance are also obtained. We illustrate the performance of convolutional network error correcting codes (CNECCs) designed for the unit-delay networks using simulations of CNECCs on an example network under a probabilistic error model.

Bounds on Probability of Error in Decision Feedback Equalizers

Evaluation of the probability of error in decision feedback equalizers is difficult due to the presence of a hard limiter in the feedback path. This paper derives the upper and lower bounds on the probability of a single error and multiple error patterns. The bounds are fairly tight. The bounds can also be used to select proper tap gains of the equalizer.

On 2-D Non-Adjacent-Error Channel Models

In this work, we consider two-dimensional (2-D) binary channels in which the 2-D error patterns are constrained so that errors cannot occur in adjacent horizontal or vertical positions. We consider probabilistic and combinatorial models for such channels. A probabilistic model is obtained from a 2-D random field defined by Roth, Siegel and Wolf (2001). Based on the conjectured ergodicity of this random field, we obtain an expression for the capacity of the 2-D non-adjacent-errors channel. We also derive an upper bound for the asymptotic coding rate in the combinatorial model.

Dynamics of nonlinear error growth and season-dependent predictability of El Nino events in the Zebiak-Cane model

With the intermediate-complexity Zebiak-Cane model, we investigate the 'spring predictability barrier' (SPB) problem for El Nino events by tracing the evolution of conditional nonlinear optimal perturbation (CNOP), where CNOP is superimposed on the El Nino events and acts as the initial error with the biggest negative effect on the El Nino prediction. We show that the evolution of CNOP-type errors has obvious seasonal dependence and yields a significant SPB, with the most severe occurring in predictions made before the boreal spring in the growth phase of El Nino. The CNOP-type errors can be classified into two types: one possessing a sea-surface-temperature anomaly pattern with negative anomalies in the equatorial central-western Pacific, positive anomalies in the equatorial eastern Pacific, and a thermocline depth anomaly pattern with positive anomalies along the Equator, and another with patterns almost opposite to those of the former type. In predictions through the spring in the growth phase of El Nino, the initial error with the worst effect on the prediction tends to be the latter type of CNOP error, whereas in predictions through the spring in the decaying phase, the initial error with the biggest negative effect on the prediction is inclined to be the former type of CNOP error. Although the linear singular vector (LSV)-type errors also have patterns similar to the CNOP-type errors, they cover a more localized area than the CNOP-type errors and cause a much smaller prediction error, yielding a less significant SPB. Random errors in the initial conditions are also superimposed on El Nino events to investigate the SPB. We find that, whenever the predictions start, the random errors neither exhibit an obvious season-dependent evolution nor yield a large prediction error, and thus may not be responsible for the SPB phenomenon for El Nino events. These results suggest that the occurrence of the SPB is closely related to particular initial error patterns. The two kinds of CNOP-type error are most likely to cause a significant SPB. They have opposite signs and, consequently, opposite growth behaviours, a result which may demonstrate two dynamical mechanisms of error growth related to SPB: in one case, the errors grow in a manner similar to El Nino; in the other, the errors develop with a tendency opposite to El Nino. The two types of CNOP error may be most likely to provide the information regarding the 'sensitive area' of El Nino-Southern Oscillation (ENSO) predictions. If these types of initial error exist in realistic ENSO predictions and if a target method or a data assimilation approach can filter them, the ENSO forecast skill may be improved. Copyright (C) 2009 Royal Meteorological Society

Year 6 students' idiosyncratic notions of unitising, reunitising and regrouping decimal number places

Having flexible notions of the unit (e.g., 26 ones can be thought of as 2.6 tens, 1 ten 16 ones, 260 tenths, etc.) should be a major focus of elementary mathematics education. However, often these powerful notions are relegated to computations where the major emphasis is on "getting the right answer" thus procedural knowledge rather than conceptual knowledge becomes the primary focus. This paper reports on 22 high-performing students' reunitising processes ascertained from individual interviews on tasks requiring unitising, reunitising and regrouping; errors were categorised to depict particular thinking strategies. The results show that, even for high-performing students, regrouping is a cognitively complex task. This paper analyses this complexity and draws inferences for teaching.

A timing-specific memory distortion effect in young children

It has been suggested that there are systematic distortions in children's memory for temporal durations, such that children's memory is not just less accurate than that of adults but qualitatively different. Experiment I replicated the memory distortion effect by demonstrating developmental change in the tendency to confuse a reference duration with one that is shorter rather than longer than it. When the long-term memory demands of the task were reduced by providing reminders of the reference duration on every trial, there were no such qualitative changes in error patterns (Experiment 2). Further evidence for developmental changes in memory distortion was found in the temporal generalization task of Experiment 3, in which stimuli were spaced logarithmically rather than linearly. In Experiment 4, a similar distortion pattern was absent in a task in which children made judgments about the pitch rather than the duration of stimuli, suggesting the effect may be specific to time estimation. (C) 2003 Elsevier Inc. All rights reserved.

A cognitive and psycholinguistic investigation of neologisms

Background: Jargon aphasia with neologisms (i.e., novel nonword utterances) is a challenging language disorder that lacks a definitive theoretical description as well as clear treatment recommendations (Marshall, 2006). Aim: The aims of this two part investigation were to determine the source of neologisms in an individual with jargon aphasia (FF), to identify potential facilitatory semantic and/or phonological cuing effects in picture naming, and to determine whether the timing of the cues relative to the target picture mediated the cuing advantage. Methods and Procedures: FF’s underlying linguistic deficits were determined using several cognitive and linguistic tests. A series of computerized naming experiments using a modified version of the 175 item-Philadelphia Naming Test (Roach, Schwartz, Martin, Grewal, & Brecher, 1996) manipulated the cue type (semantic versus phonological) and relatedness (related versus unrelated). In a follow-up experiment, the relative timing of phonological cues was manipulated to test the effect of timing on the cuing advantage. The accuracy of naming responses and error patterns were analyzed. Outcome and Results: FF’s performance on the linguistic and cognitive test battery revealed a severe naming impairment with relatively spared word and nonword repetition, auditory comprehension of words and monitoring, and fairly well preserved semantic abilities. This performance profile was used to evaluate various explanations for neologisms including a loss of phonological codes, monitoring failure, and impairments in semantic system. The primary locus of his deficit appears to involve the connection between semantics to phonology, specifically, when word production involves accessing the phonological forms following semantic access. FF showed a significant cuing advantage only for phonological cues in picture naming, particularly when the cue preceded or coincided with the onset of the target picture. Conclusions: When integrated with previous findings, the results from this study suggest that the core deficit of this and at least some other jargon aphasics is in the connection from semantics to phonology. The facilitative advantage of phonological cues could potentially be exploited in future clinical and research studies to test the effectiveness of these cues for enhancing naming performance in individuals like FF.

Using seasonal hindcasts to understand the origin of the equatorial cold tongue bias in CGCMs and its impact on ENSO

The cold equatorial SST bias in the tropical Pacific that is persistent in many coupled OAGCMs severely impacts the fidelity of the simulated climate and variability in this key region, such as the ENSO phenomenon. The classical bias analysis in these models usually concentrates on multi-decadal to centennial time series needed to obtain statistically robust features. Yet, this strategy cannot fully explain how the models errors were generated in the first place. Here, we use seasonal re-forecasts (hindcasts) to track back the origin of this cold bias. As such hindcasts are initialized close to observations, the transient drift leading to the cold bias can be analyzed to distinguish pre-existing errors from errors responding to initial ones. A time sequence of processes involved in the advent of the final mean state errors can then be proposed. We apply this strategy to the ENSEMBLES-FP6 project multi-model hindcasts of the last decades. Four of the five AOGCMs develop a persistent equatorial cold tongue bias within a few months. The associated systematic errors are first assessed separately for the warm and cold ENSO phases. We find that the models are able to reproduce either El Niño or La Niña close to observations, but not both. ENSO composites then show that the spurious equatorial cooling is maximum for El Niño years for the February and August start dates. For these events and at this time of the year, zonal wind errors in the equatorial Pacific are present from the beginning of the simulation and are hypothesized to be at the origin of the equatorial cold bias, generating too strong upwelling conditions. The systematic underestimation of the mixed layer depth in several models can also amplify the growth of the SST bias. The seminal role of these zonal wind errors is further demonstrated by carrying out ocean-only experiments forced by the AOCGCMs daily 10-meter wind. In a case study, we show that for several models, this forcing is sufficient to reproduce the main SST error patterns seen after 1 month in the AOCGCM hindcasts.

Production of definite and indefinite articles in typically developing English-speaking children and children with SLI

The present paper examines the production of definite and indefinite articles in English-speaking typically developing (TD) children and children with Specific Language Impairment (SLI). Twenty four English-speaking children with SLI (mean age: 7;5), twenty nine TD age-matched (TD-AM) children (mean age: 7;5) and eleven younger (mean age: 5;5) TD vocabulary-matched (TD-VM) children participated in a production task involving short stories without picture props based on Schafer and de Villiers (2000). Article production was examined in two different semantic contexts for the definite article, namely in the anaphoric and the bridging context. In the anaphoric condition, definiteness is established via linguistic means, whereas in the bridging condition via shared world knowledge. Indefinite article production was examined in the referential specific, non-referential predicational, and non-referential instrumental contexts. The referential specific context involves [+speaker, −hearer] knowledge and the non-referential predicational and instrumental [−speaker, −hearer] knowledge. Results showed that in the definite article contexts, all three groups performed better on the bridging compared with the anaphoric condition; in the indefinite article contexts, they had better performance on the non-referential predicational vs. the referential specific and the non-referential instrumental conditions. In terms of errors, the TD-VM children and the children with SLI produced significantly more substitutions than the TD-AM children in the definite article contexts. In the indefinite article contexts, the three groups did not differ in terms of accuracy or error patterns. The present results point towards problems in the discourse integration of entities that are part of the speaker's and hearer's knowledge in children with SLI and TD-VM controls, especially in definite articles. These problems are accentuated in the children with SLI due to their grammatical impairment and suggest that children with SLI exhibit a delayed acquisition profile.

The acquisition of verbal paradigms in Dutch and Greek L2 children: cross-linguistic differences and inflectional defaults

Previous research with children learning a second language (L2) has reported errors with verb inflection and cross-linguistic variation in accuracy and error patterns. However, owing to the cross-linguistic complexity and diversity of different verbal paradigms, the cross-linguistic effects on the nature of default forms has not been directly addressed in L2 acquisition studies. In the present study, we compared accuracy and error patterns in verbal agreement inflections in L2 children acquiring Dutch and Greek, keeping the children’s L1 constant (Turkish). Results showed that inflectional defaults in Greek follow universal predictions regarding the morphological underspecification of paradigms. However, the same universal predictions do not apply to the same extent to Dutch. It is argued that phonological properties of inflected forms should be taken into account to explain cross-linguistic differences in the acquisition of inflection. By systematically comparing patterns in child L2 Dutch and Greek, this study shows how universal mechanisms and target language properties work in tandem in the acquisition of inflectional paradigms.

Use of electropalatography in the treatment of disordered articulation following traumatic brain injury: A case study

Electropalatography (EPG) was used as a biofeedback tool in a case study of a 30-year-old male with disordered articulation following traumatic brain injury (TBI). Based on qualitative measures of the participant's intelligibility, improved articulation of the fricatives /s/ and /integral/ were selected as treatment targets. Therapy was administered three times a week for 5 weeks. Results showed that word and sentence intelligibility increased approximately 10%, and error patterns for lingual articulation indicated that fricative -> stop and other fricative errors decreased considerably. EPG measures for /s/ exhibited a significantly more anterior main focus of articulatory contact post therapy. Consonant durations were significantly longer during weeks 3 and 4, and this finding was associated with the emergence of an articulatory contact pattern with a groove rather than complete closure. This articulatory pattern appeared inconsistently and was found to vary across articulations of /s/ but also within a single consonant production. For /integral/, the amount of contact was significantly reduced post therapy and an increase in duration was noted during week 4, similar to that occurring in the production of /s/. Spatial and timing measures were more variable than in normal speakers of English and indicated a general increase in variability across weeks for both /s/ and /integral/. It was concluded that, although the correct fricative patterns appeared only intermittently during production of the consonants, there seemed to be sufficient information for the listener to be able to classify the sound as a fricative. As a part of an intervention program, visual EPG biofeedback therapy would appear to have a definite role in assisting dysarthric speakers exhibiting difficulties with lingual articulation in understanding their errors, learning how to exploit kinesthetic, and acoustic sources of feedback, and how to make appropriate adjustments in tongue articulation to increase the level of speech intelligibility.

CROSS-LINGUISTIC TRANSFER OF SPELLING SKILLS IN SPANISH-SPEAKING ADULT ESL LEARNERS

Spelling is an important literacy skill, and learning to spell is an important component of learning to write. Learners with strong spelling skills also exhibit greater reading, vocabulary, and orthographic knowledge than those with poor spelling skills (Ehri & Rosenthal, 2007; Ehri & Wilce, 1987; Rankin, Bruning, Timme, & Katkanant, 1993). English, being a deep orthography, has inconsistent sound-to-letter correspondences (Seymour, 2005; Ziegler & Goswami, 2005). This poses a great challenge for learners in gaining spelling fluency and accuracy. The purpose of the present study is to examine cross-linguistic transfer of English vowel spellings in Spanish-speaking adult ESL learners. The research participants were 129 Spanish-speaking adult ESL learners and 104 native English-speaking GED students enrolled in a community college located in the South Atlantic region of the United States. The adult ESL participants were in classes at three different levels of English proficiency: advanced, intermediate, and beginning. An experimental English spelling test was administered to both the native English-speaking and ESL participants. In addition, the adult ESL participants took the standardized spelling tests to rank their spelling skills in both English and Spanish. The data were analyzed using robust regression and Poisson regression procedures, Mann-Whitney test, and descriptive statistics. The study found that both Spanish spelling skills and English proficiency are strong predictors of English spelling skills. Spanish spelling is also a strong predictor of level of L1-influenced transfer. More proficient Spanish spellers made significantly fewer L1-influenced spelling errors than less proficient Spanish spellers. L1-influenced transfer of spelling knowledge from Spanish to English likely occurred in three vowel targets (/ɑɪ/ spelled as ae, ai, or ay, /ɑʊ/ spelled as au, and /eɪ/ spelled as e). The ESL participants and the native English-speaking participants produced highly similar error patterns of English vowel spellings when the errors did not indicate L1-influenced transfer, which implies that the two groups might follow similar trajectories of developing English spelling skills. The findings may help guide future researchers or practitioners to modify and develop instructional spelling intervention to meet the needs of adult ESL learners and help them gain English spelling competence.