Information processing using number systems with bases higher than ten (original) (raw)
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The Quarterly Journal of Experimental Psychology, 2014
How do kindergarteners solve different single-digit addition problem formats? We administered problems that differed solely on the basis of two dimensions: response type (approximate or exact), and stimulus type (nonsymbolic, i.e., dots, or symbolic, i.e., Arabic numbers). We examined how performance differs across these dimensions, and which cognitive mechanism (mental model, transcoding, or phonological storage) underlies performance in each problem format with respect to working memory (WM) resources and mental number line representations. As expected, nonsymbolic problem formats were easier than symbolic ones. The visuospatial sketchpad was the primary predictor of nonsymbolic addition. Symbolic problem formats were harder because they either required the storage and manipulation of quantitative symbols phonologically or taxed more WM resources than their nonsymbolic counterparts. In symbolic addition, WM and mental number line results showed that when an approximate response was needed, children transcoded the information to the nonsymbolic code. When an exact response was needed, however, they phonologically stored numerical information in the symbolic code. Lastly, we found that more accurate symbolic mental number line representations were related to better performance in exact addition problem formats, not the approximate ones. This study extends our understanding of the cognitive processes underlying children's simple addition skills.
Acta Psychologica, 2010
Recent research has suggested addition performance to be determined by both the need for a carry operation and problem size. Nevertheless, it has remained debatable, how these two factors are interrelated. In the current study, this question was pursued by orthogonally manipulating carry and problem size in two-digit addition verification. As the two factors interacted reliably, our results indicate that the carry effect is moderated by number magnitude processing rather than representing a purely procedural, asemantic sequence of processing steps. Moreover, it was found that the carry effect may not be a purely categorical effect but may be driven by continuous characteristics of the sum of the unit digits as well. Since the correct result of a carry problem can only be derived by integrating and updating the magnitudes of tens and units within the place-value structure of the Arabic number system, the present study provides evidence for the idea that decomposed processing of tens and units also transfers to mental arithmetic.
Arithmetic Training Does Not Improve Approximate Number System Acuity
Frontiers in Psychology, 2016
The approximate number system (ANS) is thought to support non-symbolic representations of numerical magnitudes in humans. Recently much debate has focused on the causal direction for an observed relation between ANS acuity and arithmetic fluency. Here we investigate if arithmetic training can improve ANS acuity. We show with an experimental training study consisting of six 45-min training sessions that although feedback during arithmetic training improves arithmetic performance substantially, it does not influence ANS acuity. Hence, we find no support for a causal link where symbolic arithmetic training influences ANS acuity. Further, although short-term number memory is likely involved in arithmetic tasks we did not find that short-term memory capacity for numbers, measured by a digit-span test, was effected by arithmetic training. This suggests that the improvement in arithmetic fluency may have occurred independent of short-term memory efficiency, but rather due to long-term memory processes and/or mental calculation strategy development. The theoretical implications of these findings are discussed.
Cognitive Capacity Demands of Two Addition Algorithms
Perceptual and Motor Skills, 1981
18 college students solved addition problems with either an addition algorithm that requires a written record of intermediate results and running sums or with the standard algorithm which does not require such a written record. Students who used the experimental algorithm had significantly faster reaction times to a tone than those who used the traditional algorithm. This result supports the view that arithmetic tasks are composed of several subtasks which are performed simultaneously. A capacity model of attention is suggested as an appropriate paradigm to use in investigating arithmetic processing.
A memory-based account of automatic numerosity processing
Memory & Cognition, 2005
We investigated the mechanisms responsible for the automatic processing of the numerosities represented by digits in the size congruity effect (Henik & Tzelgov, 1982). The algorithmic model assumes that relational comparisons of digit magnitudes (e.g., larger than {8,2}) create this effect. If so, congruity effects ought to require two digits. Memory-based models assume that associations between individual digits and the attributes "small" and "large" create this effect. If so, congruity effects ought only to require one digit. Contrary to the algorithmic model and consistent with memory-based models, congruity effects were just as large when subjects judged the relative physical sizes of small digits paired with letters as when they judged the relative physical sizes of two digits. This finding suggests that size congruity effects can be produced without comparison algorithms.
Novel Approach to the Learning of Various Number Systems
International Journal of Computer Applications, 2011
A number system is a set of rules and symbols used to represent a number, or any system used for naming or representing numbers is called a number system also known as numeral system. Almost everyone is familiar with decimal number system using ten digits. However digital devices especially computers use binary number system instead of decimal, using two digits i.e. 0 and 1 based on the fundamental concept of the decimal number system. Various other number systems also used this fundamental concept of decimal number system i.e. quaternary, senary, octal, duodecimal, quadrodecimal, hexadecimal and vigesimal number system using four, six, eight, twelve, fourteen, sixteen, and twenty digits respectively. The awareness and concept of various number systems, their number representation, arithmetic operations, compliments and the inter conversion of numbers belong different number system is essential for understanding of digital aspects. More over, the successful programming for digital devices require the understanding of various number systems and their inter conversion. Understanding all these number systems and particularly the inter conversion of numbers requires allot of time and techniques to expertise. In this paper the concepts of the most common number systems, their representation, arithmetic, compliments and interconversion is taken under the consideration in tabulated form. It will provide an easy understanding and practising of these number systems to understand as well as memorise them. Few of these number systems are binary, quaternary, senary, octal, decimal, duodecimal, quadrodecimal, hexadecimal and vigecimal.
Extending the Mental Number Line: A Review of MultiDigit Number Processing
Zeitschrift Fur Psychologie-journal of Psychology, 2011
Multi-digit number processing is ubiquitous in our everyday life – even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive
Number Systems, Base Conversions, and Computer Data Representation
When we write decimal (base 10) numbers, we use a positional notation system. Each digit is multiplied by an appropriate power of 10 depending on its position in the number: For example: 843 = 8 x 10 2 + 4 x 10 1 + 3 x 10 0 = 8 x 100 + 4 x 10 + 3 x 1 = 800 + 40 + 3 For whole numbers, the rightmost digit position is the one's position (10 0 = 1). The numeral in that position indicates how many ones are present in the number. The next position to the left is ten's, then hundred's, thousand's, and so on. Each digit position has a weight that is ten times the weight of the position to its right.