Making sense of the minus sign or becoming flexible in ‘negativity’

Vlassis, Joëlle

doi:10.1016/j.learninstruc.2004.06.012

Cited by 102 publications

(109 citation statements)

References 7 publications

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“…Negative numbers are thought to be conceptually difficult for both children and adults (De Cruz, 2006). Although no one has addressed the question of whether negative numbers interfere with solving arithmetic problems, there is evidence suggesting that negative terms (e.g., 6 -x = 4) might be more difficult for students to process than positive terms in algebraic equations (Peterson & Aller, 1971;Vlassis, 2004). We hypothesized that arithmetic problems with negative numbers may be processed differently because they usually require subtraction, rather than because of the negative number, per se.…”

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confidence: 99%

“…On this view, we expect a three-way interaction among format, problem size, and operation such that the format effects will be smallest for the ostensibly hardest problems (i.e., recasted large subtraction). Such an interaction is counter-intuitive if negative numbers are considered to be more difficult than positive numbers (e.g., Vlassis, 2004), but existing research on negative numbers in comparison tasks favors a componential view (Ganor-Stern & Tzelgov, 2008;Shaki & Petrusic, 2005).…”

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Negative Numbers in Simple Arithmetic

Das

LeFevre

Penner-Wilger

2010

Quarterly Journal of Experimental Psychology

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Are negative numbers processed differently than positive numbers in arithmetic problems?In two experiments, adults (N = 66) solved standard addition and subtraction problems such as 3 + 4 and 7 -4 and recasted versions that included explicit negative signs, that is, 3 -(-4), 7 + (-4), and (-4) + 7. Solution times on the recasted problems were slower than on standard problems, but the effect was much larger for addition than subtraction. The negative sign may prime subtraction in both kinds of recasted problems. Problem size effects were the same or smaller in recasted as compared to standard problems, suggesting that the recasted formats did not interfere with mental calculation. These results suggest that the underlying conceptual structure of the problem (i.e., addition vs. subtraction) is more important for solution processes than the presence of negative numbers.Negative Numbers -3 Negative Numbers in Simple ArithmeticAre negative numbers processed differently than positive numbers in arithmetic problems? Negative numbers are thought to be conceptually difficult for both children and adults (De Cruz, 2006). Although no one has addressed the question of whether negative numbers interfere with solving arithmetic problems, there is evidence suggesting that negative terms (e.g., 6 -x = 4) might be more difficult for students to process than positive terms in algebraic equations (Peterson & Aller, 1971;Vlassis, 2004). We hypothesized that arithmetic problems with negative numbers may be processed differently because they usually require subtraction, rather than because of the negative number, per se. Subtraction is slower and more error-prone than addition (Campbell & Xue, 2001;LeFevre, DeStefano, Penner-Wilger, & Daley, 2006;Seyler, Kirk, & Ashcraft, 2003). On this view, the conceptual structure of the problem (i.e., the mental operation that is required) is the critical determinant of processing difficulty rather than the presence of a negative number.In the present research, adults solved addition and subtraction problems in standard formats, that is 3 + 4 and 7 -4, and as addition or subtraction of negative numbers in recasted formats, that is, 3 -(-4), 7 + (-4) and (-4) + 7. In the recasted formats, the conceptual structure of the problem did not change but the format was manipulated to explicitly isolate a negative number. Shaki and Petrusic (2005) outlined two hypotheses about how people process negative numbers. According to the magnitude-polarity hypothesis, the magnitude of the number and the polarity (i.e., positive vs. negative) are processed independently, as distinct features of the stimulus. In contrast, according to the number-line hypothesis, people represent negative and positive numbers along a number line, with negative numbers extending to infinity on the left and positive numbers extending to infinity on the right. Accordingly, the negative or positive status of the number is integral to mental processing. Ganor-Stern and Tzelgov (2008) The most common use of negative signs in arit...

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confidence: 99%

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confidence: 99%

Negative Numbers in Simple Arithmetic

Das

LeFevre

Penner-Wilger

2010

Quarterly Journal of Experimental Psychology

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“…At the next level of learning, such as algebra, students can algebraically model correctly but incorrectly in their countdown operations (Seng, 2012). Based on students' difficulties, Vlassis (2004) suggest that the minus sign plays a major role in the development of understanding and using negative numbers.…”

Section: Activity 4: Arithmetic Operation Without Toolsmentioning

confidence: 99%

Analysis of Didactical Contracts on Teaching Mathematics: A Design Experiment on a Lesson of Negative Integers Operations

Fuadiah

Suryadi

Turmudi

2017

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This paper presents the analysis of teaching of Math that took place in the classroom to identify the characteristics of didactical contracts that occur as part of a didactical situation. The design experiment was conducted on 31 seventh grade students and a mathematics teacher on negative integer operations lessons. The researchers analyzed the ongoing learning, how the didactical situation stages evolve in the teacher-student interaction that allows the formation of new knowledge or concepts in the student and how the teacher organizes the responsibility for achieving the learning objectives. The Analysis showed that students, at an early stage, can perform negative integers operations by utilizing the basic concepts they get in primary school. This concept can bridge into a new knowledge that identifies the properties of integer count operations and builds power of mind on problem-solving. Keywords: Didactical Contract, Didactical Situation, Negative Integers Operations. AbstrakArtikel ini menganalisis pengajaran matematika yang berlangsung di dalam kelas dalam mengindentifikasi karakteristik kontrak didaktis yang terjadi sebagai bagian dari situasi didaktis. Disain eksperimen dilakukan terhadap 31 siswa kelas 7 dan seorang guru matematika pada pelajaran operasi bilangan bulat negatif. Peneliti melakukan analisis terhadap pembelajaran yang berlangsung, bagaimana tahap-tahap situasi didaktis berkembang dalam interaksi guru-siswa yang memungkinkan terbentuknya pengetahuan baru atau konsep pada siswa dan bagaimana guru mengorganisir tanggung jawab untuk mencapai tujuan pembelajaran. Analisis menunjukkan bahwa siswa, pada tahap awal, dapat melakukan operasi hitung bilangan bulat negatif dengan memanfaatkan konsep dasar yang mereka dapatkan di sekolah dasar. Konsep ini dapat menjembatani menuju pengetahuan yang baru yaitu mengidentifikasi sifat-sifat operasi hitung bilangan bulat dan membangun daya pikir pada pemecahan masalah.

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“…Interestingly, the initial belief that a proof may be correct, such as Andrew Wiles' proof of Fermat's last theorem, usually does not depend on thorough scrutiny, but on concurrence with high-level ideas long before the details are checked (Thurston, 2006). Mathematics education is organized as a stepwise progression from more elementary to complex notions, as can be witnessed in students who learn to solve equations with negative terms: they build on their already acquired ability to solve equations involving positive terms (Vlassis, 2004). By explaining the individual learning of mathematical concepts in this way, from more elementary to complex, we are faced with the problem of origin of the most elementary mathematical knowledge, which cannot be derived from earlier knowledge.…”

Section: How We Acquire Mathematical Knowledgementioning

confidence: 99%

“…In her study of adolescents' understanding of negative numbers, Vlassis (2004) noticed how 14-year-olds made surprisingly many errors on equations with negative terms. Interviews with these pupils revealed that they primarily relied on procedural rules that they had learned by heart, such as 'if both terms are negative, the sum is negative'.…”

Section: The Case Of Negative Numbersmentioning

confidence: 99%

Mathematical symbols as epistemic actions

Cruz

Smedt

2010

Synthese

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Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to express mathematical concepts-they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition.

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Making sense of the minus sign or becoming flexible in ‘negativity’

Cited by 102 publications

References 7 publications

Negative Numbers in Simple Arithmetic

Negative Numbers in Simple Arithmetic

Analysis of Didactical Contracts on Teaching Mathematics: A Design Experiment on a Lesson of Negative Integers Operations

Mathematical symbols as epistemic actions

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