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Cited by 38Mathematics Education and Teaching TechniquesMathematics Education and PedagogyCognitive and developmental aspects of mathematical skills

Relational thinking: The bridge between arithmetic and algebra

Open access
Kızıltoprak, Ayhan·2017·lnternational Electronic Journal of Elementary Education
DOI 10.26822/iejee.2017131893
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Abstract

The purpose of this study is to investigate the development of relational thinking skill, which is an important component of the transition from arithmetic to algebra, of 5th grade students. In the study, the qualitative research method of teaching experiment was used. The research data were collected from six secondary school 5th grade students by means of clinical interviews and teaching episodes. For observing the development of relational thinking, pre and post clinical interviews were also conducted before and after the eight-session teaching experiment. Qualitative analysis of the research data revealed that the relational thinking skills of all the students developed. It was also found that there was an interaction between the development of fundamental arithmetic concepts and relational thinking; that the students developed concepts related to arithmetical operations such as addend and sum; minuend, subtrahend and difference; multiplicator and product; and dividend, divisor and quotient. Moreover, students were able to use these concepts effectively although they failed to provide formal explanations about the relations between them. In addition, the students perceived the equal sign not only finding a result but also as a symbol used to establish a relation between operations and expressions.

Integrity signals

Registry

Crossref · 10.26822/iejee.2017131893

Cites

27 of 27 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 39 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    The students developed concepts for arithmetic operations, including addend and sum, minuend, subtrahend and difference, multiplicator and product, and dividend, divisor and quotient.

    “that the students developed concepts related to arithmetical operations such as addend and sum; minuend, subtrahend and difference; multiplicator and product; and dividend, divisor and quotient.”
    Quote in the paperSubstantial claim
  • Findingown data

    The development of fundamental arithmetic concepts and relational thinking were interrelated in the students.

    “It was also found that there was an interaction between the development of fundamental arithmetic concepts and relational thinking;”
    Quote in the paperSubstantial claim
  • Findingown data

    The students used these arithmetic operation concepts effectively even though they could not give formal explanations of the relations among them.

    “Moreover, students were able to use these concepts effectively although they failed to provide formal explanations about the relations between them.”
    Quote in the paperSubstantial claim
  • Findingown data

    The relational thinking skills of all six participating fifth-grade students improved over the teaching experiment.

    “Qualitative analysis of the research data revealed that the relational thinking skills of all the students developed.”
    Quote in the paperSubstantial claim
  • Findingown data

    In the preliminary interviews, only Ozan and Tülay solved the equations '+ = +' [9a] and '-= -' [9b] by focusing on the difference between numbers through relational thinking and by realizing that in subtraction there will be a similar increase or decrease in both minuend and subtrahend.

    “Among the students, only Ozan and Tülay solved the equations of " + = + " [9a] and " -= -" [9b] by focusing on the difference between the numbers via relational thinking and by realizing that in subtraction, there will be a similar increase or decrease in both minuend and subtrahend.”
    Quote in the paperSubstantial claim
  • May combine two claims. 35
  • Classified as probably not a substantial claim. 17
  • The quote only partly states this claim. 1

  • May combine two claims.

    Findingown data

    The students saw the equal sign both as a way to find a result and as a symbol for relating operations and expressions.

    “In addition, the students perceived the equal sign not only finding a result but also as a symbol used to establish a relation between operations and expressions.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation

    In the preliminary interviews of this study, the students treated the equal sign not as a symbol expressing a relation but as a signal to find a result or perform an operation, or as a symbol-syntactic indicator placed before the answer.

    “it was found in the present study during the preliminary interviews that the students did not regard the equal sign as a symbol showing a relation but as "finding a result or doing something" and "a symbol of operation or a symbol-syntactic indicator used before the answer" (Warren, 2006).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Warren, 2006 source

  • May combine two claims.

    Findingown data

    Students who could not state the relational meaning of the equal sign in the preliminary interviews gave the sign a correct meaning in the final interviews as a result of the teaching process, and they took it to convey balance rather than result.

    “it was found that the students who failed to say the relational meaning of the equal sign during the preliminary interviews were able to give correct meaning to this sign during the last interviews thanks to the teaching process conducted in the study and that they considered the sign to convey the meaning of 'balance' rather than 'result'.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For questions involving the distributive property, the students solved only the number sentence based on the relation between multiplication and addition through relational thinking and had difficulty with all the other number sentences.

    “in relation to the questions involving the distributive property, it was found that the students were able to solve only the number sentence involving the relation between multiplication and addition based on relational thinking and that they experienced difficulty in all the other number sentences.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the second sub-theme, the students did not start with result-oriented thinking for the given number sentences and searched for relations between numbers and operations, but they eventually favored doing calculations because they were not completely sure about the relations.

    “According to this sub-theme, the students did not start with result-oriented thinking for the given number sentences, and they searched for the relations between numbers and operations; however, they eventually favored doing calculations as they were not completely sure about the relations.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In the last interviews, only İrem and Semih did relational thinking for the equation '+ = -' [9c] and established relations between the numbers.

    “Only İrem and Semih did relational thinking for this question and established relations between the numbers.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In the preliminary interviews, every student recognized the equal sign and described it as 'equality of two numbers' and 'finding the result of an operation'.

    “All the students reported that they recognized the equal sign, and they referred to this sign as "equality of two numbers" and "finding the result of an operation".”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the preliminary interviews, the number sentences where result-oriented thinking dominated were those involving multiplication and division, those requiring the distributive property of multiplication over addition or subtraction, and those with operations in parentheses.

    “The preliminary interviews revealed that the number sentences dominated by result-oriented thinking were those requiring multiplication and division, those requiring use of distributive property of multiplication over addition or subtraction, and those requiring operations in parentheses.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The students in this study mostly answered the open and true/false number sentences in ways consistent with result-oriented thinking and either avoided or failed to establish a relation between the numbers and operations for the equality.

    “it was seen that the students mostly provided answers to the open and true/false number sentences as appropriate to the result-oriented thinking processes and that they avoided establishing or failed to establish a relation between the numbers and operations for the equality.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Three of the students recognized the commutative property among the fundamental arithmetic properties, and one student recognized the associative property.

    “it was another striking result that three of the students managed to recognize the commutative property among the fundamental arithmetic properties and one student was able to recognize the associative property.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In the last interviews, students who used the introductory operational processes to relational thinking failed to recognize the distributive property in some of these questions and did not understand why there were more than two boxes in some of these questions.

    “It was also found that the students who used the introductory operational processes to relational thinking failed to recognize the distributive property in some of these questions and that they did not understand why there were more than two boxes in some of these questions.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In the first sub-theme, the students found the result through result-oriented thinking, but they were able to explain the relation between the operations after finding the result.

    “The students found the result via result-oriented thinking; on the other hand, they were able to explain the relation between the operations after finding the result.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In the last interviews, students with moderate and low academic achievement had difficulty in some open number sentences (2c, 6e, 8a, 8b and 8c) and did not do result-oriented thinking in any of these questions.

    “It was seen in the last interviews that students with moderate and low levels of academic achievement had difficulty in some open number sentences (2c, 6e, 8a, 8b and 8c) and that they did not do result-oriented thinking in any of these questions.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Although most students of moderate and high achievement reasoned in a result-oriented way for addition and subtraction number sentences in the preliminary interviews, they were able to explain the relations behind the results they had obtained.

    “Although most of the students with moderate and high levels of achievement thought on result-oriented basis for the number sentences involving addition and subtraction during the preliminary interviews, it was seen that they were able to explain the relations for the results they had found.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In this study, students of moderate and high achievement had difficulty with some number sentences and did not reason in a result-oriented way in any of the number sentences.

    “in the present study, it was found that the students with moderate and high levels of achievement had difficulty in some number sentences and that they did not think on result-oriented basis in an of the number sentences.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the final interviews, to solve the questions the students focused on the common multiples of the numbers in the equation and used properties of operations for questions involving multiplication and division together.

    “During the last interviews, the students, to solve the questions, focused on the common multiples of the numbers found in the equation and used the properties of operations to solve the questions which involved multiplication and division together.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Findingown data

    The students' explanations regarding their responses were thought to belong to the theme of operational process, which could be regarded as an introduction to relational thinking and which was made up of two sub-themes: explaining the relation after finding the unknown and pre-relational thinking.

    “In this respect, the students' explanations regarding their responses were thought to belong to the theme of operational process, which could be regarded as introduction to relational thinking and which was made up of two sub-themes: Explaining the relation after finding the unknown and Pre-Relational Thinking.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the preliminary interviews, the students struggled to give different meanings of the equal sign and mostly used it to find the result of an operation.

    “However, it was seen that the students had difficulty stating different meanings of the equal sign and that they generally used it to find the result of an operation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the preliminary interviews, all the students had difficulty with division, compared the divisions on each side of the equation by calculating, and gave wrong answers.

    “The preliminary interviews also demonstrated that all the students had difficulty dealing with division, compared the divisions on each side of the equation via calculation and provided wrong answers.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In this study, students initially answered the open and true/false number sentences using a result-oriented process in the preliminary interviews, and later answered the same kind of sentences using relations between numbers and operations in the final interviews.

    “the students first responded to the open and true/false number sentences based on the result-oriented process during the preliminary interviews and that they then responded to these open and true/false number sentences based on the relations between numbers and operations in the last interviews.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In the last interviews, the students developed their understanding of arithmetic operation concepts such as addend, sum, minuend, subtrahend, difference, multiplicator, product, dividend, divisor and quotient, and they used the relations between these concepts effectively.

    “It was also seen that the students developed their understanding of such concepts related to arithmetical operations as addend, sum, minuend, subtrahend, difference, multiplicator, product, dividend, divisor and quotient and that they used the relations between these concepts effectively.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Relational thinking offers a different perspective on arithmetic and is central to teaching and learning it.

    “Koehler (2004) points out that relational thinking provides a different perspective for arithmetic and plays a key role in teaching/learning it.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Koehler (2004)

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Under Boulton et al.'s three-phase model, algebraic thinking develops in sequence, and moving to a higher phase is unhealthy if the current phase is not well developed.

    “According to this model, whose phases are called "arithmetic", "early algebra" and "algebra", it is claimed that algebraic thinking in students is developed consecutively and that transition to a higher step will not be healthy if the current step has not been developed efficiently.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Boulton et al. (2000)

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation

    Relational thinking lets students restructure arithmetic operations to alter a calculation and transform number sentences using basic arithmetic properties.

    “This key role brings about two benefits which allow students not only to restructure arithmetic operations to change the given calculation but also to transform the number sentences with the use of fundamental arithmetic properties (Koehler, 2004).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Koehler, 2004

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationreasoning

    This relational focus requires relational thinking, which is important in the development of algebraic thinking.

    “This focus requires relational thinking, which has an important place in the development of algebraic thinking.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    There is no agreement in the literature about what early algebraic thinking is or what parts it has.

    “In related literature, there is no consensus on what algebraic thinking at an early age is or on what components it may have (Cai & Moyer, 2008).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Cai & Moyer, 2008

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisecitation

    Developing basic arithmetic skills enables students to write number sentences with symbols, understand core operation properties, and represent a number in multiple forms.

    “Development of fundamental arithmetic skills allows writing down number sentences with mathematical symbols, understanding the fundamental features of operations and conceptualizing a number in a wide variety of forms (5=7-2, 5=3+2, etc.).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Number sentences in elementary school mathematics textbooks are mostly presented as operations-equation-answer, and examples supporting the use of relational thinking in number sentences are limited.

    “In related literature, there are several research results which report that number sentences in elementary school mathematics course books are mostly given as operations-equation-answer and that the number of examples supporting the use of relational thinking in number sentences is limited (Köse & Tanışlı, 2011).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Köse & Tanışlı, 2011

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductown data

    The pilot revealed time-related problems and an order from easier to harder questions.

    “In the pilot study, which lasted two days, certain difficulties were observed: the students experienced time-related problems, and the questions were in an order starting with easier questions and ending with harder ones.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisecitation

    Number sentences act as a window that reveals students' mathematical thoughts.

    “Considering the fact that number sentences act as a window that reveals students' mathematical thoughts (Carpenter et al., 2003)”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Carpenter et al., 2003

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductnone given

    The revised questions were piloted again with three other students from a different class in the same school, and the interview questions were finalized.

    “The questions which were changed were piloted again with three other students from a different class in the same school, and the clinical interview questions were finalized as in Table 1.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductown data

    The interview questions were piloted with a group similar to the study participants.

    “The interview questions arranged in line with their views were piloted with a group of individuals similar to the participants of the study.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductnone given

    Participation was voluntary, and consent was obtained from the students, their parents, and the Ministry of National Education.

    “Lastly, participation to the study was on voluntary basis, and the necessary consents of the students, their parents and the Ministry of National Education were taken.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    When students learn distributive-property relations through relational-thinking-based teaching, they can more easily recognize multiplicative relations and produce better strategies for multiplying multi-digit numbers.

    “Carpenter et al. (2005) state that when students learn the relations involving the distributive property via a method of teaching based on relational thinking, they can recognize multiplicative relations more easily and produce more effective strategies for the multiplication of multi-digit numbers.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Carpenter et al. (2005) source
    • Rests on an assumption: The distributive property is essential for understanding multiplication and for developing multiplicative reasoning.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductreasoning

    The number of questions was reduced, and a pilot was run with individuals similar to the research participants.

    “Depending on their views, the number of the questions was decreased, and a pilot application was conducted with a group of individuals with characteristics similar to those of the research participants.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    That study found that most fifth-grade students considered operations on both sides of the equation and were aware of the equal sign's relational meaning in addition to its operational meaning.

    “The study demonstrated that especially most of the fifth grade students took the operations in both sides of the equation and that they were aware of the relational meaning of the equal sign besides its operational meaning.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Rittle-Johnson et al., 2011 source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given

    In the backward analysis, the data were gathered under three main themes by two independent field experts.

    “In the backward analysis, both the clinical interviews and the teaching process were examined, and the data were gathered under three main themes by two independent field experts.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data

    The students' failure to recognize the commutative and associative properties, despite these being taught in early elementary grades, indicates that students do not internalize these fundamental arithmetic properties.

    “this situation indicates that students fail to internalize these fundamental arithmetic properties.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductnone given

    Students' names were kept confidential, and nicknames were used in reporting findings.

    “While presenting the findings, the students' names were kept confidential, and nicknames were used.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductnone given

    The researchers reached an agreement on which theme the students' responses belonged to.

    “The researchers reached an agreement on which theme the students' responses belonged to.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The constant analysis allows the researcher to form and arrange the research model in line with students' knowledge, actions and tendencies.

    “The important point in this analysis is that it allows the researcher to form and arrange the research model in line with students' knowledge, actions and tendencies (Simon, 2000).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Simon, 2000 source

  • Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

    Study design and conductown data

    Field experts reviewed the prepared interview questions.

    “Following this, field experts were asked for their views about the questions prepared.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Study design and conductcitation

    The clinical interview questions were based on studies by Koehler (2004) and Carpenter et al. (2003).

    “For the preparation of the questions to be directed during the clinical interviews, studies conducted by Koehler (2004) and Carpenter et al. (2003) were taken as basis.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites Koehler (2004)

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites Carpenter et al. (2003)

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationown data

    Students' relational thinking skills can be developed at earlier ages than those of the students in this study.

    “Based on the findings obtained in the study, it is thought that students' relational thinking skills can be developed at earlier ages.”
    Quote in the paperNot a substantial claim
44 of 44

  • The paper relies on this without support.

    The distributive property is essential for understanding multiplication and for developing multiplicative reasoning.

    “Distributive property is essential for understanding the multiplication and for developing multiplicative reasoning.”

  • The paper relies on this without support.

    Students' operational or relational understanding of the equal sign influences their relational thinking and their future learning of algebraic concepts.

    “Students' operational or relational understanding of the equal sign has influence on relational thinking and on understanding the algebraic concepts they will learn in the future.”

  • The paper relies on this without support.

    The students' verbal responses and explanations during interviews accurately reflect their underlying thinking processes.

    “The students' ways of thinking during the preliminary interviews were examined under three headings: Operational process based on relational thinking, operational process as an introduction to relational thinking, and result-oriented operational process.”

  • The paper relies on this without support.

    The observed changes in students' relational thinking are attributable to the teaching process rather than other factors.

    “In this respect, it could be stated that all the students' relational thinking skills were developed.”
4 of 4

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Preliminary-interview account overlooks Tülay's exception

    Minor

    The later account says all students used result-oriented thinking on question 2c in the preliminary interviews. The earlier account explicitly exempts Tülay from that result for question 2c.

  • A named student is absent from the participant lists

    Minor

    The account names Salih as a student who answered a preliminary-interview question, but the six students listed in the results include Semih, not Salih.

  • Banknotes cannot give all six students 20 TL

    Minor

    The listed banknotes total 80 TL: four groups worth 20 TL each. Six participants receiving 20 TL apiece would require 120 TL.

3 of 3

Cites

Flagged first, then by citations

Research works printed but not linked: 9

  1. [5]5. Boulton G. M., Lewis G., Cooper T. J., Atweh B., Pillay H. & Wills L. (2000). Readiness of algebra. In T. Nakahara & M. Koyama (Eds.), Proceeding of the 24th Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 89-105). Hiroshima, Japan: International Group for the Psychology of Mathematics Education.
  2. [6]6. Cai, J. & Moyer, P. (2008). Developing algebraic thinking in earlier grades: Some insights from international comparative studies. In C. E. Greenes (Ed.), Algebra and Algebraic thinking in School Mathematics (pp.169- 179). Reston, VA: NCTM.
  3. [7]7. Carpenter, T. P., & Franke, M. L. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra (Vol. 1, pp. 155-162). Melbourne, Australia.
  4. [15]15. Hunter, J. (2007). Relational or calculational thinking: Students solving open number equivalence problems. In. J. Watson & K. Beswick (Eds.), Proceeding of the 30th Annual Conference of the Mathematics Education Research Group of Australasia-Mathematics: Essential Research, Essential Practice vol. 1 (pp. 421- 429).
  5. [23]23. Köse, N., & Tanışlı, D. (2011). İlköğretim matematik ders kitaplarında eşit işareti ve ilişkisel düşünme [Equal sign and relational thinking in elementary mathematics textbooks]. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi (EFMED), 2(5), 251-277.
  6. [30]30. Molina, M. & Ambrose, R. (2008). From an operational to a relational conception of the equal sign. Thirds graders' developing algebraic thinking. Focus on Learning Problems in Mathematics, 30(1), 61-80.
  7. [37]37. Seo, K.-H., & Ginsburg, H. P. (2003). ''You've got to carefully read the math sentence...": Classroom context and children's interpretations of the equals sign. In A. J. Baroody & A. Dowker (Eds.), The Development of Arithmetic Concepts and Skills. Mahwah, NJ: Lawrence Erlbaum.
  8. [42]42. Warren, E. (2004). Generalising arithmetic: Suporting the process in the early years, In M. J. Hoines & A. B. Fuglestad (Eds.), Proceeding of the 28th Conference of the International Group for the Psychology of Mathematics Education (vol. 4, pp. 417-424), Bergen, Norway: International Group for the Psychology of Mathematics Education.
  9. [45]45. Watanabe, T. (2008). Algebra in elementary school: A Japanese perspective. In C. Greenes (Ed.), Algebra and Algebraic Thinking in School Math: 70th Yearbook (pp. 183-194). Reston, VA:NCTM.

Cited by

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Authors
Kızıltoprak, A.
Corresponding author
Kızıltoprak, Ayhan
Venue
lnternational Electronic Journal of Elementary Education
Publisher
lnternational Electronic Journal of Elementary Education
Published
2017-09-30
Volume
10
Issue
1
Pages
131-145
ISSN
1307-9298 (ISSN-L)
DOI
10.26822/iejee.2017131893
OpenAlex
W2761703972
Cited by
38
References
24
Citations by year
2025: 4 · 2023: 4 · 2022: 6 · 2021: 10 · 2020: 7 · 2019: 3 · 2018: 4
FWCI
13.66
Type
journal article
Language
en
Open access
Locations
  • lnternational Electronic Journal of Elementary Education (publishedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
Topics
  • Social Sciences › Education › Mathematics Education and Teaching Techniques
  • Physical Sciences › Mathematics › Applied Mathematics › Mathematics Education and Pedagogy
  • Physical Sciences › Mathematics › Statistics and Probability › Cognitive and developmental aspects of mathematical skills
Keywords
relational reasoning, transition from arithmetic to algebra, fifth grade students, arithmetic concepts, equals sign, teaching experiment, clinical interview, teaching episodes

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