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Cited by 42Mathematics Education and Teaching TechniquesMathematics Education and PedagogyMathematics Education and Programs

Generalizations of third and fifth graders within a functional approach to early algebra

Open access
Pinto, Eder, Cañadas, María C.·2021·Mathematics Education Research Journal
DOI 10.1007/s13394-019-00300-2
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Abstract

We describe 24 third (8–9 years old) and 24 fifth (10–11 years old) graders’ generalization working with the same problem involving a function. Generalizing and representing functional relationships are considered key elements in a functional approach to early algebra. Focusing on functional relationships can provide insights into how students work with two or more covarying quantities rather than isolated computations, and focusing on representations can help to identify the type of representations useful to them. The goals of this study are to (1) describe the functional relationships evidenced in students’ responses and (2) describe the representations that the students use. In addressing these research objectives, we describe student responses drawn from a Classroom Teaching Experiment in each grade. We analyzed students’ written responses to different questions designed to generalize the relationships in a problem that involves the function y = 2x + 6. Our findings illustrate that 11 third graders and 19 fifth graders provide evidence of functional relationships in their responses. Three third graders and all fifth graders generalized the relationship. We conclude that these differences may be due to the students’ previous classroom mathematical experiences, since students in higher grades would be more likely to focus on the relationships between variables, whereas third graders would focus on the details of arithmetic computations. In addition, we find that natural language is the main vehicle used to generalize in both grades. Unlike third graders, fifth graders perceive general rules from the numerical calculation and express these generalizations even when not explicitly requested to do so.

AI snapshot

Written from the paper’s full text.

TL;DR

This study describes how 24 third graders (8–9 years old) and 24 fifth graders (10–11 years old) generalized the linear function y=2x+6 in a tiles problem. More fifth graders evidenced and generalized functional relationships than third graders, with natural language the main representational vehicle in both grades.

Key findings

  • 11 third graders and 19 fifth graders provided evidence of functional relationships; 13 third graders and 5 fifth graders did not.
  • Three third graders generalized the relationship, while all 19 fifth graders who provided evidence of functional relationships generalized it. Among fifth graders, 16 generalized only when prompted in Q5, and 3 generalized in both specific-value questions and the general case.
  • Correspondence was the most frequent functional relationship; covariation was identified in 2 third graders’ answers (T03 and T12), and recursive patterning was not identified in third graders. Fifth graders showed correspondence only.
  • Natural language was the main vehicle used to generalize in both grades. Third graders who generalized did so when explicitly asked in Q5; fifth graders expressed generalizations even when not explicitly requested. Four fifth graders used algebraic notation spontaneously (F01, F05, F06, F08).

Methods

  • The study drew on a four-session Classroom Teaching Experiment (CTE) in each grade; this paper reports the last session, the tiles problem involving y=2x+6. Participants were 24 third graders (8–9 years old) and 24 fifth graders (10–11 years old) from one school in southern Spain, with medium-high socioeconomic level and no prior algebra, generalization, or function instruction.
  • Students answered a worksheet with Q1–Q4 on specific values (5, 8, 10, and 100 white tiles) and Q5 on the general case. Written responses were coded for functional relationships (recursive patterns, correspondence, covariation) and representations (natural language, pictorial, manipulative, numerical, algebraic notation, tabular, graphical).
  • The first author coded responses and the second checked them; calibration and joint coding sessions were used, with a reliability coefficient greater than 90%.

Limitations

  • Stated in the paper: analyzing written responses may be a limitation because worksheets may not capture all of students’ ideas; interviews are suggested for future research.
  • Evident from the text: the sample comes from one school with only one third-grade and one fifth-grade classroom, and the analysis focuses on a single CTE session/problem, which limits generalizability.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1007/s13394-019-00300-2

Cites

52 of 52 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

1 of 8 cited sources checked · 1 supported · 0 flagged

  • Findingown data

    All 19 fifth graders who provided evidence of a functional relationship generalized the relationship involved in the tiles problem (y=2x+6).

    “Nineteen of the 24 fifth graders provided evidence of a functional relationship; all of these generalized the relationship involved in the tiles problem (y=2x+6).”
    Quote in the paperSubstantial claim
  • Findingown data

    F01 used the same functional relationship for 8, 10, and 100 tiles (Q2, Q3, Q4).

    “This student used the same functional relationship for 8, 10, and 100 tiles (Q2, Q3 and Q4).”
    Quote in the paperNot a substantial claim
  • Findingown data

    Of the 16 students who generalized when prompted, 12 expressed the generalization in terms of a rule that would yield the function y=2x+6 in algebraic notation.

    “Of the 16 students who generalized when prompted, 12 expressed the generalization in terms of a rule that would yield the function y=2x+6 in algebraic notation (e.g., F01's answer in Figure 8).”
    Quote in the paperNot a substantial claim
  • Findingown data

    Three fifth-grade students generalized when answering questions involving specific values and the general case (F05, F06, and F08).

    “Sixteen students only generalized when answering Q5, and three students generalized when answering questions involving specific values and the general case (F05, F06, and F08).”
    Quote in the paperNot a substantial claim
  • Findingown data

    As Table 6 shows, and considering students' responses to Q1 and Q2 (5 and 8 white tiles, respectively), students tend to express the relationships in the same way, numerically and/or using natural language.

    “As Table 6 shows, and considering students' responses to Q1 and Q2 (5 and 8 white tiles, respectively), students tend to express the relationships in the same way, numerically and/or using natural language.”
    Quote in the paperNot a substantial claim
  • Classified as probably not a substantial claim. 11
  • May combine two claims. 21
  • The quote only partly states this claim. 3

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

    Interpretationcitation

    Our study adds three more criteria: the relationship between variables identified by students; the variety of representations used in addition to algebraic notation; and the type of question in which students generalized (spontaneous or prompted generalization).

    “Our study adds three more: (a) the relationship between variables identified by students; (b) the variety of representations used in addition to algebraic notation; and (c) the type of question in which students generalized (spontaneous or prompted generalization, in the terms of Pinto & Cañadas, 2018).”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites Pinto & Cañadas, 2018 source

  • May combine two claims.

    Findingown data

    The other two third-grade students who generalized in Q5 did so incorrectly; their answers referred only to the 6 tiles needed on the right and left, i.e., the constant term.

    “The other 2 students who generalized in Q5 did so incorrectly; their answers referred only to the 6 tiles needed on the right and left, i.e., the constant term in the Covariation was identified in 2 students' answers (T03 and T12).”
    Quote in the paperNot a substantial claim

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

    Findingown data

    Three students generalized in different questions on the worksheet.

    “As shown in Table 6, three students generalized in different questions on the worksheet.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingcitation

    These third-grade students were consistent in perceiving numerical regularity among variables in Q1-Q4 (their words reveal the general rule), but they could not answer when asked for the general relationship for any number (Q5).

    “These students were consistent in perceiving numerical regularity among variables in Q1-Q4 (their words reveal the general rule), but they could not answer when were asked for the general relationship for any number (Q5).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Some third-grade students only provide evidence of functional relationships when working with specific values (T3, T5, T6, T12, T13, T14, T19, and T24), while others generalize the relationship (T9, T11, and T22).

    “Some only provide evidence of functional relationships when working with specific values (T3, T5, T6, T12, T13, T14, T19, and T24), while others generalize the relationship (T9, T11, and T22).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Unlike third graders, fifth graders perceive general rules from the numerical calculation and express these generalizations even when not explicitly requested to do so.

    “Unlike third graders, fifth graders perceive general rules from the numerical calculation and express these generalizations even when not explicitly requested to do so.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    In some cases, like that of T09, students can express this generality for any value, while other students (e.g., the 8 third graders) do not express the general rule for any value.

    “In some cases, like that of T09, students can express this generality for any value, while other students (e.g., the 8 third graders) do not express the general rule for any value.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Of the 11 third graders considered, only two used pictorial representation (T19 and T24), and only one (T3) used manipulative.

    “Of the 11 third graders considered, only two used pictorial representation (T19 and T24), and only one (T3) used manipulative.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingown data

    Evidence of fifth graders' generalization was found in responses to Q1, Q2, and Q5.

    “As Table 6 shows, evidence of students' generalization was found in responses to Q1, Q2, and Q5.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingown data

    Correspondence was the most frequent functional relationship in third graders' responses.

    “Correspondence was the most frequent functional relationship in students' responses.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In Q3 and Q4, T13 defined pairs of values (a, f(a)) for the a values in each specific value (10 and 100) and established their relationship to the number of grey tiles: 26 and 206, respectively.

    “In Q3 and Q4, he defined pairs of values (a, f(a)) for the a values in each specific value (10 and 100) and established their relationship to the number of grey tiles: 26 and 206, respectively.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Three third-grade students did generalize, providing evidence of correspondence.

    “On the other hand, three students did generalize, providing evidence of correspondence.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    F06 generalized in the question that explicitly asked for this relationship (Q5), as well as in the question that asked for the relationship for a specific value (Q1).

    “F06 generalized in the question that explicitly asked for this relationship (Q5), as well as in the question that asked for the relationship for a specific value (Q1).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    Most fifth grade students generalized, while the third graders did not.

    “These findings could explain why most fifth grade students generalized, while the third graders did not.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    Students in higher grades are more likely to focus on the relationships among quantities, whereas third graders still focus on the details of arithmetic computations or pictorial representations introduced.

    “since students in higher grades would be more likely to focus on the relationships between variables, whereas thirdgraders would focus on the details of arithmetic computations.”
    • Also stated in: § Results and Discussion · hedged
      “students in higher grades are more likely to focus on the relationships among quantities, whereas third-graders still focus on the details of arithmetic computations or pictorial representations introduced.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: Students had no previous algebra knowledge.
    • Rests on an assumption: Prior to the CTE, these students had not worked with problems involving generalization and/or functions.

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

    Interpretationreasoning

    F01 generalized her responses to Q2-Q4 but did not give the general relationship because it was not requested.

    “Apparently, she generalized her responses to Q2-Q4 but did not give the general relationship because it was not requested.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    T13 perceived a regularity that extended to different specific values but was not able to represent that regularity when asked directly about generalization.

    “This approach to decomposing the number of grey tiles—the same approach used in all of the questions—could indicate that T13 perceived a regularity that extended to different specific values but was not able to represent that regularity when asked directly about generalization.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Study design and conductcitation

    The reliability coefficient for the coding of students' answers was greater than 90%, above the acceptable minimum.

    “The reliability coefficient was greater than 90%, above the acceptable minimum (Tinsley & Brown, 2000).”
    Quote in the paperSubstantial claim

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

    Premisecitation

    This situation is similar to that noted by other authors (e.g., Warren, Miller, & Cooper, 2007) who argue that students' ability to identify functional relationships grows as they work with an increasing number of specific values involving two variables.

    “This situation is similar to that noted by other authors (e.g., Warren, Miller, & Cooper, 2007) who argue that students' ability to identify functional relationships grows as they work with an increasing number of specific values involving two variables.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Warren, Miller, & Cooper, 2007

      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).

    Interpretationreasoning

    This finding was foreseeable because our study posed the problem in a way intended to elicit correspondence and covariation, which relate the dependent and independent variables.

    “This finding was foreseeable because our study posed the problem in a way intended to elicit correspondence and covariation, which relate the dependent and independent variables.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    Students' use of natural language to generalize is consistent with results reported by Radford (2018), who found that natural language is a useful vehicle for expressing general rules, as well as a useful scaffold for development of more symbolic representations.

    “These students' use of natural language is consistent with results reported by Radford (2018), who found that natural language is a useful vehicle for expressing general rules, as well as a useful scaffold for development of more symbolic representations.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Radford (2018) source

  • The quote only partly states this claim.

    Interpretationreasoning

    This difference result may explain why only three third-grade students generalized.

    “This difference result may explain why only three students generalized.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S4 (based).

    Interpretationnone given

    The introduction of pictorial or manipulative representations in students' work with functional thinking tasks could help students to understand the dynamic relationships between variables and thus serve as a first step to start students thinking about covarying quantities.

    “Second, the introduction of pictorial or manipulative representations in students' work with functional thinking tasks could help students to understand the dynamic relationships between variables and thus serve as a first step to start students thinking about covarying quantities.”
    Quote in the paperNot a substantial claim

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

    Interpretationcitation

    Such deduction should lead to useful conclusions for teaching, as these elements are present in a number of countries' Mathematics curriculums.

    “Such deduction should lead to useful conclusions for teaching, as these elements are present in a number of countries' Mathematics curriculums (Ministerio de Educación, Cultura y Ciencia, 2014).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Ministerio de Educación, Cultura y Ciencia, 2014

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

  • May combine two claims.

    Interpretationreasoning

    As the arithmetical computations enabled the third graders to answer the first questions we asked them, these students did not see the need to find the relationship between the variables mentioned.

    “As the arithmetical computations enabled the third graders to answer the first questions we asked them, these students did not see the need to find the relationship between the variables mentioned.”
    Quote in the paperSubstantial claim

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

    Study design and conductreasoning

    The analysis of the students' written responses could be one limitation of this study, as the worksheets may not capture all of the students' ideas.

    “The analysis of the students' written responses could be one limitation of this study, as the worksheets may not capture all of the students' ideas.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationown data

    In contrast to the third graders who generalized, fifth graders generalize without following a worksheet designed for this purpose.

    “In contrast to the third graders who generalized, these answers show that fifth graders generalize without following a worksheet designed for this purpose, in line with similar findings from other studies (e.g., Amit & Neria, 2007).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Amit & Neria, 2007

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

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    The tiles problem helps researchers identify the elements of generalization and functional thinking that can be deduced from elementary school students' spontaneous replies.

    “Finally, the tiles problem helps researchers identify the elements of generalization and functional thinking that can be deduced from elementary school students' spontaneous replies.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    The tiles problem enables students to use different procedures when answering the questions involving specific values until they are able to generalize.

    “First, it enables students to use different procedures when answering the questions involving specific values until they are able to generalize.”
    Quote in the paperSubstantial claim

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

    Premisecitation

    Generalization is not simple and requires time.

    “While acknowledging that generalization is not simple and requires time (Dienes, 1961), we believe that learning sequences must be designed to guide students to a general rule that is valid for different specific values, while working with different types of mathematical representation.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Dienes, 1961

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

  • May combine two claims.

    Interpretationcitation

    These students' prior mathematical experiences seem to have influenced how they attend to and relate covarying quantities, and how they perceive general rules.

    “As in reports by other authors (e.g., Carraher et al., 2008), however, these students' prior mathematical experiences seem to have influenced how they attend to and relate covarying quantities, and how they perceive general rules.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Carraher et al., 2008 source
31 of 31

  • The authors state this as an assumption.

    Functional thinking involves functional relationships, which may or may not be generalized and which may be expressed through different types of representations.

    “We therefore assume that functional thinking involves functional relationships, which may or may not be generalized and which may be expressed through different types of representations.”

  • The paper relies on this without support.

    Students had no previous algebra knowledge.

    “As Algebra is not a content introduced at elementary level in Spain, students did not have previous algebra knowledge.”

  • The paper relies on this without support.

    Prior to the CTE, these students had not worked with problems involving generalization and/or functions.

    “Prior to the CTE, these students had not worked with problems involving generalization and/or functions.”

  • The paper relies on this without support.

    Students' written responses accurately reflect their mathematical thinking and generalization abilities.

    “The analysis of the students' written responses could be one limitation of this study, as the worksheets may not capture all of the students' ideas.”

  • The paper relies on this without support.

    The tiles problem is a valid task for eliciting functional thinking and generalization in elementary students.

    “The tiles problem was specifically designed to promote functional thinking in elementary students.”
5 of 5

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.

  • Table 2 labels a ten-white-tile response as Q2

    Minor

    The worksheet defines Q2 as the question about eight white tiles; ten white tiles are asked about in Q3.

  • T03's response is attributed to the wrong figure

    Minor

    The text says Figure 4 shows T03's reasoning, whereas Figure 4 is identified as showing T21 and F15. T03's response is labeled Figure 7.

  • Third-grade representation discussion cites the fifth-grade table

    Minor

    The discussion of third graders' representations refers to Table 6, but Table 6 reports fifth graders' correspondence. Table 5 is headed as the representation table.

3 of 3

Cites

Flagged first, then by citations

1 of the 53 references couldn’t be matched to a known paper.

Research works printed but not linked: 9

  1. Blanton, M. (2008). Algebra and the elementary classroom: transforming thinking, transforming practice. Portsmouth, NH: Heinemann.
  2. Blanton, M. L. (2017). Algebraic reasoning in grades 3-5. In M. T. Battista (Ed.), Reasoning and sense making in the mathematics classroom. Grades 3–5 (pp. 67–102). NCTM: Reston, VA.
  3. Brizuela, B., & Earnest, D. (2008). Multiple notational systems and algebraic understanding: the case of the “best deal” problem. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 273–302). New York, NY: LEA.
  4. Cañadas, M. C., & Molina, M. (2016). Una aproximación al marco conceptual y principales antecedentes del pensamiento funcional en las primeras edades [Approach to the conceptual framework and background of functional thinking in the early years]. In E. Castro, E. Castro, J. L. Lupiáñez, J. F. Ruíz-Hidalgo, & M. Torralbo (Eds.), Investigación en educación matemática. Homenaje a Luis Rico (pp. 209-218). Granada, Spain: Comares.
  5. Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York, NY: Lawrence Erlbaum Associates.
  6. Küchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s understanding of mathematics: 11–16 (pp. 102–119). London: John Murray.
  7. Merino, E., Cañadas, M. C., & Molina, M. (2013). Uso de representaciones y patrones por alumnos de quinto de educación primaria en una tarea de generalización [Representations and patterns used by fifth grade students in a generalization task]. Edma 0-6: Eduación Matemática en la Infancia, 2(1), 24-40.
  8. Schifter, D., Bastable, V., Russell, S. J., Seyferth, L., & Riddle, M. (2008). Algebra in the grades K–5 classroom: learning opportunities for students and teachers. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics: seventieth yearbook of the National Council of Teachers of Mathematics (pp. 263–277). Reston, VA: NCTM.
  9. Schwartz, J. L., & Yerushalmy, M. (1992). Getting students to function on and with algebra. In E. Dubinsky & G. Harel (Eds.), The concept of function: aspects of epistemology and pedagogy (pp. 261–289). Washington, DC: Mathematical Association of America.

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Authors
Pinto, E., & Cañadas, M. C.
Affiliations
Universidad de Granada (ES)
Corresponding author
Pinto, Eder
ORCID
Venue
Mathematics Education Research Journal
Publisher
Springer Science and Business Media LLC
Published
2019-12-16
Volume
33
Issue
1
Pages
113-134
ISSN
  • 1033-2170 (ISSN-L)
  • 2211-050X
DOI
10.1007/s13394-019-00300-2
OpenAlex
W2995369338
Semantic Scholar
ea78c6bfa746ed5be8741688ca958f9d125f8256
Cited by
42
References
53
Citations by year
2026: 5 · 2025: 6 · 2024: 7 · 2023: 6 · 2022: 14 · 2021: 2 · 2020: 2
FWCI
12.15
Type
journal article
Language
en
Open access
Licence
Locations
  • Mathematics Education Research Journal (publishedVersion)
  • SUMA (Spanish Federation of Mathematics Teachers' Societies (FESPM)) (submittedVersion)
  • Institutional Repository of the University of Granada (University of Granada) (submittedVersion, cc-by-nc-nd)
  • UKM Journal Article Repository (Universiti Kebangsaan Malaysia) (submittedVersion)
Funders
  • Comisión Nacional de Investigación Científica y Tecnológica (72160307-2015)
  • National R&D Project, Spanish Ministry of Economy and Competitiveness (EDU2016-75771-P)
Topics
  • Social Sciences › Education › Mathematics Education and Teaching Techniques
  • Physical Sciences › Mathematics › Applied Mathematics › Mathematics Education and Pedagogy
  • Physical Sciences › Mathematics › Modeling and Simulation › Mathematics Education and Programs
Keywords
early algebra, generalization, functional relationships, student representation, linear functions, third grade, fifth grade students, grade level differences

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