# Difference between revisions of "Sequencing learning with multiple representations of rational numbers (Aleven, Rummel, & Rau)"

Line 25: | Line 25: | ||

=== Background & Significance === | === Background & Significance === | ||

+ | A quintessential form of coordinative learning occurs when learners work with multiple external representations (MERs) of subject matter. Accumulating evidence points towards the promise of learning with MERs (Ainsworth, Bibby, & Wood, 2002; Larkin & Simon, 1987; Seufert, 2003), and also to the need for students to make sense out of the different representations by connecting and abstracting from them (Ainsworth, 1999). | ||

+ | <br> | ||

+ | This research focuses on a difficult area of early mathematics learning: fractions. Both teachers’ experiences and research in educational psychology show that students have difficulties with fraction arithmetic and with the various representations for fractions (e.g. Brinker, 1997; Callingham & Watson, 2004; Caney & Watson, 2003; Person et al., 2004; Pitta-Pantazi, Gray & Christou, 2004). Coordinating between MERs is regarded as a key process for learning across areas of mathematics (Kilpatrick, Swafford, & Findell, 2001; NCTM, 2000), including fractions (e.g. Kieren, 1993; Moss & Case, 1999; Martinie & Bay-Williams, 2003; Thompson & Saldanha, 2003). | ||

+ | <br> | ||

+ | A number of authors have argued, based on observational studies, that MERs can lead to deeper conceptual understanding of fractions (Corwin et al., 1990; Cramer et al., 1997a, 1997b; Steiner & Stoeckling, 1997). However, we know of no experimental studies that have investigated the advantages of instruction with multiple (graphical) fraction representations over instruction that focuses on a single representation, with one exception: an in vivo experiment, in which 132 6th-grade students used four versions of CTAT-built tutors (Rau, Aleven, & Rummel, 2009). Students learning with MERs and prompted to self-explain performed best on a posttest and delayed posttest assessing procedural and conceptual knowledge of fractions. | ||

+ | <br> | ||

+ | At this point, however, we do not know enough about the circumstances that may influence the effectiveness of learning with multiple representations of fractions, a criticism that has been leveraged against the existing body of research on learning with MERs more generally (Ainsworth, 2006; Goldman, 2003). The proposed research looks at how the development of fluency with any given representation interacts with sense making across representations. First, as Ainsworth (2006) points out, being able to | ||

+ | interpret a particular type of representation is a prerequisite for learning from it. However, such <i>‘representational fluency’</i> does not just emerge by itself, but requires practice. Second, it is important that students engage in <i>sense making</i> across the different representations to relate them and integrate the information they provide (Ainsworth, 2006; Brinker, 1997; Paik, 2005; Uttal et al., 1997). According to cognitive flexibility theory (Spiro & Jehng, 1990), being presented with MERs challenges the learner to switch between different perspectives on the same concepts. Under this perspective, learning with MERs supports the development of robust – flexible and transferable – knowledge (Kaput, 1989), to the extent that learners <i>coordinate</i> between the representations, that is, cognitively link the information the MERs provide and abstract underlying conceptual knowledge. A key question is therefore whether learners should build up fluency with each representation first, before they engage in sense-making activities aimed at coordinating representations, or whether they develop more flexible knowledge when they become familiar with the different representations in parallel and continuously engage in sense making across representations. This potential conflict is inherent in designing instruction with MERs. | ||

+ | |||

=== Glossary === | === Glossary === | ||

Line 31: | Line 40: | ||

=== Research questions === | === Research questions === | ||

+ | When learning with multiple external representations, do students learn more robustly when new representations are introduced gradually, with (highly) infrequent switching between representations, or when the representations are introduced right from the start, with students switching often between representations? Or, is a transition from infrequent to frequent switching between representations most beneficial? | ||

=== Hypotheses === | === Hypotheses === | ||

+ | *We hypothesize that a mix of these two designs (i.e., an intermediate position on the continuum between highly infrequent and highly frequent switching between the representations) would be best as it allows learners to gain some experience with one representation before moving on to the next, but also facilitates making connections across representations as the (temporal) distance between representations is smaller than in the highly infrequently switching design. | ||

+ | *We hypothesize that gaining fluency with each of the representations is more important at the beginning of a tutoring session than towards the end. Therefore, we expect a sequence that transitions from infrequent to frequent switching between representations to be more effective than the extremes of the continuum between highly infrequent and highly infrequent switching between the representations. | ||

=== Dependent variables === | === Dependent variables === | ||

=== Independent Variables === | === Independent Variables === | ||

+ | *Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on: | ||

+ | **Reproduction of conceptual knowledge | ||

+ | **Reproduction of procedural knowledge | ||

+ | **Transfer of conceptual knowledge | ||

+ | **Transfer of procedural knowledge | ||

+ | |||

+ | *Log data collected during tutor use, used to assess: | ||

+ | **Learning curves | ||

+ | **Time on task | ||

+ | **Error rates | ||

+ | **Hint usage | ||

+ | **Latency of responses | ||

=== Findings === | === Findings === | ||

+ | Data collection is still in progress. | ||

=== Explanation === | === Explanation === | ||

+ | Data collection is still in progress. | ||

===Further Information=== | ===Further Information=== | ||

Line 48: | Line 74: | ||

==== References ==== | ==== References ==== | ||

+ | *Ainsworth, S. (1999). <i>Designing effective multi-representational learning environments</i> (No. 58). Nottingham: ESRC Centre for Research in Development, Instruction & Training Department of Psychology. | ||

+ | *Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. <i>Learning and Instruction, 16</i>(3), 183-198. | ||

+ | *Ainsworth, S. (in press). How do animations influence learning? In Robinson & Schraw (Eds.), <i>Current Perspectives on Cognition, Learning, and Instruction: Recent Innovations in Educational Technology that Facilitate Student Learning.</i> | ||

+ | *Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. <i>Journal of the Learning Sciences</i>, 11(1), 25-61. | ||

+ | *Bodemer, D., Ploetzner, R., Bruchmüller, K., & Häcker, S. (2005). Supporting learning with interactive multimedia through active integration of representations. <i>Instructional Science, 33</i>(1), 73-95. | ||

+ | *Bodemer, D., & Faust, U. (2006). External and mental referencing of multiple representations. <i>Computers in Human Behavior</i>, 22(1), 27-42. | ||

+ | *Brinker, L. (1997). <i>Using Structured Representations To Solve Fraction Problems: A Discussion of Seven Students' Strategies.</i> | ||

+ | *Callingham, R., & Watson, J. (2004). A Developmental Scale of Mental Computation with Part-Whole Numbers. <i>Mathematics Education Research Journal, 16</i>(2), 69-86. | ||

+ | *Caney, A., & Watson, J. M. (2003). <i>Mental Computation Strategies for Part-Whole Numbers.</i> Paper presented at the International Educational Research Conference, Auckland, New Zealand. | ||

+ | *Corwin, R. B., Russell, S. J., & Tierney, C. C. (1990). <i>Seeing fractions: A unit for the upper elementary grades.</i> Sacramento, CA: California Dept. of Education. (ED 348 211). | ||

+ | *Cramer, K., Behr, M., Post, T., & Lesh, R. (1997a). <i>Rational Number Project: Fraction Lessons for the Middle Grades: Level 1.</i> Dubuque, IA: Kendall/Hunt Publishing. | ||

+ | *Cramer, K., Behr, M., Post, T., & Lesh, R. (1997b). <i>Rational Number Project: Fraction Lessons for the Middle Grades: Level 2.</i> Dubuque, IA: Kendall/Hunt Publishing. | ||

+ | *Goldman, S. R., Mertz, D. L., & Pellegrino, J. W. (1989). Individual differences in extended practice functions and solution strategies for basic addition facts. <i>Journal of Educational Psychology, 81</i>(4), 481-496. | ||

+ | *Kaput, J.J. (1989). Linking representations in the symbolic systems of algebra. In S. Wagner & C. Kieran (Eds.), <i>Research agenda for mathematics education: Research issues in the learning and teaching of algebra</i> (pp.167-194). Reston, VA: National Council of Teachers of Mathematics. | ||

+ | *Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), <i>Rational numbers: An integration of research</i> (pp. 49-84). Hillsdale, NJ: Erlbaum. | ||

+ | *Kilpatrick, Jeremy; Swafford, Jane; Findell, Bradford (Eds.); Mathematics Learning Study Committee, National Research Council (2001). Conclusions and recommendations. In <i>Adding It Up: Helping Children Learn Mathematics</i> (pp. 407-432). Washington, D.C.: The National Academies Press. | ||

+ | *Kozma, R. B., Russell, J., Jones, T., Marx, N., & Davis, J. (1996). The use of multiple, linked representations to facilitate science understanding. In S. Vosniadou, E. De Corte, R. Glaser & H. Mandl (Eds.), <i>International perspectives on the design of technology-supported learning environments.</i> (pp. 41-60). Hillsdale, NJ, England: Lawrence Erlbaum Associates, Inc. | ||

+ | *Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. <i>Cognitive Science: A Multidisciplinary Journal, 11</i>(1), 65-100. | ||

+ | *Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children's learning of multiplication. <i>Journal of Experimental Psychology: General, 124</i>(1), 83-97. | ||

+ | *Lowe, R. K. (1999). Extrating information from an animation during complex visual learning. <i>European Journal of Psychology of Education, 14</i>(2), 225-244. | ||

+ | *Martin, T., & Schwartz, D. L. (2005). Physically Distributed Learning: Adapting and Reinterpreting Physical Environments in the Development of Fraction Concepts. <i>Cognitive Science: A Multidisciplinary Journal, 29</i>(4), 587-625. | ||

+ | *Martinie, S.L., & Bay-Williams, J.M. (2003) Investigating students’ conceptual understanding of decimal fractions using multiple representations. <i>Mathematics Teaching in the Middle School, 8</i>(5), 244-248. | ||

+ | *Millsaps, G. M., & Reed, M. K. (1998). <i>Curricula for Teaching about Fractions. ERIC Digest.</i> | ||

+ | *Moss, J., & Case, R. (1999). Developing children's understanding of the rational numbers: A new model and an experimental curriculum. <i>Journal for Research in Mathematics Education, 30</i>(2), 122-147. | ||

+ | *Moss, J. (2005). Pipes, Tubes, and Beakers: New approaches to teaching the rational-number system. In J. Brantsford & S. Donovan (Eds.), <i>How people learn: A targeted report for teachers</i> (pp. 309-349): National Academy Press. | ||

+ | *Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. <i>Contemporary Educational Psychology, 26</i>(3), 400-417. | ||

+ | *NCTM. (2000). <i>Principles and Standards for School Mathematics.</i> Reston, VA: National Council of Teachers of Mathematics. | ||

+ | *Paik, J. H. (2005). <i>Fraction concepts: A complex system of mappings.</i> ProQuest Information & Learning, US. | ||

+ | *Person, A. C., Berenson, S. B., & Greenspon, P. J. (2004). The Role of Number in Proportional Reasoning: A Prospective Teacher's Understanding. <i>International Group for the Psychology of Mathematics Education.</i> | ||

+ | *Pitta-Pantazi, D., Gray, E. M., & Christou, C. (2004). Elementary School Students' Mental Representation of Fractions. <i>International Group for the Psychology of Mathematics Education.</i> | ||

+ | *Plötzner, R., Bodemer, D., & Neudert, S. (2008). Successful and less successful use of dynamic visualizations in instructional texts. In R. K. Lowe & W. Schnotz (Eds.), <i>Learning with Animation. Research implications for design.</i> New York: Cambridge University Press. | ||

+ | *Seufert, T. (2003). Supporting coherence formation in learning from multiple representations. <i>Learning and Instruction, 13</i>(2), 227-237. | ||

+ | *Spiro, R. J. & Jehng, J. C. (1990). Cognitive flexibility and hypertext: Theory and technology for the nonlinear and multidimensional traversal of complex subject matter. In D. Nix & R. Spiro (Eds.), <i>Cognition, education, and multimedia: Exploring ideas in high technology</i> (pp. 163-205). Hillsdale, NJ: Lawrence Erlbaum Associates. | ||

+ | *Steiner, G. F., & Stoecklin, M. (1997). Fraction calculation: A didactic approach to constructing mathematical networks. <i>Learning and Instruction, 7</i>(3), 211-233. | ||

+ | *Thompson, P.W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), <i>A research companion to principles and standards for school mathematics</i> (pp. 95–113). Reston, VA: National Council of Teachers of Mathematics. | ||

+ | *Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. <i>Journal of Applied Developmental Psychology, 18</i>(1), 37-54. | ||

+ | *Witherspoon, M. L. (1993). Fractions: In Search of Meaning. Arithmetic Teacher, 40(8), 482-485. | ||

+ | *Yang, D.-C., & Reys, R. E. (2001). One fraction problem, many solution paths. <i>Mathematics Teaching in the Middle School, 7</i>(3), 164-166. |

## Revision as of 02:08, 14 May 2009

## Sequencing learning with multiple representations of rational numbers

Vincent Aleven, Nikol Rummel, and Martina Rau

### Summary Table

#### Study 1

PIs |
Vincent Aleven & Nikol Rummel |

Other Contributers |
Graduate Students: Martina Rau (CMU HCII) |

Study Start Date |
September 1st, 2008 |

Study End Date |
August 31st, 2009 |

Number of Students |
~350 |

Total Participant Hours |
~2100 |

DataShop |
Log data is uploaded and available in the DataShop |

### Abstract

We investigate a key issue in *coordinative learning*, namely, how learning with multiple external representations (MERs) should be sequenced to effectively support students’ conceptual understanding. In order to benefit from MERs, learners must attain some level of *fluency* in interpreting and manipulating the individual representations, and must also engage in *sense making* across the representations to relate them and abstract underlying concepts. The question arises how tasks involving different representations should be sequenced so that both these aspects of robust learning are realized. In particular, how frequently should students switch between representations? We focus on fractions as a challenging topic area for students in which multiple representations are often used and likely to support robust learning. This research will contribute to the literature on early mathematics learning, learning with multiple representations, and learning with intelligent tutoring systems. It will also add to the portfolio of studies in the PSLC’s coordinative learning cluster.

### Background & Significance

A quintessential form of coordinative learning occurs when learners work with multiple external representations (MERs) of subject matter. Accumulating evidence points towards the promise of learning with MERs (Ainsworth, Bibby, & Wood, 2002; Larkin & Simon, 1987; Seufert, 2003), and also to the need for students to make sense out of the different representations by connecting and abstracting from them (Ainsworth, 1999).

This research focuses on a difficult area of early mathematics learning: fractions. Both teachers’ experiences and research in educational psychology show that students have difficulties with fraction arithmetic and with the various representations for fractions (e.g. Brinker, 1997; Callingham & Watson, 2004; Caney & Watson, 2003; Person et al., 2004; Pitta-Pantazi, Gray & Christou, 2004). Coordinating between MERs is regarded as a key process for learning across areas of mathematics (Kilpatrick, Swafford, & Findell, 2001; NCTM, 2000), including fractions (e.g. Kieren, 1993; Moss & Case, 1999; Martinie & Bay-Williams, 2003; Thompson & Saldanha, 2003).

A number of authors have argued, based on observational studies, that MERs can lead to deeper conceptual understanding of fractions (Corwin et al., 1990; Cramer et al., 1997a, 1997b; Steiner & Stoeckling, 1997). However, we know of no experimental studies that have investigated the advantages of instruction with multiple (graphical) fraction representations over instruction that focuses on a single representation, with one exception: an in vivo experiment, in which 132 6th-grade students used four versions of CTAT-built tutors (Rau, Aleven, & Rummel, 2009). Students learning with MERs and prompted to self-explain performed best on a posttest and delayed posttest assessing procedural and conceptual knowledge of fractions.

At this point, however, we do not know enough about the circumstances that may influence the effectiveness of learning with multiple representations of fractions, a criticism that has been leveraged against the existing body of research on learning with MERs more generally (Ainsworth, 2006; Goldman, 2003). The proposed research looks at how the development of fluency with any given representation interacts with sense making across representations. First, as Ainsworth (2006) points out, being able to
interpret a particular type of representation is a prerequisite for learning from it. However, such *‘representational fluency’* does not just emerge by itself, but requires practice. Second, it is important that students engage in *sense making* across the different representations to relate them and integrate the information they provide (Ainsworth, 2006; Brinker, 1997; Paik, 2005; Uttal et al., 1997). According to cognitive flexibility theory (Spiro & Jehng, 1990), being presented with MERs challenges the learner to switch between different perspectives on the same concepts. Under this perspective, learning with MERs supports the development of robust – flexible and transferable – knowledge (Kaput, 1989), to the extent that learners *coordinate* between the representations, that is, cognitively link the information the MERs provide and abstract underlying conceptual knowledge. A key question is therefore whether learners should build up fluency with each representation first, before they engage in sense-making activities aimed at coordinating representations, or whether they develop more flexible knowledge when they become familiar with the different representations in parallel and continuously engage in sense making across representations. This potential conflict is inherent in designing instruction with MERs.

### Glossary

- Conceptual knowledge: knowledge about the rationale of a solution procedure
- Procedural knowledge: knowledge of the components of a correct procedure involving knowledge about step-by-step actions for solving problems

### Research questions

When learning with multiple external representations, do students learn more robustly when new representations are introduced gradually, with (highly) infrequent switching between representations, or when the representations are introduced right from the start, with students switching often between representations? Or, is a transition from infrequent to frequent switching between representations most beneficial?

### Hypotheses

- We hypothesize that a mix of these two designs (i.e., an intermediate position on the continuum between highly infrequent and highly frequent switching between the representations) would be best as it allows learners to gain some experience with one representation before moving on to the next, but also facilitates making connections across representations as the (temporal) distance between representations is smaller than in the highly infrequently switching design.
- We hypothesize that gaining fluency with each of the representations is more important at the beginning of a tutoring session than towards the end. Therefore, we expect a sequence that transitions from infrequent to frequent switching between representations to be more effective than the extremes of the continuum between highly infrequent and highly infrequent switching between the representations.

### Dependent variables

### Independent Variables

- Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:
- Reproduction of conceptual knowledge
- Reproduction of procedural knowledge
- Transfer of conceptual knowledge
- Transfer of procedural knowledge

- Log data collected during tutor use, used to assess:
- Learning curves
- Time on task
- Error rates
- Hint usage
- Latency of responses

### Findings

Data collection is still in progress.

### Explanation

Data collection is still in progress.

### Further Information

#### Connections

#### Annotated Bibliography

#### References

- Ainsworth, S. (1999).
*Designing effective multi-representational learning environments*(No. 58). Nottingham: ESRC Centre for Research in Development, Instruction & Training Department of Psychology. - Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations.
*Learning and Instruction, 16*(3), 183-198. - Ainsworth, S. (in press). How do animations influence learning? In Robinson & Schraw (Eds.),
*Current Perspectives on Cognition, Learning, and Instruction: Recent Innovations in Educational Technology that Facilitate Student Learning.* - Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics.
*Journal of the Learning Sciences*, 11(1), 25-61. - Bodemer, D., Ploetzner, R., Bruchmüller, K., & Häcker, S. (2005). Supporting learning with interactive multimedia through active integration of representations.
*Instructional Science, 33*(1), 73-95. - Bodemer, D., & Faust, U. (2006). External and mental referencing of multiple representations.
*Computers in Human Behavior*, 22(1), 27-42. - Brinker, L. (1997).
*Using Structured Representations To Solve Fraction Problems: A Discussion of Seven Students' Strategies.* - Callingham, R., & Watson, J. (2004). A Developmental Scale of Mental Computation with Part-Whole Numbers.
*Mathematics Education Research Journal, 16*(2), 69-86. - Caney, A., & Watson, J. M. (2003).
*Mental Computation Strategies for Part-Whole Numbers.*Paper presented at the International Educational Research Conference, Auckland, New Zealand. - Corwin, R. B., Russell, S. J., & Tierney, C. C. (1990).
*Seeing fractions: A unit for the upper elementary grades.*Sacramento, CA: California Dept. of Education. (ED 348 211). - Cramer, K., Behr, M., Post, T., & Lesh, R. (1997a).
*Rational Number Project: Fraction Lessons for the Middle Grades: Level 1.*Dubuque, IA: Kendall/Hunt Publishing. - Cramer, K., Behr, M., Post, T., & Lesh, R. (1997b).
*Rational Number Project: Fraction Lessons for the Middle Grades: Level 2.*Dubuque, IA: Kendall/Hunt Publishing. - Goldman, S. R., Mertz, D. L., & Pellegrino, J. W. (1989). Individual differences in extended practice functions and solution strategies for basic addition facts.
*Journal of Educational Psychology, 81*(4), 481-496. - Kaput, J.J. (1989). Linking representations in the symbolic systems of algebra. In S. Wagner & C. Kieran (Eds.),
*Research agenda for mathematics education: Research issues in the learning and teaching of algebra*(pp.167-194). Reston, VA: National Council of Teachers of Mathematics. - Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.),
*Rational numbers: An integration of research*(pp. 49-84). Hillsdale, NJ: Erlbaum. - Kilpatrick, Jeremy; Swafford, Jane; Findell, Bradford (Eds.); Mathematics Learning Study Committee, National Research Council (2001). Conclusions and recommendations. In
*Adding It Up: Helping Children Learn Mathematics*(pp. 407-432). Washington, D.C.: The National Academies Press. - Kozma, R. B., Russell, J., Jones, T., Marx, N., & Davis, J. (1996). The use of multiple, linked representations to facilitate science understanding. In S. Vosniadou, E. De Corte, R. Glaser & H. Mandl (Eds.),
*International perspectives on the design of technology-supported learning environments.*(pp. 41-60). Hillsdale, NJ, England: Lawrence Erlbaum Associates, Inc. - Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words.
*Cognitive Science: A Multidisciplinary Journal, 11*(1), 65-100. - Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children's learning of multiplication.
*Journal of Experimental Psychology: General, 124*(1), 83-97. - Lowe, R. K. (1999). Extrating information from an animation during complex visual learning.
*European Journal of Psychology of Education, 14*(2), 225-244. - Martin, T., & Schwartz, D. L. (2005). Physically Distributed Learning: Adapting and Reinterpreting Physical Environments in the Development of Fraction Concepts.
*Cognitive Science: A Multidisciplinary Journal, 29*(4), 587-625. - Martinie, S.L., & Bay-Williams, J.M. (2003) Investigating students’ conceptual understanding of decimal fractions using multiple representations.
*Mathematics Teaching in the Middle School, 8*(5), 244-248. - Millsaps, G. M., & Reed, M. K. (1998).
*Curricula for Teaching about Fractions. ERIC Digest.* - Moss, J., & Case, R. (1999). Developing children's understanding of the rational numbers: A new model and an experimental curriculum.
*Journal for Research in Mathematics Education, 30*(2), 122-147. - Moss, J. (2005). Pipes, Tubes, and Beakers: New approaches to teaching the rational-number system. In J. Brantsford & S. Donovan (Eds.),
*How people learn: A targeted report for teachers*(pp. 309-349): National Academy Press. - Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence.
*Contemporary Educational Psychology, 26*(3), 400-417. - NCTM. (2000).
*Principles and Standards for School Mathematics.*Reston, VA: National Council of Teachers of Mathematics. - Paik, J. H. (2005).
*Fraction concepts: A complex system of mappings.*ProQuest Information & Learning, US. - Person, A. C., Berenson, S. B., & Greenspon, P. J. (2004). The Role of Number in Proportional Reasoning: A Prospective Teacher's Understanding.
*International Group for the Psychology of Mathematics Education.* - Pitta-Pantazi, D., Gray, E. M., & Christou, C. (2004). Elementary School Students' Mental Representation of Fractions.
*International Group for the Psychology of Mathematics Education.* - Plötzner, R., Bodemer, D., & Neudert, S. (2008). Successful and less successful use of dynamic visualizations in instructional texts. In R. K. Lowe & W. Schnotz (Eds.),
*Learning with Animation. Research implications for design.*New York: Cambridge University Press. - Seufert, T. (2003). Supporting coherence formation in learning from multiple representations.
*Learning and Instruction, 13*(2), 227-237. - Spiro, R. J. & Jehng, J. C. (1990). Cognitive flexibility and hypertext: Theory and technology for the nonlinear and multidimensional traversal of complex subject matter. In D. Nix & R. Spiro (Eds.),
*Cognition, education, and multimedia: Exploring ideas in high technology*(pp. 163-205). Hillsdale, NJ: Lawrence Erlbaum Associates. - Steiner, G. F., & Stoecklin, M. (1997). Fraction calculation: A didactic approach to constructing mathematical networks.
*Learning and Instruction, 7*(3), 211-233. - Thompson, P.W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*(pp. 95–113). Reston, VA: National Council of Teachers of Mathematics. - Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics.
*Journal of Applied Developmental Psychology, 18*(1), 37-54. - Witherspoon, M. L. (1993). Fractions: In Search of Meaning. Arithmetic Teacher, 40(8), 482-485.
- Yang, D.-C., & Reys, R. E. (2001). One fraction problem, many solution paths.
*Mathematics Teaching in the Middle School, 7*(3), 164-166.