Difference between revisions of "Sequencing learning with multiple representations of rational numbers (Aleven, Rummel, & Rau)"
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−  == Learning with Multiple  +  == Learning with Multiple Representations in a Complex, Realworld Domain: Intelligent Tutoring Systems for Fractions == 
''Vincent Aleven, Nikol Rummel, and Martina Rau''  ''Vincent Aleven, Nikol Rummel, and Martina Rau''  
Line 8:  Line 8:  
 '''Other Contributers'''  <b>Graduate Students:</b> Martina Rau (CMU HCII)<br>   '''Other Contributers'''  <b>Graduate Students:</b> Martina Rau (CMU HCII)<br>  
    
−   '''2008 study'''  <i>N</i> =  +   '''2008 study'''  <i>N</i> = 132 6thgrade students 
    
 '''2009 study'''  <i>N</i> = 388 5th and 6thgrade students   '''2009 study'''  <i>N</i> = 388 5th and 6thgrade students  
Line 18:  Line 18:  
 '''Study End Date'''  August 31st, 2012   '''Study End Date'''  August 31st, 2012  
    
−   '''Total Number of Students to date'''  <i>N</i> =  +   '''Total Number of Students to date'''  <i>N</i> = 1210 
    
 '''Total Participant Hours'''  ~6000   '''Total Participant Hours'''  ~6000  
    
−   '''Data available in DataShop'''  [https://pslcdatashop.web.cmu.edu/DatasetInfo?datasetId=325 Dataset: Fraction Study Spring 2009 (log data only)][https://pslcdatashop.web.cmu.edu/DatasetInfo?datasetId=  +   '''Data available in DataShop'''  [https://pslcdatashop.web.cmu.edu/DatasetInfo?datasetId=325 Dataset: Fraction Study Spring 2009 (log data only)]<br>[https://pslcdatashop.web.cmu.edu/DatasetInfo?datasetId=429 Dataset: Fraction Study Spring 2010 (fractions portion only)] 
* '''Pre/Post Test Score Data:''' No  * '''Pre/Post Test Score Data:''' No  
Line 42:  Line 42:  
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 6thgrade students used four versions of CTATbuilt tutors (Rau, Aleven, & Rummel, 2009). Students learning with MERs and prompted to selfexplain performed best on a posttest and delayed posttest assessing procedural and conceptual knowledge of fractions.  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 6thgrade students used four versions of CTATbuilt tutors (Rau, Aleven, & Rummel, 2009). Students learning with MERs and prompted to selfexplain performed best on a posttest and delayed posttest assessing procedural and conceptual knowledge of fractions.  
<br>  <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).  +  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). Learning with multiple representations is challenging. An important prerequisite for benefiting from the multiplicity of multiple graphical representations is that students conceptually understand each one of them (Ainsworth, 2006). 
−  +  <br>  
−  +  When designing intelligent tutoring systems that use multiple graphical representations, designers must decide how to temporally sequence the GRs. How often should the curriculum alternate between multiple graphical representations? Practice schedules are likely to impact how students understand each GR. In particular, it may matter whether items with the same attributes (e.g., task types) are practiced in a “blocked” manner (e.g., A – A – B – B) or are interleaved with practice of other item types (e.g., A – B – A – B). Research on contextual interference shows that interleaving task types leads to better learning results than blocked practice [5, 6]. A common interpretation of this finding is that interleaved practice encourages deep processing [6]. Since students cannot hold all relevant knowledge components in working memory, they must reactivate taskspecific knowledge components as they come up again in the task sequence.  
−  +  <br>  
−  +  The presented research investigates the effect of sequencing multiple graphical representations on students' learning of fractions.  
−  +  
−  +  
=== Research questions ===  === Research questions ===  
−  +  #<i>Which task attribute(s) should designers of intelligent tutoring systems interleave?</i> Should we interleaved task types or multiple graphical representations?  
+  
+  #<i>Sequencing multiple graphical representations</i> Do students benefit most from blocked or interleaved multiple graphical representations when task types are interleaved?  
=== Hypotheses ===  === Hypotheses ===  
Line 57:  Line 57:  
*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.  *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 ===  +  === 2009 Study === 
+  ==== Dependent variables ====  
*Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:  *Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:  
−  **  +  **Representational knowledge 
−  **  +  **Operational knowledge 
−  +  
−  +  
*Log data collected during tutor use, used to assess:  *Log data collected during tutor use, used to assess:  
Line 69:  Line 68:  
**Error rates  **Error rates  
**Hint usage  **Hint usage  
−  
−  === Independent Variables ===  +  ==== Independent Variables ==== 
−  #<i>  +  #<i>Blocked representations / interleaved topics</i> – representations are blocked while topics are interleaved (students switch topics after every 18 problems) 
−  #<i>  +  #<i>Fully interleaved representations / blocked topics</i> – representations are highly interleaved (students switch representations after each problem) while topic types are blocked 
−  #<i>  +  #<i>Moderately interleaved representations / blocked topics</i> – representations are moderately interleaved (students switch representations after every three problems) while topic types are blocked 
−  #<i>  +  #<i>Increasingly interleaved representations / blocked topics</i> – the length of the blocks of representations is gradually reduced (at the beginning, students switch topics after every twelve problems, at the end they switch after each single problem) while topic types are blocked 
+  
+  ==== Findings ====  
+  We contrasted the effects of interleaving task types (while blocking multiple graphical representations) and several versions of interleaving multiple graphical representations (while blocking task types) in an ITS for fractions. Our results show an advantage for interleaving task types over interleaving multiple graphical representations.  
+  
+  === 2010 Study ===  
+  ==== Dependent variables ====  
+  *Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:  
+  **Area model problems  
+  **Number line problems  
+  **Fraction comparison  
+  **Proportional reasoning  
+  
+  *Log data collected during tutor use, used to assess:  
+  **Learning curves  
+  **Time on task  
+  **Error rates  
+  **Hint usage  
−  ===  +  ==== Independent Variables ==== 
−  +  #<i>Blocked representations </i> – students switch representations after 36 problems  
+  #<i>Moderately interleaved representations</i> – students switch representations after every six problems  
+  #<i>Fully interleaved representations</i> –students switch representations after each problem  
+  #<i>Increasingly interleaved representations</i> – the length of the blocks is gradually reduced from twelve problems at the beginning to a single problem at the end  
+  #<i>Circleonly control</i> – students work with only the circle representation  
+  #<i>Rectangleonly control</i> – students work with only the rectangle representation  
+  #<i>Numberlineonly control</i> – students work with only the number line representation  
−  
−  
−  ===  +  ==== Findings ==== 
−  +  Students in the multiple representation conditions improved their test scores from pretest to posttest, and pertained their learning gains at the delayed posttest. This was not true of the single representation conditions. We found a slight advantage for interleaving multiple representations: On different dependent measures, it was either the fully interleaved, or the increasingly interleaved condition that outperformed the remaining conditions, but never the blocked condition. Finally, the multiple representation conditions significantly outperformed the single representation conditions (including the number line only condition) on number line test items.  
−  ===  +  === Publications === 
+  *Feenstra, Laurens; Aleven, Vincent; Rummel, Nikol; & Taatgen, Nils. Multiple interactive representations for fractions learning. I10th international conference on intelligent tutoring systems (ITS), 2213. 2010.  
+  *Rau, Martina; Aleven, Vincent; Rummel, Nikol, TuncPekkan, Zelha; Pacilio, Laura. How to schedule multiple graphical representations? A classroom experiment with an intelligent tutoring system for fractions. Under review.  
+  *Rau, Martina; Aleven, Vincent; Rummel, Nikol. Blocked versus Interleaved Practice With Multiple Representations in an Intelligent Tutoring System for Fractions. 10th International Conference of Intelligent Tutoring Systems (ITS), 413422. 2010.  
+  *Rau, Martina; Aleven, Vincent; Rummel, Nikol. Intelligent Tutoring Systems with Multiple Representations and SelfExplanation Prompts Support Learning of Fractions. 14th International Conference on Artificial intelligence in Education (AIED), 441448. 2009.  
+  *TunçPekkan, Zelha; Zeylikman, Lyubov; Aleven, Vincent; Rummel, Nikol. Fifth Graders’ Conception of Fractions on Numberline Representations. The annual meeting of North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, Ohio. 2010.  
+  *TuncPekkan, Zelha; Rau, Martina; Aleven, Vincent; Rummel, Nikol. External Representations and Fractional Knowledge. Third Annual interScience of Learning Center (iSLC) Conference For Students and Postdoctoral Fellows at the Science of Learning Centers, Boston, MA. 2010.  
−  +  === References ===  
*Ainsworth, S. (1999). <i>Designing effective multirepresentational learning environments</i> (No. 58). Nottingham: ESRC Centre for Research in Development, Instruction & Training Department of Psychology.  *Ainsworth, S. (1999). <i>Designing effective multirepresentational 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), 183198.  *Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. <i>Learning and Instruction, 16</i>(3), 183198. 
Latest revision as of 11:29, 12 March 2011
Learning with Multiple Representations in a Complex, Realworld Domain: Intelligent Tutoring Systems for Fractions
Vincent Aleven, Nikol Rummel, and Martina Rau
Summary Table
PIs  Vincent Aleven & Nikol Rummel 
Other Contributers  Graduate Students: Martina Rau (CMU HCII) 
2008 study  N = 132 6thgrade students 
2009 study  N = 388 5th and 6thgrade students 
2010 study  N = 690 4th and 5thgrade students 
Study Start Date  September 1st, 2008 
Study End Date  August 31st, 2012 
Total Number of Students to date  N = 1210 
Total Participant Hours  ~6000 
Data available in DataShop  Dataset: Fraction Study Spring 2009 (log data only) Dataset: Fraction Study Spring 2010 (fractions portion only)

Abstract
We investigate a key issue in coordinative learning, namely, how learning with multiple graphical representations should bue used to effectively support students’ conceptual understanding of fractions. In a previous experiment (Rau, Aleven, & Rummel, 2009), we demonstrated that students benefit from learning with multiple graphical representations when compared to a single graphical representation, provided that they were prompted to relate the graphical representations to the symbolic representation of fractions (e.g., 1/2). In two consecutive studies, we investigated how multiple representations should be sequenced. Prior research on contextual interference has demonstrated that interleaving different types of learning tasks can foster a deep understanding of the underlying concepts. Do the same advantages apply to interleaving representations? In future studies, we plan to investigate ways to explicitly support students in relating the different graphical representations to one another. 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; PittaPantazi, 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 & BayWilliams, 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 6thgrade students used four versions of CTATbuilt tutors (Rau, Aleven, & Rummel, 2009). Students learning with MERs and prompted to selfexplain 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). Learning with multiple representations is challenging. An important prerequisite for benefiting from the multiplicity of multiple graphical representations is that students conceptually understand each one of them (Ainsworth, 2006).
When designing intelligent tutoring systems that use multiple graphical representations, designers must decide how to temporally sequence the GRs. How often should the curriculum alternate between multiple graphical representations? Practice schedules are likely to impact how students understand each GR. In particular, it may matter whether items with the same attributes (e.g., task types) are practiced in a “blocked” manner (e.g., A – A – B – B) or are interleaved with practice of other item types (e.g., A – B – A – B). Research on contextual interference shows that interleaving task types leads to better learning results than blocked practice [5, 6]. A common interpretation of this finding is that interleaved practice encourages deep processing [6]. Since students cannot hold all relevant knowledge components in working memory, they must reactivate taskspecific knowledge components as they come up again in the task sequence.
The presented research investigates the effect of sequencing multiple graphical representations on students' learning of fractions.
Research questions
 Which task attribute(s) should designers of intelligent tutoring systems interleave? Should we interleaved task types or multiple graphical representations?
 Sequencing multiple graphical representations Do students benefit most from blocked or interleaved multiple graphical representations when task types are interleaved?
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.
2009 Study
Dependent variables
 Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:
 Representational knowledge
 Operational knowledge
 Log data collected during tutor use, used to assess:
 Learning curves
 Time on task
 Error rates
 Hint usage
Independent Variables
 Blocked representations / interleaved topics – representations are blocked while topics are interleaved (students switch topics after every 18 problems)
 Fully interleaved representations / blocked topics – representations are highly interleaved (students switch representations after each problem) while topic types are blocked
 Moderately interleaved representations / blocked topics – representations are moderately interleaved (students switch representations after every three problems) while topic types are blocked
 Increasingly interleaved representations / blocked topics – the length of the blocks of representations is gradually reduced (at the beginning, students switch topics after every twelve problems, at the end they switch after each single problem) while topic types are blocked
Findings
We contrasted the effects of interleaving task types (while blocking multiple graphical representations) and several versions of interleaving multiple graphical representations (while blocking task types) in an ITS for fractions. Our results show an advantage for interleaving task types over interleaving multiple graphical representations.
2010 Study
Dependent variables
 Previously validated pretest, immediate posttest, and delayed posttest measuring student performance on:
 Area model problems
 Number line problems
 Fraction comparison
 Proportional reasoning
 Log data collected during tutor use, used to assess:
 Learning curves
 Time on task
 Error rates
 Hint usage
Independent Variables
 Blocked representations – students switch representations after 36 problems
 Moderately interleaved representations – students switch representations after every six problems
 Fully interleaved representations –students switch representations after each problem
 Increasingly interleaved representations – the length of the blocks is gradually reduced from twelve problems at the beginning to a single problem at the end
 Circleonly control – students work with only the circle representation
 Rectangleonly control – students work with only the rectangle representation
 Numberlineonly control – students work with only the number line representation
Findings
Students in the multiple representation conditions improved their test scores from pretest to posttest, and pertained their learning gains at the delayed posttest. This was not true of the single representation conditions. We found a slight advantage for interleaving multiple representations: On different dependent measures, it was either the fully interleaved, or the increasingly interleaved condition that outperformed the remaining conditions, but never the blocked condition. Finally, the multiple representation conditions significantly outperformed the single representation conditions (including the number line only condition) on number line test items.
Publications
 Feenstra, Laurens; Aleven, Vincent; Rummel, Nikol; & Taatgen, Nils. Multiple interactive representations for fractions learning. I10th international conference on intelligent tutoring systems (ITS), 2213. 2010.
 Rau, Martina; Aleven, Vincent; Rummel, Nikol, TuncPekkan, Zelha; Pacilio, Laura. How to schedule multiple graphical representations? A classroom experiment with an intelligent tutoring system for fractions. Under review.
 Rau, Martina; Aleven, Vincent; Rummel, Nikol. Blocked versus Interleaved Practice With Multiple Representations in an Intelligent Tutoring System for Fractions. 10th International Conference of Intelligent Tutoring Systems (ITS), 413422. 2010.
 Rau, Martina; Aleven, Vincent; Rummel, Nikol. Intelligent Tutoring Systems with Multiple Representations and SelfExplanation Prompts Support Learning of Fractions. 14th International Conference on Artificial intelligence in Education (AIED), 441448. 2009.
 TunçPekkan, Zelha; Zeylikman, Lyubov; Aleven, Vincent; Rummel, Nikol. Fifth Graders’ Conception of Fractions on Numberline Representations. The annual meeting of North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, Ohio. 2010.
 TuncPekkan, Zelha; Rau, Martina; Aleven, Vincent; Rummel, Nikol. External Representations and Fractional Knowledge. Third Annual interScience of Learning Center (iSLC) Conference For Students and Postdoctoral Fellows at the Science of Learning Centers, Boston, MA. 2010.
References
 Ainsworth, S. (1999). Designing effective multirepresentational 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), 183198.
 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), 2561.
 Bodemer, D., Ploetzner, R., Bruchmüller, K., & Häcker, S. (2005). Supporting learning with interactive multimedia through active integration of representations. Instructional Science, 33(1), 7395.
 Bodemer, D., & Faust, U. (2006). External and mental referencing of multiple representations. Computers in Human Behavior, 22(1), 2742.
 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 PartWhole Numbers. Mathematics Education Research Journal, 16(2), 6986.
 Caney, A., & Watson, J. M. (2003). Mental Computation Strategies for PartWhole 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), 481496.
 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.167194). 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. 4984). 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. 407432). 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 technologysupported learning environments. (pp. 4160). Hillsdale, NJ, England: Lawrence Erlbaum Associates, Inc.
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 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), 8397.
 Lowe, R. K. (1999). Extrating information from an animation during complex visual learning. European Journal of Psychology of Education, 14(2), 225244.
 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), 587625.
 Martinie, S.L., & BayWilliams, J.M. (2003) Investigating students’ conceptual understanding of decimal fractions using multiple representations. Mathematics Teaching in the Middle School, 8(5), 244248.
 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), 122147.
 Moss, J. (2005). Pipes, Tubes, and Beakers: New approaches to teaching the rationalnumber system. In J. Brantsford & S. Donovan (Eds.), How people learn: A targeted report for teachers (pp. 309349): National Academy Press.
 Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. Contemporary Educational Psychology, 26(3), 400417.
 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.
 PittaPantazi, 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), 227237.
 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. 163205). 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), 211233.
 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), 3754.
 Witherspoon, M. L. (1993). Fractions: In Search of Meaning. Arithmetic Teacher, 40(8), 482485.
 Yang, D.C., & Reys, R. E. (2001). One fraction problem, many solution paths. Mathematics Teaching in the Middle School, 7(3), 164166.