C. GODAU ET AL.

OPEN ACCESS

http://dx.doi.org/10.1207/s1532690xci0103_3

Baroody, A. J., Ginsburg, H. P., & Waxman, B. (1983). Ch ildren’s use

of mathematical structure. Journal for Research in Mathematics

Education, 14, 156-168. http://dx.doi.org/10.2307/748379

Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2002). Young children’s

understanding of addition concepts. Educational Psychology, 22, 513-

532. http://dx.doi.org/10.1080/0144341022000023608

Canobi, K. H., Reeve, R. A., & Pattison, P. E. (2003). Patterns of knowl-

edge in children’s addition. Developmental Psychology, 39, 521-534.

http://dx.doi.org/10.1037/0012-1649.39.3.521

Cowan, R., & Renton, M. (1996). Do they know what they are doing?

Children’s use of economical addition strategies and knowledge of

commutativity. Educational Psychology, 16, 407-420.

http://dx.doi.org/10.1080/0144341960160405

Dubé, A. K., & Robinson, K. M. (2010). The relationship between adults’

conceptual understanding of inversion and associativity. Canadian

Journal of Experimental Psychology/Revue Canadienne de Psy-

chologie Expérimentale, 64, 60-66.

http://dx.doi.org/10.1037/a0017756

Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and

computation. Cognition, 44, 43-74.

http://dx.doi.org/10.1016/0010-0277(92)90050-R

Gaschler, R., Vaterrodt, B., Frensch, P. A., Eichler, A., & Haider, H.

(2013). Spontaneous usage of different shortcuts based on the com-

mutativity principle. PLoS ONE, 8, Article ID: e74972.

http://dx.doi.org/10.1371/journal.pone.0074972

Gilmore, C. K., McCarthy, S. E., & Spelke, E. S. (2007). Symbolic

arithmetic knowledge without instruction. Nature, 447, 589-591.

http://dx.doi.org/10.1038/nature05850

Gilmore, C. K., McC arth y, S. E., & Spelke, E. S. (2010). Non-symbolic

arithmetic abilities and achievement in the first year of formal

schooling in mathematics. Cognition, 115, 394-406.

http://dx.doi.org/10.1016/j.cognition.2010.02.002

Godau, C., Haider, H., Hans en, S., Vaterrodt, B. , Schubert, T., Frensch ,

P. A., & Gaschler, R. (2013). Increasing the usage of an arithmetic

shortcut by offering an easy-to-find shortcut based on the same

mathematical principle. Manuscript submitted for publication.

Haider, H., & Frens ch, P. A. (1999). Eye movement during skill acqui-

sition: More evidence for the information-reduction hypothesis.

Journal of Experimental Psychology: Learning, Memory, and Cogni-

tion, 25, 172-190. http://dx.doi.org/10.1037/0278-7393.25.1.172

Hansen, S., Haider, H., Eichler, A., Gaschler, R., Godau, C., & Frensch,

P. A. (2013). Fostering formal commutativity knowledge with ap-

proximate arithmetic. Manuscript submitted for publication.

Madsen, A., Rou infa r, A., Larso n, A. M., Loschk y, L. C., & R ebello , N.

S. (2013). Can short duration visu al cues influence students’ reason-

ing and eye movements in physics problems? Physical Review Spe-

cial Topics—Physics Education Research, 9, Article ID: 020104.

http://dx.doi.org/10.1103/PhysRevSTPER.9.020104

Obersteiner, A., Reiss, K., & Ufer, S . (2013). How training on exact or

approximate mental representations of number can enhance first-

grade students’ basic number processing and arith metic skills . Learn-

ing and Instruction, 23, 125-135.

http://dx.doi.org/10.1016/j.learninstruc.2012.08.004

Resnick, L. B. (1992). From protoquantities to operators: Building mathe-

matical competence on a foundation of everyday knowledge. In G.

Leinhardt, R. P utnam, & R. A. Ha ttrup (Eds.), Ana lysis of arithmetic

for mathematics teaching. Hillsdale, NJ: L. Erlbaum Associates.

Robinson, K. M., & Dubé, A. K. (2012). Children’s use of arithmetic

shortcuts: The role of attitu des in strategy choice. Child Develop ment

Research, 2012, 10. http://dx.doi.org/10.1155/2012/459385

Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and non-

symbolic contexts: Benef its of solving problems with manipulativ es.

Journal of Educational Psychology, 101, 88-100.

http://dx.doi.org/10.1037/a0013156

Siegler, R. S., & Jenkins, E. (1989). How children discover new stra-

tegies (Vol. xiv). Hillsdale, NJ: Lawre n ce Erlbaum Asso ciates, Inc.

Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy

discoveries: A microgenetic analysis. Journal of Experimental Psy-

chology: General, 127, 377-397.

http://dx.doi.org/10.1037/0096-3445.127.4.377

Sophian, C., Harley, H., & Martin, C. S. M. (1995). Relational and

representational aspects o f early number development. Cognition and

Instruction, 13, 253-268.

http://dx.doi.org/10.1207/s1532690xci1302_4

Thomas, L. E., & Lleras, A. (2007). Moving eyes and moving thought:

On the spatial compatibility between eye movements and cognition.

Psychonomic Bulletin & Review, 14, 663-668.

http://dx.doi.org/10.3758/BF03196818

Verschaffe l, L., Luwel, K., Torb eyns , J ., & Do o ren , W. V . (2 0 09 ). C o n -

ceptualizing, investigating, and enhancing adaptive expertise in ele-

mentary mathematics educatio n. European Journal of Psychology of

Education, 24, 335-359. http://dx.doi.org/10.1007/BF03174765

Wilkins, J. L., Bar oo dy, A. J. , & Tiilikain en, S. (20 01). Kindergartn ers ’

understanding of additive commutativity within the context of word

problems. Journal of Experimental Child Psychology, 79, 23-36.

http://dx.doi.org/10.1006/jecp.2000.2580