<p>Curricular approaches to connecting subtraction to addition and fostering fluency with basic differences in grade 1</p>

Arthur J. Baroody

doi:10.30827/pna.v10i3.6087

Autores/as

Arthur J. Baroody University of Illinois at Urbana-Champaign; University of Denver, Estados Unidos

DOI:

https://doi.org/10.30827/pna.v10i3.6087

Palabras clave:

Currículos de matemáticas de primero de educación primaria, Estrategia de la sustracción como suma, Fluidez informativa, Relaciones parte-todo, Trayectorias de aprendizaje

Resumen

Six widely used US Grade 1 curricula do not adequately address the following three developmental prerequisites identified by a proposed learning trajectory for the meaningful learning of the subtraction-as-addition strategy (e.g., for 13 – 8 think “what + 8 = 13?”): (a) reverse operations (adding 8 is undone by subtracting 8); (b) common part-whole relations (5 + 8 and 13 – 8 share the same whole 13 and parts 5 and 8); and (c) the complement principle in terms of part-whole relations (if parts 5 and 8 make the whole 13, then subtracting one part from the whole leaves the other part).

Aproximaciones curriculares para conectar la sustracción con la adición y promover la fluidez con las diferencias básicas en primer curso de educación primaria

Seis currículos ampliamente usados en Estados Unidos para primero de primaria no tratan adecuadamente tres prerrequisitos identificados por una trayectoria de aprendizaje propuesta para el aprendizaje significativo de la estrategia de sustracción como adición (ejemplo, para 13 – 8 piensa “¿qué + 8 = 13?”): (a) operaciones inversas (sumar 8 se deshace restando 8); (b) relaciones parte-todo comunes (5 + 8 y 13 – 8 comparten el mismo todo 13 y las partes 5 y 8) y (c) principio de complemento en relaciones parte-todo (si 5 y 8 dan el todo 13, al restar una parte al todo se obtiene la otra parte).

Handle: http://hdl.handle.net/10481/40449

WOS-ESCI

Descargas

Los datos de descargas todavía no están disponibles.

Biografía del autor/a

Arthur J. Baroody, University of Illinois at Urbana-Champaign; University of Denver, Estados Unidos

Código ORCID

Citas

Baroody, A. J. (1984). Children's difficulties in subtraction: Some causes and questions. Journal for Research in Mathematics Education, 15, 203-213. doi:10.2307/748349

Baroody, A. J. (1988). A cognitive approach to writing instruction for children classified as mentally handicapped. Arithmetic Teacher, 36(2), 7-11.

Baroody, A. J. (1989). Kindergartners' mental addition with single-digit combinations. Journal for Research in Mathematics Education, 20, 159-172. doi: 10.2307/749280

Baroody, A. J. (1992). The development of kindergartners' mental-addition strategies. Learning and Individual Differences, 4, 215-235. doi: 10.1016/1041-6080(92)90003-W

Baroody, A. J. (1999). Children's relational knowledge of addition and subtraction. Cognition and Instruction, 17, 137-175. doi: 10.1207/S1532690XCI170201

Baroody, A. J., Eiland, M. D., Purpura, D. J., & Reid, E. E. (2012). Fostering kindergarten children's number sense. Cognition and Instruction, 30, 435-470. doi: 10.1080/07370008.2012.720152

Baroody, A. J., Eiland, M. D., Purpura, D. J., & Reid, E. E. (2013). Can computer-assisted discovery learning foster first graders' fluency with the most basic addition combinations? American Educational Research Journal, 50, 533-573. doi: 10.3102/0002831212473349

Baroody, A. J., Eiland, M. D., & Thompson, B. (2009). Fostering at-risk preschoolers' number sense. Early Education and Development, 20, 80-120. doi: 10.1080/10409280802206619

Baroody, A. J., Ginsburg, H. P., & Waxman, B. (1983). Children's use of mathematical structure. Journal for Research in Mathematics Education, 14, 156-168. doi: 10.2307/748379

Baroody, A. J., & Kaufman, L. C. (1993). The case of Lee: Assessing and remedying a numerical-writing difficulty. Teaching Exceptional Children, 25(3), 14-16.

Baroody, A. J., & Lai, M. L. (2007). Preschoolers' understanding of the addition-subtraction inverse principle: A Taiwanese sample. Mathematics Thinking and Learning, 9, 131-171. doi: 10.1080/10986060709336813

Baroody, A. J., Lai, M. L., Li, X., & Baroody, A. E. (2009). Preschoolers' understanding of subtraction-related principles. Mathematics Thinking and Learning, 11, 41-60. doi: 10.1080/10986060802583956.

Baroody, A. J., Lai, M.-L., & Mix, K. S. (2006). The development of young children's number and operation sense and its implications for early childhood education. In B. Spodek & O. Saracho (Eds.), Handbook of research on the education of young children (pp. 187-221). Mahwah, NJ: Erlbaum.

Baroody, A. J., Purpura, D. J., Eiland, M. D., & Reid, E. E. (2014). Fostering first-graders' fluency with basic subtraction and larger addition combinations via computer-assisted instruction. Cognition & Instruction, 32, 159-197. doi: 10.1080/07370008.2014.887084

Baroody, A. J., Purpura, D. J., Eiland, M. D., & Reid, E. E. (2015). The impact of highly and minimally guided discovery instruction on promoting the learning of reasoning strategies for basic add-1 and doubles combinations. Early Childhood Research Quarterly, 30, 93-105. doi: 10.1016/j.ecresq.2014.09.003

Baroody, A. J., Purpura, D. J., Eiland, M. D., Reid, E. E., & Paliwal, V. (in press). Does fostering reasoning strategies for relatively difficult basic combinations promote transfer by K-3 students? Journal of Educational Psychology.

Baroody, A. J., Torbeyns, J., & Verschaffel, L. (2009). Young children's understanding and application of subtraction-related principles: Introduction. Mathematics Thinking and Learning, 11, 2-9. doi: 10.1080/10986060802583873

Baroody, A. J., & Varma, S. (2006, November). The active construction view of basic number fact knowledge: New directions for cognitive neuroscience. Invited presentation at the Neural Basis of Mathematical Cognition Conference, Nashville, TN.

Baroody, A. J., Wilkins, J. L. M., & Tiilikainen, S. (2003). The development of children's understanding of additive commutativity: From protoquantitative concept to general concept? In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 127-160). Mahwah, NJ: Erlbaum.

Bloom, P., & Wynn, K. (1997). Linguistic cues in the acquisition of number words. Journal of Child Language, 24, 511-533. doi: 10.1017/S0305000997003188

Briars, D. J., & Larkin, J. G. (1984). An integrated model of skills in solving elementary word problems. Cognition and Instruction, 1, 245-296. doi: 10.1207/s1532690xci0103_1

Campbell, J. I. D. (2008). Subtraction by addition. Memory & Cognition, 36, 1094-1102. doi: 10.3758/MC.36.6.1094

Canobi, K. H. (2004). Individual differences in children's addition and subtraction knowledge. Cognitive Development, 19, 81-93.

Canobi, K. H. (2005). Children's profiles of addition and subtraction understanding. Journal of Experimental Child Psychology, 92, 220-246.

Canobi, K. H. (2009). Conceptual-procedural interactions in children's addition and subtraction. Journal of Experimental Child Psychology, 102, 131-149. doi: 10.1016/j.jecp.2008.07.008

Canobi, K. H., Reeve, R. A., & Patterson, P. E. (1998). The role of conceptual understanding in children's problem solving. Developmental Psychology, 34, 882-891. doi: 10.1037/0012-1649.34.5.882

Carroll, W. M., & Issacs, A. (2003). Achievement of students using the University of Chicago School Mathematics Project's Everyday Mathematics. In S. L. Senk & D. R. Thompson (Eds.), Standards-based school mathematics curricula: What are they? What do students learn? (pp. 79-108). Mahwah, NJ: Erlbaum

Council of Chief State School Officers (2010). Common core state standards: Preparing America's students for college and career. Retrieved from http://www.corestandards.org/

Dowker, A. D. (2003). Young children's estimates for addition: The zone of partial knowledge and understanding. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 35-74). Mahwah, NJ: Erlbaum.

Dowker, A. (2009). Use of derived fact strategies by children with mathematical difficulties. Cognitive Development, 24, 401-410. doi: 10.1016/j.cogdev. 2009.09.005

Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22, 206-214. doi: 10.1016/j.learninstruc.2011.11.001

Dyson, N., Jordan, N., & Glutting, J. (2013). A number sense intervention for low-income kindergartners at risk for mathematics difficulties. Journal of Learning Disabilities, 46, 166-181. doi: 10.1177/0022219411410233

Dyson, N., Jordan, N. C., & Hassinger-Das, B. (in press). Kindergarten number operations: Connecting the dots. Teaching Children Mathematics.

Eiland, M. D. (2014). Fostering fluency with basic addition and subtraction facts using computer-aided instruction (Unpublished doctoral dissertation). https://www.ideals.illinois.edu/bitstream/handle/2142/50759/Michael_Eiland.pdf?sequence=1

Ernest, P. (1985). The number line as a teaching aid. Educational Studies in Mathematics, 16, 411-424.

Fayol, M., & Thevenot, C. (2012). The use of procedural knowledge in simple addition and subtraction. Cognition, 123, 392-403. doi:10.1016/j.cognition.2012.02.008

Frye, D., Baroody, A. J., Burchinal, M. R., Carver, S., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, US Department of Education.

Fuson, K. C. (2006). Math expressions. Boston, MA: Houghton Mifflin.

Fuson, K. C. (2009). Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning, learning without teaching, or helpful learning-path teaching. Cognitive Development, 24, 343-361. doi: 10.1016/j.cogdev. 2009.09.009

Hattikudur, S., & Alibali, M. W. (2010). Learning about the equal sign: Does comparing with inequality symbols help? Journal of Experimental Child Psychology, 107, 15-30. doi: 10.1016/j.jecp.2010.03.004

Houghton Mifflin Harcourt (2015). Go math! Boston, MA: Author.

Huttenlocher, J., Jordan, N. C., & Levine, S. C. (1994). A mental model for early arithmetic. Journal of Experimental Psychology: General, 123, 284-296.

James, W. (1958). Talks to teachers on psychology: And to students on some of life's ideals. New York, NY: Norton & Company.

Jordan, N. C., Huttenlocher, J., & Levine, S. C. (1992). Differential calculation abilities in young children from middle- and low-income families. Developmental Psychology, 28, 644-653. doi: 10.1037/0012-1649.28.4.644

Jordan, N. C., Huttenlocher, J., & Levine, S. C. (1994). Assessing early arithmetic abilities: Effects of verbal and nonverbal response types on the calculation performance of middle- and low-income children. Learning and Individual Differences, 6, 413-432. doi: 10.1016/1041-6080(94)90003-5

Jordan, N. C., Glutting, J., Dyson, N., Hassinger-Das, B., & Irwin, C. (2012, April). Building kindergartners' number sense: A randomized controlled study. Paper presented at the annual meeting of the National Council of Teachers of Mathematics, Philadelphia.

Klein, J. S., & Bisanz, J. (2000). Preschoolers doing arithmetic: The concepts are willing but the working memory is weak. Canadian Journal of Experimental Psychology, 54, 105-114. doi: 10.1037/h0087333

Larson, N. (2008). Saxon math 1. Austin, TX: Harcourt Achieve. Macmillan/McGraw Hill.

Macmillan/McGraw Hill (2009). Math Connects. New York, NY: Author

Mix, K. S. (2009). How Spencer made number: First uses of the number words. Journal of Experimental Child Psychology, 102, 427-444. doi: 10.1016/j.jecp.2008.11.003

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics: Standards 2000. Reston, VA: Author.

National Mathematics Advisory Panel (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: US Department of Education.

National Research Council (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Nunes, T., Bryant, P., Hallett, D., Bell, D., & Evans, D. (2009). Teaching children about the inverse relation between addition and subtraction. Mathematical Thinking & Learning, 11, 61-78.

Palmer, A., & Baroody, A. J. (2011). Blake's development of the number words “one,” “two,” and “three.” Cognition and Instruction, 29, 265-296. doi: 10.1080/ 07370008.2011.583370

Peters, G., De Smedt, B., Torbeyns, J., Ghesquire, P., & Verschaffel, L. (2010). Adults' use of subtraction by addition. Acta Psychologica, 135, 323-329. doi: 10.1016/j.actpsy.2010.08.007

Piaget, J. (1964). Development and learning. In R. E. Ripple & V. N. Rockcastle (Eds.), Piaget rediscovered (pp. 7-20). Ithaca, NY: Cornell University.

Piaget, J. (1965). The child's conception of number. New York, NY: Norton.

Putnam, R. T., deBettencourt, L. U., & Leinhardt, G. (1990). Understanding of derived fact strategies in addition and subtraction. Cognition and Instruction, 7, 245-285. doi: 10.1207/s1532690xci0703_3

Rathmell, E. C. (1978). Using thinking strategies to teach basic facts. In M. N. Suydam & R. E. Reys (Eds.), Developing computational skills (pp. 13-50). Reston, VA: National Council of Teachers of Mathematics.

Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children's problem solving ability in arithmetic. In H. P. Ginsburg (Ed.), The Development of mathematical thinking (pp. 153-196). New York, NY: Academic Press.

Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99, 561-574. doi: 10.1037/0022-0663.99.3.561

Rittle-Johnson, B., & Star, J. R. (2011). The power of comparison in learning and instruction: Learning outcomes supported by different types of comparisons. In J. P. Mestre & B. H. Ross (Eds.), The psychology of learning and motivation: Cognition in education (Vol. 55, pp. 199-225). Oxford, United Kingdom: Academic Press.

Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.

Sarnecka, B. W., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108(3), 662-674. doi: 10.1016/j.cognition.2008.05.007

Saxe, G. B., Earnest, D., Sitabkhan, Y., Haldar, L. C., Lewis, K. E., & Zheng, Y. (2010). Supporting generative thinking about integers on number lines in elementary mathematics, Cognition & Instruction, 28, 433-474. doi: 10.1080/07370008.2010.511569

Scott Foresman-Addison Wesley (2005). Mathematics (first grade). Pearson Education. Upper Saddle River, NJ: Author.

Sophian, C., & McCorgray, P. (1994). Part-whole knowledge and early arithmetic problem-solving. Cognition and Instruction, 12, 3-33. doi: 10.1207/ s1532690xci1201_1

Sophian, C., & Vong, K. I. (1995). The parts and wholes of arithmetic story problems: Developing knowledge in the preschool years. Cognition and Instruction, 13, 469-477. doi: 10.1207/s1532690xci1303_5

Starkey, P., & Gelman, R. (1982). The development of addition and subtraction abilities prior to formal schooling in arithmetic. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 99-116). Hillsdale, NJ: Erlbaum.

Steinberg, R. M. (1985). Instruction on derived facts strategies in addition and subtraction. Journal for Research in Mathematics Education, 16, 337-355. doi: 10.2307/749356

Thompson, I. (1997). The role of counting in derived fact strategies. In I. Thompson (Ed.), Teaching and learning early number (pp. 52-61). Buckingham, United Kingdom: Open University Press.

Thornton, C. A. (1990). Solution strategies: Subtraction number facts. Educational Studies in Mathematics, 21, 241-263. doi: 10.1007/BF00305092

Thornton, C., & Toohey, M. (1985). Basic math facts: Guidelines for teaching and learning. Learning Disabilities Focus, 1, 44-57. doi: 10.2307/748382

University of Chicago School Mathematics Project (UCSMP) (2005). Everyday mathematics teacher's lesson guide (Volume 1). Columbus, OH: McGraw-Hill

Van den Heuvel-Panhuizen, M., & Treffers, A. (2009). Mathe-didactical reflections on young children's understanding and application of subtraction-related principles. Mathematical Thinking and Learning, 11, 102-112. doi: 10.1080/ 10986060802584046

Verschaffel, L., Greer, B., & De Corte, E. (2007). Whole number concepts and operations. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 557-628). Charlotte, NC: NCTM & Information Age Publishing.

Walker, D., Bajic, D., Kwak, J., & Rickard, T. C. (2014). Specificity of children's arithmetic learning. Journal of Experimental Child Psychology, 122, 62-74. doi: 10.1016/j.jecp.2013.11.018

Walker, D., Mickes, L., Bajic, D., Nailon, C. R., & Rickard, T. C. (2013). A test of two methods of arithmetic fluency training and implications for educational practice. Journal of Applied Research in Memory and Cognition, 2, 25-32. doi: 10.1016/j.jarmac.2013.0