Although basing instruction on learning trajectories (LTs) is often recommended, there is little direct evidence regarding the premise of a LT approach—that instruction should be presented (only) one LT level beyond a child’s present level. We evaluated this hypothesis in the domain of early shape composition. One group of preschoolers, who were at least two levels below the target instructional LT level, received instruction based on an empirically validated LT. The counterfactual (skip-levels) group received an equal amount of instruction focused only on the target level. At posttest, children in the LT condition exhibited significantly greater learning than children in the skip-levels condition, mainly on near-transfer items; no child-level variables were significant moderators. Implications for theory and practice are discussed.
AlonzoA. C.GotwalsA. W. (2012). Leaping forward: Next steps for learning progressions in science. In AlonzoA. C.GotwalsA. W. (Eds.), Learning progressions in science (pp. 475–490). Rotterdam, Netherlands: Sense. doi:10.1007/978-94-6091-824-7_20
2.
BaroodyA. J.ClementsD. H.SaramaJ. (2019). Teaching and learning mathematics in early childhood programs. In BrownC. P.McMullenM. B.FileN. (Eds.), Handbook of early childhood care and education (pp. 329–354). Hoboken, NJ: Wiley Blackwell.
3.
BondT. G.FoxC. M. (2007). Applying the Rasch model: Fundamental measurement in the human sciences (2nd ed.). Mahwah, NJ: Erlbaum. doi:10.1111/j.1745-3984.2003.tb01103.x
4.
CarnineD. W.JitendraA. K.SilbertJ. (1997). A descriptive analysis of mathematics curricular materials from a pedagogical perspective: A case study of fractions. Remedial and Special Education, 18, 66–81.
5.
CaseyB. M.ErkutS.CederI.YoungJ. M. (2008). Use of a storytelling context to improve girls’ and boys’ geometry skills in kindergarten. Journal of Applied Developmental Psychology, 29, 29–48.
6.
ClarkR. E.KirschnerP. A.SwellerJ. (2012). Putting students on the path to learning: The case for fully guided instruction. American Educator, 36, 6–11.
7.
ClarkeD. M.CheesemanJ.ClarkeB.GervasoniA.GronnD.HorneM.. . .SullivanP. (2001). Understanding, assessing and developing young children’s mathematical thinking: Research as a powerful tool for professional growth. In BobisJ.PerryB.MitchelmoreM. (Eds.), Numeracy and beyond: Proceedings of the 24th Annual Conference of the Mathematics Education Research Group of Australasia Incorporated (Vol. 1, pp. 9–26). Reston, Australia: MERGA.
8.
ClementsD. H.BattistaM. T.SaramaJ.SwaminathanS. (1997). Development of students’ spatial thinking in a unit on geometric motions and area. Elementary School Journal, 98, 171–186.
9.
ClementsD. H.SaramaJ. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6, 81–89. doi:10.1207/s153278 33mtl0602_1
10.
ClementsD. H.SaramaJ. (2007). Effects of a preschool mathematics curriculum: Summative research on theBuilding Blocks project. Journal for Research in Mathematics Education, 38, 136–163.
11.
ClementsD. H.SaramaJ. (2007/2013). Building blocks, Volumes 1 and 2. Columbus, OH: McGraw-Hill Education.
12.
ClementsD. H.SaramaJ. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal, 45, 443–494. doi:10.3102/0002831207312908
13.
ClementsD. H.SaramaJ. (2014a). Learning and teaching early math: The learning trajectories approach (2nd ed.). New York, NY: Routledge.
14.
ClementsD. H.SaramaJ. (2014b). Learning trajectories: Foundations for effective, research-based education. In MaloneyA. P.ConfreyJ.NguyenK. H. (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 1–30). New York, NY: Information Age.
15.
ClementsD. H.SaramaJ.BattistaM. T.SwaminathanS. (1996). Development of students’ spatial thinking in a curriculum unit on geometric motions and area. In JakubowskiE.WatkinsD.BiskeH. (Eds.), Proceedings of the 18th Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 217–222). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
16.
ClementsD. H.SaramaJ.DiBiaseA.-M. (2004). Engaging young children in mathematics: Standards for early childhood mathematics education. Mahwah, NJ: Erlbaum.
17.
ClementsD. H.SaramaJ.LiuX. (2008). Development of a measure of early mathematics achievement using the Rasch model: The Research-based Early Maths Assessment. Educational Psychology, 28, 457–482. doi:10.1080/01443410701777272
18.
ClementsD. H.SaramaJ.SpitlerM. E.LangeA. A.WolfeC. B. (2011). Mathematics learned by young children in an intervention based on learning trajectories: A large-scale cluster randomized trial. Journal for Research in Mathematics Education, 42, 127–166. doi:10.5951/jresematheduc.42.2.0127
19.
ClementsD. H.SaramaJ.WolfeC. B.Day-HessC. A. (2019). REMA—Research-based Early Mathematics Assessment. Denver, CO: Kennedy Institute, University of Denver. (Original work published 2008)
20.
ClementsD. H.WilsonD. C.SaramaJ. (2004). Young children’s composition of geometric figures: A learning trajectory. Mathematical Thinking and Learning, 6, 163–184. doi:10.1207/s15327833mtl0602_1
FantuzzoJ. W.GadsdenV. L.McDermottP. A. (2011). An integrated curriculum to improve mathematics, language, and literacy for Head Start Children. American Educational Research Journal, 48, 763–793.
23.
FryeD.BaroodyA. J.BurchinalM. R.CarverS.JordanN. C.McDowellJ. (2013). Teaching math to young children: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education.
24.
FusonK. C. (2004). Pre-K to grade 2 goals and standards: Achieving 21st century mastery for all. In ClementsD. H.SaramaJ.DiBiaseA.-M. (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 105–148). Mahwah, NJ: Erlbaum.
25.
GravemeijerK. P. E. (1999). How emergent modelsmayfoster the constitution of formal mathematics. Mathematical Thinking and Learning, 1, 155–177.
26.
HelmJ. H.KatzL. G. (2016). Young investigators: The project approach in the early years (3rd ed.). New York, NY: Teachers College Press.
27.
JordanN. C.GluttingJ.DysonN.Hassinger-DasB.IrwinC. (2012). Building kindergartners’ number sense: A randomized controlled study. Journal of Educational Psychology, 104, 647–660. doi:10.1037/a0029018
28.
KatzL. G.ChardS. C. (2000). Engaging children’s minds: The project approach (2nd ed.). Stamford, CT: Ablex.
29.
LeshR. A.YoonC. (2004). Evolving communities of mind—in which development involves several interacting and simultaneously development strands. Mathematical Thinking and Learning, 6, 205–226.
30.
LinacreJ. M. (2014). A user’s guide to Winsteps/Ministep Rasch-model computer program. Chicago IL: Winsteps.com.
31.
MaloneyA. P.ConfreyJ.NguyenK. H. (Eds.). (2014). Learning over time: Learning trajectories in mathematics education. New York, NY: Information Age.
32.
MansfieldH. M.ScottJ. (1990). Young children solving spatial problems. In BookerG.CobbP.deMendicutiT. N. (Eds.), Proceedings of the 14th annual conference of the Internation Group for the Psychology of Mathematics Education (Vol. 2, pp. 275–282). Oaxlepec, Mexico: International Group for the Psychology of Mathematics Education.
33.
MastersG. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.
34.
MuellerU.SokolB.OvertonW. F. (1999). Developmental sequences in class reasoning and propositional reasoning. Journal of Experimental Child Psychology, 74, 69–106.
35.
MurataA. (2004). Paths to learning ten-structured understanding of teen sums: Addition solution methods of Japanese grade 1 students. Cognition and Instruction, 22, 185–218.
36.
MuthénL. K.MuthénB. O. (1998/2014). Mplus user’s guide (6th ed.). Los Angeles, CA: Author.
37.
National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author.
38.
National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards. Washington, DC: Author.
39.
National Research Council. (2007). Taking science to school: Learning and teaching sciences in grades K-8. Washington, DC: National Academies Press.
40.
National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: National Academies Press. doi:10.17226/12519
41.
PiagetJ. (1964). Development and learning. In RippleR. E.RockcastleV. N. (Eds.), Piaget rediscovered (pp. 7–20). Ithaca, NY: Cornell University.
42.
RazelM.EylonB.-S. (1990). Development of visual cognition: Transfer effects of the Agam program. Journal of Applied Developmental Psychology, 11, 459–485.
43.
RazelM.EylonB.-S. (1991, July). Developing mathematics readiness in young children with the Agam Program. Paper presented at the meeting of the Fifteenth Conference of the International Group for the Psychology of Mathematics Education, Genova, Italy.
44.
ResnickL. B.FordW. W. (1981). The psychology of mathematics for instruction. Hillsdale, NJ: Erlbaum.
45.
ReynoldsA.WheatleyG. H. (1996). Elementary students’ construction and coordination of units in an area setting. Journal for Research in Mathematics Education, 27, 564–581.
46.
RhoadsC. H. (2010). The implications of “contamination” for experimental design in education. Journal of Educational and Behavioral Statistics, 36, 76–104. doi:10.3102/1076998610379133
47.
SalesC. (1994). A constructivist instructional project on developing geometric problem solving abilities using pattern blocks and tangrams with young children (Unpublished master’s thesis). University of Northern Iowa, Cedar Falls.
48.
SaramaJ.ClementsD. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.
49.
SaramaJ.ClementsD. H.VukelicE. B. (1996). The role of a computer manipulative in fostering specific psychological/mathematical processes. In JakubowskiE.WatkinsD.BiskeH. (Eds.), Proceedings of the 18th Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 567–572). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
50.
SimonM. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145. doi:10.2307/749205
51.
The Spatial Reasoning Study Group. (2015). Spatial reasoning in the early years: Principles, assertions, and speculations. New York, NY: Routledge.
52.
SteffeL. P.CobbP. (1988). Construction of arithmetical meanings and strategies. New York, NY: Springer-Verlag.
53.
SteffeL. P.ThompsonP. W.GlasersfeldE. V. (2000). Teaching experiment methodology: Underlying principles and essential elements. In KellyA. E.LeshR. A. (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Mahwah, NJ: Erlbaum.
54.
TullisP. (2011). The death of preschool. Scientific American Mind, 22(5), 36–41.
55.
VurpillotE. (1976). The visual world of the child. New York, NY: International Universities Press.
56.
WilsonM. (2009). Measuring progressions: Assessment structures underlying a learning progression. Journal of Research in Science Teaching, 46, 716–730.
57.
WrightB. D.StoneM. H. (1979). Best test design: Rasch measurement. Chicago, IL: MESA Press.
58.
WrightR. J.StangerG.StaffordA. K.MartlandJ. (2006). Teaching number in the classroom with 4-8 year olds. London, England: Paul Chapman/Russell Sage.
59.
WuH.-H. (2011). Understanding numbers in elementary school mathematics. Providence, RI: American Mathematical Society.
