Early algebraic thinking and the US mathematics standards for grades K to 5 / El pensamiento algebraico temprano y los estándares matemáticos en la Educación Primaria (6–12 años) en Estados Unidos
Restricted accessResearch articleFirst published online September, 2019
Early algebraic thinking and the US mathematics standards for grades K to 5 / El pensamiento algebraico temprano y los estándares matemáticos en la Educación Primaria (6–12 años) en Estados Unidos
We relate studies of algebraic thinking in elementary school (grades K–5, ages five to 10) to current mathematics standards in the United States. Classroom research increasingly paints a promising picture of the potential of young students to learn to think algebraically. This confirms the overall judgment behind recommendations to create a K–12 algebra strand (for ages five to 17) but leaves a host of challenges regarding the implementation of algebraic thinking activities in classrooms and, more generally, to the preparation of teachers.
Get full access to this article
View all access options for this article.
References
1.
BalacheffN. (2001). Symbolic arithmetic vs. algebra the core of a didactical dilemma. In SutherlandR.RojanoT.BellA.LinsR. (Eds.), Perspectives on school algebra (pp. 249–260). Dordrecht: Springer.
2.
BallD. L.Ferrini-MundyJ.KilpatrickJ.MilgramR. J.SchmidW.SchaarR. (2005). Reaching for common ground in K–12 mathematics education. Notices of the AMS, 52, 1055–1058.
3.
BeckmannS. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14, 42–46.
4.
BlantonM. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. New York, NY: Pearson Education.
5.
BlantonM.BrizuelaB.GardinerA. M.SawreyK.Newman-OwensA. (2015). A learning trajectory in 6-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46, 511–558.
6.
BlantonM.BrizuelaB. M.GardinerA. M.SawreyK.Newman-OwensA. (2017). A progression in first-grade children’s thinking about variable and variable notation in functional relationships. Educational Studies in Mathematics, 95, 181–202. Retrieved fromhttp://rdcu.be/osuZ
7.
BlantonM.LeviL.CritesT.DoughertyB. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3–5. Essential understanding series. Reston, VA: National Council of Teachers of Mathematics.
8.
BlantonM.StephensA.KnuthE.GardinerA. M.IslerI.KimJ.-S. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46, 39–87.
9.
BodanskiiF. (1969/1991). The formation of an algebraic method of problem solving in primary school. In DavydovV. V. (Ed.), Psychological abilities of primary school children in learning mathematics (Vol. 6, pp. 275–338). Reston, VA: National Council of Teachers of Mathematics.
10.
BoyerC. B. (1968). A history of mathematics. New York, NY. John Wiley & Sons, Inc.
11.
BrizuelaB. (2016). Variables in elementary mathematics education. The Elementary School Journal, 117, 46–71.
12.
BrizuelaB.Lara-RothS. (2002). Additive relations and function tables. Journal of Mathematical Behavior, 20, 309–319.
13.
BrizuelaB. M.BlantonM.SawreyK.Newman-OwensA.Murphy GardinerA. (2015). Children’s use of variables and variable notation to represent their algebraic ideas. Mathematical Thinking and Learning, 17, 34–63.
14.
BrizuelaB. M.EarnestD. (2008). Multiple notational systems and algebraic understanding: The case of the “best deal” problem. In KaputJ. J.CarraherD. W.BlantonM. L. (Eds.), Algebra in the early grades (pp. 373–402). Mahwah, NJ: Erlbaum & NCTM.
15.
BrizuelaB. M.SchliemannA. D. (2004). Ten-year-old students solving linear equations. For the Learning of Mathematics, 24(2), 33–40.
16.
CaiJ.KnuthE. (Eds.). (2011). Early algebraization. A Global dialogue from multiple perspectives. Berlin: Springer.
17.
CañadasM. C.BrizuelaB. M.BlantonM. (2016). Second graders articulating ideas about linear functional relationships. Journal of Mathematical Behavior, 41, 97–103.
18.
CarpenterT.LeviL. (1999). Developing conceptions of algebraic reasoning in primary grades. Paper presented at the Annual Meeting of the American Educational Research Association, Montreal, Canada.
19.
CarraherD. W.MartinezM.SchliemannA. D. (2008). Early algebra and mathematical generalization. ZDM – The International Journal on Mathematics Education (formerly Zentralblatt fur Didaktik der Mathematik), 40, 3–22.
20.
CarraherD. W.SchliemannA. D. (2007). Early algebra and algebraic reasoning. In LesterF. K.Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 669–705). Charlotte, NC: Information Age Publishing.
21.
CarraherD. W.SchliemannA. D. (2016). Powerful ideas in elementary school mathematics. In EnglishL.KirshnerD. (Eds.), Handbook of international research in mathematics education (pp. 191–218). New York, NY: Taylor and Francis.
22.
CarraherD. W.SchliemannA. D. (2018). Cultivating early algebraic reasoning. In KieranC. (Ed.), Teaching and learning algebraic thinking with 5-12- year-olds (pp. 107–138). The Global Evolution of an Emerging Field of Research and Practice. ICME-13 Monographs. Chaim: Springer International Publishing.
23.
CarraherD. W.SchliemannA. D.BrizuelaB. (2000, October). Early algebra, early arithmetic: Treating operations as functions. [Medium format: CD] Plenary presentation at XXII Meeting of the North American chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ.
24.
CarraherD. W.SchliemannA. D.BrizuelaB. (2005). Treating operations as functions. In D. W. Carraher & R. Nemirovsky (Eds.), Media and meaning. [Medium format: CD]. Monographs of the Journal for Research in Mathematics Education.
25.
CarraherD. W.SchliemannA. D.BrizuelaB.EarnestD. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37, 87–115.
26.
CarraherD. W.SchliemannA. D.SchwartzJ. (2008). Early algebra is not the same as algebra early. In KaputJ. J.CarraherD. W.BlantonM. L. (Eds.), Algebra in the early grades (pp. 235–272). Mahwah, NJ: Erlbaum & NCTM.
DavydovV. V. (1969/1991). Psychological abilities of primary school children in learning mathematics (Vol. 6). Reston, VA: National Council of Teachers of Mathematics.
29.
DoughertyB. (2008). Measure up: A quantitative view of early algebra. In KaputJ. J.CarraherD. W.BlantonM. L. (Eds.), Algebra in the early grades (pp. 389–412). Mahwah, NJ: Erlbaum & NCTM.
30.
EarnestD. (2014). Exploring functions in elementary school: Leveraging the representational context. In KarpK. (Ed.), Annual perspectives in mathematics education: Using research to improve instruction (pp. 171–179). Reston, VA: National Council of Teachers of Mathematics.
31.
FerraraF.SinclairN. (2016). An early algebra approach to pattern generalisation: Actualising the virtual through words, gestures and toilet paper. Educational Studies in Mathematics, 92(1), 1–19.
32.
Ferrini-MundyJ.LappanG.PhillipsE. (1997). Experiences with patterning. Teaching Children Mathematics, 3, 282–289.
33.
FujiiT.StephensM. (2001). Fostering understanding of algebraic generalisation through numerical expressions: The role of the quasi-variables. In ChickH.StaceyK.VincentJ.CooperJ.WarrenE. (Eds.), Proceedings of the 12th ICMI study Conference: The future of the teaching and learning of algebra (Vol 1, pp. 258–264). Melbourne: The University of Melbourne. Retrieved from https://minerva-access.unimelb.edu.au/handle/11343/35000
34.
GowersT. (Ed.). (2008). The Princeton companion to mathematics. Princeton, NJ: Princeton University Press.
35.
GreenbergJ.WalshK. (2008). No common denominator: The preparation of elementary teachers in mathematics by America’s Education Schools. [Mathematics advisory group: R. Askey, A. Chen, M. Goldenberg, R. Howe, J. Kamras, J. Milgram, R. Ramos & Y. Sagher] National Council on Teacher Quality. Retrieved fromwww.nctq.org
KaputJ. J. (1995). Long-term algebra reform: Democratizing access to big ideas. In LaCampagneB.Kaput (Eds.), The algebra initiative colloquium (pp. 37–53).Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.
39.
KaputJ. J. (2008). What is algebra? What is algebraic reasoning? In KaputJ. J.CarraherD. W.BlantonM. L. (Eds.), Algebra in the early grades (pp. 5–17). New York, NY: NCTM & Lawrence Erlbaum Associates.
40.
KaputJ. J.CarraherD. W.BlantonM. L. (Eds.). (2008). Algebra in the early grades. Mahwah, NJ: Erlbaum & NCTM.
41.
KatzV. J. (1995). The development of algebra and algebra education. In LaCampagneC. B.BlairW.KaputJ. J. (Eds.), The algebra initiative colloquium (Vol. 1, pp. 19–36). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Retrieved from https://files.eric.ed.gov/fulltext/ED385436.pdf
42.
KieranC. (2004). Algebraic thinking in the early grades: What is it. The Mathematics Educator, 8, 139–151.
43.
KieranC. (Ed.). (2018). Teaching and learning algebraic thinking with 5-12- year-olds. The global evolution of an emerging field of research and practice. In ICME-13 monographs, New York, NY: Chaim: Springer.
44.
KleinerI. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20, 282–300.
45.
LaCampagneC. B.BlairW.KaputJ. J., (Eds.). (1995). The algebra initiative colloquium: Vol 1: Plenary and reactor papers. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Vol 2: Working group papers ED385437.pdf Retrieved from https://files.eric.ed.gov/fulltext/ED385436.pdf
46.
LovelessT. (2001). A tale of two math reforms: The politics of the new math and the NCTM standards. In LovelessT. (Ed.), The great curriculum debate: How should we teach reading and math (pp. 184–209). Washington, DC: Brookings Institution.
47.
MartinezM. V.BrizuelaB. M. (2006). A third grader’s way of thinking about linear function tables. Journal of Mathematical Behavior, 25, 285–298.
48.
National Commission on Teaching and America’s Future (NCTAF). (1996). What matters most: Teaching for America’s future. New York, NY: NCTAF. Retrieved from:https://eric.ed.gov/?id=ED572506
49.
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved fromhttp://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
NCTM [National Council of Teachers of Mathematics]. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author.
52.
NCTM [National Council of Teachers of Mathematics] Algebra Working Group. (1995). Algebra in the K–12 Curriculum: Dilemmas and Possibilities (Final Report to the Board of Directors). East Lansing, MI: Michigan State University.
53.
QuineW. V. (1987). Quiddities: An intermittently philosophical dictionary. Cambridge, MA: Harvard University Press.
54.
RadfordL. (2018). The emergence of symbolic algebraic thinking in primary school. In KieranC. (Ed.), Teaching and learning algebraic thinking with 5-to 12-year-olds (pp. 3–25). Chaim: Springer.
55.
ResnickM. D. (1981). Mathematics as a science of patterns: Ontology and reference. Noûs, 15, 529–550. Special Issue on Philosophy of Mathematics. Retrieved fromhttps://www.jstor.org/stable/2214851
56.
SchliemannA. D.CarraherD. W.BrizuelaB. M. (2001). When tables become function tables. In Heuvel-PanhuizenM. V. D. (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 145–152). Utrecht: Freudenthal Institute.
57.
SchliemannA. D.CarraherD. W.BrizuelaB. M. (2007). Bringing out the algebraic character of arithmetic: From children’s ideas to classroom practice. Hillsdale, NJ: Lawrence Erlbaum.
58.
SchliemannA. D.CarraherD. W.BrizuelaB. M. (2012). Algebra in elementary school. In CoulangeL.DrouhardJ.-P. (Eds.), Enseignement de l’algèbre élémentaire: Bilan et perspectives (pp. 109–124). (Special Issue of Recherches en Didactique des Mathématiques). Grenoble: La Pensée Sauvage, Editions.
SchmittauJ. (2005). The development of algebraic thinking. ZDM, 37, 16–22.
61.
SchoenfeldA. (1995). Report of working group 1. In LaCampagneC. B.BlairW.KaputJ. J. (Eds.), The algebra initiative colloquium: Vol 2 ED385437.pdf#page=11 (pp. 11–18). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.
62.
SchoenfeldA. H.ArcaviA. (1988). On the meaning of variable. The Mathematics Teacher, 81, 420–427.
63.
SchwartzJ.YerushalmyM. (1992). Getting students to function in and with algebra. The Concept of Function: Aspects of Epistemology and Pedagogy, 25, 261–289.
64.
SteenL. A. (1988). The science of patterns. Science, 240, 611–616. doi:10.1126/science.240.4852.611
65.
SteenL. A. (1990). Pattern. In SteenL. A. (Ed.), On the shoulders of giants: New approaches to numeracy (pp. 1–10). Mathematical Sciences Education Board & National Research Council. Washington, DC: National Academy Press.
66.
Teixidor-i-BigasM.SchliemannA. D.CarraherD. W. (2013). Integrating disciplinary perspectives: The Poincaré Institute for Mathematics Education. The Mathematics Enthusiast, 10, 519–562.
67.
UsiskinZ. (1988). Conceptions of school algebra and uses of variables. In CoxfordA. F.ShulteA. P. (Eds.), The Ideas of algebra, K–12. 1988 Yearbook (pp. 7–13). Reston, VA: National Council of Teachers of Mathematics.
