Which ideas should be included in the K–12 curriculum, how they are learned, and how they should be taught have been debated for decades in multiple subjects. In this article, we offer mathematics as a case in point of how new standards-related policies may offer an opportunity for reassessment and clarification of such debates.
Purpose/Objective
Our goal was to specify instructional models associated with terms such as “reform” and “traditional”—which, in this article, we refer to as “dialogic” and “direct”—in terms of perspectives on what it means to know mathematics, how students learn mathematics, and how mathematics should be taught.
Research Design
In the spirit of “adversarial collaboration,” we hosted a series of semi-structured discussions among nationally recognized experts who hold opposing points of view on mathematics teaching and/or learning. During those discussions, the recent consensus regarding what students should learn—as represented by the Common Core State Standards for Mathematics (CCSSM)—was taken as a common goal, and additional areas of agreement and disagreement were identified and discussed. The goal was not to reach consensus but to invite representatives of different perspectives to clarify and come to agreement on how they disagree.
Findings/Results
We present two instructional models that were specified and refined over the course of those discussions and describe nine key areas that distinguish the two models: (a) the importance and role of talk; (b) the importance and role of group work; (c) the sequencing of mathematical topics; (d) the nature and ordering of mathematical instructional tasks; (e) the nature, timing, source, and purpose of feedback; (f) the emphasis on creativity (i.e., authoring one's own learning; mathematizing subject matter from reality); (g) the purpose of diagnosing student thinking; (h) the introduction and role of definitions; and (i) the nature and role of representations. Additionally, we elaborate a more nuanced description of the ongoing debate, as it pertains to particular sources of difference in perspective.
Conclusions/Recommendations
With this article, we hope to advance ongoing debates in two ways: (a) discrediting false assumptions and oversimplified conceptions of the “other side's” arguments (which can obscure both the real differences and real similarities between different models of instruction), and (b) framing the debates in a manner that allows for more thoughtful empirical investigation oriented to understanding learning in the discipline.
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References
1.
AndersonJ. R., RederL. M., & SimonH. A. (1998). Radical constructivism and cognitive psychology. In RavitchD. (Ed.), Brookings papers on education policy: 1998 (pp. 227–255). Washington, DC: Brookings Institute.
2.
AndersonJ. R., RederL. M., & SimonH. A. (2000). Applications and misapplications of cognitive psychology to mathematics instruction. Texas Education Review, 1(2), 29–49.
3.
ApplebeeA. N., LangerJ. A., NystrandM., & GamoranA. (2003). Discussion-based approaches to developing understanding: Classroom instruction and student performance in middle and high school English. American Educational Research Journal, 40(3), 685–730.
4.
BallD. L., Ferrini-MundyJ., KilpatrickJ., MilgramR. J., SchmidW., & SchaarR. (2005). Reaching for common ground in K–12 mathematics education. Notices of the American Mathematical Society, 52(9), 1055–1058.
5.
BarrowL. H. (2006). A brief history of inquiry: From Dewey to standards. Journal of Science Teacher Education, 17(3), 265–278.
6.
BoalerJ. (1999). Participation, knowledge, and beliefs: A community perspective on mathematics learning. Educational Studies in Mathematics, 40(3), 259–281.
CarmichaelS. B., MartinoG., Porter-MageeK., & WilsonW. S. (2010). The state of state standards–and the Common Core–in 2010. Washington, DC: Thomas B. Fordham Institute.
9.
CarpenterT. P., & LehrerR. (1999). Teaching and learning mathematics with understanding. In FennemaE. & RombergT. A. (Eds.), Mathematics classrooms that promote understanding (pp. 19–32). Mahwah, NJ: Lawrence Erlbaum.
10.
ClarkR. E., KirschnerP. A., & SwellerJ. (2012). Putting students on the path to learning: The case for fully guided instruction. American Educator, 36(1), 6–11.
11.
CobbP., YackelE., & WoodT. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2–33.
Common Core State Standards Initiative. (2010). Common Core State Standards for English Language Arts & Literacy in History/Social Studies, Science, and Technical Subjects. Retrieved from http://www.corestandards.org/assets/CCSSI_ELA%20Standards.pdf
ConfreyJ. (1994). Splitting, similarity, and the rate of change: A new approach to multiplication and exponential functions. In HarelG. & ConfreyJ. (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 293–332). Albany, NY: State University of New York Press.
EvansR. W. (2006). The social studies wars, now and then. Social Education, 70(5), 317–321.
18.
FullanM. (2007). The new meaning of educational change (4th ed.). New York, NY: Teachers College Press.
19.
GarelickB. (2005). An a-maze-ing approach to math: A mathematician with a child learns some politics. Education Next, 5(2), 29–36.
20.
GilovichT., MedvecV. H., & KahnemanD. (1998). Varieties of regret: A debate and partial resolution. Psychological Review, 105(3), 602–605.
21.
GravemeijerK. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105–128.
22.
GreenoJ. G. (1997). On claims that answer the wrong questions. Educational Researcher, 26(1), 5–17.
23.
GreenoJ. G. (1998). Where is teaching?Issues in Education, 4(1), 111–119.
24.
GutsteinE. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice.New York, NY: Routledge.
25.
IzumiL., & CoburnK. G. (2001). Facing the classroom challenge: Teacher quality and teacher training in California's schools of education.San Francisco, CA: Pacific Research Institute for Public Policy.
26.
KirschnerP. A., SwellerJ., & ClarkR. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.
27.
KlahrD., & NigamM. (2004). The equivalence of learning paths in early science instruction: Effects of direct instruction and discovery learning. Psychological Science, 15(10), 661–667.
28.
KleinD. (2003). A brief history of American K–12 mathematics education in the 20th century. In RoyerJ. M. (Ed.), Mathematical cognition: A volume in current perspectives on cognition, learning, and instruction (pp. 175–225). Greenwich, CT: Information Age Publishing.
29.
KleinD. (2007a). A quarter century of US “math wars” and political partisanship. Journal of the British Society for the History of Mathematics, 22(1), 22–33.
30.
KleinD. (2007b). School math books, nonsense, and the National Science Foundation. American Journal of Physics, 75(2), 101–102.
KoedingerK. R., & AlevenV. (2007). Exploring the assistance dilemma in experiments with cognitive tutors. Educational Psychology Review, 19(3), 239–264.
33.
LampertM. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(3), 29–63.
LeeH. S., & AndersonJ. R. (2013). Student learning: What has instruction got to do with it?Annual Review of Psychology, 64, 445–469.
36.
MaL. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum.
37.
MartinD. B. (2007). Beyond missionaries or cannibals:: Who should teach mathematics to African American children?. The High School Journal, 91(1), 6–28.
38.
MayerR. E. (2004). Should there be a three-strikes rule against pure discovery learning? The case for guided methods of instruction. American Psychologist, 59(1), 14–19.
39.
MellersB., HertwigR., & KahnemanD. (2001). Do frequency representations eliminate conjunction effects? An exercise in adversarial collaboration. Psychological Science, 12(4), 269–275.
40.
MosesR. P., & CobbC. E. (2001). Radical equations: Civil rights from Mississippi to the Algebra Project.Boston, MA: Beacon Press.
41.
NathanM. J., & KnuthE. J. (2003). A study of whole classroom mathematical discourse and teacher change. Cognition and Instruction, 21(2), 175–207.
42.
National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: Author.
43.
National Council of Teachers of Mathematics. (2013). Supporting the Common Core State Standards for Mathematics. Retrieved from http://www.nctm.org/ccssmposition/
44.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
45.
National Research Council. (2001). Adding it up: Helping children learn mathematics.KilpatrickJ., SwaffordJ., and FindellB. (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
46.
New Standards Primary Literacy Committee. (1999). Reading & writing grade by grade: Primary literacy standards for kindergarten through third grade (Illustrated by Garin Baker). Washington, DC: National Center on Education and the Economy.
NystrandM., & GamoranA. (1997). The big picture: Language and learning in hundreds of English lessons. In NystrandM., GamoranA., KachurR., & PrendergastC. (Eds.), Opening dialogue. Understanding the dynamics of language and learning in the English classroom (pp. 30–74). New York, NY: Teachers College Press.
49.
PearsonP. D. (2004). The reading wars. Educational Policy, 18(1), 216–252.
50.
RichardsJ., & von GlasersfeldE. (1980). Jean Piaget, psychologist of epistemology: A discussion of Rotman's Jean Piaget: Psychologist of the real. Journal for Research in Mathematics Education, 11(1), 29–36.
51.
RogoffB. (1994). Developing understanding of the idea of communities. Mind, Culture, and Activity, 1(4), 209–229.
52.
RosenshineB. (2012). Principles of instruction: Research-based strategies that all teachers should know. American Educator, 36(1), 12–19, 39.
53.
SchoenfeldA. H. (2002). Making mathematics work for all children: Issues of standards, testing, and equity. Educational Researcher, 31(1), 13–25.
54.
SchoenfeldA. H. (2004). The math wars. Education Policy, 18(1), 253–286.
55.
SenkS. L., & ThompsonD. R, (2003). School mathematics curricula: Recommendations and issues. In SenkS. L. & ThompsonD. R. (Eds.), Standards-oriented school mathematics curricula: What does the research say about student outcomes? (pp. 1–30) Mahwah, NJ: Lawrence Erlbaum.
56.
SfardA. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13.
57.
SimonM. A. (1994). Learning mathematics and learning to teach: Learning cycles in mathematics teacher education. Educational Studies in Mathematics, 26(1), 71–94.
58.
SimonM. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.
59.
SlavinR. E., LakeC., & GroffC. (2009). Effective programs in middle and high school mathematics: A best-evidence synthesis. Review of Educational Research, 79(2), 839–911.
60.
SnowC. E., BurnsM. S., & GriffinP. (Eds.) (1998). Preventing reading difficulties in young children. Washington, DC: National Academy Press.
61.
SteffeL. P., & GaleJ. (Eds.). (1995). Constructivism in education. Hillsdale, NJ: Lawrence Erlbaum.
62.
SteffeL. P., & KierenT. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25(6), 711–733.
63.
SteinM. K., EngleR. A., SmithM. S., & HughesE. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.
64.
ThompsonP. W. (2000). Radical constructivism: Reflections and directions. In SteffeL. P. & ThompsonP. W. (Eds.), Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld (pp. 412–448). London, UK: Falmer Press.
65.
TyackD., & CubanL. (1995). Tinkering toward utopia: A century of public school reform.Cambridge, MA: Harvard University Press.
66.
VenezkyR. L. (1992). Textbooks in school and society. In JacksonP. Q. (Ed.), Handbook of research on curriculum (pp. 436–462). New York, NY: Macmillan.
67.
von GlasersfeldE. (1989). Constructivism in education. In HusenT. & PostlethwaiteT. N. (Eds.), The International Encyclopedia of Education (Supplementary Volume; pp. 162–163). Oxford, UK: Pergamon.
68.
WuH. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23(3), 14–19, 50–52.
69.
WuH. (2004). Must content dictate pedagogy in mathematics education? Unpublished manuscript, Department of Mathematics, University of California, Berkeley.
