Mathematics teacher education, mathematical habits of mind, assessment tasks, teacher reflection.
||September 03, 2018
||The Author(s) 2018. This article is published with open access at www.chitkara.edu.in/publications
As schools and teachers in the U.S. fine-tune their implementation of mathematics standards promoting college and career readiness, the number of support resources continues to expand. One resource focus experiencing significant growth involves sample items and tasks asserting alignment with the college and career ready mathematical content and practice standards. Such samples regularly identify both the content standards addressed and the mathematical habits of mind that students have the potential to engage in. Consistently absent are evaluation criteria articulating how engagement and demonstration of associated mathematical practices can be assessed, concurrent with content. The authors discuss the development of rubrics that attempt to faithfully assess the integration of mathematical content and practice standards and highlight the benefits to mathematics teachers, coaches, professional developers, and mathematics teacher educators of engaging in such reflective rubric-creating activities.
As schools and teachers in the U.S. refine and enhance implementation of college and career readiness standards (Common Core State Standards for Mathematics, Nebraska’s College and Career Ready Standards for Mathematics), the number of resources supporting enactment continues to expand in both breadth and depth. One resource focus experiencing significant growth involves sample items, tasks, and assessments proclaiming alignment with such standards. Summative assessment resources typically provide stakeholders with practice tests, open access sample assessment tasks, or ‘released’ items (Partnership for the Assessment of Readiness for College and Careers, Smarter Balanced Assessment Consortium, Minnesota Department of Education). In addition, a variety of organizational and individual entities, not to mention the plethora of mathematics textbook publishers, continue to add to the inventory of available items and tasks (Dan Meyer’s Three-Act Math Tasks, Illustrative Mathematics, Achieve the Core’s Mathematics Tasks and Assessments). Such sample tasks regularly identify both the content standards addressed and the mathematical habits of mind that students have the potential to engage in (Common Core’s Standards for Mathematical Practice, Virginia Department of Education’s Mathematical Process Goals for Students). Frequently accompanying these tasks are instructional recommendations, implementation guidelines, or evaluation criteria, such as rubrics or point systems. Unfortunately, with few exceptions (Education Development Center’s Implementing the Mathematical Practice Standards; New York City Department of Education’s WeTeachNYC Library), such resources fail to explicate how students might actually engage in the indicated mathematical processes and proficiencies or what such engagement might look like. By ‘what engagement might look like’, the authors mean “to articulate what a student would need to say or write (communicate) to establish her engagement in particular mathematical habits of mind”. Finally, such resources fail to make explicit how demonstrating specific mathematical habits of mind, or lack of such demonstration, impacts student assessment. In this report, the authors put forth and discuss the development of three rubrics that attempt to coherently assess the integration of mathematics content and mathematical habits of mind. Rubric development involved a small sample of prospective and practicing secondary school mathematics teachers (teachers of students ages 14-18 years), referred to as ‘participating teachers’. Although the rubrics themselves might appear of limited value, due to their non-generic nature (their connection to specific content), the rubrics should be considered as secondary to their development—a very cognitive process for everyone involved in their construction.
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- Achieve the Core. (2018). New York: Student Achievement Partners. Retrieved from https://achievethecore.org/ Arizona Department of Education. (2014).
- Arizona’s college and career ready standards mathematics: Standards - Mathematical practices - Explanations and examples, High school grades 9–12. Retrieved from http:// www.azed.gov/standards-practices/k-12standards/ mathematics-standards/
- Brahier, D. J. (2013). Teaching secondary and middle school mathematics (4th ed.). Pearson Education USA.
- Brookhart, S. M. (1999). The art and science of classroom assessment: The missing part of pedagogy. ASHEERIC Higher Education Report (Vol. 27, No.1). The George Washington University, Graduate School of Education and Human Development. Retrieved from https://files.eric.ed.gov/fulltext/ED432937.pdf
- Conference Board of the Mathematical Sciences. (2012). Mathematical education of teachers II. American Mathematical Society and Mathematical Association of America. Retrieved from https://www.cbmsweb.org/ archive/MET2/met2.pdf
- Courtney, S. (2017). What teachers understand of model lessons. Cogent Education, 4(1), 1-22. https://doi.or g/10.1080/2331186X.2017.1296528
- Covey, S. R. (2004). The seven habits of highly effective people: Powerful lessons in personal change. Free Press/ Simon & Schuster.
- Daro, P. & Burkhardt, H. (2012). A population of assessment tasks. Journal of Mathematics Education at Teachers College, 3(1), 19-25.
- Education Development Center. (2016). Implementing the mathematical practice standards. Retrieved from http://mathpractices.edc.org/
- Glasersfeld, E.V. (1995). Radical constructivism: A way of knowing and learning. Falmer Lndon.
- Illustrative Mathematics. (n.d.). Retrieved from https:// www.illustrativemathematics.org/
- Indiana Department of Education. (2017). Indiana Academic Mathematics Standards. Retrieved from https://www.doe.in.gov/standards/mathematics
- Inside Mathematics. (2017). Retrieved from http://www. insidemathematics.org/ IXL Learning. (2018).
- IXL alignment to Common Core math standards. Retrieved from http://www.ixl.com/ standards/common-core/math
- Jonassen, D. H., & Strobel, J. (2006). Modeling for meaningful learning. In Hung, D., & Khine, M.S. (Eds.), Engaged learning with emerging technologies (pp. 1-27). Springer. https://doi.org/10.1007/1-40203669-8_1
- Kentucky Department of Education. (2017). Kentucky academic standards with targets: High school – Algebra 1. Retrieved from https://education.ky.gov/ curriculum/standards/kyacadstand/Pages/default.aspx
- Mathematics Assessment Project. (2015). Shell Center for Mathematics Education at the University of Nottingham (Shell Center), the University of California, Berkeley (UC Berkeley). Retrieved from http://map.mathshell.org/materials/index.php
- Meyer, D. (2017). Three-act math task bank. Retrieved from http://blog.mrmeyer.com/category/3acts/ Minnesota Department of Education. (2017).
- Minnesota comprehensive assessments (MCA). Retrieved from https://education.mn.gov/MDE/fam/tests/
- Moskal, B. M. (2000). Scoring rubrics: what, when and how? Practical Assessment, Research & Evaluation, 7(3).
- National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. National Governors Association Center for Best Practices, Council of Chief State School Officers. Retrieved from http://www.corestandards.org/Math/
- Nebraska Department of Education. (2015). Nebraska’s college and career ready standards for mathematics. Retrieved from https://www.education.ne.gov/math/
- New York City Department of Education. (n.d.). WeTeachNYC library. Retrieved from https://www. weteachnyc.org/resources/
- Partnership for Assessment of Readiness for College and Careers (PARCC). (2012). PARCC item development, Invitation to Negotiate (ITN) 2012-31, Appendix G: Project terms and definitions. Retrieved from http://www.myflorida.com/apps/vbs/adoc/ F28718_AppendixPagesITN201231PARCCItem DevelopmentFinal.pdf
- Partnership for Assessment of Readiness for College and Careers (PARCC). (2017a). PARCC Mathematics Evidence Table Algebra I (EOY). Retrieved from https://parcc-assessment.org/mathematics/
- Partnership for the Assessment of Readiness for College and Careers (PARCC). (2017b). PARCC model content frameworks: Mathematics, Grades 3-11 (Version 5.0). New Meridian Corporation. Retrieved from https:// parcc-assessment.org/model-content-frameworks/
- Piaget, J. (1962/2000). Commentary on Vygotsky’s criticisms of Language and thought of the child and judgment and reasoning in the child. (L. Smith, Trans.; Original work published in 1962). New Ideas in Psychology, 18(2-3), 241-259. Retrieved from https://lchcautobio.ucsd. edu/wp-content/uploads/2015/10/Piaget-19622000Commentary-on-Vygotsky.pdf
- Rosenbaum, D. A. (1972). The theory of cognitive residues: A new view of fantasy. Psychological Review, 79(6), 471486. http://dx.doi.org/10.1037/h0033548
- Salomon, G., Globerson, T., & Guterman, E. (1989). The computer as a zone of proximal development: Internalizing reading-related metacognition from a reading partner. Journal of Educational Psychology, 81(4), 620-627. http://dx.doi.org/10.1037/002206184.108.40.2060
- Salomon, G., Perkins, D. N., & Globerson, T. (1991). Partners in cognition: Extending human intelligence with intelligent technologies. Educational Researcher, 20(3), 2-9. https://doi.org/10.3102/0013189X020003002
- Schön, D. A. (1983). The reflective practitioner. New York: Basic Books.
- Smarter Balanced Assessment Consortium. (2018). The mathematics summative assessment blueprint. The Regents of the University of California. Retrieved from https://portal.
- smarterbalanced.org/library/en/ mathematics-summative-assessment-blueprint.pdf
- Smarter Balanced Assessment Consortium. (2015). Content Specifications for the summative assessment of the Common Core state standards for mathematics. The Regents of the University of California. Retrieved from https://portal.smarterbalanced.org/library/en/ mathematics-content-specifications.pdf
- Stanford Center for Assessment, Learning and Equity. (2013, September). edTPA secondary mathematics assessment handbook. Board of Trustees of the Leland Stanford Junior University. Retrieved from http://edtpa.aacte. org/
- Steffe, L. P., & Thompson, P. W. (2000). Interaction or intersubjectivity? A reply to Lerman. Journal for Research in Mathematics Education, 31(2), 191-209. https://doi.org/10.2307/749751
- Sztajn, P., Marrongelle, K. A., Smith, P., & Melton, B. L. (2012). Supporting implementation of the Common Core State Standards for Mathematics: Recommendations for professional development. Raleigh, NC: Friday Institute for Educational Innovation at North Carolina State University. Retrieved from https://pdxscholar.library.pdx.edu/cgi/ viewcontent.cgi?article=1157&context=mth_fac
- Thompson, P. W. (2013). In the absence of meaning. In K. Leatham (Ed.), Vital directions for research in mathematics education (pp. 57-93). Springer. https://doi.org/10.1007/978-1-4614-6977-3_4
- Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Se?pulveda (Eds.), Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education, (Vol 1, pp. 45-64). More?lia, Mexico: International Group for the Psychology of Mathematics Education (PME).
- Wiggins, G., & McTighe, J. (2005). Understanding by design (2nd ed.). Alexandria, VA: Association for Supervision and Curriculum Development.