This month’s blogging focus is on professional development (PD) … another source of learning for me and educational leaders is the membership in organizations and associations related to leadership, education, and the superintendency. Once such organization is the American Association of School Administrators (AASA). One of the membership benefits is access to peer-reviewed research and commentary on current trends, issues, and areas of educational concern.
As I have mentioned in other blog posts, DPS109 is focused on 5 main areas of growth, “The Big5”: Common Core State Standards, Teacher Evaluation, Technology, Organizational Culture, and the Superintendent’s Task Force for Middle Level Education. Our professional development, time, energy, community outreach, and resource allocation are focused and concentrated under the umbrella of the Big 5.
In this blog post, I’m reprinting an article published in the Spring 2014 AASA Journal of Scholarly Practice. The article is shared here as an example to the readers of scholarly materials that school leaders look to for guidance, information, and “research” in support of personal professional development. Articles like these also support organizational professional development and we leaders personally learn and grow as we support our organization’s learning and growing. Locally in DPS109, we look to professional organizations to gain greater expertise on our “Big 5”.
The article reprinted below is about mathematics and the new Common Core State Standards and support for teachers in their pedagogical growth and development. This post and the reprinted article also serve as windows into the world of a practitioner scientist whose purpose is to engage, inspire, and empower members of a school district, community, and the general blog reading public! Thank you for reading, commenting, and supporting public education!
Copyright – Articles published by the American Association of School Administrators (AASA) in the AASA Journal of Scholarship and Practice fall under the Creative Commons Attribution-Non-Commercial-NoDerivs 3.0 license policy http://creativecommons.org/licenses/by-nc-nd/3.0/). Please refer to the policy for rules about republishing, distribution, etc. In most cases our readers can copy, post, and distribute articles that appear in the AASA Journal of Scholarship and Practice, but the works must be attributed to the author(s) and the AASA Journal of Scholarship and Practice. Works can only be distributed for non-commercial/non-monetary purposes. Alteration to the appearance or content of any articles used is not allowed.
Commentary (page 38) AASA Journal of Scholarship and Practice Spring 2014
Supporting Mathematics Teachers in the Common Core Implementation
P. Holt Wilson, PhD, Assistant Professor, Department of Teacher Education and Higher Education, School of Education, University of North Carolina, Greensboro, Greensboro NC
Holly A. Downs, PhD, Assistant Professor, Educational Research Methodology Department, School of Education, University of North Carolina, Greensboro, Greensboro, NC
Abstract
Based on work with elementary grades teachers in mathematics professional development to prepare for the implementation of the Common Core State Standards for Mathematics, we offer a set of recommendations for school leaders who wish to assist teachers in adjusting their instruction to meet the challenges that the new standards present.
Key Words: Common Core State Standards, mathematics, professional development, learning progressions
Now that the Common Core State Standards have been adopted by 45 states, the District of Columbia, and four U.S. territories, schools are on the frontline in proactively shaping these changes in ways that support teachers in assisting students in meeting them. Yet monthly curriculum updates, documents that crosswalk previous standards with the new ones, the barrage of commercially available curriculum and training programs, and uncertainties of future assessments have placed school leaders in the difficult but all too familiar place of “building a plane while flying it.”
While trying to make sense of these myriad changes with incomplete information, they must still move forward in supporting teachers in preparing for these new standards. Many are left with questions: What really is different about these standards? How can I best support my teachers in the transition? In response, we draw upon our experiences from professional development, specifically from a year-long project with the teachers of two elementary schools, in preparation for the implementation of the Common Core State Standards for Mathematics (CCSS-M) (CCSSI, 2010). We describe two broad issues for school leaders to consider and offer a set of recommendations for school administrators working in similar schools to assist teachers in adjusting their instruction to meet the challenges that the new standards present.
So, What Is Different? The CCSS-Mi is comprised of two connected sets of expectations for student learning: the Standards for Mathematical Content and the Standards for Mathematical Practice. Together, they “define what students should understand and be able to do in their study of mathematics” (p. 4) and in our view represent major advances in standards-based reform in at least two distinct ways.
First, the writers began with “research-based learning progressions” to inform the priorities and sequencing of the topics that students encounter (p. 4). Using this approach, the Standards for Mathematical Content are aligned with research on mathematics learning regarding the ways children develop mathematical ideas over time (Daro, Mosher, & Corcoran, 2011).
Scholars working in the area of learning progressions point to numerous benefits, including opportunities for assessment systems
that provide instructional guidance for teachers (Battista, 2004; Confrey & Maloney, 2012) and more coherent curricular programs (Clements & Sarama, 2008).
In the classroom, emerging research indicates that knowing learning progressions supports teachers in preparing instruction that
simultaneously takes into account students’ experiences and prior knowledge, creating instructional environments more aligned with students’ likely paths of learning, assessing students with a focus on what they know (as opposed to what they do not know), and documenting common misconceptions (Edgington, 2012; Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996; Sztajn, Wilson, Edgington, & Confrey, 2011; Wilson, 2009). Thus, the Standards for Mathematical Content put into place a foundation that allows for student-centered mathematics instruction throughout their K-12 experiences.
Second, the Standards for Mathematical Practice “describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years” (CCSSI, 2010, p. 8).
These practices include behaviors and skills such as persevering in problem solving, critiquing others’ mathematical arguments, and using tools strategically. Though the inclusion of expectations that describe the processes and dispositions of mathematical proficiency is not new in the standards tradition, two national groups developing assessments aligned with the CCSS-M–-the Smarter Balanced Assessment Consortium (SBAC) and the Partnership for Assessment of College and Careers Readiness (1PARCC)–-formally include the Standards for Mathematical Practice in the frameworks for their assessment design.
Such an inclusion suggests that high-stakes tests will not only assess concepts and procedures but also promote these ways of
engaging in mathematics. There is a concern that these practices will be marginalized in classrooms (Confrey & Krupa, 2010), and
research suggests that teachers require time and resources to develop instructional routines that support the Standards for Mathematical Practice (Krupa, 2011).
From our work with elementary grades teachers, we believe that these advances–-the foundation of learning progressions and an emphasis on mathematical practices–-warrant two considerations for school leaders wishing to support teachers in the CCSS-M implementation:
1. Opportunities for teachers to learn about and engage with the learning progressions on which the new standards are designed enrich their understandings of the mathematics students bring to the classroom and how students’ understandings are likely to progress.
2. Opportunities to learn and adopt new pedagogical strategies to create nurturing environments for students to develop these mathematical practices lead to instruction that is more student-centered.
Supporting Teachers in Implementation
After the adoption of the Common Core State Standards by our state, leaders from two schools approached us to design and facilitate professional development to support elementary grades teachers in preparing for the CCSS-M implementation. Both schools were identified as “high need” by the state, using criteria that included a large percentage of economically disadvantaged students, teachers working outside of their area of licensure or holding provisional licenses, and low performance on year-end testing in reading and/or mathematics.
In response to their request, our project team created a 120-hour professional development program for elementary grades teachers to plan for the new standards. To do so, we aimed to share with teachers (1) a selection of the learning progressions that underlie the standards and (2) student-centered instructional practices that create spaces for students to experience and gain expertise with the Standards for Mathematical Practice.
Over the course of the 2011-2012 school year, our team worked with 30 teachers, 15 from each school, who demonstrated
moderate to large effect sizes on pre/post measures of content knowledge, pedagogical content knowledge, and the ability to identify and analyze student-centered instructional practices. As we reflected on the project and its success, two broad ideas emerged that we believe offer direction for school leaders wishing to assist teachers in adjusting their instruction to meet the challenges of the CCSS-M.
More Than Just Content Knowledge
Undoubtedly, the CCSS-M represents a curriculum significantly different than previous state standards, both in terms of sequencing and cognitive demand (Porter, McMaken, Hwang, & Yang, 2011), and will require that teachers teach mathematical topics with which they may be unfamiliar. Yet learning more mathematics is unlikely to assist teachers in implementing
the new standards. It has been shown that teachers’ content knowledge alone is insufficient to support student learning (Begle,
1972; Kilpatrick, Swafford, & Findell, 2001).
Instead, understandings of particular mathematics concepts that are flexible and multifaceted allow teachers to recognize and
build upon students’ prior knowledge in instruction (Ball, Thames, & Phelps, 2008), and such knowledge has been demonstrated to be a strong predictor of student achievement (Hill, Rowan, & Ball, 2005). Research on teachers’ learning about learning progressions suggests deeper content and pedagogical content knowledge in mathematics result from a focused study on students’ mathematical thinking described by the progressions (Mojica, 2010; Wilson, 2009).
Thus, rather than simply assisting teachers in learning “more math,” our work with them stressed learning about and engaging
with the mathematical ideas that students bring with them to the classroom through focusing on learning progressions. Tools such as the Progressions Documents for the Common Core Math Standards http://ime.math.arizona.edu/progressions/ and the Learning Trajectory Display of the Common Core State Standards for Mathematics posters (Confrey, Maloney, & Nguyen, 2011) were particularly useful in supporting teachers in learning to consider the mathematics of their grade level in relation to their students’ previous and future understandings rather than as a set of isolated procedures for students to apply.
For example, consider the development of multi-digit multiplication of whole numbers. Although the formalization of this idea with the familiar algorithm is delayed until Grade 5 in the CCSS-M, the new standards expect students to begin building multiplicative understandings much earlier. In Grades 1 and 2, students work with equal-sized parts, a foundational idea for
multiplication. They investigate and use properties of operations in Grades 3 and 4, gaining deeper understandings of the ways
multiplication works. Only in Grade 5 are students expected to learn and apply the formal procedure.
Without an understanding of the ways these ideas build across grades, one can imagine a well-intentioned teacher, desiring to
help his or her students, prematurely introducing the algorithm and curtailing the development of a deeper understanding of the
concept.
For the teachers with whom we worked, knowing how students’ understanding of multiplication develops across grades as
described by the learning progression assisted them in identifying the ideas that students already knew, such as repeated addition or decomposing into tens and ones, and in customizing their instruction in response to those understandings. For these teachers, the learning progressions helped them make informed instructional choices in relation to their students’ understandings and their own knowledge of content and curriculum. More Than Just “Good Teaching”
Terms like good teaching and best practices are commonly used when referring to markers of quality instruction, such as cooperative learning and formative assessment strategies. Yet the implicit, and perhaps unintended, message of these phrases is that effective instructional approaches are independent of the content being taught. Put another way, the language of “it’s just good teaching” leads many to believe that, for example, high quality mathematics instruction entails the same pedagogical strategies as effective literacy instruction and that the same
strategies for teaching mathematical procedures are appropriate for teaching mathematical concepts. Yet progress in the learning sciences suggests that this overgeneralization is misleading (cf. Sawyer, 2006). Some instructional approaches are more effective at assisting students in learning domain-specific knowledge than others.
Piaget (1950) made distinctions among different types of knowledge, two of which he called social-conventional and logical-mathematical knowledge. For social-conventional knowledge, the source of ideas is outside of the learner and must therefore be internalized from a more knowledgeable other, such as pre-reading strategies in literacy or locating continents on a globe in social studies. In mathematics, examples of this kind of knowledge include mathematical vocabulary and notation, like the word rhombus for a figure with four sides of the same length or the symbol = for denoting equivalence. Instructional practices aimed at supporting students in developing social-conventional knowledge might include direct instruction, modeling, or the gradual release of responsibility (Pearson & Gallagher, 1983) with an “I do—We do—You do” format (Fisher & Frey, 2008).
In contrast, learning concepts requires students to bring their prior knowledge to bear on a novel, problematic situation. Logical-
mathematical knowledge exists as relationships among ideas in one’s mind and must therefore be constructed by adapting one’s current understandings to address new situations, such as understanding changing states of matter or the relationship between mass and density in science. Examples of this type of knowledge in mathematics include concepts and relationships, such as an understanding of place value or the connections among arithmetic operations.
In contrast with direct instruction, student-centered mathematics instruction for developing logical-mathematical knowledge might include the use of high cognitive demand tasks to elicit multiple approaches from students (Stein, Grover, & Henningson, 1996) and the careful sequencing and connecting of these approaches through discussion (Smith & Stein, 2011) in a “Launch-Explore-Discuss” format (Smith, Bill, & Hughes, 2008).
As we worked with teachers during the project, there was often confusion and tension about selecting instructional approaches to
address particular Mathematical Content Standards. Perhaps an unintended consequence of intensive literacy initiatives that they had experienced, many of the teachers advocated using “gradual release” and “modeling” to support their students in learning mathematical concepts and wrestled with how to organize their instruction differently.
Not only are these strategies likely to be ineffective at supporting learning of logical-mathematical knowledge, we argue that these approaches provide only limited opportunities for students to gain expertise in the Mathematical Practice Standards. Over the course of the project, however, teachers began to differentiate instructional approaches that were likely to engender the types of mathematical understandings that meet the CCSS-M from others they used for other content areas.
Recommendations
Based on these two broad ideas, we offer a set of recommendations for school leaders working in similar schools wishing to assist teachers in adjusting their mathematics instruction to meet the challenges of the CCSS-M. Though we acknowledge that our suggestions are based on experiences from one year-long project with a small number of teachers from two schools with particular contexts, we contend that these recommendations may prove useful and resonate with the findings of other scholars working in the areas of professional development (e.g., Garet, Porter, Desimone, Birman, & Yoon, 2001; Heck, Banilower, Weiss, & Rosenberg, 2008; Wei, Darling-Hammond, & Adamson, 2010) and teachers’ learning of learning trajectories (Sztajn, Confrey, Wilson, & Edgington, 2012).
1. Offer and personally participate in
professional development on learning
progressions. Professional development
opportunities for teachers and school
leaders should ensure that the content
includes attention to children’s
mathematical thinking and the learning
progressions that describe its
development across grades.
2. Provide time for teachers to articulate
students’ mathematical development
across grades. School leaders should
provide time and support for cross-
grade conversations that examine and
describe their own students’
development of mathematical
understanding over time.
3. Support teachers in understanding
effective instruction for mathematics
concepts. Professional development and
instructional support for teachers should
emphasize the importance of
pedagogical strategies for teaching
mathematics concepts that allow
students to engage in the Standards for
Mathematical Practice.
4. Allocate time for cross-subject matter
discussions. Provide opportunities for
teachers to clarify the similarities and
differences of effective instructional
practices in mathematics and other
disciplines such as literacy.
5. Understand that learning new
instructional practices takes time.
Professional development should offer
scaffolded opportunities for teachers to
try new practices over extended periods
of time in their own classrooms.
6. Acknowledge examples of quality
mathematics instruction. Mark
instances of mathematics instruction
that builds upon students’ thinking in
walk-throughs and formal observations
and communicate with all faculty that
such instruction is valued.
AUTHORS’ NOTE
The authors contributed equally to the writing of this manuscript. The work on this article was supported by the U.S. Department of Education’s ESEA Title II-A Improving Teacher Quality Grants program awarded to the University of North Carolina at Greensboro. Any opinions, findings, conclusions or recommendations expressed herein are those of the authors. A special thank you to Craig Peck for his feedback on an earlier version of this paper and to members of our research group and our partners in schools: Kerri Richardson, Carol Seaman, Ana Floyd, Wendy Rich, Michelle McCullough, and Aundrea Carter.
Further, one anonymous reviewer pointed out that the CCSS are not field-tested and lack empirical support for claims related to college and career readiness. Our purpose is not to advocate for these standards but rather to share our experiences in supporting classroom teachers and school administrators in meeting standards that have been set for them.
Author Biographies
P. Holt Wilson is an assistant professor at the University of North Carolina at Greensboro. His research focuses on mathematics teachers’ knowledge, practice, and professional development.
Holly Downs is an assistant professor at the University of North Carolina at Greensboro. Her research focuses on the evaluation of multi-site educational initiatives and programs delivered via traditional and online settings, particularly from the science, technology, engineering, and mathematics (STEM) fields.
References
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