# Front Matter

# Front Matter

Title

# Developing Teachers’ Pedagogical Design Capacity for Engaging Students in the Standards for Mathematical Practice

Researcher Name and Affiliation

Raymond Johnson

University of Colorado Boulder

Author Note

Author Note

This dissertation prospectus is in partial fulfillment of requirements for a Ph.D. in Curriculum and Instruction in the School of Education at the University of Colorado Boulder. Raymond Johnson can be contacted at raymond.johnson@colorado.edu.

The research described in this proposal is based upon work supported by the National Science Foundation under Grant No. 1147590. Any opinions, findings, and conclusions or recommendations are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Abstract

Keywords

# Introduction

# Introduction

# Introduction

## Background and Problem Statement

- Describe the “loose coupling” between standards and practice
- Standards as affording opportunities
- Talk about the difficulty of scaling educational reforms and interventions

Lack of theory: Remillard’s commentary from MTaW

Remillard: “I say ‘much needed’ because, at present, the field of research on teachers’ use of mathematics curriculum materials lacks a theoretical and conceptual base. As a field, we do not have —- or have not been explicit about —- theories that underlie and explain the relationships that are central objects of study. As a result, the field has not produced a body of knowledge about the teacher—curriculum material relationship that is generalizable across teachers, materials, or contexts, or that can inform the work of policy-makers, curriculum decision-makers, and curriculum material designers in substantive ways.” (2009, p. 85)

## Background and Problem Statement

Mathematics teachers in over 40 U.S. states are adjusting their curriculum and instruction in response to the Common Core State Standards for Mathematics (CCSSM; Common Core State Standards Initiative, 2010a), the latest set of standards in what has become a 25+ year era of standards-driven education reform. Unlike most sets of state standards that mathematics teachers worked with previously, the CCSSM features a list of eight Standards for Mathematical Practice (SMPs; Common Core State Standards Initiative, 2010b) that describe not the content students should learn, but rather the nature of students’ activity as mathematical thinkers. The SMPs are derived from rich descriptions of practice in two previous standards documents: the National Council of Teachers of Mathematics’ (NCTM; 2000) Principles and Standards for School Mathematics and the National Research Council’s (2001) Adding it Up. When initial drafts of the CCSSM did not include any standards for mathematical practice, reviewers requested that they be added (Mary Kay Stein, personal communication).

The SMPs are frequently featured in presentations at mathematics education conferences, made the focus of professional development, and are the subject of books (Koestler, Felton, Bieda, & Otten, 2013; O’Connell & SanGiovanni, 2013) and articles (Billings, Coffey, Golden, & Wells, 2013; Pilgrim, 2014; Stephan, 2014) written for teachers. At a workshop conducted by the Colorado Department of Education that I attended in March of 2011, not long after the state’s adoption of the CCSSM, attention was given almost entirely to the eight SMPs rather than content standards. The 2015 NCTM Annual Meeting was similar, as SMP-related sessions were plentiful and mentions of CCSSM content standards were relatively scarce. While getting this kind of attention and promotion is helpful for encouraging adoption, high-quality student engagement in the practices is likely to happen only when teachers understand the SMPs well and design experiences for students that provide fertile ground for that engagement. There are strong reasons to believe that achieving this is far from easy: standards implementation in general presents a host of challenges, and the SMPs—grade level-agnostic descriptions occupying a scant three pages of the printed CCSSM document—are likely to be interpreted in a variety of ways where it matters most, by teachers making choices about curriculum and its enactment with students.

This prospectus describes parts of a larger study during which high school algebra teachers interpreted the SMPs while evaluating the quality of mathematical tasks. This research was undertaken in the context of a research-practice partnership focused on improving teachers’ pedagogical design capacity (Brown, 2009), their ability to mobilize their knowledge and curricular resources in ways such as perceiving the affordances of curriculum and selecting curricular resources. The partnership took the form of a design research partnership (Coburn, Penuel, & Geil, 2013), which adopted principles of design-based implementation research (Penuel, Fishman, Cheng, & Sabelli, 2011) to collaboratively design tools and routines for algebra teachers that would improve their ability to make productive adaptations to their curriculum using high-quality mathematical tasks that were aligned to the CCSSM.

# Rationale

The SMPs are described in the CCSSM as “varieties of expertise that mathematics educators at all levels should seek to develop in their students” (Common Core State Standards Initiative, 2010b, p. 6), and they rest on important prior work in the field of mathematics education. From the *Principles and Standards for School Mathematics* (2000), the SMPs distill important elements of NCTM’s *process standards*: problem solving, reasoning and proof, communication, representation, and connections. From *Adding It Up* (2001), the SMPs build upon the National Research Council’s five strands of *mathematical proficiency*: adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition towards mathematics. Taken together, these ideas inform the eight SMPs found in the CCSSM:

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

In comparison to the process standards and mathematical proficiencies provided by NCTM and the National Research Council, the SMPs are described very briefly, using fewer than 200 words of text for each of the eight practices. To illustrate, here is the complete description in the CCSSM for the second practice (SMP2), *Reason abstractly and quantitatively*:

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. (Common Core State Standards Initiative, 2010b, p. 6)

As with the other SMPs, SMP2 does not provide a grade-level or content-specific description of what is meant by *reasoning abstractly and quantitatively*. In the context of curriculum alignment, it is imaginable that many, if not most, mathematical tasks require some degree of abstraction and quantitative reasoning. Yet SMP2 stresses the ability to decontextualize as well as contextualize, which might lead some teachers to require both in a mathematical task before claiming a task is aligned to SMP2. There are also multiple ways a teacher might interpret the meaning of *contextualize*: in some curriculum it is common for students to create a real-world context for which a particular equation, graph, or table might describe a phenomenon, and teachers might see such activities as aligned with SMP2. Other teachers might argue that such tasks are really focused on mathematical modeling, described by SMP4. Teachers without experience with such curriculum might not make either connection, and look for meaning elsewhere in SMP2, such as the call for students to “probe into the referents for the symbols involved.” Teachers putting emphasis on this part of SMP2 might look for tasks that ask students to define the variables they use when writing equations, which would be a slightly different take on what it might mean for a task to be aligned with SMP2.

This kind of thought experiment highlights how different teachers, each building on different knowledge and experiences, can interpret standards differently. In a fundamental way, this difference in interpretation undermines what it means for something to be a *standard*, and raises questions about the assumptions that go into the creation of a standards document like the CCSSM. One of these assumptions is that educational standards describe a set of outcomes that can transcend geographic and institutional contexts, and that teachers and students everywhere are working in parallel towards the same goal. But as these goals take on different meanings for different people, then neither the destination nor the path to get there is as uniform as a standards document might lead one to believe. Sadler (2014) argued that attempts to codify academic achievement standards may be futile, and that agreement on the meaning of standards is unlikely without intensive work involving concrete examples of student work and a process of consensus-building.

It is unknown how teachers interpret the eight SMPs and their short descriptions, particularly in the context of choosing curriculum materials for use in instruction. While the SMPs themselves describe student activity, not curriculum, there is a sense that “the task predicts performance” (City, Elmore, Fiarman, & Teitel, 2009, p. 30; Doyle, 1983), something Jason Zimba, one of the three CCSSM authors, describes as “practice forward.” Therefore, if students are to engage in the SMPs during classroom activity, their teacher will need to identify mathematical tasks that support that engagement, and building teachers’ capacity to make those identifications should be a focus of both theoretical and practical study.

## Purpose, Aims, and Scope

Somewhere I should expose my biases: pretty standards-neutral, skeptical they really do much on their own, although they become a catalyst for changes (good and bad) people could have done without the standards

A great outcome would be a guide with tips for how to choose or modify tasks to enhance the practices, with specific tips to each practice (with data to back up that it works)

## Purpose, Aims, and Scope

This is a study of how curriculum reforms driven by educational standards can be designed to scale across a large, urban school district.

## Literature Review

- Discuss prior studies of standards-based curriculum reform and the “standards-new text-teachers use text” fidelity model typically employed
- Discuss educative curriculum materials (Ball & Cohen, Krajcik, Davis, etc.)

How much of the lit review should recap studies about curriculum modifications? (e.g., is Remillard’s work with elementary teachers and the effects of teacher editions of textbooks relevant enough to be included?)

Major sources:

- Remillard, Stein, Smith’s handbook chapter
- Mathematics Teachers at Work
- NCTM Yearbook on mathematics curriculum
- Scour Megan Taylor’s dissertation for relevant stuff

## Literature Review

## Theoretical Framework

`Don't clutter with methods-related frameworks.`

Curriculum theory

- Remillard’s model of curriculum
- van den Akker’s web of influences/objectives?
- pedagogical design capacity?

Where and what is my framework for teacher learning?

See Cobb et al. (2001), p. 118, for argument about the relationship between theory and instructional practice

## Theoretical Framework

Cobb et al. (2001) on theories: “It should be clear from this discussion that we view a theoretical idea such as that of classroom mathematical practice as a conceptual tool whose development reflects particular interests and concerns. We mention this because academic discourse about education often reflects the assumption that instructional approaches should be derived from theory in a top-down manner. The design-research cycle involves an alternative view of the relation between theory and instructional practice in which neither is taken as primary. Instead, the basic relation is one of reflexivity in which the development of theoretical ideas is driven by and remains rooted in instructional practice that is itself guided by current theoretical ideas (cf. Cobb & Bowers, 1999). From this point of view, the relevant criterion when assessing the value of a theoretical construct is whether it enables us to be more effective in supporting students’ mathematical learning.

## Research Questions

## Research Questions

# Study Design

# Study Design

Somewhere early here need to talk about the DBIR study as the larger context for my smaller, design study

DBIR is not a method, but rather a context

# Study Design

## Research Goals and Approach

- Restate practical (supporting teacher’s modifications) and theoretical (curriculum design features) goals
- McKenney:
- Practical:
*“outlines a systematic, transparent process for developing an intervention”* - Theoretical:
*“outlines a systematic, transparent process for empirical investigation”*(findings which should inform the intervention)

- Practical:

## Research Goals and Approach

Despite the attention they receive, standards are an educational means, not an end. Standards are designed to provide a structure for curriculum, instruction, and assessment, all of which should support student learning. Randomized trials might tell us if there is a causal link between standards and student learning, but despite high regard from the policy-oriented [@NationalResearchCouncil2002;@NationalMathematicsAdvisoryPanel2008] and the popular press [@Kolata2013], randomized trials should be seen as only one element of a well-designed research program [@Sloane2008].

The reality for mathematics teachers in over 40 states is that the Common Core State Standards for Mathematics (CCSSM) have been adopted, new materials and instructional strategies have emerged, and new assessments are on their way. All these things are happening in the complexity of classrooms, schools, and districts, where implementation strategies are unclear, if they exist at all.

The approach taken in this study is an example of *design-based implementation research* [@Penuel2011]. Design-based approaches in education research have been promoted for their ability to contribute to theory while engineering new processes and products in the complexity of practice [@Cobb2003design;@Burkhardt2003;@vandenAkker2006]. While much design research (sometimes called *design experiments*) is conducted at the classroom level, design-based implementation research (DBIR) expands the approach to include engineering processes and products to support teaching and learning within educational systems.

According to Penuel et al. [-@Penuel2011], DBIR features four common elements:

- A focus on persistent problems of practice from multiple shareholders’ perspectives;
- A commitment to iterative, collaborative design;
- A concern with developing theory related to both classroom learning and implementation through systematic inquiry; and
- A concern with developing capacity for sustaining change in systems.

`Where to integrate and expand on each of these?`

The goal of this research is not to establish if the CCSSM standards are effective, or to test a particular support against a control. Instead, this research has two primary goals, one practical and one theoretical. In researching and supporting teachers’ curriculum modifications due to the CCSSM SMP, a theoretical goal is to establish a theory of task design and adaptation that describes the “potential” of task features to elicit practices upon enactment. More practically, the research also should yield strategies for teachers and curriculum staff to implement and support high-quality uses of the SMP. This kind of theoretical-practical goals is well-suited for design-based implementation research.

## Methods

`Brief summary here, details below`

## Methods

### Participants and Selection

McKenney:

*Clarify the criteria for selecting participants and the research site; justify choices

- Describe any relevant triangulation of different respondent perspectives (e.g., one construct may be studied from different perspectives, such as that of the observer, the teacher, and the learner)
- Any salient characteristics of the setting (socio-economic status? demographics?

What’s worth saying about the policy climate?

Do I want to seek participation by those not already involved in the project? Is it worth it? What might I gain?

### Participants and Selection

The proposed research will be conducted within a larger DBIR project involving a partnership between university researchers and support personnel with both teachers and curriculum support staff in a large urban school district. The teachers form a “Teacher Advisory Board” (TAB) and were recruited as participant designers of a system to support an improved HS Algebra 1 curriculum. The recruitment of teachers was conducted by the district curriculum staff, whose goal was to assemble a TAB that included teachers from different schools and who had a variety of skills and experience. The TAB consisted of 11 teachers for most of the 2012-2013 school year. Most returned for the 2013-2014 school year, and several new TAB members were added. [`Be more specific after our numbers settle.`

]

During the 2012-2013 phase of the project, a DBIR approach was taken to focus on teachers’ recognition and use of high-quality mathematical tasks. Four task quality dimensions were identified: alignment to the CCSSM standards, cognitive demand [@Stein2009purple], language, and technology use. Included in those discussions were attempts to align mathematical tasks to the SMP. [*Include data from some of those discussions?*] At several points during the year, discussions between the researchers and district leaders turned to the SMP. The district leaders expressed a desire for the SMP to play a greater role in teachers’ curriculum and instruction. The research question for this study, then, is mostly district-driven, and represents the *focus on persistent problems of practice from multiple shareholders’ perspectives* aspect of DBIR. `Not sure this is in the right place.`

### Instrumentation

McKenney:

- Describe instruments, “insofar as they are known at the time of proposal writing”
- align instruments with research questions
- triangulation of instruments around constructs

Possible instruments:

- Teacher interviews and/or survey
- Audio/transcripts from TAB meetings
- Audio/transcripts from planning meetings
- Observation using IQA? (need to think about which practices are and aren’t measured by the IQA)
- Document analysis? (student work?)
- Remillard’s curriculum use/modification timelines?
- Kosko’s measure of SMP understanding (this sounded really underdeveloped at NCTM…maybe ask Karl how it’s coming along)
- Bill mentioned current attempts in NL to create measures of pedagogical design capacity?
`follow up`

Need to read the “Studying the Enacted Math Curriculum” book for both instruments and analysis ideas

### Instrumentation

### Procedures

McKenney:

- Procedures section is the most logical place to discuss the design work
- Difficult to comprehensively describe the
*intervention*at the proposal stage - Useful to describe the
*systematic*process through which the solution will be conceptualized and developed- Establishment of design requirements
- Framing of design propositions
- Careful elaboration of ideas from skeleton designs to detailed specifications, across the micro-cycles envisioned
- Describe how the process will be documented and audited
- Present any initial (integrated) design propositions that are being considered

- “A description of the design procedures can also set the stage for explaining the research procedures.” [
`Oh...they're different...`

] - Example: help readers envision involvement of different participants at various stages of the research
- Detailed account of how instruments will be administered (e.g., pairs of students will be observed, individuals interviewed, etc.)
- Describe how IRB requirements/concerns are met

### Procedures

### Data analysis

Techniques to analyze data, including units and methods of analysis aligned with questions asked and data collected

### Data Analysis

# Relevance

# Relevance

We write and push standards to drive change. If that really happens, we ought to understand how it works and make the good parts work better.

# Relevance

# Dissemination

# Dissemination

- Open!
- Shirky:
*“Publishing isn’t a job anymore. It’s a button.”* - Don’t be boring.
- If I can’t make this work seem compelling to the average math teacher, I don’t deserve to do it.

# Dissemination

# Limitations

# Limitations

- Only 1 district
- Only one subject/level
- Supports might depend on particular administrative structures/resources

# Limitations

# Work Plan

Timeline Budget

# Work Plan

# Work Plan

## Timeline

- Defend prospectus by April 1
- Collect data from TAB members in Spring 2014, revise design
- Collect more data from TAB members (summer?) on revisions
- Roll out to non-TAB members in early fall 2014, collect data
- Final design revisions by end of 2014
- Write/finish Spring 2015