# Stein, Remillard, & Smith (2007)

How Curriculum Influences Student Learning

## Detailed Summary of How Curriculum Influences Student Learning

Stein, Remillard, and Smith open their chapter with comments on the growth of research in the area of curriculum use. The 1992 NCTM Handbook did not include a chapter on this topic, but the emergence of new curricula following the 1989 NCTM Standards spurred a greater research interest in curricula and how they influence student learning. Also, the No Child Left Behind requirement that federal funds only be spent on effective curricular materials drove curriculum developers to prove their materials had a positive effect on student learning.

### Conceptual Issues, Definitions, and Boundaries

#### Multiple Meanings of Curriculum

Stein, Remillard, and Smith define curriculum as "the substance or content of teaching and learning (as distinguished from the 'how' of teaching)" (p. 321). They recognize, however, that curriculum is frequently used to describe a prescribed set of materials or content expectations described by policy documents or frameworks.

Within curriculum research, care is typically taken to describe differences in curriculum:

Opportunities for research exist both within and between each of these stages of curriculum. For example, teachers' beliefs and goals transform a written curriculum into an intended curriculum, and within the enacted curriculum exists all the complexities of classrooms full of students that shape the implementation of a lesson. Furthermore, the enacted curriculum and student learning will shape teachers' future indended curriculum. Stein, Remillard, and Smith summarize the following list of factors that mediate curriculum:

Frequently the written curriculum is improperly judged to have a direct causal relationship on student learning. When all factors are considered, "it points to the fallacy of assuming that the materials themselves are the primary agent in shaping opportunities for student learning and instead uncovers the important role played by the interpretive and interactive influences of teachers and students" (p. 323).

#### Curriculum Materials: An Evolving Concept

Like many people, Stein, Remillard, and Smith use the terms curriculum materials and textbook somewhat interchangeably, intending to refer to "printed or electronic, often published, materials designed for use by teachers and students before, during, and after mathematics instruction" (p. 323). To distinguish the two, the authors pose textbooks as a classroom resource traditionally seen as the provider of practice exercises while curriculum includes instructional guides that emphasize both pedagogy and mathematical content. Interestingly,

To many, the term curriculum materials was used to connote something akin to an "anti-textbook" because these resources offered programs of instruction that rejected the notion that learning mathematics involved completing decontextualized exercises in a book. In contrast to textbooks, which were developed and marketed by commercial publishing companies, curriculum materials tended to be designed by mathematics experts and mathematics education researchers and, prior to the late 1990s, were sold independently to a fairly small market. For most standards-based curriculum materials, students' work during instruction involves investigative projects instead of exercises found on the pages of a "student textbook." Student textbooks are replaced by thin, often consumable, student workbooks that are designed to support students investigative work. The centerpiece of most lessons is the thinking that is required to grapple with the investigative task; student work books are designed to support that thinking by providing a basis for recording, summarizing or reflecting on one’s actions and thinking." (p. 323)

In contrast to curricula of the past, like the New Math materials, modern standards-based curricula typically include pedagogical guidance. In this way they are targeted at teacher learning and not meant to be used by students directly and independent of the teacher, an approach Remillard (2000) refers to as speaking to rather than speaking through the teacher. Standards-based curricula are typically published by a commercial publisher alongside other materials, a process that can create compromises in the author's approach and blur the lines between standards-based and tradtional materials.

Some believe that the written and enacted curriculum should differ as little as possible, with teachers implementing the written text precisely as intended by the author or publisher. Others believe a text and other materials are merely a resource for teachers to use in their lessons. This belief typically views fidelity of implementation as impossible as both teachers and students will construct their own vision of the curriculum (Remillard, 2005).

#### Literature Selection and Boundaries of this Review

For this chapter, Stein, Remillard, and Smith reviewed peer-reviewed research related to the effects of textbooks on both students and teachers. Most of the research stemmed from the NSF-funded curriculum development following the publication of the 1989 NCTM Standards. As an NSF requirement, curriculum developers were required to do evaluations of their work and many of the developers were researchers with the appropriate skills and methods for determining the effectiveness of curriculum. In contrast, research by commercial publishers often focuses on marketability instead of student learning. Stein, Remillard, & Smith provide a table (p. 325) listing common curricula by grade level, funder, and if they judge it to be standards-based or conventional. The authors did not consider policy documents or curriculum frameworks, although they acknowledge the influence of such documents.

### Section One: Research on Curriculum Materials and Student Learning

Stein, Remillard, and Smith associate the growth in curriculum research on the math wars of the 1990s. While the 1989 NCTM Standards were broad enough to garner wide appeal, the standards-based curriculum that followed had to be more specific and brought more scrutiny. Critics wanted "proof" that the texts were effective, even though "conventional textbooks used at the time had little or no evidence of their effectiveness" (p. 326). Most of the studies that followed assumed that curriculum materials mattered and caused differences in what students learned, ignoring details associated with the intended and enacted curriculum.

#### Research on Content of Curriculum Materials

Most teachers rely on curriculum materials as a primary teaching tool (Grouws, Smith, & Sztajn, 2004) and the research clearly indicates that content coverage is important; students must be exposed to a mathematical topic to learn it (Hiebert & Grouws, 2007). Yet, many factors affect how well content is learned, and some curriculum materials attempt to be educative for teachers (Davis & Krajcik, 2005) as well as students.

##### What Content is Covered?

Content analyses compare curriculum materials to standards, frameworks, curriculum from other countries, or other external criteria, depending on the values of the researcher doing the comparison. The National Research Council (2004) found poor reliability in ratings depending on the rater and the various criteria available. Therefore, it is important to align one's goals and values for curriculum with the goals and values of the curriculum analysis.

Stein, Remillard, and Smith summarize the three most prominent content analyses of the decade before their chapter was written:

• Project 2061, a project of the American Association for the Advancement of Science, reviewed in the late 1990s 13 middle school textbooks and only found 4 to be satisfactory. The best of these four were Connected Mathematics, followed by Mathematics in Context, MathScape, and Middle Grades Math Thematics. No conventional texts were rated satisfactory.
• The U.S. Department of Education in 1999 used 8 criteria to find the following texts to be "exemplary:" Connected Mathematics Project (CMP), the Middle School Mathematics through Applications Project (MMAP), Cognitive Tutor, College Prepatory Mathematics (CPM), Contemporary Mathematics in Context (Core-Plus), and the Interactive Mathematics Program (IMP). Judged as "promising" were Everyday Mathematics, MathLand, Number Power, and the University of Chicago School Mathematics Program's (UCSMP) Integrated Mathematics 7-12. These ratings came under attack by mathematicians opposed to the NCTM Standards.
• The organization and website Mathematically Correct, known for its opposition to the Standards, reviewed a number of textbooks across several grade levels. Conventional textbooks were judged superior to standards-based curricula; Everyday Mathematics was the highest-rated standards-based text with a grade of "C", with all others getting a "D" or "F." Nearly all conventional curricula were given grades of "A" or "B."

The lessons learned from these reviews may say more about the reviewing methods themselves than the reviewed curriculum. Hiebert (1999) recommended that consumers determine what they value and then seek out the review whose methods most reflect those values.

##### How is Content Presented?

In this section, Stein, Remillard, and Smith get at the "pedagogical intent" (p. 330) of curricular materials, and concede that just as values and judgement play a part in content coverage, "the criteria that researchers have used to make judgments about the pedagogical intent of various curricula are necessarily related to personally held views regarding the nature of mathematics and how students learn it" (p. 330). Curricular materials might influence teachers' practice with the inclusion of features like pre-tests, formative assessments, differentiation strategies, group activities, and group discussion prompts. Stein, Remillard, and Smith identify three overarching features of how curriculum is presented:

• Order and manner of presentation. Materials vary in how they are sequenced and how explicit they are about new ideas; some texts strategically build to higher-order thinking, while others engage students in introductory activities designed to bring out student thinking.
• Balance. Texts vary in their use of competing curricular elements, such as concepts vs. procedures, calculators vs. by-hand calculation, and balance of mathematical representations. For example, Clopton, McKeown, McKeown, & Clopton (1998) from Mathematically Correct discouraged calculator use, while many reformers endorse calculator use.
• Organizational style. Some textbooks use sequenced chapters and lessons, many of which spiral (Stein & Kim, 2009) to integrate previous topics into new content with increased expectations for mastery. Stein & Kim (2009) refer to these as integral curriculum because they must be taught in sequence over time to have the intended effect. Others curricula take a modular approach (Stein & Kim, 2009), often structured around thematic units, that allows subcomponents of the curriculum to be separated and recombined in different configurations. Many standards-based curriculum are modular, but research has said little about the effectiveness of this approach.

Both Project 2061 and Mathematically Correct have reviewed how content is presented. Project 2061 used 24 instructional criteria in 7 categories, and rated highly Connected Mathematics, Mathematics in Context, MathScape, and Middle Grades Math Thematics. All conventional texts were rated unsatisfactory in the criteria of instructional support. Mathematically Correct's review focused on "quality of presentation" and "quality of student work." Connected Mathematics and Math Thematics received the lowest scores, while conventional texts (and Saxon texts in particular) rated highly.

##### The Support of Teacher Learning

Stein, Remillard, and Smith find that because standards-based curriculum focuses on students "doing mathematics" instead of the explicit instruction of skills, teachers need to provide the right classroom environment for the curriculum to have the intended effect. Often standards-based curricula is designed to help teachers with both the subject matter and how to teach it (Ball & Cohen, 1996; Davis & Krajcik, 2005). This kind of educative curricula seeks to meet these five high-level criteria (Davis & Krajcik, 2005, pp. 5-6):

1. Educative curricula could help teachers learn how to anticipate and interpret what learners may think about or do in response to instructional activities.
2. Educative curricula could support teachers’ learning of subject matter.
3. Educative curricula could help teachers consider ways to relate units during the year.
4. Educative curricula could make visible curriculum developers’ pedagogical judgments.
5. Educative curricula could promote teachers’ pedagogical design capacities or their ability to use personal resources and the supports embedded in the materials to adapt curriculum to achieve productive instructional ends (as opposed to performing “lethal mutations”) (Brown & Campione, 1996, p. 291).

Stein & Kim (2009) evaluated Everyday Mathematics and Investigations to determine how each communicated the text developers' rationale and reasoning (which Stein and Kim called transparency) and how the textbooks helped teachers predict student strategies on tasks. Stein and Kim found that Investigations was more transparent and therefore more easily implemented with fidelity, whereas tasks in Everyday Mathematics had rationales that were less clear (Stein, Kim, & Seeley, 2006).

Brown (2009) organizes curricula into resource-centric or procedure-centric. Resource-centric attempts to communicate to teachers the main ideas and curricular features to teachers, but leaves details of implementation to the teachers. Procedure-centric focuses on actions for executing lessons. Stein & Kim (2009) judged Everyday Mathematics to be procedure-centric while Investigations was resource-centric. Stein, Remillard, and Smith judge that neither resource- or procedure-centric materials are always superior, as the needs of teachers vary with experience, the nature of the adopted curricula, and the instructional skills of the teachers.

#### Examination of Student Learning from Mathematics Curriculum Materials

Because the NSF-supported curriculum projects required formative and summative evaluations, much data was collected about their influence on student learning. Many of the summative evaluations are reported in Standards-Based School Mathematics Curricula: What Are They? What Do Students Learn?, a 2003 book edited by Sharon Senk and Denisse Thompson. Findings for the new curriculum were promising: students using the new curricula commonly equaled their traditionally-taught peers on traditional measures of mathematics ability, such as computation, and exceeded the performance of their peers on tasks focused on conceptual understanding and problem solving (Putnam, 2003; Chappell, 2003; Swafford, 2003). However, some caution about these results is warranted due to differences in methodology, selection strategies for comparison groups, and possible researcher bias, as in some cases the curriculum creators personally trained the teachers in the study and/or performed the evaluations. Even so, many of the studies used standardized test data to make their comparisons, instead of creating assessments that might have favored their particular curriculum (Kilpatrick, 2003).

##### Comparative Studies Conducted by External Researchers

Stein, Remillard, and Smith point out that determining if one curriculum is better than another is more difficult than it might seem. The validity of such a study depends on having credible well-tested outcome measures with known psychometric properties. Also, it becomes difficult to compare two curricula with widely differing goals; does an evaluator only consider the goals the curricula have in common, the combined goals of both, or favor some goals over others? Also, how are comparison groups of students, classrooms, and teachers created in contexts where random assignment is not possible? Lastly, how does an evaluator account for variability in curriculum enactment? It is well-established that even teachers using the same task will vary in their approach (Stein, Grover, & Henningsen, 1996; Tarr, Chávez, Reys, & Reys, 2006). Observing these differences for a large number of teachers is resource-intensive and self-reporting from teachers is unreliable (Ball & Rowan, 2004). These and other difficulties led the National Resource Council (2004) to state that the "studies as a whole across 19 programs studied does not permit one to determine the effectiveness of individual programs with a high degree of certainty" (p. 3). Stein, Remillard, and Smith remind readers that inconclusive evidence does not necessarily mean the curricula are ineffective, but only that the NRC panel was unable to make confident judgments about effectiveness.

The What Works Clearinghouse produced a report on middle school mathematics curriculum summarizing four randomized controlled studies. One study revealed significant effects for Cognitive Tutor Algebra 1 (Morgan & Ritter, 2002) while another showed significant effects for I CAN Learn (Kirby, 2004). Both curricula are standards-based and incorporate technology. The other two studies, involving the standards-based Expert Mathematician/UCSMP and the traditional Saxon curricula failed to find significant effects.

Non-randomized studies do reveal a pattern of higher student conceptual understanding and problem solving when using standards-based curricula (e.g. Boaler, 1997; Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000; Thompson & Senk, 2001). On skill-based and procedural tasks, students using standards-based curricula frequently scored the same as peers using more conventional curricula (e.g., Riordan & Noyce, 2001; Thompson & Senk, 2001). There were exceptions to this, however, as in a study by Huntley, Rasmussen, Villarubi, Sangtong, & Fey (2000) where students using the standards-based Core-Plus Mathematics scored lower than conventionally-taught peers on skill-based Algebra problems when calculators were not allowed.

The general agreement between various studies has led some (e.g., Schoenfeld, 2002) to declare that standards-based curricula as effective. However, Stein, Remillard, and Smith remind us that goals and expected outcomes matter, as students will perform best on tests that resemble the content and approach of their textbooks. Also, the measures that claim students in one curricula outperform those using another might not apply to students of all abilities, or might not have much practical significance. Implementation also plays a larger role than many of these studies reveal. In Boaler & Staples (2008), teachers using the same curriculum reported scores that differed significantly according to their teaching approach, findings similar to Huntley, Rasmussen, Villarubi, Sangtong, & Fey (2000). Balfanz, Mac Iver, & Byrnes (2006), in a non-observational study, found that higher levels of implementation were associated to higher student achievement. Therefore, one should be cautious about making causal statements between curriculum and student achievement, as few studies reveal how the curriculum takes affect and is influenced by other variables in instruction. A study by Tarr, Reys, Reys, Chávez, Shih, & Osterlind (2008) is one of a few studies that attempt to describe these interaction, defining what they termed a "standards-based learning environment" (SBLE). The authors found that using standards-based curriculum led to higher SBLE, and students in high-SBLE classrooms outperformed those in lower-SBLE classrooms using the same standards-based curriculum, similar to results seen by Boaler & Staples (2008).

END

## Also

APA
Stein, M. K., Remillard, J. T., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–369). Charlotte, NC: Information Age.
BibTeX
@incollection{Stein2007,