Day by Day: Investigating Variation in Elementary Mathematics Instruction That Supports the Common Core

Abstract

The Common Core State Standards for Mathematics (CCSS-M) aim to engage students in complex mathematical practices, including modeling, reasoning, and argumentation. Currently, little is known about how teachers’ daily instruction supports these practices. This study draws upon data from daily logs completed by third-, fourth-, and fifth-grade mathematics teachers from 39 states to learn about students’ engagement in standards-aligned mathematical practices. We find that there are both substantial fluctuations in students’ engagement in these practices and reported cognitive demand from day to day, as well as large differences across teachers. Practices in which students engage are related to teachers’ perceptions of student ability. These findings offer a broad perspective on how CCSS-M–based instruction unfolds across a range of states and policy environments.

Keywords

educational policy hierarchical linear modeling instructional practices mathematics education

Since 2011, 45 U.S. states and the District of Columbia have formally adopted the Common Core State Standards for Mathematics (CCSS-M). Although about half of those states have made changes to their standards since that time, independent analyses suggest that key aspects of the CCSS-M have largely been retained (Achieve, 2017). One distinctive feature of the CCSS-M is the articulation of Standards for Mathematical Practice (SMP) alongside content standards at each grade level (Common Core State Standards Initiative, 2018). These practice standards focus on mathematical processes and proficiencies like problem solving, reasoning, and mathematical modeling, and they draw upon previous research (National Council of Teachers of Mathematics, 2000; National Research Council, 2001).

Of the 24 states that have made changes to their standards since adopting the Common Core, only 3—Arkansas, Florida, and Missouri—do not include any mathematical practices as part of their K–12 state standards (Achieve, 2017). Our own review also suggests that even those states that never adopted the CCSS-M also have practice standards for mathematics. That said, we do not know the extent to which teachers are integrating those practice standards into their daily instruction, although research suggests that the complex processes and proficiencies emphasized by the SMP can be challenging both for students to undertake and teachers to orchestrate (Bartell et al., 2017; Floden et al., 2017; Stein et al., 2008).

In this article, we utilize a unique dataset of instructional logs completed over a 10-day period by a sample of third- through fifth-grade mathematics teachers working in a wide variety of school and district contexts across the United States to investigate the extent to which teachers report that students are regularly engaged in complex practices exemplified by CCSS-M. We leverage the detailed data from these logs to gain insight into variation in mathematics instruction within and across teachers. We also consider whether the variation we observe in teachers’ reported practices is related to grade level or classroom characteristics. Lastly, we consider the extent to which teachers’ ratings of the cognitive challenge of their lessons is related to particular practices. This work is intended to fill a gap in the literature by shedding light on the extent to which instruction varies in a given teacher’s classroom, the extent to which variation in the standards-based mathematics practices exists across teachers in a variety of schools and districts across the United States, and the typical cognitive challenge of such practices. Although past studies have relied on logs to explore day-to-day variation in instructional practice, nearly all of these studies have used restricted samples and focused on teaching in specific school districts or localized research settings (Rowan & Correnti, 2009; Rowan et al., 2004; Walkowiak, 2018). Instructional data from a multistate, geographically diverse sample of randomly selected teachers across the United States has enabled us both to confirm results from previous, more idiosyncratic samples of teachers and provide new insights on potential grade-level effects on what teachers report about their instructional practices, as well as how teachers’ reports about their instruction appear closely related to their perceptions about the achievement level of their students. We know of no other study that is able to provide similar data for teachers across such a diverse number of states and districts.

Current Research on Standards Implementation

Annual teachers’ surveys offer some positive evidence about the extent to which teachers have taken up the instructional practices necessary to support successful implementation of CCSS-M. For example, Kane and colleagues (2016) show that teachers have fully embraced SMP and report that teachers have made considerable changes to their practice as a result of standards implementation. Additionally, according to Opfer et al. (2016), majorities of teachers report that their students engage in nearly all the SMPs outlined in CCSS-M often or daily.

Although these survey-based studies offer an optimistic picture of CCSS-M implementation, there are several reasons why we should interpret these findings cautiously. First, teachers charged with implementing CCSS-M do not always have access to or use high-quality instructional materials closely aligned with CCSS-M (Kaufman et al., 2018; McDuffie et al., 2017). Without such materials, teachers may lack guidance on how to support students to reach the high bar set by CCSS-M or how to apply standards to their instruction. Second, teachers need to have deep knowledge about their standards in order to apply them, and many teachers may not have this knowledge or access to knowledgeable colleagues who can support their work (Kaufman et al., 2018; Perry et al., 2015; Supovitz et al., 2016). Third, Powell et al. (2013) have noted that students with learning disabilities can have specific difficulties with mathematics that keep them from meeting standards, even if teachers do emphasize those standards in the classroom. As a result, teachers may need much more training and support to help students with learning disabilities to meet CCSS-M. Taken together, these three issues suggest the possibility that teachers may be indicating through one-time surveys that they are implementing aspects of CCSS-M when their students are not engaging deeply with them on a regular basis.

Need for Information on Day-to-Day Variation in CCSS-M Implementation

In addition to the above drawbacks of annual surveys, there are also some limitations to what can be captured through surveys administered only one time each academic year. Researchers have long acknowledged problems of memory when asking respondents to report retrospectively, especially over a long period of time (Groves, 1987; Mullens & Gayler, 1999). Additionally, educators responding about their instruction may hold differing definitions of particular teaching approaches among themselves and compared to those of researchers trying to measure those practices (Ball & Rowan, 2004; Hill, 2005).

Most importantly for the current study, although retrospective analyses based on annual surveys may be useful in characterizing central tendencies in mathematics instruction, it is widely accepted that some instructional practices are relatively stable from day to day, whereas others fluctuate. Asking teachers to report on instruction retrospectively potentially obscures or conceals these natural fluctuations in instructional practice that may be occurring over time (Rowan et al., 2004).

One potential method for capturing information about variation in instructional practice is instructional logs. Through logs, teachers are typically asked, at the end of each day, to record aspects of their instruction with a particular class or even a specific student, and they do so over a period of days (usually 5–20). By asking teachers to provide frequent, specific, and highly structured information about instruction, logs may provide more accurate information about typical behaviors, because they overcome many of the problems associated with asking respondents to summarize their behavior retrospectively at a single timepoint (e.g., Camburn et al., 2010; Leigh et al., 1998; Rowan & Correnti, 2009). For example, the University of Michigan’s Study of Instructional Improvement (SII) used log-based measures to characterize topic coverage and instructional practices used by elementary teachers. These log-derived measures showed strong agreement (between 50% and 86%) with measures based on structured classroom observations (Hill, 2005), although levels of agreement depended on teacher recall and on differences in the perspectives of teachers and observers. Hill (2005) specifically noted that items focused on “reform” or standards-based content could have different meanings to researchers and teachers. The Instructional Practices Log in Mathematics (IPL-M) (Walkowiak, 2018) captures practices related to CCSS-M, including students’ engagement in problem solving, making connections, and engaging in mathematics talk. According to Walkowiak (2018), measures derived from the IPL-M showed strong agreement (33%–71% exact agreement, 66%–79% within-one agreement) with measures based on structured classroom observations.

Distinct from retrospective surveys, instructional logs are also equipped to capture information about how instructional practices vary across teachers and how they vary from one day of instruction to the next. In their log research, Walkowiak (2018) found median intraclass correlation coefficients (ICCs) of .29, suggesting that teachers vary considerably in their instructional practices from day to day. These ICCs also imply that logs can be used to make reliable distinctions among teachers based on instructional practice when administered over a sufficient number of days. Similar findings have been found in other instructional log studies and across school subject areas (Adams et al., 2017; Rowan & Correnti, 2009).

Research Questions

This study uses instructional logs administered to a multistate sample of third- through fifth-grade mathematics teachers (N = 334) to answer the following research questions:

1) What teacher-reported mathematical practices characterize mathematics instruction in third through fifth grade?

2) How much variation exists in teacher-reported mathematical practices used for instruction in third through fifth grade? And to what extent is this variation related to classroom characteristics?

3) According to teachers, how cognitively challenging are the mathematical tasks in which their third- through fifth-grade students are engaged? And is there a relationship between the cognitive challenge of the tasks in which teachers engage their students and the mathematical practices teachers indicate using in their instruction?

Data and Methods

The American Teacher Panel

The instructional logs discussed in this article were completed by a subsample of third- through fifth-grade mathematics teachers who are members of a nationally representative survey panel of K–12 public school teachers (American Teacher Panel, or ATP), which employs a probability sampling design (sampling within each state), with oversampling of specific subgroups of teachers (e.g., elementary school teachers, high school mathematics teachers, and teachers in urban schools), as well as oversampling of teachers from 24 states, to permit analyses at a state level. Technical details on the sampling design and realized ATP sample can be found in Robbins and Grant (2020). Since 2015, nationally representative subpanels of teachers from the full panel are asked to respond to numerous online survey requests each year.

Analytic Sample

To form the subpanel for this study, the ATP reached out to a nationally representative sample of 2,673 K–12 teachers from all subjects with a screening questionnaire to determine who would be eligible and willing to complete the instructional logs. Based on data teachers provided when they enrolled in the ATP, we surmised that 588 of the 2,673 teachers taught third- through fifth-grade mathematics and were thus eligible to complete the logs.¹ As a part of the screening questionnaire, the 2,673 teachers were informed about the log administration process and told that they would be asked to complete 10 instructional logs for their first math class each week that was composed of third, fourth or fifth graders. This was defined as the “target math class” for each teacher. Of the 2,673 teachers who received the screening questionnaire, 419 indicated they provided mathematics instruction to students in Grades 3 through 5, and 334 completed instructional logs. We thus surmise that between 56% and 80% of third- through fifth-grade mathematics teachers who were part of the nationally representative sample responded to our instructional logs.²

Descriptive information about the composition of the classrooms and schools for both the teachers who completed the instructional logs and the other third- through fifth-grade mathematics teachers in ATP subsample suggests that the participating teachers were largely working in similar school and classroom contexts (Table 1), although there were some statistically significant differences in terms of class size (Grade 4), percentage of students enrolled with IEPs (Grade 5), and the representation of rural schools (Grade 3).

Table 1

Sample Descriptive Statistics

	Grade 3		Grade 4		Grade 5
	Sample	ATP	Sample	ATP	Sample	ATP
Classroom composition
Number of students taught (mean)	26.5	26.3	33.1	24.5^a	36.1	35.3
English Language Learners (%)	14.4	22.8	14.6	22.8	17.2	21.9
Individual Education Program (%)	16.8	29.5	17.0	29.5	32.5	48.3^a
School composition
Minority enrollment (%)	58.0	60.3	58.4	60.3	57.8	68.5
Majority Free or Reduced Price Lunch eligible (%)	56.5	58.1	56.9	58.1	60.5	61.4
City	33.0	30.8	33.3	30.8	23.6	27.3
Rural	25.3	5.0^a	10.6	30.8	27.4	20.5
N	99	20	94	13	106	44

Note. Seven individuals responded that they taught multigrade classes.

Statistically significant differences (p < .05) between the analytic sample and the rest of the eligible ATP respondents.

We reiterate here that although teachers who completed instructional logs were generally not significantly different from those participating in the larger nationally representative ATP, the sample is not nationally representative, and the results presented in this article cannot be used to infer population trends. However, our sample was drawn from a randomly selected ATP of teachers across the United States who responded to the logs and, thus, provides a unique opportunity to study instructional practice in a diverse set of geographical and school-based contexts. Participating teachers worked in 314 schools in 240 school districts in 39 states (with somewhat larger samples of teachers in California, Louisiana, New Mexico, and New York).³ To our knowledge, these instructional logs are the only existing source of information about day-to-day instructional practices among mathematics teachers working in multiple states across the United States and thus present a unique perspective on how standards adoption may influence teaching and learning.

Instruments and Measures

ATP instructional logs

The ATP instructional log was a self-administered, online questionnaire. The log asked teachers to report on the instruction provided to a target math class, which (i.e., the first math class that was taught each week composed of third, fourth, and/or fifth graders). The log contained more than 50 items, which were a mix of checklist, forced choice, and Likert-type items. Our article focuses on teachers’ responses to items addressing the instructional strategies and practices that teachers used during the target math class. In order to ease administrative burden and to ensure that the log questionnaires could be fielded to all participants in roughly the same time window, teachers were given flexibility about where in the development of a unit they began completing instructional logs. The 334 participating teachers completed 3,162 instructional logs. Of the 334 teachers, 289 completed all 10 logs, and 319 completed 5 or more logs.

Measures of instructional practices

Items (Table 2) used to measure teachers’ practices were drawn from the IPL-M (Walkowiak, 2018). Evidence supporting the validity and reliability of measures derived from the IPL-M are available in Walkowiak (2018). These items were measured using 3-point Likert-type items (1 = no time or almost no time, 2 = some time [less than half the lesson], 3 = considerable time [half the lesson or more]).

Table 2

Practices Addressed in Instructional Logs

Instructional Log Item	Related Standard for Mathematical Practice
a) How much time did the students participate in the following types of activities?
Review mathematics content previously covered
Review or practice math facts that they have memorized
Review recent mathematics homework
b) How much time did the students participate in the following types of classroom activities?
Connect math topic of lesson to another math topic
Connect math topic of lesson to another subject (e.g., social studies)
Work on problem(s) that have multiple answers or multiple solution methods
Perform tasks focused on math procedures
Take a test, quiz, or other assessment
Complete worksheets or problem sets from text
Work on mathematics homework
Solve problems at the board
Use flashcards, games, or computer activities to improve recall or skill
Listen to me explain the steps to a procedure or work examples on the board
Read from a textbook to learn information
Apply or connect math topic of lesson to a “real world” situation or idea	Model with math
Perform tasks requiring new, unfamiliar ideas or methods	Make sense of problems and persevere in solving them
Use hands-on tools to explore mathematical ideas or to solve problems	Use appropriate tools strategically
c) How much time did the students participate in the following activities through oral discussion or writing?
Answer recall questions that require a few words
Demonstrate different ways to solve a problem	Make sense of problems and persevere in solving
Write explanations of mathematical ideas, solutions, or methods	Make sense of problems
Explain their thinking about mathematics problems	Make sense of problems
Make explicit connections between concrete and abstract representations	Reason abstractly and quantitatively
Prove that a solution is valid or that a method works for all similar cases	Construct viable arguments and critique reasoning of othersLook for and make use of structure
Discuss ideas, problems, solutions, or methods with other students	Construct viable arguments
Restate another student’s ideas in different words	Construct viable arguments
Pose questions to other students or the teacher about the mathematics	Construct viable arguments
Similarities and differences among mathematical representations or solution methods	Look for and express regularity in repeated reasoningLook for and make sure of structure

The practices included in the instructional logs were not developed to directly measure student engagement in the SMP that are part of the CCSS-M. Instead, the practices were intended to measure five key dimensions of mathematics instruction grounded in research, including research that supports the SMP: problem solving, connections, procedural instruction, math talk, and use of multiple representations. We identify particular practice items from the instructional logs that also reflect language within particular SMPs, as framed in CCSS-M documents (Table 2) (Common Core State Standards Initiative, 2018).⁴

Measures of cognitive challenge

In addition to reporting on the specific activities and instructional practices used in the target class, teachers were asked to report on the degree to which students worked on “cognitively challenging tasks” on a given day (on a 4-point scale from 1 = not at all to 4 = to a great extent). Kaufman and colleagues (2019) used this item in a previous study and found a significant correlation between teachers’ responses on it for a single just-taught lesson and a cognitive challenge score from an independent observer for the same lesson ( $ρ = 0.28$ , p < .05), which provides some validity evidence for using this item in our instructional log.

Measures of class composition and school context

Information about school context was derived by merging the ATP data with Common Core of Data (CCD), which is collected annually from all local and state education agencies in the United States by the National Center for Education Statistics. CCD includes information on the percentage of students eligible for Free or Reduced Price Lunch (FRPL), percent minority enrollment, and school urbanicity. Information about classroom composition was derived from self-reported information included in the log. Teachers were asked to report the number of students enrolled in the target class, the number of students designed as English Language Learners (ELL), the number of students designated as having Individual Education Program (IEPs), and the percentage of students they perceive to be achieving below grade level in mathematics (0%−25%, 26%−50%, 51%−75%, or 76%−100%).

Analytic Methods

In order to address the first research question about the practices that characterize typical instruction in our sample, we examined the average percentage of teachers choosing each response category on each of the 10 instructional days. We first calculated the percentage of teachers responding in each category on each day, and then averaged those percentages over days and teachers. We did this both overall (for all respondents) as well as separately by grade level.

In order to address the second research question (exploring variation in instruction across teachers), we used hierarchical linear models that assumed days of instruction were nested within teachers and included teacher random effects. For item (y), we expressed the response of teacher j on day i as

y_{i j} = α + u_{0 j} + e_{i j},

(1)

where $α$ was an overall item mean and $u_{0 j}$ and $e_{i j}$ were teacher and residual random effects, with mean zero and variance $τ_{0}$ and $σ^{2}$ , respectively. In order to explore how instructional practice varied within teachers (across the 10 days) and between teachers, we treated the outcome variables as continuous. Treating the variable as continuous allowed us to estimate the proportion of variance in reported instructional practice use that was between teachers (i.e., an intraclass correlation).

To explore the extent to which instructional practices were related to classroom characteristics, including grade level and the percentage of students perceived by teachers to be achieving below grade level in mathematics, we used modified versions of the model in Equation (1) to account for the fact that the survey response options have an ordering, but the distance between these options is not uniform. Specifically, we used cumulative link mixed models (Christensen, 2018), which allowed us to model the outcomes under the assumption that there is an underlying continuum (in this case time) that is partitioned into ordered categories. These models also allowed us to account for the fact that instructional days are nested within teachers. The models incorporated grade-level fixed effects and student achievement-level fixed effects.

In all models, we controlled for school-level minority enrollment, FRPL eligibility, and urbanicity. The fixed-effect estimates from the cumulative link models can be converted into odds ratios to aid in interpretation (Christensen, 2018), which were used to determine whether there were grade- or achievement-based differences in the odds that specific instructional practices were used. Negative model coefficients suggest that an outcome is less likely, compared to the reference group, and positive coefficients suggest that an outcome is more likely, compared to the reference group, given all other variables are held constant. For grade level, we took third grade as the reference group; and for achievement level, we created a dichotomous variable, comparing the highest category (76%–100% low achieving) to all other classrooms. The Benjamini-Hochberg method was used to control the false discovery rate (Benjamini & Hochberg, 1995).

In order to address the third research question (exploring the cognitive demand of tasks across teachers), we used models similar to Equation (1) with reports of student engagement in cognitively challenging tasks as the outcome to decompose variance within and between teachers. In order to examine the relationships between cognitive demand and a teacher’s instructional practices, we calculated the Pearson correlations between the 10-day average reports of instructional practice use and cognitive demand.

Results

In this section, we present results corresponding to each of the three research questions, which consider (a) practices characterizing typical mathematics instruction; (b) variation in practices, along with associations between variation and classroom characteristics; and (c) cognitive challenge of practices in which teachers report engaging their students.

Practices That Characterize Typical Instruction

We begin by addressing the first research question: What teacher-reported mathematical practices characterize typical mathematics instruction in third through fifth grade? According to the data, teachers reported spending a considerable amount of time reviewing content. Table 3 shows the average percentage of teachers responding about their practices in each response category over the 10 instructional days. Nearly one-third of teachers reported spending considerable time reviewing mathematics content previously covered, and nearly all teachers (92.47%) reported spending at least some time doing so. Additionally, almost four-fifths of all teachers reported spending at least some time performing tasks focused on math procedures and applying or connecting mathematics to a “real world” situation or idea. Similar percentages reported spending at least some time having students discuss ideas, problems, solutions or methods; demonstrating different ways to solve a problem; or explaining their thinking about mathematics problems.

Table 3

Average Percentage of Teachers Reporting That Students Participated in Classroom Activities

	Considerable Time	Some Time	No Time or Almost No Time
(a) How much time did the students participate in the following types of activities?
Review mathematics content previously covered	33.02%	59.45%	7.53%
Review or practice math facts that they have memorized	10.06%	46.49%	43.45%
Review recent mathematics homework	4.19%	33.23%	62.58%
(b) How much time did the students participate in the following types of classroom activities?
Connect math topic of lesson to another math topic	19.48%	53.51%	27.01%
Connect math topic of lesson to another subject (e.g., social studies)	4.29%	19.56%	76.15%
Work on problem(s) that have multiple answers or multiple solution methods	23.45%	39.71%	36.84%
Perform tasks focused on math procedures	31.77%	48.25%	19.98%
Take a test, quiz, or other assessment	11.03%	20.23%	68.75%
Complete worksheets or problem sets from text	13.00%	48.40%	38.61%
Work on mathematics homework	2.44%	19.14%	78.41%
Solve problems at the board	11.05%	45.80%	43.15%
Use flashcards, games, or computer activities to improve recall or skill	5.53%	32.50%	61.96%
Listen to me explain the steps to a procedure or work examples on the board	8.68%	57.14%	34.18%
Read from a textbook to learn information	0.88%	13.27%	85.85%
Apply or connect math topic of lesson to a “real world” situation or idea	31.54%	47.16%	21.29%
Perform tasks requiring new, unfamiliar ideas or methods	20.87%	40.61%	38.52%
Use hands-on tools to explore mathematical ideas or to solve problems	18.82%	32.22%	48.95%
(c) How much time did the students participate in the following activities through oral discussion or writing
Answer recall questions that require a few words	5.50%	38.79%	55.71%
Demonstrate different ways to solve a problem	29.05%	49.68%	21.27%
Write explanations of mathematical ideas, solutions, or methods	22.46%	48.72%	28.82%
Explain their thinking about mathematics problems	23.57%	57.86%	18.57%
Make explicit connections between concrete and abstract representations	21.39%	46.39%	32.21%
Prove that a solution is valid or that a method works for all similar cases	16.72%	41.13%	42.15%
Discuss ideas, problems, solutions, or methods with other students	22.10%	55.32%	22.58%
Restate another student’s ideas in different words	8.28%	39.60%	52.12%
Pose questions to other students or the teacher about the mathematics	8.94%	44.45%	46.61%
Similarities and differences among mathematical representations or solution methods	13.80%	42.36%	43.85%

Note. Practices related to the Common Core State Standards for Mathematical Practice are in italics.

Most commonly, teachers who reported spending considerable time reviewing mathematics content previously covered also did not report considerable time engaged in any activity other than review (10% of all teachers). A much smaller percentage of teachers reported spending a considerable amount of instructional time reviewing in conjunction with using hands-on tools to ideas or to solve problems (3%), having students solve problems at the board (2.5%), having students perform tasks requiring new unfamiliar ideas or methods (2%), or reviewing or practicing memorized math facts (2%).

On the other hand, four of the other prevalently reported practices—applying or connecting mathematics to a “real world” situation or idea; discussing ideas, problems, solutions, or methods; demonstrating different ways to solve a problem; and explaining their thinking about mathematics problems—are closely aligned with the CCSS-M SMPs of Modeling With Math, Constructing Viable Arguments, and Making Sense of Problems and Persevering in Solving Them, respectively.

Variation in Instructional Practices

Next, we turn to the second set of research questions: How much variation exists in teacher-reported mathematical practices used for instruction in third through fifth grade? And to what extent is this variation related to classroom characteristics?

The logs show that instruction varied considerably across days and across teachers. What is more, the logs show that mathematics instruction varied systematically across grades, and also varied systematically based on teacher perceptions of the achievement levels of their students. Table 4 displays estimated between-teacher and within-teacher variance components for each of the reported instructional practices. On average, approximately two thirds of the variance in reported use of instructional practices was across days within teachers, suggesting that the specific constellation of instructional practices used by a teacher varies from lesson to lesson. For example, although the average teacher reported spending between some time and considerable time reviewing mathematics content that was already covered, nearly one third of the teachers had some days where they reported spending no time or almost no time reviewing, and some days where they reported spending considerable time reviewing. Only 11% of teachers consistently reported spending the same amount of time reviewing on every day. The day-to-day variation in the amount of time spent in review activities may reflect a cycle of development and review over the course of a unit. However, it is also possible that over the course of 10 instructional days, some teachers administered unit tests or other summative assessments, and the periods of review were more prevalent in lessons that are closer to these assessments.

Table 4

Estimated Variance Components of Instructional Practices

	Between Teacher	Within Teacher	ICC
(a) How much time did the students participate in the following types of activities?
Review mathematics content previously covered	.08	.26	.24
Review or practice math facts that they have memorized	.16	.26	.38
Review recent mathematics homework	.14	.19	.42
(b) How much time did the students participate in the following types of classroom activities?
Connect math topic of lesson to another math topic	.16	.29	.36
Connect math topic of lesson to another subject (e.g., social studies)	.11	.18	.38
Work on problem(s) that have multiple answers or multiple solution methods	.19	.39	.33
Perform tasks focused on math procedures	.16	.34	.32
Take a test, quiz, or other assessment	.06	.40	.13
Complete worksheets or problem sets from text	.13	.32	.29
Work on mathematics homework	.10	.13	.43
Solve problems at the board	.17	.27	.39
Use flashcards, games, or computer activities to improve recall or skill	.13	.23	.36
Listen to me explain the steps to a procedure or work examples on the board	.10	.27	.27
Read from a textbook to learn information	.05	.09	.36
Apply or connect math topic of lesson to a “real world” situation or idea	.17	.34	.33
Perform tasks requiring new, unfamiliar ideas or methods	.13	.43	.23
Use hands-on tools to explore mathematical ideas or to solve problems	.23	.36	.39
(c) How much time did the students participate in the following activities through oral discussion or writing
Answer recall questions that require a few words	.14	.22	.25
Demonstrate different ways to solve a problem	.15	.34	.31
Write explanations of mathematical ideas, solutions, or methods	.20	.31	.39
Explain their thinking about mathematics problems	.14	.28	.33
Make explicit connections between concrete and abstract representations	.22	.31	.42
Prove that a solution is valid or that a method works for all similar cases	.21	.31	.40
Discuss ideas, problems, solutions, or methods with other students	.12	.32	.22
Restate another student’s ideas in different words	.15	.26	.37
Pose questions to other students or the teacher about the mathematics	.14	.27	.34
Similarities and differences among mathematical representations or solution methods	.17	.32	.35

Note. Practices related to the Common Core State Standards for Mathematical Practice are in italics. ICC = intraclass correlation coefficient.

Similar patterns were present for the items related to the SMP, suggesting that although teachers were addressing practice standards in their instruction, these practice standards were addressed inconsistently, and the extent to which these practice standards were the focus of instruction varied from lesson to lesson. There are several different plausible explanations for this inconsistency in teachers’ instruction related to practice standards. On the one hand, it may be that teachers were not systematically attending to the practice standards or were unsure of how to integrate them into certain kinds of lessons. Or it may be that over the development of a unit, some lessons lent themselves more readily to certain practice standards. For example, a unit may be launched by giving students a task that requires them to use new, unfamiliar ideas or methods as a way of motivating the new material. But students may not be asked to compare and contrast solution methods until later in a unit, after students have had some time to explore and familiarize themselves with a new method.

Although within-teacher variability was substantial, the fact that one third of the variance in reported use of instructional practices was between teachers suggests that there are meaningful and substantive differences among teachers and that distinctions can be made among teachers based on their 10 days of reports. For example, on average, across the 10 days, some teachers reported very little time having students perform tasks that required new, unfamiliar ideas or methods, and some teachers reported considerable time on such tasks.

Table 5 shows the fixed effects estimates for cumulative link mixed models predicting item scores by grade level and teacher perceptions of class achievement. Compared to third-grade students, fourth- and fifth-grade students were reportedly less likely to spend considerable time reviewing math facts that they had memorized, and fifth-grade students were reportedly more likely to review homework. In contrast, students in higher grades were less likely to read from a textbook.

Table 5

Estimated Fixed Effects for Grade Level and Perceived Class Achievement From Cumulative Link Mixed Models

	Grade		Low Achieving
	Fourth	Fifth	Low Achieving
	Est. (SE)	Est. (SE)	Est. (SE)
(a) How much time did the students participate in the following types of activities?
Review mathematics content previously covered	−0.11 (0.19)	0.10 (0.20)	0.64** (0.23)
Review or practice math facts that they have memorized	−0.92** (0.27)	−0.89** (0.29)	0.47 (0.34)
Review recent mathematics homework	0.27 (0.25)	0.89** (0.28)	−1.33** (0.43)
(b) How much time did the students participate in the following types of classroom activities?
Connect math topic of lesson to another math topic	−0.43 (0.22)	−0.31 (0.23)	−0.57 (0.27)
Connect math topic of lesson to another subject (e.g., social studies)	−0.18 (0.24)	−0.11 (0.25)	−0.82* (0.29)
Work on problem(s) that have multiple answers or multiple solution methods	−0.58 (0.33)	−0.70 (0.35)	−0.89 (0.42)
Perform tasks focused on math procedures	0.02 (0.22)	−0.23 (0.22)	−0.89* (0.26)
Take a test, quiz, or other assessment	0.00 (0.23)	0.26 (0.23)	0.28 (0.28)
Complete worksheets or problem sets from text	0.29 (0.18)	−0.15 (0.19)	−0.60* (0.22)
Work on mathematics homework	−0.12 (0.17)	−0.14 (0.17)	−0.14 (0.2)
Solve problems at the board	−0.11 (0.22)	0.17 (0.23)	−0.61 (0.27)
Use flashcards, games, or computer activities to improve recall or skill	0.26 (0.37)	0.94 (0.38)	−0.62 (0.47)
Listen to me explain the steps to a procedure or work examples on the board	0.00 (0.26)	0.07 (0.27)	−0.06 (0.32)
Read from a textbook to learn information	−0.88* (0.28)	−0.96* (0.29)	0.20 (0.35)
Apply or connect math topic of lesson to a “real world” situation or idea	−0.72* (0.26)	−1.38*** (0.27)	−0.08 (0.33)
Perform tasks requiring new, unfamiliar ideas or methods	0.02 (0.21)	0.12 (0.21)	−0.48 (0.25)
Use hands-on tools to explore mathematical ideas or to solve problems	−0.97 (0.41)	0.29 (0.40)	−0.73 (0.51)
(c) How much time did the students participate in the following activities through oral discussion or writing
Answer recall questions that require a few words	−0.07 (0.19)	−0.40 (0.19)	−1.03*** (0.25)
Demonstrate different ways to solve a problem	−0.25 (0.26)	−0.16 (0.27)	−0.80* (0.32)
Write explanations of mathematical ideas, solutions, or methods	−0.07 (0.26)	−0.42 (0.28)	−0.88* (0.32)
Explain their thinking about mathematics problems	−0.18 (0.25)	−0.23 (0.27)	−1.00** (0.32)
Make explicit connections between concrete and abstract representations	0.13 (0.21)	0.08 (0.21)	−0.63* (0.25)
Prove that a solution is valid or that a method works for all similar cases	−0.19 (0.23)	−0.03 (0.20)	−0.47 (0.28)
Discuss ideas, problems, solutions, or methods with other students	−0.21 (0.26)	−0.19 (0.26)	−0.37 (0.31)
Restate another student’s ideas in different words	−0.11 (0.23)	−0.14 (0.24)	−0.62* (0.28)
Pose questions to other students or the teacher about the mathematics	0.03 (0.24)	0.33 (0.25)	−0.62* (0.29)
Similarities and differences among mathematical representations or solution methods	−0.57 (0.28)	−0.32 (0.29)	0.28 (0.34)

Note. For grade level regressions, the reference group is third grade; and for the achievement regressions, the reference group is 0%–75% achieving below level. All significance tests adjusted for multiple comparisons. Practices related to the Common Core State Standards for Mathematical Practice are in italics. Not shown in the table are coefficients for covariates including school-level minority enrollment, Free or Reduced Price Lunch eligibility, and urbanicity.

p < .05. **p < .01. ***p < .001.

In general, students were not more likely to engage in practices related to the SMP in higher grades than in lower grades, according to teachers’ reports. However, students in higher grades were reportedly less likely to apply or connect a math topic to a “real world” situation or idea. To guide interpretation of these coefficients, an estimated model parameter of −1.38 suggests that the odds of fifth-grade teachers reporting spending “considerable time” applying or connect a math topic to a “real world” situation or idea (versus “some time” or “no time or almost no time”) were 75% lower than third-grade teachers, holding all other covariates constant.

Compared to classrooms where teachers reported working with fewer below-grade-level students, those teachers with more below-grade-level students were reportedly more likely to spend considerable time reviewing mathematics content previously covered: The odds of these teachers reporting spending “considerable time” reviewing content previously covered (versus “some time” or “no time or almost no time”) were around 90% higher than other teachers, holding all other covariates constant. Teachers working with more below-grade-level students were also less likely to spend considerable time reviewing homework: The odds of reporting spending “considerable time” reviewing homework (versus “some time” or “no time or almost no time”) were around 74% lower than the reference group, holding all other covariates constant. Teachers with more below-grade-level students were also less likely to be engaged in several activities related to the practice standards, including having students demonstrate different ways to solve problems; writing explanations of mathematical ideas, solutions, or methods; making explicit connections; restating another student’s ideas in different words; and posing questions to other students or the teacher about the mathematics. On average, the odds of teachers with more below-grade-level students reporting spending “considerable time” on these activities (versus “some time” or “no time or almost no time”) were around 50% lower than the reference group, holding all other covariates constant.

There are several competing hypotheses that could explain these relationships. First, it could be that learning tasks were differentially assigned to students based on teacher beliefs about student ability. Prior research has found that “mathematically rich” tasks that engage students in instruction requiring reasoning and creativity, the coordination of multiple concepts and methods, and the application of skills into novel contexts are often reserved for students that are perceived as high achieving, whereas students perceived as lower achieving spend more time focusing on the development and practice of basic skills (Stipek et al., 2001). At the same time, some research indicates that instructional practices are differentially effective at fostering student understanding and promoting student achievement for students at different achievement levels (Gamoran et al., 1997; Sowell, 1989), and teachers may be making different instructional choices for students at different achievement levels to support their improvement.

Variation in Cognitive Challenge

Finally, we turn to the third set of research questions: How cognitively challenging are the mathematical tasks in which third- through fifth-grade students are reportedly engaged? Is there a relationship between the cognitive challenge of the tasks and the mathematical practices teachers indicate using? The logs show that cognitive challenge varied considerably across days and across teachers. What is more, the logs show that cognitive challenge was systematically associated with the use of specific instructional practices.

On average, approximately 69% of the variance in reported cognitive challenge was across days within teacher (ICC = .31). Although the average teacher reported that students worked on cognitively challenging task between a small and moderate degree, only 5% of teachers consistently reported the same amount of cognitive challenge in each class. Twice that number reported at least one class with a considerable degree of cognitive challenge and at least one class with no cognitive challenge at all. However, even with a large amount of within-teacher variability, there were meaningful differences between teachers in terms of their reported use of cognitively challenging tasks. At the 25th percentile of the distribution, the mean rating of the extent to which students worked on cognitively challenging tasks was 2.50. This means that 25% of teachers used cognitively challenging tasks to a moderate degree, a small degree, or not at all. On the other hand, at the 75th percentile of the distribution, the average rating was 3.20. This means that 25% of teachers used cognitively challenging tasks to a moderate or considerable degree or higher. Additionally, the use of cognitively challenging tasks was related to student achievement. Teachers with higher percentages of below-grade-level students were less likely, on average, to report that students worked on cognitively challenging tasks.

There were also some noteworthy relationships between teacher ratings of cognitive challenge and reported uses of instructional practices, consistent with research and theory. The strongest correlations were between teacher ratings of cognitive challenge and instructional practices that were theorized to be more cognitively demanding for students, including those tied to the SMP. For example, there were robust correlations among cognitive challenge and the extent to which students were asked to make explicit connections between concrete and abstract representations ( $ρ = 0.45$ ); discuss ideas, problems, solutions, or methods with other students ( $ρ = 0.44$ ); apply or connect math to a “real world” situation or idea ( $ρ = 0.43$ ); work on problems that have multiple answers or solutions ( $ρ = 0.42$ ); demonstrate different ways to solve a problem ( $ρ = 0.41$ ); and explain thinking about mathematics ( $ρ = 0.41$ ). On the other hand, the smallest correlations were among cognitive challenge and the use of flashcards, games, or computer activities to improve recall ( $ρ = - 0.02$ ); reviewing memorized math facts ( $ρ = 0.01$ ); reading from a textbook ( $ρ = 0.03$ ); completing worksheets from the text ( $ρ = 0.04$ ); reviewing math content previously covered ( $ρ = 0.04$ ); and answering recall questions that require a few words ( $ρ = 0.05$ ).

Summary and Discussion

The objective of this article was to investigate the extent to which students were being engaged in complex practices exemplified by CCSS-M and the SMP on a daily basis, according to teachers’ reports. We addressed this objective using a unique set of log data from teachers across a diverse number of states and districts. We know of no other study that has employed log data across such a diverse number of states and districts.

This article makes two distinct contributions to the literature. First, several of our findings confirm results from past research based on surveys and smaller scale studies. Second, several findings provide new insights and nuance regarding the extent to which teachers are engaging their students in practice standards.

As noted in prior research, we find that review is a prominent feature of many classrooms, with the vast majority of teachers reporting at least some time spent reviewing content previously covered. Using data from the Trends in International Mathematics and Science Study (TIMSS), Stigler and Hiebert (1999) found that in the United States, on average, more than half the time in each class is spent reviewing previously learned content, and that it is fairly common for teachers to spend an entire lesson reviewing. We also find that there are meaningful and substantive differences among teachers, in terms of their practices, with at least some teachers consistently reporting very little engagement in certain practices—like performing tasks that require new or unfamiliar ideas—compared to others. The magnitude of these between-teacher differences are similar to those reported in other research using instructional logs in elementary classrooms (Adams et al., 2017; Rowan & Correnti, 2009; Rowan et al., 2004; Walkowiak, 2018).

This article also offers some new findings that are facilitated by the use of instructional logs to capture day-to-day teaching variation. First, contrary to past research suggesting that grade-level effects account for only a small portion of the variance across teachers (Rowan et al., 2004), we found a significant proportion of the variation in some instructional practices could be attributed to grade level. We also found that teachers who reported higher percentages of low-achieving students in their classrooms were much less likely to engage in many standards-aligned practices compared to their counterparts who reported lower percentages of low-achieving students. This is consistent with research suggesting that learning tasks are often differentially assigned to students based on teacher beliefs about student ability (Oakes, 1996; Stipek et al., 2001). However, we do not know of other instructional log or other research of teachers in such a variety of contexts demonstrating the extent to which standards-aligned practices are undertaken on a day-to-day basis depending on teachers’ perceptions about the achievement levels of their students.

Additionally, although some research—mainly based on one-time surveys—suggests that most teachers engage their students in practice standards fairly regularly (e.g., Opfer et al., 2016), teachers’ responses to instructional logs in this study provided a somewhat different picture. Specifically, the logs indicated that mathematics teachers spent “considerable time” on areas related to practice standards in one third of the lessons in our sample and reported spending “no time” on such areas for between one quarter to nearly one half of the lessons in our sample. These data imply that teachers may not be engaging with the SMP to the extent that they may report doing so in one-time surveys and that instructional logs may provide a clearer picture of day-to-day variation in teachers’ instructional practices.

Finally, because this article probes into the instructional practices in the context of CCSS-M adoption and implementation, using an instructional log that is sensitive to the SMP, it also offers a new perspective on contemporary elementary mathematics instruction across the United States. Although the CCSS-M are thought to be more cognitively demanding than prior standards and require teachers to focus more on using instructional strategies that promote deep conceptual understanding (Porter et al., 2011), empirical evidence supporting this idea is scarce. In the current study, we found some empirical evidence supporting this claim: There were substantial correlations between cognitive challenge and many practices aligned with the SMP.

Our study has some limitations. First, although participating teachers came from states across the United States, participation in the study was voluntary, and therefore responses do not constitute a nationally representative sample. Nonetheless, teachers from a wide variety of state contexts participated in the study and, thus, might provide some sense of the variation in mathematics instructional practices across the country. Second, although instructional logs may be less susceptible to response biases that impact one-time surveys, logs are still based on teachers’ self-report and may not completely reflect the actual practices in which teachers engage in their classrooms. And finally, the language we used to describe particular practices may not be the same language that teachers would use to describe those practices. Thus, their reports in regard to a particular practice may not be aligned with what that practice signals to researchers or other readers.

Implications

These findings represent one of the first attempts to provide information about daily mathematics classroom practices in the context of CCSS-M adoption among K–12 public school teachers working in geographically diverse contexts across the United States. As such, we believe that these findings have several implications for research and practice.

First, our work primarily implies that the SMP may be challenging for teachers to undertake when they are serving students at lower achievement levels and/or that they may need much more support to do so. In our instructional logs, we asked teachers about the materials they were using in the classroom. Yet we did not uncover any robust associations between materials and practice, although our findings are limited by our sample size. This study thus raises questions about what materials and resources could support engagement in practice standards among lower achieving students.

This study also suggests that the SMP—and associated instructional practices—present more cognitive challenge than other more traditional practices. Specifically, our research indicated that when teachers reported engaging in practices aligned with practice standards, they also often reported that their lessons were more cognitively challenging for students. In contrast, practices without clear alignment to standards typically were not significantly associated with our measure of cognitive challenge. Further studies might explore which particular practices present cognitive challenges to students most consistently, and which do not.

Overall, our investigation suggests we need to dig much more deeply into the instruction happening in mathematics classrooms. For example, our research suggests that many teachers—and particularly those who perceive that they teach more low-achieving students—spend a lot of time reviewing previously covered mathematics content. Students who spend most of their time reviewing previously covered content might be at risk to fall behind even more quickly, particularly if they are also not engaging in ambitious practices like those reflected in state practice standards. Additional research might shed light upon the nature of teachers’ review lessons and the extent to which that review is actually useful and beneficial to all students in a classroom.

Taken together, our findings imply that teachers need additional support (e.g., more professional development and support through resources and instructional materials) to implement the SMP, especially those teachers with low-achieving students. Without support to do so, teachers who perceive their students as lower achieving may be less likely to engage their students in cognitively demanding activities and more likely to repeat content, thereby lowering the potential mathematical achievement of their students.

Footnotes

Notes

Authors

JONATHAN D. SCHWEIG, PhD, is a social scientist at the RAND Corporation, 1776 Main Street, Santa Monica, CA 90401; jschweig@rand.org . His research focuses on education policy and the measurement of instructional practices, classroom and school climate, and social and emotional competencies.

JULIA H. KAUFMAN, PhD, is a senior policy researcher at the RAND Corporation, 1776 Main Street, Santa Monica, CA 90401; jkaufman@rand.org . Her research focuses on understanding teachers’ instruction and how state and school district policies support instructional improvement.

V. DARLEEN OPFER, PhD, is vice president of RAND Education and Labor and the distinguished chair in education policy at the RAND Corporation, 1776 Main Street, Santa Monica, CA 90401; Darleen_Opfer@rand.org . Her research focuses on understanding the policies and school conditions that support improvements in teaching and learning.

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