Measuring Middle School Students’ Algebra Readiness

Understanding Number Systems

Foundational to students’ ability to reason algebraically is their knowledge of number systems and the relationships between those systems. Deepening students’ understanding of natural numbers, whole numbers, integers, and rational numbers supports them in making connections between number systems and mathematical operations (e.g., subtraction and negative integers). Specifically, students’ understanding of fractions in upper elementary and middle grades is a significant predictor of later achievement in high school algebra and mathematics even when controlling for general intellectual ability, other mathematics content knowledge, and family background (Siegler et al., 2012). Because numeracy and, more specifically, understanding rational numbers are critical for success in algebra (NMAP, 2008), students need to develop a solid conceptual understanding of number systems in the middle grades.

Place value and magnitude are central concepts that support students’ understanding of number systems (Wu, 2001). However, reading numbers correctly using place value language does not guarantee that students understand the value of each numeral (Milgram, 2005). Decomposing and composing numbers into their component values (374 = 300 + 70 + 4) deepens students’ knowledge of number (Gersten et al., 2009) and assists them in representing numbers in a variety of ways. Place value and magnitude concepts also assist students in making connections between number systems. For example, when locating and ordering rational numbers on a number line, students begin to understand that any whole number can be represented as a rational number and that multiple representations of a number may exist within a system (e.g., fractions and decimals). Understanding representations of the magnitude of whole numbers (Booth & Siegler, 2008) and fractions (Siegler, Thompson, & Schneider, 2011) is significantly related to students’ overall mathematics achievement, as well as performance on whole number and fraction arithmetic, respectively. Because of the critical link between students’ understanding of these two central concepts (i.e., place value and magnitude) and their knowledge of number systems, and the critical link between students’ knowledge of number systems and their performance in algebra, it follows that students’ understanding of place value and magnitude concepts is a prerequisite skill for algebraic proficiency.

Facility With Basic Number Properties

In addition to understanding number systems, students’ knowledge and application of number properties affects their algebra readiness and their ability to reason algebraically. These number properties include the distributive property, the commutative, associative, identity, and inverse properties of addition and multiplication, and mathematical equality. Equality can be considered one of the most central ideas students must understand to reason algebraically (Geary et al., 2008). However, many students have misconceptions about the equal sign, some identifying it as an operator. This stems from early mathematics experiences where the equal sign often indicates to students that they needed to perform an operation (e.g., 40 + 8 = ___). Viewing the equal sign in this manner may prevent students from developing essential algebra-readiness skills (Kieran, 1981). Conceptually understanding equality prepares students to properly manipulate equations.

In addition, being able to flexibly apply number properties enables students to efficiently solve contextualized or unfamiliar problems and apply and understand mathematical algorithms (Geary et al., 2008). Decomposing and composing numbers, which relies heavily on number properties, enables students to work flexibly with numbers during procedural computations, understand the mechanisms behind mathematical algorithms, and generalize their understandings to abstract situations. Furthermore, the ability to manipulate numbers using these properties lays the foundation for the manipulation of symbols in algebra.

Understanding and Application of Algorithms

To efficiently perform mathematical operations, such as addition, subtraction, multiplication, and division, mathematical algorithms are commonly used. A mathematical algorithm is generally considered to be a computational process or sequence of specific steps that can be used to solve a specific task (Wu, 2010). Although many algorithms, such as the traditional long-division algorithm, contain multiple steps, others are less formal and less complex, such as counting on to add two numbers (Geary et al., 2008). Proficiency in executing algorithms, specifically whole number division, significantly predicts upper elementary and middle school students’ later achievement in high school algebra and mathematics (Siegler et al., 2012).

Understanding why algorithms work and being able to apply algorithms enables students to transfer their mathematical knowledge to new situations and contexts (Siegler et al., 2011). As stated previously, students’ knowledge of basic number properties can serve as an anchor for understanding the conceptual underpinnings of algorithms. As students develop fluency with algorithms, they are able to use their working memory to solve more complex problems instead of having to focus on procedural computations.

Development of these skills, however, is not an intuitive process and, as recent achievement data suggest, may not be occurring in upper elementary and middle school classrooms. Multiple reasons may explain these results; however, as noted in the NMAP (2008), students need additional instructional support to extend their understanding of whole numbers to other number systems (including rational numbers and integers), solidify their understanding of properties of whole numbers, and generalize their knowledge of operations to rational numbers. For teachers to monitor students’ progress in these concepts and respond with timely and individualized instructional changes, technically adequate assessment systems are needed. The purpose of this article is to document the design of and technical adequacy data for three measures intended to provide teachers with timely data that can be used to help make informed instructional decisions to support students’ development of algebra-readiness skills and knowledge.

Content Review

To determine whether the ARPM measures appropriately represent the critical foundations of algebra, content-related evidence was evaluated. Three mathematics education experts and three mathematics teachers reviewed the test protocols for alignment with mathematical skills and knowledge needed for success in algebra.

Pilot Test Procedures

Participants

Students were recruited at the classroom level from a convenience sample of schools in two states. Schools with which the researchers had established relationships were invited to participate. Participants included 154 Grade 6 students, 173 Grade 7 students, and 250 Grade 8 students from four public middle schools. School 1 was a public charter school located in an urban community in the south; 100% of the mathematics classes participated. School 2 was located in a rural community in the south; 100% of the mathematics classes participated. School 3 was located in a suburban community in the Pacific Northwest; five mathematics classes serving students in Grades 7 and 8 participated. School 4 was a suburban middle school in the Pacific Northwest; four Grade 8 mathematics classes participated.

Demographic information was collected from students’ self-report. Due to the incomplete records we received and unreliable nature of self-report data, school-level demographic data obtained from the states’ department of education websites are presented in Table 1. Because students were recruited at the classroom level, we anticipated that the demographic characteristics of the sample closely matched the demographics of the school; however, it was not possible to verify this assumption. The study was conducted at the end of the spring semester after the state testing windows had closed. During the study year, all schools achieved satisfactory performance ratings in mathematics by the state education agency.

OPEN IN VIEWER

Table 1.

Demographic Characteristics for Participants’ Schools Obtained From the States’ Department of Education.

School	Geography	Location	n	Caucasian (%)	Hispanic (%)	African American (%)	Other (%)	Special education (%)	English learners (%)	Econ disadv (%)	Met stds (%)
1 a	Urban	South	264	—	43	57	0	8	7	58	86
2	Rural	South	307	76	21	1	2	7	2	29	92
3	Suburban	PNW	498	81	12	2	5	19	6	57	83
4	Suburban	PNW	586	71	19	2	8	18	9	61	72

Note. Econ disadv = economically disadvantaged; met stds = met standards in mathematics; PNW = Pacific Northwest.

a

This school serves students in Grades 5 to 8. These data are inclusive of all students.

Measures

Sample items for each measure are included in Figure 1.

OPEN IN VIEWER

Figure 1.

Sample student directions and items for the Algebra Readiness Progress-Monitoring measures in Grade 6.

Number Properties (NP) algebra-readiness measure

The NP measure assesses students’ ability to recognize and use number properties to efficiently solve arithmetic problems. In each of the 54 items included on this measure, students evaluate two expressions and then fill in the blank with the correct symbol (greater than, less than, or equal sign) to make the statement true. Answers are scored dichotomously as correct or incorrect.

The content sampled for the measure is based on the National Council of Teachers of Mathematics (NCTM; 2006) Curriculum Focal Points and is aligned with the CCSS-M. Specifically, the CCSS-M requires that students in Grades 6 to 8 know, apply, and use number properties to generate equivalent expressions (e.g., 6.EE.3, 6.NS.4, 7.EE.4, 7.NS.1, 8.NS.1; EE stands for “Expressions and Equations,” and NS stands for “The Number System”). In accordance with these standards, the number properties assessed across Grades 6 to 8 are the commutative, associative, inverse, and identity properties of addition and multiplication as well as the distributive property. As the NCTM Curriculum Focal Points and the CCSS-M for Grades 6 to 8 emphasize students’ facility with whole numbers, rational numbers, and integers, the NP measure also emphasizes these number systems.

Quantity Discrimination (QD) algebra-readiness measure

The QD measure assesses students’ ability to quickly recognize magnitude differences within and between grade-level-appropriate number systems. In each of the 54 items included on this measure, students evaluate two numbers and circle the number that is greater. Items are forced choice between two options and scored dichotomously.

Pertinent content for this measure was cross-referenced using the NCTM Curriculum Focal Points and the CCSS-M for Grades 6 to 8. Students in these grades should be able to compare, order, and determine equivalence among numbers in a variety of number systems (e.g., 6.NS.6, 6.NS.7, 7.NS.1, 8.NS.2). This measure tests students conceptual knowledge about magnitude within and among number systems, including whole numbers, fractions, and decimals at Grade 6; integers and rational numbers at Grade 7; and whole, rational, and irrational numbers, and negative numbers across all number systems at Grade 8.

Proportional Reasoning (PR) algebra-readiness measure

Students’ ability to read, order, and compare proportions is assessed using grade-level-appropriate representations in the PR measure. In each of the 54 items included on this measure, students evaluate the proportion and select the symbol (greater than, less than, or equal sign) that makes the statement true. Items are scored dichotomously.

In accordance with the NCTM Curriculum Focal Points and the CCSS-M, students must be able to extend their work with ratio to recognize and solve proportions (e.g., 6.RP.1, 7.RP.1, 8.EE.5; RP stands for “Ratios and Proportional Relationships”). In this measure, students compare proportions represented in percent notation (percent of 100 or percent of a whole) or number notation expressed as a part of a whole (e.g., 8 out of 32) or a fraction of a whole (e.g., 3/4 of 16).

In addition to being aligned with content standards from the NCTM Curriculum Focal Points and the CCSS-M, all three of the measures incorporate Standards for Mathematical Practice from the CCSS-M. More specifically, these measures require that students “Make sense of problems and persevere in solving them,” “Reason abstractly and quantitatively,” and “Look for and make use of structure.”

Content Representation of ARPM Measures

We asked mathematics education experts and middle school mathematics teachers to thoroughly review the measures to determine whether the assessed skills and knowledge adequately represent the critical foundations of algebra. The mathematics education experts rated the PR and QD measures as extremely accurate, both mathematically (M = 4.0, SD = 0) and symbolically (M = 4.0, SD = 0). One reviewer rated one NP form as mostly accurate because of a formatting issue; all other NP forms were rated as extremely accurate overall. The average rating for mathematical accuracy of NP was 3.9 (SD = 0.3) and of symbolic notation was 4.0 (SD = 0).

On most dimensions, the middle school mathematics teachers rated the measures as mostly to extremely appropriate for students in the assigned grade level. The average rating for instructional relevance of the content across the measures and grades was 3.2 (SD = 0.8) on a 4-point scale. The NP measure received the highest rating for instructional relevance across grades (M = 3.4, SD = 0.7). PR had the lowest rating for instructional relevance with an average score of 3.0 (SD = 0.9); teachers perceived proportional reasoning as less instructionally relevant for students in Grade 8 as compared with Grades 6 to 7. The average rating for sensitivity to growth was 3.7 (SD = 0.7). The majority of the comments regarding the content were due to variations in state content standards (i.e., Texas Essential Knowledge and Skills) as compared with the NCTM Curriculum Focal Points and the CCSS-M. Teachers also noted that grade-level content could be added to the measures, such as negative rational numbers on the Grade 7 QD measure and irrational numbers on the Grade 8 NP measure. Teachers provided feedback regarding the test layout and format, requesting additional space between items. One teacher noted that there should be an equal distribution of correct responses as greater than, less than, and equal to.

Technical Adequacy of ARPM Measures

Pilot test data were analyzed using the Rasch model with Winsteps statistical software (Linacre, 2013). During the calibration process, items across parallel forms of the same measure were equated using a concurrent equating method (Hanson & Beguin, 2002). Common anchor items were used to link the forms so that all items within an ARPM measure were calibrated to the same scale. Using this equating design, subsequent performance on a subset of items can be meaningfully compared with performance on another subset of items.

Prior to examining item difficulty parameters and goodness-of-fit statistics, model assumptions were evaluated. The Rasch model assumes that students’ responses to one item are not dependent on their responses to another and that items represent a unidimensional construct (Hambleton et al., 1991). Principal Components Analyses (PCA) of the residual score matrix were planned as a common analytic approach to evaluate dimensionality for Rasch analyses. Before conducting PCA for each measure, Bartlett’s Sphericity test and Kaiser–Meyer–Olkin (KMO) index were evaluated to determine whether the data were appropriate for PCA. Bartlett’s Sphericity test evaluates whether the residual correlation matrix diverges from the identity matrix, and the KMO measure of sampling adequacy is used to evaluate the magnitude of the partial correlations among variables. For all ARPM measures, the Bartlett’s Sphericity test and KMO index indicated that the residual matrices were not suitable for modeling PCA. These results may be due to the sparse data matrices or the large number of items per measure. In the absence of a test for dimensionality, we cannot conclude that the ARPM measures are unidimensional. Noteworthy as an indication of unidimensionality, the mean correlations among items in the residual matrices for the ARPM measures were approximately zero, indicating that the Rasch model had accounted for most of the covariance in the items (Chou & Wang, 2010).

Item difficulty is plotted on the logit scale with typical values ranging from −3.0 to +3.0. The range of item difficulty parameters for the ARPM measures was evaluated to determine whether the items represent a broad spectrum of algebraic reasoning abilities. Table 2 presents the item statistics for each measure. Although Rasch analyses using Winsteps constrains the mean to zero, the standard deviation indicates a moderate range of item difficulty estimates. Levels of skewness and kurtosis were within acceptable limits (<±1). For all grades, the NP measures had the greatest range in item difficulty and include the most difficult items. The QD measures had the narrowest range with the least difficult items. The PR measures had a moderate range of item difficulty.

OPEN IN VIEWER

Table 2.

Item Statistics for Each ARPM Measure by Grade.

	Measure
Statistic	NP			QD			PR
Grade	6	7	8	6	7	8	6	7	8
Average n (SD)	112 (9)	158 (6)	232 (4)	109 (9)	155 (5)	223 (6)	110 (9)	155 (6)	227 (5)
Cronbach’s alpha	.97	.94	.93	.95	.95	.93	.95	.93	.92
Mean item difficulty (SD)	0.00 (1.16)	0.00 (1.00)	0.00 (1.26)	0.00 (0.74)	0.00 (0.77)	0.00 (0.78)	0.00 (1.37)	0.00 (0.94)	0.00 (0.89)
Max item difficulty	3.80	2.41	3.22	1.59	1.46	1.77	2.42	1.82	1.80
Min item difficulty	−2.22	−2.32	−1.95	−1.57	−2.16	−1.42	−2.33	−1.75	−2.13
Average mean-square infit (SD)	0.98 (0.13)	0.99 (0.12)	0.98 (0.11)	1.00 (0.12)	1.00 (0.08)	1.00 (0.06)	0.95 (0.20)	1.00 (0.21)	0.99 (0.29)
Average mean-square outfit (SD)	0.95 (0.41)	0.96 (0.40)	0.98 (0.30)	0.96 (0.42)	1.01 (0.29)	0.98 (0.17)	0.97 (0.46)	0.97 (0.36)	0.99 (0.41)
Skewness	0.07	−0.12	0.02	0.77	0.80	0.80	−0.35	1.03	0.47
Kurtosis	−0.07	−0.10	0.21	0.25	0.40	0.62	0.46	0.83	0.81
Mean person ability estimate (SD)	1.76 (1.05)	1.83 (1.04)	1.40 (1.09)	1.40 (1.04)	1.40 (1.04)	1.49 (0.96)	1.32 (1.02)	1.14 (1.11)	0.49 (1.05)
Skewness	1.01	0.07	0.64	−0.03	−0.38	0.11	−0.07	−0.19	−0.47
Kurtosis	0.93	−0.45	−0.47	−0.54	−0.30	−0.70	−1.42	−1.25	−0.40

Note. ARPM = Algebra Readiness Progress Monitoring; NP = Number Properties; QD = Quantity Discrimination; PR = Proportional Reasoning.

The Test Information Function (TIF) describes how precise the ability estimates can be measured at various points along the ability scale (Embretson & Reise, 2000) and is conveniently represented graphically. The peak of the curve indicates the point on the distribution at which ability estimates are most precise. For the ARPM measures, the shape of the TIFs indicate that the ARPM measures provided the most precise information between −2.0 and +2.0 (see Figure 2 for the TIFs for each grade by ARPM measure). Ability estimates at the extremes of the scale were less precisely estimated with the ARPM measures.

OPEN IN VIEWER

Figure 2.

Test information functions for each ARPM measure by grade.

The mean person ability estimates were greater than the mean item difficulty estimates and ranged from +0.49 to +1.83 (see Table 2). These data indicate that the average ability of the participants was high, possibly due to the administration of the ARPM measures at the end of the academic year. The distributions of the person ability estimates were approximately normal, with levels of skewness and kurtosis with acceptable limits (<±1) for all but the PR measures in Grades 7 and 8.

Goodness of fit with the Rasch model indicates the level of correspondence between the item difficulty parameter and the expected student performance. Model fit was determined by examining the Mean Squares Infit and Outfit statistics. Infit estimates the correspondence that is proximal to the item difficulty estimate. Outfit examines correspondence that is distal to the item difficulty estimate. To be productive for measurement, mean square values should range from 0.5 to 1.5; mean square values near 1.0 indicate little distortion of the measurement system (Linacre, 2002). The average Mean Square Infit and Outfit statistics for each ARPM measure were near 1.0. Seventeen percent (n = 82) of the total number of items were classified as misfitting across the nine ARPM measures (see Table 3). Additional research is needed to understand why these items were misfitting; however, they may have been susceptible to error associated with guessing behavior.

OPEN IN VIEWER

Table 3.

Excluded Items for Each ARPM Measure by Grade.

Measure	Grade	Total number of items	Number of misfitting items	Number of items with unacceptable correlation	Total number of items omitted (%)
NP	6	54	9	2 Unique (1 shared)	11 (20.4)
	7	54	8	0 Unique (4 shared)	8 (14.8)
	8	54	7	0 Unique (1 shared)	7 (13.0)
QD	6	54	9	0 Unique (4 shared)	9 (16.7)
	7	54	6	1 Unique (3 shared)	7 (13.0)
	8	54	0	3 Unique (0 shared)	3 (5.6)
PR	6	54	15	3 Unique (6 shared)	18 (33.3)
	7	54	6	4 Unique (5 shared)	10 (18.5)
	8	54	9	0 Unique (9 shared)	9 (16.7)

Note. ARPM = Algebra Readiness Progress Monitoring; NP = Number Properties; QD = Quantity Discrimination; PR = Proportional Reasoning.

Internal consistency reliability was evaluated using Cronbach’s alpha (see Table 2). Coefficients ranged from .92 to .97. Because low item-total correlations can affect internal consistency (Tan, 2009), 46 items with item-total correlations below .20 were excluded. Seventy-two percent (n = 33) of the items with unacceptable item-total correlations were also classified as misfitting. The numbers of items with unacceptable correlations are reported for each ARPM measure in Table 3.

Correlation Across Measures

To evaluate the relationship between the ARPM measures, we correlated the students’ ability estimates across the three ARPM measures within grade. Correlations ranged from r = .43 to .63. Table 4 displays the correlation matrix.

OPEN IN VIEWER

Table 4.

Correlation Matrix for the ARPM Measures by Grade for Convergent Evidence for Validity.

	Grade 6			Grade 7			Grade 8
Measure	1	2	3	1	2	3	1	2	3
1. NP	–	.52	.43	–	.61	.57	–	.49	.45
2. QD	.52	–	.50	.61	–	.63	.49	–	.53
3. PR	.43	.50	–	.57	.63	–	.45	.53	–

Note. ARPM = Algebra Readiness Progress Monitoring; NP = Number Properties; QD = Quantity Discrimination; PR = Proportional Reasoning.

Assessing Algebra-Readiness Content

The ARPM measures focus on critical foundations of algebra, including conceptual understanding of rational numbers, facility with and the ability to reason using proportions, and application of number properties. These skills align with recommendations established by the NMAP (2008) for supporting students’ algebra readiness. Competence with these concepts and skills provides students with foundational knowledge for abstract reasoning and lawfully manipulating and operating on numbers and symbols, thereby establishing the foundation for algebra and advanced mathematics (Milgram, 2005). However, students who are not demonstrating growth in these skills may be at risk for not being successful in algebra.

Designed to serve as a robust indicator, the ARPM measures assess content that is not directly taught in the curriculum but is broadly representative of the skills and knowledge students need to be successful in algebra. Feedback from mathematics education experts and teachers provides initial content-related evidence for the ARPM measures. The experts rated the measures as extremely accurate in their representation of critical foundations of algebra. Middle school mathematics teachers identified the measures as mostly instructionally relevant. PR in Grade 8 had the lowest rating for instructional relevance. One possible explanation for this finding is the shift in the CCSS-M in Grade 8 to graphing proportional relationships and understanding the linear equations. Although not directly assessed in the ARPM measures, students’ ability to graph and recognize linear equations presupposes proficiency with understanding proportions. Overall, in considering the utility of the measures for monitoring growth, the expert reviewers overwhelmingly rated the measures as extremely useful.

Adequately Measuring Algebra Readiness

To provide instructionally relevant and useful data that is sensitive to growth in student learning, specific technical requirements must be met. Items should adequately and reliably capture students’ knowledge and skills in relevant content. Measures should differentiate between high and low performers (Foegen, 2008). Small changes in student learning should be discernable over time (Kelley et al., 2008).

Technical adequacy data gathered for the ARPM measures indicate that these measures may be feasible for providing teachers with meaningful data about students’ development of algebra-readiness knowledge and skills in Grades 6 to 8. Rasch model fit statistics and item-total correlations indicate that a majority of the items across the ARPM measures performed as predicted by the IRT model and met the criteria needed to be productive for measurement. Strong Cronbach’s alpha coefficients indicate acceptable internal consistency reliability estimates. Item statistics for the ARPM measures indicate that the item difficulty parameters were broadly distributed across the ability scale yet are sufficiently clustered between −2.0 and +2.0 to provide precise estimates of students’ abilities.

For all ARPM measures, the dispersion of the ability estimates were approximately normal and were centered above +1.0 on the logit scale. These findings indicate that the average ability of the students was greater than the average difficulty of the items. Because the pilot study was conducted at the end of the year with intact classrooms and participants were allowed an extended period of time to complete the ARPM measures, these findings are expected and desired. Had the distribution of the person ability estimates during an untimed administration been centered at zero, implementation of timed measures would likely result in a floor effect (clustering of scores at the lower end of the distribution), thereby limiting the utility of the measures for monitoring growth.

Contributing to Algebra Readiness Skills and Knowledge

CBMs that are designed as robust indicators assess aspects of the targeted domain that are independent of the annual curriculum. The ARPM measures were designed as robust indicators of students’ algebra readiness; together, the content measures students’ ability to synthesize and integrate their mathematical knowledge and skills to demonstrate their progress in thinking algebraically and understanding complex mathematical processes. Moderate correlations between the ARPM measures indicate that student performance across the ARPM measures is positively related. Additional evidence is needed to support the claim that the ARPM measures serve as a robust indicator of algebra-readiness knowledge and skills.

Future Directions and Limitations

The present studies provide important yet incomplete data about the utility of the ARPM measures for providing teachers with meaningful data about middle school students’ algebra readiness. In this article, we provide theoretical evidence about the components of the assessed construct, as well as content-related evidence documenting the sufficiency by which we instantiated this construct. Moreover, we provided initial technical adequacy data supporting the feasibility of the ARPM measures in assessing the construct. However, additional data are needed to justify implementation of the ARPM measures for making instructional decisions.

To support teachers’ use of the ARPM measures to inform instruction, data from the pilot study were used to create three parallel forms of each measure. Using the item statistics obtained from the Rasch analyses, forms were created to include items with comparable difficulty parameters. Additional items are needed to create multiple forms to provide progress-monitoring data. Once these forms are created, studies will be necessary to examine the reliability of the slope, acceptable rates of improvement, and the sensitivity of the measures to monitoring growth. Empirical data are also needed from studies examining the relation between results on the ARPM measures and other criterion measures. Specifically, concurrent-related evidence is needed to evaluate whether results on the ARPM measures correlate with results on other measures of algebra readiness. Although there is a dearth of technically adequate measures of algebra readiness, available measures such as the Iowa Algebra Readiness Assessment (Schoen & Ansley, 2006), which serves as a placement measure, may be suitable. Importantly, predictive-related evidence is needed to determine whether results on the ARPM measures are predictive of students’ future performance in algebra.

Once sufficient alternate forms of the ARPM measures are available and properly scaled using the Rasch model, scale scores can be generated from students’ performance on each measure (Embretson & Reise, 2000). Because the ARPM measures are intended to serve as a robust indicator of algebra readiness, scores from the three ARPM measures should not be disaggregated. Instead, a composite score will be generated to represent the combined knowledge and skills students need to be successful in algebra. Data can be plotted over time to evaluate students’ progress toward proficiency in algebra-readiness knowledge and skills. Similar to other progress-monitoring systems, teachers can use these data to guide their instructional decisions such as adjusting their instructional practices if students are not making sufficient progress or increasing students’ instructional goals if they are making adequate growth.

Several limitations affect the observed findings. First, limited demographic information was available on the participants. Although accurate estimates of item difficulty parameters are obtainable through IRT with any reason-able subset sample of the population, additional details about the representativeness of the sample for the ARPM measures are needed. Demographic data would allow further analysis of item functioning for subgroup populations. Specifically, differential item functioning (DIF) analyses are needed to identify potentially biased items. Second, tests for the dimensionality assumption of IRT were not possible given the structure of the data set. Although evidence suggests that the Rasch model accounted for most of the covariance in the items, additional analyses are needed to verify essential unidimensionality with subsequent collections of data. Third, and finally, item statistics were analyzed from an untimed pilot administration of the ARPM measures. Using mixture IRT models may be useful for examining the impact of timed administration on item statistics (Bolt et al., 2002).

Implications for Practice

The results of this research have several key implications for practitioners working in middle school settings. First, based on the theoretical assertions about indicators of algebra readiness as well as empirical data collected from a rigorous content review and pilot test, the ARPM measures are feasible and adequately assess foundational skills needed for success in algebra. Although further development is needed, practitioners could use these measures to obtain meaningful and trustworthy information about students’ proficiency in algebra-readiness knowledge and skills. Second, further specifying key components of the mathematics curriculum that may support students’ readiness for algebra provides educators with additional information to consider when prioritizing the content of their instruction. Students’ conceptual understanding of rational numbers, facility with and the ability to reason using proportions, and application of number properties are associated with critical foundations of algebra and may serve as key levers within the middle school curriculum to improve performance. Additional research is needed to evaluate the relationship between these content components and subsequent success in algebra; however, practitioners may consider incorporating these concepts in their instruction. Third, and finally, with further development of an integrated assessment delivery system, data generated from the ARPM measures may help teachers make timely and informed instructional decisions. Data can be used to identify students who need varying levels of instructional support to be ready for algebra and monitor their response to differentiated instructional opportunities. As such, these measures may be implemented within a systems-level initiative designed to improve students’ preparedness for algebra.