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IES Grant

Title: Arithmetical and Cognitive Antecedents and Concomitants of Algebraic Skill
Center: NCER Year: 2011
Principal Investigator: Cirino, Paul Awardee: University of Houston
Program: Cognition and Student Learning      [Program Details]
Award Period: 2011–2015 Award Amount: $1,468,996
Type: Exploration Award Number: R305A110067

Co-Principal Investigators: Tammy Tolar (University of Houston), Lynn Fuchs (Vanderbilt University)

Purpose: Algebra is a prerequisite for access in science, technology, engineering, and mathematics. Yet many high school students do not take or are unable to complete a course in algebra, and few high school students are proficient at multi-step problem solving and algebra. In order to provide a basis for improving students' preparation for this essential skill, it is important that we identify the arithmetical or other cognitive factors that either limit or facilitate algebraic problem solving. However, there is limited empirical work that quantifies skills and abilities relevant for algebra. For example, the team will explore if algebraic proficiency comprises both procedural skill and conceptual knowledge. Each of these skills are separate and measureable, and although they are related, these types of abilities have overlapping and different sets of immediate and earlier predictors. The researchers will address this gap by conducting a series of studies that explore students' conceptual and procedural knowledge of algebra and related concepts.

Project Activities: The researchers will conduct a series of experimental studies to determine the effects of learning procedural and conceptual aspects of algebra in varying orders (e.g., concurrently, or being taught procedural skills before conceptual knowledge). They will also conduct descriptive studies that explore factors such as the effects of arithmetic knowledge and working memory on algebraic skill. Additionally, they will use data collected from a previous grant to conduct a predictive longitudinal study. One set of data to be used in this study was collected while the students were in 3rd, 4th, and 5th grade. This project will collect additional data from these students when they enroll in first year algebra during 8th or 9th grade and will explore the degree to which the mathematical knowledge and skills measured in 3rd through 5th grade predicts their algebra performance.

Products: The proposed study will provide empirical evidence as to the specific knowledge and skills that are predictive of algebra achievement and address how early such predictors are evident (e.g., whether high school algebra achievement can be predicted from 5th grade performance). The researchers will also publish scholarly reports of findings.

Structured Abstract

Setting: The research will be conducted in both laboratory settings and public schools in the Houston, Texas and the Nashville, Tennessee areas.

Population: Primary data will be collected from 8th and 9th graders in first-year algebra courses. The longitudinal data is comprised of data collected from the Nashville cohorts when they were in 3rd, 4th, and 5th grade, which will be linked to the students' performance in their 8th or 9th grade first-year algebra courses.

Research Design and Methods: During all 4 years, the researchers will conduct descriptive studies. In these studies, they will administer numerous tests to measure students' math skills and other cognitive skills (e.g., working memory capacity). Tests relating to algebraic ability will be administered within the first week of a student's first-year algebra class. The remaining tests, such as those pertaining to arithmetic ability and working memory capacity, will be administered individually and outside of class by the third week of classes. The data from these studies will be linked to available longitudinal data collected under a previous grant. This data tracked the students over their 3rd, 4th, and 5th grade years. Additionally, students in the 6th grade will be recruited during the first year to create another longitudinal cohort. These students will be tested when they begin their first-year algebra courses when they reach 8th or 9th grade.

During the second and third year of the grant, the researchers will conduct two experiments with 6th grade students in Houston. Students will receive algebra instruction in one of three sequences over two different sessions, each session being comprised of two days. They will receive either procedural instruction only and then conceptual instruction, conceptual instruction only and then procedural, or a mixture of each in both sessions. Pre- and post-test measures will be taken to see if there are differences in gains depending on the order in which procedural and conceptual knowledge about algebra are taught.

Control Condition: In experiments 1 and 2, the students who receive blended instruction (i.e., both procedural and conceptual instruction concurrently over the two sessions) will serve as the control condition for the other instruction conditions( i.e., procedural then conceptual and conceptual then procedural).

Key Measures: The key measures include assessments designed by the research team to assess procedural and conceptual algebraic knowledge along with the following pre-existing measures: the Woodcock Johnson III (Math Fluency, Oral Comprehension, and Visual-Auditory Learning), the Number Facility factor in the Educational Testing Service Kit of Factor-Referenced Cognitive Tests (Subtraction and Multiplication Test), Double Digit Mixed Estimation Fluency Test, Fraction Competency Test, Diagnostic Assessment of Proportional Reasoning, Number Line Estimation, the Test of Memory and Learning-2 (TOMAL-2), Digits Backward subtest, the Letters Backwards Automated Symmetry Span, and the Revised Vandenberg & Kuse Mental Rotations Test.

Data Analytic Strategy: Data from the experimental studies will be analyzed using repeated measure analyses of variance. The data from the descriptive studies will be analyzed using several methods, including confirmatory factor analysis, diagnostic classification modeling, item response theory modeling, hierarchical regression, structural equation modeling or path modeling, latent class analysis , and growth curve modeling.


Journal article, monograph, or newsletter

Cirino, P.T., Tolar, T.D., Fuchs, L.S., and Huston-Warren, E (2016). Cognitive and Numerosity Predictors of Mathematical Skills in Middle School. Journal of Experimental Child Psychology, 145: 95–119.

Fuchs, L.S., Gilbert, J.K., Powell, S.R., Cirino, P.T., Fuchs, D., Hamlett, C.L., Seethaler, P.M., and Tolar, T.D. (2016). The Role of Cognitive Processes, Foundational Math Skill, and Calculation Accuracy and Fluency in Word-Problem Solving versus Prealgebraic Knowledge. Developmental Psychology, 52(12): 2085–2098.