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Overviews of Research – Mathematics

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• Research on Grades 3 to 5 Teaching and Learning
• Research on Grades 6 to 8 Teaching and Learning
• Research on Teaching with Problem-Based Instructional Tasks    
• Research on NCTM-Standards-Based (NSF-Funded) Curricula, K-12    
• Research on Teaching and Learning Fractions

Iowa Content Network

 

Overview – Research on Pre-Kindergarten through Grade 2 Teaching and Learning

It is essential that students have a strong foundation in mathematical content and the development of mathematical processes. Research results related to improving students’ achievement in mathematics in prekindergarten through grade 2 may be summarized into three categories: curriculum materials (textbooks), supplementary programs, and teaching strategies.

A set of curriculum materials that has been found to be effective is Everyday Mathematics (Fuson, 2000). An integral component of Everyday Mathematics is the focus on problem-based instruction. Other problem-based or problem-centered mathematics programs have indicated positive results in achievement (Wood & Sellers, 1997; Hiebert & Wearne, 1993).

A supplementary program that indicated positive results is Cognitively Guided Instruction (CGI) (Jacobson & Lehrer, 2000;Carpenter et al, 1999; Villasenor & Kepner, 1993; Fennema & Carpenter, 1996). Another supplementary program that indicated significant improvements in achievement is the “Explicit Schema-Based Strategy” (Jitendra et al, 1998). These programs include a training component for teachers on teaching strategies for helping students solve word problems.

Teaching strategies with positive results include the use of peer-mediated instruction (Fuchs et al, 1997) and peer-assisted learning (Fuchs et al, 1995). The emphasis was on students working with each other. Use of manipulatives is another teaching strategy showing improved student achievement; these strategies included using the empty number line (Klein et al, 1998), base-ten blocks (Fuson & Briars, 1990), and other manipulatives (Burton, 1992). Using self-generated drawings for solving word problems was yet another effective teaching strategy (van Essen et al, 1990).

In addition, a longitudinal study indicated that small class sizes in grades K- 3 made a difference in student achievement gains that persisted through grade nine (Nye et al, 2001).

The research studies supporting this overview are found in the following table. (Go to mathematics_k2.htm to examine a table of studies.)

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Overview – Research on Grades 3 to 5 Teaching and Learning

Based on Individual Studies Reviewed by the Iowa Content Network and Prominent Published Reviews of Collections of Research

Nearly three-fourths of U.S. fourth graders report liking mathematics. To maintain their enthusiasm, the mathematics in grades 3-5 must be interesting and understandable with instruction designed so the students are actively engaged in making sense of mathematics. Three central mathematical themes for grades 3-5 are identified in the Principles and Standards for School Mathematics. These themes–multiplicative reasoning, equivalence, and computational fluency–are interwoven across content areas (NCTM, 2000).

A number of studies evaluated the content strands in the grade 3-5 curriculum (algebra, data analysis and probability, geometry, measurement, and number and operations). These studies compared NCTM-standards-based to traditional curriculum; in all cases the students in the standards-based curriculum outperformed the students studying the traditional curriculum. (See NCTM-Standards-based Overview). In two studies, greater achievement was reported using a standards-based curriculum with Accelerated Mathematics rather than implementing just the standards-based curriculum (Ysseldyke et al. 2003, Spicuzza et al. 2001).

A longitudinal study indicated that small class sizes in grades K-3 made a difference in student achievement gains that persisted through grade nine (Nye et al. 2001).

The number of studies that involved only algebra, data analysis and probability, geometry, and measurement were limited. There were a large number of studies that involved evaluation of number and operations; some of these studies involved general teaching strategies that may or may not be transferable to other topics.

Algebra, Data Analysis, and Probability   
There were no separate studies related to algebra, data analysis, and probability.

Geometry and Measurement
In addition to the studies reported above, one study (Carroll, 1998) reported a growth in students’ geometric reasoning by evaluating Van Hiele levels as a result of implementing Everyday Math in contrast to a traditional curriculum.

Reasoning with respect to one-, two- and three-dimensions adds to the complexities of developing proficiency for linear, area and volume measurements. The development of conceptual understandings and procedural competence in measurement are closely related and need to be thoughtfully integrated in instruction (Lehrer 2003; Kilpatrick et al. 2001). In one study students who received conceptual instruction for 3 days did just as well as students who had the same instruction plus an additional 5 days of rote learning for area and perimeter topics (Pesek et al. 2000). A class culture of inquiry, problem solving, and sense making was effective in helping students develop volume concepts through a layering strategy (Battista 1999).

Number and Operations
Multiplicative reasoning, identified in the Principles and Standards for School Mathematics as a central theme for grades 3-5, develops slowly (Clark et al. 1996). As students are provided more experiences, they develop more sophisticated strategies for unitizing and partitioning, part of multiplicative reasoning (Lamon 1996).

Studies involving basic facts and whole number algorithms showed no differences between standards-based curricula and traditional curricula in regular or inclusion classrooms. In studies that evaluated word problems, more complex computations, and student explanations, results favored the standards-based groups (Wood et al. 2003; Mokros, 2003). For the development of rational numbers, results for average and lower ability students favored curricula that were based on conceptual understanding, including multiple representations, benchmarks, and connections (Cramer, et al. 2002; Moss, 1999; Wearne 1990).

Positive results with respect to greater accuracy, more efficient strategies, better conceptual understanding, and higher scores for standardized achievement tests were also found for problem-centered curricula or curriculum based on real world problems as compared to traditional textbooks (Anghileri et al. 2002; Wood et al. 1997). (Also, see overview of research for using problem-based instructional tasks.)

Teaching explicit strategies to solve word problems or to develop more sophisticated thinking strategies resulted in positive differences for average and low ability students (Fuchs et al. 2003; Darch et al. 2001; Hohn et al. 2002; Pogrow 1995).

Other effective teaching strategies include promoting more student responsibility for learning by having students set and regulate their learning goals and having them work in pairs (Fuchs et al. 1997; Fuchs et al. 1995; Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, Hosp, and Jancek 2003; Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, and Schroeder 2003).

Professional development makes a difference. Student test scores improved significantly when teachers participated in staff development while implementing standards-based curricula in contrast to no staff development during implementation (Battistich et al. 2003). Gains in student achievement also appeared to be directly related to changes in teacher instruction and beliefs as a result of Cognitively Guided Instruction (CGI), a teacher development program (Fennema et al. 1996).

The research supporting this overview falls into two categories – prominent published reviews of collections of research, and individual studies. These two categories are presented below.

1. Prominent Published Reviews of Collections of Research These nationally-recognized reviews of substantial bodies of research have been written by experts in the field and reviewed by peers.

2. Individual Studies (Go to mathematics_35.htm to examine a table of studies.)

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Overview – Research on Grades 6 to 8 Teaching and Learning

Based on Individual Studies Reviewed by the Iowa Content Network and Prominent Published Reviews of Collections of Research

The traditional U. S. middle grades mathematics curriculum has been limited in depth and breadth and the weakest area in both international and domestic comparisons with respect to mathematics proficiency is middle school students’ performance in geometry (NCTM, 2000). In order to strengthen students’ backgrounds and increase their options in life and career opportunities, it is recommended in the Principles and Standards for School Mathematics that a significant amount of both algebra and geometry need to be integrated in the middle grades curriculum for all students (NCTM, 2000).

A number of studies evaluated the content strands in the grade 6-8 curriculum (algebra, data analysis and probability, geometry, measurement, and number and operations); these studies compared NCTM standards-based to traditional curriculum. In all cases the test scores of students in the standards-based curriculum was comparable or exceeded test scores of students studying the traditional curriculum. (See NCTM Standards-based Summary).

Three studies focused on ability grouping. When comparing homogeneous to heterogeneous groupings, low ability or average students were more likely to make gains in heterogeneous classes. Performances of high ability students were not effected. (Hunt 1996; Linchevski et al. 1998; Leonard 2001). These results are consistent with summaries on the effects of ability grouping including data from international studies that support the assumption that all students can learn mathematics and work within heterogeneously grouped classes (Kilpatrick 2001).

Greater student gains were the result of teaching strategies that included presenting problem solving situations in context using videos, organizing instruction around big ideas, teaching explicit strategies, guiding through scaffolding, and including review that requires students to apply their knowledge (Arthurs et al.1998; Bottge, 1999; Grossen 2000). Other factors contributing to student gains included teachers who had acquired a mathematics major; who participated in professional development in developing higher-order thinking strategies, and who implemented teaching strategies such as hands-on learning and higher-order thinking skills (Wenglisky 2002).

The number of studies that involved only algebra, data analysis and probability, geometry, and measurement were limited to nonexistent (although several studies addressed all strands together). There were a larger number of studies that involved evaluation of number and operations.

Algebra
Only one study involved only algebra; no differences in achievement were found between the Saxon Algebra text and conventional textbooks on linear combinations; the control group (conventional textbooks) performed significantly better on one subscale measuring definitions and theory. The experimental group had a significantly more positive attitude in every area except for study habits (Johnson et al. 1987).

Geometry
In addition to the studies reported above, one study (Carroll 1998) reported a growth in students’ geometric reasoning by evaluating van Hiele levels as a result of implementing Everyday Math in contrast to a traditional curriculum. The van Hieles’ theory of levels of geometric reasoning provides a framework for curriculum and teaching and most traditional textbooks do not provide a curriculum that requires students to develop higher levels of geometric reasoning across the grades (Clements 2003).

Number and Operations
Mental computation is not only a practical skill but it also helps students develop number and operation sense; in recent decades this topic has not been included in the U.S. traditional curriculum (Kilpatrick 2001). Mental computation skills were enhanced for groups using the standards-based curriculum, Everyday Math (Carroll 1996). Another study found that mental computation skills improved when representations were presented through a computer environment (Ainsworth et al. 2002).

Research on rational numbers relates student difficulties to weak conceptual understandings of rational numbers and a lack of connecting fractions and decimals. Approaches such as standards-based curriculum that build on students’ informal understandings and incorporate representations and contexts offer more promise than rule-based programs (Kilpatrick 2001). Student achievement on fractions was enhanced when teachers were committed to implementing a reform curriculum in contrast to teachers who had no interest (Stipek et al. 1998). See Summaries on Fractions and Problem-based Instruction.

Another study that focused on student understanding involved proportional reasoning. Students made greater gains in proportional reasoning when studying curriculum where they developed their own concepts and procedures (Ben-Chaim et al. 1998).

With respect to the role of calculators and instruction, the factor in greater achievement in solving problems was learning problem-solving strategies rather than calculator use. Additionally, students who used calculators had more positive attitudes towards math and their computation scores were not significantly different (Szetela et al. 1998).

The research supporting this overview falls into two categories – prominent published reviews of collections of research, and individual studies. These two categories are presented below.

1. Prominent Published Reviews of Collections of Research

These nationally-recognized reviews of substantial bodies of research have been written by experts in the field and reviewed by peers.

2. Individual Studies (Go to mathematics_68.htm to examine a table of studies.)

Overview – Research on Teaching with Problem-Based Instructional Tasks

Based on individual studies reviewed by the Iowa Content Network and prominent published reviews of collections of research 

The use of problem-based instructional tasks as an instructional strategy to increase student achievement in mathematics is strongly supported by research. This strategy is also referred to as teaching through problem solving, using a problem-based approach, or a problem-centered approach. The strategy is implemented by getting students actively engaged in solving, explaining, and reasoning about rich problems focused on important mathematics. This strategy focuses on the teaching paradigm of “understand and apply,” rather than the traditional paradigm of “memorize and practice.”

Research suggests that teaching mathematics through problem solving is both possible and effective. For example: "What do the findings from research suggest about the feasibility and efficacy of teaching mathematics through problem solving? The research reviewed herein suggests both the feasibility and efficacy of of such approaches" (Stein, Boaler, & Silver, 2003, pp.255-256). “Problem solving should be the site in which all of the strands of mathematics proficiency converge” (Kilpatrick, Swafford, & Findell, 2001, p. 421). Strands referred to here are not the content strands, but rather the strands, or types, of mathematical knowledge such as concepts, skills, and problem solving. More precisely, the strands are: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Thus, research suggests that problem solving should be the context in which all these types of mathematical knowledge should be taught.

The research supporting the use of problem-based instructional tasks falls into three categories – (1) prominent published reviews of collections of research, (2) research on problem-based textbooks (such as the NSF-funded, NCTM-Standards-based curricula), and (3) individual studies of problem-solving instructional interventions. These three categories are presented below.

1. Prominent Published Reviews of Collections of Research about Teaching through Problem Solving These nationally-recognized reviews of substantial bodies of research have been written by experts in the field and reviewed by peers.

2. Individual Studies of Problem-Solving Instructional Interventions

The following table contains entries from the main tables on the Content Network site.

Study Author
& Title

Design Rating
Strategy, Subjects, Results
Description of
Strategy/Program

NCTM Math Standards,
Math Topics,
Grades

Study:
Hiebert & Wearne.

Title: Instructional Tasks, Classroom Discourse, and Students' Learning in Second-Grade Arithmetic.

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Strategy: Problem-based instructional tasks and solicited student discourse
Subjects: 135 second-grade students from six classrooms in one rural/suburban school. Students had a mean achievement level slightly above average.
Results: Results showed that working more problems does not lead to higher performance, neither does requiring fewer problems worked mean higher performance. The authors suggest that in mathematics classrooms, certain kinds of instructional tasks and discourse encourage more productive ways of thinking and increase achievement.

Description: Relationships between teaching and learning mathematics were examined in six second-grade classrooms. Teaching was evaluated by examining tasks presented to the students and the nature of the classroom discourse. Students were assessed on place value understanding, routine computation, and novel computation.

NCTM Math Standard:
• Number & Operations

Math Topics: Place value, multi-digit addition and subtraction of whole numbers

Grade: 2nd Grade

Study:
Villasenor & Kepner.

Arithmetic from a problem-solving perspective: An urban implementation.

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 Strategy: Cognitively Guided Instruction - GCI
Subjects: Experimental and control groups comprised of 144 first grade students from 11 public schools and one private school; 57-99% minority children.
Results: The experimental classroom students used advanced strategies in solving word problems significantly more than the control group. The experimental classroom students scored significantly higher on all three post-tests than the control group. Effect Size = +6.63 in favor of CGI.
Description: CGI is a staff development program that builds on the knowledge that students already have and helps them analyze their own thinking. Teachers encourage their students to build on their natural problem-solving strategies by encouraging them to listen to each other, ask questions, and explain how they solve problems.

NCTM Math Standards
• Number & Operations

Topic:
• Arithmetic word problems

1st Grade

Study:
Wood & Sellers.

Deepening the analysis: longitudinal assessment of a problem-centered mathematics program.

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Strategy:  Longitudinal assessment of a problem-centered mathematics program.
Subjects: Students in grades two, three, and four.
School district has a predominantly white student population although the background of the students consists of a wide range of socioeconomic backgrounds.
Results:  After 2 years in problem-centered classes, students have significantly higher achievement on standardized achievement measures, better conceptual understanding, and more task-oriented beliefs for learning mathematics than those in textbook instruction. These results remain even after the problem-centered students spend a year in a textbook-based program.

Description: Based on data from earlier studies, this experiment examined cognitive models that guide student activities.  In addition to instructional activities, the classroom setting included pair interactions and total group interactions. The study made a longitudinal analysis of arithmetical achievement of children in a problem-centered mathematics program as opposed to students in a traditional textbook-based program.

NCTM Math Standards:
• Number & Operations

Topics: Computation

2nd to 4th Grade

Study:
Jones, Thornton, Putt, Hill, Mogill, Rich, VanZoest.

Multi-digit number sense: A framework for instruction and assessment.

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Strategy: A Framework for Multi-Digit Number Sense Using Four Constructs: Counting, Partitioning, Grouping and Number Relationships
Subjects: Validation Study: Six randomly selected students in 1st and six in 2nd grade.
Implementation Study: 40 first grade students. Students were placed in two groups. There was no control group.
Results: Performances of the students were not significantly different for the two groups in the implementation study.

Description: Teachers were encouraged to: (1) use the multi-digit number sense framework to assess and build on students' understanding, (2) present challenging problems to the students, (3) guide students to construct their own solutions to the problems, (4) maximize opportunities for pairs of students to engage in collaborative problem solving, and (5) encourage students to negotiate one or more suitable solutions to the problems.

NCTM Math Standards:
• Number & Operations

Topic:
• Number sense

1st and 2nd Grades

Study:
Wenglisky.

How schools matter: The link between teacher classroom practices and student academic performance.

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Strategy: This study links teacher classroom practices and achievement on the NAEP eighth grade mathematics assessment.
Subjects: 7,146 8th grade students who took the 1996 NAEP Mathematics Assessment
Results: The study found a positive relationship between student achievement on NAEP and teachers with a major in mathematics, professional development in higher-order thinking skills and diversity, using hands-on learning and higher-order thinking skills in the classroom. The influence of these teacher qualities was greater than the influence of SES as a student characteristic on students taking the NAEP assessment.

Description: Using various classroom practices such as writing about math, solving real world problems, having students work with objects, having students talk about math, working in small groups, etc.

NCTM math standards:
• Algebra
• Geometry
• Data Analysis &
  Probability
• Measurement
• Number &   Operations

Math topics:
All topics

8th Grade

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3. Research on Problem-Based Textbooks (such as the NSF-funded, NCTM-Standards-Based curricula):

Study Author
& Title

Design Rating
Strategy, Subjects, Results
Description of
Strategy/Program

NCTM Math Standards,
Math Topics,
Grades

Study:
Carter et al.

Student learning and achievement with Math trailblazers.

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 Strategy: Math Trailblazers
Subjects: Third grade classrooms from eight Chicago-area schools. Six schools were Chicago public schools and two schools were located in a middle-class suburb of Chicago. The city schools were comprised of 98% to 100% minority students. Sixty-one to eighty-one per cent of the subjects were low income students. In the suburban schools 12% were minority students, predominantly Hispanic, and approximately 13% were low income.
Results: After first year of implementation, 6 out of 8 Math Trailblazer schools had a higher percentage of students meeting or exceeding the state goals than their historical averages on the IGAP. By the end of the second year, all 8 schools were at levels above their historical averages regardless of the initial level of student achievement. 		Description: Math Trailblazers includes the following mathematical strands: number and operations, including estimation; geometry and spatial sense; measurement; data analysis, statistics, and probability; fractions and decimals; and patterns, functions, and algebra. Problem-solving contexts support student learning in all these areas. A distinctive feature of the curriculum is the use of Teaching Integrated Mathematics and Science (TIMS) Project Laboratory Investigations that involve the use of a scientific method to study classification, length, area, volume, and mass in all grades. Speed and density are also studied in the fifth grade.

NCTM Math Standards:
• Algebra
• Geometry
• Data Analysis &
Probability
• Measurement
• Number & Operations

Topics:
All

3rd Grade

Study:
Carroll and Isaacs.

Achievement of students using the University of Chicago School Mathematics Project's Everyday mathematics.

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Strategy:
Everyday Mathematics?

Subjects:
Longitudinal study (from first grade to fifth grade) started with 496 first-grade students in five school districts.
1,885 third graders from suburban Chicago
78 fifth grade students in four classes;
246 sixth grade students in six classes;
166 fifth grade students in six districts, 4 in Illinois, 1 each in Pennsylvania and Minnesota (three districts were suburban, two rural and one urban);
Longitudinal study (from first grade to fifth grade) started with 496 first-grade students in five school districts.

Results:
Overall, Everyday Mathematics students had a mean score of 47% correct compared with 24% correct for control group. Everyday Mathematics students' mean score on the Comprehensive Testing Program was at the 94th percentile compared to the control group at the 85th percentile.

Description:
In kindergarten EM the focus is on manipulative activities that form a basis of understanding symbols. In grades 1-3 the concepts learned in previous grades are extended with special attention on mental and symbolic arithmetic, measurement, geometry, data (collection and use), and algebra. A strong emphasis is placed on formulating and solving "number stories" with real world information in science, geography, and other areas. The curriculum for grades 4-6 employs mathematical modeling of everyday events and projects while building on the mathematical strands introduced in previous grades by blending them with science, geography, sports, and architecture.

NCTM math standards:
• Geometry
• Data Analysis &  
  Probability
• Measurement
• Number & Operations

Math Topics:
Basic facts, mental and symbolic arithmetic, collection and use of data, computation, logical thinking

K to 6th Grades

Study:
Fuson.

Achievement results for second and third graders using the standards-based curriculum.

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 Strategy: Everyday Mathematics (EM)
Subjects: (Study One) Experimental Group - 343 second grade students in 22 classrooms in 11 schools including urban, suburban and rural or small-town schools. Range of SES; two classes were Spanish-speaking bilingual classes.
Control Group - 29 second graders attending a middle to upper class school in San Francisco and 33 Japanese second graders attending a middle-class public school in Tokyo.
(Study Two) Experimental Group - 236 third graders
Control Group - 1,800 students as a subset of 18,033 third graders who took the NAEP and answered all questions
Results: (Study One) On the mathematic achievement test, EM students scored between the Japanese and the US comparison students with the Japanese students scoring significantly higher than the EM students on the six most advanced items. The Everyday Math students were above the national norms for multiple digit addition and at the norm for multiple digit subtraction. 		Description: Students work in small groups or pairs exploring mathematical ideas. Students build their informal knowledge by making connections to everyday experiences. Teachers are advised to use manipulatives in order to scaffold students' thinking during problem solving and discussions. Students build conceptual understanding of number and operations by creating and solving story problems. Paper and pencil, in addition to mental, activities are designed to enable students to develop conceptual understandings of the operations and the standard multi-digit algorithms. Students are encouraged to invent and discuss their own solution methods.

NCTM Math Standards:
• Geometry
Data Analysis &
Probability
• Measurement
• Number & Operations

Topics:
• Multiple digit addition and subtraction,
• Negative numbers,
• Functions,
• Fractions,
• Mental computation,
• Geometry

2nd and 3rd Grades

 

Study:
Mokros.

Learning to reason numerically: The impact of Investigations.

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 Strategy: Investigations in Number, Data, and Space.
Subjects: (First Study) 56 third grade and 40 fourth grade students from diverse backgrounds located in Massachusetts schools in urban suburban and rural communities
(Second Study) 46 second grade students
(Third Study) 125 fourth grade students
Results: The three studies cited show that students using the Investigations program perform as well as students in traditional curricula classrooms on basic facts and algorithms with the four operations and may even perform better on difficult computations. Investigations students perform better than their counterparts from other curricula with respect to word problems, more complex calculations embedded in word problems, and problems that involved explaining how an operation worked.
 Description: The program has six major goals: (1) To provide meaningful mathematical problems for students that are based on (a) important mathematical ideas, (b) are addressed to a wide range of students, (c) require students to think mathematically, and (d) encourage the use of different strategies by students with different learning styles; (2) to develop powerful mathematical thinking, explanation, justification, and demonstration; (3) to encourage sustained thinking by focusing on a small set of significant problems within each unit; (4) to provide both coherence and depth in mathematical content; (5) to support teacher learning; and (6) to connect students of all abilities to mathematics.

NCTM Math Standards:
• Number & Operations

Topics:
• Basic facts, addition, subtraction, division, multiplication, and word problems.

2nd to 4th Grades

Study:
Carroll, W.

Results of third-grade students in a reform curriculum on the Illinois State Mathematics Test.

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Strategy: Everyday Mathematics, University of Chicago School Mathematics Project (UCSMP).
Subjects: 1885 third-graders from twenty-six schools in nine greater Chicago metropolitan area districts were in the UCSMP group; the comparison groups were nonusers of UCSMP in suburban Cook County and the state of Illinois. Fourteen of the twenty-six schools had used the curriculum since kindergarten. The remaining twelve schools adopted the curriculum during the 1991-92 or 1992-93 school year.
Results: All 26 UCSMP schools scored well above the average state score and three schools scored below the suburban Cook County Schools. The fourteen schools who had implemented UCSMP since kindergarten scored 75 points higher than the state scores - median of 343 as compared to 268. For these UCSMP schools 54% exceeded state standards and 2% did not meet state standards.

Description: The UCSMP is a reform curriculum that incorporates small group work to explore mathematics in real life contexts and incorporates calculators and manipulatives. Students are encouraged to use these tools or invent strategies to solve problems and share solutions as part of class discussions.

NCTM math standards:
• Algebra
• Geometry
• Data Analysis &
  Probability
• Measurement
• Number &   Operations

Math topics:
All topics

3rd Grade

Study:
Woodward & Baxter.

The effects of an innovative approach to mathematics on academically low-achieving students in inclusive settings.

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Strategy:  Everyday Mathematics.
Subjects: 205 third graders; of particular interest was a subset of students who had learning disabilities or were low achieving in mathematics.
Results:   Results of the Iowa Test of Basic Skills (ITBS) concepts subtest favored the intervention group. ITBS results for the low-achieving students indicated non-significant differences. An individually administered test of problem-solving abilities strongly favored the intervention group.

Description: Everyday Mathematics is an elementary curriculum developed by the University of Chicago School Mathematics Project.  It de-emphasizes computation in order to include more topics in mathematics at a greater depth than in a traditional curriculum.  It is a student-centered curriculum that has a problem-solving approach with an emphasis on conceptual development and number sense.  Students are taught using a variety of tools including manipulatives and calculators.

NCTM math standards:
• Number & Operations

All Topics

3rd Grade

Study:
Carroll.

Geometric knowledge of middle school students in a reform-based mathematics curriculum.

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Strategy: Everyday Math
Subjects: Experiment group: 109 ethnically diverse sixth grade heterogeneously grouped students;
76 ethnically diverse fifth grade heterogeneously grouped students from rural, suburban and urban districts.
Control group: 137 sixth graders matched to experiment group on location, demographics including SES;
91 fifth graders matched to experiment group on location, demographics including SES
Results: On all measures, EveryDay Math students outperformed the traditional curriculum students.
Sixth grade effect Size = +0.64
Fifth grade effect Size = +0.73

Description: This study is a quasi-experimental design. The intervention was an on-going treatment (the EveryDay Math curriculum) to which students had not been randomly assigned. Rather, the students had been either exposed to the UCSMP or a traditional curriculum since kindergarten and were being assessed as to their level of geometry thinking (according to the van Hiele theory) after those experiences

NCTM math standards:
• Geometry
• Number & Operations

Math Topic(s):
Geometric reasoning

5th & 6th grades

Study:
Romberg & Shafer.

Mathematics in Context - Preliminary evidence about student outcomes.

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Strategy: Mathematics in Context (MiC)
Subjects: 17 schools from 2 suburban and 2 urban districts
54 teachers in grades 5-8, 80% have used MiC curriculum
2225 students in the program 1,2, or 3 years
Results: The degree of implementation of the Mathematics in Context curriculum varied throughout the four districts. According to data, students in the MiC program outperformed students in the conventional program on Number, Geometry, Algebra, and Statistics/Probability. However, the percentages of correct responses varied among all groups.

Description: Mathematics in Context is a standards-based curriculum for grades 5-8 designed to help students progress from informal to formal mathematical reasoning in number, geometry (and measurement), algebra, statistics, and probability. The curriculum focuses on placing students in realistic situations that they must resolve. During their resolution process, students progress from informal notions toward formal mathematical reasoning and representations to model and solve non-routine problems. Throughout the curriculum, students develop conceptual knowledge first and re-visit it as necessary.

NCTM math standards:
• Algebra
• Geometry
• Data Analysis &
  Probability
• Measurement
• Number &   Operations

Math topics:
All topics

5th to 8th Grades

 

Study:
Riordan & Noyce.

The impact of two standards-based mathematics curricula on student achievement in Massachusetts ["standards-based" instruction is identified as Connected Math (cmp) and/or Everyday Math (em)].

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Strategy: Connected Math (CMP) and Everyday Math (EM)
Subjects: Standards-Based" Instruction Traditional Instruction Matched
EM1 n=48 schools 4+ years using EM n=51 schools
EM2 n=19 2-3 years using EM n=27
CMP1 n=1 4+ years using CMP n=4
CMP2 n=20 2-3 years using CMP n=30
Results: Massachusetts’ students in the standard-based programs performed significantly better on the 1999 statewide mathematics tests than did Massachusetts’ students in traditional programs attending matched comparison schools.
Effect Size: Late implementation of EM= +0.15
Early implementation of EM and CMP= +0.34

 Description: Standards-type curricula are problem-oriented providing a broad range of mathematical topics with a focus on concept development and student discussion prior to written work.

NCTM math standards:
• Algebra
• Geometry
• Data Analysis &
  Probability
• Measurement
• Number &   Operations

Math topics:
All topics

4th & 8th Grades

 

Study:
Ridgway, Zawojewski, Hoover & Lambdin.

Student attainment in the Connected Mathematics curriculum.

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 Strategy: Connected Math (CMP)
Subjects: 5 Midwest, 2 West, and 2 East schools, 2 CMP classes and 1 nonCMP in each school
CMP Non CMP
6th grade 338 6th grade 162
seventh grade 627 seventh grade 234
8th grade 820 8th grade 275
Results: All one-year gains in the Balanced Assessment (BA) test were statistically significant. No one-year gains in ITBS (Iowa Test of Basic Skills) test were statistically significant, but sixth graders who started out behind their counterparts finished ahead of non-CMP students by the eighth grade.
Effect Size:
  BA test   ITBS
6th grade +0.16 -.55
seventh grade +0.58 -.17
8th grade +0.96 +0.31

Description: A problem-oriented curriculum which provides a broad range of mathematical topics with a focus on concept development and student discussion prior to written work.

NCTM math standards:
• Algebra
• Geometry
• Data Analysis &
  Probability
• Measurement
• Number &   Operations

Math topics:
All topics

6th to 8th Grades

Study:
Reys, Reys, Lapan and Holliday.

Assessing the impact of standards-based middle grades mathematics curriculum materials on student achievement.

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Strategy: Standards-based middle grades mathematics curriculum. Subjects: Two thousand eighth grade students from six school districts. Students were matched for prior mathematics achievement, SES (free and reduced lunch), the same building configuration, and geographic location.
Results: After two years in a standards-based curriculum, the treatment group equaled or exceeded the comparison group. Students in each of the three districts using standards-based curriculum scored significantly higher in two content strands (algebra, and data analysis and probability).

Description: Eighth grade students used standards-based mathematics curriculum materials (specifically MathThematics and Connected Mathematics) for at least two years.

NCTM Math Standards:
• Algebra
• Geometry
• Data Analysis &
  Probability
• Number &   Operations

Math Topic(s):
All topics, including discrete mathematics.

6th to 8th grades

Study:
Billstein & Williamson.

Middle grades MATH thematics: The STEM project.

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Strategy: MATH Thematics - STEM Project
Subjects: Study 1 treatment group: 115 midwestern MATH Thematics 6th graders in middle-class suburban school.
Study 1 control group: 46 non-MATH Thematics students from another school district.
Study 2: longitudinal; rural western school; 2 groups of 9th graders tested at different times (1993 & 1997 ITEDS). 1993 9th graders had not completed the MATH Thematics curriculum, whereas1997 9th graders had completed MATH Thematics curriculum for grades 6 - 8.
Study 3: Suburban 6th & 8th graders at a MATH Thematics field-test site had used MATH Thematics at the seventh grade level for 4 years.
Results: Study 1: no significant difference.
Study 2: 1993 Non MATH-Thematic students ranked 44th percentile on ITEDs; 1997 MATH Thematic students ranked 89th percentile on ITEDs.
Study 3: Treatment students scored 12% higher on ITBS.
Study 4: After statistical corrections, 14 of 42 CRT objectives favored MATH-Thematic students, 1 of 42 CRT objectives favored control students.

Description: A Standards-based curriculum for grades 6, 7, and 8 characterized as problem-oriented with a focus on concept development and student discussion prior to written work. Students are expected to actively do mathematics by investigating, discovering, and applying mathematics to new situations. Students must be able to communicate their ideas effectively through cooperative groups and whole-class discussions.

NCTM math standards:
• Algebra
• Geometry
• Data Analysis &
  Probability
• Measurement
• Number &   Operations

Math topics:
All topics

6th to 8th Grades

 

Study:
Ben-Chaim et al.

Proportional reasoning among seventh grade students with different curricular experiences.

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 Strategy: Connected Mathematics Program (CMP)
Subjects: 187 eighth graders, 6 classes in Michigan, 1 in San Diego and 1 in Pittsburgh
128 seventh graders, 3 classes in Michigan, 1 in Toledo, 1 in San Diego, 1 in Pittsburgh
Results: Seventh graders outperformed control students 53% to 28% on collections task and on each individual problem

Description: The treatment group was taught using CMP materials for one school year. In this reform curriculum, students are encouraged to construct their own conceptual and procedural knowledge through collaborative problem-solving activities. The data for this study were generated by five proportion problems focusing on rate or population density.

NCTM math standards:
• Number & Operations

Math Topic(s):
Rational numbers, proportionality (rate, density, ratio)

seventh and 8th grades

Study:
Senk.

Effects of the UCSMP secondary school curriculum on students' achievement.

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Strategy: University of Chicago School Mathematics Project (UCSMP)
Subjects: Study 1: Three districts. Site A, one large urban district magnet school, grades 7-12, (1/3 minority). Sites B and C (affluent suburban areas with largely white populations).
Study 2: Study 2: Three private coed schools, mostly middle to upper income families; six public schools (two in mid-sized cities, two in suburbs, and two magnet schools in urban areas).
Study 3: Study 3: Students (n = 139) from four schools, one in Indiana with an African American majority; two in Oregon with white majorities; and one middle class, suburban school in South Carolina, (25% African American).
Study 4: Study 4: Included four schools. One suburban Atlanta school, predominantly white; one rural school with predominantly white population; one small town Mississippi school, 25% white and 75% African American; and one school in an affluent area of Philadelphia with all white student population.
Results: Study 1: Standardized test results were mixed. At sites A and B, UCSMP students outperformed all but one of their comparison groups at statistically significant levels. At site C, course level comparison group outperformed UCSMP students.
Study 2: PDM students' posttest scores were comparable to a calculus sample and higher than the precalculus sample from the SIMS.
Study 3: Results indicated that classes using UCSMP Geometry outperformed comparison classes on the test assessing reasoning, transformations, and applications not covered on the standardized test.
Study 4: Results indicated that classes using UCSMP Advanced Algebra outperformed comparison classes on the Multiple-Choice Test.

Description:

UCSMP is composed of courses called Transitional Mathematics (Algebra; Geometry; Advanced Algebra; Functions, Statistics, and Trigonometry; and Precalculus and Discrete Mathematics). Algebra uses variables to develop linear, exponential, and quadratic patterns. Geometry presents coordinates, transformations, measurement formulas, along with work involving writing proofs. Advanced Algebra studies functions, equations, and inequalities. The Functions, Statistics, and Trigonometry course integrates statistical and algebraic concepts. Precalculus and Discrete Mathematics incorporate many topics required for success in Calculus and Computer Science. The content strands of applied mathematics, prealgebra, and elementary geometry are woven throughout the six courses.

NCTM Math Standards
• Algebra
• Geometry

All Math Topic(s)

7th to 12th grades

 

Study:
Schoen & Hirsch.

The Core-Plus Mathematics Project: Perspectives and student achievement.

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 Strategy: Core-Plus Mathematics Project
Subjects: 64 high schools in 11 states, cross-section of diverse students from urban, suburban, and rural communities
Results: Results showed that all CPMP groups performed significantly better on solving mathematical tasks set in context than the comparison groups. The comparison groups performed significantly better with paper-and-pencil procedures at the end of Course 1, but the results were not significant for Course 2.
Effect Size on ITED-Q
Course 1 ES=+.19
Course 2 ES=+.04
 Description: The CPMP curricula consist of a single core sequence for college-bound and employment-bound students during the first three years of high school that embody the content, processes, and teaching principles recommended by the NCTM standards. Building connections among topics is the primary focus. Strands are also connected by thinking mathematically, such as visual thinking, recursive thinking, searching for and explaining patterns, making and checking conjectures, reasoning with multiple representations, inventing mathematics, and providing arguments and proofs. Fundamental themes of data, representation, shape, and change also provide connections across strands.

Curriculum development principles include: (1) mathematics as an active science of patterns, (2) problems that include a context for developing student understanding of mathematics, (3) the processes of exploration and experimentation necessarily precede and complement theory, and (4) the use of graphics calculators and other technology as tools for developing mathematical understanding and for solving authentic problems. Pedagogical principles include the importance of students' sense making of mathematics and real-life contexts. The instructional materials are designed to reflect the importance of collaborative learning, social interaction, and communication.

NCTM Math Standards
• Algebra
• Geometry
• Data Analysis &
  Probability
• Number &   Operations

Math Topic(s):
Symmetry, functions, matrices, data analysis, and curve fitting.

9th to 12th grade.

Study:
Thompson & Senk.

Title: The effects of curriculum on achievement in second-year algebra: The example of the University of Chicago School Mathematics Project.

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Strategy: Skills and concepts learned and applied using the University of Chicago School Mathematics Project (UCSMP) Advanced Algebra textbook
Subjects: 306 students in heterogeneous classes studying second-year algebra in four high schools. The high schools selected were in a White middle-class suburb of Atlanta, a rural area that is becoming a suburb of Chicago, a small semi-rural community in Mississippi, and an affluent suburb of Philadelphia. Of the students, 19% were in Grade 10, 76% in Grade 11, and 5% in Grade 12. Additionally, 84% were Caucasian, 3% African American, 1% Hispanic, while the remainder were classified as other or unknown.
Results: Students using the UCSMP curriculum significantly outperformed students in the comparison curriculum (p=0.0014) on all items of the post-test. However, analysis of the items all students in the study had the opportunity to learn did not indicate a significant difference (p=0.108). Performance on the eight skill items in this last analysis was comparable for the two curricula.

Description: UCSMP is a curriculum that uses reading and problem solving, realistic applications, technology (graphing calculators and/or computers), a multidimensional approach understanding, and an instructional format featuring continual review combined with a modified mastery-learning strategy. It emphasizes understanding of concepts through multiple representations, realistic contexts, and the use of technology. There is less emphasis on skills than in a traditional curriculum. The instructional method often uses small-group explorations and extended projects, both involving writing about mathematics.

NCTM Math Standard :
• Algebra

Math Topics: Linear equations, binomial multiplication, graphs of quadratic equations, exponential equations, the equation for the inverse-square function, recursion

Grades: 10th to 12th Grade

Study:
M. A. Huntley et al

Title: Effects of standards-based mathematics education: A study of the Core-Plus Mathematics Project algebra and functions strand

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Strategy: Core-Plus Mathematics Project (CPMP)
Subjects: Third year high school students from six U.S. schools (two of the sites were in the Southeast, two in the Midwest, one in the South, and one in the Northwest). The number of students ranged from 90 - 180 per site with a maximum of 593 students tested at the conclusion of the study.
Results: Students in the experimental group outperformed students in the control group on items that involved applying algebraic concepts to solving contextual problems. On items that assessed symbol manipulation, the control group outperformed the experimental group.

Description: The Core-Plus Mathematics Project (CPMP) is a four-year integrated high school mathematics curriculum. Major concepts are developed through investigating concepts in the context of applied problems and mathematical modeling. In particular, algebraic concepts are developed using graphic, numeric, and symbolic representations. Graphing calculators are integral, promoting connections among the forms of representation, encouraging new methods of problem solving, and decreasing the need for symbolic manipulation procedures. Since this study was carried in 1997 the curriculum has been revised (in 2003 and in press) to enhance the in-context approach and also include more work with procedural algebraic skills.

NCTM Math Standard:
• Algebra

Math Topics: Functions, algebraic expressions, systems of equations

Grade: 11th Grade

Study:
Webb.

The impact of the Interactive Mathematics Program on student learning.

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Strategy: Interactive Mathematics Program (IMP)
Subjects: Study 1: 1121 students' transcripts from 3 California high schools (1 Middle-Sized Area = 2075 students, 1 Metro Area=2000 students, 1 City Area=1600 students)
Study 2: 116 high scoring students (divided equally between treatment and control group) from the Metro Area high school.
Study 3: 115 ninth grade, 184 tenth grade, 62 eleventh grade students
Study 4: 136 ninth grade, 174 tenth grade students
Study 5: 62 ninth grade, 80 tenth grade students
Results: Study 1: Total number of students taking more than three years of college-qualifying math courses was significantly higher for the treatment group than the control group.
Study 2: No significance on SAT math test, but there was a significant difference favoring the IMP students in both math and overall GPA.
Study 3: The mean scores for the IMP 9th - 11th graders were significantly higher than the control group.
Study 4: The mean scores for the IMP 9th and 10th graders were significantly higher than the control group.
Study 5: At grade 9, the mean score of IMP students was significantly lower than the control group in a traditional Algebra course enhanced with a unit on statistics. There was no significant difference for the 10th graders.

Description:
IMP is a sequence of college preparatory courses designed for grades 9-12. It is organized into five to eight week units that focus on a central problem or theme. It incorporates technology instruction and applications throughout the program. Students solve both routine and non-routine problems, using calculators. Students are encouraged to work cooperatively. They learn to experiment, investigate, ask questions, make and test conjectures, reflect, and accurately communicate their ideas and conclusions. Students are assigned to work on "Problems of the Week" up to five days. They are encouraged to function as independent learners as a result of using multiple sources of information, including their teachers, the textbook, classmates, and other resources.

NCTM Math Standards
• Algebra
• Geometry
• Data Analysis &   Probability

Math Topic(s):
Algebra, geometry, trigonometry, probability, statistics, discrete mathematics, and matrix algebra.

9th to 12th grades

 

Study:
Lott et al.

Curriculum and assessment in SIMMS Integrated Mathematics.

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Strategy: SIMMS Integrated Math
Subjects: High school students (n = 125-280) in intact classes in five studies (four in Montana and one in El Paso). SIMMS programs levels 1, 2, 4, and 6 versus non-SIMMS programs.
Results: Montana: On all levels no significant difference on PSAT. Level 1 - On project constructed questions called End-of-Year-Tasks (EOYT), SIMMS students scored significantly higher on the common item. Level 2 - No significant difference on EOYT.
El Paso: no significant difference on PSAT nor EYOT

 Description:
SIMMS Integrated Math is a complete mathematics program for grades 9-12. It is constructed in six levels each consisting of one year of work. Levels 1 and 2 provide basic mathematical literacy. Levels 3 and 5 are for students with non-mathematical or non-scientific aspirations, whereas levels 4 and 6 prepare students to complete a post-secondary mathematics curriculum.

The program has the following characteristics, it is (1) integrated and interdisciplinary, (2) problem-centered and applications based, (3) technology based, (4) sensitive to multiple perspectives and the negative effects of bias and stereotyping, (5) multi-modal to accommodate multiple learning styles.

Mathematical modeling is the basic thread of the curriculum, knitting together mathematics and other subjects. Students organize, relate, interpret, justify, evaluate, summarize, and communicate ideas as they explore topics and make inquiries about real-world and mathematical concepts. They confront complex issues by "mathematizing" real-world problems, interpret and communicate solutions, and ultimately reflect on their own performance.

NCTM Math Standards
• Algebra
• Geometry
• Data Analysis &
  Probability

Math Topic(s):
Basic math concepts, pre-algebra, algebra, geometry, and college-prep.

9th - 12th grades

Study:
Cichon & Ellis.

The effects of Math Connections on student achievement, confidence, and perception.

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 Strategy: Math Connections (MC)
Subjects: Summary of several studies
Results: On CAPT, PSAT and SAT, MC students scored as well or better than nonparticipants. Students in classes with MC trained teachers have greater confidence in learning mathematics and see the usefulness of mathematics to a greater degree than non-participants.
Case study results of Hispanic and African-American at-risk students in an inner-city school suggest that all students can successfully meet the challenge of a mathematically rigorous course.		 Description: Math Connections is presented to students as a powerful tool for interacting with their environment. Its primary goals are to: (1) bridge the worlds of education, students, and business through mathematics; (2) increase mathematical power of all students; (3) empower students for their own learning; (4) develop a core curriculum that reflects the standards of the National Council of Teachers of Mathematics (NCTM); and (5) empower teachers to meet NCTM's Professional Standards for Teaching Mathematics.

Math Connections enables students to investigate a concept in order to recognize patterns that can be presented as a formula. Each year of the curriculum is based on a theme that connects and unifies mathematical topics. The theme for Year 1 is Data, Numbers, and Patterns. Students use data, linear equations, and graphing calculators to forecast events. Year 2's theme, Shapes in Space, focuses on properties and measurement of figures. The principles of congruence and triangulation of polygons are, also, examined. Students use matrices to solve linear equations and learn to calculate the volumes of a cone and a sphere. Mathematical Models, is the theme for Year 3. At this time, logic and the properties of an axiomatic system are introduced.

NCTM Math Standards
• Algebra
• Geometry
• Data Analysis &
  Probability

All Math Topic(s)

9th to 12th grades

 

Study:
O' Callaghan

Computer-Intensive Algebra and students' conceptual knowledge of functions.

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Strategy:
Computer-Intensive Algebra (CIA)

Subjects:
Eighty volunteer students with lower-middle to middle SES status enrolled in a college algebra course.

Results:
Treatment subjects achieved gains in the concepts of modeling, translating, and interpreting as they relate to functions. There was no significant difference in groups in regard to the concept of reification.

Description:
CIA teaches college algebra using computer technology and with a focus on real-world situations.

NCTM Math Standard:
• Algebra

Math Topics: College algebra, functions, modeling.

12th grade

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Overview – Research on NCTM-Standards-Based (NSF-Funded) Curricula, K-12

Numerous projects, funded by the National Science Foundation in the 1990s, developed K-12 curriculum materials, which are designed to include more important mathematics and enhance student learning. These curricula are referred to variously as Standards-based curricula, NSF curricula, alternative curricula, or reform curricula. Key content and equity issues leading to reform were concerns that the U.S. mathematics curriculum was not internationally competitive, the high school curriculum focused on math/science-based college preparation at the expense of mathematics for all students and all potential college majors, and large numbers of K-12 students failed or stopped studying mathematics with a disproportionate number of nonwhite students in this category (Schoenfeld, 2002).

An NCTM-Standards-based curriculum is cohesive and comprehensive; it includes the content strands of number and operations, algebra, geometry, measurement, and probability and data analysis (Trafton, et al., 2001). Connections are developed across grade levels, and across and within topics so that mathematics is perceived as being coherent and an integrated whole. Connections to real life contexts are the basis of mathematical tasks so that applications become part of understanding and interpreting math. Standards-based materials develop topics in depth and promote sense making, as the focus is on conceptual development and understanding. Standards-based materials engage students through intriguing tasks; the learning is student-centered with problem solving as a vehicle for both learning and teaching mathematics. Multiple and connected representations are used to explore and develop concepts. Calculators and computing technology are used as tools for learning.

It should be noted when reviewing research results that the goals for these alternative Standards-based curricula are not the same as the traditional textbook programs. The goals for the new curricula are much more ambitious mathematically, including deep conceptual understanding, powerful problem solving ability, and solid skill proficiency. From both domestic studies and international comparisons, it is reasonable to set more ambitious goals for our students and students can understand significant mathematics without sacrificing skill proficiency (Hiebert, 2003).

Conceptual Understanding and Problem Solving – In K-12