Among the value-added services Carnegie Learning provides to schools and school districts is help in preparing and writing grant proposals that reflect a partnership between schools, districts, and Carnegie Learning.
In addition, Carnegie Learning is providing the following information about Carnegie Learning and its products. Schools and districts can use this information when preparing proposals or when beginning to plan for grant programs. Descriptions of Carnegie Learning’s products and services are presented in greater detail in the Executive Summary on this website or through the company’s product publications.
Carnegie Learning, Inc.
Independent studies across a large sample of Carnegie Learning implementations show significant increases in student test scores, student attitudes toward math, and student critical thinking skills. Building on its reputation for excellence, Carnegie Learning continues to receive recognition.
Scientifically Based Research
Because Carnegie Learning’s pedagogy is scientifically based, its curricula are increasingly at the forefront of high-school transformation initiatives Carnegie Learning’s math programs are rooted in more than two decades of cognitive science research at Carnegie Mellon University on how students think, learn, and apply new knowledge in mathematics. The results of this research formed the foundation for development of Carnegie Learning’s Cognitive Tutor® software, a unique modeling technology that teaches students to think mathematically.
Carnegie Learning’s Cognitive Tutor software is based on Dr. John R. Anderson’s ACT-R theory of learning and performance, which differentiates between “tacit performance” knowledge and “static verbalized” knowledge. According to ACT-R theory, performance knowledge can only be learned by doing, not by listening or watching. ACT-R theory calculates such performance knowledge using a series of “if-then” rules that associate internal goals and external cues with new internal goals and external actions.
Using this theory, Carnegie Learning built a cognitive model of student problem solving by writing “if-then” rules that captured students’ various strategies and their common misconceptions. The result is Cognitive Tutor software, which follows students through their problem-solving activities via “model tracing,” which identifies the “just-in-time” assistance a student needs.
The cognitive model also traces increases in students’ knowledge across a range of problem-solving activities, dynamically updating estimates of how well students know each skill. These estimates are used to select what problems to present next, and to adjust pacing for each individual student. This individualized problem-generating process continues until all essential facts and concepts are mastered—providing a conceptual approach to math learning that students master naturally.
Vertically Aligned Curricula
With Cognitive Tutor as the design structure, all Carnegie Learning courseware can meet students exactly where they are and bring them to where they need to be. The software monitors students’ knowledge on a minute-by-minute basis and provides students with personalized learning paths. Each unit begins with a look ahead—a preview of the new objectives to be mastered. Each unit ends with a look back—a review of math concepts. It is this recycling of skills that moves students forward, step by step, to higher math skill levels.
The vertically aligned curricula begin with Bridge to Algebra, an algebra readiness curriculum. Bridge to Algebra creates an important link between basic math and high-school algebra. Students practice math facts while learning conceptually until they are individually ready to move to the next level of learning. Bridge to Algebra touches on, teaches thoroughly, and then moves beyond drills introduced in earlier levels and younger grades. With Bridge to Algebra “bridging” middle- and high-school math skill levels, students can then progress to Carnegie Learning’s Algebra 1, Geometry, and Algebra II coursework.
- With a focus on mathematics literacy and problem solving, Carnegie Learning’s Bridge to Algebra covers the five middle-school content strands identified in NCTM and most state standards.
- Carnegie Learning’s Algebra I curriculum helps to direct a student’s intuitive problem-solving abilities toward understanding and mastering higher-order mathematical concepts.
- Carnegie Learning's Geometry courseware helps students understand geometric concepts and build critical skills in spatial reasoning.
- Carnegie Learning’s Algebra II prepares students for higher levels of mathematical modeling.
All courses include the Cognitive Tutor integrated software; the course curricula (write-in textbooks, supplemental assignment sheets, built-in assessments, and activities); users’ guides to accompany the software; teachers’ guides, initial teacher training, ongoing professional development, and technical support.
While Carnegie Learning’s curricula have proven effective with all levels of student achievement, it has shown strong student achievement outcomes for those students who have little chance of success, as presumed from their test scores and preparation. It is these students who are the target population for many of the public and private funding opportunities.
Integrated Math Series
Carnegie Learning's Integrated Math curricula is a complete, three-course series that integrates numeric, algebraic, geometric, and statistical curricula to help students analyze and solve challenging problems. This course sequence was designed for those states that require districts to use an integrated approach to teaching math.
Math Prep
Currently, close to half of the states require high-school students to pass exit exams in order to earn a high-school diploma. Many students are unlikely to pass these exams. Carnegie Learning’s Math Prep is a new and exciting research-based course developed specifically to improve student performance on state exit exams. The course is flexible in its design, and can be effectively used over a full year or as a supplemental intervention.
Math Prep’s national edition is based on NCTM standards. The course can also be customized to reflect a state’s standards and exit examination requirements.
Evaluation and assessment
Assessment is built directly into Carnegie Learning’s Cognitive Tutor curricula to measure basic functional skills and facts as well as the higher-order reasoning and problem-solving skills recommended in the NCTM standards. Based on logical scope and sequence, each course advances only as fast or as slowly as an individual student masters continually assessed facts. For each skill, a problem situation tests a student’s ability in applying a basic fact and his or her conceptual application of that skill to additional problem situations. Representational tests directly examine each student’s success in translating among multiple math representations, such as symbols, graphs, and tables.
Each module moves forward as directed by algorithms based on the individual student’s mastery (or non-mastery) of math facts and concepts. Students who answer problems correctly will be moved forward more rapidly through modules, moving with greater speed toward problems at higher cognitive levels. By contrast, students who do not answer problems correctly, demonstrating that they have mastered facts and concepts, will be given another (and another and another) series of math problems until all necessary skills addressing these same facts and concepts are learned.
Along with new problems, students get new hints that help them address the skills needed to learn. Like a “tutor in a box,” each individual student has a customized learning program that addresses his/her beginning level of skill, creates fluency in the use of math concepts, and reinforces facts needed to master these concepts. The software program constantly assesses each student’s skills while adapting programmatically to the mastery level of each student.
In addition to immediate software-based reinforcement for students, teachers can also visually review strand achievement levels for each of their students and be able to identify those who need a personal touch, a kind word, or a human voice in addition to more targeted time on task. Teachers, therefore, can adjust classroom content and assignments to provide remediation or acceleration as needed to groups of students or to individuals.
This ability to prompt both software and educator responsiveness to all students’ individual needs is a strength of Carnegie Learning’s math programs. Carnegie Learning’s Cognitive Tutor curricula present math facts with age-appropriate problems that let high-school students “practice” math in a manner different from that presented in lower grades and in other curricular designs.
Using Carnegie Learning’s Cognitive Tutor program, students gain the confidence they need to learn math fundamentally and conceptually, and avoid the label of “remedial” learning so often accompanying the very structure of other program approaches.
