Mathematics and Science Partnerships February 14, 2008

2 Presidential Executive Order April 2006 The Panel will advise the President and the Secretary of Education on the best use of scientifically based research to advance the teaching and learning of mathematics, with a specific focus on preparation for and success in algebra.

3 What Concerns Led to the President’s Order? National prosperity and safety in international context –Role of mathematics national well-being –Gathering Storm –Workforce of the future Options for individuals and families –College admission and graduation –Candidacy for technical workforce –Earning power –Adaptability

4 Math Proficiency of US Students International comparisons Low fractions of proficiency on NAEP Falling proficiency at higher grades Heavy remedial demand upon entry into college Achievement gap Algebra as a gateway

5 Overview Task Groups: –Conceptual Knowledge and Skills –Learning Processes –Instructional Practices –Teachers –Assessment Subcommittees: –Standards of Evidence –Survey of Algebra Teachers –Instructional Materials Reviewed 16,000 research studies and related documents. Gathered public testimony from 63 organizations. 11 Panel meetings to date.

6 Evidence Guidelines Executive Order –Marshal the best scientific evidence. –Review research relating to proven-effective and evidence-based mathematics instruction. What is the best scientific evidence? –3 broad categories of quality. Highest quality = high internal and external validity. Promising or suggestive = has limitations. Opinion = values, impressions, or weak evidence.

7 Findings of the Task Groups Conceptual Knowledge and Skills Task Group: The Major Topics of School Algebra Covering all of school algebra traditionally extending over two courses, Algebra I and Algebra II Symbols and Expressions Linear Equations Quadratic Equations Functions Algebra of Polynomials Combinatorics and Finite Probability

8 Findings of the Task Groups Conceptual Knowledge and Skills Task Group: Critical Foundations for Success in Algebra Fluency with Whole Numbers Fluency with Fractions Particular Aspects of Geometry and Measurement

9 Fluency with Whole Numbers 1.By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers. 2.By the end of Grade 5, students should be proficient with multiplication and division of whole numbers. Fluency with Fractions 1.By the end of Grade 4, students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals. 2.By the end of Grade 5, students should be proficient with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals. 3.By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimals. 4.By the end of Grade 6, students should proficient with all operations involving positive and negative integers. Benchmarks for the Critical Foundations

10 Fluency with Fractions (cont.) 5.By the end of Grade 7, students should be proficient with all operations involving positive and negative fractions 6.By the end of Grade 7 students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality. Geometry and Measurement 1.By the end of Grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e. trapezoids). 2.By the end of Grade 6, students should be able to analyze the properties of two dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three-dimensional shapes and solve problems involving surface area and volume. 3.By the end of Grade 7, students should understand relationships involving similar triangles. Benchmarks for the Critical Foundations

11 Recommendations of the Task Group Conceptual Knowledge and Skills Task Group: 1.The Task Group recommends that school algebra be consistently understood in terms of the list of Major Topics of School Algebra (MTSA). 2. The MTSA, accompanied by a description of the mathematical connections among these topics, should be the main focus of Algebra I and Algebra II: –Standards in state curriculum frameworks –Courses –Textbooks –End-of-course assessments

12 Recommendations of the Task Group Conceptual Knowledge and Skills Task Group: 3. Proficiency with whole numbers, fractions, and particular aspects of geometry are the Critical Foundations of Algebra. - Emphasis on these essential concepts and skills must be provided at the elementary and middle grade levels. 4. Use Benchmarks for the Critical Foundations to guide: - classroom curricula - mathematics instruction - state assessments 5. Teach for mastery in a few topics, instead of mile-wide-inch- deep curriculum: - Follow a coherent progression, with emphasis on mastery of key topics. - Avoid an approach that continually revisits topics year after year without closure.

13 Findings of the Task Groups Learning Processes Task Group: 1.Scientific knowledge on learning and cognition needs to be applied to the classroom to improve student achievement. 2. It is well documented that limits in the ability to keep multiple things in mind (working-memory) can hinder performance in mathematics: - Practice can offset this through automatic recall, which results in less information to keep in mind and frees attention for other aspects of problem solving. - Learning is most effective when practice is combined with instruction on related concepts. - Conceptual understanding promotes transfer of learning to new problems and better long-term retention.

14 Findings of the Task Groups Learning Processes Task Group: 3. Most children develop considerable knowledge of numbers and other aspects of mathematics before they begin kindergarten. - Children from families with the combination of low incomes, low levels of parental education, and single parents often have less mathematical knowledge when they begin school than do children from more advantaged backgrounds. This tends to hinder their learning for years to come. - There are promising interventions that can substantially improve the mathematical knowledge of these young children before they enter kindergarten.

15 Findings of the Task Groups Learning Processes Task Group: 4.U.S. students' performance in solving arithmetic problems with speed and efficiency lags behind that of students in a number of other countries. Empirical studies comparing the U.S. and China suggest the lag is related to differences in the frequency and organization of problem practice. 5.Fractions, fundamental to the learning of algebra, are formally introduced in elementary school, yet remain difficult for many U.S. adults. 6.The errors students make when they solve algebra problems reveal a poor grasp of the principles of arithmetic, mathematical equality, and the procedures for transforming equations.

16 Learning Processes Task Group : 6.The development of conceptual knowledge, factual knowledge, problem solving, and procedural skill is intertwined, each facilitating learning of the others. To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, factual knowledge and problem solving skills. 7.Children’s goals and beliefs about learning are related to their mathematics performance. Experimental studies have demonstrated that children’s beliefs about the relative importance of effort and ability can be changed. Related research demonstrates the engagement and sense of efficacy for Black and Hispanic students can be increased in mathematical learning contexts. Teachers and other educational leaders should consistently help students and parents to understand that an increased emphasis on the importance of effort is related to improved mathematics grades. Findings of the Task Groups

17 Findings of the Task Groups Instructional Practices Task Group: 1.Formative assessment enhances mathematics achievement, particularly when: - Information is used to determine focus of instruction. - Expert teachers offer advice. - Computer-assisted instruction or peer tutoring is a component. 2. Research on students who are low achievers, have difficulties in mathematics, or have learning disabilities related to mathematics tells us that the effective practice includes: - Explicit methods of instruction available on a regular basis. - Clear problem solving models. - Carefully orchestrated examples/ sequences of examples. - Concrete objects to understand abstract representations and notation. - Participatory thinking aloud by students and teachers.

18 Findings of the Task Groups Instructional Practices Task Group: 3. The use of real-world problems as an instructional approach, in comparison to typical instruction, shows promise for improving students‘ performance on real-world problem solving tasks. - This has been shown in only a few areas of mathematics and at certain grade levels, using very specific “real-world” problem approaches. - When ideas are taught using “real-world” contexts then students’ performance on assessments involving similar “real-world” problems is improved. 4. Use of technology shows promise when: - Computer-assisted instruction supports drill and practice - Well designed tutorials are delivered through computer-assisted instruction - Learning is supported by the careful, targeted application of computer programming - More research is needed.

19 Findings of the Task Groups Instructional Practices Task Group: 5. All-encompassing recommendations that instruction should be child-centered or teacher-directed are not supported by research. 6. Mathematically precocious students with sufficient motivation appear to be able to learn mathematics successfully at a much higher rate than normally- paced students, with no harm to their learning.

20 Findings of the Task Groups Teachers Task Group: Teachers Make a Difference 1.Evidence shows that a substantial part of the variability in student achievement gains is due to who the teacher is. 2.Less clear from the evidence is exactly what it is about particular teachers--what they know and do--that makes them more effective.

21 Teachers Task Group: Teacher Content Knowledge 1.Research on the relationship between teachers’ mathematical knowledge and students’ achievement confirms the importance of teachers’ content knowledge. However, existing research does not reveal the specific mathematical knowledge and instructional skills needed for effective teaching, especially at the elementary and middle school level. 2.Direct assessments of teachers’ actual mathematical knowledge provide the strongest indication of a relation between teachers’ content knowledge and their students’ achievement. More precise measures are needed to specify in greater detail the relationship among elementary and middle-school teachers’ mathematical knowledge, their instructional skill, and students’ learning. Findings of the Task Groups

22 Findings of the Task Groups Teachers Task Group: Alternative Pathways into Teaching 1.Alternative pathways into teaching are not significantly different from one another in terms of the impact on teachers' knowledge and effectiveness. Teacher Incentives 1.Salary differential between teaching and other technical fields is large. 2.Exit rate of math and science teachers is greater than other teachers. 3.Location-based pay may keep experienced teachers in high- need schools. 4.Performance pay for teachers may enhance students’ achievement.

23 Findings of the Task Groups Teachers Task Group: Elementary Mathematics Specialist Teachers The task group recommends that research be conducted on the use of full-time mathematics teachers in elementary schools. These would be teachers with strong knowledge of mathematics who would teach mathematics full-time to several classrooms of students, rather than teaching many subjects to one class, as is typical in most elementary classrooms. This recommendation for research is based on the task group’s findings about the importance of teachers’ mathematical knowledge.

24 Findings of the Task Groups Assessment Task Group: Two Main Recommendations: 1. NAEP and state tests must focus on the mathematics that students should learn, with scores reported and tracked over time. 2. States and NAEP need to develop better quality control and oversight procedures to ensure that test items: - Are of the highest quality. - Measure what is intended. - Do not include design or wording problems that provide unintended sources of difficulties.

25 Findings of the Subcommittees Survey of Algebra Teachers Subcommittee: Mid-April – June 29th, 2007 collection period. Sample of 1,000 randomly chosen public school Algebra I teachers; 743 teachers responded. –Questions on student preparation, motivation, work habits, and skills. Asked teachers about insights on curriculum and instruction, and what would help all math teachers do a better job.

26 Findings of the Subcommittees Survey of Algebra Teachers Subcommittee: Main findings from the survey: 1. Ratings of student preparation: inadequate. 2. Ratings of curriculum & instruction: good. 3. Views on major challenges of teaching Algebra I: unmotivated students. Implications: –Attention to pre-algebra math is needed to: Remedy the specific skill deficiencies. Identify ways in which negative attitudes toward mathematics are developed.

27 Findings of the Subcommittees 1.U. S. mathematics textbooks are far too long -- often pages. Mathematics textbooks are much smaller in many nations with higher mathematics achievement than the U.S. Excessive length makes our books unnecessarily expensive and tends to undermine coherence and focus. 2.Publishers must ensure the mathematical accuracy of their materials. Engage mathematicians, as well as mathematics educators, in writing, editing, and reviewing these materials. Instructional Materials Subcommittee

28 Timeline to Final Report Final working meeting of the Panel took place on December 14 and 15 in Baltimore, MD. Draft Final Report submitted to U.S. Department of Education for review. Final Report delivered to the President by February 28, –Task Group and Subcommittee reports will be published with the Final Report.

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