In 2006, the United States President appointed the National Math Advisory Panel to advise on the best use of scientifically based research to advance the teaching and learning of mathematics. The Panel consisted of mathematicians, math educators, and cognitive psychologists who were charged with recommending ways in which to improve mathematics achievement for all students. The NMAP’s Final Report: Foundations for Success was published in March, 2008. The Panel’s findings regarding mathematics instruction and learning, and math education research, are equally relevant for Canadian students and teachers. Below, we quote some of the key findings of the NMAP.
Pg. xvi: The Panel took consistent note of the President’s emphasis on “the best available scientific evidence” and set a high bar for admitting research results into consideration. In essence, the Panel required the work to have been carried out in a way that manifested rigor and could support generalization at the level of significance to policy… In all, the Panel reviewed more than 16, 000 research publications.
From Pg. xvi – xxvii:
1) A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.
By the term proficiency, the Panel means that students should understand key concepts, achieve automaticity as appropriate (e.g., with addition and related subtraction facts), develop flexible, accurate, and automatic execution of the standard algorithms, and use these competencies to solve problems.
10) To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem solving skills. Debates regarding the relative importance of these aspects of mathematical knowledge are misguided. These capabilities are mutually supportive, each facilitating learning of the others. Teachers should emphasize these interrelations; taken together, conceptual understanding of mathematical operations, fluent execution of procedures, and fast access to number combinations jointly support effective and efficient problem solving.
11) Computational proficiency with whole number operations is dependent
on sufficient and appropriate practice to develop automatic recall of addition and related subtraction facts, and of multiplication and related division facts. It also requires fluency with the standard algorithms for addition, subtraction, multiplication, and division. Additionally it requires a solid understanding of core concepts, such as the commutative,distributive, and associative properties. Although the learning of concepts and algorithms reinforce one another, each is also dependent on different types of experiences, including practice.
14) Children’s goals and beliefs about learning are related to their mathematics performance. Experimental studies have demonstrated that changing children’s beliefs from a focus on ability to a focus on effort increases their engagement in mathematics learning, which in turn improves mathematics outcomes: When children believe that their efforts to learn make them “smarter,” they show greater persistence in mathematics learning. …
15) Teachers and developers of instructional materials sometimes assume that students need to be a certain age to learn certain mathematical ideas. However, a major research finding is that what is developmentally appropriate is largely contingent on prior opportunities to learn. Claims based on theories that children of particular ages cannot learn certain content because they are “too young,” “not in the appropriate stage,” or “not ready” have consistently been shown to be wrong. Nor are claims justified that children cannot learn particular ideas because their brains are insufficiently developed, even if they possess the prerequisite knowledge for
learning the ideas.
Teachers and Teacher Education
16) Teachers who consistently produce significant gains in students’ mathematics achievement can be identified using value-added analyses (analyses that examine individual students’ achievement gains as function of the teacher). The impact on students’ mathematics learning is compounded if students have a series of these more effective teachers.
17) Research on the relationship between teachers’ mathematical knowledge
and students’ achievement confirms the importance of teachers’ content
knowledge. It is self-evident that teachers cannot teach what they do not know. … Direct assessments of teachers’ actual mathematical knowledge provide the strongest indication of a relation between teachers’ content knowledge and their students’ achievement.
19) The mathematics preparation of elementary and middle school teachers
must be strengthened as one means for improving teachers’ effectiveness in
the classroom. This includes preservice teacher education, early career support, and professional development programs. A critical component of this recommendation is that teachers be given ample opportunities to learn mathematics for teaching. That is, teachers must know in detail and from a more advanced perspective the mathematical content they are responsible for teaching and the connections of that content to other important mathematics, both prior to and beyond the level they are assigned to teach.
20) The Panel 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 Panel’s findings about the importance of teachers’ mathematical knowledge. The use of teachers who have specialized in elementary mathematics teaching could be a practical alternative to increasing all elementary teachers’ content knowledge (a problem of huge scale) by focusing the need for expertise on fewer teachers.
21) Schools and teacher education programs should develop or draw on a variety of carefully evaluated methods to attract and prepare teacher candidates who are mathematically knowledgeable and to equip them with the skills to help students learn mathematics.
23) All-encompassing recommendations that instruction should be entirely “student centered” or “teacher directed” are not supported by research. If such recommendations exist, they should be rescinded. If they are being considered, they should be avoided. High-quality research does not support the exclusive use of either approach.
25) Teachers’ regular use of formative assessment improves their students’ learning, especially if teachers have additional guidance on using the assessment to design and to individualize instruction. Although research to date has only involved one type of formative assessment (that based on items sampled from the major curriculum objectives for the year, based on state standards), the results are sufficiently promising that the Panel recommends regular use of formative assessment for students in the elementary grades.
33) Publishers must ensure the mathematical accuracy of their materials. Those involved with developing mathematics textbooks and related instructional materials need to engage mathematicians, as well as mathematics educators, at all stages of writing, editing, and reviewing these materials.
Research Policies and Mechanisms
40) As in all fields of education, the large quantity of studies gathered in literature searches on important topics in mathematics education is reduced appreciably once contemporary criteria for rigor and generalizability are applied. Therefore, the Panel recommends that governmental agencies that fund research give priority not only to increasing the supply of research that addresses mathematics education but also to ensuring that such projects meet stringent methodological criteria, with an emphasis on the support of studies that incorporate randomized controlled designs (i.e., designs where students, classrooms, or schools are randomly assigned to conditions and studied under carefully controlled circumstances) or methodologically rigorous quasi-experimental designs. These studies must possess adequate statistical power, which will require
Practice and standard algorithms
Pg. 26: Debates regarding the relative importance of conceptual knowledge,
procedural skills (e.g., the standard algorithms), and the commitment of addition,
subtraction, multiplication, and division facts to long-term memory are
misguided. These capabilities are mutually supportive, each facilitating learning
of the others. Conceptual understanding of mathematical operations, fluent
execution of procedures, and fast access to number combinations together support effective and efficient problem solving.
Computational facility with whole number operations rests on the automatic recall of addition and related subtraction facts, and of multiplication and related division facts. It requires fluency with the standard algorithms for addition, subtraction, multiplication, and division. Fluent use of the algorithms not only depends on the automatic recall of number facts but also reinforces it.
Pg. 30: For all content areas, practice allows students to achieve automaticity of basic skills—the fast, accurate, and effortless processing of content information—which frees up working memory for more complex aspects of problem solving.