Welcome to the Math Literacy Website!

In order for students at DCE to be considered mathematically literate, they should be fluent in their basic skills, understand core mathematical concepts, and be able to apply their knowledge to other problems that they will face in their lives. The goal of this website is to provide resources and support to parents and students to help our students achieve this goal. For further explanation about the goal of math proficiency, see the following research article. If you have any comments or questions about the D.C.Everest math program or this website, please contact me.

Tony Pickar, Math Curriculum Coordinator D. C. Everest School District 6500 Alderson Street Weston, WI 54476 (715) 359-6561 ext. 4250 (715) 355-7220 fax tpickar@dce.k12.wi.us

National Research Council (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swatford, and B. Findell (Eds.) Mathematics Learning Study Committee, Center for Education, Division of Behavioral Sciences and Education.

• Excerpts: Mathematical Proficiency: five strands: (p 5)
• conceptual understanding—comprehension of mathematical concepts, operations, and relations
• procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
• strategic competence—ability to formulate, represent, and solve mathematical problems
• adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
• productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

• … the rational numbers present a major challenge, in part because rational numbers are represented in several ways (e.g., common fractions and decimal fractions) and used in many ways (e.g., as parts of regions and sets, as ratios, as quotients). There are numerous properties for students to learn, including the significant fact that the two numbers that compose the common fractions (numerator and denominator) are related through multiplication and division, not addition. (p 7)
• … proportional reasoning…has been described as the capstone of elementary school arithmetic. Proportional reasoning is sophisticated and complex; it needs to develop over many years. (p 8)
• The quality of instruction also depends on how students engage with learning tasks. Students must link their informal knowledge and experience to mathematical abstractions. (p 9)
• The integrated and balanced development of all five strands of mathematical proficiency…should guide the teaching and learning of school mathematics. Instruction should not be based on extreme positions that students learn, on one hand, solely by internalizing what a teacher or book says or, on the other hand, solely by inventing mathematics on their own. (p 11)

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students…know more than isolated facts and methods. Students often understand before they can verbalize that understanding. (p 118)

Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently….Connected with procedural fluency is knowledge of ways to estimate the result of a procedure. It is not as critical as it once was that students develop speed or efficiency in calculating with large numbers by hand. (p 121)

• …once students have learned procedures without understanding, it can be difficult to get them to engage in activities to help them understand the reasons underlying that procedure…Further, when students learn a procedure without understanding, they need extensive practice so as not to forget the steps. (pp 122-3)
• When skills are learned without understanding, they are learned as isolated bits of knowledge. Learning new topics then becomes harder since there is no network of previously learned concepts and skills to link a new topic to. This practice leads to a compartmentalization of procedures that can become quite extreme, so that students believe that even slightly different problems require different procedures. (p 125)

Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them…Although in school, students are often presented with clearly specified problems to solve, outside of school they encounter situations in which part of the difficulty is to figure out exactly what the problem is…(T)hey are likely to need practice in problem formulation as well as in problem solving. (p 124)

With a formulated problem in hand, the student’s first step in solving it is to represent it mathematically in some fashion, whether numerically, symbolically, verbally, or graphically…In building a problem model, students need to be alert to the quantities of the problem. (pp 124-5)

Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations…Many conceptions of mathematical reasoning have been confined to formal proof and other forms of deductive reasoning. Our notion of adaptive reasoning is much broader, including not only informal explanation and justification but also intuitive and inductive reasoning based on pattern, analogy, and metaphor. As one researcher put it, "The human ability to find analogical correspondences is a powerful reasoning mechanism." Analogical reasoning, metaphors, and mental and physical representations are "tools to think with," often serving as sources of hypotheses, sources of problem solving operations and techniques, and aids to learning and transfer.,, (p 129)

Research suggests that students are able to display reasoning ability when three conditions are met: they have a sufficient knowledge base, the task is understandable and motivating, and the context is familiar and comfortable…One manifestation of adaptive reasoning is the ability to justify one’s work…in the sense of "provide sufficient reason for." (p 130)

Productive disposition refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics. (p 131)

http://www2.lab.brown.edu/investigations/spotlight/archive/may01.html

Ten Research Findings from "Adding It Up" Wendy Gulley
Center for the Enhancement of Science and Mathematics Education (CESAME)
Northeastern University

Finding #1: Knowledge learned with understanding provides a foundation for generating new knowledge for solving unfamiliar problems. (Bransford et al., 1999). For example, students who understand place value and other multidigit number concepts are more likely than students without such understanding to invent their own procedures for multicolumn subtraction that others have presented to them. (Hiebert and Wearne, 1996)

Finding #2: In the brain, the way that new knowledge is organized and connected to previous knowledge is critical to the ability to retrieve and apply that knowledge. Learning with understanding leads to better organization and connections in the brain than does memorizing. (Donovan et al., 1999). When students learn without understanding they learn isolated bits of knowledge. Learning new topics is then more difficult because there is no network of previously learned concepts to link a new topic to. (Saxe, 1990)

Finding #3: A good conceptual understanding of place value in the base-ten system supports multidigit computational fluency, accurate mental arithmetic, and flexibility with numbers. (several references cited here, including Fuson, 1990)

Finding #4: Justifying and explaining ideas improves students’ reasoning skills and their conceptual understanding. (Maher and Martino, 1996)

Finding #5: Teachers need to expand the study of data beyond just graphing data. Four key processes are describing, organizing, representing, and analyzing data. (Shaughnessy et al., 1996)

Finding #6: Elementary students are capable of learning more geometry than is usually taught. Given enough early opportunities to learn about geometric figures, by the end of second grade they should be able to identify a wide range of examples and nonexamples of geometric figures, classify, describe, draw, and visualize shapes; and describe and compare shapes based on their attributes. (Clements, 2000)

Finding #7: Students emerging from traditional elementary school arithmetic have developed habits that make the study of algebra more difficult. For example, they have an orientation to execute operations rather than to use them to represent relationships, which leads to the use of the equal sign to announce a result rather than to signify equality. They also have trouble moving from an addition statement written horizontally to its equivalent subtraction statement (e.g., writing 35 + 42 = 77 as 35 = 77 – 42, or writing x + 42 = 77 as x = 77 – 42). (Thompson et al., 1994)

Finding #8: The most effective sets of activities for teaching about rational numbers spend time at the outset helping students develop meaning for the different forms of representation. Students work with multiple physical models for rational numbers as well as pictures, realistic contexts, and verbal descriptions. Time is spent helping students connect these supports with the written symbols for rational numbers. (Several references, including Cramer et al., in press)

Finding #9: How a teacher views mathematics and its learning affects his/her teaching practice, which in turn affects what students learn and how they view themselves as mathematics learners. (Thompson, 1992)

Finding #10: When class norms allow for students to feel comfortable doing mathematics and sharing their ideas with others, students see themselves as capable of understanding. (Cobb et al., 1995).