Critical Issue: Aligning and Articulating Standards Across the Mathematics Curriculum

ISSUE: Students often do not see any relationship between the mathematics they learn in school and the other subjects they study. As a result, many students do not see or understand the connection between mathematics and their daily lives. Some students may excel at performing mathematical functions, but there is evidence that they cannot apply those skills to solve everyday problems. Still other students find their K-12 mathematics instruction experiences to be fragmented or unnecessarily repetitious.

With the development of the Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989), schools have a new opportunity to ensure that students are engaged in full and coherent mathematical development. Educators can use the NCTM Standards as a framework to articulate mathematics experiences from kindergarten through high school. The NCTM Standards, which emphasize problem solving and connections between mathematics and other disciplines as well as students' daily lives, also can be used to stimulate the aligning of mathematics standards with schoolwide goals for meaningful and engaged learning and with goals of other disciplines addressed by schools.

OVERVIEW: New information about the learning process should guide schools as they move toward a more coherent math curriculum. For example, a growing research base supports the idea that students learn mathematical concepts best when the concepts are presented gradually over time. Educators must be sure to provide the conceptual base and experiences needed so that students will be able to understand new and more difficult concepts.

Research also has verified the importance of building on students' prior knowledge when helping them learn new concepts. This approach verifies not only the importance of articulating students' math experiences from kindergarten through grade 12 but also the importance of aligning students' math experiences with their other experiences both inside and outside school. Educators should keep in mind that the development of a child involves multiple settings--the home, the neighborhood, the school, and the workplace. People learn and grow in all of these settings. Students of all ages construct meaning about themselves and their world out of personal experiences, including the influences of culture (Caine and Caine, 1991; Beane, 1995). Learning is enhanced when curriculum and instruction integrate student experiences with the development of meaning. Iran-Nejad, McKeachie, and Berliner (1990) state, "The more meaningful, the more deeply or elaborately processed, the more situated in context, and the more rooted in cultural, background, cognitive, and personal knowledge an event is, the more readily it is understood, learned, and remembered" (p. 511).

Dorothy Strong, Director of the Bureau of Mathematics for the Chicago Public Schools, talks about how culturally relevant connections in mathematical content make lessons more interesting and applicable for students. [216k audio file] Excerpted from the video series Schools That Work: The Research Advantage, videoconference #2, Children as Problem Solvers (NCREL, 1991). A text transcript is available.

Because it develops in multiple settings, math ability can be used to solve problems in many different contexts and environments. Students should experience the ways that mathematical approaches to issues and problems can help them think through topics and problems in their other classes. Mathematics should not be isolated from other school subjects or the everyday lives of students.

Math educators should take a broad perspective when supporting the growth of mathematical thinkers. Students, when looking over the span of their entire school career, should sense that they have come to understand key mathematical concepts with an increasing depth and richness through the interconnection of curriculum with the rest of their lives. They should see their overall experience in mathematics as coherent and progressively developmental, not fragmented and repetitious.

GOALS: Mathematics education should encompass the following goals:

• All mathematics students are actively engaged in meaningful learning experiences that focus on building mathematical connections.
• All teachers are involved in ongoing learning and professional development focused on developing curriculum and enhancing instructional practices incorporating the NCTM Standards.
• All administrators are exercising instructional leadership in fostering a unifying vision of mathematics teaching and learning. They support and encourage mathematics teachers at all grade levels to work with each other, with teachers in other content areas, and with community members toward the goal of providing meaningful and connected mathematics learning experiences for all students.
• All parents and community members are meaningfully engaged in helping the school community rethink the learning and teaching of mathematics. They make decisions about the curriculum, instructional approaches, and appropriate assessment. They provide resources for supporting this new connected view of learning and teaching in mathematics, and they help students to see how ideas taught in mathematics classes are useful in everyday life.

ACTION OPTIONS:

Students can do the following things in order to become better learners of mathematics:

• Students should identify and make connections between mathematics and their own lives, including their work, their play, and broader issues in their community. They should not view mathematics as a subject that is dealt with only in school.
• Students should pursue opportunities to use mathematics in other curriculum areas.
• Students should be able to recognize, understand, and use relationships between different mathematical areas.

Teachers can do the following things in order to help students find meaning and connections within mathematics:

• Teachers should develop a comprehensive mathematics curriculum while considering the full range of concepts and instruction processes that are outlined in the NCTM Standards.
• Teachers should understand how mathematical ideas are learned over the course of time.
• Teachers should find innovative and creative ways to help students connect to the already existing network of ideas and experiences, thereby teaching mathematics in the context of meaning and application. Teachers should teach much more than procedures.

Tom Carpenter, Professor of Curriculum and Instruction at the University of Wisconsin-Madison, describes the importance of linking classroom material to the kinds of knowledge that students learn outside of school. [234k audio file] Excerpted from the video series Schools That Work: The Research Advantage, videoconference #2, Children as Problem Solvers (NCREL, 1991). A text transcript is available.

• Teachers should use technology to enhance, extend, and connect classroom experiences within and outside the classroom.
• Teachers should design alternative assessments, including performance-based tasks, that are consistent with the instructional goals and concept-oriented aims of an integrated and connected mathematics curriculum.
• Teachers should design mathematics writing projects that are based on issues and questions justifying the mathematics and that will increase student involvement in the classroom.
• Teachers should seek opportunities to work with others within and across grade/age levels at planning interdisciplinary curriculum units. These units might be designed for teaching a particular unifying theme or content topic.
• Teachers should participate in the writing and/or development of integrated problem-solving curriculum units for use throughout the year. In planning these units, teachers can use a yearlong chart that displays the agenda and goals for each quarter's mathematics instruction. Other components of this system can include a chart showing the progression of students' problem-solving strategies through the grades, and a cross-reference chart depicting how the material in the mathematics curriculum will relate to what is being studied in the other subject areas. These methods can be used by teachers when creating curriculum and instruction plans for all areas of mathematics.

Administrators can do the following things to support the alignment and articulation of standards in mathematics toward improved student learning.

• Administrators should be aware of of current educational research and should implement the changes necessary for creating coherent mathematics curricula, instruction, and assessment.
• Administrators should help communicate a new vision of mathematics that is consistent with a schoolwide view of engaged learning.
• Administrators should provide the leadership and resources necessary to support teachers who are working on curriculum alignment throughout the school system. Such resources could include professional development for teachers and other activities that support changing how mathematics is taught.
• Administrators should create different working models to support collegial planning and cooperation within the school community.
• Administrators should garner the support and engagement of parents and community members in implementing the new vision of mathematics.

Parents and community members can help enhance student learning in mathematics by doing several things:

• Parents can support teachers and schools in creating a vision of more meaningful learning. Such a vision will help guide the development of curricula, instruction, and assessment to reflect the NCTM Standards.
• Parents can support teachers and schools to create more meaningful and connected mathematics curriculum materials. They also can support local efforts to secure funds for these educational reforms.

Susan Gehn, a first-grade teacher at Cottage Grove Elementary School in Cottage Grove, Wisconsin, talks about how the parents of her students have supported her efforts to use different methods for teaching mathematics. [594k QuickTime slide show] Excerpted from the video series Schools That Work: The Research Advantage, videoconference #2, Children as Problem Solvers (NCREL, 1991).

• Video
• Video with captions (audio description available)
• Text transcript (fastest download)

• Community members can help to create partnerships between schools, businesses, and community facilities by serving as mentors and providing experience-based learning opportunities.
• Community members can provide learning opportunities that connect and align with classroom instruction, allowing students to utilize reasoning and problem-solving skills.

IMPLEMENTATION PITFALLS: School systems that have moved from central control of issues to local control, or "site-based" management, at individual schools may have trouble articulating standards between the different sites. A primary goal of local control is to increase teacher participation in decisionmaking and the subsequent implementation of new curriculum and evaluation standards. The downside of this initiative often is a lack of communication, articulation, and alignment of goals and philosophies between the different schools in the district.

One solution to this problem is to have staff from all the schools in the system participate in the planning and initiatives at every school. Another strategy for facilitating communication between schools in the district is to employ K-12 subject-area consultants. Such consultants can provide individual schools with direct assistance and also can connect curricula across schools when common issues surface.

Another obstacle to strong communication, alignment, and articulation in grades K-12 is that the preservice preparations of elementary, middle, and high school teachers often are very different. Elementary teachers usually receive a "student-centered" orientation and extensive training in child development, while high school teachers are more known for having subject-matter expertise. All teachers from all school districts need to discuss curricula, instructional strategies, and assessment practices to make sure the methodologies and goals of individual teachers do not conflict and hinder the mathematical education of the students.

Parents and community members might be suspicious of alignment and integration efforts if such efforts conflict with their own school experiences. Parents may want school records to show credit for particular courses so that college admissions officers will know what specific mathematic classes the children have taken.

One way to respond to these parental concerns is for schools to move toward evaluation systems based on performance and competency. In such systems, credit and grades are based on competence demonstrated by student performance (or by a set of performances in a portfolio of student work) instead of mere enrollment in a course. This type of system promotes an integrated curriculum. It can be used for teaching and learning mathematical concepts in settings beyond the school.

DIFFERENT POINTS OF VIEW: Some mathematics educators say that the curriculum should be moved forward. They argue that algebra should be taught to all students in eighth grade. Supporters of this view believe that a more rigorous curriculum will make students more competitive internationally.

Other educators believe that a more valuable goal for the middle school curricula is to develop a foundation for algebra while concentrating heavily on problem solving, geometry, probability, statistics, and measurement. The rationale for this view is that students need to explore the basic building blocks of algebra before they enter high school in order to make a smooth transition into the algebra curriculum.

Another question causing debate is whether there should be content specialists in elementary schools. Several mathematics professional groups recommend that mathematics specialists be placed at the head of elementary school programs. These specialists would oversee the entire mathematics instructional program, work with small groups of students, teach advanced math topics or special needs classes, and work with other classroom teachers. NCTM's 1984 position statement on Mathematics Leaders in Elementary/Middle Schools describes this view and the rationale behind it.

Opponents of the idea believe the role of the specialist is contrary to the goal of building mathematical connections-- especially interdisciplinary and real-life associations--because the specialist might be too specifically orientated to mathematics. In addition, most states currently have no certifications or endorsements for elementary school mathematics specialists. Changing the situation would require commitment from state certification agencies, state legislatures, schools, and the professional mathematics community.

ILLUSTRATIVE CASES:

Problem Solving and Critical Thinking in Mathematics: A K-8 Professional Development and Interdisciplinary Curriculum Development Project

CONTACTS:

The EQUALS Project
Lawrence Hall of Science
University of California
Berkeley, CA 94720
510-652-1823; fax 510-643-5757
e-mail: equals@maillink.berkeley.edu
WWW: http://equals./lhs.berkeley.edu

Mathematical Association of America
1529 18th St. N.W.
Washington, DC 20036
202-387-5200; fax 202-265-2384
e-mail: maahq@maa.org

Mathematical Sciences Education Board
2101 Constitution Ave. N.W.
Washington, DC 20418
202-334-2000

Midwest Consortium for Mathematics and Science Education
North Central Regional Educational Laboratory
1120 Diehl Road, Suite 200
Naperville, IL 60563-1486
(630) 649-6500, fax (630) 649-7600
e-mail: info@ncrel.org

National Council of Teachers of Mathematics
1906 Association Drive
Reston, VA 22091-1593
703-620-9840
Fax: 703-476-2970
e-mail: InfoCentral@nctm.org
WWW: http://www.nctm.org

References

Date posted: 1995