A major focus of the current mathematics and science education reforms is on developing "literacy;" that is, helping students to understand and use the languages and ideas of mathematics and science in reasoning, communicating, and solving problems. In many ways, these standards documents are far more voluminous and complex than any scope and sequence in place in school systems today. But these documents are meant to be used as frameworks which provide guidance in education reform - they are not the definitive sources articulating to teachers how education reform must occur in their classrooms.
Our plan in this discussion is to lay out the components of mathematics and science literacy as set down in the major reform documents and then, using selected how-to articles, to show how strategies and activities tried by math and science teachers have been used, or can be used, to promote math and science literacy among students. For pragmatic reasons only, our discussions often focus either on mathematics or science reform recommendations and examples. In doing this, we do not mean to imply that the elements of literacy in these disciplines are somehow separate or different. In fact, the separate discussions show how both the mathematics and science education communities, coming from different directions at different points in time, independently arrived at similar positions and many of the same recommendations regarding the ideas of literacy.
In support of this discussion of the components of literacy, we also provide samples of resources, materials, and services that teachers might find useful in promoting mathematics and science literacy in their classrooms. The how-to articles are meant to be quick-reads that can be applied or adapted to classrooms directly. These articles are included to make it easier to decide which ones might be of special interest. Other articles and documents are intended as sources of a more general background. These documents provide some of the research bases and rationales behind some of the reform recommendations. Finally, we have included other references and information on databases which are not directly cited in the discussion but might prove valuable as additional sources of classroom ideas.
During the last decade, the mathematics education community appeared to lack clear focus and a sense of direction. Although many conferences were held, papers written, and reports produced, there was not a general consensus regarding which direction mathematics education should head.
The Standards offer an organization of important mathematical topics and abilities by grade-level groups (Kindergarten - grade 4, grades 5 - 8, and grades 9 - 12). Throughout the Standards the emphasis is: "knowing" mathematics is "doing" mathematics. Knowledge should emerge from problem situations so that students have a strong conceptual basis for reconstructing their knowledge at a later time. Furthermore, problem solving situations develop mathematical literacy by: (a) providing motivation for developing concepts by establishing a "need to know;" (b) providing opportunities to read, write, discuss, and explore mathematical ideas; and (c) providing opportunities to make conjectures, test, and build arguments about a conjecture's validity. In short, the Standards describes a new curriculum for school mathematics in which students learn more, and often different, mathematics and in which methods of mathematics instruction are significantly different.
This notion of what a mathematics-literate American learner might be is parallel to that of a science-literate student. Similarly to the mathematics education community, science educators and scientists also began to grapple with the lackluster performance of our students relative to students world-wide.
Both the science education and mathematics education communities have in common the image of what discipline literacy requires for this century. Both have pushed the concept of literacy beyond that used in the reading literature. Whereas in reading it is common to find literacy defined by functional grade level performance, in science and mathematics the essence of literacy is increasing sophistication over the course of schooling. This represents a fundamental shift from literacy as a status notion to one of literacy as relative to the context of knowing - that is, the real world, the domains of the discipline, and specific applications. For teachers, this non-static notion of literacy presents considerable challenge and opportunity. And, when presented with standards documents that are not linear, not sequential, and not hierarchical in their recommendations, the teacher or teacher-advocate has the responsibility of translating the image statements into instructional materials, textbooks, learning activities, and pedagogical practices.
A major premise of both the mathematics and science standards is that what a student thinks, knows, and can do is greatly dependent upon how the student learned it. Research across a variety of disciplines indicates that students may learn best when they construct their own understanding of the material. This implies that teachers do not, and cannot, pass understanding to their students; instead, teachers can only engage students in activities from which students construct their own meaning. In short, learning is an individual activity fostered within the social context of teaching. This does not imply, however, that students must always "reinvent the wheel." For example, basic computation and algorithms were invented precisely so that people would not have to count on their fingers and toes to solve each problem. Formulas in science serve similar practical purposes. However, such activities should not dominate the mathematics or the science curriculum. Furthermore, computational procedures should be developed in contexts so that students perceive them as tools for solving problems not as problems to be solved.
In the mathematics and science reform literature, meaningful learning is promoted when students actively inquire. Inquiry in the reformed mathematics and science classrooms is more than just doing activities; it involves interacting with peers, teachers, people outside of the classroom and the school, and all kinds of resources. In the inquiry classroom prescribed in the reform literature, students work collaboratively on problems that are engaging and relevant; they ask questions; they access and use information from a variety of resources; and they challenge the ideas of others. Teachers, in turn, challenge their students about their observations, hypotheses, explanations, procedures, and evidence. We refer to this interactive kind of inquiry as "Inquiry" (i.e., inquiry with a capital I and in italics) to emphasize the importance of interacting orally and in writing as recommended in the reform movements. Inquiry is not restricted by student age, content, or context. Students at the earliest grade levels can use and develop the skills of mathematical and scientific Inquiry.
There should be a lot of investigation and debate going on in the science classroom. Inquiry in science is characterized by its demand for evidence, reliance on a blend of logic and imagination, expectation that scientists try to identify and avoid bias, rejection of authoritarianism, and recognition that science is a complex social activity. These characteristics of scientific inquiry are translated into instructional goals and standards in the reform literature. Similarly in mathematics, the theme of Inquiry is manifested in the standards and in the instructional activities. Whole-class discussions can provide students opportunities to synthesize, evaluate, and summarize strategies, ideas, and/or hypotheses. Small group discussions can provide opportunities to discuss and exchange ideas with peers, and individual work can help students to develop confidence in their own mathematical abilities. Different instructional approaches and activities such as those which develop students' Inquiry abilities will be discussed in the following sections.
In both mathematics and science, Inquiry can be "issues-based." This approach heightens the interest level, and therefore, the engagement of students. Throughout the Standards the importance of connecting mathematics to real-world problems (and hence utilizing an issues-based approach in teaching) is emphasized. Real-world problems with 'messy' numbers or too much or not enough information or that have multiple solutions, each with different consequences, will better prepare students to solve problems they are likely to encounter in their daily lives. The key to the effectiveness of the issues-based approach is to use a learning prompt which is appropriate and interesting to the learners. Teachers must also take care to use open-ended tasks for which there are multiple correct solutions. These open-ended tasks will then promote experimentation and exploration on the part of each student and will avoid the recall of particular facts, algorithms, or procedures.
An example of a series of issues-based lessons is Mathematics in Baseball in which students work in small groups investigating baseball statistics as well as other aspects of the game. Stimuli for small group discussions are provided in the article, which encourage students to exchange ideas, offer and receive constructive criticism, develop and test hypotheses, and make and correct mistakes in their small groups. Another example of a teaching module utilizing the issues-based approach is Involve the Community which confronts misconceptions students may have about the usefulness of mathematics and science in their own lives outside of school. Students go into the local community and interview someone to find out how that person uses mathematics on the job. Students are then responsible for developing mathematical problems described during their interviews, scheduling the person to speak to class, and writing a term paper concerning what they learned during their interview.
In science, the unity of perspective is not as evident. There are those who interpret science literacy to mean that life skills and citizenship are the key elements, not the rigorous scholarship or mastery of any specific science content or processes. This is countered by those who advocate for a focus on conceptual learning in the context of real-world problem solving.
Those who advocate for a life-skills and citizenship approach to science instruction are exemplified by Hurd's statement, "Modern science is driven more by societal needs than by theory." This societal perspective with its emphasis on life learning and citizenship places greater value on "knowing how" than on "knowing that" in defining science literacy.
The issues-based, societal perspective is the basis of the Science-Technology-Society approach. With its learn the science you need to know when you have a need to know it philosophy, there is no such thing as a fixed science curriculum or mandatory content or set of process skills. Students in an classroom identify personal, school, or community problems and issues and work collaboratively on a solution, learning and using appropriate science content and skills in the process of solving the problem or resolving the issue. Proponents claim that the approach develops science literacy more effectively than a content-driven curriculum because the problems are real and the learning is relevant to the students. The "medium is the message;" concepts and principles of science, however well-learned outside the context of a societal or personal issue, are not science at all, or at least not the kind of science worth learning. Teachers have reported numerous issues-based instructional activities across a range of issues.
Advocates of conceptual change learning view the elements of understandings and habits of mind in the definition of science literacy as key and argue for Inquiry that promotes meaningful learning of critical content and process knowledge in science. Teachers emphasize the importance of both the students and the teacher knowing what the student already knows and uses in everyday life situations and applications to engage students and provide context for learning science concepts and processes.
The major difference between the issues-based Inquiry and Inquiry for conceptual change is that specific ideas are targeted for instruction. For example, in science, playground equipment and amusement park rides are used to explore basic laws of force and motion. The properties of liquids and gases are investigated when students make their own carbonated soft drinks; or the life cycle of a common house fly is learned by studying the droppings of the classroom guinea pig. Teachers have used state-of-the-art technologies such as high-speed trains, entrepreneurial interests of students, "who-dunnit" detective scenarios and The Great Tape Robbery, and even current hit movies to capture student interest and to teach for conceptual change with regard to basic science concepts.
Similarly, important mathematics content is described throughout the Standards. For example, all three grade-level divisions include probability and/or statistics standard(s) as well as a geometry standard; and two grade-level divisions include measurement, estimation, algebra, and functions standards. As in science, Inquiry is a central theme in classroom instruction: throughout the Standards, verbs such as explore, justify, represent, solve, construct, discuss, investigate, describe, develop, and predict are used to convey this active physical and mental involvement of children in learning the content of the curriculum.
As in conceptual change learning in science, specific ideas, skills, and/or mathematical concepts can be targeted for instruction. For example, teachers have used common materials such as popcorn for developing data analysis skills, calculators to discover number patterns and hone estimation skills. Teachers have integrated math and art to develop geometric concepts, and math and science to develop geometric concepts and measurement and estimation skills.
In another lesson, students learn to apply probability models as well as use simulations to estimate probabilities concerning boy/girl birth ratios and the average number of children in a family. And student development of spatial imagery is targeted in the lesson Promoting Visual Imagery in Young Pupils. The magazines are useful resources for Inquiry for conceptual change-type instructional activities.
The challenge of managing Inquiry learning environments without sacrificing intellectual vigor is not insignificant. Student-centered learning is grounded in moderating the Inquiry-based classroom, which is prompted with exploration and stimulated with manipulatives in a way that is connected to the students' real world. By reflecting on students' growth in the disciplines, teachers will understand what pedagogical techniques work well to move students along on their learning journey.
Because the standards themselves represent the possibilities for instructional focus rather than the requirements for instructional focus, the teacher is placed in the important leadership role of selecting the optimal content to engage each particular group of students in the work of the discipline. The selection process must take into account local content goals, learning goals that ensure that the "habits of mind" of the disciplines are reinforced, resource availability, and the interest levels and developmental characteristics of the students. In short, optimizing learning in mathematics and science is not an algorithmic process.
The role of teacher as facilitator of learning begins to take on real meaning as the standards are implemented. And teachers seeking a cookbook for effective mathematics and science teaching will be sorely disappointed.
In clear and unequivocal ways, the role of the teacher as implementer of either the science or mathematics standards becomes more important in defining the learning journey for students than ever before. Because the standards documents are to be used as frameworks to guide mathematics and science education reform, teachers' professional judgment becomes more and more powerful as a force in defining the schooling experience for students. For this reason, those teachers who choose to or are chosen to teach mathematics and science must have an in-depth understanding of that which they are teaching.
In order for teachers to make decisions about what, when, and how to teach science and mathematics, they must have a rich understanding of the content and appreciate how knowledge in a content area is created, organized, linked to other disciplines and applied to real-world settings.
Teachers who effectively use the standards documents to guide daily instructional decisions must have specialized knowledge of how to teach the content (i.e., content-pedagogy), and they must recognize misconceptions and background knowledge that may make growing sophistication problematic. They must, of course, also be able to modify and reorganize to meet the needs of all learners.
The emphasis in the critical response skills is on argument and evidence. We contend that some or all of these "symptoms" should be used more or less as "ground rules" for Inquiry in (and out of) the classroom. For example, making activities such as checking that statements (both oral and written) do not intermingle fact and opinion or that celebrities aren't used as authorities in arguments part of what routinely happens in science class will develop the "habits of mind" so valued by reform proponents and reinforce the use of these same habits of mind beyond the classroom walls and school years.
Teachers who use an issues-based approach usually have little trouble creating an Inquiry environment. Personal and societal issues (e.g., landfills, toxic waste, AIDS, pollution) are readily controversial and lend themselves to investigation and argument. It is generally not difficult to find students who will take opposite sides of an issue or classrooms of students to find public groups with opposing positions. Teachers have used a variety of issues-based topics with great success, for example, investigations of ecological problems; Integrating Science, Mathematics, and Environmental Education Resource and Guidelines; The Curriculum File; Computerized Simulation as an Inquiry Tool, or problems dealing with death and aging; Debates: Verbal Encounters in the Science Classroom. What is challenging for the teacher who utilizes an issues-based approach is to keep students focused on those aspects of the argument that can be resolved with scientific and/or mathematical evidence, and/or which utilize scientific and/or mathematical reasoning, and to minimize those aspects of the argument that are strictly emotional, political, and/or personal. One must also keep in mind that just because an activity is issues-based does not imply that students will use or develop Inquiry skills. Students must generate and evaluate arguments on the basis of the scientific and/or mathematical evidence they gather and evaluate their conjectures using the rigorous standards of mathematical and scientific inquiry.
In classrooms where the emphasis is on understanding specified content and process outcomes, controversies are not at all obvious to students (or to most teachers) and any disagreements that arise are usually not as sensational as they are in an issues-based classroom activities. But the potential for argument and for debate nonetheless exists. Facilitating Inquiry when there is no "hot" social issue requires that the focus shift to the controversy embedded in science and mathematical ideas themselves. The alternative conceptions and/or misconceptions that students hold in a given topic or area are excellent sources of controversial ideas that can be investigated. For example, controversies from the history of science such as the phlogiston theory of heat or the geo-centric model of our solar system - can be used to stimulate Inquiry for conceptual change. Common misconceptions of mathematical concepts can be also confronted and explored, as can common student mathematical errors.
The strategy of simply suspending judgment or withholding the correct answer is very effective at stimulating discussion. Challenging students who hold different ideas to produce evidence of their respective positions is one way to stimulate debate, discussion, and investigation and turn routine lessons into Inquiry sessions. Simple modifications of cookbook activities (e.g., adding an open-ended question or posing an extra-credit question) or project work that requires students to work collaboratively is another effective way of facilitating Inquiry.
Inquiry can also be facilitated directly by using variations of the "student-teaching-students" idea. A strategy that has been around for centuries but still is effective today is "cross-age tutoring." But tutoring must be done in a hands-on rich environment so that legitimate Inquiry can take place. When older students inquire with younger students, both benefit from the experience.
As it does in any language, the ability to communicate well requires more than the knowledge of vocabulary and grammatical rules - one must also be fluent in using the language in both speaking and writing. Both the mathematics and science standards identify learning to communicate mathematically as an important goal for all students. As students communicate their ideas, they learn to clarify, refine, and consolidate their thinking. In other words, communication helps students to enhance their understanding of mathematics.
Learning the language of mathematics or science is not a simple task. For young children, representing is an important way of communicating mathematical ideas. Physical models can be used to represent and develop mathematical concepts. Furthermore, with young children the connections between thought and spoken word are usually stronger than those between thoughts and written symbols. Thus, children should be encouraged to relate their everyday language to mathematical language and symbols and to verbalize their thoughts and thinking processes.
As students progress in school, their mathematics communications should become increasingly sophisticated, that is, become more formal and symbolic. The introduction and use of technical symbolism should, however, evolve as a natural extension and refinement on the students' own language. Moreover, great care must be taken to ensure that students are aware of the connections between mathematical concepts and symbols, otherwise students are likely to view symbols as disparate, empty objects which are to be memorized and/or manipulated. All students should be provided opportunities to listen to, read about, write about, speak about, and reflect upon their mathematical ideas. It is not enough for students to merely write a response to an exercise or to "show all their work" on a problem. It is equally important that students be able to explain how they arrived at their responses as well as describe the difficulties they encountered during problem-solving processes. Students must constantly be encouraged to clarify, paraphrase, or elaborate on their mathematical ideas and relationships. These are means by which students enhance their mathematical understanding and teachers monitor their students' mathematical progress and understanding.
In reformed mathematics and science classrooms, literacy means being able to express oneself, defend one's ideas, and critically analyze claims both orally and in writing. Journaling, logging, and keeping a portfolio are as much a part of the reformed science and mathematics classroom regimen as they are of any arts or humanities classroom. Portfolio assessment strategies are especially effective in promoting Inquiry. When students have to explain, argue, and reflect on their work rather than simply to select responses, answer questions, and complete standard form assignments, both their writing and Inquiry skills are enhanced. Portfolio scoring rubrics based on evidence and logic of argument communicate to students the value and importance of Inquiry skills.
Student problems and misunderstanding can be revealed and corrected. Teachers can use student writing to identify which instructional techniques did/did not work and modify their techniques accordingly. Furthermore, through student writing teachers can examine what students have learned versus what those students think they have learned and use this information in assisting students in developing their metacognitive skills ("knowing how to learn"). However, it should be noted that the utilization of information gained from student writing is dependent upon the quality of both the writing prompts and the teachers. Teachers must collect, read, and give feedback to students frequently. And teachers must be ready to receive and use constructive (and perhaps not-so-constructive) criticism.
The move to emphasize patterns of argument and thought in the language of mathematics and science Inquiry has payoff potential across the curriculum. For example, there is strong evidence that analyzing the language and layout of good expository material enhances the general reading, comprehension, and critical thinking skills of younger students. High school science and mathematics teachers report great improvement in creative writing skills, critical thinking, student attitudes toward the subjects, and conceptual understanding when students keep journals and are encouraged to compose creative writing reports (e.g., case histories and resumes) in place of standard laboratory reports or other conventional tasks. However, it should be noted that attempts by teachers to translate results of alternate assessments, journal writings, and other creative writing into letter grades can be difficult and that racial differences may affect students' performance on open-ended items on standardized tests as compared to multiple-choice items was found.
The current reform agenda has the potential to dramatically alter the experiences that children have in science and mathematics classrooms in America. The standards documents themselves set the tone for a new understanding of science and mathematics literacy for all Americans. They present an image of the classroom that is Inquiry-oriented, activity-based, and engaging. The role of the teacher changes from that of disseminator of information to one of a mentor-scholar as children present ideas, challenge ideas, and reconceptualize these ideas.
This shift in the image of what a learning environment should look like calls upon teachers to take risks and to incorporate new instructional strategies into established pedagogical practices. The standards themselves, as a replacement for a scope and sequence or hierarchical curriculum, challenge the teacher to make professional judgments about what the appropriate content and context vehicles are for each group of students to maximize their learning. Teachers must take on instructional leadership roles. To do this, they must have both content and content-pedagogical knowledge. Background in the disciplines of mathematics and science is essential to support an effective Inquiry-oriented, student-centered classroom consistent with the standards documents. This expertise is also essential to the appropriate and reasonable assessment of whether or not students are becoming increasingly sophisticated in their understanding of important science and mathematics. This view of what quality work is in mathematics or science does not come from the standards. This understanding can and should come only from the expert professional classroom teacher. Thus, the ultimate challenge of the reform goals for classroom practice is for teachers to increase in their understanding of mathematics and science so that they have the appropriate frame of reference for identifying appropriate learning goals, selecting instructional resources to support students' construction of meaning, sequencing and pacing the activities in the learning environment to support learning, and monitoring the status of students' journey towards science and mathematics literacy.
Jeff C. Palmer is a teacher, success coach, trainer, Certified Master of Web Copywriting and founder of https://Ebookschoice.com. Jeff is a prolific writer, Senior Research Associate and Infopreneur having written many eBooks, articles and special reports.
