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What is Paso Partners - Integrating Mathematics, Science and Language: An Instructional Program?

Integrating Mathematics, Science and Language: An Instructional Program is a two-volume curriculum and resources guide developed by Paso Partners - a partnership of three public schools, an institution of higher education, and SEDL specialists.

On this page
- Assumptions Underlying the Materials
- Structure of the Guide
- Language Activities Related the Mathematics and Science Processes
- List and Recommended Sequence of K-3 Integrated Units

The resource is designed to help elementary school teachers organize their classrooms and instructional activities in order to increase achievement of Hispanic primary-grade children whose first language is not English. The guide offers a curriculum plan, instructional strategies and activities, suggested teacher and student materials and assessment procedures that focus on the acquisition of:
  • higher-order thinking skills to apply newly learned knowledge and understanding;
  • understanding of relations between mathematics and science concepts;
  • knowledge, i.e., specific items of information and understanding of relevant concepts; and
  • language to gain and communicate knowledge and understanding.

Motivational strategies and materials compatible with the students' own social and cultural environment are incorporated into the instructional materials to develop and enhance positive attitudes and values toward mathematics, science and language learning.

Spanish language translation: Accompanying each complete unit in English is a Spanish version of background information for the teacher, as well as a Spanish version of the formal introductory portion of the lesson cycle.

Assumptions Underlying the Materials

A number of assumptions about teaching and learning have guided the development of the materials.

Assumptions about Learning

  1. All children, even the very young, learn mathematics and science concepts by developing cognitive structures through interactions with the environment.
  2. In the process of learning mathematics and science, students "experience" instructional activities as an integrated whole, i.e., as an affective, cognitive and relevant activity.
  3. Language development is an integral aspect of the acquisition of mathematics and science concepts and skills. It becomes an even greater factor in cognitive growth and development for children whose first language is not the same as the language of school instruction. Effective learning occurs when the student acquires language in the context of academic instruction as well as in social interaction.
  4. Children learn mathematics and science constructively, i.e., children build or construct meaning by using their own experience and previous knowledge as a guide.
  5. Children acquire language within the context of everyday experience. Language concepts and skills are not learned in isolation, but rather as a consequence of interaction within a setting that is compatible with the experiential and cultural background of the students.
  6. Students construct concepts through experiences that involve using manipulatives, pictures, verbal interactions and other models representing the concepts to learn.
  7. Mental structures effectively develop through educational activities that allow students to explore, investigate, apply and solve problems related to "tentative constructs" that students modify during the learning process.
  8. In learning mathematics and science, as well as in acquiring and developing language, the students assimilate experiences into a construct that is available to them through subjective representation. However, the meaning of the representation must be consistent with experience, with the meaning of related constructs and with conventional meanings constructed by others.
Assumptions about Teaching
  1. The design and the implementation of an effective instruction activity include cognitive, affective and relevant aspects of the social and cultural context in which the science, mathematics and language concepts develop.
  2. Teachers help create effective and appropriate mathematics, science and language constructs through a variety of approaches that include:
    • spontaneous opportunities that provide and provoke suitable questions, conflicts, material and explanations to induce inquiry;
    • inductive and deductive sequences that provide students relevant examples to help them extract the common features and important ideas of a concept or generalization; and
    • pragmatic or practical opportunities for students to grapple with and solve real-world problems that students discuss with their peers and the teacher in order to verify and affirm their thinking.
  3. To assist students in developing mathematics, science and language constructs, teachers provide many carefully selected and structured examples that facilitate abstraction of common features to form a concept. Also, teachers present interesting and challenging problems. Teachers use manipulatives, pictures, graphs and verbal interactions to support and encourage learning.
  4. Teachers facilitate acquisition of mathematics and science concepts by children whose first language is not English through appropriate language development strategies that assume a language-rich environment in which students may use either the home language (e.g., Spanish) or English or both to communicate knowledge and understanding.
  5. For children whose first language is not English, teachers give specific attention to the development of specific concepts (science and mathematics, in this case) within the overall context of both Spanish and English language development.

Structure of the Guide

The guide is bound into two volumes. Volume One contains materials for use in Kindergarten and Grade One. Materials in Volume Two are for use with students in Grades Two and Three. Depending on the students' academic backgrounds and local curriculum expectations, the materials for each grade level may provide a full academic year of instruction. Each volume contains an introductory section and three units for each grade level.

Structure of each Unit
Each unit is designed to assist teachers in offering up-to-date science and mathematics content, along with appropriate language usage, through teaching and learning strategies that will excite children about the world of mathematics, science and language. The selection and arrangement of the material is planned to engage children's natural inquisitive nature and to stimulate them to investigate, explore and learn. Teachers are helped to create dissonance in familiar situations in order to stimulate questioning, hypothesizing, exploring and problem solving.

Each unit contains three types of materials: (1) unit overview materials and background information for the teacher, (2) the lessons and (3) an annotated bibliography and list of teacher reference/resource materials.

Spanish language translation. Preceding each complete unit in English is a Spanish version of background information for the teacher, as well as a Spanish version of the formal introductory portion of the lesson cycle.

Unit overview materials and background information for the teacher. Presented first in the unit is a recommended list of content and/or skills students should have as Prior Knowledge before initiating unit activities. Next Specific Mathematics, Science and Language Objectives are listed followed by a Topic Concept Web. The web shows relationships among the various science content elements that teachers will present in the unit. In turn, the web prompts the identification of two major ideas, one in science and one in mathematics, that the class will develop in each lesson. It also encourages teachers to view teaching as providing children opportunities to develop cognitive structures that are more global and complex than those that students can demonstrate by performance on objective-defined tasks. Therefore, the application, or problem-solving, phase of the lessons takes on a specific character and increased importance - it allows the student and the teacher to look for dimensions in understanding that go beyond the level that can be universally required of all students. There is no vertical or horizontal "cap" or "ceiling" in thinking that circumscribes the students' progress.

Next is a list of key Vocabulary items, in both English and Spanish, that the teacher will use in presenting the unit. The students will gain an understanding of the terms and may incorporate some, or most, of them into their active vocabularies.

The Teacher Background Information section, which follows the Vocabulary section, contains science and mathematics content. This content, also in both English and Spanish, is provided as a ready reference for teachers to draw upon as they implement the unit.

Next is The Lesson Focus that lists each of the Big Ideas presented in each of the lessons. Each Big Idea is stated as an overarching concept, or principle, in science and/or mathematics that generates the lesson activities. The Big Idea is what each student is to construct. The construct has many other ideas that relate to it, both in mathematics and science, thus forming a web of ideas. The construct, however, develops within a language context - either in English or Spanish - in order to formalize the concept. Once assimilated, the Big Idea can facilitate students' future learning in related content areas. Thus, the Lesson Focus, together with the array of objectives, gives the teacher a view of the extent and direction of development of the Big Idea in each lesson.

Following The Lesson Focus is an Objectives Grid displaying the unit objectives by content area and by lesson activity. Objectives, in and of themselves, cannot dictate the scope of the instruction. Learning takes place when the students "experience" instructional activities as an integrated whole, i.e., as an affective, cognitive and relevant activity. Thus, the grid serves to provide direction and indicators of student progress. The objectives are used to develop assessment procedures by which to measure, in part, student achievement.

Lesson Design
Each lesson design assists the teacher in developing the Big Ideas selected for a given lesson. The term "lesson" as used in this guide means a set of activities selected to teach the Big Ideas. It is not meant to convey the notion that the material included in a "lesson" is to be taught within a single period of time on any given day. One "lesson" may extend over several days.
Each lesson provides the instructional context and the activities for the students to acquire the concepts, or build the constructs, contained in the lesson's Big Ideas. The lesson does suggest a sequence in which to implement the activities, but there is no "single" sequence or a given time limit in which to present the unit. Indeed, a number of the units require previous preparation on the part of the teacher, and in some cases on the part of the students. Some units, for example, require the students to collect, organize and summarize data and then to apply their findings. This process may require a period of three or four weeks. Nonetheless, prior to initiating the unit, teachers should construct an overall and day-to-day schedule for the implementation of the unit.
The lesson's content develops through a process that reflects a cycle. The process moves through various phases of the learning cycle. Learning cycles to facilitate the organization of science and mathematics instruction have been proposed for some years; many cycles incorporate an inquiry approach to learning with emphasis on problem solving. Typically, a learning cycle includes an experimentation phase during which the learner actively experiments with concrete materials to develop, or "construct", an idea. Although scholars vary in their opinions as to the required nature, design and number of such phases, all include at least three phases: experimentation, concept introduction and development, and application.

The Lesson Cycle
For the purpose of this guide, a five-phase lesson cycle has been employed:

  1. Encountering the Idea
  2. Exploring the Idea
  3. Getting the Idea
  4. Organizing the Idea
  5. Applying the Idea
Each phase of the cycle is described briefly below.

Encountering the Idea, or developing a "readiness" state, is the first phase in the cycle. During this time the teacher provides a background, or enabling structures, to facilitate the development of "new constructs." This phase of the teaching cycle is important for students whose early childhood experiences may not have been sufficiently varied to provide them with some of the necessary underlying concepts on which to build the Big Ideas that the lesson promotes. Therefore, this cycle shapes a backdrop on which to develop the new ideas. Additionally, the readiness activities alert the students to the direction of the lesson by providing provocative questions and conflicting situations designed to bring the students into an exploration perspective.

Because language development is a fundamental co-requisite for learning mathematics and science concepts, processes and skills, many of the lessons begin with literature (e.g., oral stories, children's books) and discussion activities that set the stage for posing questions and presenting conflicting situations related to the mathematics and science Big Ideas that are the focus of the lesson. The use of well-selected literature, in addition to being an effective tool in language development, is an effective motivational strategy. Other language development strategies are presented below in the section, Language Activities Related to Mathematics and Science Processes.

Exploring the Idea, or experimentation, is the phase in which learners are involved with concrete or familiar materials in activities designed to have them encounter new information that they can assimilate in their attempt to find responses to the questions posed earlier and/or to hypothesize a resolution to the conflicting situation presented. During this stage, the learner explores the new ideas through the use of materials in learning centers, with the teacher providing relatively little structure. As students realize that there are new ideas they have not dealt with previously and that produce some confusion, doubt or interest, they discuss among themselves and with the teacher what these ideas may mean. At this point, the teacher moves the students into the next phase of the cycle.

Getting the Idea, or concept introduction and development, is the phase in which the teacher helps the learners assimilate and accommodate the new information into a new structure that signifies the development of a new understanding. The students begin to work with new words conveying the new concepts. They work with new ideas in many different ways to ensure that a new idea is valid. The main emphasis during this phase is to see what is happening. What do we know? How do we know this is true? How can we explain this? Students may want to brainstorm and ask related questions, or they may choose to go back to the exploration or experimentation phase to validate the new ideas.

Organizing the Idea is the phase in which the students consciously consider the new ideas in their own right. They attempt to understand a new idea as a whole. New terminology, notation and symbols are introduced at this time. Students may then express their ideas and opinions through a variety of activities.

During this phase, the students may relate the new ideas to associated ideas in other areas of subject matter. They make new connections, generalizations and abstractions. They may decide that the best manner to organize and communicate the new ideas is through charts, tables, number sentences, graphs, diagrams or verbal and written explanations. Thus, the information is organized in a logical and quantitative manner. The students may report the results of their experiments, observations, conclusions and interpretations to the class. Students may to do additional reading or listening to tapes. Once the students have grasped the concepts, they are ready for the application phase of the lesson.

Applying the Idea is the phase in which students develop a broad grasp of the concepts. In this phase the students relate the new ideas to their own world - to something "real" - and to associated ideas in other areas of subject matter. They are then able to solve problems and answer related questions. They may also formulate their own problems.

Assessment of Student Achievement is ongoing on an informal basis throughout the lesson through teacher observation of the students' interactions and behaviors. Assessment strategies are provided in the final phase of each lesson or unit to assist the teacher in determining the extent to which the students have grasped the Big Ideas presented in a given lesson and/or unit.

Language Activities Related to Mathematic and Science Processes

Because language development is a fundamental co-requisite for learning mathematics and science concepts, processes and skills, the lessons in many instances begin with literature (e.g., stories, books) and discussion activities that set the stage for posing questions and presenting conflicting situations related to the Big Ideas in mathematics and science that are the focus of the lesson.

Language development strategies specifically related to mathematic and science processes were incorporated into the lessons. Some examples of these are described briefly below.

Sequencing. The students tell or write a story, indicating the sequence of events by using ordinal numbers. They may also use such words as "then", "next", and "finally" to show sequence. The students may take a nature walk around the school and report their observations in order of occurrence.

Questioning. In the initial stage of a unit the students may list, in the form of questions, information that they would like to have about the topic. As they proceed through the unit and gather further information, they may record answers to the questions that they formulated.

Comparing/contrasting. Student may design and make charts, graphs or diagrams that compare or contrast two concepts. For example, the students may use Venn diagrams to compare and contrast spiders with insects.

One-to-one correspondence/counting. In comparing objects, students use comparative adjectives (e.g., "longer", "shorter", "bigger", "smaller"). In comparing groups or sets in preparation for counting, the students begin to use the notion of "more than" and "less than." In making these comparisons, they may compare two groups physically by laying them side by side. In increasing the accuracy of their statements, students can say, for example, "The tiger cage in the zoo has three tigers, and the bear cage has six bears; the zoo has more bears than tigers." They can put three tigers alongside six bears, show that the three tigers are "tied" with three bears and that there are three extra bears. They conclude that there are three more bears than tigers, and that six is three more than three.

Predicting/hypothesizing. During the initial stage of a unit, and after the students have listed the questions that they would like to answer, they hypothesize answers or solutions to as many of the questions or problems as they can. During the implementation of the unit, they explore hypotheses and confirm or reject them as they gather evidence. The students verbalize their reasons for confirming or rejecting the hypotheses.

Validating/persuading. During problem-solving sessions, the students study the nature or character of the evidence they can use to confirm or reject a hypothesis. They suggest reasons why in some cases one negative example is sufficient to reject a hypothesis, while in other cases several positive examples are not sufficient to confirm or reject a hypothesis.

Conferring. Students ask for a conference with the teacher and/or other students to discuss or exchange opinions about an important, a difficult or a complex matter. For example, a student is preparing to write in her journal but needs clarification about an idea. She asks the teacher to meet her at the "conference table" (which is inaccessible to other students for the duration of the conference) in order to discuss her ideas prior to writing about them in her journal. The student may ask that another student join the conference, particularly if the students have done the work collaboratively. The student initiates the conference, gives it direction and decides when the purpose of the conference has been met. A student may also request a conference for the purpose of assessing her achievement or progress.

List and Recommended Sequence of K- 3 Integrated Units

Grade K and 1 Integrated Units

     Grade K
          Five Sense
     Grade 1
          Plants and Seeds
          The Human Body
          Good Health

Grade 2 and 3 Integrated Units

     Grade 2
          Sun and Stars
     Grade 3
          Simple Machines
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