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
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:
- 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
- higher-order thinking skills to apply newly learned knowledge and
- understanding of relations between mathematics and science concepts;
- knowledge, i.e., specific items of information and understanding of relevant
- 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
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.
A number of assumptions about teaching and learning have guided the development
of the materials.
Assumptions about Learning
Assumptions about Teaching
- All children, even the very young, learn mathematics and science concepts
by developing cognitive structures through interactions with the environment.
- 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.
- 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
- Children learn mathematics and science constructively, i.e., children build
or construct meaning by using their own experience and previous knowledge as a
- 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.
- Students construct concepts through experiences that involve using
manipulatives, pictures, verbal interactions and other models representing the
concepts to learn.
- 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.
- 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.
- 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.
- 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
- 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.
- 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.
- 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.
- 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
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
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.
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
The Lesson Cycle
For the purpose of this guide, a five-phase lesson cycle has been employed:
Each phase of the cycle is described briefly below.
- Encountering the Idea
- Exploring the Idea
- Getting the Idea
- Organizing the Idea
- Applying the Idea
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
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.
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
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
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
Plants and Seeds
The Human Body
Grade 2 and 3 Integrated Units
Sun and Stars