astronomy in nrce

8/21/2019 Astronomy in NRCE

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Review of Astronomy Education Research

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Astronomy in the National Science Education Standards

Manuscript published in

Journal of Geoscience Education

vol. 48(1), pp. 39-45 (2000)

Jeffrey Paul Adams

Timothy Frederick Slater

Montana State University

Department of Physics

Bozeman, MT 59717

Tel. 406 994 7835 (JA)

Tel. 406 994 1693 (TS)

Fax. 406 994 4452

Email: adams @physics.montana.edu

Email: [email protected]

About the Authors: Jeff Adams and Tim Slater team-teach Introductory Astronomy

at Montana State University. They have just finished their first text-book that

includes innovative small learning group activities for the large lecture course

titled, Mysteries of the Sky: Activities for Collaborative Learning Groups,

through Kendall Hunt Publishing.


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Astronomy in the National Science Education Standards

ABSTRACT

The NRC National Science Education Standards provide a framework to design curriculum for K-12

astronomy education. At grades K-4, students should be learning about the objects and motions in the

sky from a geocentric perspective. At grades 5-8, students should be learning about the motions of the

solar system from a heliocentric perspective. At grades 9-12, students should be learning about stellar

evolution and the structure of the Universe. In support of instruction, an extensive review of the

literature demonstrates that existing research on student learning addresses only a small subset of the

astronomy objectives prescribed by the NSES and does not address age-appropriate conceptual

development. A need exists to develop and rigorously assess a collection of age-appropriate

assessment instruments based on research into student understanding of fundamental astronomical

concepts. These instruments would serve to clearly define the expected cognitive levels of the specific

NSES objectives and provide a means of assessing curricular materials that claim to be aligned with the

NSES. Because of the complex nature of these concepts, the scientific community must be active

participants in this process.

INTRODUCTION

The 1996 National Science Education Standards (NSES) proposed by the National Research

Council aggressively outline the rules for effective classroom instruction, age-appropriate guidelines for

curriculum materials development, authentic assessment procedures, and professional development

programs for teachers (NRC, 1996). In addition to the emphasis on describing science through unifying


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concepts and processes, the NSES provide specific learning objectives. In terms of K-12 astronomy, a

careful overview of the NSES suggests eleven major astronomy objectives (see Table 1). These

objectives include describing the objects and motions of the sky (grades K-4), the characteristics of

gravity and the solar system (grades 5-8), and the origin and evolution of stars, galaxies, and the

Universe (grades 9-12). These objectives are found in both the earth/space science content strand and

the themes of unifying concepts and processes. However, the recognition of astronomy as an important

part of reformed science education is only the first step toward improving the quality and quantity of

astronomy education in K-12 classrooms.

Now that the NSES have clearly articulated the content standards, the next challenge is to

develop age-appropriate assessment instruments that both define the cognitive level of the standards and

critically assess astronomy education materials and professional programs that claim to be aligned with

the NSES. Carefully constructed assessment instruments are necessary to define what it means for

students to know a stated objective at different levels. A knowledge-level assessment would require,

for instance, that students state: Sky objects have properties, locations, and movements that can be

observed and described. This would indicate nothing but an ability to memorize. A synthesis-level

assessment might ask students to describe the motions of objects in the sky as viewed from Mars.

Experience in the physics education community has demonstrated that how we choose to assess student

learning helps to define the objective itself. Such assessment must necessarily be derived from an

ongoing educational research effort at understanding the common preconceptions held by students and

the age-readiness of students to understand important ideas at various cognitive levels. These

assessment instruments will fundamentally influence Standards-based curriculum development.


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Recently, several nationally recognized projects have begun materials development under the

banner of national-standards-based astronomy education. Rigorous work in physics education has

demonstrated that instructional interventions can be effectively created only by using sound educational

research as a guide for curriculum development (Shaffer & McDermott, 1992b). As a result, there is a

timely need for a rubric for determining the degree of alignment of such materials with the NSES and

their effectiveness in achieving desired learning outcomes (as defined by the assessment). The same

applies to federally funded teacher-enhancement programs in astronomy education.

This article provides: a summary of the NSES astronomy objectives; a review of the astronomy

education literature, focusing specifically on preconceptions, misconceptions, nave conceptions, and

alternative frameworks in K-12 astronomy; a discussion the impact of assessment procedures; and a

call for a focused research agenda for astronomy educators. This article thus provides perspective and

guidance for teachers of astronomy, curriculum developers, and science education researchers.

NSES Astronomy Objectives

A careful review of the NSES content strands reveals numerous places where astronomy and

space science concepts are either explicitly or implicitly described as an important part of a students

science education. Explicitly, in the earth and space sciences content strand, the K-12 astronomy

objectives can be grossly organized into eleven age-appropriate conceptual objectives, which are

shown in Table 1. A discussion of the equally important unifying concepts and processes in science,

science as inquiry, relationships to life and physical science, science and technology, nature of scientific

knowledge, and the social perspective of science is beyond the scope of this manuscript.

Grades K-4 Objectives. From the perspective of age-appropriate instruction, geometrical and

abstract astronomy in the early grades is indeed problematic. The NSES suggest that the goals of


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astronomy education at the K-4 level should focus on describing the properties, locations, and motions

of sky objects from a geocentric perspective. These sky objects include the Sun, Moon, stars, clouds,

birds, and airplanes. Some parents might believe that a child who can recite 101 fun facts about the

planets is impressive, but the NSES focus on the underlying processes and themes of science instead of

facts. This is not to imply that memorized factual knowledge is not importantplanet names can be,

and should be, learned in the same way as the names of farm animals, the multiplication tables, and the

months of the year. However, knowing the names and the order of the planets at the knowledge level is

only the first step in an astronomical knowledge base and therefore must not form the core of the

learning objectives. Instead, as has been done with NSES, the objectives should be framed so students

can be expected to fully participate in the process of science at an age-appropriate level. For example,

at the K-4 level, it is appropriate for students to observe and chart the changing phases of the Moon.

To memorize the phase cycle would disengage students from the scientific process and be wholly

inappropriate; it would be equally inappropriate though for K-4 students to be expected to create

mental models of the abstract geometry of the Sun-Earth-Moon system. Likewise, seasons should be

studied from a geocentric perspective with an emphasis on student observations. For instance, K-4

students should understand that winter results from the Sun being low in the sky as opposed to a tilted-

Earth explanation.

Grades 5-8 Objectives. In the culture of US education, the last formal astronomy for the vast

majority of students is in grades 8 or 9 in the context of an earth-science curriculum. It is at this level

that the NSES expect students to describe the solar system from a heliocentric (Sun-centered)

perspective. As shown in Table 1, students should be using the heliocentric motion of the solar system

to explain the phenomena of day/night, seasons, eclipses, and lunar phases. At this stage, students


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should be comparing and contrasting the solar system objects in terms of terrestrial/non-terrestrial,

satellite systems, rotation/revolution, size/mass, and additional characteristics that students find

important.

The NSES suggest that the concept of gravity should be introduced at the 5-8 level. Students

should have an accurate model of a spherical Earth where gravity holds people to Earths surface and

explains tides. Through a heliocentric study of solar-system motions, students should understand that

gravity causes the planets to orbit in nearly circular orbits as described by Keplers Laws. This

conceptual foundation of gravity is revisited in secondary level objectives when describing the

characteristics of galaxies and star clusters

Grades 9-12 Objectives. The four secondary-level astronomy objectives focus on the

observation, origin, evolution, and characteristics of the Universe beyond the solar system. Possibly

even controversial in some schools, the NSES call for students to understand current scientific

explanations of the origin and evolution of the Earth (multiple atmospheres), the Solar System (nebular

hypothesis), the elements (nucleosynthesis), and the Universe (the Big Bang). Comprehension of the

observational and theoretical evidence for exotic objects such as neutron stars and black holes is indeed

abstract; yet, these are the same topics that provide cover stories for popular newsmagazines. In many

ways, todays students are more familiar with these news stories than basic naked-eye astronomy.

Interestingly, except for portions of eight and ninth grade earth-science and the very small

percentage of students taking high school physics or, even rarer, astronomy, in grades 11 or 12, there

currently appears to be little astronomy taught in US schools at the secondary level. The NSES provide

a needed impetus for increasing the amount of astronomy included in the secondary curriculum.

The Importance of Assessment


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A highly pertinent example of how an assessment instrument can redefine what it means to

know the meaning of a concept is found in the closely related field of physics education, where a

single test known as the Force Concept Inventory (FCI) has catalyzed a complete rethinking of what it

means for students to know Newtons Laws of Motion (Hestenes, Wells, and Swackhammer, 1992).

The great attention afforded this 29-item (30 items in the revised version) multiple-choice test derives

from four factors: (1) there is widespread agreement within the community as to the importance of the

content being assessed; (2) the test items are deceptively simple leading most instructors to greatly

overestimate the likely success rate of their students; (3) the results are highly consistent across a large

constituency ranging from high school classes to courses for physics majors at large research institutions;

and (4) student responses are highly resistant to traditional instruction. Although investigators within the

growing community of physics educators were well aware that correct solutions to traditional problems

often masked deep-seated misconceptions (Arons, 1983; Arons, 1984; Caramaza, McCloskey &

Green, 1981; Clement, 1982; Goldberg & McDermott, 1986; Hallound & Hestenes, 1985;

McDermott, 1984; Trowbridge & McDermott, 1980), the FCI brought this startling fact to the attention

of a much wider audience of physics educators (Mazur, 1997). This has resulted in a number of

important curriculum reform efforts whose success has been, at least in part, demonstrated through

much improved FCI gain scores, when compared to traditional instruction (Hake, 1996). What is

important is that a carefully constructed, highly validated assessment instrument based on sound

research into student understanding has provided guidance in defining the level of understanding that

should be encouraged through appropriate instruction. Although there has been some enlightening work

done concerning misconceptions in astronomy (see below), there are many gaps in the current research


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base that must be closed before age-appropriate assessment instruments can be fully developed to

define the NSES astronomy objectives and assess programs purporting to address them.

Review of the Literature

Given the popularity of astronomy, it is somewhat surprising to find the amount of research in

astronomy education to be limited compared to the quickly growing volume of conceptual research in

physics education. What does exist though is highly provocative and demonstrates the need for

additional research. This review is organized in terms of the NSES astronomy objectives summarized in

Table 1.

Objects and Changes in the Ear th and Sky. To date, most of the educational research into

student astronomy misconceptions concerning sky objects and motions has been based on a heliocentric

perspective. As described in the next section, enormous attention has historically been paid to

explanations and models for lunar phases and the seasons. The most often noted study of astronomy

misconceptions was conducted in the1980s by Sadler (1992). Often referred to as the Project

STAR study, in reference to the NSF Instructional Materials Development project initiated following

the study, Sadler found an overwhelming number of high school students could not correctly answer

simple multiple-choice questions regarding the predictable motions of the day and night sky.

Schneps interviewed 23 adults standing in line at a Harvard commencement ceremony in a

project that is known as the Private Universe video (Schneps, 1987). He found that only two of

twenty-three adults could adequately explain why it is hotter in the summer than in the winter; the most

cited reason was the Earths distance from the Sun. It seems that this conception is universal. Atwood

found that of 49 elementary-education majors, 38 could not adequately explain the causes of the

seasons in a written narrative (Atwood & Atwood, 1996). Of the 49 students, 18 cited distance to the


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Sun as being important, 10 described the Earths northern hemisphere being closer due to tilt, and the

remaining describing a variety of reasons such as Earths spin or the jet stream. Atwoods study went

on to show that 42 of the 49 could not verbally explain the causes of the seasons using models either.

Several reports in the literature confirm Sadlers results on inaccurate conceptions from a

heliocentric perspective (Philips, 1991). However, there is little work describing student conceptions of

the geocentric sky. Sadlers survey-study reported both that students often think the Moons phases

are caused by clouds and that students do not understand the nightly motions of the stars. Working

independently, Reed (1972) and Chamblis (1990) found that students working with celestial spheres or

in planetarium environments could substantially improve their geocentric understanding of the sky.

Rollins and others found that only 79% of Texas high school students could adequately answer

questions about concepts of day and night (Rollins, Denton, & Janke, 1983; Frayer, Schween-Ghatala

& Klausmeier (1972).

Ear th i n the Solar System. According to the NSES, students in the upper elementary grades

should be changing from viewing the Universe in a geocentric perspective to a heliocentric perspective.

Sadlers study, although very well known, was not the first study of its kind. In 1976, Nussbaum and

Novak (1976) developed a research design to investigate the progression of student notions of the

Earth. It appears that student conceptions of gravity are closely tied to a conception of a spherical Earth

(Vosniadou & Brewer, 1989). Treagust and Smith (1989) interviewed twenty-four grade-10 students

and, from their interviews, developed a questionnaire administered to 113 students. They found that

students think gravity is affected by temperature, gravity is selective about what and when it affects, and

gravity is stronger at great distances in order to pull things back. They also found that students believe

that planets with slow or no rotation have little or no gravity as do planets that are far from the Sun. The


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facts were confirmed by Osborne and Gilbert (1980)Philips (1991) reports that students think that

gravity needs air to exist. Additional studies demonstrated that these conceptions are consistent across

a variety of cultural and educational backgrounds (Nussbaum, 1979; Mali & Howe, 1979; Sneider &

Pulos, 1983; Baxter, 1989).

Lunar phases are traditionally difficult for many students to understand. On the basis of studies

of 76 elementary-education majors, Callison and Wright (1993) reported that students who use physical

models to explain the Sun-Earth-Moon system had significant categorical shifts from pretest to posttest

but that students who were taught to only use mental models did not improve significantly. Interestingly,

no significant correlation between spatial ability and model development was found. As part of a larger

study, Baxter found that students often think Moon phases are caused by Earth shadows (Baxter,

1989). This idea was confirmed in independent studies by Skam (1994) and Dai (1991).

There have been few studies regarding student conceptions of the solar system. It is generally

accepted that students, and adults, often have difficulty grasping the size and layout of the solar system.

As part of a larger study, Slater (1993) reported that some students think that there are hundreds of

stars in the solar system, that our Sun will become a black hole, that the asteroid belt is densely

populated, that the space shuttle goes to the moon each week, and that comets and meteors look the

same in the sky. Vosniadou (1992) argues convincingly that students can easily accept that our Sun is

hot but not that our Sun is a star because the Sun as a star concept is too far removed from direct

experience. In fact, she reports that most students think that, of the two most prominent objects in the

sky, the Earths Moon is more like a star than the Sun is.

The Ori gin and Evolut ion of the Ear th and Uni verse. There is an apparent lack of

educational research in student conceptions or beliefs regarding these objectives. Roettger (1998)


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recently reported that there are some significant differences between novice and expert views of the

Universe. She found that about 80% of experts she interviewed described the Universe by starting with

galaxies and clusters of galaxies and reducing down to the Solar System and Earth with strong emphasis

on the vast emptiness of the Universe. Students, in contrast, do not appear to have a consistent

conception of the Universe and, when asked to describe the Universe, often seem to be at a loss for

words. Philips (1991) reports that adults think that the Universe contains only the planets in our Solar

System and that the Universe is static. Students believe the planets and Sun formed directly from the

Big Bang. Lightman and Sadler (1993) report that only 25% of high school students can adequately

read an HR diagram following instruction.

A Proposed Research Agenda

To date, the majority of the current research base is aimed at the heliocentric astronomy

objectives, which appear in the NSES at the middle-school level. There is little research reported on

the NSES astronomy objectives targeted at either the K-4 or 9-12 grades and, what research has been

done, does not generally address the age-appropriate nature of the concepts. The NSES call for a

geocentric perspective of describing the objects and motions in the sky at the early grades, yet there is

almost no work reported in this area.

A large contingent of the physics education research community have successfully adopted a

reductionist approach to assessment and curriculum development (Trowbridge & McDermott, 1980;

Trowbridge & McDermott, 1981; Lawson & McDermott, 1987; Goldberg & McDermott, 1987;

McDermott, Rosenquist & van Zee, 1987). Their approach has been to select particular concept

areas, develop interview protocols to understand the range of student conceptions, and then create

short instructional interventions, or tutorials, to directly help students replace poor conceptions with


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more advanced ideas (Shaffer & McDermott, 1987a). In respect to this proven approach in physics

education, the astronomy education community needs to consider using such an approach to each of the

NSES astronomy objectives.

The first step would be to develop an in-depth understanding of student conceptions at the

various grade levels. At least initially, interviews likely need to be conducted using a widely accepted,

systematic interview protocol. Only after such work is completed, will it be possible to create research-

informed learning experiences, materials, and programs to effectively address particular student

misconceptions. Only then will it be feasible to develop a series of widely accepted, summative

assessment measures through which to view both NSES-aligned materials and student performance in

astronomyan ambitious task certainly requiring several years of dedicated work by many individuals

to complete. The most difficult aspect is that studies on the age-appropriate nature of concepts must be

conducted with K-12 students in the corresponding age groups.

As summarized in Table 1, the first task for the lower grades is to evaluate student geocentric

conceptions of the day and night sky. Because of an increasing distance between youth and the

environment, the results of such a study are likely to be very different than studies would have found

thirty years ago. As teachers note that it is very difficult to get students to return to school activities in

the evening or complete individual homework assignments, developing appropriate learning experiences

beyond a single visit to a planetarium is indeed a formidable challenge.

For the upper grades, almost no work has been done on student concepts of the origin,

characteristics, and evolution of the solar system, stars, or the Universe. This research focus will have

to start from a very small knowledge base. The particular challenge for the upper grades is to identify

the higher-order thinking that is expected because, given the remote nature the systems under study, this


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is the area where it would in fact be easiest for teachers to revert to an emphasis on lower knowledge-

level objectives.

Discussion

The most severe limitation of the current research base is that the studies described here do not

correlate well with the learning objectives stated in the NSES nor do they correlate well with the age-

appropriate nature of the objectives. Furthermore, a majority of the literature reviewed for this

manuscript focuses on heliocentric descriptions of seasons and lunar phases. There have been few

studies of student misconceptions reported on the bulk of the astronomy learning objectives as viewed

through the filter of the NSES. There is however, some promising work in student assessment strategies

currently underway (Zeilik et. al, 1997; Sadler, 1997).

The astronomy education community needs to consider using such an approach to EACH of the

NSES astronomy objectives to develop: (1) an understanding of student conceptions; (2) appropriate

learning experiences; and most importantly, (3) a series of assessment measures through which to view

both NSES aligned materials and student performance in astronomy.

In very general terms, there are four principle stages for science-education-improvement plans:

(1) setting high standards and clear expectations for students; (2) measuring student achievement,

attitudes, and an opportunity to learn using several different indicators; (3) reporting and synthesizing

progress information that is used to determine the extent to which student and school expectations are

being met; and (4) revising and improving pedagogy, materials, and strategies to meet community

expectations and student needs. As the NSES have clearly stated the astronomy standards for our

nations youth, the next logical step is to develop a strategy for measuring the effectiveness of

instructional materials and teacher-enhancement programs that purport to be aligned with the NSES.


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However, this step can not be completed until needed educational research has fully considered the

conceptual nuances of each objective listed in Table 1.

Recently, Morrow (1998) has advanced the PAPERS evaluation rubric as an appropriate

assessment rubric for astronomy education programs. In this checklist scheme: "P" - Pedagogy is

aligned with NSES; "A" - Accuracy is confirmed by participating scientists; "P" - Practicality for use in

the classroom is confirmed by teacher field-testing; "E" - Excitement is evident in students who use the

materials; "R" - Relevant materials are presented in the larger context of science; and "S" - Standards

are achieved by actually measuring an increase in NSES content knowledge of participating students or

teachers. It is this last item, the NSES-based content knowledge, that needs tools and strategies

resulting from rigorous educational research. This need should be, and can be, met with a concerted

effort within the astronomy-education research community. At such time, likely three to five years

hence, it will be possible to develop and implement a fully compliant and widely accepted summative

evaluation protocol for standards-based astronomy education.

The goal of this ambitious research agenda is to develop strategies for evaluating astronomy

education materials and programs for students and teachers under the umbrella of the NSES. It is still

unclear what form such instruments will take; however, it will certainly be a robust combination of an

alignment checklist, qualitative evidence of an opportunity to learn age-appropriate topics, authentic

assessment samples, and several quantitative instruments that directly measures student achievement at

different conceptual levels. However, it is critical that the development of valid assessment rubrics be

founded on a strong research base.


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Table 1: NRC-NSES Astronomy Objectives and Corresponding Literature Review

Objective Research to Date

1. Sky objects have properties, locations, and movements that can be

observed and described. [K-4, p. 134]

Sadler (1990) demonstrated prolific sky motion misconceptions among

both students and adults. Philips (1991) reports that students think

planets cannot be seen without a telescope or do not move.

2.The sun provides the light and heat necessary to maintain the

temperature of the earth. [K-4, p. 134]

Rollins, Denton & Janke (1983) found only 59% of high school seniors

related Suns energy to higher order concepts and principles as major

source of energy powering Earths phenomena.3. Objects in the sky have patterns of movement. The sun, for example,

appears to move across the sky in the same way every day, but its path

changes slowly over the seasons. The moon moves across the sky on a

daily basis much like the sun. The observable shape of the moon changes

from day to day in a cycle that lasts about a month. [K-4, p. 134]

Schneps 1987 Private Universe video informed scientific

community that even college educated adults and students have

embraced misconceptions for cause of seasons and changing

appearance of Moon that are not always revealed unless in depth

interviews are used. Students think Moon phases are caused by

Earths shadow (Baxter 1989) or clouds (Sadler, 1990) and seasons

due to distance to Sun (Philips, 1991).

4. The earth is the third planet from the sun in a system that includes the

moon, the sun, eight other planets and their moons, and smaller objects,

such as asteroids and comets. The sun, an average star, is the central and

largest body in the solar system. [5-8, p. 160]

Vosniadou (1992) argues that students can easily accept that our Sun

is hot but not that our Sun is a star because the Sun as a star is too far

removed from experience. Students think Earths Moon is more like a

star than our Sun.

5. Most objects in the solar systems are in regular and predictable motion.

those motions explain such phenomena as the day, the year, phases of the

moon, and eclipses. [5-8, p. 160]

Callison & Wright (1993) report that using physical models to

represent solar system motions are more effective than mental

models. Students spatial ability is not apparently related to

knowledge of Earth/Sun/Moon relationships. Vosniadou (1992) found

many different student models for day/night cycle.

6. Gravity is the force that keeps planets in orbit around the sun and

governs the rest of the motion in the solar system. Gravity alone holds us

to the earth's surface and explains the phenomena of the tides. [5-8, p.

161]

Nussbaum & Novak (1976) and Vosniadou & Brewer (1989) found

students misconceptions of gravity are related to student conceptions

of Earths shape. Vosniadou (1992) found all student conceptions of

Earths shape could be categorized into six mental models. Treagust

& Smith (1989) and Osborne & Gilbert (1980) found students believe

planets gravity decrease with slow rotation rates or with increasing

distance from Sun.


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7. The sun is the major source of energy for phenomena on the earth's

surface, such as growth of plants, winds, ocean currents, and the water

cycle. Seasons result from variations in the amount of Sun's energy

hitting the surface, due to the tilt of the earth's rotation on its axis and the

length of the day. [5-8, p. 161]

Sadler (1992) convincingly showed students think seasons result of

Sun/Earth distance even when able to identify Earths tilt as important

variable. Rollins, Denton & Janke (1983) found only 59% of high

school seniors related Suns energy to higher order concepts and

principles as major source of energy powering Earths phenomena.

8. The sun, the earth, and the rest of the solar system formed from a

nebular cloud of dust and gas 4.6 billion years ago. The early earth wasvery different from the planet we live on today. [9-12, p. 189]

No directly related research found. Philips (1993) reports that adults

think that the Universe contains only the planets in our solar systemand that the Universe is static. Students believe the planets and Sun

of the solar system formed from the Big Bang.

9. The origin of the universe remains one of the greatest questions in

science. The "big bang" theory places the origin between 10 and 20 billion

years ago, when the universe began in a hot dense state; according to this

theory, the universe has been expanding ever since. [9-12, p. 190]

No research found. Recently, Roettger (1998) reported that

professional scientists and non-scientists have widely different

descriptions of the Universe.

10. Early in the history of the universe, matter, primarily the light atoms

hydrogen and helium, clumped together by gravitational attraction to form

countless trillions of stars. Billions of galaxies, each of which is a

gravitationally bound cluster of billions of stars, now form most of the

visible mass in the universe. [9-12, p. 190]

No strongly related research found. Peripherally, Treagust & Smith

(1989) found that students think gravity is affected by temperature.

Philips (1991) reports gravity needs air to exist.

11. Stars produce energy from nuclear reactions, primarily the fusion of

hydrogen to form helium. These and other processes in stars have led to

the formation of all the other elements. [9-12, p. 190]

No directly related research found. Peripherally, Sadler and Lightman

and Sadler (1993) reported only 25% can adequately read an HR

diagram and showed that teachers grossly overestimate the

knowledge of students even after instruction.


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astronomy in nrce

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