This document analyzes how 29 high school and introductory college chemistry textbooks introduce, define, and explain the mole concept. The analysis focused on how the mole is defined, what atomic concepts are introduced beforehand, if Avogadro's constant is presented as experimental, and the context for introducing moles. The study found that textbooks most commonly define moles as Avogadro's number or in terms of carbon-12 atoms, introduce atomic concepts beforehand when using the carbon-12 definition, note the experimental basis of Avogadro's constant, and introduce moles in relation to counting particles too small to weigh directly.

1. JOURNAL OF RESEARCH IN SCIENCE TEACHING VOL. zyxw
30, NO. 4, PP. 321-337 (1993) zy
A Content Analysis of the Presentation of the Mole Concept in
Chemistry Textbooks
John R. Staver zyxw
Centerfor Science Education, Kansas State University, Manhattan, Kansas zyx
66506
Andrew T. Lumpe
Department zyxwvut
o
f Curriculum and Educational Technology, University o
f Toledo,
Toledo. Ohio 43606
Abstract
The goal of this study was to examine the means used by textbook authors to introduce,
define, and explain the mole concept in high school and introductory college chemistry textbooks.
The analysis was framed by four questions: zyxwv
1. How is the mole defined?
2. What concepts about the atom are introduced prior to the mole?
3. Is Avogadro’s constant presented as an experimentally determined value?
4. What is the context for introducing the mole?
Twenty-nine high school and introductory college level chemistry texts were examined.
After independent reading of appropriate sections of each text, discussion of differences, second
or third readings of texts, and subsequent discussions, both authors reach 100% agreement
concerning the results. Major conclusions were
1. Two ways of defining the mole dominate the texts. One way defines the mole
as Avogadro’s number (6.02 X particles; the other method defines the
mole in terms of carbon-12.
2. All texts that present a definition in terms of C-12 introduce and define concepts
about the atom prior to introducing the mole.
3. Most texts at all levels point out that the value 6.02 X loz3is an experimentally
determined quantity.
4. Nearly all texts discuss the mole in relation to the problem of finding a way to
count particles that are too small to be directly weighed. Most texts also use a
familiar counting unit, such as the dozen, to introduce the mole by analogy.
Four issues were discussed: (a) the defining attributes of the mole concept itself and the
cognitive requirements for comprehending the two most frequently used definitions; (b) the
connection between the definition of the mole presented in the text and the concepts about atoms
that are introduced before the mole concept is developed; (c) the experimental zyx
natureof Avogadro’s
number; and (d) the context or setting for developing the mole concept. zyxw
0 1993 by the National Association for Research in Science Teaching
Published by John Wiley & Sons, Inc. CCC 0022-4308/93/040321-17

2. 322 STAVER AND LUMPE zyxwv
The Mole Concept and Chemistry
Amount of substance is not only a physical quantity, it represents one of seven
standard units of measurement defined by the International System (SI). Named the zy
mole, the SI unit of measurement for amount of substance is one of the most fundamental
concepts in science. According to Kotz and hrcell (1987), the SI definition of the
mole is: zyxwvuts
“. . . the amount of substance that contains as many entities as there are in
exactly 0.012 kg of carbon-I2 (12 g of C-12 atoms)” (p. 1-21). The SI definition of
the mole has been accepted by both the International Union of Pure and Applied
Chemistry (IUPAC) and the International Federation of Clinical Chemistry (IFCC)
(Lehmann,Worth, & Zinder, 1988).The mole concept therefore is not only a fundamental
unit of measurement, but it is also an important foundation for more complex chemical
concepts such as stoichiometry, concentration of solutions, the equilibrium constant,
and pH. If students are to construct accurate frameworks of complex chemicalconcepts
and successfully solve quantitative problems involving such notions, then they must
have a clearly defined and well-connected conceptualization of the mole concept. The
mole concept, then, is a cornerstone to building a successful understanding of domain-
specific knowledge in chemistry. If students fail to understand the mole concept, and
many do fail, it is likely that their chemical problem-solving ability will be severely
limited. Moreover, such difficultieswill surface early on, as the mole concept typically
is introduced early in high school and freshman college chemistry.
Background and Related Research
Four issues guide the research reported here and our continuing program of inquiry
on the mole concept. The first issueconcernsthe cognitiverequirementsfor comprehending
definitionsof the mole concept. The second issue deals with the frameworkof concepts
that surround the mole as well as the position of the mole within the conceptual
framework. The third issue focuses on the empirical nature of Avogadro’s number.
The fourth issue centers on the context for introducing and developing the mole concept.
Related research is presented in the following paragraphs, and the conclusions of the
research reported here are examined with respect to these issues in the discussion
section.
Beginning chemistry students typically have three sources of outside information;
the textbook, the teacher, and other students, when attempting to construct an under-
standing of a concept. A full understanding of students’ difficultiesand successes will
result only from comprehensive examinations of not only each source alone, but also
the nature and extent of interactions among all sources in classroom settingsas students
first encounter the mole concept. In the study reported here, we focused on the first
source, the textbook, as part of our long-term program of research on the mole concept.
Whereas the focus is necessary, it is also limiting in that the results and conclusions
are based only on the authors’ analysis of material in the texts examined. Issues such
as how students use a text, what methods teachers employ to introduce the mole
concept, and the nature of dynamic interactions among text, teacher, and student in
classrooms are subjects for further research.
Researchers in chemical education such as Herron (1990) and Abraham (1990)
state that the major impecllments to successfulchemicalproblem solving zyx
are(a) insufficient
understanding of the concepts involved, (b) use of memorized algorithms or rules,

3. CONTENT ANALYSIS 323
and (c) inability to transfer understanding between the atomic/molecular and the mac-
roscopic levels in solving problems. The mole concept presents potential difficulties
for students in all the just-mentioned areas. zyxw
As defined by SI, the mole is rooted in
absmct concepts of the atomic/molecularlevel. Moreover, the experimentallydetermined
value of the mole, 6.02 zyxwvu
X is too large to be comprehended in a concrete manner. zy
A cursory degree of understanding can be attained by using algorithms, but in-depth
comprehension requires a level of consideration well beyond memorizing formal pro-
cedures. Finally, the mole represents a vehicle for moving between the macroscopic
and atomic/molecularlevels, but the vehicle itself is difficultto grasp, let alonecommand.
Specificstudies in chemical problem solving show that studentsfrequently struggle
during their initial encounters with the mole and its applicationin stoichiometry.Gabel,
Sherwood, and Enochs (1984) found that a majority of students utilize algorithms to
solve problems and do not understand the concepts central to the problems themselves.
Herron and Greenbowe (1986) demonstrated that students often memorize rules in
attempting to solve problems.
Using written quizzes and exams, Staver (1989) found that students commonly
harbor misconceptions about the mole that hinder stoichiometryproblemsolving. These
misconceptions often relate to the definition of the mole that the studenthas memorized
and uses in problem solving. For example, students often define the mole as a fixed
number equal to 6.02 X Moreover, students frequently exhibited great confusion
concerning why the mass of a single elementary particle in atomic mass units is
numerically equal to the molar mass of the substance in grams (e.g., one molecule of
ammonia, NH3 = 17.0 amu; one mole of NH3 = 17.0 grams).
Utilizing think-aloud interviews with beginning chemistry students, Staver and
Lumpe (1990) furtherinvestigatedstudents’understandingof thejust-mentionednumerical
identity relation. Staver and Lumpe found that a major stumblingpoint in mole concept
related problem solving centers on students’ comprehension of the numerical identity
relation.
Reports clearly suggest that science texts represent a major source of information
for students. According to Stake and Easley (1978), 90% of all science teachers use
science textbooks 90% of the time. Brandwein (1981) observed that a majority of
science teaching revolves around the use of specific science terms that are introduced
in the text. Moreover, little emphasis is placed on comprehension or use of these
science terms. It seems patent that the presentation of the mole concept in the textbook
represents an important source of knowledge for students’ comprehension of the mole
concept itself, as well as the numerical identity relationship. However, a search of the
literature in chemical education revealed only one study with a focus on textbook
analysis of the mole concept. Cervellati, Montuschi, Perugini, Grimellini-Tomasini,
and Balandi (1982) examined 13 secondary chemistry texts commonly used in Italian
schools. According to Cervellati and his associates, the term zyxw
mole was only a synonym
for gram-molecule in a majority of texts; some texts gave incorrect definitions; one
text presented a contradiction of definitions. These authors criticized eight texts for
failing to link the mole concept with the concept of a standard number of particles.
They noted that only three texts provided a correct definition for the mole and linked
it to the definition of Avogadro’s number. Whereas all books presented the value of
Avogadro’s number, its experimental determination was not emphasized. Cervellati et
al. also found that the texts gave little attention to the use of the mole concept in
stoichiometry; most texts provided only rules to be memorized for problem solving.
Finally, these researchers administered a diagnostic exam designed to reveal students’

4. 324 STAVER AND zyxwvu
LUMPE
misconceptions about the mole. The results of students’ responses suggest that their
misconceptions were in some cases related to misleading definitions presented in the
textbooks.
Purpose
Our purpose in this article is to examine the means used by textbook authors to
introduce, define, and explain the mole concept in high school and introductorycollege
chemistry textbooks. We framed the analysis with four questions:
1. How is the mole defined?
2. What concepts about the atom are introduced prior to the mole?
3. Is Avogadro’s constant presented as zyxwv
an experimentally determined value? zyx
4. What is the context for introducing the mole?
Method zyxwv
Texts
Thirty-six high school and introductory college level chemistry texts were initially
selected for analysis. However, chemistry texts are revised every four or five years in
order to remain current and competitive. For example, GeneralChemistrywith Qualitative
Analysis (Holtzclaw, Robinson, zyxwvu
& Odom, 1991) was published initially in 1957, then
revised in 1963, 1968, 1972, 1976, 1980, 1984, and 1988. We decided that no text
with a copyright date earlier than 1986 should be included in the study. Therefore,
five texts published prior to 1986 were removed from the list.
Initial examination of the texts revealed that one text, Foundations o
f Chemistry
(Toon, Ellis, Doyle, Ivanco, & Percival, 1990), does not include the term mole in its
glossary, nor does it introduce or define the mole concept. Examination of the preface
revealed that this text is designed for use by high school students who have already
taken one year of chemistry. This text was excluded from further analysis.
Further examination found that two texts, Preparatory Chemistry (Stoker, 1990b)
and Introduction to Chemical Principles (Stoker, 1990a), are in fact identical throughout
the first 15chapters, which include the mole concept.Introductionto ChemicalPrinciples
contains 4 additional chapters, whereas 15 chapters represent the entire content of
Preparatory Chemistry. Introduction toChemicalPrinciples was excluded from further
analysis, leaving a total of 29 texts for examination.
Classijication o
f Texts
Each text was assigned to one of three categories according to statements in the
text preface, other introductory parts, and the text itself. The categories are (a) high
school, (b) preparatory college level for students with little mathematics or chemistry
backgrounds, and (c) introductory college level for students with substantial mathematics
and chemistry backgrounds. A list of the books examined is presented in Tables l(a),
l(b), and l(c).
Analysis of Texts
Content analysis techniques were used to examine the texts. Berelson (cited in
Borg & Gall, 1989) states that content analysis is useful to describe content of com-

5. CONTENT zyxwvu
ANALYSIS 325 zy
Table 1(a) zyxwvutsrq
High School Chemistry Texts Examined*
Title Publisher Date
Addison-Wesley Chemistry (Wilbraham, Staley, Addison-Wesley 1990
Chemistry: A Modern Course (Smoot, Smith, zyxwv
& Merrill 1990
Chemistry in the Community (ACS, 1988) KendalVHunt 1988
Chemistry Today I (Whitman, Zinck, & Nalepa, 1988
Heath Chemistry (Herron et al., 1987) Heath 1987
Modern Chemistry (Tzimopoulos, Metcalfe, Holt, Rinehart, & Winston 1990
Prentice-Hall Chemistry: The Study o
f Matter Prentice-Hall 1989
Simpson, & Matta, 1990)
Price, 1990)
Prentice-Hall of Canada
1988)
Williams, & Castka, 1990)
(Dorin, Demmin, & Gabel, 1989)
* Complete bibliographies can be found in the reference list.
munications. Budd, Thorp, and Donohew (1967) point out that content analysis is
useful in making predictions about written material, the receiver of material, and
relationships between the material and the receiver. Moreover, problems of bias are
lessened because written communications are nonreactive (Borg & Gall, 1989).
The four just-mentioned questions framed the analysis, and specific parts of text
were examined in order to answer the following questions:
1. What concepts about the atom are introduced prior to the mole?
To answer this question we read those chapters of the text prior to the introduction of
the mole concept that focused on concepts relevant to the atomic/molecular level,
concepts that form a foundation for the mole concept, such as the three-particle model
of an atom, atomic number, mass number, atomic mass, isotope, and the relative mass
system of atomic mass.
2. How is the mole defined?
3. How is Avogadro’s constant presented?
Table l(b)
Preparatory College Chemistry Texts Examined*
Title Publisher
~
Basic Chemical Principles (Peters, 1988)
Basic Concepts o
f Chemistry (Sherman, Sherman, &
Chemistry: An Introduction (Slabaugh & Seager, 1988)
Introduction to Chemistry (Dickson, 1987)
Introductory Chemistry: A Foundation (Zumdahl, 1990)
Preparatory Chemistry (Stoker, 1990b)
Understanding Chemistry (Herron, 1986)
Russikoff, 1988)
Saunders
Houghton Mifflin
West
Wiley
Heath
Macmillan
Random House
Date
1988
1988
1988
1987
1990
1990
1986
* Complete bibliographies can be found in the reference list.

6. 326 zyxwvuts
STAVER AND LUMPE zyxwv
Table l(c) zyxwvutsrq
Introductory College Chemistry Texts Examined*
Title Publisher Date
ChemisQ (Gdlespie, Hmphreys, Baird, zyxwvu
& Robinson, 1986)
Chemistry (Chang, 1988)
Chemistry (Zumdahl, 1989)
Chemistry: An Experimental Science (Bodner & Pardue,
Chemistry and Chemical Reactivify (Kotz & Purcell, 1987)
Chemistry: The Central Science (Brown & LeMay, 1988)
Chemistry: Principles and Reactions (Masterson & Hurley,
Chemical Principles (Masterson, Slowinski, & Stanitski,
General Chemistry (Chang, 1986)
General Chemistry (Ebbing, 1987)
General Chemistry (Brescia, Arents, Meislich, & Turk,
General Chemistry (Petrucci, 1989)
General Chemistry with Qualitative Analysis (Holtzclaw et
Introduction to General, Organic, and Biochemistry (Bet-
P??nciples o
f Modern Chemistry (Oxtoby & Nachtrieb, 1986)
* Complete bibliographies can be found in the reference list.
1989)
1989)
1985)
1988)
al., 1991)
telheim & March, 1988)
Allyn and Bacon
Random House
Heath
Wiley
Saunders
Prentice-Hall
Saunders
Saunders
Random House
Houghton Mifflin
Harcourt, Brace,
Jovanovich
Macmillan
Heath
Saunders
Saunders
1986
1988
1989
1989
1987
1988
1989
1985
1986
1987
1988
1989
1991
1988
1986 zy
4. What is the context for introducing the mole? To answer these questions we
read those sections of the chapter in which the mole concept was first introduced,
defined, and applied, as well as the glossary definition of the mole.
We examined the texts in small blocks, taking four to six texts from a category
at a time. Each text was read independently by both authors, who made extensive
written notes. Following the first reading for a block of texts, the authors met to
examine the extent of agreement and disagreement, using their notes to focus the
discussions. Points of accord were discussed but required no further reading. With
respect to issues of dissent for a specific text, each author independently read the
disputed text a second time, again taking notes, then met again to discuss further the
disputed points. Most discords were resolved after a second reading, but a few points
required a third independent reading and discussion. All points of disagreement were
eventually resolved, and the authors reached total (10%) agreement for each text.
Results
The results arepresented as groups of statementsbeginningwith sevenhigh school
level texts. The first group of results is also summarized in Table 2.
1. Five (Table 2- 1,2,3,5,7) out of seven high school texts define the mole as
6.02 X loz3particles; all five that so define the mole also refer to the value as

7. Table 2 zyxwvutsrqponm
Results zyxwvutsrqpo
o
f Content Analysis o
f High School Texts
States that States that
Basic atomic Refers to 6.02 Avogadro’s Avogadro’s
concepts Defines mole zyxwvuts
x zyxwv
loz3as number is constant
introduced Defines mole in terms of Defines mole Avogadro’s experimentally was
prior to the according to numerical as 6.02 x loz3 number/ determined/ discovered/
Title (author) mole carbon-12 equivalency particles constant measured determined
1.
2.
3.
4. zyxwvutsrqp
5.
6. zyxwvutsrqp
7.
Addison-Wesley Chemistry X X
(Wilbraham et al., 1990)
Chemistry: A Modern
Course (Smootet al., 1990)
(ACS, 1988)
man et al., 1988)
Heath Chemistry (Herron et
al., 1987)
mopoulos et al., 1990)
TheStudy o
f Matter (Dorin
et al., 1989)
ChemistryintheCommunity X
Chemistry Today 1 (Whit- X X
Modern Chemistry (Tzi- X X
Prentice-Hall Chemistry: X
X X X
X X X
X X
X X
X X
X x
X X X
X

8. 328 STAVER AND zyxwvu
LUMPE
Avogadro’s number or constant. Three of these texts (1,2,7)mention that Avo-
gadro’s number is an experimentally determined value.
2. Two high school texts (4,6) define the mole only as an amount of substance
having the same number of particles as the number of C-12 atoms in exactly
12gramsof C-12. Both texts introduce Avogadro’snumber immediatelyfollowing
the mole; both texts take note of its experimental nature.
3. One text (1) defines the mole as 6.02 zyxwv
X loz3particles, then defines the mole
again according to carbon-12. This text points out the experimental nature of
Avogadro’s number.
4. Five (1,3,4,6,7) out of seven texts introduce concepts of the atom such as the
three-particle model, atomic and mass number, atomic mass, and the relative
mass system of atomic masses prior to introducing the mole concept. The two
texts that do not introduce all of these concepts define the mole only as 6.02
x loz3particles.
The second group of results pertains to seven preparatory college level texts that
presume little or no previous background in chemistry. These results are summarized
in Table 3.
1. Two (Tables 3-2,7) out of seven texts define the mole only as 6.02 X loz3
particles; both books introduce atomic mass and relative mass prior to the mole.
One text (7) mentions that Avogadro’s number was experimentally determined;
the other text (2) notes only that it was discovered.
particles, then redefines the mole
according to the number of atoms in 12 grams of C-12. This text introduces
the three-particle model, atomic and mass number, atomic mass, isotope, and
the system of relative mass prior to the mole. This text further points out the
experimental nature of Avogadro’s number.
3. Two texts (1,4) define the mole as the amount that contains the same number
of particles as there are atoms of C-12 in exactly 12 grams of C-12; both texts
introduce the three-particle model, mass number, atomic number, atomic mass,
isotope, and the system of relative masses prior to the mole. Both texts mention
that Avogadro’s number was experimentally determined.
4. One text (5) defines the mole as the number of carbon atoms in 12.01 grams
of carbon. This text introduces the three-particle model, mass number, atomic
number, atomic mass, isotope, and the system of relative masses prior to the
mole. It notes the experimental nature of Avogadro’s number.
5. One text (3) defines the mole as the number of particles in a sample with a
mass in grams numerically equal to the mass of a single particle; it introduces
the three-particle model, atomic number, mass number, isotope, and relative
mass prior to the mole. It states that Avogadro’s number was experimentally
determined.
6. Six (1,3-7) of the seven texts mention specifically the experimental nature of
Avogadro’s number; one text (2) mentions that scientists discovered Avogadro’s
number but does not specifically mention that it was done by experiment.
2. One text (6) defines the mole as 6.02 X
The third group of results focuses on 15introductorycollege level texts that assume
that students have some previous background in chemistry. These results are summarized
in Table 4.
1. Five (Table 4-7,8,11,14,15) out of 15 texts define the mole as Avogadro’s
number, 6.02 X loz3,of particles; these texts present the three-particle model,

9. Table 3 zyxwvutsrqponm
Results o
f Content Analysis of College Preparatory Texts
Title (author)
Refers to 6.02 States that States that
Basic atomic zyxwvutsrqp
X Avogadro’s Avogadro’s
concepts Defines mole as number is constant
introduced Defines mole in terms of Defines mole as Avogadro’s experimentally was
prior to the according to numerical 6.02 X number/ determinedl discovered/
mole carbon-12 equivalency particles constant measured determined
1.
2.
3. zyxwvutsrqpo
4. zyxwvutsrqponm
5.
6.
7.
Basic Chemical Principles X
(Peters, 1988)
Basic Concepts o
f Chem- zyxwvutsrq
istry (Sherman et al., 1988)
(Slabaugh & Seager, 1988)
(Dickson, 1987)
Foundation (Zumdahl,
1990)
(Stoker, 1990b)
Understanding Chemistry
(Herron, 1986)
Chemistry:An Introduction X
Introduction to Chemistry zyxwvutsrqpo
X
Introductory Chemistry: A X
Preparatory Chemistry X
X
X
X
X
X X
X X X
X X
X X
X x
X X X
X X
w
N
D

10. W
W zy
0 z
Table zyxwvutsrqpo
4 zyxwvutsrqpon
Results o
f Content Analysis o
f College Texts
States that States that
Basic atomic Avogadro's Avogadro's
concepts Defines mole Refers to 6.02 zyxwv
X number is constant
introduced Defines mole in terms of Defines mole as experimentally was
prior to the according to numerical as 6.02 X Avogadro's determined/ discovered/
Title (author) mole carbon-12 equivalency particles numberkonstant measured determined
1. zyxwvutsrqponmlkjih
2.
3.
4. zyxwvutsrqponm
5.
6.
Chemistry (Gillespie et X X
Chemistry (Chang, 1988) X X
Chemistry (Zumdahl, X X
Chemistry: An Experi- X X
Chemistry and Chemical X X
Chemistry: The Central X X
al., 1986)
1989)
mental Science (Bodner
et al., 1989)
Reactivity (Kotz & Pur-
cell, 1987)
Science (Brown &
LeMay, 1988)
X X
X X
X X
X X
X
X
9
z
U

11. CONTENT ANALYSIS 33 zy
1
X zyxwvut
x x x
x x x x
x x
x x x
x x
x x x x
x x x x
x x x x
X X
X
x x
x x x x

12. 332 STAVER AND zyxwvu
LUMPE
atomic number, mass number, atomic mass, isotope, and the relative system
of atomic masses prior to the mole.
2. The remaining 10 texts (1-6,9,10,12,13) define the mole as an amount of
substance with a number of particles equal to the number of amounts in exactly
12 grams of C-12; 9 texts (1-6,9.10,12) introduce the three-particle model,
atomic number, mass number, atomic mass, isotope, and the relative system
of atomic masses prior to the mole; 1 text (13) introduces the three-particle
model, atomic number, mass number, atomic mass, isotope, and the atomic
mass unit but does not discuss the system of relative masses prior to the mole.
3. burteen (1-9,ll- 15)out of 15texts note that Avogadro’s number is experimental
in nature; some texts describe specific experiments for determining the value.
1 text (10) states that Avogadro’s number has been determined and performs a
calculation but does not emphasize its experimental character.
The final results are common to nearly all 29 texts. Most books mention that the
mole provides an indirect method for counting fundamental particles in samples large
enough to work with in the laboratory. The high school and preparatory level college
texts devote more space to such explanations. Second, most books make comparisons
between Avogadro’s number as a counting unit and more familiar counting units such
as the dozen.
Conclusions
The four questions that framed our analysis can now be answered.
1. How is the mole defined? Two ways of defining the mole dominate the texts
analyzed. The mole defined as 6.02 zyxwv
X particles is most frequent among
the high school texts. The mole defined in terms of C-12 is presented in a
minority of the high school texts but represents the majority definition in college
level texts.
2. What concepts about the atom are introduced prior to the mole? All texts that
present a definition in terms of C-12 introduce and define concepts about the
atom prior to introducing the mole. Concepts introduced by such texts are the
three-particle model of the atom, atomic number, mass number, atomic mass,
isotope, and the system of relative masses.
3. Is Avogadro’s constant presented as zyxwv
an experimentally determined value? The
results show that a sizable majority of texts at all levels point out that the value zyx
6.02 zyxwvut
X loz3is an experimentally determined quantity.
4. what is the context for introducing the mole? Almost all texts examined discuss
the mole in relation to the problem of finding a way to count particles that are
too small to be directly weighed. Many texts also use a familiar counting unit,
such as the dozen, to introduce the mole by analogy.
Discussion
Four issues require discussion. The first issue concerns the defining attributes of
the mole concept itself and the cognitive requirements for comprehending the two
most frequently used definitions. To examine this issue, we must first attend to the
fundamental nature of concepts themselves. Lawson, Abraham, and Renner (1989)
define a concept as a pattern of regularity named by a term and set forth three types

13. CONTENTANALYSIS 333 zy
of concepts: (a) by apprehension, (b) descriptive, and (c) theoretical. Concept by
apprehension refers to the derivation of immediate meaning from the environment.
Colors such as red or green are immediately apprehended from the environment.
Descriptive concepts stem from perceived relations of events and objects. Learners
must constructa pattern from experienceswith the environment,but such constructions
may be tested by direct experience. Examples are concepts such as earlier, older, and
shorter. Theoretical concepts also stem from perceived relations but the attributes
themselvesare not perceptible. The purpose of theoreticalconceptsis to explainevents
that have no directly perceivable causes (Lawson et al., 1989).
Consider first the definition of the mole as Avogadro’s number, 6.02 zyx
X of
particles. On the surface, this definition seems rather simple; one mole is 6.02 x
of anything. But can high school and college freshmen directly perceive 6.02 X
We think not. In fact, we view this definition as a theoretical concept. Lawson et al.
(1989) point out that abduction, or reasoning by analogy, plays a central role in the
formation of theoretical concepts. Abduction is used extensively by text authors to
establish a connection between 6.02 X loz3and a more familiar counting unit, for
example, the dozen. Consider this passage from zyxw
Heath Chemistry (Herron, Kukla,
Schrader, Erickson, & DiSpezio, 1987).
A dozen is a convenient unit for expressing a frequently used quantity. However,
one or two dozen atoms are too small to be seen with even the most powerful
microscope. The term mole is used to talk about a number of atoms, molecules,
ions, or electrons, just as a dozen is used to talk about a number of eggs, oranges,
or doughnuts. (p. 98)
Nearly all high school and collegepreparatorytextbooks in this analysisincluded such
analogies. Our classification of this definition as a theoretical concept is founded on
an understandingof concepts as delineatedby Lawson et al. (1989) and in the frequent
use of abduction in the prose of the textbooks that we examined.
Consider next the definition of the mole in terms of C-12: “. ..the amount of
substance in a system that contains as many elementary units (atoms, molecules, or
formula units) as there are 126Catoms in exactly 12.00000 grams of 12&” (Stoker,
1990b, p. 279). The unit is the atom, molecule, or ion; none is directly perceivable.
C-12representsa specificisotope, another theoreticalconcept. The mole is an unknown
number (at least to students who encounter this definition first) equal to the number
of C-12 atoms in a specific amount of C-12. According to Lawson et al. (1989), these
defining attributes of the mole are clearly theoretical. Both definitions, then, are
theoretical in nature. Using the Lawson et al. (1989) vocabulary and meaning, we
believe that learners need to be reflective rather than intuitive thinkers to comprehend
either definition.
The second issue deals with the connection between the definition of the mole
presented in the text and the concepts about atoms that are introduced before the mole
concept is developed. If text authors define the mole accordingto carbon-12, then, in
order to lay a proper foundationfor such a definition, they must introduce and develop
concepts such as the three-particle model of the atom, atomic number, mass number,
atomic mass, isotope, and the relative system of atomic mass in atomic mass units.
Each book that fits this condition does introduce at least these concepts prior to
introducing the mole. If text authors define the mole in terms of Avogadro’s number

14. 334 STAVER AND zyxwvu
LUMPE
(6.02 zyxwvuts
X of particles, then no such precondition exists. It is interesting to note
that three of the five high school texts and all introductory college level texts that use
only Avogadro’s number as a definition present the just-mentioned concepts about
atoms prior to the mole. But neither of the two college preparatory texts present all
such concepts about the atom prior to introducing the mole.
The third issue centers on the experimental nature of Avogadro’s number. Readers
should recall the Staver & Lumpe (1990) finding that students often believed 6.02 X
loz3to be a fixed number, and the Cervellati et al. (1982) result that Italian chemistry
texts did not emphasize the empirical nature of Avogadro’s number. Clearly, this
misconception is not due to the absence of this point in text prose. Most texts point
out that Avogadro’snumber is an experimentallydetermined value. A workinghypothesis,
however, is that use of familiar fixed counting units such as the dozen may play a
central role in this misconception. Our clear recollection is that the amount of prose
devoted to development of the mole through familiar analogies is far greater than the
amount devoted to the experimental nature of Avogadro’s number. Learners may be
focusing on analogies in text due to the larger amount of space devoted to them and
largely ignoring the experimental nature of Avogadro’s number due to the single line
or so of space concerning that issue. Nowhere in our analysis did we find a discussion
contrasting the experimental character of Avogadro’s number with the fixed nature of
familiar counting units. Frequently, we found passages describing their likenesses.
Consider this line from zyxwvut
Chemistry in zyxwvu
the Community (American Chemical Society
[ACS], 1988) “You are familiar with the counting units ‘pair’ or ‘dozen’. Just as one
dozen water molecules means 12 water molecules, to a chemist one mole of water
molecules means 6.02 X loz3molecules”(p. 11 zyxw
1). We need to rememberthat analogies
rarely fit perfectly. Therefore, textbook authors need to contrast as well as compare,
thereby pointing out the limits as well as the similarities of analogous concepts.
The final issue deals with the context or setting for developing the mole concept.
Two aspects stand out in the setting for development of the mole concept. One is the
use of analogies, which we have already discussed. The second is introducing the
mole as an answer to the problem of counting objects far too small to be counted
directly. Consider the following passage from Chemistry in the Community:
One formula unit of copper(1)sulfide plus two molecules of oxygen react to
give two formula units of copperzyxwv
(II) oxide and one molecule of sulfur dioxide.
This information is not too useful to a metal refinery owner who wants to know
how much sulfur dioxide air pollutant will be released by roasting a certain amount
of copper (I) sulfide ore. Chemists have devised a counting unit called the mole
. . . that helps the refiner solve his problem. (p. 111)
Most books mention the counting by weighing problem, with the high school and
college preparatory texts devoting more space compared to the college texts. But is
the counting by weighing context as meaningful to beginners as it is to experts? The
intent is to provide an advance organizer, but students’ poor conceptual understanding
and their inability to transfer understanding, problems noted by Herron (1990) and
Abraham (1990),may prevent them from using beneficial strategies, thereby hindering
the value of the counting by weighing context for some students.
Gamer (1990) argues that a theory of settings is useful in explaining why learners
use, or fail to use, strategies in learning. She points out that strategies are goal driven

15. CONTENT zyxwvu
ANALYSIS zyxwv
335
and that goals vary across settings. Moreover, at least five reasons exist for failure to
use strategies: “1) poor cognitive monitoring; 2) primitive routines that yield a product;
3) a meager knowledge base; 4)attributions and classroom goals that do not support
strategy use; and 5) minimal transfer” (Garner, 1990, p. 517). Further research is
needed to clarify this point as well as other issues. For example, does the use of fixed,
familiar counting units encourage construction of the mole as a fixed number? Are
students’ primitive routines insufficient?What strategies, then, are sufficient?Researchers
must examine these as well as other questionsto better understand and remove difficulties
associated with teaching and learning the mole concept.
Implications for Teaching
The four issues that guide the research reported here also provide a framework
for improving the teaching of the mole concept in high school and college chemistry.
Readers should note that the issues are presented in a different order compared to prior
sections of the article. First, the cognitive requirements of both definitions most hquently
developed in textbooks are very high, largely due to their abstract, theoretical nature.
Thus, students whose learning is best characterized as concrete and intuitive rather
than abstract and reflective may have great difficulty with the mole concept. Chemistry
teachers need to utilize concrete activities when introducing the mole in order to help
concrete, intuitive learners begin to grasp the concept. Closely connected is a second
issue, the context and setting within which the mole is developed. Teachers should
emphasize analogies involving concrete, familiar concepts, pointing out to students
the extent to which familiar analogies such as the dozen apply to the mole, and
particularly where they do not apply. Moreover, teachers should be aware that the
settings found most frequently in most textbooks may affect individual learners in
different ways as they construct an understanding of the mole concept. The third issue
stems from limitations of the second issue. When introducing the mole, teachers must
take care to develop Avogadro’s constant as an empirical value with uncertainty,
contrasting it with most concrete analogies, such as the dozen, which are based on
fixed, counted values. The final issue concerns the conceptual framework and sequence
of concept development. Chemistry teachers as decision makers must consider the
global goal of the course itself, how the mole concept fits into the overall goal, what
definition of the mole yields the best fit, then what textbook best facilitates students’
progress toward the goal.
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