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Summary Key Concepts Case Studies: Re

ﬂect and Evaluate

Critical Thinking

What Is Critical Thinking?

Application: Fostering Critical Thinking

Thinking Skills and Dispositions

What Are Higher-order Thinking Skills?

What Are Thinking Dispositions?

Outline Learning Goals

Explain the difference between a higher-order thinking skill and a thinking disposition, and discuss why both skills

and dispositions are important.

Explain what critical thinking means.

Identify

ﬁve instructional strategies that can be used to foster critical thinking.

Problem Solving

What Is Problem Solving?

Obstacles to Successful Problem Solving

Application: Teaching Problem-solving Strategies

ﬁne problem solving and explain the difference between a well-deﬁned problem and an ill-deﬁned problem.

Discuss the role of algorithms, heuristics, the IDEAL approach, and problem-based learning in teaching problem

solving.

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THINKING SKILLS AND DISPOSITIONS

Socrates challenged the loose thinking of the youth of his day by
asking questions such as “What is the evidence?” and “If this is true,
does it not follow that other things must also be true?”
He promoted an approach that disciplined his students’ thinking and
guarded against the human tendency to accept fallacious arguments
and draw unwarranted conclusions (Resnick, 1987). Today, teachers
in every grade level and every discipline often ask themselves the
same question: “How can I get my students to think?” Educational
and professional success requires the development of thinking skills
and a consistent internal motivation to use those skills in appropriate
situations (Facione, Facione, & Giancarlo, 2000).

What Are Higher-order Thinking Skills?

Higher-order thinking involves complex cognitive processes that
transform and apply our knowledge, skills, and ideas. Norman R. F.
Maier (1933, 1937) used the term reasoning or productive behavior
to describe higher-order thinking and described lower-order thinking
as learned behavior or reproductive thinking. He demonstrated
experimentally that the two are qualitatively different types of
behavior patterns. Learning the multiplication tables through repeated
practice is an example of lower-order thinking—reproducing a
behavior previously observed or practiced.
Higher-order thinking moves beyond reproducing previous learning
and draws on analysis, synthesis, and evaluation skills (Lewis &
Smith, 1993). For example, a student may know how to compute the
area of a rectangle but may be faced instead with a problem that asks
for the area of a parallelogram. If the student is able to see how to
convert the parallelogram to a rectangle of the same size and
proportion, he has produced new knowledge from the integration of
previous learning experiences. Simple repetition of a previous
behavior is insufﬁcient. The student has to transform and apply
previous learning in a new way.

The idea that thinking can be divided into higher and lower levels

was elaborated on by Benjamin Bloom and colleagues in their
Taxonomy of Educational Objectives, usually called Bloom’s
taxonomy (Anderson & Krathwohl, 2001; Bloom, Englehart, Frost,
Hill, & Krathwohl, 1956). Table 14.1 provides additional distinctions
between lower-order and higher-order thinking. Higher-order
thinking skills frequently are interwoven with basic skills during the
teaching and learning process (Resnick, 1987; Shaw, 1983). For

example, in order for children to understand what they read, they
have to be able to make inferences and to use knowledge or
information that goes beyond the written text; thus, teaching reading
involves interweaving both basic reading skills and higher-order
thinking processes.

Many have argued that in order to compete in the twenty-ﬁrst

century, American students need to be taught a curriculum that
balances core knowledge—such as math, science, and reading—with
instruction in how to think—such as critical thinking, problem
solving, and making connections between ideas (Wallis & Steptoe,
2006). Enhancing the quality of thinking of all children is important,
but it may have particular signiﬁcance for those minority students
who historically have not performed as well as their more
economically advantaged peers (Armour-Thomas, Bruno, & Allen,
2006). A curriculum that emphasizes higher-order thinking skills has
been found to substantially increase the math and reading
comprehension scores of economically disadvantaged students
(Pogrow, 2005).

Thinking Skills. Effective teachers encourage their students to be critical
thinkers and problem solvers.

Bloom

’s taxonomy:

See page 360.

TA B L E 1 4 .1

A Comparison of Lower-order

and Higher-order Thinking

Lower-order Thinking

Higher-order Thinking

Reproductive behavior Productive behavior Repeating past

experiences Integrating past experiences Routine or mechanical

application of previously acquired information

Interpreting, analyzing, or

otherwise manipulating information

Recalling information Manipulating information Knowledge,

comprehension, and application Analysis, synthesis, and

evaluation

Sources: Bartlett, 1958; Bloom et al., 1956; Maier, 1933; Marzano, 1993; Newman, 1990.

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Although learning theorists see the development of higher-order
thinking as an important goal for all students, teachers often believe
that stimulating higher-order thinking is appropriate only for
high-achieving students (Torff, 2005). According to this view,
low-achieving students are, by and large, unable to deal with tasks
that require higher-order thinking skills and thus should be spared the
frustration generated by engaging in such tasks. This rationale for
reserving higher-level thinking for high achievers is not supported in
the research literature, however. Rather, research strongly suggests
that teachers should encourage students of all academic levels to
engage in tasks that involve higher-order thinking skills (Miri, David,
& Uri, 2007; Zohar & Dori, 2003).

What Are Thinking Dispositions?

What sets good thinkers apart is not simply superior cognitive ability
or particular skills; rather, it is their “tendencies to explore, to inquire,
to seek clarity, to take intellectual risks, to think critically and
imaginatively” (Tishman, Jay, & Perkins, 1992, p. 2). These
tendencies can be called thinking dispositions. Teaching for thinking
involves nurturing dispositions such as (Facione et al., 2000;
Tishman, Jay, & Perkins, 1992):

truth-seeking, a desire to understand clearly, to seek connections and

explanations;

open-mindedness, the tendency to explore alternative views, to generate

multiple options;

analytical thinking, the urge for precision, organization, thoroughness, and

accuracy;

systematic planning, the drive to set goals, to make and execute plans, and to

envision outcomes;

intellectual curiosity, the tendency to wonder, probe, and ﬁnd problems; a zest

for inquiry;

conﬁ dence in the use of reasons and evidence, or the tendency to

question the given, to demand justiﬁcation, and to weigh and assess
reasons and;

metacognition, the tendency to be aware of and monitor the ﬂow

of one’s own thinking and the ability to exercise mature judgment.

Empirical research has shown that thinking skills and

thinking dispositions are two distinct entities (Ennis, 1996; Perkins,
Jay, & Tishman, 1993;

Taube, 1997). A thinking skill is a cognitive strategy, whereas a
thinking disposition is a personal attribute (Dewey, 1933). There

is    a    difference    between teaching a thinking skill and motivating
students to cultivate a curious, inquisitive nature in which thinking
skills are used consistently (Fisher & Scriven, 1997). Skills and
dispositions are mutually reinforcing and thus should be explicitly
taught and modeled together (King    &    Kitchener,    1995).
Teachers have a responsibility not only to promote thinking skills,
but also to motivate    students    to    make    higher-order thinking a
habit.

What intellectual dispositions have become consistent features
of the way you think? In what areas do you have thinking skills
but lack the motivation to use them?

CRITICAL THINKING

The ideal critical thinker is habitually inquisitive, well-informed,
trustful of

Metacognition: See page 214.

Signi

ﬁcance of Thinking Skills. The development

of higher-order thinking skills is associated with
substantial increases in the math and reading
comprehension scores of minority students who are
economically disadvantaged.

Motivation:

See page 256.

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rea son, open-minded, ﬂexible, fair-minded in evaluation, honest
in facing personal biases, prudent in making judgments, willing
to reconsider, clear about issues, orderly in complex matters,
diligent in seeking relevant information, reasonable in the
selection of criteria, focused in inquiry, and persistent in seeking
results which are as precise as the subject and circumstances of

inquiry permit. (APA, 1990, p. 3)

What Is Critical Thinking?

Critical thinking is the process of evaluating the accuracy and worth
of information and lines of reasoning. A disposition toward critical
thinking could be characterized as the consistent internal motivation
to use critical thinking skills to decide what to believe and do
(Facione et al., 2000). A critical thinker not only is capable of
reﬂecting, exploring, and analyzing, but chooses to think in advanced,
sophisticated ways (Esterlee & Clurman, 1993). We use aspects of
critical thinking all the time in tasks such as comparing food labels to
see which foods are most nutritious, deciding which candidate to vote
for in an election, and evaluating advertising claims. In any instance
of critical thinking or reasoning, at least one question is at issue. For
example, in an elementary classroom, students might be given a set of
objects and asked to form and test a hypothesis about whether each
object will sink or ﬂoat. During the critical thinking process, students
deconstruct a problem, issue, or argument using guidelines such as
those listed here (Marzano et al., 1988; Paul & Elder, 2006).

Frame of reference or points of view involved: Clearly identify

the point of view from which a problem is expressed.

Assumptions made: Identify what is assumed or taken for granted in thinking about the issue.

Central concepts and ideas involved: Identify the most important ideas that are relevant to the issue.

Principles or theories used: Identify the principles or theories

used to support an argument. Clarify them, question them,
consider alternatives, and apply theories precisely and
appropriately.

Evidence, data, or reasons given: Identify lines of reasoning

and the evidence on which the reasoning is based. Use logic in
trying to determine whether a statement or an argument has a solid
basis in fact. Identify contradictions.

Interpretations or claims made: Examine whether the

interpretations or claims made are valid and grounded in evidence.

Inferences made: Rationally argue in favor of the inferences

being made. Formulate and consider possible objections to
inferences.

Potential implications and consequences: Figure out the

implications and consequences of a line of reasoning or course of
action.

Let’s think about these skills within the context of reading a piece of
literature. As a critical thinker, the reader might consider the points of
view of different characters, identify central themes in the book, look
for evidence to support assertions being made, and consider
alternative endings, noting the possible implications these would have
for the various characters.

Critical thinking abilities emerge gradually (King & Kitchener,

2002; Pillow, 2002). The development of critical thinking proceeds in

stages that reﬂect an increasing ability to analyze one’s own thinking
with a view toward improving it (Paul & Elder, 2006). In the initial
stage, individuals may be completely unaware of any signiﬁcant
problems in their thinking. Once they are faced with problems in their
thinking (through self-discovery or through direct challenge of their
ideas and beliefs by someone else), they may try to improve. At the
next stage, they recognize the need for regular practice and take
advantage of ways to practice good thinking habits. In the ﬁnal stage,
critical thinking habits become second nature (automatic) as the
individual becomes a “master thinker” (Paul & Elder, 2006).
Qualitative dimensions that reﬂect an individual’s skill in critical
thinking include clarity, accuracy, precision, relevance, depth,
breadth, and use of logical reasoning (Paul, 1990; Paul & Elder,
2006), as shown in Table 14.2.

Application: Fostering Critical Thinking

The ﬁrst step in fostering critical thinking in the classroom is to make
students aware of what it means to think critically. Teachers can have
students examine the lives and works of individuals who were critical
thinkers or have them examine and discuss their own thinking
processes. After students have

Automaticity: See page 230.

Cognitive thinking is similar to Piaget

’s concept of cognitive

equilibrium: See page 120.

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TA B L E 1 4 . 2

Qualitative

Dimensions of Critical Thought

Dimension Description

Clarity If a student

’s statement is unclear, we cannot really tell

much else about the level of thinking re

ﬂected because we don’t

know what the student is trying to say.
Accuracy A statement can be clear but not accurate, as in

“ Most dogs weigh

more than
300 pounds.

”

Precision A statement can be clear and accurate but not precise.
For example,

“Jack is overweight.” We don’t know if Jack is 2

pounds overweight or 200 pounds overweight.
Relevance A statement can be clear, accurate, and precise but
irrelevant to the issue at hand.
Depth A statement can be clear, precise, accurate, and relevant but still be
super

ﬁcial.

For example, the statement

“Just say no,” often used to

discourage kids from using drugs, lacks depth in response to the
very complex issues of drug use and peer pressure.
Breadth A line of reasoning can display all the dimensions already listed but lack
breadth.
For example, a political argument may look at a question only
from the liberal standpoint and ignore the conservative view.
Logic Students may pull together a combination of thoughts that
are contradictory or do not make any sense. Their thinking in this
case is illogical.

Source: Paul & Elder, 2006.

been given the opportunity to identify the characteristics of critical
thinking, they can be encouraged to monitor their own thinking for
evidence of these characteristics. When a basic foundation for critical
thinking has been established, teachers can use speciﬁc instructional
strategies to help students think critically. These include questioning
during class discussion, writing techniques, hypothesis testing,
inductive and deductive reasoning, and argument analysis. Let’s
examine each of these.

QUESTIONING IN CLASS DISCUSSION

Questioning is one of the most frequently used methods for sparking
critical thinking and analysis (Marzano, 1993). Class discussion,
which provides a logical venue for introducing different types of
questioning, can take three main forms (Paul, Binker, Martin, &
Adamson, 1989):

Spontaneous discussion can provide a model of listening critically and

exploring personal beliefs.

It is especially useful when students become interested in a topic,
when they raise an important issue during class, or when they are
just on the brink of grasping an idea.

In an exploratory discussion, the teacher raises questions in order

to assess students’ prior knowledge and values and to uncover their
beliefs or biases. Exploratory discussion also can be used to ﬁgure
out where students’ thinking is fuzzy or unclear.

Issue-speciﬁc discussion is used to “explore an issue or concept

in depth, evaluate thoughts and perspectives, distinguish the known
from the unknown, and synthesize relevant factors and knowledge”
(Paul et al., 1989, p. 28).

Table 14.3 provides general questions that can be used effectively in
any of these discussion formats. In order to maximize critical thinking
during class discussion, teachers need to provide sufﬁcient wait time
by pausing for several seconds after posing a question to give students
time to think before they are called on to respond (Tobin, 1987).
When teachers wait an average of at least 3 seconds after

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TA B L E 1 4 . 3

Question Sets That Can Be

Used Effectively in Class Discussions

Question type

Questions

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Clari

ﬁcation What do you mean by

___________

How does

___________

relate to

___________

Could you give me an example?
Why do you say that?

Assumptions All of your reasoning seems to be based on the idea that

___________

Why have you based your reasoning on

___________

rather than

__________

Is that always the case? Why do you think
that assumption holds here? What could we
assume instead? How would that change our

conclusions?

Reasons and evidence

What is your reason for saying that?
What other information do we need to know?
Is there reason to doubt the evidence or the
sources of our information? What led you to
that conclusion?
What would convince you otherwise?

Viewpoints or perspectives

You seem to be approaching the issue from

_____________

perspective. Why

have you chosen this rather than that
perspective?
How might other groups/people respond to this issue?
What would someone who disagrees say?
How are Ken

’s and Roxanne’s ideas alike? How are they different?

Implications and consequences

What are you implying by your statement?
But if that happened, what else would
happen as a result? What effect would that
have?

Adapted from Paul, Binker, Martin, & Adamson, 1989.

posing a question, the result is greater student-to-student interaction
during learning and increased student participation in class
discussions (Honea, 1982; Swift & Gooding, 1983).

In recent years, researchers have explored the use of online

discussion formats as a vehicle for higher-order thinking (Dutton,
Dutton, & Perry, 2002; Garrison, Anderson, & Archer, 2001).
Content analyses of online discussions have shown that students’
messages are lengthy and cognitively deep, incorporating critical
thinking skills such as inference and judgment as well as
metacognitive strategies related to reﬂecting on experience and
self-awareness (Hara, Bonk, & Angeli, 2000). While there are
advantages to holding discussions in either setting (face-to-face or
online), well-structured online discussions can expand the time spent
focusing on course objectives and allow more time for reﬂection
(Meyer, 2003). In order to extend the uses of discussion, teachers can
(Pierce, Lemke, & Smith, 1988):

break down an initial exploratory discussion about a complex

issue into simpler parts and have students choose aspects they
want to research or explore, or

have students write summaries immediately after a discussion,

allowing them to work together to ﬁll in gaps, provide
clariﬁcation, or add new thoughts or questions.

WRITING TECHNIQUES

Although the importance of writing as a basic skill has always been
recognized, it was Raymond Nickerson (1984) who ﬁrst noted the
value of writing as a tool for enhancing higher-order thinking.
Composing a piece of writing is a complex task that involves

planning, reviewing, weighing alternatives, and making critical
decisions (Marzano, 1995; Scardamalia & Bereiter, 1986). A great
advantage of writing is its versatility—teachers can utilize writing in
virtually every content area (Martin, 1987). Journal writing is a
format commonly used in classrooms to enhance students’
understanding

Writing skills: See page 222.

Metacognition: See page 214.

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Deductive Reasoning. Detective Sherlock Holmes said,

“If you eliminate the

impossible, whatever remains

—however improbable—must be the truth.” Like

Holmes, students reason deductively, narrowing the possibilities and drawing
conclusions based on the available evidence.

Argument Analysis.
In debate, students use critical thinking to present well-supported arguments
and to evaluate points made by their opponents.

of content, elucidate students’ thinking processes in a way that
teachers can react to, and provide opportunities for self-reﬂection
(Marzano, 1993).

HYPOTHESIS TESTING

Hypothesis testing involves examining research data and results to
determine what conclusions reasonably can be drawn to support or
refute a stated hypothesis. In an elementary classroom, a teacher
might organize students into groups and present each group with a
tray of materials to use in making a small light bulb work. The
students would begin by forming hypotheses about how the
materials could be used in combination to make their light bulb turn
on and would follow up by actually testing their hypotheses.
Hypothesis testing is not only germane to science activities. For
example, students in a literature class might be asked to form
hypotheses about how they think a story will end based on what they
have read so far and then be asked to compare their hypotheses with
the actual ending.

INDUCTIVE REASONING AND DEDUCTIVE REASONING

Teachers can provide opportunities for students to practice inductive
reasoning, logical thinking that moves from speciﬁc examples to
formulation of a general principle. For example, a teacher might
present a set of souvenirs from different countries and have students
work together to discover a general principle that describes what all
the items have in common. Students also can beneﬁt from activities
that require them to use deductive reasoning, a form of logical
thinking that moves from the general to the speciﬁc. For example, if
a student is presented with several characteristics of Native
Americans and asked to identify which tribe the characteristics best
describe, the student must use deductive reasoning. When Sherlock
Holmes takes a general set of clues and pieces them together to ﬁnd
the solution to a crime, he is using deductive reasoning—moving
from the general to the speciﬁc.

ARGUMENT ANALYSIS

Argument analysis involves challenging students to evaluate reasons
critically in order to discriminate between those that support a
particular conclusion and those that do not. For example, a teacher
might organize a class debate on the merits of wearing school
uniforms, with one side presenting arguments for the wearing of
uniforms and the other side presenting arguments against. Each side
not only would present its own arguments, but also would analyze the
arguments presented by its opponents in order to decide which were
valid in support of the conclusion being drawn.

While speciﬁc teaching strategies are useful, the intellectual

climate of the classroom is equally important in fostering critical
thinking. Students need an open, stimulating, supportive climate in
which they are encouraged to explore and express opinions, examine
alternative positions on controversial topics, and justify their beliefs
(Gough, 1991). See Box 14.1 for a summary of tips for creating a
classroom environment conducive to critical thinking.

Think about the classrooms you have spent time in over the
years, and identify speci

ﬁc classroom events or assignments

that fostered critical thinking. How does the promotion of critical
thinking vary by grade level or by discipline?

PROBLEM SOLVING

Critical thinking is an important part of deﬁning and solving
problems. In everyday situations, we are called on to solve problems
of various levels of complexity. Let’s consider what the
problem-solving process involves.

What Is Problem Solving?

A problem, quite simply, is any situation in which you are trying to
reach some goal and you need to

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BOX 14.1

Tips for Creating a Classroom Environment Conducive to Critical
Thinking

Reward good critical thinking.

Model critical thinking skills and dispositions.

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Explain different thinking strategies and focus students

’ attention

on important aspects of critical thought (e.g., examining
evidence).

Challenge poor critical thinking.

Actively engage students in critical thinking by providing diverse contexts for practicing

reasoning skills.

Create a climate of reasoned inquiry through assignments, class

discussions, and collaborative learning activities.

Encourage metacognition by teaching students how to examine their own thinking

processes.

Sources: Facione, Facione, & Giancarlo, 2000; Tishman, Perkins, & Jay, 1995.

ﬁnd a means to do so. It has a starting point, a goal (desired outcome),
and one or more paths for achieving that goal. Problem solving is the
means we use to reach a goal in spite of an obstacle or obstacles.
Some problems have clear goals and solutions; others don’t.
Problems can range from well deﬁned to ill deﬁned, depending on a
variety of problem characteristics (Hamilton & Ghatala, 1994). In
well-deﬁned problems, a goal is clearly stated, all the information
needed to solve the problem is available, and only one correct
solution exists. For example, a kindergartener who needs to match the
word two to its numeral faces a well-deﬁned problem with a clear
goal and a single answer. An illdeﬁned problem is one in which the
desired goal may be unclear, information needed to solve the problem
is missing, and/or several possible solutions exist. High school juniors
who are participating in a discussion of First Amendment rights in a

U.S. history course face an ill-deﬁned problem because the speciﬁc
goals are unclear, many important facts may not be in evidence, and
there probably is more than one “right” answer.

Problem solving requires a complex range of skills that develop at

different rates. Children are naturally curious and often will try to
ﬁgure out solutions to problems on their own; however, their
strategies may not be the most effective or efﬁcient ways to approach
the problem. Preschoolers are able to use available information
appropriately in problem solving, but they often fail to identify or
retrieve useful information from memory. Upper elementary students
not only can use current information effectively but also can draw on
prior knowledge to aid in understanding and solving a problem
(Kemler, 1978). Older children remember what did or did not work
on previous occasions and permanently reject hypotheses that failed,
whereas young children are less likely to reﬂect on how well their
previous strategies worked and may continue to use an ineffective
strategy (Carr & Biddlecomb, 1998; Davidson & Sternberg, 1998).

Research on novice and expert problem solvers provides further

information about how students’ approaches to problem solving may
differ based on their level of experience. Novice or inexperienced
problem solvers tend to apply problem-solving strategies mindlessly,
without any real understanding of what they are doing or why they
are doing it (Carr & Biddlecomb, 1998; Davidson & Sternberg,
1998). Expert problem solvers are more likely to (Bruning, Schraw,
Norby, & Ronning, 2004; Chi, Glaser, & Farr, 1988):

recognize potential problems,

perceive meaningful patterns in the information they are given,

perform tasks quickly and with few errors,

hold a larger quantity of information in working and long-term memory,

take time to carefully analyze a problem before implementing a solution, and

monitor their own performance and make adjustments.

Obstacles to Successful Problem Solving

Problem solving may break down due to the inexperience of the
problem-solver or as a result of several cognitive obstacles. Let’s
examine each of these in turn.

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Functional ﬁxedness is the inability to use objects or tools in a new way

(Dunker,
1945). For example, you need to draw a straight line and have a ruler
nearby; however, you are thinking of the ruler only as an instrument
for measuring length, so you don’t use it as a straightedge.

Response set is our tendency to respond to events or situations

in the way that is most familiar to us. Consider this problem: You
are presented with Figure 14.1 and are asked to connect all the dots
with four lines or fewer without lifting your pen. The solution to this
problem (found in Figure 14.2) requires breaking your response set

and drawing lines that extend beyond the mental border you
instinctively visualize around these dots.

Belief perseverance is the tendency to hold onto our beliefs

even in the face of contradictory evidence. For example, students
tend to be overconﬁdent about how quickly they can complete a
writing assignment. Even when the writing process takes twice as
long as they expect, students tend to remain overly conﬁdent and fail
to adjust their estimates for future writing assignments (Buehler,
Grifﬁth, & Ross, 1994).

The common theme in all these obstacles is lack of ﬂexibility. A rigid
mind-set narrows possibilities, while an open mind increases the
chance that you will be able to reframe a problem in a way that
suggests a workable solution.

Application: Teaching Problem-solving Strategies

Teachers can enhance students’ problem-solving skills by teaching
and modeling speciﬁc strategies that students will use frequently
within a particular area, by providing students with general rules of
thumb that may work in a variety of contexts, and by presenting
varied contexts and opportunities for students to practice their
problem-solving skills.

ALGORITHMS AND HEURISTICS

Solutions to some problems can be found by using an algorithm, a
prescribed sequence of steps for achieving a goal. For example, if you
want to calculate the area of a cylinder and know its dimensions, you
can apply a speciﬁc formula that will give you the correct answer.
The problem-solving process may take a little time, but the solution is
clearly achievable if the appropriate algorithm is chosen and its steps
are followed accurately.

Not all problems come with directions to follow. In the absence of

speciﬁc directions, you might need to use a heuristic to solve the
problem at hand. A heuristic is a general problem-solving strategy
that might lead to a right answer or to a solution that usually is
reasonably close to the best possible answer. It’s a rule of thumb, an
educated guess, or common sense. The difference between algorithms
and heuristics basically comes down to formal steps versus an
informal rule of thumb, or accuracy versus approximation. Students
are using heuristics when they guess a close answer to a
multiplication problem, rounding to the nearest large unit (e.g., 19
times 18 is about 400); shoppers are using heuristics when they
examine the price tag as an indicator of which electronic device
probably has the most features. Heuristics help us narrow down
possible solutions to ﬁnd one that works (Stanovich & West, 2000).

    Let’s consider three common heuristics:
1.      Means-end analysis is a heuristic in which the main
problem-solving goal is divided into subgoals. For example, the
main goal of developing a classroom management    plan    for    your
ﬁrst    year    of    teaching could be broken down into subgoals of
deﬁning your rules, outlining your classroom procedures, arranging
your classroom in an orderly way, and developing a schedule to help
your students understand the classroom routine.

2.    The working-backward strategy is a systematic approach in
which you start with the ﬁnal goal and think    backward    to
identify    the    steps    necessary    to reach that goal. For instance,
students who have a major term paper due in three weeks can work
back-

Figure 14.1. Mental Set
Problem. Assume that you are given a sheet of paper showing these nine
dots. Connect the nine dots with four straight lines. You must draw all four lines
without lifting your pencil from the paper. You may not fold, cut, or tear the
paper in any way.

Lack of Flexibility. A rigid, in

ﬂexible mind-set is an obstacle to effective

problem solving.

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TA B L E 1 4 . 4

Guidelines for Instruction in the Use of Algorithms and Heuristics

Use of algorithms

Describe and demonstrate speci

ﬁc algorithms.

Module 14 :

Critical

Thinking and Problem Solving

Work on a particular problem together, and talk students through each step of the algorithm.

Help students learn to check their work and to catch errors in their applications of an algorithm.

Ask students to explain what they are doing as they work through a problem, or ask them to show their

written work. This makes it easier for the teacher to pinpoint speci

ﬁc misunderstandings or errors and

provide students with corrective feedback.

ﬁne the situations in which the algorithms should be used. This step is

essential. In math, for example, students may memorize formulas for calculating
the area of different shapes; however, if they do not understand when to apply a
particular formula, they will not be able to solve math problems correctly.

Use of heuristics

Teach students how to better de

ﬁne ill-deﬁned problems. This may involve

taking a large, vague problem and breaking it down into several smaller
problems or steps to be tackled.

Teach students how to distinguish essential

from nonessential information as they gather data to solve a problem.

Show

students where and how to

ﬁnd any missing information they need to solve a

problem.

Give students the opportunity to solve problems in groups. Group

work can be fertile ground for problem solving if students have the opportunity to
share ideas, model different approaches for one another, and discuss the merits
of different approaches to solving a problem.

Sources: Atkinson, Derry, Renkle, & Wortham, 2000; Renkle & Atkinson, 2003; Rogoff, 2003.

ward to ﬁgure out how much time they will need to allow for each step in the research and writing process and set
intermediate deadlines for themselves.

3. Analogical thinking limits the search for solutions to situations that are most similar to the one at hand. A class

that is analyzing sources of pollution in a local river might consider the sources of pollution in other waterways in
the region for comparison.
Explicit instruction in the use of algorithms and heuristics is very useful in helping students become better

problem solvers (Dominowski, 1998; Kramarski & Mevarech, 2003). Table 14.4 provides guidelines for instruction
in the use of each of these problem-solving methods. In the real world, students are faced with many problems that
are ill-deﬁned and are not solvable by a clear algorithm. Teachers therefore need to pay particular attention to
providing students with authentic learning experiences that give practice in solving complex problems (Resnick,
1988; Sternberg et al., 2000).

You may not have heard the terms algorithm and heuristic prior to reading this module, but you undoubtedly
have used these problem-solving strategies many times. In what situations have you used an algorithm to
solve a problem? When have you used a heuristic?

GENERAL AND SPECIFIC STRATEGIES

Problem-solving strategies often are speciﬁc to a particular content area (e.g., formulas for calculating area or
perimeter; strategies for identifying common grammar or punctuation errors when editing a paper). Although
content area often is very important, certain problem-solving strategies tend to be helpful across a variety of contexts
(Davidson & Sternberg, 1998; Dominowski, 1998). John Bransford and Barry Stein (1993) use the acronym IDEAL
to identify ﬁve important steps found in many different problem-solving approaches: (1) identify the problem, (2)
deﬁne goals, (3) explore

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cluster four

cognitive processes

TA B L E 1 4 .5

IDEAL Problem-solving Steps

Step Description

I Identify problems and opportunities. Identifying that a problem exists is a critical

ﬁrst step in the

problem-solving process. D De

ﬁne goals and represent the problem. Success in

problem solving often depends on how we represent the problem. This step
involves focusing on relevant information, understanding what the problem is
asking, and activating prior knowledge schemas that might relate to the problem.

E Explore possible strategies. Look at possible options and strategies that might be used to solve

the problem.

A Anticipate outcomes and act. Consider possible consequences of different strategies proposed in

the previous step, and implement the chosen strategy.

L Look back and learn. After implementing a strategy, evaluate whether the problem is solved

effectively.

Source: Bransford & Stein, 1993.

possible strategies, (4) anticipate outcomes, and (5) look back and learn. Table 14.5 describes each of these steps.
People seem to move back and forth from general strategies, such as those described in the IDEAL
method, to content-speciﬁc approaches, depending on the needs of the situation and people’s level of expertise. In an
area in which we have little experience, we are more likely to rely on general knowledge and problem-solving
strategies; however, as we develop expertise within a content area, we are more likely to apply speciﬁc strategies
that we know are effective (Alexander, 1996).

PROBLEM-BASED LEARNING

Problem-based learning (PBL) is experiential, hands-on learning organized

around the investigation and resolution of messy, real-world problems (Torp & Sage, 2002). In this approach,
students are engaged problem

Start

Figure 14.2: Solution to Mental Set Problem. If you had never seen this problem before, you probably approached it with a
speci

ﬁc set of expectations— you assumed the four lines had to remain within the perimeter of the nine dots. To solve the problem,

you must go beyond the perimeter of the dots.

Problem-based Learning. Experiential, hands-on learning is centered around real-world problems.

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solvers who identify the root problem and the conditions needed for a
good solution, pursue meaning and understanding, and become
active, self-directed learners (Hmelo-Silver, 2004). Imagine that you
want to teach your students how to identify minerals. A traditional
lesson might involve providing students with several tools (e.g.,
mineral key, pictures of crystal shapes, streak plate, glass plate, nail,
penny, weak acid, hand lens), a step-by-step demonstration of how to
use the tools, and an activity in which students are to identify the
minerals based on the demonstrated procedure. In a PBL unit
designed to cover the same curriculum, you might tell your students
that they are playing the roles of geologists. Their task is to identify
the minerals at a couple of local sites in order to facilitate the
modiﬁcation of local zoning ordinances. Tools and various minerals
are made available, and the students work in small groups to identify
the minerals. The lesson content in each case is similar; however, the
PBL lesson is designed to engage students’ curiosity and tie their
learning to a real-life context.

What elements of problem-based learning can you identify in this textbook?

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key concepts

Summary

Explain the difference between a higher-order thinking skill and a thinking disposition, and
discuss why both skills and dispositions are important. Empirical research has shown that thinking
skills and thinking dispositions are two distinct entities. Higher-order thinking skills re

ﬂect an

individual

’s ability to manipulate and transform information in order to solve problems or make decisions.

Thinking dispositions re

ﬂect an individual’s consistent internal motivation to use thinking skills. Having the

ability to engage in higher-order thinking does not guarantee the disposition to do so; therefore, teachers
must understand how to build intellectual skills in students as well as how to foster the dispositions to use
those skills on a regular basis.

Explain what critical thinking means. Critical thinking is the process of evaluating the accuracy and
worth of information and lines of reasoning. During the critical thinking process, students deconstruct a
problem or an issue to identify and carefully consider characteristics such as the frames of reference or
points of view involved, the assumptions being made, the evidence or reasons being advanced, the
validity of the interpretations and claims being made, and the implications and consequences that follow
from a particular decision.
Identify

ﬁve instructional strategies that can be used to foster critical thinking. Teachers can use

one of

ﬁve popular and effective strategies for promoting critical thinking. (1) Questioning techniques help

identify students

’ prior knowledge, clarify values, and uncover students’ beliefs or biases. (2) Writing

techniques can be used in virtually every content area and can challenge students to plan, review, weigh
alternatives, and make critical decisions. (3) Hypothesis testing involves examination of research data
and results to determine what conclusions reasonably can be drawn to support or refute a stated
hypothesis. (4) Inductive and deductive reasoning are forms of logical thinking that ask
students to move from speci

ﬁc examples to formulating a general principle (inductive) or from a general

set of clues to a speci

ﬁc solution (deductive). (5) Argument analysis challenges students to evaluate

reasons critically in order to discriminate between those that support a particular conclusion and those
that do not.
De

ﬁne problem solving and explain the difference between a well-deﬁned problem and an

ill-de

ﬁned problem. A problem is any situation in which you are trying to reach some goal and need to

ﬁnd a means to do so. Problem solving has a starting point, a goal (desired outcome), and one or more
paths for achieving that goal. A well-de

ﬁned problem is one in which a goal is clearly stated, all the

information needed to solve the problem is available, and only one correct solution exists. An ill-de

ﬁned

problem is one in which the desired goal is unclear, information needed to solve the problem is missing,
and/or several possible solutions to the problem exist.
Discuss the role of algorithms, heuristics, the IDEAL approach, and problem-based learning in
teaching problem solving. An algorithm is a prescribed sequence of steps for achieving a goal.
A heuristic is a general problem-solving strategy that might lead to a right answer or to a solution that
usually is reasonably close to the best possible answer. The difference between algorithms and heuristics
basically comes down to formal steps versus an informal rule of thumb, or accuracy versus
approximation. The acronym IDEAL identi

ﬁes ﬁve important steps found in many different

problem-solving approaches: (1) identify the problem, (2) de

ﬁne the goals, (3) explore possible strategies,

(4) anticipate outcomes, and (5) look back and learn. Problem-based learning (PBL) is experiential,
hands-on learning organized around the investigation and resolution of messy, real-world problems.

Key Concepts

algorithm analogical thinking argument analysis belief perseverance critical thinking deductive reasoning exploratory

discussion functional

ﬁxedness

heuristic higher-order thinking hypothesis testing ill-de

ﬁned problem inductive reasoning issue-speciﬁc discussion

means-ends analysis problem

problem-based learning (PBL) problem solving response set spontaneous discussion thinking dispositions wait time

well-de

ﬁned problems working-backward strategy

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255

Case Studies:

Refl ect and Evaluate

Early Childhood:

“Air”

These questions refer to the case study on page 206.

1. What thinking dispositions did Barb foster in her classroom?
How would you describe the motivation and engagement level of
her students?
2. Describe how Barb used questioning techniques to promote
critical thinking during the experiment with the cork and the cup.

3. How did Barb challenge her students to think critically during the discussion of kite aerodynamics?
4. Was the kite project an example of a well-de

ﬁned or an ill-deﬁned problem? Explain.

5. Which steps of the IDEAL problem-solving sequence did you notice during the kite project?

Elementary School:

“Reading About Pirates”

These questions refer to the case study on page 208.
1. How would you describe Kiana

’s thinking dispositions during the reading rotations?

2. Which features of higher-order thinking, if any, did you see
represented in the learning events that transpired in Ian

’s

classroom?
3. Was Ian

’s assignment for the students completing the short

story booklets at their seats bene

ﬁcial, or could he have given them

a more thought-provoking assignment? Explain.

4. How could Ian incorporate aspects of critical thinking into his reading-group activities?
5. What steps of the IDEAL problem-solving model did you see used in this case?

Middle School:

“King Washington”

These questions refer to the case study on page 210.
1. Which thinking dispositions did you see represented in the learning events that transpired in

Tom

’s classroom?

2. How could Tom

’s discussion of George Washington promote

students

’ critical thinking about the concept of leadership?

3. What dimensions of critical thought should Tom consider when
evaluating his students

’ thinking during the Continental Congress

activity?
4. Does the Continental Congress activity present students with a
well-de

ﬁned or an ill-deﬁned problem to solve? Explain.

5. Is the Continental Congress activity an example of
problem-based learning as de

ﬁned in your reading? Explain.

High School:

“I Don’t Understand”

These questions refer to the case study on page 212.

1. What thinking dispositions could So Yoon have tried to enhance
in order to facilitate students

’ learning of the new math concept?

2. How might belief perseverance have interfered with students

’

decision making about how to approach the math problems?

3. Did the type of problems So Yoon assigned require the use of algorithms or the use of heuristics?

Explain.

4. What steps from the IDEAL model could have been utilized by So Yoon

’s students?

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