Authentic Tasks And Mathematical Problem Solving

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Authentic Tasks and Mathematical Problem Solving
David K. Pugalee, Patricia Douville, Corey R. Lock, Josephine Wallace
University of North Carolina at Charlotte
Approaching mathematics teaching from a humanistic approach involves dynamic interactions supporting
meaningful and relevant mathematics learning. A primary goal of such learning is to prepare students to
solve everyday problems. Authentic tasks are a critical tool in developing the level of mathematical
understanding and conceptualizing indicative of mathematical literacy. Authentic tasks can be
characterized by in-depth analysis along four dimensions: thinking and reasoning, discourse,
mathematical tools, and attitudes and dispositions. Each of these dimensions, collectively and singularly,
must be an explicit focus of the mathematics that engages the students. This paper will present a model
demonstrating how these dimensions contribute to a balanced instructional and curricular focus that is at
the heart of humanistic mathematics education.
In order to support students’ development of mathematical literacy, a standards-based mathematics
curriculum should provide for experiences that differ from those indicative of the traditional, typical
classroom. Such a curriculum provides opportunities for the students to explore important mathematical
topics in the form of situational problems that actively engage students. These lessons, grounded in real-
life problems, include four dimensions:
• Thinking and reasoning – engaging in such activities as gathering data, exploring, investigating,
interpreting, reasoning, modeling designing, analyzing, formulating hypotheses, using trial and error,
generalizing, and verifying outcomes
• Discourse – engaging in individual, small group, and whole class interactions; role of language and
interactions in the construction of mathematical meaning
• Mathematical tools – using symbol systems such as tables, graphs, and drawings and technological tools
such as calculators, computers, and manipulatives
• Attitudes and disposit ions – developing self-regulation, persistence, reflection, and enthusiasm (Pandey,
1990).
Mathematics that is embedded in authentic tasks frequently mix hard and soft data, provide students
opportunities to approach the problem with multiple approaches and reach different conclusions grounded
in sound mathematical reasoning are the types of tasks that are typical in life and work. While such tasks
frequently do not appear to be ‘school’ mathematics problems, they serve as an effective means of
developing important mathematical understanding while also supporting the development of basic skills.
Thinking and Reasoning
Students’ experiences with mathematics should support the development of their ability to reason and use
logic progressing on a continuum from concrete to formal to abstract. Tasks should build inductive
reasoning skills through making conjectures based on generalizations from patterns in observations. In
addition, students should employ deductive reasoning in testing those conjectures. In all cases, students’
thinking and reasoning is enhanced when they are members of classroom communities that require them
to explore, make conjectures, test their ideas, defend the validity of the outcomes, and convince others of
their approach. Authentic tasks provide the necessary framework for integrating both mathematics
content and process objectives into meaningful learning situations. Consider the thinking and reasoning
possible in the following example:
This lesson emerged when the school had a tornado drill and students discovered that the hallway was too small to
fit them all. Students investigate the concept of area by figuring out how many people will fit in areas in the school
building. Specifically, students investigate how many of them will fit in the following areas: (1) their classroom, (2)
a hallway, (3) a lobby, and (4) the cafeteria tables. Students are divided into groups of four and assigned to one of
the three areas. The groups choose the materials they want to use and are accompanied to their designated areas by
classroom volunteers. The groups estimate the number of students who will fit in an area and then develop a strategy
for determining the exact number. During their investigations, students organize and record their solutions and use
measuring, counting, and addition to find the total number of people. At the end of the lesson, the class reconvenes
and students share their strategies and results. (Annenberg/CPB, 2002).
Thinking and reasoning should be an explicit element in teaching mathematics. The United
Kingdom’s Cognitive Acceleration through Mathematics Education project is an example of a successful
program with theoretical bases in social psychology and developmental psychology. The lesson provides
appropriate mediation using cognitive conflict to engage students in higher levels of mathematical

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thinking (Mok & Johnson, 2001). On-going research shows that CAME supports metacognitive
development through raising one’s awareness of how they learn and solve problems. Explicit focus on
reasoning and thinking supports the intent of the NCTM standards (NCTM, 2000) that advocate students
should recognize reasoning and proof as fundamental to mathematics, make and investigate mathematical
conjectures, develop and evaluate mathematical arguments, and select and use various types of reasoning.
Experience with mathematical modes of though builds cognitive capacity to read critically, identify
fallacies, detect bias, assess risk, and suggest alternatives (National Research Council, 1989). Authentic
tasks must include substantive emphasis on developing such habits of thought.
Discourse In mathematical communities “we continually and actively build and rebuild our worlds not
just through language, but language used in tandem with actions, interactions, non-linguistic symbol
systems, objects, tools, technologies, and distinctive ways of thinking, valuing, feeling, and believing.”
(Gee, 2000, p. 11) Well-balanced mathematics instruction allows students to explore the big ideas of
mathematics in a real-world context. For example, students in one middle school classroom used a model
of similar triangles to find the estimated distance of a new road connecting two-cities. The task required
students to employ a mathematical model but also realize that in this real-world context meant that their
responses were ‘best’ estimates of the actual distance. More importantly, the classroom experiences
reinforced mathematical reasoning and communication. Students were asked to write about thei
r solution
processes. Peer evaluations of the responses supplied feedback on the mathematical ‘soundness’ of the
approaches and the extent to which good mathematical communication had been modeled. The
importance of discourse is evident as students in this classroom engaged in tasks embedded in
experiences requiring them to justify their mathematical reasoning and communicate such thinking to
their peers and teacher (Pugalee, 2001; Lajoie, 1999).
One means of promoting discourse communities involves a three-phase approach applicable to
group discussions. In the first phase, the intent is to get as many student ideas out in the open as possible.
What do you think? Why? are good prompts to get the discussion ‘geared-up’. In the second phase, the
class moves to comparing and evaluating ideas. Open discussion is encouraged, but the teacher focuses
on the content of the discussions so that the discussion can be facilitated to promote mathematical
understanding. Is your idea the same as …? What do you think about…? are ways to structure this level
of the discussion. The third or focusing phase of the discussion focuses the range of ideas. Teachers
filter and direct ideas to help the class progress on the issue (Sherin, 2000). Teachers of mathematics
should develop tasks that support the use of a variety of forms of discourse, including oral, written,
pictorial, symbolic, and graphic. Teachers should take an active role in advancing and organizing
discourse so student learning is maximized, including encouraging students to make conjectures so their
ideas can be assessed and others can consider the reasonableness of the ideas. Student communication
should include multiple modalities including written and oral work. Various structures provide
collaborative and cooperative experiences where student listen to and react to others, clarifying their own
thinking while asking questions that develop standards for mathematical reasoning and communication
(Pacific Regional Education Laboratory, 1996).
Mathematical Tools Authentic tasks allow students to explore multiple modes of representation in
experientially real contexts using a variety of technological tools. Abrams (2001) argues that school
mathematics has ignored the development of mathematical models in applied settings resulting in a lack
of skills such as abilities to analyze units, make choices among multiple representations, and recognize
common structures. He advocates moving away from traditional school mathematics as self-contained to
curricula that emphasizes mathematics as a tool for solving important problems from other disciplines.
NCTM (1991) advocates the use of various tools to aid students in obtaining mathematical power:
computers, calculators, and other technology; concrete materials usedas models; pictures, diagrams,
tables, and graphs; invented and conventional terms and symbols; metaphors, analogies, and stories;
written hypotheses, explanations, and arguments; oral presentations and dramatizations.
Gravemeijer, Cobb, Bowers, and Whitenack (2000) describe one approach (formulated by Kaput,
1994) that provides a bridge from formal mathematical symbolizations through the use of computer-based
representation systems. Instead of using ready-made graphs of time and distance, students would use
their own authentic experiences to understand symbolizing. MathCars provides a computer-simulated

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driving experience that creates distance-time graphs, velocity-time graphs, and data tables of a simulated
trip. Various mathematical symbolizations are linked to dashboard displays. Such investigations allow
everyday experiences of motion to be linked to formal graphical representations.
The Middle-school Mathematics through Applications Program (MMAP, see http://mmap.wested.org/.) is
a good example of how tools and authentic problem solving provide central curricular tenets. The
project-based materials use various software tools and engage students with real-world problems
requiring them to apply the mathematics they have learned. The units emphasize proportional reasoning,
algebraic expressions, and functions as well as statistics, probability, measurement, and geometry. One of
the goals of the program is to increase students’ conceptual understanding of mathematics, to improve
their ability to usestandard symbolic notation, and to improve their mathematical communication skills.
In one unit on Antarctica, students play the roles of architects designing living space for themselves or a
client or the design of a research station (depending on grade level). Projects such as these emphasize the
role of mathematical tools in developing students’ capacity to explore, investigate, and formulate
conclusions through explorations in problem situations grounded in real world situations.
Attitudes and Dispositions
Attitudes and dispositions have historically not been considered in problem solving research.
Many students experience “‘math phobia’, a disconnect between the mathematics and its applications, and
mathematics getting in the way of learning (inward view of math, mathematics for its own sake, too
abstract, fuzziness, etc.)” (Henderson et. al., 2001). (Lajoie (1999) who worked with the K-12 Authentic
Statistics Project, designed to prepare students for decision making in the real work, found that “when
students work on tasks they find engaging and are expected not merely to give the ‘right’ answers but to
understand the statistical relationships they use and manipulate, they own that knowledge, stay interested
in the mathematics, and do not fear working on problems in new contexts.” (p. 131). In one problem at
the middle school level, students calculated their pulse rate, used computer software to analyze and chart
class data, and engaged in discussions about differences in the data representations. Likewise, an
elementary project reported that students engaged in an authentic task involving water quality monitoring
felt that they had done important work and were proud of their accomplishments. Scientists who
reviewed the culminating presentations related that they were impressed by the quality of the students’
thinking as well as their presentation skills (Vye et. al., 1998).
DeBellis and Goldin (1997) posit that beliefs, attitudes, emotions, and values interplay with
cognition and as such can either facilitate or hinder monitoring during problem solving. While emotions
are more temporary, the others have more stability serving as self-regulating structures. Affective
pathways are either positive or negative and impact problem-solving behavior. If a positive pathway is
experienced initially during problem solving experiences, curiosity may serve as a motivating factor
leading the individual to a deeper understanding of the problem and the enactment of exploratory
heuristics. Frustration may lead to an impasse resulting in an ineffective revision of strategies.
Experiencing a negative pathway may lead to bafflement and the individual resolving to use ‘safe’
procedures rather than exploration. Should these efforts fail, the initial frustration may result in anxiety
and reliance on others, including teachers and peers, as authorities as well as to avoidance behaviors.
Expert problem solvers handle affective issues better. Whereas some students feel they should follow
established procedures during problem solving, others employ originality and self-assertiveness. While
‘good” problem solvers enact responses that are productive when faced with insufficient understanding,
other students resort to guessing or using likely, and often inappropriate, procedures perhaps because they
feel that their personal mathematical knowledge is deficient.
NCTM’s position statement, “Guiding Students’ Attitudes and Decisions Regarding their
Mathematics Education” (see www.nctm.org), elaborates on what school programs should do to foster
positive attitudes and dispositions. The elements are consistent with those explicated throughout this
paper regarding the dimensions of effective authentic learning tasks. Teachers should build on the
curiosity and eagerness to learn expressed by elementary students by using real
-life experiences
emphasizing the value and usefulness of mathematics. Middle school students need encouragement that
fosters confidence in their ability to make sense of mathematics. Real-life examples help this age group
make connections between mathematics and their future educational and vocational options. Secondary

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students should recognize the relationships between academic and career choices. Students’ beliefs in the relevance
and value of mathematics should be reinforced, as should their views about their ability to "do" mathematics, and
their need to be life long learners. All students should be provided with current information on the relationship of
what they are learning and future options, the increasing number of career opportunities available through
mathematical study, and how careers in other field depend on mathematical knowledge.
Conclusions Authentic tasks are important in the development of students’ mathematical literacy and their ability
to engage in mathematics applied in diverse fields as adult members of society. In considering the nature of such
important tasks, four dimensions were described: thinking and reasoning, discourse, mathematical tools, and
attitudes and dispositions. Another way to view the importance of these elements is through a model of
mathematical literacy (Pugalee, 1999). Problem solving, representing, manipulating, and
reasoning are represented in the outer circle of the model. These are the processes of being mathematically literate
and are essential components to address in developing and/or selecting authentic mathematical tasks. The enablers
of communication, technology, and values play a crucial and central role as depicted in the model. These enablers
facilitate the “doing of mathematics”. Each of the circles is interdependent underscoring the complex and intricate
nature of mathematics. Authentic tasks should be a substantive part of mathematics problem solving; however, such
tasks must provide extensive connections to mathematics content and processes that develop mathematical
understanding and power. The characteristics of such problems as elucidated in this paper provide a framework for
guiding the design and implementation of such tasks into mathematics classrooms.
References
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