Quantitative Thinking!
Quantitative Thinking!
Quantitative Thinking!
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CHAPTER 19
KATHLEEN A. CRAMER
University of Wisconsin, River Falls
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In the following sections we will examine both issues. We will explain the
importance of conceptual and procedural knowledge in teaching
quantitative understandings to children, and the role of representation in
the acquisition of mathematical concepts. We also will discuss current
thinking related to the question of what kinds of skills should be taught
and the present and future influences of technology on the scope,
sequencing, and presentation of mathematics curricula.
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Here are a few examples showing how concept knowledge and procedural
knowledge differ:
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Questions about the nature of knowledge, the nature of one who knows
(or the one who learns), the "transmitability" (or lack thereof) of
knowledge and conditions under which learning most effectively occurs
have persisted since 500 B.C. A problem results from a view of knowledge
which requires a match between the cognitive structures themselves and
what those structures are supposed to represent. The difficulty here is
that it is never possible to determine how well our (or others') mental
structures represent what they are intended to represent, for such
assessment "lies forever on the other side of our experimental interface"
(Von Glaserfeld, 1987). Since we can never step outside of ourselves and
achieve a truly objective perspective, a different view of what it is "to
know" is required, one which is not based on a correspondence with
reality.
Such a view was stated by Osiander in 1627 in his retort to the critics of
Copernicus' revolutionary idea that the Earth was not the center of the
universe. He said, "There is no need for these hypotheses to be true, or
even to be at all like the truth; rather one thing is sufficient for them -
that they yield calculations which agree with the observations" (Popper,
1968).
This second conception of knowledge, one that fits our observations, has
profound implications for education and instruction and for the
organization of experience. Piaget (1952) characterized this situation as
follows: "Intelligence organizes the world by organizing itself." Thus, an
individual's experiences and their subsequent reorganization become the
beginning, middle, and end points of conceptual development and the
concomitant evolution of intelligent action.
But how can one directly experience a number which is, by its very
nature, an abstraction? The answer is that one cannot, but it is possible
to experience representations of it. Representation is, therefore, a crucial
component in the development of mathematical understanding and
quantitative thinking. Without it, mathematics would be totally abstract,
largely philosophical, and probably inaccessible to the majority of the
populace. With it, mathematical ideas can be modeled, important
relationships explicated, and understandings fostered through a careful
construction and sequencing of appropriate experiences and observations.
It is currently held that it is the translation between different
representations of mathematical ideas, and the translations between
common experience and the abstract symbolic representation of those
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The Rational Number Project (Post, Behr, Lesh, & Wachsmuth, 1985), in a
series of research projects supported by the National Science Foundation,
has, over the past decade, utilized this framework in the development of
its own theory-based instructional units. This project corroborated the
interactive nature of student approaches and has found the use of
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For example, the concept of adding fractions can taught using several
translations. A translation from written symbol to manipulative mode can
be shown by asking a child to model the sum 1/2 + 1/4 with manipulative
materials, such as fraction circles. Similarly, using the illustration shown
in Figure 19.3, translation from pictorial mode to written symbols can be
achieved by asking the child to write a number sentence for each step in
the addition problem (1/2 + 1/4; 1/2 = 2/4; 2/4 + 1/4 = 3/4).
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The report also stresses the need for practical, "hands on" activities to
develop mathematical concepts before emphasizing the practice of skills
and procedures. Instruction that includes problem solving, applications,
and an interdisciplinary approach to mathematics instruction was also
recommended.
For the schools to emphasize facility with paper and pencil computation at
the expense of higher level cognitive processes at a time when calculators
and computers are commonplace tools in the adult world is reactionary.
To emphasize computation when international studies show that American
children rank in the lower half of nearly every mathematics category
compared to children in other leading industrial nations is unacceptable.
For example, in the Second International Mathematics Study
(International Association for the Evaluation of Educational Achievement
[IEA], 1984), eighth grade children in the United States ranked below
students in Japan and Canada in every category assessed. In
measurement and geometry, U.S. children scored in the bottom quarter
of 20 developed countries. In a more recent international study
comparing performance of children in comparable cities in Minnesota,
Japan, and China, the highest average score of an American fifth grade
class assessed was below the average score of the lowest Japanese fifth
grade class assessed (Stevenson, Shin-Ling, & Stigler, 1986).
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of Mathematics, 1987).
The NCSM (1988) has recently updated the list of basic skills. The new
position paper lists 12 critical areas of mathematical competence for all
students. Problem solving, applying mathematics to everyday situations,
alertness to reasonableness of results, estimation, geometry, and
measurement will be reaffirmed as basic skills. Five new categories are:
(a) mathematical reasoning (b) communicating mathematical ideas, (c)
algebraic thinking, (d) statistics, and (e) probability. The spirit the earlier
report remains intact. Additional positions relating to instruction include
the following:
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QUANTITATIVE THINKING
What is meant by mental arithmetic skills? Here are two examples. The
first is taken from Hope's article in the 1986 Yearbook of the NCTM.
Consider the problem 99 x 8. An unskilled mental calculator will be tied
into the tedious paper and pencil algorithm to find the answer: Nine times
eight is 72. Record the two and carry the seven. Nine times eight is 72
plus seven is 79. The answer is 792. A skilled mental calculator sees 99 x
8 as one group of eight less than 100 x 8. Eight hundred minus eight is
792. A skilled mental calculator could compute 25 x 480 simply by using
the number relationship: 25 = 100/4. Instead of calculating with paper
and pencil, the problem computes mentally as 100 x 480 = 48000/4 =
12000. There are other types of calculations, and with practice, children
can learn and develop their own techniques. Flexibility is key. Note in the
examples above that the focus is on the relationships between the
numbers, described previously as conceptual knowledge.
Research has shown that children who are quick to learn the basic
arithmetic facts create efficient mental strategies for obtaining answers.
For example, a child's thought process behind 8 + 7 might be: "I know 8
+ 8 is 16 so 8 + 7 must be one less, 15." A quick way to find the product
of 8 and 7 involves this type of thinking: "4 x 7 = 28; I have twice as
many groups of 7 so 8 x 7 is 28 + 28."
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Percentage responding
*1) 32 7%
2) 31.33; 311/3 etc. 16%
3) 25%
4) wrong operation 20%
5) no answer 32%
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Adding two positive numbers yields a quantity larger than either added.
(1/2 + 1/3 cannot equal 2/5 because 2/5 is less than 1/2, one of my
original addends.) Dividing by a whole number yields a quantity less than
the dividend, but division by an amount less than 1 yields a larger
quantity.
INFLUENCES OF TECHNOLOGY
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The second question raised by this technology is, "How should the way
mathematics is taught be changed, given the availability of calculators
and computers?" The NCTM recommends that mathematics programs
take full advantage of the power of calculators and computers at all grade
levels (NCTM, 1980). Using calculators in problem-solving situations
allows all students to participate in vastly expanded types of activities.
Lack of computational ability need not separate students into those who
work on low level skills and those who have the opportunity to work on
higher level activities. Tedious calculations need not limit the type of
problems students are given. Applied problems with real data need not be
avoided since the calculator can do the messy work. Using calculators in
problem-solving situations is not limited to the mathematics classroom.
Data collected in science experiments, and making predictions or
understanding trends in the social sciences can all be processed with
calculators.
Calculators are not just for number crunching; calculators also can be
used to develop new mathematics concepts and to reinforce previously
learned concepts. Below are three activities which use a calculator to
reinforce and extend a basic concept. The first deals with place value, the
second productivity, and the third relates to conservation of natural
resources.
1. Enter 6,425 on your calculator using only the numbers 1000, 100,
10, 1 and + , = keys.
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Software programs have been created that present problems and monitor
student performance. These allow both teacher and students to keep
track of progress. Drill and practice programs do not teach new skills;
they simply provide opportunity for the practice of learned skills.
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Conclusion
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During the past two decades enormous changes have occurred in the
scope and depth of the research in mathematics education. New
organizations, steady (although insufficient) federal funding, and the
evolution of the concept of cooperative research has appreciably
improved the extent to which we understand how students learn
mathematical and quantitative ideas. Large-scale and long-term projects
have emerged dealing with how children learn early number concepts,
geometry, and rational number concepts (fraction, ratio, decimal); how
children develop concepts of multiplication and division and learn
estimation strategies and processes; the influence of sex-related variables
on mathematics performance; the impact of calculators and computers;
and other aspects of thinking and concept development. There is still
much that is not fully understood, but real progress has been made.
Schools have not kept abreast of such progress. In fact, mathematics
curricula and methods have not changed very much at all in the last 20
years. Paper and pencil calculations and text-related worksheets still
dominate in the mathematics classroom.
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References
Charles, R., & Lester, F. (1982). Teaching problem solving: What, why and
how. Palo Alto, CA: Dale Seymour Publications.
Hill, S. (1979, March). The basics in mathematics: More than the third R.
Newsletter: National Council of Teachers of Mathematics, I5 (3), 4.
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Lindquist, M. L., Carpenter, T. P., Silver, E. A., & Mathews W. (1983). The
third national mathematics assessment: Results and implications for
elementary and middle school. Arithmetic Teacher, 31 (4), 14-19.
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Post, T. R., Behr, M., Lesh, R., & Wachsmuth, I. (1985) Selected results
from the Rational Number Project. Proceedings of the Ninth Psychology of
Mathematics Education Conference (pp. 342-351). Antwerp, The
Netherlands: International Group for the Psychology of Mathematics
Education.
Reys, R., Bestgen, B., Rybolt, J., & Wyatt, J. W. (1982) Processes used by
good computational estimators. Journal for Research in Mathematics
Education, 13 (3), 103-201.
Reys, R., Suydam, M., & Lindquist, M. (1984). Helping children learn
mathematics. Englewood Cliffs, NJ: Prentice Hall.
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Annotated Bibliography
Charles, R., & Lester, F. (1982). Teaching problem solving: What, why and
how. Palo Alto, CA: Dale Seymour Publications. This very readable booklet
addresses issues associated with mathematical problem solving,
specifica1ly, why problem solving is important and how to integrate
problem solving into the mathematics program. Ideas come directly from
experiences with teachers and children at several grade levels.
Reys, R., Suydam, M., & Lindquist, M. (1984). Helping children learn
mathematics. Englewood Cliffs, NJ: Prentice-Hall. A methods book for
teachers of elementary school mathematics. The first section provides
information on the changing mathematics curriculum and how children
learn mathematics. The second section discusses strategies and teaching
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