Table Of ContentDOCUMENT RESUME
SE 063 389
ED 441 664
AUTHOR
Kaput, James J.
Transforming Algebra from an Engine of Inequity to an Engine
TITLE
of Mathematical Power by "Algebrafying" the K-12 Curriculum.
National Center for Improving Student Learning and
INSTITUTION
Achievement in Mathematics and Science, Dartmouth, MA.
National Science Foundation, Arlington, VA.; Office of
SPONS AGENCY
Educational Research and Improvement (ED), Washington, DC.
PUB DATE
2000-00-00
NOTE
21p.
CONTRACT
R117610002; R305A6007; RED9353507; RED9619102
Opinion Papers (120)
PUB TYPE
MF01/PC01 Plus Postage.
EDRS PRICE
*Algebra; *Educational Change; Elementary Secondary
DESCRIPTORS
Education; Learning Strategies; *Mathematics Curriculum;
*Modern Mathematics; Teaching Methods
ABSTRACT
This paper asserts that the key to algebra reform is to
integrate algebraic reasoning across all grades and all topics, to
"algebrafy" school mathematics. The distinction is made between algebra "the
institution" and algebra "the web of knowledge and skill," which is also
clarified. Finally, suggestions are made as to how the educational community
might work towards a genuine algebra for all. The appendix provides concrete,
classroom-based illustrations of the different aspects of algebra at the
elementary grade level. (Contains 11 references.)
(CCM)
Reproductions supplied by EDRS are the best that can be made
from the original document.
Transforming Algebra from an Engine of Inequity to an Engine of
Mathematical Power By "Algebrafying" the K-12 Curriculum'
James J. Kaput, University of Massachusetts-Dartmouth
4-1
We begin with two assumptions. First, just as algebra has acted as a constricted gateway to
significant mathematics and all that follows from mastery of that mathematics, algebra reform is the
gateway to K-12 mathematics reform for the next century. Second, by acknowledging the
several different aspects of algebra and their roots in younger children's mathematical activity, a
deeply reformed algebra is not only possible, but very achievable within our current capacity for
change.
The key to algebra reform is integrating algebraic reasoning across all grades and all topics - to
"algebrafy" school mathematics. This integration solves three major problems:
1. It opens curricular space for 21st century mathematics desperately needed at the secondary
level, space locked up by the 19th century high school national curriculum now in place.
2. It adds a new level of coherence, depth, and power to school mathematics, both as
curriculum and as a habit of mind.
late,
3. It eliminates the most pernicious curricular element of today's school mathematics
abrupt, isolated and superficial high school algebra courses.
A strands approach to algebra that begins early also fits well with an inclusive, big-idea strands
oriented approach to the curriculum at large, contrasting with the layer cake-filter structure that
delays and ultimately denies access to powerful ideas for all but the few. An algebrafied K-12
curriculum helps democratize access to powerful ideas.
Our discussions of algebra must be as honest and clear as possible. To this end, it helps to
distinguish Algebra the Institution from Algebra the Web of Knowledge and Skill that we want
students to develop in school, so that criticisms of the former are not heard as statements about the
latter. Algebra the Institution is a peculiarly American enterprise embodying the standard courses,
textbooks, tests, remediation industry, and their associated economic arrangements, as well as the
supporting intellectual and social infrastructure of course and workplace prerequisites, cultural
expectations relating success in algebra to intellectual ability and academic promise, special
interests, relations between levels of schooling, and so on. Exhortation for and legislation of
Algebra For All tacitly assume the viability and legitimacy of this Institution. But
this algebra is the disease for which it purports to be the cure!
It alienates even nominally successful students from genuine mathematical experience, prevents
real reform, and acts as an engine of inequity for egregiously many students, especially those who
are the less advantaged of our society.
Our challenge is to create an implementable alternative to this inimical Institution, to transform an
engine of inequity to an engine of mathematical power. This paper will first contextualize our
situation historically, second, clarify what we mean by Algebra the Web of Knowledge and Skill -
and third suggest how we might work towards a genuine algebra for all. An
what is algebra?
Appendix provides concrete, classroom based illustrations of the different aspects of algebra at the
elementary grade level.
The work described in this paper was supported by grants from the Department of Education (R117G10002 &
R305A6007) and the National Science Foundation (RED 9353507 & RED 9619102). The views expressed,
however, are those of the author.
U.S. DEPARTMENT OF EDUCATION
Office of Educational Research and Improvement
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Much More Math for Many More People
We sometimes lose sight of the magnitude of the changes that are taking place across the
generations. Centuries-long trends, both in the evolution of the subject matter and in the
demographics of those who need to learn that subject matter are not only continuing, they are
accelerating. Shop-keeper arithmetic was sufficient for more than 90% of the population until this
century. Nowadays, we want 100% of the population to graduate from secondary school of one
kind or another, and hear calls for all students to obtain at least a two year college education
(although we have difficulty raising actual secondary graduation rates above 75%). Contrast these
expectations with the fact that the percentage of the 17-18 year old population cohort
taking AP calculus today (about 3.5%) equals the percentage of the population
graduating from high school a century ago! So, to contextualize the contemporary calls
for "Algebra for All" we ask what percent of the population was expected to learn algebra a century
ago? Perhaps 2% - mainly boys who were socially, economically, culturally and ethnically very
similar to their teachers. And we further ask if either the curricula, the texts, or the pedagogies
have changed in a way that might deliver algebra to a mightily more diverse 98% who were
previously absolved of algebra learning? (We are assuming all students are still in school in 8th or
9th grade.) The answer to these questions is plainly NO! And this is one reason for the current
attention to algebra reform.
Another reason for deep algebra reform has to do with the future. We have every reason to believe
that the mathematics of the next century will be more different from today's than today's is
different from that of the 18th century. The primary reason for this is the emergence of the
computer medium, a medium in which new mathematical forms flourish. For example, Dynamic
Geometry systems such as the Geometer's Sketchpad and Cabri Geometry are simply not possible
in the static inert medium of pencil and paper, but, when coupled with appropriate educational
activities, provide entirely new mathematical experiences for the student (as do similar systems for
sophisticated users and builders of mathematics). Of more direct relevance is the iterative
mathematics supported in the computer medium that takes the form of dynamical systems, for
example, and their use in modeling nonlinear phenomena. This mathematics and the nonlinear
science that grows with it is exploding in importance as we end the 20th century. Indeed, the
nature of science is undergoing a profound transformation, whereby the complex phenomena that
were in principle ignored by classical methods are now the focus of intense study across many
physical sciences, life sciences and social sciences, as well as applications in
different sciences
engineering, economics, and elsewhere. The rapid iterative computation that computers make
possible is coupled to highly visual displays of complex data that results, as in the now-familiar
fractal and Mandelbrot graphics, for example. Similar things could be said about changes in
statistics and probability, as bootstrapping and other resampling methods are rapidly replacing
complex and cumbersome traditional formulas, and new interactive visual displays of data are
becoming mainstream tools.
A direct implication of these changes is the need for more room at the secondary level for the
mathematics for the next century that is already blooming profusely, mainly outside of schools.
An accompanying implication is the need to do much more mathematics in K-8. Jointly, these
implications suggest that we call no longer afford the inefficient, curriculum and resource gobbling
high school (Institutional) algebra courses that dominate curricula and expectations today, and that
we must instead integrate a larger, more modem and powerful algebra throughout K-8
mathematics. I will now try to be a bit clearer on what kind of algebra we mean.
Five Forms of Algebraic Reasoning
In my view, algebraic reasoning is a complex composite of five interrelated forms of reasoning.
The first two of these underlie all the others, the next two constitute topic strands in the curriculum,
and the last reflects algebra as a web of languages - its linguistic side. All five richly interact
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conceptually as well as in activity - to understand this algebra is to make connections, abstractions
and generalizations. All five can and should be started early.
1. (Kernel) Algebra as Generalizing and Formalizing Patterns & Constraints, especially,
but not exclusively, Algebra as Generalized Arithmetic Reasoning and Algebra as
Generalized Quantitative Reasoning
2. (Kernel) Algebra as Syntactically-Guided Manipulation of Formalisms
3. (Topic-strand) Algebra as the Study of Structures and Systems Abstracted from
Computations and Relations
4. (Topic-strand) Algebra as the Study of Functions, Relations, and Joint Variation
5. (Language aspect) Algebra as a Cluster of (a) Modeling and (b) Phenomena-Controlling
Languages
Figure 1 is intended to provide an image of how the forms of reasoning overlap and interrelate.
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Figure 1: Five Aspects of Algebra
Forms (1) and (2) underlie all the others, with (1) based both within and outside of mathematics,
and (2) done in conjunction with (1). It is difficult to point to mathematical activity that does not
Algebrafying School Mathematics
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involve generalizing and formalizing in a central way. It is one of the features of thinking that
makes it mathematical. Also, the actions one performs with formalisms identified as (2), the kinds
of manipulations that dominate current algebra courses, should typically occur as the result of prior
formalizing of situations and phenomena, so that they can be related to those situations and
phenomena. And the formalisms may be of many different types, not merely variables over sets of
familiar numbers (or transcendentals over some field). Furthermore, it is possible that the
manipulation can yield general patterns and structures at another level of generalizing and
formalizing - which is the essence of the third, structural, form of algebraic reasoning. In order to
them, which leads to
use or communicate generalizations, one needs languages in which to express
(5).which in turn permeates all the others. While (3) is a school mathematics topic strand
occurring nowadays mainly at the advanced levels, it is also an important growing domain of
mathematics in its own right - abstract algebra. On the other hand, topic strand (4) functions, is
general
more a school mathematics domain, and lives in the world of mathematics more as a
breadth of this school
purpose conceptual tool rather than a branch of mathematics. One hint of the
algebra is the fact that (3) and (4) lie on opposite sides of a deep boundary in mathematics
separating algebra and analysis. Both appear in school algebra.
Traditional school algebra focuses on (2) at the expense of all the others. And while calls for a
functions approach to algebra went ignored for almost a century, some of our contemporaries tend
to see (4) as all of school algebra. But the list suggests that algebra is more than functions,
although the idea of function is an extremely powerful organizer of mathematical activity across
topics and grade levels. But so are all the other forms of algebra listed, which is exactly why
algebra can play the key role across K-12 mathematics that I and others suggest. This wider view
of algebra emphasizes its deep, but varied, connections with all of mathematics.
Connections to the Framework's View of Algebra
The analysis of algebra offered here is fully compatible with that offered in the Algebra Framework
appearing elsewhere in this document. My call for Integration is another way of emphasizing the
role of Contextual Settings offered in the Framework. And there is a close connection between the
five forms described above and the four Organizing Themes described in the Framework. The
most obvious connections are between my two Topic Strands, Structures and Systems (3), and
Functions, Relations and Joint Variation (4), and, respectively, the Themes of Structure and of
Functions and Relations. The two Themes of Modeling, and Language and Representation, are
embodied in my "Web of Modeling and Phenomena Controlling Languages" (5). I chose to
identify two essential aspects, or forms of algebraic reasoning as "kernels" that underlie the rest
Generalizing and Formalizing, and the Manipulation of Formalisms. These are embodied in each
of the Framework Themes as well. Given the complexity and richness of algebra both as a tool of
thought and as a object of study we should expect differences in descriptions, and perhaps be
surprised at the similarity of the two offered here. But, of course, real differences and diversity
in the
can be expected to appear in their realization in curriculum and their implementation
classroom. We can expect and should welcome wide variation in how algebra can be integrated in
the K-12 curriculum for all students.
Algebra Before Acne anti the Role of Teachers
The language aspect of algebra supports both early and integrated algebra. Early because people
require repeated use of a language over an extended amount of time to become fluent in its use.
Indeed, "algebra before acne" is more than a flippant phrase - language learned before puberty is
learned without an accent and is deeply integrated with one's patterns of thinking. And algebra
learning must be integrated with the learning of other mathematics because, to learn a language
people need to use it to express something significant to them, such as the quantitative relationships
arising inside mathematics (for example that occur in arithmetic and geometry) and outside
mathematics when we use it to model our world.
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As the examples in the Appendix indicate, appropriate instructional materials can "seed" each
aspect of algebra listed in relatively ordinary elementary mathematical activity. One key is that
teachers need to be able to identify and nurture these roots of algebraic reasoning in forms that
appear very different from what is deemed "algebra" under the auspices of the Algebra the
Institution. For example, generalization is initially expressed using ordinary language, intonation,
and gesture rather than through the use of formal symbolism. These identification and nurturing
skills require teacher-development focused on student thinking rather than skills with traditional
formalisms. The other key is that these beginnings go somewhere important mathematically,
both in terms of growth in notational competence and in terms of the significance of the big ideas
that these notations are used to express, which in turn will require carefully designed classroom
materials to help guide the way.
The new approaches will begin in familiar circumstances, but will lead to new tools, unprecedented
introduced by
applications, populations of students traditionally not targeted to learn algebra
teachers traditionally not educated to teach algebra - neither the old algebra nor some new version.
This route involves generalizing and expressing that generality using increasingly formal
languages, where the generalizing begins in arithmetic, in modeling situations, in geometry, and in
virtually all the mathematics that can or should appear in the elementary grades.
The next generation of mathematics education reform begins with current reforms
and the elementary school teachers and classrooms of today, but its ultimate
success depends on success in this algebrafication of school mathematics.
Mathematical power as defined in the Standards lies in this direction.
Appendix: Classroom-Based Examples of Early Algebraic Reasoning2
Introduction
The following examples are based in actual student work and language, with a few examples of
student MIS-understanding based in the traditional curriculum. They focus on the different forms
of algebraic reasoning outlined in the paper above, taking illustrations from across many grade
levels and mathematical topic areas. Most of the illustrations are adapted from a book being
prepared by the author, Employing Children's Natural Powers to Build Algebraic Reasoning in
the Content of Elementary Mathematics. We emphasize where we need to go rather than where
symbol
we are, or have been. One key aspect of the examples below that contrasts with traditional
manipulation algebra is the frequent opportunity to reflect on or articulate their knowledge to others
strings of
as opposed to concentrating on remembering procedures that they can only know as
the intensive study of the last three letters of the alphabet. A second key aspect is the
symbols
that
way teachers build on students' naturally occurring linguistic and cognitive powers in ways
put a premium on active sense-making and understanding.
1. Algebra as Generalizing and Formalizing Patterns & Constraints
Generalization and formalization are an intrinsic feature of much mathematical activity, and the
mathematical systems and situational contexts in which generalization and formalization can be
2 would like to thank Virginia Bastable, Deborah Schifter, Dolores Strom, Rich Lehrer, Guershon Harel, Margie
Pligge, Mary Spence, Cornelia Tierney, and Steve Monk for sharing their vivid examples.
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done are everywhere. Indeed, it is difficult to point to mathematical activity that does not involve
generalizing and formalizing in a central way. Perhaps pure computational arithmetic of the sort
that dominates elementary school mathematics, the kinds of counting and sorting involved in
combinatorics, and pure spatial visualization are candidates of mathematical activity that do not
emphasize generalizing and formalizing. Also, the actions one performs with formalisms,
identified as the second kernel aspect of algebra, are typically not generalizing and formalizing per
also possible that the
se, but typically occur as the direct or indirect result of prior formalizing. It is
manipulation can yield general patterns and structures at another level of generalizing and
formalizing - which is the essence of the third, structural, aspect of algebra in our list.
Generalizing involves deliberately extending the range of one's reasoning or communication
beyond the case or cases considered, explicitly identifying and exposing commonality across
relations
cases, or lifting the reasoning or communication to a level where patterns across and
themselves.
among cases or situations become the focus, rather than the cases or situations
Appropriately expressed, the patterns, procedures, relations, structures, etc., can become the
objects of reasoning or communication. But in order to use or communicate generalizations, one
needs languages in which to express them, which, for a young child who does not yet possess a
formal language, may mean using spoken natural language. Here, intonation and gesture may be
used to communicate the intention that a statement about a particular case be read or heard as
representing a general class of statements. In this case identifying the intended generality may
require a skilled and attentive ear, the ear of teachers who have had experience in listening carefully
to children.
We distinguish two sources of generalization and formalization: (a) reasoning and communicating
in mathematics proper, usually beginning in arithmetic, and (b) reasoning and communicating in
situations that are based outside mathematics but are subject to mathematization, usually beginning
in quantitative reasoning. In a sense, this is a bogus distinction if one believes, as I do, that all
mathematics arises from experience and becomes mathematical upon appropriate activity and
processing. However, if the starting point for the generalizing and formalizing is in previously
mathematized experience, then I would argue that it falls in "mathematics proper," whereas, if it
starts in a situation experienced as yet-to-be mathematized, then I would say that its source is
outside "mathematics proper," and is based in phenomena or situations.
The distinction is especially problematic in the early years, where mathematical activity takes very
concrete forms and is often tightly linked to situations that give rise to the mathematical activity.
Nonetheless, even here a distinction seems worthwhile. A student who is generalizing patterns in
with objects and
sequences of numbers in a hundreds table or multiplication table is working
relations already conceived as mathematical. On the other hand, consider a student who is
comparing differences in prices between cashews (expensive) and peanuts (cheap) for two
different brands, A and B. If the A-brand cashew-peanut difference is bigger than the B-brand
difference, and she claims that a small increase in the price of B-brand peanuts will not change the
outcome of the comparison, then I would regard her as generalizing from her conception of the
situation rather than from within mathematics proper. Later, she might model the same situation
using algebraic differences and inequalities, writing something analogous to
A-a > B-b implies A-a > B-b+x if x>0 (and perhaps x<B-b).
In this case, she would be reasoning within mathematics if she took the necessity of the implication
worked with the
as following from properties of the number system she was working in. If she
inequality apart from modeling any situation and tried to prove the implication using number
system properties, then she would clearly be working strictly within mathematics. On the other
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Algebrafying School Mathematics
7
hand, we would expect that her conceptions of the differences and inequalities were rooted in
conceptual activities based in situations experienced as meaningful.
Examples of Early Generalizing and Formalizing:
The following situations were observed and documented by Virginia Bastable and Deborah
Schifter (Bastable & Schifter, in preparation). The first involves a third grade class in which the
teacher asks how many pencils are there in three cases, each of which contains twelve pencils.
After the class arrived at a repeated addition 12+12+12 solution, the teacher showed how the result
could be seen as a 3X12 multiplication. She expected to move on to a series of problems of this
type, but a student noted that each 12 could be decomposed into two 6's, so the answer could be
described as 6+6+6+6+6+6+6+6, or six sixes, which could be written as 6X6. Another student
observed that each 6 could be thought of as two threes, which led to 12X3 as another way of
expressing 36. "And at this, Anna exclaimed, 'Wow, we have found a lot of things that equal
thirty-six. Oh look! This one is the backwards of our first one, 3 x 12.'" Anna's observation led
to an extended investigation, described by Bastable & Schifter as follows.
The children then continued to find more ways to break apart and group the numbers
to total 36. Looking at the column of twelve 3s, Steve offered that if you circle three
3s, you end up with four groups, giving you 4x9=36. At this, Joe declared, "And so
we can add another one to the list because if 4 x 9 = 36 then 9 x 4 = 36 too."
Anna objected to this last claim, asking, "Does that always work? I mean, saying
each one backwards will you always get the same answer?" [Virginia] Brown [the
teacher] responded, "That's an interesting question. What do you think?" Anna
replied, "I'm not sure. It seems to, but I can't tell if it would always work. I mean
for all numbers."
For homework, Brown asked the class to think about ways to prove or disprove
Anna's question. The next day various children explained their thinking by noting
number pairs such as 3 x 4 and 4 x 3. While some children used manipulatives to
illustrate their examples, Anna was not totally convinced. "But I'm still not sure it
would work for all numbers." The teacher decided to table the question but to
continue to explore multiplication by introducing arrays.
Two weeks later, Brown reminded the children of Anna's question: "Can anyone
think of a way to use arrays to prove that the answer to a multiplication equation
would be the same no matter which way it was stated?"
The class thought about this for a while--some alone, others with partners --until
Lauren timidly raised her hand. "I think I can prove it." Lauren held up 3 sticks of 7
Unifix cubes. "See, in this array I have three 7s. Now watch. I take this array,"
picking up the three 7-sticks, "and put it on top of this array." She turned them ninety
therm nn cpvpn q-sticks shP had previnncly arranged "And lnnk,
they fit exactly. So 3x7 equals 7x3 and there's 21 in both. No matter which equation
you do it for, it will always fit exactly."
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Algebrafying School Mathematics
At the end of Lauren's explanation, Jeremy, who had been listening intently, could
hardly contain himself. He said that Lauren's demonstration had given him an idea
"I'll use the same equation as Lauren, but I'll
for an even clearer way to prove it.
only need one of the sets of sticks. I'll use this one." He picked up the three sticks
which had seven in each stick. "When you look at it this way," holding the sticks up
vertically, "you have three 7s." Then he turned the sticks sideways. "But this way
So this one array shows both 7x3 or 3x7."
you have seven 3s. See? .
.
.
Anna nodded her head. Although Lauren and Jeremy had demonstrated with a 3x7
array, the representation convinced her of the general claim. "That's a really good
way to show it, and so was Lauren's. It would have to work for all numbers."
This example illustrates students actively generalizing before they have a formal language in which
to express their generalizations. They are using a variety of notational devices in combination with
natural language in a social context. Importantly, the questions of certainty and justification arise
notations. The
as an integral aspect of generalizing, and are interwoven in the use of the different
basic issue voiced by Anna was the range of the generalization - does it hold for all numbers? The
students are using the cubes and sticks to generate their ideas, to show one another their thinking,
and to justify their claims. The mathematical claims are clearly theirs rather than the teacher's. It
would trivialize this account and its contents to think of it merely as the children developing the
concept of commutativity of multiplication (of natural numbers), because the very idea of
multiplication is being built (although only two aspects, repeated addition and array models), as is
the idea of mathematical justification and proof. While the episode began in a concrete situation, it
pencils and cases were simply a stepping off point
quickly became a mathematical exploration
that (inadvertently) led the students to grouping and decomposition of whole numbers. Eventually,
it led back to concrete arrays being used to exhibit equivalence of alternative groupings. The
invariance of the "amount," first under alternate groupings of 21 and then under alternate
orientations of the same grouping (offered by Jeremy), is made concrete in the physical arrays.
"It would have to work for
And in the end, the generality is not only realized, but is made explicit
all numbers." It is easy to imagine that this property might be given a more formal expression
later, first perhaps as "box times circle = circle times box" where numerals could be written in the
spaces, and then even later as aXb = bXa.
Another aspect of this situation deserves attention: the fact that the generalizing took place in what
most teachers would regard as the normal course of mathematical concept development in an
"ordinary" mathematics classroom ("ordinary" in the sense of fitting the NCTM Professional
Teaching Standards - this is clearly an excellent teacher doing a good job). It was certainly not
traditional symbol manipulation algebra - it was a series of arithmetic lessons where generalizing
rather than computing was at the center. But why is generality so important? Because that's what
makes these concepts mathematical!
The Bastahle /Schifter paper (in preparation) from which the above case was taken includes several
examples of such episodes across grades 1-6 involving properties of numbers (odd-even, zero)
and operations, extensions to other number systems beyond the natural numbers, and so on.
Many important questions remain unanswered about these activities and how to organize them,
including the roles of language and special notations, how to discern generality in students'
informal utterances, what the interplay between generalizing and justification might be, what the
role of concrete situations may be, and so on.
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2. Algebra as Syntactically-Guided Manipulation of (Opaque) Formalisms
The tremendous power of algebraic symbolism and its syntax that we use to guide our
manipulation of it is behind the prodigious development of modern science and technology. When
dealing with formalisms, whether they be traditional algebraic ones or more exotic ones, the
attention is on the symbols and syntactical rules for manipulating them rather than what they may
stand for. However, it is also possible to act on formalisms "semantically," where your actions are
guided by what you believe the symbols to stand for. Indeed, most fruitful use of symbols
involves alternating between actions on symbols without reference to what they might stand for
and then interpreting the results semantically - that is, in terms of what the symbols stand for. This
dual status of "looking at" vs. "looking through" symbols is reflected in Figure 2.
Figure 2: Looking AT vs. Looking THROUGH Symbols
As is widely appreciated, much of the traditional power of formalisms arises from the internally
consistent, referent-free operations that they afford. These free the user to operate on relationships
far more complex than could be managed if the user needed to attend to what the symbols and
transformations stood for. One suspends attention to meaning and focuses on the symbols
themselves. To paraphrase Bertrand Russell. "Aleebra allows you to think less and less about
more and more."
The problem has been that our traditional algebra curriculum has concentrated on the "less and
less" part, resulting in alienation from meaning and even from mathematics itself for many
students, who practice endless rules for symbol manipulation and come to believe that this is what
mathematics is. The power of using the form of a mathematical statement as a basis for reasoning
is lost as students lose the connection with the quantitative relationships that the symbols might
stand for. Research has provided many examples of the difficulties that students have been led
for example, believing that (a
into, quite often due to over-generalizing patterns such as linearity
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