Algebra in the Early Grades
A Skeptic’s Guide
A Skeptic’s Guide to Algebra in the Early Grades
The ideas presented here are a summary of ideas presented in an NCTM publication entitled Algebra
in the Early Grades Edited by James J. Kaput, David W. Carraher and Maria L. Blanton. Copyright
2008 by Taylor & Francis Group, LLC.
Are Young Children Ready For Algebra?
One of the arguments opponents of algebra in the early grades is that of developmental
readiness. This is the notion that certain ideas and concepts can be learned only when required
mental structures or prerequisite concepts are in place. There is some strength to this argument if
you observe the language development of children. A one year old toddler will not learn to read or
write no matter the efforts pressed toward that end. “There is simply too much knowledge about
spoken language skills that children need to acquire before one can expect them to master reading
and writing.” (J. Kaput, D. Carraher and M. Blanton, 2008)
James J. Kaput set aside the question of whether the ideas are inherently too abstract for
young minds or whether they simply rest on a knowledge base too extensive to be acquired in a short
time span. Instead, he asked whether there are any algebraic concepts, ideas, and methods within the
reach of young learners. This brings to mind two questions, one is addressed in the article by James
Kaput. First, is there enough to justify the systematic inclusion of algebra in early mathematics
curricula. The basis of the whole book is that there exists enough algebraic concepts to be included.
The second question is whether these algebraic concepts are of enough value to be included
in the early mathematics curricula. To address this question, we must evaluate the purpose of such
an inclusion. What is the value of studying algebraic concepts and methods in the early years? This
is not addressed in this article or the first chapter. Instead, the argument of what is algebra and
algebraic thinking is addressed.
An important concept to include in an evaluation of whether young children are ready for
algebra is that of representational systems in mathematics. Representational systems include place
value notation, graphs, tables, and algebraic script. These systems were developed over centuries
through a process of evolution. Students are able to discover the concepts of mathematics, but the
conventions of representational systems cannot be discovered. For example, it is possible to
discover the concept of “greater than” through an experience of comparing quantities of similar
objects. They can determine a greater quantity over another quantity through their experience. The
greater than “>” symbol is a convention that communicates the idea of one quantity being greater
than a quantity of another. Although there is a sense or reason to the symbol, there is little to no
chance that a person will discover the symbol from all the possible marks that could be made with a
pencil.
Shouldn’t Kids Learn Basic Skills First?
There are those who argue that even if young children are not developmentally hampered
from learning algebra, this instruction should wait until arithmetic is well mastered. They believe
that algebra must follow arithmetic. This is the basis of traditional mathematics instruction which
down plays the generalizations associated with the computational aspects of elementary mathematics
instruction. An important question to ask in this argument is, “Isn’t arithmetic based upon the rules
of algebra?” James J. Kaput stresses the importance of higher order thinking skills such as
generalization. He further stressed that it is not necessary to diminish the importance of routine
skills such as computational proficiency for higher order thinking skills.
Can Teachers Add Algebra To An Already Bulging Early Mathematics Curriculum?
This question implies the following:

The curriculum is presently full; there is no room for anything else.

To add another topic, some existing topics must either be dropped or covered in less depth

It is impossible to eliminate existing topics; it is inadvisable to cover the present topics in less
depth
Therefore, algebra cannot be added to the early grades mathematics curriculum. This ignores the
idea that algebraic concepts in the early grades does not necessarily make the elementary curriculum
bigger. Algebraic concepts can help address the current content more deeply. It is argued in this
article that the list of key topics in an “algebrafied” curriculum would actually be shorter than that of
existing curriculum because of the more connected mathematical experience for children.
To Be Truly Algebraic, Shouldn’t The Activity Involve The Syntactic Rules Of Algebra?
According to the editors of the book, early algebra researchers tend to take a broad view of
symbolic reasoning. We are presented with the case of a student who states, “For any temperature
you give in degrees Celsius, I can find the temperature in degrees Fahrenheit by multiplying by 5/9
and adding 32.” This student has expressed a function that can be expressed as f (n)  n(5 / 9)  32 .
Both expressions are generalizations of a relationship.
Of course the algebraic script has some distinct advantages over the spoken language. It’s
succinct and well suited to further analysis and derivations. The advantage of the spoken expression
is that it is able to convey information about the thermal context that is missing in the algebraic
expression. This spoken language serves as an important starting point. The strength of using a
known language is the ability to describe algebraic concepts. This is important to the underlying
goal of early algebra which is to help them see and express generalities in their experiences with
mathematics. Initially, the language of generalizations is often broadly defined and evolves toward
more specialized forms such as algebraic script and Cartesian graphs.
Is Algebra The Same As Generalization? If So, Isn’t It Everywhere?
It seems unreasonable to restrict the expression algebraic thinking to occasions on which
students use algebraic notation. Such a narrow conception of algebraic activity obscures the
relations between algebra and early mathematical thinking, leaving mathematics educators with little
insight into how the learning of algebraic notation can build on existing skills. Rather than trying to
parse precise boundaries of algebraic generalization, the goal of elementary grades mathematics
should be to harness the innate powers in the particular context of number, quantity, visualization,
and so forth, so that children engage in generalizing as a mathematical activity. Sometimes, this
challenges us to reexamine deeply held beliefs about children’s mathematical powers.
Questions for this Section:
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What do you consider algebra and algebraic thinking?
Can you determine any link between your definition of algebra and arithmetic?
What is your comfort level with algebra?
Could your comfort level be a block or bridge to algebraic thinking in your classroom?
What is the difference between algebra concepts and algebra conventions?
What is the role of generalizations in elementary math