Texas Mathematics Teacher Fall/Winter 2019 - 6

Algebra: Searching for Meaning
Tori Hudgins, Graduate Student

ne of my most formative experiences as a preservice teacher was confrontation with an example
of meaning in algebra. In a conference presentation,
Campbell (2009) was multiplying pairs of whole numbers
that had a difference of two. For example, one times
three, two times four, three times five, and so on. He
noticed when he started making a list of their products
that the answers were remarkable (3, 8, 15, etc.) because
they were each exactly one less than a perfect square. In
fact, the product seemed to be always one less than the
perfect square of the whole number halfway between the
multiplicand and multiplier. He set out to investigate
whether this was always true; he labeled the whole
number halfway between the multiplicand and multiplier
as x, wrote the product as (x-1)(x+1), and quickly realized
that this generalization was indeed true, as multiplication
of these binomials yields x² - 1. (See Figure 1)

x-1

x+1

(x-1)(x+1)

x2-1

(1)(3)=3

22-1=3

(2)(4)=8

32-1=8

(3)(5)=15

42-1=15

(4)(6)=24

52-1=24

Figure 1. Examples of multiplying pairs of whole
numbers with a difference of two compared to
squaring the whole number halfway between them
and subtracting one.
The astonishing part of this example, to the speaker and
to myself, was not that the pattern did hold for all whole
numbers, and not even that this same algebraic proof
worked generally for all pairs of real numbers with a
difference of two and the real number halfway between
them. Rather, we were stunned by the realization that the
proof was something so algebraically simple that most
beginning algebra students could recognize the truth of
it. But would they? Even for students that are skilled at
multiplying two binomials, the profound numerical truths
imparted by algebraic expressions often go unnoticed.
While successful teaching requires understanding
students' unique struggles with and backgrounds in
learning mathematics, it is also vital to consider how our
teaching habits may help or hinder students' perception
of meaning. The Mathematics Teaching Practices in
NCTM's Principles to Actions (2014, p. 10) encourage us,
for example, to "implement tasks that promote reasoning
and problem solving" and to "build procedural fluency
from conceptual understanding" (emphasis added). If
we aim to focus on these positive actions, we must also
recognize our established behaviors that deter or distract
from a sense-making focus. How do we unintentionally
de-emphasize meaning to our students, and how can
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| Fall/Winter 2019

we begin to move our language habits towards sensemaking? The following questions, a selection of those
raised by a qualitative study of algebra teachers (Harel,
Fuller, & Rabin, 2008), can help us evaluate how we
teach for meaning in algebra and beyond.
Who has the authority to determine correctness of an
answer or statement?
Once, after teaching a lesson on curve sketching using
calculus techniques, a student asked me if we would
have been allowed to use the y-intercept in graphing
the relevant function. This certainly could have been a
tactful way to point out that I had indeed not mentioned
it in this particular example, but students ask genuine
permission questions of this sort frequently. Shouldn't
a calculus student be capable of asking oneself, "Is this
function defined on a domain including zero?" and if
so, conclude that the function will have a y-intercept
and include it? They need not appeal to my authority
to validate their usage of algebra. However, students
can obtain an external-validation mindset from teachers'
unintentional language and response patterns. When
we teach with phrases such as, "it just so happens," "it
turns out," "because that's what it is," "because it fits
the definition," or any classroom-version of "because I
said so," we can unintentionally imply to students that
mathematics is arbitrary. Thus, the only way to know if
an answer is correct is to confirm it with an expert. While
collaborative community work is a fantastic strategy for
success (Zakaria, Chin, & Daud, 2010), these phrases
and approaches to problem-solving can cover up
meaning that might otherwise help students understand
and verify their own knowledge without needing to
check in with their teachers. This also supports their
development of a positive mathematical identity and
seeing themselves as 'doers' of mathematics (Aguirre,
Mayfield-Ingram, & Martin, 2013).
An example of success that I have had in moving
students towards reliance on their own sense-making
abilities is in teaching writing linear equations with
ratios. I have used the following verbal description
to support this move: Let x be the number of female
students and y be the number of male students. There
are three times as many female students as male
students. Having taught linear equations at both the
secondary and post-secondary levels, I have found that
students often understand the truth of the sentence
- that one group is three times as large as the other.
Often when students write the statement on their own,
many will take the order of the nouns in the sentence
and write 3x = y. This is a common misunderstanding
(Clement, 1982). But someone will usually point out
without prompting that x = 3y is also an option, and then
we discuss the question: Which group is supposed to
be larger/smaller? Students usually already understand
and believe that the female group is larger than the
male group. Then we discuss: Which group size must
be multiplied by three, in order for the group sizes to be
equal? Of course, the smaller one does, i.e., the males.
Texas Mathematics Teacher

Texas Mathematics Teacher Fall/Winter 2019

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