Texas Mathematics Teacher Fall/Winter 2019 - 7

Algebra: Searching for Meaning

Once we have reasoned through these relationships
together, most students are comfortable in proceeding
to pick the second equation, x = 3y. At no point in this
discussion does the teacher need to validate that second
choice, at least not until after the students have already
decided it is the accurate option. Moreover, this process
of understanding the sentence and comparing quantities
is straightforward and, through practice, reproducible
without teacher verification of choice. By considering
meaning, students are just as capable of exercising
mathematical authority as they consider their teachers
to be.
How do we avoid student confusion of procedure with
meaning?
A large portion of classroom time is spent teaching
procedures in order to enable students to efficiently solve
complex problems. While procedural fluency may be
necessary for efficiency, students often miss the purpose
for the procedure and can sometimes confuse a broad idea
with a specific process.
An example of this confusion arises each time I teach
undergraduate freshmen how to factor polynomials. A
college freshman has likely learned factoring in algebra
and used it in geometry, algebra II, pre-calculus, and
possibly even calculus courses. They know how to list the
factors of a natural number and nod in agreement when
I state that factoring means the expression of a quantity
with multiplication. But it is clear that several students
have not connected the ideas of meaning in general
factoring to the specific process of factoring quadratic
trinomials; after a discussion of meaning, several still
exclaim in recognition the first time I write two pairs of
parentheses on the board. They recognize the procedure
for factoring a specific type of polynomial but have often
missed or lost sight of the connection to what factoring a
quantity means when doing this particular procedure.
Teaching for meaning can remain the focus even while
teaching a procedure. As an example, let us focus on
writing linear equations and systems of equations, as
it is possible for students to state correct equations but
completely misunderstand what the variable(s) represent
and why the equations were right. If the focus in
teaching the writing of equations is on why an equation
makes sense, students can still write them correctly and
potentially gain higher levels of confidence as to their
correctness (as they also transfer mathematical authority
to themselves).

"

Consider the following problem, taken from Lial,
Hungerford, Holcomb, and Mullins (2019):
Maria Martinelli purchased two plots of land for
$120,000. On the first plot, she made a profit of
15%, and she lost 10% on the second. Her total
profit was $5500. How much did she pay for
each piece of land?.
If the majority of writing time is spent on identifying
the number of variables (we need to identify purchase
prices for two plots of land) and meticulously labeling
these variables (e.g., x = price in dollars for the first plot of
land, y = price in dollars for the second plot of land), then
information can be sorted by type (dollar amounts spent,
dollar amount of total profit, and ratios for determining
profit on individual land plots). This organization results
in limiting the number of possible equations to those
that make sense because units move to the forefront of
consideration. While we may know we want to end up
with linear equations, I have found that when the method
to obtain them focuses on determining correct quantitative
relationships rather than about matching a form from
previous examples, my students have more success
working independently and generalizing their knowledge
to broader contexts.
In what contexts do we refer to procedures as
"shortcuts"?
The idea of a "shortcut" is relative to the learner. For
example, if I wish to travel from Waco to Austin, driving
on Interstate 35 provides the quickest route and it is
reasonable to refer to I-35 as a "shortcut". But there are
many situations in which it makes no sense to refer to I-35
as a "shortcut" - for instance, when traveling from Waco
to Houston. In that case, driving on I-35 will significantly
lengthen the trip, regardless of whether we are measuring
distance or time.
It's important that we consider context when we promote
"shortcut" ideas to students, and that we teach them how
to consider the context for themselves. Harel et al. (2008)
share a classroom anecdote in which a secondary teacher
refers to cross-multiplication as a "shortcut" for the
following problem:

x 10
=
3 4

While procedural fluency may be necessary for efficiency,
students often miss the purpose for the procedure and can
sometimes confuse a broad idea with a specific process.

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Fall/Winter 2019 | 7


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Texas Mathematics Teacher Fall/Winter 2019

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http://www.brightcopy.net/allen/txmt/68-02
http://www.brightcopy.net/allen/txmt/68-01
http://www.brightcopy.net/allen/txmt/67-01
http://www.brightcopy.net/allen/txmt/66-02
http://www.brightcopy.net/allen/txmt/66-01
http://www.brightcopy.net/allen/txmt/65-02
http://www.brightcopy.net/allen/txmt/65-01
http://www.brightcopy.net/allen/txmt/64-02
https://www.nxtbook.com/allen/txmt/64-1
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