Texas Mathematics Teacher Fall/Winter 2019 - 8

Algebra: Searching for Meaning
This makes sense if we consider a "shortcut" here to
mean that we eliminate fractions faster than in alternate
methods (although that elimination is debatable, as
the answer itself will be a fraction). However, if we
want a solution that isolates x the fastest, it seems that
multiplying both sides by 3 would be the appropriate
"shortcut." If we want to understand the answer quickly,
it might be fastest to convert the right-hand side to 2.5 and
consider what quantity needs to be divided by 3 in order
to obtain 2.5 (which can be manipulated like a quarter of
a dollar, if students are comfortable with decimal ideas).
It may be that by "shortcut," we simply mean the least
number of steps. In that case, for a problem this simple,
each of the methods is approximately the same length.
While cross-multiplication can act as a solving "shortcut"
in many contexts, it does matter in what way we consider
a "shortcut" to be beneficial, and it does matter with
which problem we attempt to demonstrate its usefulness.
Whether a "shortcut" is actually short and efficient is
contextual.
Another algebra example of context occurs when solving
quadratic equations standard form.. Which method is
generally a "shortcut" for solving quadratic equations:
factoring to solve or using the quadratic formula? One
hint I often give students is that factoring will be faster if
it applies, but that if the coefficients are large, it may be
faster to go straight to the formula. But those hints are not
broad enough to cover all contexts! Consider:

The first equation factors into (13x+432)(72x-109)=0,
negating my first hint. Just because it will factor, doesn't
mean that getting there will be efficient. Similarly, the
second equation has coefficients that probably seem large
in an algebra setting, yet any algebra student should be
able to rewrite it as 100(3x+5)(x-4)=0.
Closing
As we consider questions of meaning, we are enabled to
make mathematically sound decisions in our work. In
learning to consider meaning, procedures can be extended
into broader contexts and viewed as tools in more
complex problems. Considering meaning within context
can move us from rigid thinking about a problem and its
solution method to more creative or efficient solutions.
As we hope to see these abilities grow and flourish within
our students, we must foster discussion of these ideas by
considering our own methods, flexibility, and language.
When we repeatedly come back to the question of
meaning in our teaching, our students are encouraged to
constantly return to it as well.

936x2 + 29687x - 47088 = 0
300x2 - 700x - 2000 = 0

Tori Hudgins, Graduate Student * Tori_Hudgins@baylor.edu
Mathematics Department * Baylor University

Reference
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Campbell, L. (2009, April). An Algebraic "Whack on the Side of the Head." Presented at the 87th Annual NCTM Meeting
and Exposition, Washington, D.C.
Clement, J. (1982). Algebra Word Problem Solutions: Thought Processes Underlying a Common Misconception. Journal for
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Harel, G., Fuller, E., & Rabin, J. (2008). Attention to meaning by algebra teachers. Journal of Mathematical Behavior, 27,
116-127.
Lial, M. L., Hungerford, T. W., Holcomb, J. P., & Mullins, B. (2019). Mathematics with applications in the management,
natural, and social sciences.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all.
Reston, VA: Author.
Zakaria, E., Chin, L. C., & Daud, Y. (2010). The effects of cooperative learning on students' mathematics achievement and
attitude towards mathematics. Journal of Social Sciences, 6(2), 272-275.

| Fall/Winter 2019

Texas Mathematics Teacher

Texas Mathematics Teacher Fall/Winter 2019

Table of Contents for the Digital Edition of Texas Mathematics Teacher Fall/Winter 2019