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Q
uantitative
R
easoning &
P
roblem
S
olving
18
© 2014 Pacific Crest
The Challenge
Not setting expectations for generalizing mathematical
knowledge
The Solution
Step 11
of the LPM (transfer/application) is the process of learning to use contexts that both differ from
the original problem/learning context and also span the types of applications of the learning in order
to build the ability to apply the learning in a general sense. A learner accomplished this by first listing
and addressing all variables in the knowledge structure and possible conditions within the contexts of
the experience. Then the learner constructs the hardest problem that includes all the variables and as
much of the difficult condition as the learner is able to. This will sensitize the learner with respect to what
he or she must focus on in the future, with respect to that learning. This enables the learner to use the
knowledge without being prompted by someone else, but by the context itself.
The Challenge
Lack of self-confidence and self-belief in learning
Mathematics
The Solution
Step 13
of the LPM (self-assessment) is critical when it comes to building self-confidence and self-belief
for learning mathematics. Each time you identify a new strength you have in learning mathematics, your
confidence increases. The increase in confidence is even greater when you identify an area for improve-
ment and do the work to turn that area into a strength. Each new insight that you gain about learning
mathematics makes mathematics that much more enjoyable, rewarding, and a familiar part of your life
.
O
ops
! A
voiding
C
ommon
E
rrors
●
Not valuing knowledge for its utility in solving problems
Example
: In chemistry, a student can memorize
the combined gas law (most commonly
represented as
pV = kT
) in all three
typical formats, shown at right, or
simply learn one format and use the
ability to apply the technique of isolating a variable to change the format as needed.
volume
pressure
temperature
kT
kT
pV
V
p
T
p
V
k
=
=
=
Why?
Mathematics concepts and ideas are everywhere and if you find a personal use for
new knowledge immediately, it will become part of your understanding and toolset
and not simply something you once studied and knew, but no longer use.
●
Not taking time to generalize knowledge from a set of problems that have been worked out
Example
: In Newton’s second law of motion (
F = ma
), what happens to the force (
F
) as the
acceleration (
a
) changes and what happens to the force as the mass (
m
) changes?
Why?
It is through generalization that future transfer becomes possible because you can
then generally predict the behavior of force with any changing mass or acceleration.
A
re You Ready?
Before continuing, you should be able to ...
I can...
OR
Here’s my question...
articulate what metacognition has to do
with learning and applying mathematics