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The Practice
11. Generalize
your understanding to different contexts
Description
When/How
to Apply
At the end of each learning cycle, 1) List how the knowledge varies 2) Test acceptable ranges
3) Look for special cases 4) Identify contexts for use
Generalizing is the process of listing the key issues dealing with a piece of knowledge, what parameters
can change, which values are the most difficult to address, and whether there is any context that causes
problems. The goal is to make sure that there is no variable or context that you haven’t yet addressed
after playing with “What if”.
Examples
Adding Mixed Numbers
In the example of:
2 1 3 6
3 4
+
what can we change?
We can change the whole numbers (3 and 6). We can change the denominators (3 and 4).
We can change the fractions (2/3 and 1/4) so that they add up to more than 1 which would
require carrying and borrowing when validating. But those three things are the only aspects
of the problem that can change. Therefore, once we know we can add whole numbers, add
fractions and convert improper fractions to mixed numbers (and visa versa), we can generalize
our knowledge to any context that requires us to add mixed numbers.
Converting Fractions to Percents
In converting fractions to percents, we can determine the
processes involved: 1) Division of a denominator into the numerator of a fraction and 2) converting
a decimal into a percent with 3) a defined number of digits represented. Dividing two numbers
is a skill learned a long time ago, converting decimals to percents has also been mastered, and
the rounding of decimals is a familiar procedure. Thus, we can convert all fractions to percents
and present the answer to a given number of decimal places.
The Practice
12. Change Perspectives
Description
When/How
to Apply
When you’re stuck and
making no headway,
1) Change or transform the representation
2) Explain to someone else 3) Ask an expert for his or her perspective
Often in mathematics we come to a brick wall and are stuck. Pounding your head against this wall is
great for building persistence, but changing strategies is better if your goal is to solve the problem. A
wonderful strategy is to see the things in a different light. This might mean making the change from
looking at it symbolically to graphically or changing a set of numbers to relative percentages. The clarity
of one’s own thinking can often be improved by just explaining your understanding to someone else.
When you’re well and truly stuck, asking an expert to share his or her insights into a problem is an
excellent way to change perspectives and move forward.
Examples
Substitution of Values
If asked to factor
4
2
5 6
x x
+ +
, you can determine that the expression
is not quadratic. However, if you make a substitution of
2
y x
=
, then the expression becomes
the quadratic expression
2
5 6
y y
+ +
which is easily factored.
Table of Values
If asked to determine the distance between two functions
5
y
x
=
and
3
8
y x
=
over a domain of
x
= .5 to 1.5, you may find that using a table of values with intervals of 0.1
provides the best way forward.
1.4 Best Practices for Learning Mathematics