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37
The Practice
5. Analyze the Differences
as a way to follow the logic
Description
When/How
to Apply
When analyzing models or examples, 1) Find the differences 2) Explain what is going on & why
Seeing differences, articulating these differences, and then studying how things change
because
of the
differences is central to learning mathematics. Testing one’s understanding by varying things (conditions,
values, parameters, etc.) and seeing the impact of these differences increases understanding. When
analyzing an example, knowing what is different between steps helps us to understand the action
of each step. The complement of what we already know is that which we don’t know
—
in the same
way, what we are currently working to learn is the complement of our prior knowledge. Differences
across problems (what things can change from problem to problem) is a key component of generalizing
knowledge.
Examples
Long Division and Decimals
With long division, what was the difference between problems
with and without decimals?
Can we convert
2.34 254.12
into whole number division?
Yes, by multiplying it by a form of 1:
2.34 100 234
254.12 100 25412
=
The only difference remaining is potentially changing the representation of the remainder.
Division of Fractions
Going back to the previous example of dividing fractions, what is the
difference between the division of a fraction by whole number and the division of a fraction by a
fraction?
With the division of fractions, the following progression of problem types is fairly typical:
Problem 1
( )
3
1
2
whole number in
the numerator
Problem 2
1
2
3
whole number in
the denominator
Problem 3
3
2
5
3
improper fractions
Problem 4
12
2
13
3
mixed numbers
Understanding what it is that makes these problems different puts you well on your way to truly
understanding dividing fractions as well as generalizing the division of fractions across possible
contexts.
1.4 Best Practices for Learning Mathematics