Page 38 - qrps

Quantitative Reasoning & Problem Solving
38 © 2014 Pacific Crest
The Practice 6. Ask “Why?” to go from just doing to understanding
Description
When/How
to Apply Every time you are confused, 1) determine what confused you 2) determine why it confused you
Mathematics is knowing what you can do but perhaps more importantly, why you can do it. This is the
heart of mathematical thinking and reasoning. Because mathematics is logical, we can always justify
and prove that we can do what we are doing. Learning this skill makes learning mathematics easier.
Asking “Why?” is the key to making the move from following a process or rule to understanding a
process or rule: You don’t understand until you understand why.
Examples
Division of Fractions In the division of fractions, almost everyone was taught to convert division
into multiplication by just flipping the denominator. If you ask the people why, the most common
response is, “That’s what they told me to do.” And it works — if you follow this rule, you can
divide fractions. What you don’t have is any understanding of why the process works and where/
when it might not.
Let’s gain that understanding: We can multiply the entire fraction by 1 and 1 can be represented
by the reciprocal of the denominator divided by itself. When this multiplication is carried out in
the denominator, we get a value of 1 for that denominator, which will yield the numerator itself,
carrying out that division. What we now have left in the numerator, is the original numerator
multiplied by the reciprocal of the denominator — just as if we had flipped the denominator
and multiplied it by the numerator. Understanding the logic of that process — the why — has
elevated our level of knowledge from Level 1 (memorization) to Level 2 (teaching).
Multiplication of Exponents With respect to the multiplication of exponents, most of us learned
to just add the exponent values and with division of exponents, to just subtract the exponent
values. We know what to do, but how many of us know why?
The Practice 7. Understand before Applying so you don’t spin your wheels
Description
When/How
to Apply
During each learning cycle, 1) Find the key points 2) Pick examples you understand
3) Present it to another 4) Verify they understand
It is a common practice complete many homework problems to increase the understanding of what has
been explained in class. Spending time on trying to apply knowledge without full understanding very
often leads to frustration and wheel spinning. Your understanding should be clearer than “foggy” or “hazy”
before you begin the application phase, if you plan to make progress. The degree of clarity in thinking can
be improved by revisiting Critical Thinking Questions and making sure that your level of understanding is
such that you can teach a process to another before you proceed to the application stage.
Examples
Division of Fractions In the division of fractions, you should understand why multiplying by
the ratio of reciprocals works. This allows you to easily address problems which have whole
numbers or common factors in them because you know the relationship between multiplication
and division and how to effectively connect them for the purpose of validation.
Long Division and Decimals What is the relationship between the divisor, dividend, and the
quotient, with respect to powers of ten? When you adjust the decimal points what are you really
doing? Once you understand the reasoning behind these processes and connections, there is no
relationship between a given divisor and dividend that you won’t be able to address effectively.