Several years ago, when my son and I were working on multiplying fractions, we were doing some work with multiplying a whole number by a fraction, specifically a proper fraction, where the numerator is less than the denominator. The example was 5/8 x 12, and the focus of the lesson was supposed to be that in order to multiply a fraction and a whole number, you change the whole number into its equivalent improper fraction, in this case, 12/1, then solve. (Cross cancel, multiply the numerators, multiply the denominators, get your answer).
Okay, so he had no problems with the process. What did bother him was when he got the answer and saw that it was less than what he started with. At first, this didn’t make sense to him because, as he put it, “Doesn’t multiplication mean more?”
So we talked about it, and with a little more contemplation:
 he understood that, yes, when you multiply whole numbers, you get more (except when one of the factors is 1 or 0
 he remembered that when you multiply by 1 (which is like multiplying by a fraction with the same numerator and denominator) the number stays the same
 by Jove, he then realized that when you multiply by a fraction that’s less than 1, you’re taking less than 1 of what you started with, so the answer is smaller.
 he then extended the thinking a bit more by relating that when you multiply by an improper fraction (where the numerator is bigger than the denominator) you’re multiplying by more than 1, so the answer would be bigger
Now, I realize this example demonstrates a rudimentary understanding of numbers, but, to me, this also illustrates a problem I think kids can run into when it comes to learning math: the difference between knowing how to perform the steps of a particular process versus understanding why an answer does or does not make sense. Sometimes, we can learn a process mechanically without thinking about what we're doing in concrete terms. Maybe the pondering comes later – or maybe it never comes at all.
I suspect this may be one reason why kids that do great in basic math and prealgebra seemingly hit a brick wall when they get to Algebra. Could it be because when studying Algebra, you may learn to mechanically follow steps to solve complex equations without really comprehending the underlying logic? Or maybe some kids never get a good enough grasp of the basics to sufficiently tackle “higher math?”
I’m not sure where the breakdown occurs, but I do know that some people have a great facility with numbers (I am, unfortunately, not one of these people), while others may struggle all their lives with math, and others are somewhere in between. Is the facility with numbers from having been taught well, a result of some innate trait, a combination? How do you determine if your child has a sufficient understanding of math principles? Do you assess with worksheets and tests or use another method? Please leave a comment. And, if you need some free resources for learning fractions, decimals, percents, ratios and proportions, take a look at these:
MathMammoth: videos and worksheets on a variety of fraction topics
Fraction Video Tutorials: on reading, writing and reducing fractions
Math Games, Videos, Worksheets: on fractions, decimals, and percentages
Dining Out: Here are free activities you can use for a coop or group. Get the kids working with fractions, decimals and percents by figuring tax, tips and discounts when ordering food at restaurants.
Fraction Worksheets and Printables
Math Antics: Free videos with access to some free worsheets.
Yummy Math: This site stands out from many on the Web, in that it focuses on relating math to real life. There’s more emphasis on concepts and critical thinking than on memorization of steps. For more on the YummyMath philosophy, read this post.
Math Snacks: Short animations and games for teaching math concepts in grades 3  8. Includes accompanying worksheets (with answer keys).
And for additional math resources, visit our math page.
