Learning fractions, according to the guidelines of the National Council for Mathematics Teachers, is supposed to start gradually over the first few grades and culminate in the ability to formally manipulate fractions by the sixth grade. It is the start of an important sequence which leads to the idea that there are numbers smaller than one, and to the introduction of percentages and ratios. Fractions constitute the formal introduction to the rational number line. In case you have forgotten, the rational numbers are numbers that can be expressed as the ratio of two integers, like 3/2, which is also 1 1/2, a number halfway between the integers one and two. Fractions are key to how American tape measures are made, marked at intervals of 1/16th of an inch, with 1/4 and 1/2 inch markings emphasized.
Unfortunately, learning fractions is the process that causes many students to begin to "hate" mathematics, to begin to see it as something difficult that they don't really need to learn. It is frequently the point where students begin studying not to learn the subject but to pass the tests. Thereafter, they forget what they have learned as soon as they no longer need the information. The proof is that, in spite of repeated practice in finding the "lowest common denominator," the majority of adults in this country do not know how to add two simple fractions of unequal bases (e.g., 1/2 + 1/3 ). This is in spite of the fact that the majority were able to do this in middle and high school.
It is the concept of fractions that students fail to learn, according to Hecht (2007) -- which is why they subsequently forget or misapply the procedures. Conceptual knowledge of fractions is defined as "the awareness of what the fraction symbols mean and the ability to represent fractions in multiple ways."
Learning the multiplication tables can be usefully contrasted to learning fractions. Souza (2008) argues that this is the first skill that students encounter in mathematics which does not arise naturally (or easily). That is, multiplication is the first skill for which there is no obvious evolutionary pressure to learn, and consequently no brain circuits are specialized for the task (as there are to assist counting, and adding and subtracting integers). Many students have difficulty learning the multiplication tables as a consequence, and many do not learn "all" of them. Students with dyscalculia usually have particular difficulty remembering such facts. Yet most adults do remember the multiplication facts that they were able to memorize, though they probably now use calculators to do any necessary calculations. More important, nearly all students learn the concept of multiplication, and have no trouble remembering it as adults.
Not so with fractions. When faced with adding two fractions, the majority of adults may remember vaguely that there is some procedure they are supposed to apply like finding the "lowest common denominator," but may not remember how apply it correctly. Most do not remember how to multiply or divide two fractions, or why they should want to do so. I have had students who had no trouble simplifying complex algebraic expressions involving division but who did not know how to do multiplication of such an expression by 1/2, or what it meant.
Students appear to fail to learn the basic concepts of fractions when they first encounter them, instead becoming overwhelmed with learning the procedures for dealing with them. Later the procedures, without the underlying understanding, become lost or jumbled together.
Math anxiety. Even more troubling is the fact that many cases of math anxiety can be traced to the student's first encounter with fractions. Math anxiety is a particularly important problem because it can cause people, who would otherwise have no particular trouble learning math, to avoid learning it and to avoid careers which require math knowledge. Math anxiety is comparatively rare in the first few grades, increasing about the time that formal operations with fractions is introduced, i.e., around sixth grade (Ashcroft, et al, 2007).
Finally, having trouble with fractions and beginning to find math "hard" or to "hate" math from that point on means that many students start just going through the motions with subsequent topics in math. They learn poorly the concepts involved with ratios and percentages, for example, topics which are encountered throughout adult life.
Worksheets. As a tutor, I find that many advanced students stumble when they encounter fractions in problems, or at least a problem that looks like it includes fractions (like multiplying by 1/2). I find it sometimes useful to go back to review the basic concepts of fractions, to review what adding, multiplying and dividing fractions means, and the relationship of fractions to the rational number line. I have included in this post several worksheets which can be used for this purpose. In most browsers, these pages can be printed, one at a time, by clicking on them and choosing file/print. Note: These review sheets are condensed, and are intended to be gone over line by line with a tutor or someone who understands fractions well.
Did Sam have enough butter for his cake, or not? It might be a long way to the store, and you could save him a trip.
References and Resources.
Current standards for math by grade can be found in the website of the National Council for Teachers of Mathematics.
Ashcroft, et al., "Is Math Anxiety a Mathematical Learning Disability," in Why is Math So Hard for Some Children, Berch and Mazzoco, Eds (2007).
Hecht, et al., "Fraction Skills and Proportional Reasoning," in Why is Math So Hard for Some Children, Berch and Mazzoco, Eds (2007).
D. Souza, How the Brain Learns Mathematics (2008).
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