When we first begin to work computationally with fractions, we are often challenged in our ways of thinking about numbers. Let me illustrate with a few examples. When we multiply two whole numbers bigger than one, the product is larger than our factors (2 x 5 = 10 and 10 is larger than 2 or 5). When we divide two whole numbers bigger than one, the quotient is smaller than the dividend (10 / 2 = 5 and 5 is less than 10).
As novice learners of mathematics, the ideas "when I multiply, the number gets larger" and "when I divide the number gets smaller" rarely get challenged. That is, perhaps, until the novice learner begins computations with rational numbers. Rational numbers are a bit of an odd beast, as I will show you with a few examples. When multiplying two rational numbers between zero and one, our product is smaller than our factors: 1/2 x 1/4 = 1/8, and 1/8 is less than 1/2 or 1/4. If we divide two positive rational numbers between zero and one, the quotient is larger than the dividend: (1/2) / (1/4) = 2, and 2 is larger than 1/2 or 1/4. Notice that this kind of conceptual thinking is counter to how we think about whole number multiplication. When multiplying as novice learners, we expect our numbers to get larger; and when dividing as novice learners, we expect our numbers to get smaller.
When someone incorrectly assumes that multiplying 1/2 x 1/4 yields a larger number than 1/2 or 1/4, we call this a direction of effect error. We use the same term for an individual who incorrectly assumes dividing two rational numbers between zero and one yields a smaller number.
It is here that I would like to point out an interesting paper that was recently published by Robert S. Siegler and Hugues Lortie-Forgues entitled Conceptual Knowledge of Fraction Arithmetic. The full article can be found here, but I would like to give a brief summary of its findings as it relates to the direction of effect error and conceptualization of fractions.
The study can be summarized into three distinct parts. In each part, three areas regarding fractions were studied:
(1) fraction arithmetic (called procedural knowledge), in which the subjects were asked to perform computations involving two positive fractions (adding, subtracting, multiplying or dividing)
(2) the direction of effect error (called conceptual knowledge), in which the subjects were given a statement of the form a/b * c/d > a/b and they were asked to determine if the statement was true or false (here the * represents one of the four standard operations +, -, x, /)
(3) magnitude of whole numbers and fractions, in which subjects were asked to label certain whole numbers and fractions on a number line to determine their accuracy
In the first part, 41 pre-service teachers from Quebec were asked to perform the three tasks listed above. The pre-service teachers had excellent knowledge of fraction magnitude, and excellent procedural/conceptual knowledge of addition and subtraction. However, their conceptual knowledge of multiplication and division was weak, as well as their procedural knowledge of division. Results are given in Table 1:
In the second part, 59 6th and 8th grade students from the greater Pittsburgh area were asked to perform the three tasks listed above. The students had excellent knowledge of fraction magnitude, and excellent procedural knowledge of all four operations with fractions. However, their conceptual knowledge of multiplication and division was weak. Results are given in Table 2:
In the third part, 17 undergraduate students in STEM fields were asked to perform the three tasks listed above. The undergraduate students were fluent in both conceptual and procedural knowledge of fractions in all areas. Results are given in Table 3:
DISCUSSION OF THE STUDY
Now, while the study does suffer from a couple drawbacks, it does raise a few interesting discussion points. The first I want to highlight is that the grade 6 and 8 students had roughly the same procedural knowledge and conceptual knowledge as the pre-service teachers. This to me is interesting and may account for why the general population is so fraction-adverse: perhaps many of our elementary school teachers actually have roughly the same base knowledge about fractions as their students (or less, as was the case in this study). Since the teachers themselves never truly conceptually understand fractions, how can we expect them to pass on conceptual understanding of fractions to their students? In this respect, fraction arithmetic gets placed into its own box in our long-term memory that is mutually exclusive to whole number arithmetic.
The second point that I would like to bring up is that those students who were in STEM fields had exceptional conceptual and procedural understanding of fraction arithmetic. I believe this is due to the fact that arithmetic with rational numbers must be mastered at an early point in their careers. One cannot be successful in calculus and analysis without a firm grasp of rational numbers, as these ideas lead to the more abstract ideas of rational expressions and rational functions. Also, these students see arithmetic with rational expressions every semester of every year - they have had many hours to perfect their knowledge. Compared to our pre-service teachers, I would guess that our STEM undergraduates see several hundreds of more hours working with rational expressions.
CONCLUDING REMARKS
So how can we use the ideas put forth in this study to inform our practice? First, faculties of education should recruit undergraduate students who have strong procedural and conceptual abilities, as they will make very strong teachers. Faculties of education often have very low entrance requirements, which runs counter to my personal beliefs. If we want good educators, shouldn't we set our bar higher? I also often I hear the argument that students who have higher-level education in mathematics do not make good teachers. This is clearly a false statement. Those who have spent more time studying mathematics have much more consolidation of conceptual knowledge when compared to those who have spent less time studying mathematics.
Secondly, we should encourage our pre-service teachers to spend more time studying mathematics with mathematicians - not by math educators in faculties of education. Since our math educators in faculties of education have not spent as much time studying mathematics, they will likely suffer from the same drawbacks as the pre-service teachers in the above-mentioned study. However, graduates of faculties of science will have spent many hours perfecting conceptual ideas related to elementary school mathematics. If we want our pre-service teachers to be better math teachers, faculties of education have to work more closely with their colleagues in mathematics to develop math content courses that are relevant and designed to deepen the knowledge of our future teachers. And these courses need to be delivered by well-trained individuals with a strong mathematics background.