Research over the last decade has revealed that most students tend to over-generalize mathematical rules. For example, children commonly bring to the study of decimals concepts from their work with whole numbers, such as “division always makes smaller” and “multiplication always makes bigger”. These concepts – though clearly true for whole numbers – lead to misconceptions and confusion with fractional and decimal computations. Another common misunderstanding is that ‘you cannot divide a smaller number by a larger one”.

Further misconceptions can develop from rote learning of multiplication and division procedures with decimals themselves. For example, the procedure of moving the decimal point of a divisor to the right to make a whole number, can lead to the erroneous conclusion that the divisor must be a whole number. Another example is the “invert and multiply” rule which many children interpret as “you can’t divide by a fraction”.

Overcoming misconceptions

In their study of Israeli and U.S. students, Anna O. Graber and Dina Tirosh recognized that overcoming mathematical misconceptions is a difficult task. Therefore, their study, in addition to describing common misconceptions, attempted to identify concepts which facilitate the development of good skills and accurate understandings about decimals.

A total of 60 fourth and fifth grade students (30 American and 30 Israeli) from different math classes representing different levels of achievement, were interviewed. Major areas investigated in these interviews were: (1) understanding of terminology, notation, definition and properties, (2) models for working with whole numbers, (3) facility with decimals, and (4) writing numerical expressions for word problems. Each interview lasted 20-25 minutes. A simple worksheet was provided for each student. Comparison between the countries was not the emphasis in this study. Nevertheless, the sequence of math instruction was found to be similar in Israel and the U.S.

### Misconceptions more common with division

Students in this study possessed some knowledge that would be helpful in building an understanding of operations with decimals. Most students were able to give a definition and identify properties of multiplication and division. They were also able to create concrete or pictorial representations (area models) of these computations with whole numbers.

However, this study did confirm previous findings which demonstrate that students develop common misconceptions about division and multiplication that are counterproductive for operations with decimals. Misconceptions were more common with division than multiplication. In addition, these students showed little knowledge of decimal notation except in the form of money, which was often taught and understood as something separate from decimals. Also, most students did not understand that a fraction is a statement of division. Students lacked an understanding of the connection between fractions and decimals (e.g. that .5 is another way to write 1/2). Moreover, U.S. students had little experience writing word problems for numerical expressions of multiplication or division.

### Recommendations

One commonly recommended instructional technique is to introduce new ideas in a manner that encourages students to built on their existing knowledge. Data from these interviews indicates, however, that many children do not have in their repertoires the knowledge and skills necessary for developing a thorough understanding of computation with decimals. Graeber and Tirosh suggest, therefore, that before students are asked to deal with the multiplication and division of decimals in fifth grade, it is essential that they should:

1. exhibit a good understanding of the meaning of decimal notation

2. be able to interpret division phrases using more than one model.

3. understand that a fraction is another way to indicate division, and

4. be able to easily translate between fractions and decimals.

In order to develop this necessary understanding, Graeber and Tirosh suggest instructional activities, such as:

1. introducing students in the early grades to the process of estimating whether the answer to a division word problem is greater than, less than or equal to one,

2. providing opportunities for students to compare and contrast the varied definitions, models and understandings of multiplication and division,

3. relating decimal notation to conrete embodiments and to currency notation (not treating decimal and monetary amounts as separate topics, even if the textbook does),

4. using decimal notation and common fraction notation to perform the same calculations and comparing results,

5. using the area model for multiplication of whole numbers prior to using it to illustrate the multiplication of decimals greater than or less than one, and

6. using manipulatives, such as decimal squares and money, to illustrate problems.

### Use of word problems to assess understanding

Students can demonstrate good computational skills and still lack an adequate understanding of these operations. But, lacking an adequate understanding inhibits the development of concepts necessary for computation with decimals. These researchers suggest that tasks, such as writing word problems and writing numerical expressions to solve word problems, can be used to evaluate the students’ understanding of operations with whole numbers. Trying to prevent the development of beliefs, such as “multiplication makes bigger” and “division makes smaller” may not be practical. These notions seem to evolve naturally during years of working with whole numbers. However, teachers can help students modify these beliefs. Unfortunately, Graeber and Tirosh concede, textbooks rarely provide suggestions for teaching counterintuitive facts, such as “multiplication sometimes makes smaller”. Currently, teachers are faced with providing such instruction on their own.

*“Insights Fourth and Fifth Graders Bring to Multiplication and Division with Decimals” Educational Studies in Mathematics, December 1990, Volume 21, Number 6, pp. 565-588.*

**Published in ERN March/April 1991 Volume 4 Number 2**