Does Classroom Instruction Stick to Textbooks? A Case Study of Fraction Division

Abstract

In this chapter, we examined the consistency between textbook and its implementation in classrooms. By investigating how two selected Chinese teachers taught fraction division over four consecutive lessons, and making use of an existing study on the treatments of the same content unit in textbooks, it was found that the sample teachers essentially adopted their textbooks. The teachers put great effort into developing students’ understanding of the meaning of fraction division and justifying why the algorithm of fraction division works by employing a problem-based approach and using multiple representations. They followed the textbooks regarding the conceptualization of concepts and algorithms, the topic coverage, the sequence of content presentation, the approach to developing the concepts and algorithms, and the selection of problems and exercises. Meanwhile, the teachers also demonstrated certain flexibility in constructing their own problems for introducing new knowledge and consolidating the learned knowledge. Finally, the authors argued that the Chinese strategies of adopting textbooks might be attributed to their teaching culture and professional development practice.

Appendix: Brief Description of Teacher B’s Lessons

Lesson 1

Two methods of fraction division were discussed via a word problem. The teacher asked students to state the meaning of fraction division and the relationship between multiplication and division. After explicitly expressing that the meaning of fraction division was the same as the meaning of whole number division, and fraction division was the inverse operation of fraction multiplication, the teacher led the class to discuss the algorithm of dividing a fraction by a whole number.

Then, the teacher asked students to express the algorithm for dividing a fraction by a whole number. To practice this algorithm, students posed several problems related to dividing a fraction by a whole number (e.g., $\frac{2}{7}\div 3$, $\frac{4}{9}\div 2$) and discussed their solutions and justification in terms of two classifications (i.e., when the numerator is divisible by the divisor and when it is not). For example, students explained why the following procedure worked: $\frac{4}{9}\div 2=\frac{4\div 2}{9}=\frac{2}{9}$. Students explained the procedure according to the meaning of fraction and whole number division. In order to help students understand why dividing a fraction by a whole number is equal to the fraction times the reciprocal of the whole number, the teacher organized a hands-on demonstration activity: one student was asked to classify 12 magnetic pads into 3 equal groups, and another student was asked to take away one third of the 12 magnetic pads.

Through comparing the two methods of arranging magnetic blocks, students realized that dividing a fraction by a whole number was equal to the fraction times the reciprocal of the whole number. Then, three types of exercise problems were organized: questions for oral answers, word application problems, and competition problems.

Lesson 2

Beginning with a word problem, the class explored the meaning and algorithm of dividing a fraction by a fraction. The problem was used to recall the method of using a diagram to represent the quantitative relationship between a standard (unit) quantity, partial rate, and partial quantity (similar to Fig. 3). The teacher presented another word problem as follows: If a train runs 45 km in 3/4 hours, how far does it run per hour? By using a similar diagram, students found three solutions to the problem and justified $45\div \frac{3}{4}=45\times \frac{4}{3}$.

Based on this discussion, students discovered the algorithm of dividing a whole number by a fraction. Immediately, the teacher assigned a similar word problem for students to solve, and students presented their three solutions on a small board.

Lesson 3

The lesson began with a review of dividing fractions by whole numbers and dividing whole numbers by fractions. The teacher presented one word problem (i.e., There is a red silk strip measuring 9 over 10 meter in length. If making one Chinese tie requires 3 over 10 of a red silk strip, how many Chinese ties can be made using the strip? How can this problem be expressed numerically? The answer to the question resulted in the following numerical expression: $\frac{9}{10}\div \frac{3}{10}= \frac{9}{10}\times \frac{10}{3}=3$. Then, the teacher asked students to generalize this rule by providing another concrete example. Finally, the rule of fraction division was synthesized in general:

Dividing a number A by a number B is equal to the number A times the reciprocal of the number B (B≠0).

After that, students worked on several different types of exercises: basic exercises, comparing sizes of two expressions (e.g., $\frac{1}{2}\div\frac{3}{5}=\frac{1}{2}\div\frac{5}{3}$, $\frac{2}{5}\times \frac{1}{5}<\frac{2}{5}\div\frac{1}{5}$), open-ended problems, and word problem solutions (e.g., $\frac{1}{3}x=\frac{4}{9}$, $5x=\frac{4}{9}$).

Lesson 4

After reviewing the rules of fraction division, the teacher presented several fraction division expressions that included at least one mixed number (e.g., $\frac{7}{8}\div1\frac{5}{6}$; $4\frac{2}{7}\div1\frac{11}{14}$). Students worked on these problems individually and shared their solutions (some corrections were made). Then, the rule for division of mixed numbers was summarized: first transforming the mixed number to an improper fraction, then using the rule of fraction division.

Then, some exercises from the textbook were assigned to four student groups to be solved, and the results were checked in class. After that, the class discussed two sets of computation problems to make the following observations: (1) When dividing by a fraction less than 1, the quotients will increase, and when dividing by a fraction larger than 1, the quotients will decrease; (2) When the denominators are the same, the larger the numerator is, the larger the fraction is. On the other hand, when the numerators are the same, the larger the denominator is, the smaller the fraction is.