The structure of prospective kindergarten teachers’ proportional reasoning

Abstract

Lamon (Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers, 2nd edn. Lawrence Erlbaum Associates, Mahwah, 2005) claimed that the development of proportional reasoning relies on various kinds of understanding and thinking processes. The critical components suggested were individuals’ understanding of the rational number subconstructs, unitizing, quantities and covariance, relative thinking, measurement and “reasoning up and down”. In this study, we empirically tested a theoretical model based on the one suggested by Lamon (Teaching fractions and ratios for understanding. Essential content knowledge and instructional strategies for teachers, 2nd edn. Lawrence Erlbaum Associates, Mahwah, 2005), as well as an extended model which included an additional component of solving missing value proportional problems. Data were collected from 238 prospective kindergarten teachers. To a great extent, the data provided support for the extended model. These findings allow us to make some first speculations regarding the knowledge that prospective kindergarten teachers possess in regard to proportional reasoning and the types of processes that might be emphasized during their education.

Appendix

Representative samples of the tasks used in the test.

1. Sharing/Comparing:

Answer the following question using this picture (Fig. 3).

Fig. 3

A set of 18 spots

Can you see thirds? How many spots are in $\frac{2}{3}$ of the set?

(Lamon 1999, p. 73)

2. Ratio:

There are 100 seats in a theatre, with 30 in the balcony and 70 on the main floor. Eighty tickets were sold for the matinee performance, including all of the seats on the main floor. What is the ratio of balcony seats to seats on the floor?

(Lamon 2005, p. 198)

3. Operator:

The teacher asked Nicholas to make some photocopies. Nicholas made a mistake and pressed the button that reduces the size of each copy by $\frac{3}{4}$. By how much should Nicholas increase each of the reduced copies so that the original size is reproduced?

(modified from Lamon 2005, p. 150)

4. Measure:

Locate $\frac{3}{4}$ on this number line (Fig. 4).

Fig. 4

Number line

(modified from Lamon 1999, p. 118)

5. Quotient:

Four people are going to share three identical pepperoni pizzas. How much will each person get? Draw a picture showing what each person’s share will look like (Fig. 5).

Fig. 5

Three pizzas and four people

(modified from Lamon 2005, p. 139)

6. Measurement:

Write two fractions between the given fractions: $\frac{1}{6}$ and $\frac{1}{5}$. Show how you worked.

(Lamon 2005, p. 122)

7. Unitizing:

The box of Bites costs €3.36 and the box of Bits costs €2.64. Which cereal is the better buy? (Fig. 6)

Fig. 6

Two boxes, their weight and prices

(Lamon 2005, p. 81)

8. Relative thinking:

Sam and Jason, two-third graders, commented on the following pictures (Fig. 7):

Fig. 7

Rectangles cut into pieces

Sam said that $\frac{7}{7}$ is larger because there are more pieces.

Jason said that $\frac{4}{4}$ is larger because the pieces are larger.

What do you think?

(Lamon 1999, p.18–19)

9. Quantities and covariance:

Dianne ran laps every day. Draw a conclusion about her running speed today, if she ran fewer laps in the same amount of time as she did yesterday. Circle the correct answer.

• A. Today she ran faster than yesterday.

• B. Yesterday she ran faster than today.

• C. Today she ran as fast as yesterday.

• D. The information given is not adequate for me to answer the question.

(modified from Lamon 2005, p. 61)

10. Reasoning up and down:

The shaded portion of this picture represents $3\,\frac{2}{3}$. How much do the 4 small rectangles represent? (Fig. 8)

Fig. 8

Shaded rectangles

(Lamon 2005, p. 73)

11. Missing value proportional problems:

John used 15 boxes of paint to paint 18 chairs. How many chairs will he paint with 25 boxes of paint?

(modified from Hart 1984; Karplus et al. 1983; Lamon 2005)