Difficulties in initial algebra learning in Indonesia

Abstract

Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students’ achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students’ difficulties in algebra. In order to do so, a literature study was carried out on students’ difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.

Appendices

Appendix 1

• 1. There is a number and if 14 is added to it, then it is equal to 60. Find the number!

• 2. Robert builds a step pattern using squares. Here are the stages he follows.

  

  As you can see, he uses one square for stage 1, three squares for stage 2, and six for stage 3. How many squares should he use for the fourth stage?

• 3. Solve for x :  3(x − 5) = 2x  − 7

• 4. Solve for x :  4x + 7 < 15.

• 5. Amir and Tono together have Rp 30,000. If Amir’s amount of money is Rp 4,000 more than Tono’s, find each of their amounts.

• 6. The objects on the scale make it balance exactly. On the left pan, there is a 1 kg weight (mass) and half a brick. On the right pan, there is one brick. What is the weight (mass) of one brick?

  

  • (a) 0.5 kg

  • (b) 1 kg

  • (c) 2 kg

  • (d) 3 kg

• 7. If 4(x + 5) = 80, then x = ...

• 8. Solve for x: 5x + 2 ≥ 10 − 3x

• 9. The picture shows the footprints of a man walking.

  

  The pace length p is the distance between the rear of two consecutive footprints. For men, the formula, $\frac{n}{P}=140,$ gives an approximate relationship between n and p where,

  n = number of steps per minute, and

  p = pace length in meters.

  • (a) If the formula applies to Heiko’s walking and Heiko takes 70 steps per minute, what is Heiko’s pace length? Show your work.

  • (b) Bernard knows his pace length is 0.80 m. The formula applies to Bernard’s walking. Calculate Bernard’s walking speed in meters per minute and in kilometers per hour. Show your working out.

• 10. If x − y = 5 and $\frac{x}{2}=3$, what is the value of y?

  • (a) 6

  • (b) 1

  • (c) −1

  • (d) −7

• 11. Solve for x: x − 9 = 13.

• 12. A cube with the edge (x + 2) cm will be made. If the skeleton of the cube is made from a wire that is not longer than 180 cm, find the boundaries of the edge.

• 13. A rectangle has length and width (3x − 4) cm and (x + 1) cm, respectively: (a) Write a formula for its perimeter; (b) If the perimeter of the rectangle is 34 cm, find the area of the rectangle.

• 14. The sum of three consecutive positive integers is not greater than 63. Find boundaries for each of possible numbers.

• 15. If L = 4 when K = 6 and M = 24, which of the following is true?

  • (a) $L=\frac{M}{K}$

  • (b) $L=\frac{K}{M}$

  • (c) L = KM

  • (d) L = K + M

  • (e) L = M − K

• 16. Solve for x : 3x + 5 = 17 − x

Note: tasks 2 and 9 are taken from PISA 2006

(http://www.oecd.org/dataoecd/14/10/38709418.pdf); tasks 6, 7, 10, and 15 are taken from TIMSS 2003 (http://timss.bc.edu/PDF/T03_RELEASED_M8.pdf); other tasks are from Indonesian mathematics textbook series.

Appendix 2

Table 11 Observed difficulties in written test: frequencies and percentages (all students, total 33 × 4 + 18 × 5 = 222 cases)

Table 12 Observed difficulties in written test of interviewed students: frequencies and percentages (total 19 × 4 + 18 × 5 = 166 cases)

Table 13 Observed difficulties in the interviews: frequencies and percentages (total 19 × 4 + 18 × 5 = 166 cases)