Examining the preparatory effects of problem generation and solution generation on learning from instruction

Abstract

The goal of this paper is to isolate the preparatory effects of problem-generation from solution generation in problem-posing contexts, and their underlying mechanisms on learning from instruction. Using a randomized-controlled design, students were assigned to one of two conditions: (a) problem-posing with solution generation, where they generated problems and solutions to a novel situation, or (b) problem-posing without solution generation, where they generated only problems. All students then received instruction on a novel math concept. Findings revealed that problem-posing with solution generation prior to instruction resulted in significantly better conceptual knowledge, without any significant difference in procedural knowledge and transfer. Although solution generation prior to instruction plays a critical role in the development of conceptual understanding, which is necessary for transfer, generating problems plays an equally critical role in transfer. Implications for learning and instruction are discussed.

Appendix: additional description of instruction phase

The approximate timeline for the activities in the instruction phase were as follows:

• 1. First 10 min The teacher engaged students in a qualitative examination of Problem 1 to disambiguate the concept of mean from SD, and to ensure that students understood the concept of SD qualitatively first before learning its quantitative formulation.

  Problem 1: Math grades on six tests for students, S1–S3

  S1:

  A, B, A, B, A, B

  S2:

  C, C, C, C, C, C

  S3:

  A, E, A, E, A, E

  (Problem 1 qualitatively contrasted the mean from the spread in a data. It was designed to emphasize that S1 is better than S2 but S2 is more consistent than S1. Also that S1 is not only better than S3 but also more consistent. Finally, S2 and S3 may have the same average grade but S2 is more consistent.)

• 2. Next 20 minutes The teacher modeled and explained the solution to Problem 2 by using a step-by-step procedure for calculating SD. Each student was also provided with this step-by-step procedure typed on an A4 sheet of paper.

  Problem 2: Marks on five tests out of 20 for students S1 and S2

  S1:

  12, 13, 14, 15, 16

  S2:

  12, 14, 14, 14, 16

  (Problem 2 was designed to contrast how two data sets with the same mean and range can have different SDs.)

Printed step-by-step procedure for calculating SD given to students:

xx

1. Calculate the mean	$\mathop x\limits^{ - }$
2. Calculate the deviation between each point and the mean, and square this difference	$(x - \overline{x} )^{2} \quad$
3. Sum the squared deviation	$\sum {(x - \overline{x} )^{2} \quad }$
4. Take the average, i.e., divide the sum of the squared differences by the total number of values	$\frac{{\sum {(x - \overline{x} )^{2} \quad } }}{N}$
5. Take the square root of the average of the deviations. The square root of the average deviations is the standard deviation	$\sqrt {\frac{{\sum {(x - \overline{x} )^{2} \quad } }}{N}}$

• 3. Next 15 min the teacher gave students 5 min to solve Problem 3 on their own. Students were allowed to use the procedure sheet. After the 5 min were up, the teacher invited students to share their solutions, discussed and explained the solution, and provided corrective feedback where needed.

  Problem 3: Runs scored in five innings by two batsmen B1 and B2

  B1:

  20, 40, 60, 80, 100

  B2:

  0, 40, 60, 80, 120

  (Problem 3 was designed to contrast how changing the range affects the spread. Said another way, even though B1 and B2 have the same mean, their SD is different because the end points of B2 are further away from the mean, and the range is greater.)

• 4. The last 10 min The teacher gave students 5 min to solve Problem 4 on their own but without the procedure sheet this time. Student solutions on Problem 4 were collected for analysis. The teacher then discussed and explained the solution.

  Problem 4: Daily temperature over 5 days in two cities C1 and C2

  C1:

  36, 38, 40, 42, 44

  C2:

  36, 38, 40, 42, 54

  (Problem 4 was designed to contrast data sets with and without an outlier. C1 and C2 are exactly the same except 44 is replaced with 54.)

The remaining 5 min were left as buffer to be used as appropriate throughout the instruction phase.