Abstract
In a study with ninthgrade mathematics students on learning the concept of variance, students experienced either direct instruction (DI) or productive failure (PF), wherein they were first asked to generate a quantitative index for variance without any guidance before receiving DI on the concept. Whereas DI students relied only on the canonical formulation of variance taught to them, PF students generated a diversity of formulations for variance but were unsuccessful in developing the canonical formulation. On the posttest however, PF students significantly outperformed DI students on conceptual understanding and transfer without compromising procedural fluency. These results challenge the claim that there is little efficacy in having learners solve problems targeting concepts that are novel to them, and that DI needs to happen before learners should solve problems on their own.
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Notes
 1.
Initially, a condition (PF vs. DI) by teacher (teacher A vs. teacher B) MANCOVA was carried out with prior knowledge as covariate. However, the main and interaction effects of teacher were not significant. Hence, the teacher (or class) factor was collapsed, and a more straightforward MANCOVA with condition as the sole betweensubjects factor has been reported.
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Appendices
Appendix A: The complex problem scenario
Mr. Fergusson, Mr. Merino, and Mr. Eriksson are the mangers of the Supreme Football Club. They are on the lookout for a new striker, and after a long search, they shortlisted three potential players: Mike Arwen, Dave Backhand, and Ivan Right. All strikers asked for the same salary, so the managers agreed that they should base their decisions on the players’ performance in the Premier League for the last 20 years. Table 4 shows the number of goals that each striker had scored between 1988 and 2007.
The managers agreed that the player they hire should be a consistent performer. They decided that they should approach this decision mathematically, and would want a formula for calculating the consistency of performance for each player. This formula should apply to all players and help provide a fair comparison. The managers decided to get your help.
Please come up with a formula for consistency and show which player is the most consistent striker. Show all working and calculations on the paper provided.
Appendix B: Examples of pretest items
Central tendencies
The table below shows the timing (in minutes) for a 2.4 km run for 40 students in Class 2E1. Calculate the mean, median and mode of the timing of Class 2E1.
Distributions
The heart rate per minute of a group of 20 adults is displayed in the dot diagram below. For example, 3 adults have a rate of 60 beats per minute. Based on this data set, how many individuals from a similar group of 40 adults would be expected to have a heart rate of 90 beats or more per minute?
Dot diagram for the heart rate per minute for a group of 20 adults
Variance
The owners of two cinemas, A and B, argue that their respective cinema enjoys a more consistent attendance. They collected the daily attendance of their cinemas for 11 days. The results of their data collection are shown below.
Cinema A  Cinema B  

Day 1  69  61 
Day 2  70  65 
Day 3  75  91 
Day 4  52  55 
Day 5  57  58 
Day 6  92  95 
Day 7  71  67 
Day 8  73  81 
Day 9  74  89 
Day 10  72  70 
Day 11  87  93 
Based on the above attendance data and statistics, which cinema do you think enjoys a more consistent attendance? Please explain mathematically and show your working.
Appendix C: Examples of posttest items
Procedural fluency item 1
Q1. Marks scored by 10 students on a test on statistics are shown below. As a measure of the variance, calculate the standard deviation of the test scores.
Conceptual understanding item 1
Q2. For Q1, one student came up with another measure of variance by taking the average of the sum of the difference between adjacent scores as shown below:\( \frac{{\left( {50  30} \right) + \left( {50  50} \right) + \left( {55  50} \right) + \left( {60  55} \right) + \left( {60  60} \right) + \left( {60  60} \right) + \left( {70  60} \right) + \left( {80  70} \right) + \left( {90  80} \right)}}{10  1} \) \( = 6.67 \)
How does the student’s measure of variance compare with the standard deviation as a measure of variance? Which one is better? Please explain your answer.
Procedural fluency item 2
In preparing for the Youth Olympics in 2010, the Ministry of Community, Youth and Sports had to decide the month in which to hold the games. They narrowed their options to July and August, and decided to examine rainfall data for ten randomly selected days in July and August in 2007 to make a choice. The amounts of rainfall (in millimeters) for the 2 months are shown below.
Day  Rainfall in July (mm)  Rainfall in August (mm) 

Week 1, Day 1  32  25 
Week 1, Day 3  35  31 
Week 2, Day 2  35  35 
Week 2, Day 4  37  37 
Week 2, Day 7  37  37 
Week 3, Day 2  37  37 
Week 3, Day 5  38  38 
Week 3, Day 7  39  39 
Week 4, Day 5  40  42 
Week 4, Day 6  40  49 

i.
Based on the information, which month should the Ministry choose, given that they would want a month that has a consistently low amount of rainfall?
Conceptual Understanding Item 2

ii.
A few days later, the Ministry relooked at the data and realized that they made a mistake for the figure recorded Week 4, Day 6 in July. Instead for 40 mm, the rainfall should be 60 mm. Given this new figure, which month should the Ministry choose now, if they want one that has a consistently low amount of rainfall?
Transfer item
Two Secondary Four students were nominated for the “Best Science Student” award for 2009. Muthu Kumaran is the top Physics student, while Alicia Kuan is the top Chemistry student for 2009. The table below shows the Physics and Chemistry top scorers between 1998 and 2009, with their scores presented in ascending order.
Top physics scorers for the past 12 years  Top chemistry scorers for the past 12 years  

Name  Year  Score  Name  Year  Score 
Yap Pei Ling  2006  81  Lim Jen Yi  1998  80 
Cho Ying Ming  1999  83  Charissa Tan  2001  81 
Bala Ayanan  2001  83  Allan Wu  2000  83 
Mohammad Azhar  2000  84  Ali Salim  2002  85 
Matilda Tay  2002  84  Derick Chan  1999  89 
Louis Ho  2005  85  David Tan  2003  90 
Tham Jing Ling  2004  85  Abdul Basher  2005  90 
Jodie Ang  1998  85  Fredrick Chay  2004  94 
Jeremy Goh  2003  85  Linda Siew  2006  95 
Chee Haw Ren  2006  85  Terry Lee  2008  96 
Susan Teo  2005  86  Low Ming Lee  2007  98 
Muthu Kumaran  2009  94  Alicia Kwan  2009  99 
Mean  85  Mean  90 
Both Muthu and Alicia are the best performers in their respective subjects for the past 12 years. Because there is only one “Best Science Student” award, who do you think deserves the award more? Please explain your decision mathematically and show your working.
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Kapur, M. Productive failure in learning the concept of variance. Instr Sci 40, 651–672 (2012). https://doi.org/10.1007/s1125101292096
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Keywords
 Problem solving
 Productive failure
 Multiple representations
 Mathematics
 Classroombased research