### Relationships and differences between problem solving and problem posing

Although problem solving and problem posing differ, they are not entirely different cognitive activities but are closely related. Several researchers have experimentally confirmed that problem-solving ability and problem-posing performance are correlated and that problem posing positively influences problem solving (Bernardo 2001; Ellerton 1986; Nikata and Shimada 2005; Silver and Cai 1996). Problem posing offers many benefits: For example, it enhances problem-solving ability and the grasp of mathematical concepts, generates diverse and flexible thinking, alerts both teachers and learners to misunderstandings, and improves learners’ attitudes and confidence in mathematics (English 1998; Silver 1994). Although problem posing is rarely adopted in general education owing to certain constraints in practical classrooms, it is as critical a skill as problem solving.

Problem solving and problem posing differ, of course, in the features and formats of their tasks. Problem solving is a comprehension task, by which a learner extracts a mathematical structure from given information and reaches a correct answer. In contrast, problem posing is a production task that requires generation of information and its synthesis. Learners show difficulty in problem posing even if they can easily solve the problems. Akay and Boz (2009) asked prospective science teachers to respond to questionnaires about problem posing after participation in a course oriented to mathematical problem posing. The prospective teachers responded that problem posing was difficult because of its nature (e.g., not knowing the steps of problem posing), their abilities (not being creative), or lack of mathematical knowledge (having difficulties understanding abstractions) although they were not novices but had been trained as teachers.

Base (A_{1}): I bought some 60-yen oranges and 120-yen apples for 1020 yen. The total number of oranges and apples was 12. How many oranges and apples did I buy?

Solution:

Let *x* denote the number of oranges and *y* denote the number of apples.

*x* + *y* = 12

60*x* + 120*y* = 1020

According to the equations above, *x* = 7 and *y* = 5.

We investigated problems posed by novices to understand the difficulties they encounter in problem posing (Kojima et al. 2010). Undergraduates were asked to generate new problems from problems initially presented as bases. The bases were simple word problems easily solved by equations. The undergraduates were then encouraged to generate problems as varied and unique as possible. The variety of problems they posed was evaluated according to the four categories shown in Fig. 1, indicating similarities in the situations and solutions between each of their problems and the bases. Category I/I indicates problems that are almost the same as the bases, D/I indicates problems generated by altering the situations of the bases, I/D indicates problems generated by altering the solutions, and D/D indicates problems generated by combining alterations in both situations and solutions. Figure 2 presents examples of problems posed in each category that were solved by simultaneous equations. The results confirmed that the undergraduates posed many problems in categories I/I and D/I and few problems in I/D. They also revealed that D/I problems with situations different from the bases were appropriately composed. On the other hand, problems in I/D and D/D, where solutions differed from the bases, were relatively simple and inappropriate. Although the bases were elementary problems, many of the posed problems were simpler than the bases. These results indicate that the novices could generate novel situations, but failed to create new solutions in problem posing; thus, even if they can easily solve problems, undergraduates have difficulty in posing new problems. Therefore, because problem posing is more difficult than problem solving, it requires additional support.

Computational support of learning by problem posing has already been developed in various domains (Barak and Rafaeli 2004; Hirashima et al. 2007; Hirashima et al. 2010; Hirai et al. 2009; Takagi and Teshigawara 2006; Yu et al. 2005). However, such computational support focuses mainly on improving performance of comprehension tasks through problem posing, such as understanding domain knowledge or procedures in problem solving. Some studies empirically analyzed problems posed by learners (e.g., Cankoy 2014; English 1998; Leung 1997; Yu and Wu 2013); however, these studies have not addressed learning from examples in problem posing.

### The effects of examples in problem posing

The research field of Intelligent Tutoring Systems/Artificial Intelligence in Education has long addressed learning from examples. Interactive scaffolding that enhances learning from examples has been implemented, and its effects have been discussed (e.g., Conati and VanLehn 2000; Koedinger and Aleven 2007; Schwonke et al. 2009; McLaren and Isotani 2011). However, the central issue in such research is basically limited to problem solving and does not include problem posing.

Even so, some studies have addressed learning from examples in problem posing. Hsiao et al. (2013) experimentally confirmed the effect of seeing worked examples on problems posed by undergraduates in the business mathematics domain. The undergraduates posed problems with a web-based learning management system in three homework exercises after lecture classes. In each exercise, half of the students were provided two problems as examples solved through concepts or formulae learned in the lecture classes. The results demonstrated the effects of the examples: undergraduates who provided examples posed fewer problems not oriented to what they had learned in the lecture classes than those who provided no examples. Hsiao et al. also examined problems posed by undergraduates in terms of complexities. However, the examples’ effects on the problems’ complexities were limited—the examples did not expand the average complexity of each posed problem. Because Hsiao et al. provided each as a worked example, the undergraduates must have read only its solution, indicating that they learned the example through comprehension tasks.

We implemented a support system to facilitate learners’ posing of diverse problems by using examples (Kojima and Miwa 2008). In the system, learners engage in the same task as the one described above (Kojima et al. 2010). They pose new problems and input the texts and equations of their solutions into the system. The system automatically understands the situations and solutions in the problems and evaluates their variety. It can also present learners with problems as examples to provide hints for idea generation. The variety of learners’ problems is evaluated, and the presentation of examples is controlled on the four-category basis shown in Fig. 1. Experimental evaluations of the system confirmed that to some extent, it could facilitate learners’ posing of diverse problems. The number of problems posed in the I/I category decreased and those in D/I and D/D increased after the learners had posed problems with the system, and the system showed them various examples belonging to D/I and I/D. However, the presentation of examples did not increase the number of problems in I/D. The lack of problem posing in I/D was consistent with the results obtained by Kojima et al. (2010).

Although the system presents examples to learners and prompts them to compare the base with their posed problems, it does not give any instructions on how to learn from the examples. The examples are merely shown to the learners. We have not examined how the learners actually learned from the presented examples: the learners may have simply read the presented examples. In other words, the learners may have understood the examples through performing a comprehension task. The comprehension of examples may have helped in generating various situations; however, it may not have necessarily facilitated understanding of the solution structures. For learners to adequately study the solutions from examples and transfer that knowledge to their problem posing, further support must be introduced. Because problem posing is a production task, it effectively allows a learner to examine each example through a productive activity.

### Learning activities of examples in problem posing

#### Learning by solving examples

Solving examples and understanding the solution is of course one of the most popular activities in mathematical learning. As mentioned above, however, learning by solving may not be effective in improving the composition of solutions in learner problem posing because problem solving differs from problem posing.

#### Learning by reproducing examples

We designed a method of learning from examples through imitation, a learning activity adopted in productive task domains (Kojima et al. 2013). Imitation—the method by which learners reproduce existing example works—has long been adopted as a major learning activity in the domains of creative generation, such as art and music. The relationship between imitation and creation has been consistently noted in such domains and the effects of imitation have been documented. For example, Ishibashi and Okada (2006) argue that imitating examples can prompt imitators’ understanding of examples and their conceptual background; imitation facilitates a creative performance by imitators. In their experiment, subjects were engaged in an artistic drawing task before and after they created copies of a presented example. Results showed that the subjects deeply understood the example through its imitation, and understanding the example then elicited understanding of the subjects’ own expressions.

Based on this insight, we implemented a system for learning by reproducing examples in problem posing (Kojima et al. 2013) as an enhancement of the system previously described (Kojima and Miwa 2008). Learning by reproduction of an example allows learners to understand the ideas used in formulating the example from the viewpoint of the poser.

Figure 3 indicates the basic framework for learning by reproducing examples. In learning with the system, a learner is required to pose new problems from an initially given base. The learner is also presented with problems as examples, each generated by altering the base. When a learner studies an example generated from the base, the system hides the example itself and shows its generation process information to indicate how it was generated (bold black arrows in Fig. 3). Generation process information also includes sufficient information to reproduce the example. The learner generates a problem identical to the example by reproducing alteration of the base as indicated in generation process information (Fig. 3(a)). This prevents the learner from merely duplicating the characters and symbols that compose the text and solution of the example. From a poser’s viewpoint, this learning activity can facilitate understanding of the essential ideas used to generate the example, particularly those for composing a solution. The learner then transfers what is learned through reproduction into the posing of new problems (Fig. 3(b)).

We experimentally verified that learning by reproducing an example facilitated problem posing through directly adopting ideas used in the example’s generation (Kojima et al. 2013). However, we have not yet confirmed whether such learning can foster composition of solutions in the learner’s own problem posing.

#### Learning by evaluating examples

Some computational systems for supporting learning by problem posing (e.g., Barak and Rafaeli 2004; Hirai et al. 2009; Takagi and Teshigawara, 2006; Yu et al. 2005) adopt problem evaluation among learners as an activity in addition to problem posing. Experiments have shown that learning through such activities improves learning performance as well as the quality of learner problems. These studies basically designed the systems from the viewpoint of collaborative learning and focused on improving understanding of domain knowledge through problem posing. Although it is empirically confirmed that evaluations of problems posed by learners had predictive effects on the problems’ qualities (Yu and Wu 2013), these studies have not immediately produced evidence about the cognitive impacts of a learner evaluating activity on problem posing by the learners themselves.

Evaluation is a process involved in the creative generation of ideas or products. The importance of evaluative skills in creativity has been documented (Runco and Chand 1994). Furthermore, the effects of evaluating examples on the evaluator’s idea generation have been empirically demonstrated. Lonergan et al. (2004) experimentally observed that evaluation of examples according to certain standards improved the originality and feasibility of ideas generated by the evaluators, depending on the qualities of the examples and standards. Therefore, evaluation of existing ideas or products can be regarded as a production task because evaluation is a cognitive activity that can contribute to creative generation.

According to the above-mentioned studies, we experimentally investigated the effects of learning from an example on solution composition for problem posing. In the investigation, we studied the learning activities of reproducing and evaluating an example. To examine differences between comprehension and production tasks, we also studied the effects of learning by solving the same example. Because novice learners pose few such problems as examples, the investigation used an I/D problem as an example of a problem having a solution more complex than the base. As mentioned above, it is important to foster posing such problems because composing novel solutions is necessary but difficult, whereas generation of new situations is easy.