To thrive in today’s globalized and technologically-driven landscape, students need to develop the ability to think not only critically but also innovatively. Therefore, there is an increasing emphasis on the need to engage students in authentic ** problem solving** situations that can foster these critical cognitive skills. In recent research in mathematical education, an increasing amount of evidence has also showed that getting students to generate and explore solutions to solve novel problems

**to instruction**

*prior***them to learn better from subsequent direct instruction of that targeted concept.**

*prepares*A learning design that aligns to the current educational need to develop students’ critical and inventive thinking capacities that are relevant to the 21^{st} century is** Productive Failure **(**PF**). This evidence based constructivist pedagogy entails the design of conditions for learners to persist in generating and exploring representations and solution methods (RSMs) for solving complex, novel problems. Though such a process may initially lead to failure to generate canonical RSMs, it has a hidden efficacy that is germane for learning provided an appropriate form of instructional intervention follows that can consolidate and assemble student-generated RSMs into canonical RSMs**.**

### Implementing a Productive Failure unit

Unlike the traditional, prevailing method of direct instruction, whereby students were first taught a new concept and its related procedures before solving problems related to the concept, the **PF **pedagogical design ** reverses** the sequence. As shown in Figure 1,

**PF**first engages students in solving a complex problem first and teaches them the canonical concept and procedures. Students will typically fail to generate or discover the correct solution(s) by themselves during the problem solving process. However, to the extent that students are able to use their prior knowledge to generate sub-optimal or even incorrect solutions to the problem, the process can be productive in preparing them to learn better from the subsequent instruction.

An example of a PF complex problem is shown in Figure 2. The targeted concept to be taught from the problem is the measure of consistency, Standard Deviation (SD). Prior to learning the concept of SD, students first work on the problem to invent various measures to assess which striker is the most consistent. The teacher then leverages the student solutions to teach the canonical concept of SD, before giving them practice problems to work on.

**Productive Failure: Beneficial to the learning of new concepts**

Compared to the traditional direct instruction approach whereby students are first taught the targeted concept and given practice problems later to work on, the **PF** learning design use the complex problem as a *preparatory *activity before the teaching of the targeted concept. Through several experiments, we have demonstrated how the sequence of problem-solving followed by instruction can be more productive for learning new mathematical concepts than vice versa.

Since its inception in 2007, the **PF** research team has impacted more than 7500 students and 100 teachers of Primary to Junior College levels from 14 Singapore schools. Results consistently show that students in the **PF** classrooms significantly outperformed their counterparts in the traditional direct instruction classrooms on conceptual understanding without compromising procedural knowledge. In addition, the **PF** students demonstrated significant transfer gains in flexibly adapting their knowledge to learn new concepts on their own.

An example of the comparison of the post-test performance for the SD unit is found in Figure 3. As shown, Secondary 3 students who learnt the concept of SD via PF outperformed their direct instruction counterparts on items related to conceptual understanding and transfer without compromising on procedural fluency.

**Benefits of using Productive Failure in your classrooms**

*Potential Benefits to your Students*

*Potential Benefits to your Students*

The study presents design opportunities for students prior to the learning of new mathematical concepts. It affords students opportunities to develop critical and inventive thinking, persevere amidst failures and challenges, and collaborate with others. Developing the competence in problem solving and posing will also be important in a globalized and technologically world, where the ability to anticipate and invent problems that will put one at an edge above those who can solve them.

*Professional Development for Teachers*

*Professional Development for Teachers*

The study will also expose mathematics teachers to new but empirically tested learning designs in the teaching of mathematical concepts. We can run professional development workshops for teachers interested in implementing **PF** units in their schools.

*Bringing Action Research to your School*

*Bringing Action Research to your School*

From the workshops, teachers will be exposed to the problems that we have designed, but may also gain insight to the design principles behind the **PF** tasks. Teachers can make use of what they have learnt in the workshops and work with their departments to incorporate **PF** into the action research efforts in their schools.