When learning a new maths concept, should learners be taught the concept first and then solve problems, or solve problems first and then be taught the concept?

The first option is relatively straightforward and one that most of us can relate to: you learn and then you apply what you have learnt. The second option is not as straightforward: if you don't even know the concept, what is the value of getting you to solve problems on that concept? Obviously, the chances are you are not going to be able to solve the problem. Why, then, design for people to fail at solving problems before they have learnt the concepts required to solve those problems? Unless, of course, it is conceivable that, under certain conditions, designing for people to fail in the process of learning something new could well lead to deeper learning.

Let me give an example from my research. Suppose one wants to teach students a new maths concept - variance, for example. Variance is a measure of how much data deviate from their average. The first and intuitive option is simply to teach students the concept of variance, show them how it is applied to solve problems and then get them to solve problems on their own. This method is known as direct instruction, and is prevalent in schools around the world.

The second, but not so intuitive, option is to give them a carefully designed problem on the concept of variance and ask them to generate as many solutions as they can. Note that not only are we asking students to solve a problem before learning the requisite concepts, but also to design multiple solutions; in fact, as many as they can. Chances are they will fail to generate the ''correct'' solution(s) found in most textbooks. However, to the extent that students are able to come up with sub-optimal or even incorrect solutions to the problem, the process can be productive in preparing them to learn better from the teaching that follows. I call this method productive failure.

By failure, I mean that students will typically not be able to generate or discover the standard or correct solution(s) by themselves. By productive, I mean that this failure can be turned into deep learning, provided the teacher can build on students' ideas and solutions, and teach them the concept proper.

It turns out that learning theory gives us reason to believe in the effectiveness of both methods.

Direct instruction reduces our working-memory load. Reducing working-memory load is important because we have a limited capacity and if we strain or overload it, we are not going to learn anything. Because direct instruction helps manage this capacity better, it should lead to better learning. From this perspective, productive failure would be an ill-advised method because jumping into problem solving without having learnt the required concepts is a sure-fire recipe for overloading your working memory.

Yet, there is a reason to believe in productive failure. When students produce sub-optimal or incorrect solutions, their prior knowledge is being activated. Such activation of what we already know is critical when learning something new because it allows us to integrate new knowledge with what we already know. This, in turn, should lead to better learning.

So, how does one choose between the two methods? We run experiments: randomised-controlled experiments and noisy, messy classroom-based experiments. This is the kind of research I have been carrying out during the past eight years in maths classrooms in India and Singapore.

What have I found? In several comparisons, both methods have led to high levels of basic knowledge about the targeted concept. However, productive-failure students invariably demonstrate significantly deeper conceptual understanding and ability to transfer what was learnt to novel problems than students of direct instruction. Interestingly, we have evidence that the greater the number of sub-optimal or incorrect solutions students produce, the more they seem to learn. In other words, the more times they failed to produce the correct solution, the more they learnt when their teacher taught them the concept proper.

Although these findings challenge the conventional wisdom and practice of direct instruction to teach new concepts, this is not to suggest that direct instruction is poor. Clearly, direct instruction is able to deliver high levels of basic knowledge and skills. But if the aim of teaching and learning is to go beyond the basics and engender deeper conceptual understanding and the ability to transfer knowledge flexibly to new situations, then it seems that designing for a certain level of failure (as opposed to minimising it) in the learning phase may well be productive for learning in the long run.

**Manu Kapur is an associate professor of curriculum, teaching and learning at the National Institute of Education in Singapore, and was a keynote speaker at a recent International Conference of the Learning Sciences at the University of Sydney.**