## Learning By Doing

Let's get this party started with a fun little puzzle. You may have seen this on your favorite social media platform [1].

## Struggling Productively

## Sources of Struggle

**Prior Knowledge:**First, we may not have all of the prerequisite knowledge to solve the problem. Once we recognize this fact (hopefully earlier rather than later!), then we can treat it as a quest to find the missing information.

**Incorrect Assumptions**: Another reason why we might struggle is because we've made an incorrect assumption. If you assume that a fox and wolf weigh the same, then you will eventually run into a road-block when solving the above problem.

**Unnecessary Problem Constraints**: We might struggle because we imposed an unnecessary constraint on the problem. This often happens while solving an insight problem. For example, in the classic nine-dot problem, problem solvers unnecessarily add the constraint that they are not allowed to go beyond the edge of the box.

**Flawed Representation**: Finally, we may have chosen a flawed or limited representation. We've seen time and again how important representations are for solving problems. For example, students fail to calculate the area of a parallelogram when presented outside of the canonical orientation (i.e., laying flat along the long side).

## What makes struggling "productive?"

According to James Hiebert and Douglas Grouws's book chapter [2],**productive struggle**happens when a student is working on a problem just outside of their current ability level. This is related to Lev Vygotsky's idea of the "zone of proximal development" (see Fig. 1).

Figure 1. Vygotsky's three zones, with the middle as the "zone of proximal development." |

*systems of equations*is a powerful problem-solving technique. However, if you first let them try to figure out how to calculate the weight of the chicken, fox, and wolf, then they might see the

**utility**of systems of equations.

**impasse**in problem solving might force a student to confront the possibility that there is something wrong with their understanding [3].

**discovery**involved. For example, there are are (at least) three key insights when solving this animal weight problem: using variables (x, y, & z) instead of animals, isolating a variable for each of the three known weights, and finally substituting the isolated variables into the other equations. Having a key insight or making a discovery is highly motivating.

**conceptual**understanding. For any of the three key insights, there is a rich conversation that can connect back to knowledge that a student already possesses (e.g., the concept of a "variable," isolating a variable, variable substitution, and mathematical equivalence).

## The Classroom Connection

Struggling doesn't have to be fraught with negative emotions. In some contexts, struggling is actually kind of fun. Think about the last video game you played. Games are specifically designed to cause you to struggle. In fact, there is some research to suggest that players actually enjoy dying (the ultimate failure!) in first-person shooter games more than shooting other players [4]. Another example is a well-written mystery. You may struggle to figure out "who dun it," but it is an entirely enjoyable experience. It would be beneficial to everyone if academic tasks that cause us to struggle to be structured in a way that is more like a game, puzzle, or mystery.Share and Enjoy!

Dr. Bob

## Going Beyond the Information Given

[1] I adapted this problem from Sara Van Der Werf's blog, and you can follow her on twitter @saravdwerf. This might also be a good time to link back to our prior conversation about problem isomorphs.*Second handbook of research on mathematics teaching and learning*, 371-404.

*Learning issues for intelligent tutoring systems*(pp. 19-41). Springer, New York, NY.

*Emotion, 8*(1), 114-120. https://doi.org/10.1037/1528-3542.8.1.114