# Active Engagement in Problem Solving

Actively engage with the problem by asking questions and drawing diagrams. We can reason within the model. We also test our logic using the model. We can also prove or disprove the statements we make about the system by reasoning within our model. Counterexamples can be used to disprove generalizations made from observations.

The very first diagram you come up with may not give you any insights, it is therefore important to draw several diagrams. It is easy to test your ideas and refine your diagrams at this stage.

## Harnessing Questions, Models, and Visualizations

Problems can often be overwhelming due to their complexity, size, or the amount of information involved. However, these can become manageable and solvable when approached in an active and engaging manner. This engagement can take several forms such as questioning the problem, drawing diagrams, and using models. Here’s how:

**Active Engagement through Questioning:**

Questions are a way to interact with problems. Asking:

- What if?
- Why?
- How?

and so forth can help dig deeper into the problem and reveal underlying patterns or principles. For instance, if you’re trying to solve a math problem, you might ask:

- What happens if I change this number?
- Why is this equation set up this way?
- How does this theorem apply here?

The answers to these questions can guide your thought process and help you approach the problem from various angles.

**Using Models for Reasoning:**

Models are simplified representations of systems or phenomena. They provide a structured framework within which we can reason and understand the problem. With a model, we can make statements or hypotheses about the system and test their validity.

Models are simplified representations of systems or phenomena in the field of computer science. They provide a structured framework within which we can reason and understand the problem. With a model, we can make statements or hypotheses about the system and test their validity.

For instance, if you’re studying the performance of a new database management system in a company, you might create a model that represents the flow of data, operations on the data, and the underlying architecture. This model allows you to reason about how data is accessed, updated, and managed in the system, and how the new database management system might affect these operations.

Furthermore, the model could help visualize how different users interact with the data, how the system handles concurrency and what kind of bottlenecks could occur. This understanding could further help in optimizing the system and making it more efficient.

**Counterexamples to Test Generalizations:**

Observations often lead to generalizations. For example, seeing a pattern in a few instances might lead you to believe that the pattern applies universally. Counterexamples are a powerful tool to challenge these generalizations. If a generalization is true, it must hold for all cases. Hence, finding even one instance where it doesn’t hold (a counterexample) disproves the generalization. This approach is common in mathematics and logic but can be applied to problem-solving in any domain.

**Multiple Diagrams for Deeper Insights:**

Drawing diagrams is a form of visualization that provides a tangible way to interact with problems. Diagrams can reveal structures, relationships, and processes that might be hard to grasp in a purely abstract or verbal form. It’s important to create multiple diagrams as your understanding evolves. The first diagram you draw might not capture all the relevant aspects or might misrepresent some aspects due to your initial lack of understanding. As you gain more insights into the problem, you can refine your diagrams or create new ones to better represent your understanding. This iterative process of drawing, refining, and redrawing can lead to deeper insights and help solve the problem.

## Active Engagement

Let’s delve deeper into the concept of “Active Engagement in Problem Solving”.

### Asking Questions:

Asking questions is a crucial part of problem-solving. By asking probing questions about the problem, you engage your mind and stimulate thinking. Questions might include:

- “What information do I have?”
- “What am I trying to find or achieve?”
- “What are my constraints?”
- “What strategies can I use?”
- “What assumptions am I making, and are they justified?”
- “How can I break this problem down into smaller, manageable parts?”

For example, let’s consider a problem where you have to calculate the shortest path from one location to another on a map. Questions you might ask could include:

- What are the possible paths?
- What distances or obstacles are associated with each path?
- How can I calculate the total distance for a given path?

### Drawing Diagrams and Building Models:

Visualizing the problem can often provide new insights. This might involve drawing a diagram, building a physical model, or even visualizing the problem in your mind.

In our shortest path example, you might start by drawing a diagram of the map, with nodes representing locations and edges representing paths between locations. You might then use a strategy such as Dijkstra’s algorithm to calculate the shortest path. By actively engaging with the problem in this way, you can often see solutions that would not be apparent otherwise.

### Reasoning within the Model

Once you have a visual model of the problem, you can start to reason within this model. This might involve making deductions, looking for patterns, experimenting with different approaches, and testing out ideas.

In our example, you might start to reason within your diagram by working out the total distance for various paths and comparing them. You might then notice patterns, such as that the shortest path often involves going to the closest unvisited node, which is the principle behind Dijkstra’s algorithm.

### Refining the Model

Your initial model or diagram is often just a starting point. As you engage with the problem more deeply, you will often find that you need to refine your model. This might involve adding more detail, changing the way you’ve represented certain elements, or even completely rethinking your approach.

In our shortest path example, you might start to refine your diagram by adding information about the distances between nodes, and by marking nodes as “visited” as you incorporate them into your path.

### Testing Ideas

Finally, active problem-solving involves testing your ideas. This might involve working through examples, making predictions and checking whether they hold true, and looking for counterexamples that disprove your ideas.

In our example, you might test Dijkstra’s algorithm by using it to solve a few different shortest path problems and checking whether it indeed gives the correct answer. If it doesn’t, that could provide a valuable counterexample that helps you refine the algorithm or your understanding of the problem.

Active engagement in problem-solving isn’t a linear process. You’ll often find yourself going back and forth between different steps, and that’s a normal part of the process. The important thing is to stay engaged, keep asking questions, and keep testing and refining your ideas.