Current research has contradicted Piaget's original notion that children do not possess much cognitive ability until they are a year old. Instead, it suggests that children as young as 4 months have a basic understanding of physics. For instance, a four-month-old can recognize that solid objects cannot pass through other solid objects and that objects roll down slopes instead of up.

Problems involving structure usually require figuring out the relationship between two elements. Take this analogy problem: women are to men as sweets are to __________. The problem requires you to figure out the implied (i.e. not explicit) relationship between women and men and apply it to sweets. Structural problems are primarily based on relationships between parts.

Problems associated with arranging involve moving parts to fulfill a certain criterion. This manifests in many forms. Puzzles, anagrams, and block building are all forms of arrangement tasks. In the example provided in the answer, the letters are appropriately arranged to make the word “dancing.”

Problems involving transformation involve making a series of steps towards a certain goal. A good example of this kind of problem is a farmer who needs to bring his chicken, fox, and seeds to the other side of the river. He cannot leave the chicken and seeds or the fox and the chicken alone. In a series of steps, he manages to bring all of them across. In other words, he figured out the steps necessary to complete the goal and executed them.  

There are many resources at our disposal to solve problems. We can use trial and error, algorithms, deductive reasoning, inductive reasoning, dialectical reasoning, heuristics, forming sub-goals, and similar problems to help us approach the problem at hand. No single one method is measurably better than the other but some are more appropriate for particular problems.

Trial and error is an appropriate approach to a problem when the number of possible solutions is relatively small. If number of possible solutions were large, then using trial and error would potentially take too long and be ineffective. If someone is presented with five light switches, they might try each one until they find the correct one. If someone is presented with 300 light switches, then trial error might not be the best option.

All of these answers are examples of algorithms. Algorithms are problems involving a step-by-step procedure to get one goal. This is not restricted to just math as one might think. People use algorithms in day-to-day life whether baking cookies using a particular recipe or driving somewhere by using directions. 

Deductive and inductive reasoning differ in the path used to reach their respective conclusions. In deductive reasoning, the researcher starts with a general statement or hypothesis and examines possibilities in order to reach a specific conclusion. On the other hand, in inductive reasoning, the researcher attempts to make generalizations from specific occurrences. 

A heuristic is something commonly used in our reasoning as a general "rule of thumb". These are rules that hopefully guide use to the solution we need. For instance, a student taking a test wants to get a good grade. In order to reach that goal, they may follow the general "rule of thumb" (i.e. heuristic) of doing the easy questions first so they can use their focus for the more difficult problems.

Dialectical reasoning involves going back and forth between opposing points of view in order to come up with a satisfactory solution. This involves fully engaging in each side of the problem. This could be as simple as a person making a pros and cons list for what trip they want to take or as complex as a politician considering all positive and negative consequences of a bill being passed.