A "cognitive schema" is a collection of knowledge, beliefs, assumptions, associations, and expectations about aspects of the world we are not familiar with. When we are heavily familiarized with something, we may not need a cognitive schema to process it. If we are thinking about an aspect of the world we are foreign to we may need the assistance of a cognitive schema. For instance, we might form a cognitive schema around New York based off of what we've read, seen, heard, etc. Then when we actually go to New York, we might end up using this cognitive schema to filter the information they're receiving.

Lev Vygotsky posited that children’s sociocultural environment plays a huge role in how they develop cognitively as well as their acquisition of language. Unlike Piaget's view, Vygotsky did not believe children went through series of stages in their linguistic and cognitive development. Their development depends on interactions with adults, cultural norms, and their environmental circumstances. He also pointed out that children use language to control their own behavior.

Cognition is a form of thinking that involves mental activities such as understanding, problem solving, and decision making. This is when we are fully engaged with our minds, using our full mental capacity to innovate, weigh options, and tease out problems. Cognition, through its inherently innovative nature, makes creativity possible.

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.