I hear people talk a lot about the importance of context in mathematics....usually, I think, for all the wrong reasons. There is a common misconception that if we show students how they can "use math" that they will find it more enjoyable and see the value in learning it in school. I'm pretty sure the problem isn't that students really want to know how this stuff is valuable in usage. The problem, I think, is that when the subject is relegated to procedures/facts/tricks it often disrupts the way a student naturally thinks.

Applied Math

By it's very name, this approach implies that we are "applying" something. Usually, there is a mathematical topic presented and then problems are selected in which students have to apply what they have learned to solve these problems. Sometimes, the problem might come first and then topics are presented (as facts) to provide students with the "tools" necessary to solve a problem ("necessitating" content). The focus here is on math as a tool to solve a problem. The assumption is that the mathematics exists before, and inspires, the problem.

Doing Math

Doing math is an act of creation by students. Usually, there is a problem/task/situation that requires students to think in the form of reflective abstraction. Mathematical "facts" arise as generalizations students make by looking for patterns and consistencies. These facts might, then, be explored in the abstract. The focus here is on problems as an entry to creating mathematics. The assumption is that the problem exists before, and inspires, some mathematics (I say "some" because although that may be anticipated by the teacher, it is ultimately defined by the students).

As I see it, yes, context is important but not to show students how they can use math. The context is important to show students how their mathematics is a natural extension of how they think and live in the world. Constance Kamii says it best:

"Most math educators think about verbal problems (word problems) as applications of computational 'skills,' rather than as the beginning point that eventually leads to generalized computation, without content, context, or practical purpose."

Kamii, C. (1985). Young children reinvent arithmetic: Implications of Piaget's theory. New York, NY: Teachers College Press.
I have been thinking a lot recently about the subtleties of problem-based approaches to math education. The following is a gross oversimplification, but I think it will illustrate the essence of what I am interested in. Let's compare two well-know approaches, Interactive Mathematics Program and Exeter Math:

Interactive Mathematics Program (IMP)
The Game of Pig - Year 1

MODEL: the unit starts with a "unit question/problem" and then smaller sub-questions (sometimes out of context of the unit question) are explored to deepen understanding before returning to the unit question again at the end of the unit.

Exeter Math
Year 1

MODEL: students are given a set of problems that are, more or less, completely unrelated. Each smaller problem stands on its own; it does not tie in to a larger context.

EXAMPLE: This unit follows the following progression:

1. Students are introduced to the unit question by playing games of Pig and thinking about strategy.

2. The first section detours from the unit question to help students "define" probability. There are investigations about gambler's fallacy, experimental versus theoretical probability, and measuring probability between 0 and 1.

3. The second section introduces "rug diagrams" as a way to represent probability. There are some investigations about this and the end of the section ties rug diagrams back to coin and dice games.

4. The third section looks at how things play out "in the long run." It involves investigations about the law of large numbers and expected value.

5. The last section looks at a simplified version of the game of Pig before returning to the original unit question.

EXAMPLE: Here are some problems, in sequence, from the Year 1 Exeter problem set:

Clearly, there are advantages and disadvantages to both approaches. What are they? Do you prefer one over the other? Why? In short, "What's the Difference?!?"
We're just wrapping up our "Infinite Possibilities" unit in which students explored the area of the Koch Snowflake. It came to my attention that Khan Academy has a set of videos on the same problem we have been tackling. So, if it exists online and we did it in class…"what's the difference:"

Khan Academy

1. Khan opens with this shot (a continuation of a previous video) in which he had shown how to calculate the area of an equilateral triangle in terms of its side length.

Infinite Possibilities

1. We open with an introduction to the figure. Students explore how it is changing at each iteration, attempt to solve for the area, and identify what they think we will need/what will be important.

2. Khan begins to organize information by showing the viewer that you can set up a table that will compare the number of sides to the overall area.

2. We pursue the "meaning" of area and investigate these irregular shapes as a way of developing the concept of total area as a sum of parts (or taking things apart and putting them back together).

3. Here Khan starts to show the viewer a pattern that is emerging. He points out that the number of sides is multiplied by four at each successive iteration and that the side length of the new triangle is 1/3 the length of the previous.

3. We look at how the area of the "new" triangle is changing at each iteration. 

4. Khan has now shown the calculations for the first four iterations. Each iteration is color coded. He has also written the terms of the sequence in a way that helps the viewer recognize patters.

4. Next we were concerned with looking at the total area at each iteration. As a class, students brainstormed what to include in a chart and then worked in teams to create it, complete it, look for patterns, and create any generalizations.

5. Khan has identified the portion of the series that is geometric and is demonstrating the calculation of an infinite geometric series.
5. Our next move was an introduction to sequences. Students attempted to look for patterns to help make predictions and used structure in the visuals to make connections to any generalizations.

6. Khan asks for a virtual drumroll as he has solved the problem and boxes the answer in magenta at the bottom right.

6. We debate about whether or not Tiger will get the "stupid" ball in the hole, which leads to an investigation of partial sums, convergence, and infinite series.

I'll leave you with a series of questions:
1. The two approaches tackle the same question. Are students learning the same things in each?
2. What does each method imply about the definition of mathematics?
3. How would you choose to measure/evaluate student progress in each approach? What is valued?
This one has been bugging me and I really wish I had more readers because I think there is a valuable discussion here:

What is the perimeter?

How long is the coastline?

The perplexing question here is essentially the same. This guy talks a lot about the quickness with which we move to abstraction. I'm wondering if maybe that applies here. I like the abstract because, in terms of creating a unit with a nice resolution, it easily comes full circle. I'm not sure you can say the same about the coastline. So…"what's the difference?!?"