It also strikes me that the reason why we’ve been teaching it the other way around for so long (start from “here’s some math” and then “here’s a sucky question we can ask about it”) is that as teachers we DO start from content objectives that need to be taught, and planning backwards is hard. I don’t have good supplies. I don’t have enough time.

Here’s an example. You have a pizza cut into 8 pieces but only 6 pieces are are left. Four friends want to share the pizza evenly.

If you ask the question, “How much pizza does each person get?”, a student might figure: Give each person one piece. There are two pieces left. Cut each of those in half and give a half piece to each friend. Each person gets one and a half slices. (You could push students further to come up with 3/16.)

But if the question is phrased, “What fraction of the pizza will each person get?” Then you might get students to use fraction division. (6/8) / 4 = 3/16. The context might be real world but the solution will be artificial. Who does fraction division in the real world? It’s hard to think of genuine contexts. I usually end up doing some combination of addition, subtraction and multiplication before I pull out fraction division.

So I think how we encourage students to use their problem solving skills in a way that is natural and intuitive is also an important component of Real World Math. It’s not just about context.

(That said we do want to encourage students to grow their toolbox of math skills that feel comfortable and intuitive. We certainly don’t want our high school students to continue adding repeatedly instead of using multiplication.)

What is motivating?

– Acceptance. It this a problem that is accepted in my social environment to waste time upon? This many be connected to the realworldness of a problem.

– Reward. Do I earn respect for the solution I might get?

– Mastership. Is it a nice problem to try my skills on? Can I solve it? Maybe this is the strongest motivation. You can replace “I” by “we”.

Assume you tell your parents about this problem and the reaction is no reaction, plus someone in the class has solved it before, and you do not really understand the problem, nor do you have any clue on it. That’s a bad problem.

Sorry. You were asking about which presentation is more real world. Does not matter! All that matters is that you create an environment for motivation.

B. My working definition of “real-world” (although I try to avoid the term whenever possible because of it’s ambiguity), would be something like:

Real-world
adj.
1. (applied to a skill, idea, theory, or procedure) Useful to most people outside the academic discipline from which the skill, idea, or procedure originates.
The Pythagorean Theorem is a real-world theory which is useful in a wide range of contexts outside geometry or mathematics (e.g. architecture, physics, computer science, etc. — its uses in algebra and trigonometry would not apply because those are within the broad discipline of mathematics).
2. (applied to specific student assignments) Reflects a plausible application of an idea/skill/theory/procedure to a situation outside the discipline in which it originates (e.g. a real-world problem which teaches the Pythagorean theorem must be a believable problem that a non-professional-mathematician would reasonably encounter and try to solve using the theorem and not some other method).

C. None of the above problems qualify as real-world.

Problems 1 and 2 are definitely interesting, and would be more effective if combined (multiple representations of the same idea). They (especially combined) help me understand the arcane vocabulary of geometry, appreciate the beauty of the relationship between area formulas, and apply algebra in solving a geometry problem (I would probably solve them by setting up the area formulae as a system of equations).

Problem 3 is less interesting to me because it is highly implausible, distracting (why are we putting candy in geometric figures?) and off-putting (you are insulting my intelligence by thinking I will only be interested in geometry if candy is involved).
It is also less effective at teaching geometry because it involves discrete integers which are only loosely related to the formulae for area of a circle and square.