Messy problem solving

Problem solving - what is that anyway?

Framing a problem in a "mathematics" class has always been unsettling for me.  Sure, students can collect data prompted by a video that helps students see interesting problems that I see, or a problem that I basically just develop for students like the best third post, or problems of prediction that students arrive at when playing with Angry Birds, or problems that perhaps involve multiple different topics in mathematics at once like the sinkhole situation from a few years ago.

A while ago I thought about (and continued to think about after that) at the different ways of framing problems.  At the time, I thought of it as analyzing in the differences in act 1's, since, to me, it was about the way of packaging a problem -- or packaging the potentials for developing problems.

But that is only a small part of problem solving.  I've also thought about (and continue to think about) the rest of the processes of problem solving.  Although in the sentence it was "the" processes, I don't think there are set processes for solving problems.

The idea that we aren't exploring the problems in its entirety bugs me.  There are much more to solving problems besides the involvement of mathematics (and of course, this involvement and ability to think through different lenses is important).

Recently our class tackled a problem which was mostly framed for them.  It went something like the following (I forget the exact wording):

It is 15 years in the future, and Abed (used a student name during class, but this one is a pseudonym) has just been promoted to become a supervisor for an international company that sells medical equipment.  His first job was to recommend the amounts that the engineers would need to develop for each product which his division was responsible for.
While he was accounting for inventory from the previous year, he found something strange about one of the products that their company sold: the other products all had the number of equipments developed recorded, except for this one.  This particular product was designed for home use of individual cancer treatment.  All of this product, except for the demos, were all sold out.
What should his recommendation for the engineers be in terms of number of products?  What does he need to consider?

After discussion and figuring out what information Abed could find, I let them know the values that he found:

• That product cost $400000 to develop in total
• The company had a revenue of $560000
• 1000 of that particular product was used for demos
• It seems that the company made $10000 profit per each of that particular product

We then decided that we need to know how many products were developed last year, in order to help us make a recommendation.

The set up was challenging for them.  A lot of them tried different ways of creating an equality to help them figure out what's going on.  Eventually we settled on a rational equation which then helped us figure out the number of products made the  year before.

The value ends up being approximately (or exactly... I can't remember now) 1054 

But that's only a tiny piece of the puzzle.  My task for them is to give me a recommendation.  The following conversation ensued when I asked for their thoughts.

The first group that shared their recommendation gave me something that went something like this:

"We think that he should recommend that the engineers develop 60 products for the next year.  Clearly the 1000 demos was not worth it because the 54 products that were on sale were all sold out.  This way we would maximize the profits while still maintaining the same customers from the previous year"

A second group objected to this recommendation, and said something like:

"But wait, we don't know if we will have the same customers.  I mean, these are for cancer patients right?" 

Another group chimed in:

"We think we should still have some products on demo for the next year"

And another group:

"There are still so many things we don't know though" 

They then looked to me to say something.  I asked them what other things might be helpful for us to make a good recommendation so Abed doesn't get fired at his new position?  Students thought about a variety of things, and the following were the factors that they believed that Abed should consider:

• Does the equipment work?
• Is there a new version this year compared to last year?
• Is it easy to use?
• Who bought them? What countries? was it mainly hospitals for trial?
• To what extent does the equipment work?
• How long does it take to plan and make the equipment?
• Does it need software?
• Was last year the first year it was developed?
• Where were the demos done? how is it done?

And the list goes on.

We used this as another opportunity to talk about problem solving.  How it is messy, cyclical, non-linear, and requires thinking at every step.  In order to make good decisions, we need to think seriously about other factors as well - and not just happy with a number that we've gotten.

Great conversations.