# Problem Solving

In case you’ve been living under a rock for the past few years, you’ll know there’s something called “common core mathematics”, and that many states had adopted it while others have rejected it to produce their own state standards.  I won’t talk about the standards here (other than to say the biggest difference between Common Core and Your State Name Here Standards is the name).  Instead, I’ll talk about something that is part of the standards:  Problem Solving.

Unfortunately, mathematicians (and by extension, math educators) are terrible at producing names.   Chemists talk about adiabetic processes; biologists talk about glycophosphlipids; geologists talk about regoliths, and so on.  Meanwhile, mathematicians talk about sets, rings, fields, continuity, surfaces.  The difference is that mathematicians use these words in very specific ways.  One way to tell a mathematician is to see them cringe every time someone refers to a group of people…

So what about problem solving?  To ordinary people, this is a problem:  “Find $2398174 \times 139871$.  But in the context of mathematics education, this is not a problem.  There’s no commonly accepted word for it, which is too bad (how you speak influences how you think:  see Neither Borrower Nor Lender Be),  so I propose the name “Task.”  Finding this product is a task:  You know how to do it, and it’s a question of following a set of steps to get the answer.

If $2398174 \times 139871$ is not a problem, then what is?  It might come as a surprise, but this can be a problem: Find $4 \times 3$.  It all depends on context.  If you know the multiplication fact $4 \times 3 = 12$, then this is not a problem:  it is a task (specifically, the task of recalling what $4 \times 3$ is equal to).  On the other hand, if you don’t know what $4 \times 3$ is equal to, then this is a real problem.

So how can you solve this problem?  One way is to wait patiently until someone whispers in your ear “$4 \times 3 = 12$.”  But this relies on waiting for someone to give you the answer, and (from a broader societal perspective) programs you to believe what you are told instead of thinking for yourself; I’ve noted elsewhere that one of the reasons higher mathematics is important for a free society is that it develops the habit of questioning what you are told.

Instead of waiting for someone to give you the answer, you can try to solve the problem.  In this case, the problem solving might go something like this:

• We’ve defined $a \times b$ to be the sum of $a$ $b$s, so $4 \times 3$ is the sum of four 3s.
• This means $4 \times 3 = 3 + 3 + 3 + 3$.
• But I know how to add:  $3 + 3 + 3 + 3 = 12$.
• So $4 \times 3 = 12$.

Ideally, the last thought is “Cool!  I can figure out mathematics on my own and not need to wait until someone tells me what to do!”

What’s important to understand is this:  Problem solving is a skill,and like all skills, it gets better the more you practice.  But you only get one chance to solve a problem.  That is to say, once you’ve solved the problem, then no variation on the same problem gives you a chance to problem solve, and you will never again have the chance to solve the problem.

(Admittedly, it’s possible to find a new solution to a problem.  But basic arithmetic has been around for thousands of years, so finding a new solution to the problem of multiplication is very difficult…indeed, finding a new solution to any problem is something that could earn you an advanced degree in mathematics)

Thus, once you’ve solved the problem of finding $4 \times 3 = 12$, then $9 \times 7$ is not a problem:  you’ve figured out how to solve it (in this case, as the sum of nine 7s).  In fact, once you’ve figured out $4 \times 3 = 12$ this way, then you know how to solve $2398174 \times 139871$and this question is not a problem anymore!

What does this mean?  Consider a traditional elementary school math lesson which shows students how to multiply two numbers using the standard algorithm.  The instant students are shown how to multiply two numbers using the standard algorithm, they lose forever the opportunity to solve the problem of multiplication.  They will never get another opportunity to solve the problem of multiplication.

This requires a substantial shift in viewpoint in how we teach students mathematics.  Traditionally, students have been programmed to apply certain Standard Algorithms for basic arithmetic operations.  This is ideal…if we want to create students who are programmed.  However, those who can only follow a program will be doomed when they confront something outside of their programming:  such students will never progress beyond their teachers.

Instead, we need to create students who are able to solve problems.   The only way to do that is to give them the opportunity to solve problems…which means holding off introduction of the standard arithmetic algorithms for as long as possible.

You might be concerned that this means students won’t be able to multiply $43 \times 15$ without a calculator.  And that’s a legitimate concern.  However, let’s consider:

• A student who hasn’t learned the standard algorithm for multiplying two 2-digit numbers can’t multiply $43 \times 15$ without a calculator:  they have no way to even begin to answer this question.
• A student who has solved the problem of multiplication can find $43 \times 15$ without a calculator:  they can add forty-three 15s together.

The problem of $43 \times 15$ exists independent of the student’s knowledge of multiplication.  The difference is the student who’s done problem solving will be able to solve the problem; the student who’s only learned how to apply algorithms will only be able to solve the problem they have an algorithm for.