Dealing with Hard Problems

If I were asked, “What is the most important idea or skill you can learn in math class?”, the answer I would probably give is HOW TO DEAL WITH HARD PROBLEMS. Here’s a link to an excellent article on dealing with hard problems:



Big Ideas in Mathematics

You open a math textbook and see the following problem:

1/3 + x = 1/8x + 1/5

There’s a lot going on.  Maybe you remember a teacher telling you about “two-step equations”, “getting x alone”, and “getting rid of the denominator”.  Maybe you remember what to do first.  Or maybe you don’t.

It’s hard to memorize a bunch of random facts and procedures.  That’s why when I think about math, I think about big ideas first.  Then, I ask myself how the little details fit inside these big ideas.  In the next few paragraphs, I’ll talk about some big ideas that students are likely to encounter in math class.  See if you can find at least one of these big ideas in each topic you study.

Big Idea 1: Equal things are interchangeable (they can be switched).

Robert went to the park to catch a Dragonite.”  Robert has a nickname, Bob.  So I could just as easily write “Bob went to the park to catch a Dragonite”, and the meaning of the sentence doesn’t change.

In math, we see problems like this:

Find the sum of 1/5 and 1/10

Like Robert, “1/5” has a nickname.  Actually, 1/5 has an infinite amount of nicknames!  One of these nicknames is “2/10”.  With this, we can change how the problem looks like, without actually changing the problem:

Find the sum of 2/10 and 1/10

Now that the problem looks easier, it can be more easily be solved: when you add two “tenths” and one “tenth” together, you have three “tenths”.  And so the answer is 3/10.

If two things are the same, you can replace one of them with the other.  Many math problems boil down to cleverly replacing parts of a problem with something equal to make the problem easier to figure out.

Big Idea 2: If two things are equal and you do the SAME thing to both of them, they stay equal. If you do different things, they don’t stay equal.

Doublemint Gum and Dubble Bubble Gum cost the same amount of money.  One day, the cost of both brands of gum increased by 50 cents (atrocious!).  Do both brands of gum still cost the same amount of money.  Absolutely!

 In math, the problems look like this:

Solve for x:

3x + 1 = 2

The equal sign says that “3x + 1” and “2” are equal.  As long as I treat them the same, I can do anything I want to them and they will stay equal.

Let’s just play around by adding the number 5 to each expression.  We get this:

3x + 1 + 5 = 2 + 5

3x + 6 = 7

Was there any point to adding 5 to both expressions?  No, not really!  It didn’t help us solve for x.  All it did was show that the same value of x which makes “3x + 1” equal to 5 also makes “3x + 6” equal to 7.

But was there any harm in what we did?  No!  Unlike a chemistry experiment where you might burn your eyebrows off if you mess around too much, you can goof around in math, take back anything you’ve done, and try again.

Let’s try the original problem again.  To solve for x, we’re going to be clever and make the equation look easier by subtracting both expressions by the number 1.  Here’s what happens:

3x + 1 – 1 = 2 – 1

3x = 1

Ah, this equation looks much simpler!  (Ask yourself: has the problem actually changed at all?)  And now that the problem looks much simpler, we can use common sense to figure out that x has to be 1/3!

If two things are the same, treating them the same means that they stay the same.  If you treat them differently, they don’t stay the same and you’ve changed the problem completely.  

Many math problems boil down to manipulating two equal things so that the problem stays the same but looks easier to solve, which makes it easier to solve.

Big Idea 3: The same thing can be represented in multiple ways.  In other words, two things can look different but be the same thing. 

In Big Idea 1 and Big Idea 2, we played around with symbols that look different but are actually the same.

For example, 1/5 and 2/10 look different but they represent the same amount of pizza, or brownies, or whatever you’re breaking up into a fraction.

Likewise, getting a 85% on a test is the same as the following:

  • 85/100 correct
  • 0.85 correct
  • 8.5/10 correct
  • 1 – 0.15 correct
  • 15% wrong
  • 100% – 85% wrong
  • Eh, you get the point

Here’s another example: 3x + 1 = 2.  Even though “3x + 1” and “2” look very different, they represent the same quantity when x is equal to 1/3.

Here’s a final example: choose any number, and double it.  For example if we chose 5, we would get 10.  Or if we chose 13, we would get 26.  Represent this idea using a table, graph, and equation.  Notice that each representation looks very different but is actually the exact same idea!

The same thing can take on many different forms.  Switching from one form to another can make problems easier to think about and solve.

Why Study Math? The Habits of Doing Mathematics

The most common question students have about mathematics is “when will I ever use this?”  This is a tough question to answer, even for mathematics teachers.  The reality is that there is no single, correct reason to study mathematics.  And while the most common response is, “mathematics can be used in your job / the real world”, I find this to be the most unconvincing response.

Instead of focusing on mathematics as a potential tool to use at a job or in the real world, it might be better to view math class as an opportunity to develop mathematical habits, which can come in handy when doing things outside of mathematics.  Below are four mathematical habits that students can learn in a mathematics class.  I’ve adopted / taken text from this blog by a PhD student in mathematics.

Discussing Definitions

A primary skill that mathematical people develop is fluidity with definitions. There’s a lot more to this than it sounds at first. What I mean by this is that mathematical people obsess over the best and most useful meaning of every word they use.

Let’s start with a mathematical example first, one which has some relationship to real life, the word “random.”  Randomness as a concept has plagued mathematics for much of its recent history because it’s difficult to nail down a precise definition of what it means for an event to be “random”.  Which is more “random”: flipping a coin 20 times and getting 20 heads in a row, or flipping a coin 20 times and getting HTHHTHHHTTTHTHHTHHTH?  Most people would say that the second situation is much more random than the first.  But the reality is that any coin flip has a 50% chance of resulting in a heads, so both situations represent equally “random” occurrences.  Or do they?  Before we can agree on whether the two situations are really equally “random”, we need to agree on exactly what we mean by “random”.

Okay, so mathematical people think about definitions very closely.  How can thinking about definitions help people in real life?  Imagine an average citizen is listening to the news and hears a politician say, “We have strong evidence of weapons of mass destruction in Iraq.”  If they had a good mathematics education they will ask, “What exactly do you mean by strong evidence and weapons of mass destruction?”  And, the crucial follow-up question, does the definition provided justify the proposed response, starting a war?  If you don’t understand the definition, you can’t make an informed voting decision.  And unfortunately many people don’t stop to think about the definitions of the words they hear, instead making choices that actually aren’t good for them or their community.

Producing Examples and Counterexamples

Mathematical people spend a lot of time coming up with examples and counterexamples to various claims.  For example, maybe you conjecture that the Earth is the center of the universe.  Then you try to come up with examples of the object that satisfy the claim. Maybe you make a toy model that shows how the Earth could be the center of the universe, if the universe were as simple as the toy.  Or you could try to go make some measurements involving the sun and moon and come up with evidence that the claim is false, that actually the Earth revolves around the sun.  In either case you’ve learned something about your original question, and you’ve done so by creating examples and counterexamples.

Being able to generate interesting and useful examples and counterexamples is a pillar of useful discourse. If you’ve ever seen a lawyer argue before a judge, you’ll see that most of the arguments are giving examples and counterexamples related to previous legal cases.  This mindset has also had countless applications to physics, engineering, and computer science.

But a subtler part of this is that mathematical people, by virtue of having made so many wrong, stupid, and false examples and counterexamples over their careers, are the least likely to blindly accept claims based on a strong voice and cultural assumptions.  In other words, studying mathematics is a fantastic way to build a healthy sense of skepticism.  This is just as useful for engineers as it is for plumbers, nurses, and garbage collectors.

Being Wrong Often and Admitting It

Two mathematicians, Isabel and Griffin, are discussing a mathematical claim in front of a blackboard.  Isabel thinks the claim is true, and she is vigorously arguing with Griffin, who thinks it is false.  Ten minutes later they have completely switched sides, and now Isabel thinks it’s false while Griffin thinks it’s true.

Scenarios like this happen all the time, but only in the context of mathematics.  The only reason it can happen is because both mathematicians, regardless of who is actually right, is not only willing to accept they’re wrong, but eager enough to radically switch sides when they see the potential for a flaw in their argument.

Unlike so many areas of life, mathematics is just the search for truth.  It’s about putting your personal pride or embarrassment aside for the sake of insight.

Teasing Apart the Assumptions Behind a Claim

Mathematics is actually a very unclear subject.  We like to think of mathematics as perfectly clear, that there definitive right and wrong answers.  To some extent this is true.  But there are many unresolved problems in mathematics and many questions in mathematics over which experts disagree.

As such, when a mathematical person makes a claim in mathematics, other mathematical people will go back to the basics and ask questions like, “What do this person’s words mean?”, “Why is this person making this claim?”, “How is this claim important?”, and “Where is this claim leading?”.

These are the same questions that one should ask when evaluating a political candidates claims about social policy, economics, and war.  Teasing apart claims should be part of everyone’s mental toolkit, as every claim has hidden assumptions that, if uncovered, has the potential to show that the claim is not as strong as you initially thought.