math – Experience Has Made Me Rich

Open-Mindedness

Recently I have been thinking a lot about why so many people seem inconvincible of certain things which I hold to be true. And while I could certainly make a list of some of these things, that is not the intention of this blog entry. Instead, I have been reflecting on open-mindedness and wanted to share.

Many people – myself included – often enter into discourse with someone of a differing opinion with the intention of convincing them to change their mind. For example, maybe your friend Paul thinks all trees in your neighborhood that are taller than 12 feet should be pruned to 12 feet or less, so as not to obstruct anyone’s view of the lakefront. You know that he’s clearly wrong! You get into a discussion. Only it’s not really a discussion – it’s an argument each of you is trying to win. Maybe out of frustration you start incorporating personal attacks. Maybe you get so angry at Paul’s refusal to capitulate, as well as the horrible things he is saying about you, that it ends your friendship. Maybe in the middle of the night, Paul prunes all of your tall trees. Maybe the next night you erect a 30 foot statue on your lawn directly in Paul’s line of sight to the lake … and so on.

It’s sad, and you don’t even like the statue, but what choice is there? Paul must be taught a lesson.

I wish this was hyperbole. Sadly, it is not. And the conclusion is clearly suboptimal.

Well … let me construct a basis for discussion with some (hopefully) fair assumptions. In doing so I’m going to have to use a little bit of math terminology, and it occurs to me that some people might not know precisely what I mean, or even be put-off by some of my more mathematical references. If you think this might be the case, I ask you to bear with me. The concepts and symbols I use are the best way for me to illustrate my point, and I’ve included here a bit of a math lesson, in case it is not something you’ve encountered in your life – it will clarify some of the words and concepts I use for the rest of this article. Of course, if you feel there’s no need for you to read this section, by all means scroll past it and keep reading (I won’t feel bad).

Some Math concepts

Sets
Mathematicians like to talk about collections of values that are somehow related, and when they do, they use the word set. We use curly brackets to list the objects (known as elements) of a set. So for example the set $F=\{apple, orange, banana, kiwi, peach, nectarine\}$ is a set I have named $F$ , and just so you know, it is the set containing all the fruits I might bring to work with me in my lunch. A subset of a set $S$ is another set that only contains elements from $S$ . So for example $M=\{apple, kiwi\}$ is the set of fruits I brought to work in my lunch on Monday, and is a subset of $F$ . On the other hand, $A=\{apple, pineapple, banana\}$ is not a subset of $F$ .

A Little Bit of Algebra (Apologies to the Arithmophobic)
Consider this simple algebra equation:
$\displaystyle 3x+4y=7$
The $x$ and $y$ are understood to be symbolic of numbers, but the use of symbols mean that they vary – which is to say, they are variable. The equation is a statement. In this particular statement,
$x = 1, y=1$
would be a valid solution (i.e., the equation becomes true), since
$3\times 1 + 4\times 1=7$ .
So would
$x = 5, y=-2$ ,
since
$3\times 5 + 4\times (-2)=7$ .
However
$x = 5, y=2$
would not be a solution (i.e., the equation becomes false), since
$3\times 5 + 4\times 2=23$ ,
which is not 7.

Statements
In math and philosophy, a statement is a sentence that must either be true or false (but not both, and not maybe). Very often the truth value (i.e., “true” or “false”) of the statement depends on values for variables contained in the statement. The algebra equation above is a statement. Another example is the statement “I like cheese”, which contains two variables: “I”, and “cheese”. If the “I” refers to “Rich Dlin” (i.e., it is me speaking and not you), and the “cheese” refers to “Havarti”, then the statement is true. If the “I” is “Rich Dlin”, and the “cheese” is “Cambozola”, the statement (I promise you) is false. Notice that if the “cheese” were to refer to “gingerbread cookie” the statement would be nonsense, since “gingerbread cookie” is not a cheese – even though it is true that I like gingerbread cookies, it is irrelevant in the context of this statement. A mathematician would say “gingerbread cookie” is not an element of the set of all cheeses. Going back to the algebra example, {(1,1),(3,-2)} is a subset of the set of solutions to the equation given. The actual set has an infinite number of solutions in it, but that’s more than I need to talk about here. What I will say is that the truth value of the statement “Three times John’s favorite number plus four times Gail’s favorite number will yield seven” is:

True if (“John’s favorite number“, “Gail’s favorite number“) belongs to the set of solutions of 3x + 4y = 7,

False if (“John’s favorite number“, “Gail’s favorite number“) does not belong to the set of solutions of 3x + 4y = 7, and

Nonsense if, for example, John claims his favorite number is “cinnamon“. Be on the lookout for nonsense – it is surprisingly common.

The Assumptions
Ok. Welcome back. Here are the assumptions I was talking about:

All questions have a right answer …
… when the answer is justified properly with a well framed statement.
The truth value of the statement may be subject to variables that change which answer is correct, but with a fixed set of values for the variables, there is a right answer. For example, the question “Should all trees taller than 12 feet in our neighborhood be pruned?” could be answered “Yes”, justified with the statement “It is unacceptable for some trees in our neighborhood to block sight lines to the lakefront”. Note that here the answer to the question is “yes” if the statement is true, and “no” if the statement is false, and may reasonably depend on whether or not the tree is also so wide, or part of a grove, as to make it impossible for a resident to see the lakefront from a different angle standing on the same property. It may also depend on whether 12 feet is a reasonable height with respect to whether or not sight lines get blocked. In this case these variables need to be introduced into the statement, or else agreed upon as not being variable.

The right answer may well not be knowable …
… even with the variable values fixed – which doesn’t mean there is no right answer!
As an example, consider the question “How many humans are alive on Earth right now?”

The number changes many times in a short span of time. So the truth value of the answer depends on what time it is indexed to.
The answer is subject to a definition of “alive”, and the answers to whether or not some organisms are living humans are in dispute.
“On” Earth needs to be defined. If I am in an airplane, am I on Earth? What if I am in low orbit?
However there is an answer, if we fix the variables.
There is currently no way, even with the variables fixed, to know the answer.

Knowing the truth is inherently valuable.
This is a big one. Many people demonstrate by their behavior that they do not adhere to this assumption. A simple example is the person who refuses to go to the doctor about a problem because they are afraid of what they might find out. In some ways, not wanting to know the truth is a human quality, especially in situations where a false belief has spawned an entire tree of values and beliefs we have been living by. If the root belief is false, what happens to the tree?

When it Comes to Truth, What We Want Doesn’t Matter
So with these assumptions, my position is that for any belief I hold, I am either right or wrong, and that I may not be able to tell. So then what am I to make of someone who disagrees? Can I immediately conclude that they are wrong? Clearly not. However I freely admit I want them to be wrong, so that I don’t have to be. After all, being wrong has some negative implications. On a fairly benign end it means I have been somehow deluded, which injures my pride. On an extreme end it may mean I have to discard an entire tree of conclusions that were premised on my error, leaving behind a buzzing hive of uncomfortable questions and observations about my previous behavior. But if the root belief is actually wrong, what choice do I really have? Since it is rooted in falsehood, the whole tree is an illusion anyway.

Here is a hard truth: What we want has nothing to do with what is true. I want there to be peace in the Middle East. But there is not peace in the Middle East, and no amount of wishing on my part, no matter how fervent, can alter the truth value of this or any other statement. On the other hand, what is true can and should definitely impact what I want. What we all want.

Ok. Here is another statement that is tautologically true: In the set of things I hold to be true, some might be false. And from a probability perspective, I am also comfortable saying that in the set of things I hold to be true, some are true, and some are false. I want to say “most are true and some are false”, but I am honestly not sure I have a reasonable argument to claim that, so we’ll leave it there as a desire more than a fact.

Shades of Gray
Now I will focus on statements where the truth depends on fixing values for the variables in the statement., which to me is the core of the shades of gray argument: In cases where there is a continuum of possibilities between true and false, almost everything in the set of things I hold to be true lies somewhere within the boundaries of the continuum, rather than on one of the ends.

Here a philosopher or mathematician might (and should!) argue that there can be no continuum between true and false, since those are binary options. My response is that I am talking about a sphere of reasonable answers centered on the truth, where anything outside the sphere is easily demonstrated to be false (or worse, nonsense), but things get a little more touchy inside the sphere. This is a consequence of my point about the truth of a statement depending on fixing values for variables the statement depends upon. To elaborate on this, I am going to define something called an assumption set.

Assumption Set
Suppose a statement depends on a set of variables. For example, consider the statement “Running is good for you.” The truth of this is not absolute. It depends on some variables:

How much running (the quantity of the running)?
How intense (the quality of the running)?
What preconditions do you have that running would exacerbate (e.g, bad knees, asthma, heart problems)?
Where do you plan to do your running (road, track, beach)?
and many more.

So before we could discuss whether the statement is true, we would have to fix values for these variables. I call these fixed values the assumption set. So for example an assumption set for this statement could be
$R=\{45 minutes per day, at 80\% of maximum heart rate, \{sensitive to sunlight, plantar fasciitis\}, track\}$ .
Notice that one of the elements (the preconditions) in this assumption set is itself a set – that’s completely acceptable. On the whole, I would judge this assumption set to be a reasonable one – which is to say, the elements of the set have a probability associated with them that makes them not unexpected in the context of discussing the claim that “Running is good for you.”
Another assumption set could be
$S=\{15 hours per day, at 120\% of maximum heart rate, \{multiple hip replacements, torn Achilles tendon\}, Interstate Highways\}$ .
On the whole, I would judge this assumption set to be very unreasonable – which is to say, it is highly improbable that this would be an assumption set on which the claim “Running is good for you” would be a relevant discussion.

Reasonable Answers (Approximately True?)
A reasonable answer to a question can be defined as a statement that is true when evaluated with a plausible assumption set. That is to say, the assumption set is comprised of elements that have probabilities high enough that if we observed them we would not be surprised. In situations where the variables are in constant flux, the approximate truth value of a statement may be argued as the one that holds given the most likely assumption set. In cases like this, we may generalize a statement as true, while being willing to challenge it in the face of a game-changing assumption set. We maybe won’t talk about who gets to define “plausible”, even though there are times when that becomes the most relevant thing.

Arguing(?) With an Open Mind
Here I have chosen to use the word “arguing”, even though in truth I prefer the word “discussing”. That’s because most people seem to think that discussions between people in disagreement need to be arguments. I disagree. Remember the assumption that we are not right about everything? And remember the assumption that knowing the truth is inherently valuable? These two should premise every discussion we enter into. So when discussing the answers to questions, or the truth about statements, we need to do our best to remember that what we are trying to do is get as close to the center of the sphere as possible, because that is a good thing to do, and because we may not be there yet.

Of course, we all think we are closer than an opponent. If not, we wouldn’t be having the discussion in the first place. But keeping in mind that if two people are in disagreement, one of them must be wrong, a productive conversation is one where at the end of it the parties have converged on something they both hold to be as close to true as they can see getting. When this happens, the world gets a win. I’ll list some techniques for true open-mindedness.

Discussing With an Open Mind

Remember that you might be wrong.
Put another way, be willing to change your mind, or adjust the approximate truth of what you believe.
See, you believe that you are probably right. You may even believe that you are certainly right (although for the truly reflective, certainty is a pretty difficult thing to attain). But your opponent has the same thoughts. Both of you probably have many reasons for these. And they probably have a lot to do with assumption sets, and which one of you is applying the most plausible set. Sometimes the discussion is not about the truth of the statement but on the plausibility of the assumption set. Keep that in mind. Yours may be the less plausible. Or maybe both assumption sets are equally plausible, in which case the statement can be split into two (or more) more detailed statements that include some of the differing assumptions explicitly. But keep in mind that emotional attachment to an assumption set can and will blind you to the plausibility of an alternate set, and ultimately cause you to refute a statement with unreasonable (even fanatical) obstinacy.
Have higher expectations for yourself than you do for your opponent.
This means you need to challenge yourself to inspect the assumptions and claims of yourself and your opponent objectively, even if they are not doing the same thing. When you do this – and do it out loud – they hear that. Look at elements of the assumption sets and objectively evaluate their probability. Also evaluate whether they change the truth value of the statement or not. And be prepared to evaluate whether or not they render the statement as nonsense – this happens surprisingly often but it’s not obvious until it is isolated. Discussing things this way models a behavior that is necessary for the two of you to converge on a conclusion you both agree with. And if you are consistent with it, your opponent will often adopt the same style, if only because they think this is the way to convince you they are right.
Thank your opponent, regardless of the outcome.
I don’t mean this as a politeness. I mean this in the most sincere sense. Any opportunity we get to reflect on our set of beliefs is valuable. Sometimes your opponent and you will converge. Sometimes you will not, and they leave the exchange completely unmoved, perhaps even claiming “victory”. This is sad, since the only true victory would be a convergence of opinion, but ultimately it is not relevant to your own experience. Make it so that if you have moved on a topic, it is because you discovered something you were not considering, or were considering incorrectly, and now you are closer to the center of the sphere of truth. If you do not move, make it because you were not presented with any strong evidence that you needed to. In either case your beliefs will have been strengthened in some way, either because you changed to something as a result of new insight, or because you were challenged in some way, and it was unsuccessful. For this you have your opponent to thank.

How to Spot Real Open-Mindedness
Many people claim to be open-minded. It may be true, or it may be a trick (some people say it so that when you fail to convince them of something it will prove they were right). True open-mindedness doesn’t mean you are ready to believe anything. It means you are willing to change your mind when presented with evidence that objectively compels you to do so. If you know of (or are) someone who has changed their mind in the moment, during rational discourse, but who was fairly slow to do so, they are probably the type of person I am describing. This goes back to my point that we are probably not right about everything we believe. Which means mind-changing can occur. Which means if you’ve seen it occur, it occurred in someone with an open mind.

Thanks for reading,

Rich

Switch the Side, Switch the Sign?!!?

You can always judge my level of incredulity by my combination of punctuation. Two exclamation marks bookended by two question marks is a high level indeed. It’s the DEFCON 1 of incredulity. It comes from the way I’ve seen a lot of my students conditioned in algebra. It’s extremely sad. I’ll explain, but I have to start with a story about something that happened last week.

Thursday night I came home from play rehearsal and got the debrief about the household goings-on from my wife. Turns out my daughter, my 11-year old angel of happiness, was crying for something like 2 hours while I was gone. Dads out there with daughters (and I guess daughters with dads?) will know that there’s a special thing going on between dads and their girls. I think the best quote I ever read to describe it was this one:

“Certain is it that there is no kind of affection so purely angelic as of a father to a daughter. In love to our wives there is desire; to our sons, ambition; but to our daughters there is something which there are no words to express.”
– Joseph Addison

So when I find out that my daughter was so miserable, a part of me curls up in the fetal position and cries too. But you want to know what made it worse? What made it even more upsetting? The source of her pain was … algebra. Algebra! Her first algebra. A person’s first exposure to algebra should be special. It should be life-defining. It should be a cherished memory that warms you on command. A polished jewel of contentment forever residing at the center of your soul. And yet here’s my daughter, crying for 2 hours, and it was because of algebra. And I wasn’t home.

Talk about two ways to break my heart. Unacceptable!

So I dug a little deeper. My wife told me that my daughter was doing question after question, getting them right, and crying that she didn’t get it. My wife, no slouch in the math department for sure, was explaining the process, but for some reason it wasn’t getting through. My daughter just kept insisting she didn’t get it, all the while getting questions correct. How does this make sense? In what universe can a child keep getting questions right and through tears insist she doesn’t understand? The answer is because she was just following orders. The old “Switch the side, switch the sign” gambit. Consider this question she was working on:

x + 3 = 7

x = 7 – 3

x = 4

Correct, right? And easy too? So why was she so miserable?

It’s because she had no idea why she was doing what she was doing, or what any of it meant, and certainly no way to tell if her answer was right. She was right by accident, and because she was following a bunch of rules. In short, she wasn’t doing any math at all. She was, for all intents and purposes (or for all “intensive purposes” if you’re one of those people who hears sayings but never sees them written down) a trained monkey repeating a task. And nobody wants that … except for maybe the monkey because they get a lot of rewards for stuff like that … but no human wants that. And my daughter is exceedingly human.

You see, here’s what she was taught (or at the very least, to give her teacher the benefit of the doubt, it’s what she thought she was taught): To get the variable alone you have to move the number to the other side. If it’s plus you do minus and if it’s times you do divide. If it’s minus you do plus and if it’s divide you do times. Ouch. So much wrong with this I don’t even know how to start. But I do know that it’s taught this way so often that I have students who think that’s what algebra is. And I know of one teacher who used to have her students repeat the mantra “Switch the side, switch the sign.” I used to ask those students what they do if it’s multiplication or division. They said that’s when you don’t switch the sign. Ooooookay then.

So here’s the thing. When students are taught “rules” for solving equations they are not learning math. They are learning algorithms. An algorithm is a sequence of steps you follow to complete a specific task. You don’t need to know why, and in fact the reason algorithms are so powerful and so common is because you don’t need to know why. Long division is an excellent example. Many of us remember how to use long division to get the answer to 97654 divided by 7. But how many of us know why it works? The algorithm was designed to turn humans into calculators so that mathematicians didn’t have to do the tedious work. In WW1 there were literally rooms full of people – called calculators – who would do repetitive tedious calculations assigned to them by codebreakers. The codebreakers knew why the calculations were required, but the volume of work to do it was so vast that if the breakers themselves were to do the work they’d never decode a single message. So calculators were invented. They were people. A lot of them. And they were good at algorithms. But they didn’t know any math. Of course nowadays we have little computers that do the same job, but the concept is the same. A computer doesn’t think – it only follows instructions. It is excellent at executing algorithms. BUT THAT ISN’T MATH!!!

Ok. Back to the algebra, and my daughter. Saturday morning we were sitting in the family room, still in our pajamas, and Phineas and Ferb had just ended. The time was right. I told her I was going to help her with algebra but first she needed to forget everything she’d learned thus far. She happily agreed and put the misery in some invisible incinerator. Ahhh. Square one. Then we had this conversation:

Me: “Imagine I split your class up into groups of 2. Pick a partner.”
Her: “Alyssa!” (this was meant to be obvious to me)
Me: “Ok, now every pair has to pick one person for the blue team and one for the red.” (blue is her favourite colour – I’m not a rookie)
Her: “Blue!”
Me: “Ok, we’re going to play a game. Blue team goes first. Here’s the game. Pick a number but not a hard one. Don’t tell red team what it is. Now your job is to give Alyssa one hint, and if she gets it right you both get a point. Otherwise nobody scores.” (yeah, lame game, I know – but there are points and a blue team so she’s right on board)
Her: “Ok. My hint is it’s my favourite number.” (I saw this coming a mile away, and had a plan)
Me: “Right, Ok. But here’s the thing. You want to be SURE Alyssa will get it right. What if she can’t remember what your favourite number is? You want to give her a clue that will work for sure. And no using your number in the clue!”
Her: “It’s my birthday.” (Ha! I saw that coming too)
Me: “What if Alyssa forgot your birthday?”
Her: “How could she? She’s coming to my party!”
Me: “Good point. But what if she thinks your birthday is on a different day than the party? After all it is. You want to be sure she’ll guess your number, so make the hint foolproof.”
Her: “Ok, I get it. My hint is my number is 5 less than 10.”
Me: “Awesome! Ok, I’m Alyssa. Is it 5?”
Her: “You knew that already because you know my favourite number.”
Me: “Good point. Ok, here, it’s red team’s turn. If you increase my number by 7 you get 12.”
Her: “Hey you can’t pick the same number as me!” (Success!!!!)
Me: “Yes I can I can pick any number I want. Ok your turn.”
Her: “If you cut my number in half you get 10” (ooooh, nice one)
Me: “20?”
Her: “Yes! Ok, give me one now!”
Me: “If you multiply my number by 2 and then add 1, you get 13.”
Her: “6?”
Me: “6? Why 6?”
Her: “Because 6 x 2 is 12 and 12 + 1 is 13.”
Me: “Nice! Ok, you go.”
Her: “If you add 67354 to my number you get 90543.”
Me: “Ummm, are you going to know if I got it right?”
Her: “No. But you always get it right so I want to know the answer.” (Love her – I always get it right? She needs to talk to my wife!)
Me: “How am I supposed to get it? Those numbers are huge!”
Her: “Just do 90543 minus 67354!”
Me: “Right. I knew that. Ok it’s 23189.” (I rock at doing subtraction in my head – blows her away every time – she checked with a calculator)
Her: “Yes. Nice one Daddy.”

This went on for a long time. She really liked the game. At some point she realized that I was not keeping score and she got mad at me. Then she realized that it would always be a tie so she said it was not a good points system. We spent some time coming up with a better points system. She came up with something. It was fairly convoluted and had to do with blue team’s ability to do an aerial cartwheel so I lost, but I’m comfortable with that.

Then I told her a story about a dude from a long time ago named al-Khwarizmi, who wrote a book called “Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala“ (no, I don’t have the name of the book memorized – I always have to look it up – but I always know the al-Jabr part). I said the name of the book is a pain to say so a lot of people just called it al-Jabr. I told her that al-Khwarizmi’s book was all about ways to answer questions like the one in the game if you were not good enough to do it in your head. Take for example the clue

“If you increase my number by 7 you get 12”

al-K (that’s what his peeps called him I bet) would have said the clue slightly differently. He would have said

“Suppose thing, increased by 7 ducats, results in 12 ducats.” (take a moment to explain that ducats are kind of like dollars)

Then he would have said that you can solve it by reversing the increase, and conclude

“Therefore thing is 12 ducats reduced by 7 ducats, which is to say 5 ducats”

Then we did a few that way. I’d write “Thing, multiplied by 6, results in 18 ducats” and she’d write “Therefore thing is 18 divided by 6, which is to say 3 ducats.”

Now some people might not believe that you can ask and expect an 11-year old to use language like this, but I’ve never understood why people would think that. Speak to them this way and they will listen, understand, and respond in kind. It’s what we’re wired to do. It’s how we learn to communicate.

I should mention that at one point she asked me if al-K called his book al-Jabr because it sounds like algebra. I told her that algebra sounds like al-Jabr because of that book! That al-K invented algebra. She thought that was super cool but also wanted to know how I could know such a thing. She was amazed to find out that I studied some history in my life. Daddy points scored.

Ok. So this gets tedious right? My daughter agreed. It’s too much writing. So then I told her about a dude named Rene Descartes who really liked al-K’s methods, but was too lazy to write it all out that way. So he’d look at the sentence

“Suppose thing, increased by 7 ducats, results in 12 ducats.”

And he’d say for example that “thing” is too many letters to write, but it’s important since it’s the number we’re trying to get people to guess. So Descartes chose the minimum number of letters possible. One. I let her choose the letter. She chose “m”. She always chooses “m” when letter-choosing is the task at hand. Then I told her that “increased by” is a pain to write out also, and asked her what she thought Descartes would say instead. She wrote down “+”. Then I said what’s “results in”? She wrote “=”. And Voila! She had written

m + 7 = 12

Then without me saying anything else, she said “Oh, so then Descartes would write m = 12 – 7! Which is 5!”.

And honestly, with that the lesson was done. I gave her about 8 more equations to solve in the Descartes style, and she got them all. We never once discussed rules, and we never once switched a bloody sign.

And there were no tears.

Maybe I’ve rescued the algebra memory.

Thanks for reading,

Rich