Math Word Problems are Problematic

Mark Twain said: Never let school interfere with your education.

Here’s a math riddle:

“Peter has 21 fewer marbles than Nancy. If Peter has 43 marbles, how many marbles does Nancy have?”

The first sentence requires me to do some linguistic fiddling. There is an implication that both Peter and Nancy possess marbles – but it is not directly stated. The second sentence begins with “If”, which means the primary grammatical elements in the question are postponed until the end. Let’s re-phrase this riddle to say:

“Peter and Nancy each have a bag of marbles. Peter has 43 marbles in his bag. Peter has 21 fewer marbles than Nancy has. How many marbles does Nancy have in her bag?”

….

This might make the riddle easier to solve. Or it might not. Either way, I can say for sure that all this wordy bullshit is irrelevant to the actual math.

Math, like Music, is a Universal Language

Now consider what it would be like if you were naturally talented in math, and you were faced with a math riddle expressed in English…but English were not your first language. You may have to spend more time on the question, and you may make some critical mistakes. The subtleties of one language may not translate to another language, causing you to trip up.

We are playing with words here. Now, playing with words is fine; it’s part of how we learn to speak, listen, read, and write. In fact, playing with words that have mathematical content is a good exercise. But this should not come into play for testing students on math skills. The problem (as always) is in the testing.

Here’s another one:

“Sue has two pencils. She spends one hour at the store and buys three more pencils. How many pencils does Sue have in all”.

WTF does “spends one hour in the store” mean? Is this just narrative fluff, or is there some clever hint in there?

If I had been presented with this problem as a young student, I would have spent some time mulling over “spends one hour at the store”. However, this is irrelevant and unrelated to the answer.

How to Obfuscate Mathematical Thinking With Clever Language

For dyslexic students, students who learn through action (kinesthetic learners), students who are visual thinkers, and students who learn best by building things, this wordsmithing can be a recipe for failure.

In the real world of adults getting things done and making a living, math is rarely experienced in the form of clever riddles. Math – at its best – is manifested deep within the texture of our daily actions.

Here’s another one:

“You have 24 cookies and want to share them equally with 6 people. How many cookies would each person get?”

Let’s think about this. I “have” 24 cookies. (That’s a lot of cookies – why would I have so many cookies?) I “want” to share them with 6 people. Okay. I have a desire to share cookies. So far so good. I’m a generous guy! But then the second sentence appears unrelated: “How many cookies would each person get?” Wait a minute: am I about to give these cookies to these people? And what exactly does “equally” mean?

I know it may seem trivial for me to analyze these details. As an adult I know what this sentence means. But as a young student, I may not have had the full vocabulary or grammatical wherewithal to jump right to an answer. Also, as a “narrative learner”, I would have really wanted to make sure I understood the characters involved, their motivations, etc. I could imagine getting easily get swept up by the storyline (simple as it is).

In short, by working out the characters of this story and their motivations, I may not actually be doing math: I might be engaging in language craft and storytelling. Which is great! But this should not interfere with my being tested on my innate math skills.

Here’s another:

“Kennedy had 10 apples. She gave some to John. Now she has 2 apples left. How many apples did she give to John?”

The tense of this little story jumps back and forth between past and present. At age 55, I am now quite facile with language, but when I was 10, I would have had to put in some effort into parsing these shifts in tense.

In fact, my language skills were quite poor when I was 10, and this had an impact on all my school subjects (not just math). Later in life, after I had escaped school and actually started to gain some relevant skills, MIT offered me an opportunity to earn a Master’s degree. They did not ask me any math riddles. MIT knows better than that.

Language

One might argue that language skills are fundamental and important for learning most anything. That’s accurate. Reading, writing, speaking, and listening are fundamentally useful, and the better you are at language, the better you are likely to become at most other skills.

If this is the case, we might conclude that mixing grammatical sentence structure with mathematical logic is a valuable skill.

Indeed.

But school curriculum designers should not confuse the ability to parse a cleverly-crafted sentence with one’s innate mathematical abilities.

The problem, as always, is with TESTING.

I’ll close with this:

An Open Letter to the Education System: Please Stop Destroying Students

Quantum Physics Has a Language Problem

I have become interested in theories of mind and all the new thinking at the intersection of physics and consciousness. So when I set out to read The Self-Aware Universe by Amit Goswami, I hoped to get a better sense of how quantum physics relates to mind.

Didn’t happen.

I also didn’t get any major insights about “action at a distance“. And most of all, I did not get any deeper insights on the idea that the act of observation can change the physical world. I’ve known about quantum mechanics for a while – enough to have a casual conversation over beer – or more likely – over a joint. But I expected that Goswami would help me get to the next level of understanding. I read the words, I followed the logic…

…but nothing ever got much farther than a few centimeters into my brain. There was no gut feeling – no somatic resolution.

Now, to be sure, I wasn’t expecting epiphanies to come tumbling out. After all, Richard Feynman famously said, “If you think you understand quantum mechanics, you don’t understand quantum mechanics.”

So, I was appropriately prepared for the difficulty of the subject matter.

What the Hell is a “Quantum Object” Anyway?

Sean Carroll says that physical theories:

“…aren’t supposed to have ambiguities … the very first thing we ask about them is that they be clearly defined. Quantum mechanics, despite all its undeniable successes, isn’t there yet.”

The main problem with explanations of quantum physics is the choice of words.

The terms “observation”, and “measurement” have particular meanings in the physicist’s lab, where a scientist might be trying to gather data on the behavior of a single photon.

Truly not something that most of us experience in daily life. Even the sight of a faint star in the night sky involves a hell of a lot of photons. And one second of this experience is actually a really long time.

But…a single photon?

I wonder if the scientist in the lab actually “experiences” a photon anyway. How does one “experience” a photon? And what does it mean to “measure” or “observe” something as fleeting and tiny as a subatomic particle?

Sean Carroll again:

“There is no consensus within the physics community about what really constitutes an observation (or “measurement”) in quantum mechanics, nor on what happens when an observation occurs.”

Another problematic term is “quantum object”. The word “object” is very familiar in classical physics. But it invites contradiction and cognitive dissonance when applied to phenomena on the quantum level.

While reading explanations on quantum physics, I become optimistic: I feel as if I am about to get a picture of why certain puzzling phenomena are true. Authors use familiar narratives and metaphors that I have direct experience with, but what they are illustrating are observations in a physics lab where fleeting subatomic particles exhibit paradoxical behaviors. These carefully-orchestrated observations that only happen in expensive laboratories are hardly the stuff of everyday experience.

And then they start talking about cats in boxes – right after telling us that cats and boxes are VERY DIFFERENT than subatomic particles.

Thanks!

By the way…apparently, it IS possible to experience the effects of quantum physics in your own home:

I just love the fact that styrofoam cups were used in this experiment.

Can Quantum Physics Ever Really Be “Explained?”

Because our sense organs and brains are optimized to deal with things on a human scale, it’s difficult for us to think about things as small as atoms (where quantum physics really matters) or as big as galaxies (where relativity really matters).

As I set out to write this article, I did some searching and noticed right away that a lot of people have pointed out that quantum physics has a language problem. And so here is where I bow out, and let the real experts speak…

Is there a Language Problem with Quantum Physics?

The Copenhagen Interpretation

So, You’re Not a Physicist…

Quantum Physics and Human Language

What If There’s a Way to Explain Quantum Physics Without the Probabilistic Weirdness?

Maybe classical clockwork can explain quantum weirdness

Programming Languages Need Nouns and Verbs

I created the following Grammar Lesson many years ago:

Like many people my age, my first programming language was BASIC. Next I learned Pascal, which I found to be extremely expressive. Learning C was difficult, because it required me to be closer to the metal.

Graduating to C++ made a positive difference. Object-oriented programming affords ways to encapsulate the aspects of the code that are close to the metal, allowing one to ascend to higher levels of abstraction, and express the things that really matter (I realize many programmers would take issue with this – claiming that hardware matters a lot).

Having since learned Java, and then later…JavaScript, I have come to the opinion that the more like natural language I can make my code, the happier I am.

Opinions vary of course, and that’s a good thing. Many programmers don’t like verbosity. Opinions vary on strong vs. weak typed languages. The list goes on. It’s good to have different languages to accommodate differing work styles and technical needs.

But…

if you believe that artificial languages (i.e., programming languages) need to be organic, evolvable, plastic, adaptable, and expressive (like natural language, only precise and resistant to ambiguity and interpretation), what’s the right balance?

Should Programs Look Like Math?

Should software programs be reduced to elegant, terse, math-like expressions, stripped of all fat and carbohydrates? Many math-happy coders would say yes. Some programmers prefer declarative languages over procedural languages. As you can probably guess, I prefer procedural languages.

Is software math or poetry? Is software machine or language?

I think it could – and should – be all of these.

Sarah Mei has an opinion. She says that Programming is Not Math.

Programming with Nouns and Verbs

First: let me just make a request of all programmers out there. When you are trying to come up with a name for a function, PLEASE include a verb. Functions DO things. Like any other kind of language, your code will grow in a healthy way within the ecology of human communicators if you make it appropriately expressive.

Don’t believe me? Wait until you’ve lived through several companies and watched a codebase try to survive through three generations of developers. Poorly-communicating software, put into the wrong hands, can set off a pathological chain of events, ending in ultimate demise. Healthy communication keeps marriages and friendships from breaking down. The same is true of software.

Many have pontificated on the subject of software having nouns and verbs. For instance, Matt’s Blog promotes programming with nouns and verbs.

And according to John MacIntyre, “Take your requirements and circle all the nouns, those are your classes. Then underline all the adjectives, those are your properties. Then highlight all your verbs, those are your methods”.

When I read code, I unconsciously look for the verbs and nouns to understand it.

When I can’t identify any nouns or verbs, when I can’t figure out “who” is doing “what” to “whom”, I become cranky, and prone to breaking things around me. Imagine having to read a novel where all the verbs look like nouns and all the nouns look like verbs. It would make you cranky, right?

The human brain is wired for nouns and verbs, and each is processed in a different cortical region.

There are two entities in the universe that use software:

(1) Computers, and (2) Humans.

Computers run software. Human communicate with it.

-Jeffrey