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

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Intelligence is NOT One-Dimensional

Why do so many people, including science writers, talk about intelligence as if it could be measured on a one-dimensional yardstick?

In “How We Evolve” Benjamin Phelan discusses the work of Bruce Lahn, who did controversial research on genetic differences among human populations that are correlated with brain size and brain function. At one point, discussing natural selection in contemporary humans, Phelan states, “…if intelligence is still under selection, that could mean that some populations at this very moment are slightly smarter than others – that, perhaps, some ethnicities are slightly smarter than others.”

Phelan is wise to be cautious and skeptical in how he reports on this subject. Basically I think this is a great article. But, like so many other writers, he makes an error in his choice of words. The use of the term “smarter”, is misguided…it is moot. The very notion that any group of humans could be “smarter” than another group is unfounded.

I would bet that this kind of misguided language has caused further aggravation to an already controversial subject.

I made the image above to express my understanding of intelligence as having several components, or modalities, with interpersonal included at the left. This shows just three modes, plotted in a cube – but there are many others (see below). We could see certain disorders, such as autism, dyslexia, and Williams Syndrome as examples of extreme imbalances in the mix of intelligences. An autistic savant might be plotted at the lower right, while a Williams might be plotted at the far left. Most of us have relatively normal balances, with plenty of mild variation. And NOBODY has super-powers in all modalities, as indicated by the absence of people in the upper-right corner.

There’s Really No Such Thing as “Smarter”

The term “smarter” is even less applicable when used in relation to technology. In the article “Is Google Making Us Stupid?“, Nicholas Carr quotes Larry Page in a speech, as saying:

“The ultimate search engine is something as smart as people – or smarter”. 

I applaud the goal of making better search engines. But software cannot and should not be measured against humans in terms of intelligence. I will repeat what I have said in other blog posts: intelligence (both human and artificial) is

MULTI-DIMENSIONAL

Changing our language to reflect this fact would alleviate so many of the conflicted debates we are hearing about the “dangers of AI“.

 

Are we over-thinking the dangers of AI?

Artificial Intelligence comes in many forms – just as natural intelligence comes in many forms within the animal kingdom and among human populations. The diversity of intelligence in technology is what keeps us safe from a runaway AI monster.

Diversity is healthy.

Now, why am I making such a big deal about a little bit of language? I am making a big deal because this little bit of language is the tip of an ugly iceberg: it is the cause of discrimination in the tech industry; it is the cause of discrimination in general; it is the reason people still use the IQ test, which falsely reduces one’s intelligence to a single number, so that person A can be called “smarter” than person B. And person B can be called “smarter” than person C.

IQ is not just a flawed concept: it is counter-productive.

The notion of IQ is MISLEADING.

Howard Gardner proposed several kinds of intelligences. Among the intelligence modalities associated with Gardner’s theories are:

Musical–rhythmic and harmonic
Visual–spatial
Verbal–linguistic
Logical–mathematical
Bodily–kinesthetic
Interpersonal
Intrapersonal
Naturalistic
Existential

We could easily add more, or combine some of these. We might also include “emotional”, “symbolic”, and “narrative“.

I would even add “dyslexic” (usually considered a disorder but increasingly recognized as associated with certain skills that are advantageous in many situations).

Maybe I’m just playing with semantics – maybe I’m just being a language wonk. But I don’t think so. I think the language we use to describe ourselves and others has a major effect on how we think and how we act. Changing the way we talk about intelligence could have a positive trickle-own effect on things as widespread as public policy, education, racism, scientific research, and…gosh, just about everything else.

We’re all SMART.

SMART is multidimensional.

IS “ARTIFICIAL LIFE GAME” AN OXYMORON?

(This is a re-posting from Self Animated Systems)

Artificial Life (Alife) began with a colorful collection of biologists, robot engineers, computer scientists, artists, and philosophers. It is a cross-disciplinary field, although many believe that biologists have gotten the upper-hand on the agendas of Alife. This highly-nuanced debate is alluded to in this article.

…

Games

What better way to get a feel for the magical phenomenon of life than through simulation games! (You might argue that spending time in nature is the best way to get a feel for life; I would suggest that a combination of time with nature and time with well-crafted simulations is a great way to get deep intuition. And I would also recommend reading great books like The Ancestor’s Tale :)

Simulation games can help build intuition on subjects like adaptation, evolution, symbiosis, inheritance, swarming behavior, food chains….the list goes on.

On the more abstract end of the spectrum are simulation-like interactive experiences involving semi-autonomous visual stuff (or sound) that generates novelty. Kinetic art that you can touch, influence, and witness lifelike dynamics can be more than just aesthetic and intellectually stimulating.

These interactive experiences can also build intuition and insight about the underlying forces of nature that come together to oppose the direction of entropy (that ever-present tendency for things in the universe to decay).

On the less-abstract end of the spectrum, we have virtual pets and avatars (a subject I discussed in a keynote at VISIGRAPP).

“Hierarchy Hinders” –  Lesson from Spore

Will Wright, the designer of Spore, is a celebrated simulation-style game designer who introduced many Alife concepts in the “Sim” series of games. Many of us worried that his epicSpore would encounter some challenges, considering that Maxis had been acquired by Electronic Arts. The Sims was quite successful, but Spore fell short of expectations. Turns out there is a huge difference between building a digital dollhouse game and building a game about evolving lifeforms.

Also, mega-game corporations have their share of social hierarchy, with well-paid executives at the top and sweat shop animators and code monkeys at the bottom. Hierarchy (of any kind) is generally not friendly to artificial life.

For blockbuster games, there are expectations of reliable, somewhat repeatable behavior, highly-crafted game levels, player challenges, scoring, etc. Managing expectations for artificial life-based games is problematic. It’s also hard to market a game which is essentially a bunch of game-mechanics rolled into one. Each sub-game features a different “level of emergence” (see the graph below for reference). Spore presents several slices of emergent reality, with significant gaps in-between. Spore may have also suffered partly due to overhyped marketing.

Artificial Life is naturally and inherently unpredictable. It is close cousins with chaos theory, fractals, emergence, and uh…life itself.

Emergence

At the right is a graph I drew which shows how an Alife simulation (or any emergent system) creates novelty, creativity, adaptation, and emergent behavior. This emergence grows out of the base level inputs into the system. At the bottom are atoms, molecules, and bio-chemistry. Simulated protein-folding for discovering new drugs might be an example of a simulation that explores the space of possibilities and essentially pushes up to a higher level (protein-folding creates the 3-dimensional structure that makes complex life possible).

The middle level might represent some evolutionary simulation whereby new populations emerge that find a novel way to survive within a fitness landscape. On the higher level, we might place artificial intelligence, where basic rules of language, logic, perception, and internal modeling of the world might produce intelligent behavior.

In all cases, there is some level of emergence that takes the simulation to a higher level. The more emergence, the more the simulation is able to exhibit behaviors on the higher level. What is the best level of reality to create an artificial life game? And how much emergence is needed for it to be truly considered “artificial life”?

Out Of Control

Can a mega-corporation like Electronic Arts give birth to a truly open-ended artificial life game? Alife is all about emergence. An Alife engineer or artist expects the unexpected. Surprise equals success. And the more unexpected, the better. Surprise, emergent novelty, and the unexpected – these are not easy things to manage…or to build a brand around – at least not in the traditional way.

Maybe the best way to make an artificial life game is to spread the primordial soup out into the world, and allow “crowdsourced evolution” of emergent lifeforms.  OpenWorm comes to mind as a creative use of crowdsourcing.

What if we replaced traditional marketing with something that grows organically within the culture of users? What if, in addition to planting the seeds of evolvable creatures, we also planted the seeds of an emergent culture of users? This is not an unfamiliar kind problem to many internet startups.

Are you a fan of artificial life-based games? God games? Simulations for emergence? What is your opinion of Spore, and the Sims games that preceded it?

This is a subject that I have personally been interested in for my entire career. I think there are still unanswered questions. And I also think that there is a new genre of artificial game that is just waiting to be invented…

…or evolved in the wild.

Onward and Upward.

-Jeffrey

Our Colorful Mathematics Revolution

Education bureaucrats are trying to gently and safely tweak a broken system so that fewer students fail math.

Meanwhile, a colorful revolution is taking shape outside the walls of a crumbling institution. A populist movement in creative math is empowering an unlikely crowd.

Authors of Wikipedia math pages aren’t contributing to this populist movement. They are intent on impressing each other; competing to see who can reduce a mathematical concept to its most accurate, most precise (and least comprehensible) definition.

A debate rages on a “new way” to do subtraction. Oh does it rage. But step back from that debate and consider that these tricks, algorithms, processes, hacks, become less relevant as new tools take their place. When calculators entered into the classroom, something started to change. That change is still underway.

Do students no longer need to learn to do math by hand? No. But calculators (and computers) have changed the landscape.

Rogue amateur mathematicians, computer artists, DIY makers, and generative music composers are creating beautiful works of mathematical expression at a high rate – and sharing them at an even higher rate. This is a characteristic trait of the “new power“.

Technology

(1) Computers are better at number-crunching than we are. If used appropriately, they can allow us to apply our wonderfully-creative human minds to significant pattern-finding and problems that we are well-suited to solve.

(2) Computer animation, generative music, data visualization, and other digitally-enhanced tools of creativity and analysis are becoming more accessible and powerful – they are helping people create mathematically-oriented experiences that not only delight the senses, but express deep mathematical concepts. And they also help us do work.

(3) The internet is enabling a new generation of talented people (amateurs and professionals) to exchange mathematical ideas, discoveries, and explanations at a rate that could never be achieved via the ponderous machinations of university funding, publishing, and teaching. There will never be another Euler. Mathematical ideas now spread through thousands of minds and percolate within hours. It is becoming increasingly difficult to trace the origins of an idea. Is this good or bad? I don’t know. It’s the new reality.

Five things You Need to Know About the Future of Math

According to Jordan Shapiro:

1. Math education is stuck in the 19th Century.
2. Yesterday’s math class won’t prepare you for tomorrow’s jobs.
3. Numbers and variables are NOT the foundation of math.
4. We can cross the Symbol Barrier.
5. We need to know math’s limitations.

We can (and will – and should) debate how math should be taught. Whether the “symbol barrier” is a actually a barrier, and whether memorizing the multiplication tables is necessary, no one can ignore the seismic changes that are rumbling underfoot.

-Jeffrey