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.

Bucky Fuller, Where are You? (On the Boxiness of Corporate Employment)

“Okay, but…if you had to choose between calling yourself a designer or calling yourself an engineer, which would you choose?”

Specialists and Generalists

I have often needed a specialist to do a specific task for me. This is normal. Specialists have a role in the economy and one could argue (along with Adam Smith) that specialization is the very basis of economy.

But too much specialization comes at a cost to innovative tech companies…and to creative individuals. Especially now, and increasingly – into the future…

Here’s an article in the Harvard Business Review on that topic:

Nourishing My Inner Bucky

Interviewers have often asked me how I rank myself in terms of software engineering skill. As if there were a one-dimensional yardstick upon which all engineers can place themselves.

When one is evaluated with a one-dimensional yardstick, one usually ends up with a low grade.

For the same reason that there are multiple dimensions to intelligence, why not use more than one yardstick to evaluate an engineer?

The space that lies between all these one-dimensional yardsticks yields great connective knowledge. This is the domain of the COMPREHENSIVIST.

I lament the boxiness of the standard company recruiting process – even within companies that claim to employ people who think outside the box (like Google). Here’s a Google employee admitting to their deplorable interview process); “Pablo writes that his best skill is product design, but that his Google recruiters only showed interest in his ability to code.”

We hear of how generalists and right-brain thinkers are in such demand these days.

Bullshit. When it comes to finding employment in companies, we are still confronted with an array of boxes, and we are still expected to show how well we fit into (one) of them. Consider Linked-In.

My Linked-In profile has the following as my “industry”:

SHIPBUILDING

Why did I choose Shipbuilding? LinkedIn REQUIRES that I choose ONLY ONE of the industries from its list, and it DOES NOT allow me to choose more than one industry. Shipbuilding was the furthest thing I could find from what I do. Instead of trying to use a single box to characterize myself, I prefer to go in the opposite direction.

Linked-In = Boxed-In

Now I want to say a few things about being an older person who has faced difficulty fitting into the workforce.

We Are All Multi-Dimensional – Increasingly as we Age

Experienced (i.e., older) programmer/innovator/designers should be contributing more of those intangibles to the tech industry that Google is so bad at seeking out.

The tech industry has a fundamental problem: software plays an increasing role in people’s lives. The world’s population is aging. Young engineers who know the latest buzzwords of the last five years are hired quickly and eagerly. An aging population tries to keep up with fast-changing software interfaces. And more and more of this aging population consists of software engineers who have something the young programmers don’t have: wisdom, experience, perspective.

We are exactly what Silicon Valley needs.

No one in particular is to blame for ageism in high-tech startups. The problem does not stem from any particular favoritism of young people: it is due to the short-sightedness of the tech industry, and the emphasis on the quick-thinking, risk-taking attributes associated with youth.

People who are professionally multi-dimensional should play a key role in human-centered software design. The cultural divide, identified by C. P. Snow in 1959, is still with us. Boxes breed boxes. That’s why we’re in the box we’re in.

-Jeffrey

Hunter Gatherer Programmer

Which side of the corporate corpus callosum are you on?

Ian McGilchrist gave a nice lecture, animated by RSA Animate about the “Divided Brain” – and how it created Western society. The mediating/inhibiting influence of the corpus collsum between the two brain hemispheres has become weakened, allowing the logical, linear left to dominate over the sensory, panoramic nature of the right.

The high tech culture of programmers has become, in my mind, the epitome of society’s left brain, and it is ghettoizing the right brain.

Let me toss out an idea: Programming skill shouldn’t be based on how good one is at manipulating numbers. Programming skill should primarily be about finding (and making) patterns, seeing connections, and using metaphors.

Computers are famously good with numbers, memory, and repetition, so why should programmers have to be good at these things too? Originally, when the computer age was young, programmers had to be sort of computer-like, just in order to build the damn things. I would contend that the culture really needs to change now that software runs so much of our lives. Programmers should be spending more time engaging in meta-math: creative pattern-finding, and building tools that match our human-like thinking; the thinking that comes from brains evolved to hunt, gather, play, explore, and build.

Our lives are increasingly dependent on software. I believe the wrong people are writing the software that runs our lives. The priests of high tech are extremely good at linear thinking (and often also at manipulating money and laws). Programmers are generally good at computation, and holding many levels of complex logic in their heads. These people have a high tolerance for software complexity.

No wonder software is so hard to use.

Hunter-gatherer skills deal with a different kind of complexity – the kind of complexity that characterizes the nonlinear world we live in. It requires all of our senses: sight, sound, touch, smell, balance…all merged unconsciously to form intuition. This intuition gathers environmental clues and builds context. These skills are ignored in most of our interactions with software. We are required to remember volumes of passwords and navigate geeky user interfaces with poor affordances. Many of these interfaces change every few months.

There is an under-appreciated range of people working on the periphery of the high-tech software industry. They know that there is a problem; they know they can help make it better. But they are on the wrong side of the corporate corpus callosum. They are in the ghetto. In order to make the situation better, they need to be empowered. They need to be on the inside. 

Being dyslexic, poor at math, slow at solving puzzles, distracted, and easily frustrated with nonintuitive tools should not keep people from participating in the software development process. In fact, I think these are the very people who are most needed. These are the people who will make software interfaces resonate with humanity at large.