When ‘smart’ calculators make us dumb

Mathematics is not made to be mechanised

What does the ‘smart’ in smart calculators actually mean? And does it make us humans any smarter?

PhotoMath alleges to be “the world’s smartest camera calculator and math assistant!” That statement alone qualifies it as the perfect case study.

Impressive speed and accuracy

The app is delightfully simple to use; so relevant in the age of instant multimedia. You just need to jot down a maths problem and hover the camera over your scribbles. The app does the rest: when I wrote down a pair of simultaneous equations, PhotoMath snapped the symbols and delivered a solution with impressive speed and accuracy. So far so good.

Next up:

+ = 100

PhotoMath yields the goods once more, sketching the geometric representation of my equation:

It was left to me to infer that the solutions correspond to the points on the circumference — the assistant might have made this more explicit. But let’s not split hairs.

A faltering knowledge base

Time to take it up a dimension:

+ = 216

A slightly devilish play on my part as I know full well, courtesy of Fermat’s Last Theorem, that this equation has no non-zero integer solutions. What does PhotoMath have to say?

“We cannot solve this problem yet. But we will soon!”

(Is this the royal ‘we’?)

My virtual assistant is beginning to exude a touch of, dare I say it, complacency. I hope it knows not to bother looking for non-zero integers. The innocent student may be led to believe that equations always have solutions (and easily computable ones at that) — a falsehood that PhotoMath is dangerously close to implying.

PhotoMath does seem able to deal with truth statements. Type in “6 < 7 “ and it will return “true”. Try “6 > 7” and it will tell you you’re bonkers (actually it won’t. The world’s smartest assistant is also without personality. A timid “false” is all you get.)

Here’s a statement I know to be false:

√2 ∈ Q

These squiggles say simply that the square root of two belongs to the set of all rational numbers, Q. In even simpler terms, this would mean that √2 can be expressed as a fraction. In fact, it can’t. Mathematicians know it and the proof is well worth a peruse. PhotoMath, with all its smarts, will surely know too, right? It again defaults to:

“We cannot solve this problem yet. But we will soon!”

(Why not just confess, “I don’t understand this yet”?)

By now it’s clear that PhotoMath has a severely limited knowledge base. The website confirms as much, listing the handful of representations that the app can handle, including “operations”, various types of “equations” and, for calculus, “derivatives, integrals”. A litany of procedurally oriented tasks: the calculator can only operate within a narrow strand of mathematics. It is only as smart as the limitations of automation allow.

PhotoMath’s restricted knowledge base is not its main pitfall. The main threat of ‘smart’ calculators is that they reduce mathematics to a shallow, computational form. The killer blow struck when I fed PhotoMath one of my favourite problems:

Mechanising my favourite problem

Which of the numbers e^π and π^e is larger?

A jolly question, because the answer is not immediately obvious: one of the two constants in each term is just shy of three, the other just in excess. Without a calculator at hand, ascertaining the relative size of each term is no trivial matter.

The particular solution that brings me delight involves logarithms (always a good bet when manipulating powers). The problem was presented to me in an interview, where I was first prompted to sketch the function log (x) / x (where log denotes the natural logarithm). A nice question in itself, requiring analytical thinking to examine the most important and interesting behaviours of the given function. Basic calculus helps to identify a maximum at the point x = e and, by considering small and large values of x, we find asymptotes along the two axes. Piecing it all together gives:

Next comes the question of which is larger, e^π or π^e. Can you figure out how to make use of the graph we’ve just sketched? The key insight is that because the logarithm is an increasing function (that is, it gets larger as you move along the x-axis),

e^π > π^e if and only if log (e^π) > log (π^e)

So to ascertain which of e^π and π^e is larger, we only need to compare log (e^π) and log (π^e). But, by standard properties of the logarithm, log (e^π) = π and log (π^e) = e*log (π) which means we are just comparing π with e*log (π). So

e^π > π^e if and only if π > e*log (π)

We need to somehow relate these terms back to our graph. It should strike you as noteworthy that the function log (x) / x obtains a maximum at x = e. In particular, this means its value at x = e is larger than its value at x = π. More explicitly:

log (e) / e > log (π) / π

which can be manipulated into the following expression (just multiply both sides by e*π and note that log (e) = 1):

π > e*log (π)

Tracing back our steps, we have just shown that e^π > π^e.

I belaboured the details to highlight the multidimensional scope of this problem; first confronting your sense of approximation, then some graph sketching, and finally the analytical method of comparing the logarithmic values. An hour spent dwelling on this single problem would not be an hour wasted.

And what does PhotoMath make of it all? Just ask it: try writing e^π > π^e and you will see, within milliseconds, that it is confirmed “true”. How does PhotoMath know? The only way it can — by computing the values (in fact, it relies on an approximation, but sheds no light on how) and comparing their size.

A discovery robbed. And yes, I must charge my phone.

A tale of two mathematics

Contrast the two approaches: one is a journey through mathematical reasoning. The other is brutish computation. One arouses our liveliest curiosities; the other dulls it into submission. One guides you towards discovery while the other robs you of it.

Even as Photomath fills its knowledge gaps and converges towards omniscience, its ways of knowing mathematics will be limited by the absence of mathematical reasoning. PhotoMath is a particular brand of ‘smart’, and one that makes for a woeful assistant.

Technologies like PhotoMath come wrapped in the promise that they will augment our human intelligence by liberating us from computation. There’s no doubt that, in the right hands, tools like PhotoMath can be put to good use. Dan Meyer, after detailing the app’s many pitfalls, ironically wished it success on the grounds that it could make mundane, repetitive tasks redundant (we can hope). Some educators are biting, rethinking homework tasks in light of ready-made solutions.

My main concern is that the ubiquity of solution-grabbing tools, when left unchecked in the hands of students seeking the quickest answer, will mechanise large chunks of mathematics, denying us the fruits of rigour, reasoning and elegance. Without deliberate application, they reduce us to their own computational selves.

An effective ‘assistant’ would guard against this. They would behave in ways that complement our modes of thinking rather than dilute it. They would engender exploratory learning experiences rather than reaching for blunt outputs. They would empower educators with explicit guidance on how to integrate these tools within their practice.

That sounds like the smarter play to me.

I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.

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