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The Case for Resilient Math

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Reaching the Square of Orthodox Resource Optimization, Vincent did what he had fantasized about for months: he began drawing a pelican directly onto the bleak, renewable, and emission-neutral bioslabs.

“Sluggish,” Vincent murmured as three minutes passed. “It took them three minutes to register me.”

Two workers from a Productive Unit approached. “You are violating the Ministry’s Optimized Restoration Window strategy. Having fun like this is prohibited. Explain yourself immediately or you will be prosecuted as a Protruder.”

“I am well aware,” Vincent declared. “But you won’t stop me. I am a reflective citizen!”

The workers burst into synchronized laughter. Without further dialogue, they processed Vincent into the Improvement and Correction Facility. There, he was sentenced to a mandatory Self-Objectification Course via extreme social media exposure, designed to permanently normalize his Perpetual Anxiety Index.

thinker Image source: Picryl

Idolic Erosion

Even though the story of Vincent is unlikely to happen in reality, we seem to live under a constant pressure of economic output and efficiency. That reality is stressful in and of itself, only to be amplified by the rapidly expanding industry of generative AI.

There are even companies, such as Microsoft, that consider it appropriate to threaten their own employees with losing their jobs in the near future.

To measure how well future generations of workers will fit the role of spreadsheet managers1, many tests and assessment frameworks have been created. One of them is the Programme for International Student Assessment (PISA), operating under the OECD:

Student performance (PISA) shows the extent to which students have acquired key knowledge and skills essential for full participation in social and economic life.

In the context of mathematics education, we are often met with ambitious reforms that introduce real-life skill applications, or competences. These, in turn, closely relate to the notion of mathematical literacy, which the OECD defines as:

… students’ ability to reason mathematically and to formulate, employ, and interpret mathematics to solve problems in a variety of real-world contexts.

Arguably, feel-good definitions of mathematical literacy are not actually that far removed from emotional intelligence: we also lack a consensus on what mathematical literacy even means and how it should be measured. As a result, we end up in a debate where a multitude of people use the same words but mean completely different things.

The tension between potentially worthwhile ideals and the reality of their execution was analyzed in 2009 research describing this exact discrepancy. Most of the mathematical literacy exam questions required factual information and mechanical computations, falling far short of “problems in a real-world context.” The paper also suggests that the structure of the test itself outlines the methods students should use to solve a given problem.

Country rankings in mathematical literacy are just one of the many ways politicians and governments justify their policies or the “unsuitability” of the previous government. PISA initially attracted a great deal of media attention because it demonstrated that we can easily assess and measure students from different countries by reducing them to a number. Naturally, such comparisons lack the social or cultural nuance that should be of high importance in pedagogy.

vanity fair Image source: Picryl

This touches on one of the biggest paradoxes of “practical education”: by definition, mathematical literacy, as defined by the OECD, cannot be standardized. Even if we were to define mathematical literacy merely as an approximating indicator of mathematical capabilities within a single cultural context2, we still face physical limitations.

Brave Real World

Any real-world example is real to only a few people3. Financial literacy is a brilliant example of this. Not every teenager wants to be involved in “computing the loan total, given the interest rate of X and a loan amount of Y.”

Apart from apathy—which is admittedly not a reliable indicator of pedagogical utility—we are also subconsciously telling students that percentages are used for money, or worse, that percentages are used when I owe someone something. Is insisting on realistic examples that are often anything but realistic, including the contrived loan example, really a goal worth pursuing?

If your answer is:

Don’t be silly, you know what purpose such examples serve: the educational system should show you the basics you can further improve and elaborate on.

…then we are entering a perpetual debate over what is basic:

  1. How often should we define what constitutes basic? Is someone responsible for gatekeeping the correct level of basicness? Is there one basic set of knowledge for everyone?
  2. How and where can you learn why a specific piece of information or competence is considered basic?
  3. Is basicness related to religion, ethnicity, culture, and the language being used, and if so, how?
  4. Why is learning the basics of high value, specifically? How can we verify that we have covered all the basics?
  5. Are there basic pieces of knowledge that do not necessarily apply to the real world? In other words, are we equating basic with immediately practical?
  6. Once you are out of the formal educational system and have no authority telling you what is basic, what purpose does having basic knowledge serve? Are you meant or even obliged to expand it of your own free will—which you didn’t have much chance to exercise at school—or is it now okay to forget some of the basic things, since maintaining them would otherwise imply an enormous cognitive overhead?

Insisting on learning only by “touching grass” also assumes that “grass-touching is the same for everyone” and that learning is linear in nature: students first discover A, then reflect on B to give a presentation on C, ending up with competences X and Y. As effective and promising as that approach sounds, mathematics education is anything but linear.

From my own tutoring experience, every student comes to me from a different background, with varying habits and ways of thinking4. Even if every student had the same background, mathematics is by evolution non-linear: there are many abstract concepts that a student must temporarily take for granted to eventually understand them properly, meaning they often have to jump back and forth.

Evading The Specificity Snare

As we have seen, mathematical literacy and insisting on immediate practicality get us nowhere. What might work, then?

Now, you might be tempted to say:

The right approach is somewhere in the middle: a bit of theory and a bit of the real world is the golden rule.

But in my opinion, that is precisely the source of failure for many reforms.

We shouldn’t model mathematics education around a single “practicality” knob, tuning it to our immediate needs. Instead, we should offer educational resources that disregard a direct connection to the world around us; practicality should be considered before that, when we are deciding which skills and competences to focus on.5

Apprenticeship training certainly still has its place, but arguably only for highly specific specializations within the education sector. The real goal should be to get students motivated and engaged, thereby slowly building transferable competences and knowledge. Over-specifying math to one fake scenario kills any chance of transfer, whereas abstraction at least gives the concept a chance to remain flexible in the future.

young pilot Image source: Picryl

At the same time, we should avoid pompousness and vanity by stating that:

“We should abandon merchant math once and for all.”

That risks ending up the same way as PISA et al., so let’s start smaller.

One of the tasks of modern mathematics education is to engage students in ways of thinking they would be reluctant to explore on their own. That is not to say math teachers should force students into doing things they don’t want to because “teacher knows better”—quite the opposite. At the same time, formal education is a brilliant opportunity to practice skills that are difficult to attain through sheer willpower and determination alone.

It is completely fine to solve problems that are completely “impractical and useless” as long as they support overall problem solving, decomposition, and abstraction.

We should also expose students to the same concept in as many different ways as possible. Only then will they be capable of isolating the “kernel” of a mathematical object and applying it to a problem they haven’t seen before. A mechanical approach to mathematics, e.g., solving dozens of identical equations, is usually already frowned upon, but we shouldn’t stop there.6

Mainly thanks to Zbyněk Kubáček, I am also a big proponent of providing meaningful historical context, which can highlight the initial struggles and blind alleys that mathematicians themselves had to deal with. We often forget that learning from the mistakes of others, and learning how something isn’t done, can be an invaluable lesson too, if applied carefully.

Happy learning!

  1. Spreadsheet management is slowly being transformed into the undeterministic and esoteric art of “agentic workflow engineering,” but that’s a topic for a different discussion. ↩︎

  2. Admittedly, this is an honorable and undeniably complex undertaking. What I am critiquing is the real-world unfeasibility, which, given the goal of PISA, is quite ironic. ↩︎

  3. This dreamed-of reality very often makes sense only to the author of the material, which I have abundant personal experience with. ↩︎

  4. Which is the way it should be, as long as the variance is not detrimental to the student. ↩︎

  5. I am not suggesting there is one panacea that will save us all; I am also confident that there are teachers and educators who try to escape this seemingly endless schism. What I am implying is that either overreliance on mathematics standardization or excessive emphasis on practicality overlooks the messiness of the real world, which both extremes fail to address. ↩︎

  6. The Pythagorean theorem is a great example. As soon as a student encounters a right triangle embedded within a larger context, rather than the obligatory standalone “ABC” triangle, they often fail to even consider applying the theorem as a possible pathway. ↩︎

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