## Background

On February 6, 2024, the Washington Post published an article entitled, The trouble with schools is too much math, by Travis Meiers, Opinion Editor (available to those with access to the Washington Post at:

https://s2.washingtonpost.com/1d1e5b0/65c8b6e40372493a92d4f8e/597309cc9bbc0f1cdcf087fc/3/45/65c8b6e40372493a92d4f18e).

The opinion argues that the math typically taught in secondary school is “useless” to most people. The opinion asks rhetorically,

“Who needs to know about ‘imaginary numbers,’the ‘Pythagorean theorem’ or the ‘quadratic equation’”? In answer to its own question, the opinion avers: Not the author and not many others, because “[o]nly 22 percent of the nation’s workers use any math more advanced than fractions, and they typically occupy technical or skilled positions. That means more than three-fourths of the population spends painful years in school futzing with numbers when they could be learning something more useful.” While a few use math “to make bridges safe, to predict the weather, to land spacecraft on the moon and Mars — you get the idea,” for most, math is “just numbers for the sake of numbers,” an “alphabet soup” of “formulas,” “equations,” and “compilations crammed into our heads in high school long enough to pass a math test, then promptly forgotten.”

Instead of math, the opinion argues, students should be taught “applied logic,” which “grows from the same mental tree as algebra and geometry but lacks the distracting foliage of numbers and formulas.” The article contends that unlike math, “applied logic” is useful becausemit “teaches us how to trace a claim back to its underlying premises and to test each link in a chain of thought for unsupported assumptions or fallacies.”

In an email introducing the article (available to those with access to the Post at https://www.washingtonpost.com/opinions/2024/02/06/end-useless-math-requirements/), the writer insists, “I’m not here to bash math.” He knows “how important it is for innovation,

infrastructure and impressive-looking blackboards.” He adds that his “shoulder chip” against math does not “spring from a mathematical ineptitude.” Indeed, he protests, he “took advanced math for four years,” including “precalculus,” if only so that he “could test out of college algebra.” With this impressive, unassailable, mathematical resume, the writer feels confident in

concluding that most of the mathematics taught in high school is “useless.” It makes him “queasy all over again just thinking about it.”

## Response

#### The Trouble With Schools Is Too Much Education,

##### a response to the editorial opinion:

The Trouble With Schools Is Too Much Math.

##### By: Lee Kraftchick, Attorney, Arbitrator and Mathematical Tourist

B.S. Mathematics 1974; J.D. 1980.

I’m not here to bash the Post’s recent opinion piece, *The trouble with schools is too much math*, for the columnist is surely an honorable man. I wish only to follow the opinion’s reasoning to its logical conclusion, for as the opinion advocates, “applied logic” is what our schools should teach in lieu of math and we, as friends, Americans, and countrymen, must employ in the analysis of issues.

As the opinion explains, “logic teaches us to trace a claim back to its underlying premises and to test each link in a chain of thought for unsupported assumptions or fallacies.” As an illustration of how this works, let us use “applied logic” to analyze the opinion itself.

“Underlying premises” (called “axioms” or “postulates” in mathematics) are the starting points from which to reason logically to a conclusion. The Post opinion begins with four

“underlying premises”: (1) mathematics is all about numbers or “numeracy”; (2) mathematics requires nothing more than memorizing and mechanically applying formulas, which are

“crammed into our heads in high school long enough to pass a math test, then promptly forgotten”; (3) most math is of no use to the vast majority of people (“only 22 percent of the nation’s workers use any math more advanced than fractions”); and (4) schools should only teach subjects that have practical use to most people (instead of forcing students to spend “painful years in school futzing with numbers” schools should teach “something more useful.”).

No serious scientist or engineer, let alone a mathematician, would agree that (1) mathematics is just about numbers or (2) primarily involves the memorization of formulas.* Definitions of mathematics vary, but all agree it includes not only numbers, but the study of space, structure, patterns, connections, reasoning logically and proving propositions. Many topics in math do not use numbers at all, others only minimally, including geometry,

mathematical logic, abstract algebra (rings, groups, vector spaces, etc.), graph theory, topology, and set theory. Memorization of established formulas may help to solve routine numerical problems, but it is no substitute for the logical reasoning that underlies the essence of most mathematics, logical proofs.

But again, I am not here to bash the Post opinion, so let us accept its underlying premises for the sake of argument, despite their plain mischaracterization of mathematics. We must then

ask, “Where do these premises lead, using the ‘applied logic’ the opinion advocates?”

Premises (3) and (4) tell us that we should not waste time in school teaching anything that will not later be of practical use in the lives of most people. There is no logical reason why these

broad premises should be limited to the teaching of mathematics alone. We must therefore consider which other subjects typically taught in high school can be eliminated under the opinion’s underlying premises.

We begin with a list of some commonly taught subjects:

- Science, including astronomy, biology, chemistry, physics, and paleontology
- Literature, including fiction and poetry, from Shakespeare and other classics to modern novels, and creative writing
- Visual arts, including drawing, painting, and sculpting
- Performing arts, including singing, dancing, acting, and playing an instrument
- Physical education, including exercise and sports from baseball to volleyball
- Foreign languages
- History and social studies
- English spelling and grammar

We can quickly and easily eliminate many of these subjects as of no practical use to most people (applying the opinion’s 22% standard). Few people use what they learn in science; who

remembers the elements of the periodic table, the precise relationships between gravity, mass, inertia, acceleration and speed, the properties of light and radio waves, the structure of DNA, the names of the moons of Jupiter, or how planetary orbits are calculated? I do remember the names of a few species of dinosaurs, but they seldom come up in my day-to-day life. Who remembers much from Shakespeare or other classical literature, let alone uses what they remember regularly? Few people make a living in the visual or performing arts. We may enjoy them, but

did we need to learn about them in high school? How many people would be ignorant of the pictures, music, movies, and television shows they like if they had not been exposed to them in school? Similarly, many people enjoy sports not because they learned about them in Phys. Ed., but because they played them after school with family and friends. My father taught me how to

ride a bike, swim, throw, kick and catch a ball, etc., not my Phys. Ed. coaches. Most people have no use for a second language; English suffices for better than 95% of the American population.

History? Interesting to some perhaps, but how much history do most of us remember? We were required to learn about the Magna Carta, the French revolution, the Renaissance, and the War of 1812, and memorize such things as the state capitols and the names of all American presidents. The information learned was soon forgotten and of no value to future employment anyway.

Moreover, anyone interested in these subjects can learn all they desire from readily available modern resources. As the opinion helpfully notes, for mathematical questions, “Siri is available 24/7 to provide the answer.” The same is true of questions in other areas. If you do not know the answer to a science, literature or history question, just ask Siri, look in Wikipedia, do a Google search, or post the question on Quora or Reddit. Better yet, we now have AI, such as ChatGPT, which can answer most any question in an easily accessible narrative form. No need to learn or memorize anything; it’s all there in the palm of your hand. Why clutter up your brain with useless information if it is just the touch of a few buttons (or a voice command) away?

This leaves just English spelling and grammar. Useful to some certainly, but likely to no more than the number who find mathematics useful. How many people spend their day writing

more than the occasional letter (other than a few journalists, of course)? The number (if you will forgive my use of math for a moment) is far short of 22% of the population. And, just as there

are simple, handheld devices to do mathematical calculations, we now have spell-check and Grammarly to correct our writing for us. If that proves too taxing, just ask ChatGPT to write whatever you need.

Following the opinion’s applied logic standard and statistical threshold, we can now safely eliminate the vast majority of what is presently taught in American high schools, at a great savings in both time and public resources. We may need just the three R’s, Readin’, Ritin’ and ‘Rithmetic, and not even much of them, certainly nothing advanced. We can easily learn all we need well before high school.

Starting with the opinion’s “underlying premises” and using “applied logic”, we have reached a plainly unacceptable, indeed absurd, result. In mathematics and “applied logic,” when

“underlying premises” lead to such results, the premises are rejected. This method of logical reasoning is known as a “proof by contradiction” or “reductio ad absurdum.” Where can one

possibly learn about such a useful tool of logic? If my calculations are correct, it is in high school math classes such as geometry. Those who “understand the thought processes” of mathematicians would recognize the obvious logical flaws in the opinion and promptly dismiss its facile conclusion.

Have I unduly bashed the Post’s opinion? Certainly not intentionally, for the Post’s editors are honorable, eminently logical (applied, of course, not mathematical) people, who are

“comfortable with spirited dialogue about what’s best for our society,” and who encourage the use of logic to “test each link in a chain of thought for unsupported assumptions or fallacies.”

Next time, perhaps, the Post’s opinion editors will follow their own advice when they do not know the answer to a math question and just ask Siri.** Which leads to another intriguing

question: Is the trouble with journalism today too much opinion writing without consideration of “underlying premises” and “applied logic”?

##### Footnotes

*No need to take my word for it, just ask ChatGPT:

**Question: Is mathematics mostly memorization?**

**Chat GPT Answer:**

Mathematics is not primarily about memorization; it’s about understanding concepts, logical reasoning, problem-solving, and applying techniques to various situations. While there are certain foundational facts, formulas, and methods that students may need to memorize, the emphasis in mathematics education is typically on developing a deep understanding of why mathematical concepts work the way they do, rather than just rote memorization.

In early education, there might be some memorization involved, such as learning basic arithmetic facts or multiplication tables. However, as students progress, the focus shifts more towards understanding the underlying principles and developing critical thinking skills to solve complex problems.

Memorization can play a role in mathematics, particularly in remembering key definitions, theorems, and formulas, but it’s usually just a small part of the overall learning process. The goal is to equip students with the skills and knowledge necessary to analyze problems, formulate strategies, and apply mathematical reasoning to a wide range of situations.

** I took the liberty of asking ChatGPT the question the opinion raises, instead of Siri, because itis not limited to Apple users and more thorough:

**Question: Why should mathematics beyond simple arithmetic be taught in school?**

**ChatGPT Answer:**

Mathematics beyond simple arithmetic should be taught in schools for several reasons:

1. Critical Thinking Skills: Advanced mathematics develops critical thinking skills such as problem-solving, logic, reasoning, and analytical thinking. These skills are essential for success in many areas of life, including academics, careers, and everyday decision-making.

2. Preparation for Higher Education and Careers: Many fields, such as science, engineering, computer science, economics, and finance, require a strong foundation in advanced mathematics. By teaching mathematics beyond arithmetic in school, students are better prepared for higher education and future careers in these fields.

3. Understanding of the World: Mathematics is the language of science and provides a framework for understanding the world around us. Concepts such as algebra, geometry, trigonometry, and calculus help students comprehend and describe natural phenomena, technological advancements, and social systems.

4. Creativity and Innovation: Advanced mathematics encourages creativity and innovation. It allows students to explore complex problems, develop new ideas, and find novel solutions. Fields like pure mathematics often involve conjectures, proofs, and discoveries that push the boundaries of human knowledge.

5. Cognitive Development: Studying advanced mathematics promotes cognitive development by challenging students to think abstractly, make connections between different concepts, and develop mental flexibility. These cognitive skills

are valuable for lifelong learning and intellectual growth.

6. Real-world Applications: Advanced mathematics has numerous real-world applications in various industries and everyday life. For example, statistics and probability are used in fields like medicine, business, and sports; calculus is used

in physics, engineering, and economics; and geometry is used in architecture, art, and design.

7. Problem-solving Skills: Advanced mathematics teaches students how to approach complex problems systematically, break them down into manageable parts, and apply different strategies to solve them. These problem-solving skills

are transferable to many other areas of study and life.

Overall, teaching mathematics beyond simple arithmetic in schools is essential for developing well-rounded individuals who are equipped with the skills and knowledge needed to succeed in an increasingly complex and interconnected

world.

If AI is “available 24/7” to answer questions, do newspapers still need opinion editors? With access to all the news sources available for free on the internet, do we even need newspapers?