Attend to Precision. Why?
by Keith Devlin @KeithDevlin@fediscience.org
The James Webb Space Telescope. Using mathematics, NASA was able to position and assemble it remotely at a precise, stable location in space one million miles from Earth. NASA image.
The Common Core State Standards for Mathematics (CCSSM) document leads off with a list of eight basic practices for doing and using mathematics from which the rest of the Standards follow by increasingly narrowing the focus to meet the needs of classroom education. Called the Mathematical Practices, they were developed by experts who set out to answer the question, “What basic practices are essential for students to master in order to be equipped for life and work in the 21st Century?” Here they are:
MP1 Make sense of problems and persevere in solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
MP8 Look for and express regularity in repeated reasoning.
Practice MP6 is (the main part of) the title of this month’s post. And my question is, why is it one of the (just) eight basic practices the CCSSM developers decided to present?
That not the same question as “Is it a good idea to attend to precision?” To be one of the MP, a practice has to be demonstrably an essential feature of doing mathematics in today’s world. (That eliminates “Getting an A on the algebra test” as an adequate answer.)
So how would you answer that question?
This is not a hypothetical puzzle. I was (essentially) asked this question some time ago by a lawyer in a long-running civil case involving an alleged inaccuracy in a corporate contract. I’ll give you part of the much longer and more expansive reply I supplied. (I had to respond to various arguments that had already been made in the trial, and I gave several reasons why it is, not just important but essential to attend to precision.) I’ll summarize three of those reasons here. (I also supplied citations, which I’ll omit in this short essay.)
1. HISTORICAL REASON
To the best of our knowledge, based on archeological evidence, mathematics began around 10,000 years ago in Sumeria, motivated by legal and financial needs to keep records of property ownership and trade as their society started to become more complex and organized. Their efforts to introduce order to, and regulation of, property ownership and trade went along with what we could now call a monetary system. (So, numbers, money, and the regulatory systems around them, all grew together in tandem—along with written language, it appears.)
Ever since then, as mathematics has grown and developed, a main driving force was always to obtain more precision in various aspects of our lives. The introduction of modern (i.e., “Hindu-Arabic”) arithmetic into northern Italy in the early 13th Century gave rise, in that one region and in a matter of a few decades, to banks, modern accounting methods, the insurance industry, and the formation of large, international trading organizations, all of which rapidly spread to all of Northern Europe. Such is the power for society of the precision and accuracy provided by mathematics when it is put in the hands of the People. Mathematics always promised, and has repeatedly delivered, accurate information that could be trusted.
Why does this argue for precision? The entire framework of mathematics arose, and was developed and refined, to provide ever greater precision and accuracy in human activities, in the first instance to support trade and commerce and a financial system, and in due course to support the development of modern science and several generations of technologies.
Achieving that accuracy is hard; even today many people find math difficult to learn, let alone master. Yet the payoff, to society and the individual, is undoubtedly worth it.
Today, the precision and accuracy that mathematics can provide was illustrated in recent times by NASA being able to design and build the James Webb Space Telescope and send it to a precise location in space one million miles from Earth (the L2 point), where it was remotely “unpacked” and deployed, studying our surrounding universe.
Back home on earth, we all of us benefit from the precision and accuracy of mathematics every minute of every day, since the phenomenal accuracy of mathematics underlies all of our science, technology, engineering, medicine, and our financial systems.
We are so used to benefiting from that accuracy, that most people rarely reflect on it. We just know that mathematics gives us secure, accurate, trustworthy information. No other human framework gives us that degree of exactitude and reliability. Provided, aways, that those of us who develop and certify that mathematics do our jobs correctly, and those who make use of that precious, powerful tool use it wisely and faithfully.
When utilizing a tool that, for 10,000 years, Humankind has come to regard as providing true, precise, accurate, and unambiguous information, careless application (and on occasion nefarious application) can mislead, sometimes with damaging or even lethal consequences.
In making any use of mathematics, just as important as making sure that any statements of mathematical values are accurate and that any and all calculations are mathematically correct, is making sure the mathematics correctly connects to the real-world situation it is being applied to. Failure to do so can on occasion be catastrophic.
The Mars Climate Orbiter. In 1999, NASA lost this $125-million craft due to a simple math error made by the engineers. NASA image.
For example, in 1999, NASA lost its $125-million Mars Climate Orbiter due to a simple math error made by the engineers.
The error was not in any of the calculations. The advanced mathematical computations were perfect. It was in the units being used. By using the wrong units, the mathematics did not connect accurately to the world; a middle-school-level error.
The engineers should, of course, have spotted the error, but they didn’t. Scientists and engineers, in particular, are so comfortable with mathematics that, provided they trust the source, they tend to believe mathematical results. In this case, their trusted source, namely, other members of the globally distributed team, were using different units.
2. EDUCATIONAL REASON
The US Education System establishes standards for mathematics and language education that, in the case of mathematics, put considerable emphasis on educational goals that ensure, inter alia, that when people use mathematics, they pay particular attention to precision, including precision with how the mathematics connects to the context in which it is being used.
Teachers are tasked to ensure that when students learn math, part of what they learn is that it must be used with care. We ask them to do so because, as a nation we recognize the crucial importance of the precision and accuracy that comes from mathematical proficiency. In the US, the most recent standards, the Common Core State Standards (CCSS), described earlier, were established by an Act of Congress in 2010, following an initiative by a bipartisan group of governors, education experts and philanthropists.
The CCSS lay out yearly benchmarks for the nation’s schools from Kindergarten through Grade 12, in mathematics and language arts. The fact that mathematics is the only discipline outside of language to be covered shows the importance America attaches to mathematical proficiency. America quite literally depends upon having a mathematically proficient workforce and citizenry, and it knows it. The CCSSMP set out the practices required to achieve that.
Note that these are practices; not curriculum topics that apply only in the school math classroom. They lay out the basic principles of human reasoning that are important when using mathematics to make decisions and act in today’s world.
The Mathematical Practices are viewed as so fundamental, that they are presented right at the top of the first page of the CCSS-Mathematics website.
The CCSSMP make clear that not only is mastery of mathematics itself—the mathematical toolbox, if you will—important, but so too is skill in making practical use of mathematics in various aspects of our lives.
Mathematical Practice MP4 is of particular relevance in this (MP6-focused) essay.
MP4 refers to the way mathematics is used in practice out in the world. In making any use of mathematics, just as important as making sure that any statements of mathematical values are accurate and that any and all calculations are mathematically correct, is making sure the mathematics correctly connects to the real-world situation it is being applied to, as the above example of the Mars Climate Orbiter illustrates.
Why does this argue for precision? Taken together, Mathematical Practices 4 and 6 imply that mathematics should be applied with care, depending on the individual circumstances. While the Common Core as a whole is a lengthy list of mathematical standards for future citizens to meet, there are just eight top-level Mathematical Practices, and two of the eight make it clear that using mathematics properly includes taking care to ensure all the numbers, percentages, formulas, equations, and computations handled when using mathematics connect properly and accurately to the world.
3. THE NATURE OF MATHEMATICAL TRUTH
Mathematics achieves its precision not by human agreement but by its inner logic.
A major part of what makes mathematics such a secure foundation for society, especially its power to give all the precision and accuracy you need, whatever the circumstances, is that mathematical truth, uniquely, is not an issue of human opinion or choice or everyday practice.
The method to establish truth was established by the ancient Greeks around 350bce, namely truth is established by rigorous proof from accepted, carefully crafted basic assumptions (“axioms”). Opinion or preference, either individual or social, plays no role. (Other than selecting the initial assumptions, that are few in number, are available for all to see, and have stood the test of time.)
Wall Street, the financial heart of New York City. Its buildings rest safely on Manhattan’s deep rock base. Its activities depend on the stability and accuracy of mathematics.
In this regard, mathematics is different from the natural sciences. In physics, for example, “truth” amounts to “based on the best observational or experimental evidence we have available today.” Fortunately for society, we have today a considerable amount of evidence to support physics, but the natural sciences cannot, and hence never will, provide the certitude of mathematical truth.
Why does this argue for precision? That incredibly powerful, internal mathematical certitude is all to naught if the mathematics is not applied correctly in the world. Any tool is only as good as the person who uses it. A tool that offers unlimited precision and accuracy in regards to its inner machinery has to be used with appropriate care.
Even with the best intentions, mistakes arise when inputting data (for the mathematics to act on) and acting on output data (from that mathematics), as occurred with the NASA satellite. Humans make errors. When they do, society rightly holds responsible the individual(s) who choose to use the tool and then, for whatever reason, misuse it when they do so. (Failure to “attend to precision” can sometimes have legal consequences. So if you favor big-stick educational techniques—I don’t—you could add that to your list of reasons.)
CONCLUSION?
The above list could go on. (Mine did, and there were other reasons I could have given that were less directly connected to the legal case I was focusing on.) The overall point, though, is that precision is crucial to mathematics for the tautological reason that the initial development of mathematics, and much of its subsequent development, was driven by the needs of human societies for limitless precision.
And mathematics is by far our best response to that need to date.