Measurement is an essential part of math. Quantities encountered in the real world are usually one of two things: a count or a measurement. Counting is what we learn first about math, before doing anything else with numbers. Counting is a way of measuring a quantity of discrete objects, but it is different from other forms of measurement. Counting requires no comparison to a standard and is therefore unitless. For many other properties of things in the real world, units are necessary. This includes (potentially) non-discrete quantities such as time, length, area, volume, cost, mass, speed, force, temperature, pressure, density, flow rate, saturation, solubility, elasticity, roughness, energy, luminosity, efficiency, loudness, accuracy, frequency, location, orientation, rotation, sweetness, harmfulness, productivity, growth, hydration, balance, consistency, viscosity, vibration, comprehensibility, relatedness, and so on.

As humans we have an understanding that measurement is never perfect— it’s physically impossible to be, even. Measuring requires the use of a tool, whether that be a physical object, a part of one’s own body, a computer system, etc. Every tool has a specific limit of precision: a difference in quantity below which the the tool cannot reliably distinguish. There are different approaches to reporting a measurement based on the resolution of the measuring tool.

First, most straightforwardly, one can simply report the smallest unit on the tool itself. If a meter stick has its smallest markings representing millimeters, then a reported measurement could be (for example) 0.628 meters, or 62.8 centimeters, or 628 millimeters, all of which have the same number of significant figures. Second, a different approach I have seen used in science classes is to estimate one digit past the smallest unit on the tool, since it can often be discerned if the actual quantity is closer to one mark or the next. This introduces an acceptable amount of uncertainty without exaggerating the precision. Finally, the most sophisticated method is to use the previous approach while also explicitly reporting how much uncertainty there is, e.g. 62.83 ± 0.05 mm.

In my experience with school math as both a student and an educator, I have most often seen the first two approaches. Typically this has to be discussed when asking students to perform measurements, though sometimes it is glossed over. That being said, when it comes to math class, most consideration of precision tends to be dropped as soon as calculations begin. The exception is discussions about rounding.

Students are often correctly taught to avoid rounding intermediate calculations and round only the final answer. The method of rounding is also virtually always taught explicitly. However, when it comes to how many digits to round to, this is often specified as a particular number of digits to the right of the decimal point, ignoring trailing zeros. This makes things easier, but it is mathematically nonsensical.

Reported measurement	Digits past decimal point	Sig figs
0.628 m	3	3
62.8 cm	1	3
628 mm	0	3
62.800 cm	3	5
628.000 mm	3	6

Why, when teaching about anything involving measurement, should we suddenly abandon mathematical rigor once calculations happen?

The true answer, I think, is because significant figures can be very challenging for students whereas reading off a measurement correctly is relatively easy. Additionally, this comes up a lot when students are not measuring things themselves, such as in word problems. If the focus is on some other topic, like scale factors, educators generally want to be able to keep the focus on that topic. It would be undesirable to take time away to explain something different.

My reflexive and admittedly blunt response to this perspective is too bad. We as educators don’t get to bend math to our will to make it more palatable to students, as much as we might like to. We have an obligation to teach math as it really is, honestly and rigorously. Moreover, it doesn’t benefit students to stay hyperfocused on one topic at a time. Ideas in math are fundamentally interconnected. We also have to meet students where they’re at, meaning if they lack a prerequisite understanding or skill we must address that first.

On the more positive side, this is a problem with a clear solution. If students are consistently asked to use significant figures when dealing with measurements, they will develop an understanding of their use. It would be far less likely to be a barrier to discussing a different topic, since it won’t be something new and challenging.

That being said, it is also very difficult to change curricula and to find time to teach everything that is important. I’m not suggesting an individual teacher can solve this situation on their own. It requires a paradigm shift in math education to start taking the idea of precision seriously. I think it would be worthwhile, too, since measurement is one of the few things learned in math that people really do use on a daily basis in their everyday lives. People read the speedometer on their car, people weigh themselves, people measure things around the house for purposes of home improvement or interior decorating, people measure ingredients, people read measurements on food labels and lightbulbs and PC specs, and so on. Many people get by with only a vague sense of what these measurements are telling them, without the amount of precision or uncertainty crossing their mind at all. This impairs people’s ability to make mathematically valid judgments.

“But,” one might protest, “this is the domain of science class.” Why? And are science classes doing a good enough job with this? Virtually everyone I have ever spoken to about it, student or adult, has expressed confusion about, and strong dislike of, significant figures. Not being a science educator myself, I can only imagine what the problem might be. My best guess is that, despite students being consistently asked to use them, significant figures aren’t being given the time they need to be developed as a concept in themselves. It also doesn’t seem to me like there is any reason why science class should be devoting a lot of time to this. Fundamental properties of quantity and measurement are math’s domain.