What value is there in requiring students to declare the dimensions of an answer when it is already clear from context?

Question

When I was in late primary and middle school (east coast US, early 1990's), we were assigned a lot of word problems of the following general form:

Mary has eight self-sealing stem bolts. She sells half to John and then finds an additional self-sealing stem bolt on the playground. How many self-sealing stem bolts does she now have?

The obvious way to solve this is to show:

8 - (8/2) + 1 = 5

My teachers expressed to me in no uncertain terms that this was absolutely wrong. The correct answer was:

8 self-sealing stem bolts - ( 8 self-sealing stem bolts / 2 ) + 1 self-sealing stem bolt = 5 self-sealing stem bolts

Sometimes, if we failed to fully clarify the units of the answer, the teacher would mock us by saying things like, "Five? Five what? Five elephants? You need to be more specific."

This always struck me as redundant and pedantic. The wording of the question indicates that any response must be addressed in terms of self-sealing stem bolts and not any other unit (e.g. elephants, crystals, spiral notebooks, or planet-killing photon cannons) to qualify as a reasonable answer, so an answer of "5" is in fact perfectly clear and unambiguous when taken in context with the question asked and not as a lone number floating out their in the void of context-free and content-free nothingness.

Certainly, if I was asked today on the job, "Hey, how many vials of delta-omega varuonic acid do we have left on the shelf?", I would give an answer like, "Hey, yeah, looks like five.", rather than pedantically wasting everyone's time by saying, "The answer to your question is that there are five vials of delta-omega varuonic acid on the shelf."

I am aware that tracking units throughout is useful when doing dimensional analysis, but the vast majority of problems that we were assigned involved only one dimension or unit.

A small number of teachers went even beyond this and required all answers to word problems to be expressed in complete sentences. For them, an answer of "5 self-sealing stem bolts" wasn't good enough, the correct answer was "Mary has 5 self-sealing stem bolts.", but discussing that is a question for another day.

So, I can come up with the following cases in which requiring students to disclose, declare, or identify the units of their steps or answer might be considered pedagogically relevant:

• The problem inherently requires dimensional analysis/unit conversion in such a way as to require the student to track and record the units in order to identify the correct mathematical operations to perform.
• Identifying the units of the answer is part of the problem itself rather than stated within the question. For example, asking students "What is Ann's speed?" may require them to recognize the difference between units of speed, time, and distance, and so having the student identify which unit is the speed unit demonstrates learning.
• The problem is worded in such a way as to give the student choice over which units to give the answer in. For example, a question requiring the student to "Provide the length" might allow the student to choose whether to express the length in terms of meters, centimeters, inches, furlongs, or FIFA-regulation soccer field lengths, and thus the grader must know which units the student chose in order to grade their answer.

Other than the cases above, is there pedagogical value in requiring students to explicitly state the units of answers when the appropriate units are already clear from context?

Answer 1

In my experience with remedial-level community-college students (USA), it is simply never the case that the units are trivially "clear from context". I can easily see some of my former students, when prompted for units in the OP's example, saying any of either "people", "playgrounds", "feet", "seconds" (depending on what they most recently saw as an example before the test), etc.

Scanning some actual remedial arithmetic tests from a few years ago, noting that every test I gave had a word problem like this, I'd say that 80% of the students were never able to specify any units at all, even though this was a direct and repeated piece of instruction. Secondarily, as soon as there is more than one unit of anything in the problem, it's close to a random guess which one they pick for an answer. Simplified example: "In 2010 there were 1000 people in a school, in 2020 there were 1500 people, what was the percentage increase?" can result in answers with units of years, people, or percent.

I would never suggest carrying a very long units name through every expression of a calculation. But generally for word problems I do ask for the answer as a complete sentence with units. While such units may seem obvious to you or I, they are not for everyone, and students regularly get them incorrect -- so I feel a responsibility to test and give correction where needed on that issue.

For the complete-sentence aspect, part of my thinking is that such communication is not necessarily verbal (such as in the OP's example), but perhaps more likely to be via email, text messaging, a written report, etc., and so students should have an opportunity to exercise and sharpen that skill before entering a workplace.

Answer 2

The legitimate purpose of this is trying to get the student to actually read the question and take note of the fact that there is indeed a context.

Many students approach mathematical exercises as "set of instructions to carry out" rather than "problem to be solved". Subsequently, they dislike "word problems", because they just obfuscate the instructions. They might just scan the question for numbers, and then guess some arithmetic operations to carry out on those numbers.

Now of course requesting to state the unit can just lead to the student also scanning the question for what might be the unit, rather than ever conceptualizing what is going on. Moreover, my suspicion is that for at least some teachers insisting on this, providing the unit is just part of whatever arcane set of instructions apply, rather than any actual understanding of the issue mentioned above.

Answer 3

Different STEM fields have different conventions about this. These conventions are adapted to the needs of the field, so in general if the work you're doing has the flavor of field X, you will probably benefit from doing what's conventional in field X -- the convention was probably adopted by those people because it made their work either more efficient or less prone to error.

The field that has the most clearly defined conventions is physics, where the vast majority of numbers have units, and there is a variety of units in use for the same thing. So for instance, one typically uses the kilogram as the base unit for mass these days, but it wouldn't be surprising to find the gram used in some contexts (and the gram is the non-prefixed unit). The physics convention is spelled out in style guides for journals and is also used consistently in writing such as freshman physics textbooks. The convention is basically that you always write the unit explicitly. In this convention, you do not (usually) write units if the number is dimensionless, so, e.g., it would be fine to write 5 rather than 5 self-sealing stem bolts -- but the latter would be fine if it gives better readability in context.

In math, the most common convention is to treat the units as part of the definition of the variable. "Let ρ be the density in units of g/cm3. [calculations...] So we find that ρ=17.38."

In engineering, you see more of a mix of people switching back and forth between the physicsy style and the mathy style. Often engineers are dealing with some awkward mixture of SI and non-SI units, so for example you'll see a lot of equations that look like $A=0.13784bc^2/\sqrt{d}$, where the pure SI version would be, say, $A=3\pi bc^2/\sqrt{d}$, but they're measuring $d$ in bushels or something.

So if you're teaching schoolkids, I would suggest that you teach them always to write explicit units on things that have units taken from some arbitrarily defined human-constructed system of measurement, such as the SI. In those contexts, experience shows that it really isn't a good idea just to not write the units, on the theory that the units have already been defined by context. Failing to do so leads to errors, such as the famous Mars probe that crashed because someone didn't convert meters to feet.

Related to this is the use of units as a tool for reasoning. It's extremely common for people to get all the way to college without really having a clear idea of the conceptual meanings of multiplication and division. If they need to solve a problem, and they haven't been told what algorithm to use, they don't know how to tell whether to multiply or divide. One way of building their reasoning ability is to get them in the habit of thinking about units. Then if they have 10 pounds/in2 and 100 in2, they have the mental tools to be able to figure out that multiplying would be what would give you a force in units of pounds.

Answer 4

I think it is ridiculous to keep repeating "self-sealing stem bolts" over and over in order to maintain units throughout a computation. But there is important benefit in keeping correct units. Practically, it is just a natural part of being precise, which may have safety consequences (e.g. Gimli glider). For this reason alone, I would say there is no clear benefit in having incorrect units.

Yet, there is a far better alternative to what you said your teachers expected you to write:

Let $n$ be the initial number of self-sealing stem bolts that Mary had, and $r$ be the final number. Then $r = n-n·\frac12+1 = 8-8·\frac12+1 = 8-4+1 = 5$.

No units have obscured the mathematics, but the solution is precise!

Answer 5

Apologies for jumping to early college. It turns out that the problem persists, if you don't get into the habit of (at least mentally) keeping track of dimensions/units/whatnot.

I have been battling with the consequences of not doing that (or rather, students adding fitting units as an afterthought without checking whether the appropriate units actually resulted from the calculation). One of my pedagogical tricks is to insist on "reality checks" like this. If you are calculating a volume, and get an answer in square meters, you should immediately infer that something went horribly wrong rather than simply keep the numerical value and switch to cubic meters. That will never wash.

I expect to run into this, again, a few week from now. I will be teaching a first course on vector calculus to undergrads. I predict that ten per cent of them will fail to include the Jacobian, when calculating a some volume using polar coordinates. If they do that in their personal assignment, I will deduct a little (only little, I'm a nice guy) from their score and direct them to a video I prepared just for them. Sorry, it's in Finnish. Anyway the title is the (purposefully) patronizing "Why I must not forget to include the Jacobian?" In there I show that if you don't, you will get a formula like $A=2\pi R$ for the area of a circle of radius $R$ (call it a disk if you want to). A bell should be ringing, but some have not been educated enough to notice how this formula produces wrong units. Sigh.

Answer 6

Sometimes the problem has only one unit, but the answer has different units. A cube has a face with an area of 9 square inches. Find the volume. The answer would of course be in cubic inches. The fact that only one unit is in the problem doesn't make the choice of units easy.

I have always insisted that students write units. Depending on the difficulty I would take off more or less partial credit if they left the units off. I did this because students don't always have the judgement to decide when they need to make the units clear. As my geometry problem is meant to show, the units aren't always clear from the problem.

Sometimes the units might be clear. Consider: John has 9 apples. Kim has three times as many. How many more apples does Kim have than John? A student might say 18. In that case, I would gently remind the student that we need to specify 18 what. But I would never say 18 what? 18 elephants?? That's just insulting.

My students learned to specify what the units were and I'm sure it helped them in the long run.