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Background:

My friend told me that when she was in elementary school, the teacher would ask questions like "If you have $6$ apples and eat $2$ of them, how many apples do you have left?" A kid in the class answered $4$, to which the teacher replied "$4$ what? $4$ elephants?"

Seemingly the teacher wanted to emphasize the importance of using units.

My friend told me that this led her to believe for some time that unitless numbers were meaningless, e.g. that calculating the area of a rectangle with side lengths $2$ and $3$ was meaningless.

I have also been wondering whether one should write calculations as

$6 \text{ apples} - 2 \text{ apples} = 4 \text{ apples}$

$20 \text{ cakes} / 5 \text{ cakes per person} = 4 \text{ persons}$

or as

$6-4 = 2$

$20/5 = 4$

Question:

What does research indicate about how one should treat units in elementary school? Hopefully some of the concerns above have been addressed quite directly in some research paper.

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  • $\begingroup$ Related: Quantity as the basic concept for applied mathematics $\endgroup$ May 31 at 0:39
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    $\begingroup$ I disagree with the teacher. The answer automatically had units of apples from the phrasing of the question. If the student said “four apples”, then there were four apples apples which would be meaningless. $\endgroup$
    – paw88789
    May 31 at 3:44
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    $\begingroup$ @paw88789: I strongly agree (and I also like Ben Crowell's answer). FYI, issues like this are extremely important in high stakes standardized test questions, and for some tests all word problems involving measurements are provided explicit units, otherwise the question could be challenged as having multiple keys (among other things). Also, each of the first few problems here would not survive a challenge of having an incorrect key if on an actual test (and thus cost the company over $100K if not a pretest item). $\endgroup$ May 31 at 4:56
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    $\begingroup$ It's sad that the teacher in this case accidentally confused and thus hindered a student who seemed to intuitively grasp the abstraction. Seems like a non-ideal approach. $\endgroup$
    – bob
    Jun 1 at 15:45
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There is a distinction to be made between numbers that depend for their meaning on an arbitrary standard of measurement and those that don't. If I ask how many states there are in the US, the answer is 50. If I ask how far it is to your dentist, and you say 15, then that's just wrong. 15 miles? 15 blocks? 15 minutes? This is pretty standardized as usage in science and engineering, which matches usage in ordinary speech.

I don't think this is an issue that can be settled by research. Research can't tell us whether poetry should rhyme, whether men should wear skirts, or whether it's OK to put a 30 amp fuse on a circuit that's only wired to be able to handle 20 amps. These are matters of taste, convention, and utility. The standard usage for units in science and engineering is based on a combination of taste, convention, and utility.

Students at the elementary school level should of course be taught to write numbers with units when it's normal and correct to do so. In addition, kids at this age need to absorb a conceptual understanding of the meaning of multiplication and division. Writing fake units like "5 apples" or "5 cakes per person" is a nice way to help them develop competence at reasoning about what operations make sense, because they can check that the result of their calculation has units that make sense, as in your cakes-people example. However, it should be taught to them as a method of checking, keeping track, and deciding what makes sense -- not as an arbitrary rule.

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    $\begingroup$ -1 for "this can't be settled by research". Of course it can! Take two groups students, teach one one way, teach one the other way, wait 15 years, see which group understands units better... $\endgroup$
    – user253751
    May 31 at 8:55
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    $\begingroup$ @JimmyT. that's why you try to use large random groups, as much as practical. No different from any other study. Since you teach entire classes at a time, and members of a class have much in common with each other, you would probably design the study around classes rather than students. $\endgroup$
    – user253751
    May 31 at 9:17
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    $\begingroup$ @BenCrowell: "whether it's OK to put a 30 amp fuse on a circuit that's only wired to be able to handle 20 amps" has been studied and researched and proven to cause more house fires. Electrical codes are empirical - they are arrived at by examining things that didn't work (things that caused damage or loss of life) then making rules to minimize or eliminate the risk. $\endgroup$
    – JRE
    May 31 at 13:16
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    $\begingroup$ Why do you call apples and cakes/person "fake units"? $\endgroup$ May 31 at 13:20
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    $\begingroup$ @NoName It is splitting hairs, but I think when you ask "How many?" the answer is a unitless number. "How many miles to the store?" has the answer "10". "What is the distance to the store?" has the answer "10 miles" or "16 kilometers". $\endgroup$ May 31 at 22:13
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The CRA approach says to move from concrete to pictorial to abstract concepts. The idea of numbers without units or identifiers is treating numbers as an abstraction, and a positive goal. You start with the concrete examples - 3 apples plus 2 apples is 5 apples, 3 pebbles plus two pebbles is 5 pebbles, 3 people plus 2 people is 5 people, and spot the pattern that lets you say 3 plus 2 is 5. What is "3" on its own? Three whats? The point is "3" represents a pattern with an empty slot for the unit/type of object into which you can insert anything.

When there is only one entity under discussion being counted, it is perfectly legitimate (and a positive achievement) to use the abstraction and let context fill it in. The answer to "If you have 6 apples and eat 2 of them, how many apples do you have left?" is actually 4, and not 4 apples, because the units to use are specified in the question. If you had asked "If you have 6 apples and eat 2 of them, what do you have left?", then the correct answer would be 4 apples. There is a subtle distinction that kids pick up automatically from usage - in this case, it sounds like the kid who answered the question understood the rules better than the teacher did.

What I think the teacher is trying to address is the tendency to overdo the abstraction, and abstract in situations where it is not safe to do so. The most common situations are where there are several different units in use, or where you are writing numbers for someone who might not know the context.

If you have six boxes of eggs (with six eggs to a box) and you drop two of them, how many do you have left? We have two different units: eggs, and boxes. Did you drop two eggs or two boxes? Are you asking how many eggs or how many boxes are left? You can no longer assume a unit, and must specify at every step.

The rules over when units are or are not required are complicated. Sometimes you can set the formal rules out explicitly - in which case you need to get them right, and explain all the exceptions and corner cases too, or cause more confusion when usage conflicts with the rules you've just given. Or you can teach them by example, using units when needed, and not when you don't, and let children's natural language acquisition skills deduce the rules subconsciously. That's usually easier and more accurate, but often results in knowledge that people can't explain - they just know it "feels wrong" to say it a certain way.

As mathematics gets more formal and precise, all this intuitive encoding of knowledge starts to get in the way, and people lose track of the complicated manipulations being applied subconsciously and get confused about what concepts they're using. Being careful about units (and more generally, about types) is one way to resolve this confusion. But it is a stage of abstraction beyond the abstraction of pure numbers.

You have to teach it in stages. First you count concrete objects - 3 apples, 3 pebbles, 3 people. Then you recognise the pattern, and abstract them all as a "3". Only when the children are able to do that reliably, do you start to explore the multi-unit ambiguous situations requiring more precision, and put the units back in. And at a much, much later stage of education, you can start to pull apart the different types and the rules governing them more formally (cardinal numbers, ordinal numbers, rationals, real numbers, vectors, torsors, groups, dimensional analysis, etc.). It may help that later learning if you give some indication early on that these are really different types of things that we are tacitly mixing up together, so it doesn't come as such a shock later, but it just confuses things to try to explain everything at once too early, or to skip over important intermediate stages.

I can't tell from what you write whether the teacher is trying to prevent students using the abstraction at all, or whether having taught that stage, they are moving on to the next step of explaining the exceptions where you do need to keep the concrete types. The "how many apples do you have left?" question is not a good example, but I don't know if that's what the teacher actually used, or is your friend's paraphrase.

But I would say that children need to learn both approaches. They need to be able to abstract pure numbers from a concrete situation, and they also need to be able to keep and manipulate units and types in more complex multi-unit situations. The latter is a significantly more advanced stage of understanding, that requires a firm grasp of the abstraction process first. This isn't based on any pedagogical research - it's based on the structure of the mathematical concepts being taught. It's not that one approach is better than the other and should be the only one taught, it's that you need to teach them both, one after the other.

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  • $\begingroup$ I can relate to much of what you say, but I don't understand how you arrive at the conclusion that dealing with units "is a significantly more advanced stage of understanding, that requires a firm grasp of the abstraction process first." My intuition would tell me that it's easier for a kid to understand the concept of 3 apples or 5 meters, then to understand the concept of the pure numbers 3 and 5. $\endgroup$ Jun 1 at 6:51
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    $\begingroup$ @MichaelBächtold I think what OP's saying is that units themselves are an abstraction, and a higher level one than numbers. Sure "3 apples" and "5 meters" are (or rather can be) concrete, but none of "3", "5" or "meter" are. "Apple" is an edge case, because you can point to an apple, but the idea of using "apple" as a unit is an abstraction, because real apples are all different. Why not use "Granny Smith" or "Red Delicious" or "Cosmic Crisp" instead? $\endgroup$
    – No Name
    Jun 1 at 11:19
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    $\begingroup$ @MichaelBächtold I never said "one apple" was more abstract than "one", only that "apple" (as the unit, not the object) was. "One apple" is a very concrete thing, especially if you point at the apple in question. And since apples-the-objects are concrete things, it's harder to realize that apples-the-units even can be abstract, referring to the platonic ideal of "apple" as opposed to any particular apple or apples. I'd also argue it doesn't matter for "apples", so long as we agree on the definition of "apples". Meters and other measures are a whole other kettle of fish. $\endgroup$
    – No Name
    Jun 1 at 12:29
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    $\begingroup$ I was about to pen a lengthy comment, but your answer has taken the words out of my mouth! In general, when solving problems where units will “interact” to form new ones (e.g., rate problems, problems involving conversions), or where a quantity is variously expressed in multiple measurement units, I strongly encourage students to keep track of the units by displaying them all quantities’ units (or fake units) throughout the working, both as guidance (of the operations to perform) and as checks. $\endgroup$
    – Ryan G
    Jun 1 at 16:41
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    $\begingroup$ I completely agree that "four" is a correct answer and probably the most common one by quite a bit, but I disagree that "four apples' is incorrect. Nobody would be pedantic enough to parse that as "four apples apples" and get confused. Well, except the OP's friend's teacher maybe! $\endgroup$
    – Thierry
    Jun 1 at 22:37

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