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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.

Justin Hancock
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    Related: Quantity as the basic concept for applied mathematics – JRN May 31 '21 at 00:39
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    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. – paw88789 May 31 '21 at 03:44
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    @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). – Dave L Renfro May 31 '21 at 04:56

2 Answers2

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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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    -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... – user253751 May 31 '21 at 08:55
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    @user253751 How do you exclude 15 years of all the other influences that affected their understanding but wasn't the same for both groups? – Jimmy T. May 31 '21 at 09:08
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    @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. – user253751 May 31 '21 at 09:17
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    @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. – JRE May 31 '21 at 13:16
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    Why do you call apples and cakes/person "fake units"? – Steven Gubkin May 31 '21 at 13:20
  • @user15245 it's not like big studies have never been done. Remember that the "wait 15 years" step costs nothing – user253751 May 31 '21 at 13:48
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    Technically the answer to "how many states does the US have" has units (states). It's just that the unit is stated in the question and English allows ellipsis in this context. – No Name May 31 '21 at 19:36
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    @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". – Steven Gubkin May 31 '21 at 22:13
  • @StevenGubkin Because "apple", "cake" are ambiguous units of measurement e.g. two unequal apples still each constitute "1 apple". – hiccups Jun 01 '21 at 04:56
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    @hiccups That is how all units work. The first mile and the second mile are not the same mile: they occupy different physical locations. This difference is meaningful in some circumstances (if you are biking, then the inclination of these two different miles might be relevant to you). We always "quotient out" irrelevancies when we choose a unit, and this is a large part of the power of units. – Steven Gubkin Jun 01 '21 at 11:39
  • @StevenGubkin I don't agree. The difference in your example comparing two physical stretches of "one mile" comes from the uncertainty in the measurement ("one"), and not from an inherent ambiguity in the definition of the unit itself ("mile"). – hiccups Jun 02 '21 at 03:08
  • @hiccups 1) "How many does Sally have?" 2) "Sally has 3 and bought 2. How many does Sally have?" 3) "Sally has 3 one-pound bags of flour bought 2 two-pound bags of flour. How many does Sally have?" – user3067860 Jun 02 '21 at 16:10
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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.

Nullius in Verba
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    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. – Michael Bächtold Jun 01 '21 at 06:51
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    @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? – No Name Jun 01 '21 at 11:19
  • @NoName In which sense is "one apple" a higher level abstraction that the number "one"? – Michael Bächtold Jun 01 '21 at 11:31
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    @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. – No Name Jun 01 '21 at 12:29
  • @NoName to me "one apple" is not a concrete apple. But independent of that, you are not explaining why the number "one" is at a lower level of abstraction than the unit "apple". – Michael Bächtold Jun 01 '21 at 15:11
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    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. – ryang Jun 01 '21 at 16:41
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    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! – Thierry Jun 01 '21 at 22:37
  • @MichaelBächtold Yes, the number one is considered a higher abstraction than 1 apple. In this system apples are fully concrete. We can hold them, or at least imagine holding them. Concrete = easy. Next come shapes. "3 triangles" is partly abstract since triangles are an abstraction. Pure numbers are then considered fully abstract, and thus the most difficult. The terminology is confusing at first, but makes a kind of internal sense. – Owen Reynolds Jun 01 '21 at 22:43
  • @OwenReynolds that's similar to how think if it, but seems opposite to what NoName and the OP say. Let me ask another question for clarification: what's the difference between "one apple" (or "an apple") and what No Name calls the "unit apple"? I suspect children learn early on to deal with phrases like "Do you want an apple?" even though there's no concrete apple to point at. How to distinguish the apple in that question from the unit "apple"? Regarding other units: we can point at a meter length, or experience an hour time. In which sense are these units more abstract than the number 1? – Michael Bächtold Jun 02 '21 at 06:41
  • @MichaelBächtold I don't see any "unit apple" in the answer. I've seen this method and never heard of that concept. Now, counting 5 identical apple-icons would be slightly more abstract than 5 counting 5 "real-looking" apple drawings. But that's just the Concrete<->Abstract scale. As for meters, the assumption is that measuring is always more abstract: in a 5-meter long garden hose, you need to imagine the meters and understand no overlaps are allowed as we "make" those 5 meters. – Owen Reynolds Jun 02 '21 at 14:50
  • If physics was a mandatory class and started in middle school, it all would be ironed out there. In physics you always use units, and dimensional analisys is a standard practice if not to invent a new formula than to verify that the formula you use is sensible. Unlike "pure" math where you add 2 and 3 and get 5, in physics you must be careful not to treat miles as kilometers, or your missile will miss the target. In fact, abstract 2 and 3 are not abstract, the only reason units may be omitted is because these numbers are plotted on the same number line: "1" is the length of a unit segment. – Rusty Core Jun 16 '21 at 17:03
  • Lot of good stuff here. But: "There is a subtle distinction that kids pick up automatically from usage..." is not generally true. The majority of my (community college) students never pick up on a subtleties like that, unless given direct instruction and assessment. – Daniel R. Collins Jun 17 '21 at 14:34