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"Suppose you bought four boxes of pencils having five pencils in each, how many pencils do you have altogether?" — "Nine." — "How come?" — "Because 4 plus 5 is 9." — "But you cannot add boxes to pencils!" — "Why?"

Indeed, why? Why you can multiply boxes and pencils, but cannot add? This is sort of self-evident for most adults, but how do you explain it to an elementary-school student?

I came up with an approach calling a box a "bunch of things" (Common Core likes to use the word "group"), so a box by itself has no meaning, what does have meaning is that it groups, combines, ties together several things that we are actually interested in, say pencils. If each box combines the same amount of things, then we can define and use multiplication as quick addition of the same number of things.

Similar approach can be used to explain why you cannot add two tens of flowers and five flowers as 2 + 5 = 7, because it is not clear seven of what we are getting. First, we "unbunch" two tens into one, two, three, ..., twenty flowers, then add five flowers to them, so we can count them, twenty five flowers. It just so happens that by having ten-based positional system we can simply write 5 to the right of 2 to get the correct number, it won't work if we had two dozen flowers instead of two tens.

Another phrase commonly used is that you can add "like things", things that are similar. All this kinda works, but does not feel perfect, does not seem rigorous, persuasive enough. Does anyone have a better idea, approach, script to explain to kids why adding apples to apples is ok, but adding apples to apple crates is not?

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    $\begingroup$ Funnily enough, my (then) two year old son absolutely refused to add cars and tractors. There's three cars, and five tractors. You can't add cars and tractors together, they're different things! :) $\endgroup$ – Luaan Aug 16 at 11:45
  • $\begingroup$ This doesn't really answer your question, but in actual practice when explaining these things, I'd use pictures or schematic diagrams. Draw 4 squares to represent the boxes and then draw 5 line segments (the pencils) inside each square, then ask how many pencils there are. Well, you add them up. But since your adding 5 four times, then multiply. Regarding the last sentence, this part will vary a lot with how much actually needs to be said, as it depends on their background with multiplication as repeated addition. If you have too many boxes to draw, use simpler examples, then generalize. $\endgroup$ – Dave L Renfro Aug 16 at 13:13
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    $\begingroup$ As far as actually explaining why adding boxes to pencils doesn't work but multiplying does, the reason of course is unit-based (you have 5 pencils/box, not 5 pencils), but this isn't something you can use with an elementary school student. I think it's probably fine to just explain with examples that the (wrong) method doesn't work, because at this stage (and indeed, for many adults for many things $\ldots),$ that's probably sufficient. For the rare super-inquisitive student who wants something deeper, well that needs to be fairly personalized to the student. $\endgroup$ – Dave L Renfro Aug 16 at 13:19
  • $\begingroup$ If you have a square grid of granola bars (real world example) it is much faster to count the edges and multiply than count each singular bar. $\endgroup$ – Beshoy Hanna Aug 16 at 16:39
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Roughly speaking, adding apples to apples is factoring.

One can reasonably interpret "four apples" as the result of multiplying the number "four" with the unit "apple" — in fact, I would even argue that you should think in this way.

So, when you add like things, what you are actually doing is invoking the distributive law to factor the unit out:

$$ 4 \cdot \text{apple} + 5 \cdot \text{apple} = (4 + 5) \cdot \text{apple} $$

But when you add unlike things, such as

$$ 4 \cdot \text{apple} + 5 \cdot \text{oranges} $$

there's nothing to factor out: no further simplification is possible. Note, however, that if we're willing to forget information, and just say apples and oranges are both fruits, we can then make a substitution

$$ 4 \cdot \text{fruit} + 5 \cdot \text{fruit} = (4 + 5) \cdot \text{fruit}$$

and then simplify.

The examples you consider are similar. If we add

$$ 4 \cdot \text{apple} + 5 \cdot \text{<crate of ten apples>} $$

we can't factor anything out. However, if we're willing to ignore the containers, we do have an identity:

$$ 10 \cdot \text{apple} = \text{<crate of ten apples>} $$

This means we can make a substitution into the original formula

$$\begin{align} 4 \cdot \text{apple} + 5 \cdot \text{<crate of ten apples>} &= 4 \cdot \text{apple} + 5 \cdot (10 \cdot \text{apple}) \\&= 4 \cdot \text{apple} + (5 \cdot 10) \cdot \text{apple} \\&= (4 + 5 \cdot 10) \cdot \text{apple} \\&= 54 \cdot \text{apple} \end{align}$$

so we also get the explanation of why you should multiply $5 \cdot 10$. (of course, you can actually compute the arithmetic anywhere in the calculation you like, rather than waiting until the end as I did)

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  • $\begingroup$ Marvelous! Taking your and gaeguri's answers and mixing stuff from some other answers in, I can build a decent argument. $\endgroup$ – Rusty Core Aug 19 at 15:42
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    $\begingroup$ And then you can take that idea even farther, and end up with tensor products. $\endgroup$ – Xander Henderson Aug 20 at 0:57
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Here is where it helps to get more concrete instead of more general. Have the student draw a picture of the problem and similar problems. First, you demonstrate drawing one box of pencils (a square) with five pencils inside (perhaps five tally marks) and make sure they understand the picture. Then ask them to draw four boxes of pencils (four squares) each with five pencils inside. When you ask them how many pencils there are at this point (by saying the original question again, and tying it to their drawing), they should get a correct answer (perhaps encourage them to skip-count if they are not good at multiplying yet), and at that point you should be able to ask them why they think adding 4 pencils + 5 boxes doesn't work to answer the question. Their self-explanation will probably be much better at making sense to them than any way we try to do it, because they will have processed it in context of what they already know about adding. You can always help clarify their wording at this point, and help them refine their statement so that they get at the crux of the issue: "pencils" and "boxes of pencils" are different "wholes."

To take it a step further if they still have a hard time explaining it, have them draw three pencils plus two boxes of pencils and repeat the process. How many pencils are there? Why didn't 3 + 2 work? Draw 3 pencils + 2 pencils. How is this drawing different than 3 pencils + 2 boxes of pencils?

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    $\begingroup$ Also a great answer; maybe it would be even better if you just let the kids draw the boxes ("Okay, now draw four pencil boxes. Draw five pencils in each of the boxes. How many pencils are there in total?"). More advanced maths students/teachers tend to get too absorbed in the abstractness of maths - but in the end, it's all there for some final, real purpose - make it real, and it's much easier to understand. $\endgroup$ – Luaan Aug 16 at 12:02
  • $\begingroup$ @Luaan: I scanned though the comments and answers before writing the two comments I made under the question and I didn't notice that you had already described the drawing 4 boxes with 5 pencils in each idea that I gave in my first comment. $\endgroup$ – Dave L Renfro Aug 16 at 13:23
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Why you can multiply boxes and pencils, but cannot add?

In this case, you're multiplying pencils-per-box with boxes. The units cancel and you're left with pencils. Teach students to write fractions with units, and cancel accordingly, just as with numbers.

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    $\begingroup$ Maybe this should really be a comment, as it is really taking issue with X in "if you can do X, why can't you do Y". $\endgroup$ – Nick C Aug 15 at 18:32
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    $\begingroup$ I don't see much of a conceptual difference between "cannot add pencils to boxes" and "cannot add pencils to pencils per box" that I can use for an elementary student to explain and convince him. $\endgroup$ – Rusty Core Aug 15 at 18:48
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    $\begingroup$ @RustyCore: for older students, units and dimensional analysis are great tools to learn to make sure you're multiplying vs. adding the right things, or if you should be dividing. My first memory of seeing this was I think high school chemistry where my teacher used this routinely in evaluating formulas. But it might be worth trying for younger kids, like asking "adding 4 what to 5 pencils in each box?" might illuminate the problem. Actual fractions might or might not help, IDK. $\endgroup$ – Peter Cordes Aug 16 at 10:13
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    $\begingroup$ This goes to the core of the problem, but I don't think it's a good answer for an elementary student. Heck, many adults have trouble working with units properly :) If you want to use this explanation, you should also show how you'd actually explain this to the elementary students in question. $\endgroup$ – Luaan Aug 16 at 11:51
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    $\begingroup$ @PeterCordes +1 Dimensional analysis. $\endgroup$ – Micah Epps Aug 16 at 15:57
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The thing is, it is possible to add 4 boxes to 5 pencils.

4 boxes + 5 pencils = 9 things.

So start by showing the student what they can do with addition, and what its real-world application is. But then remind them what you really want to know - how many pencils are in the boxes (which they can easily count to see it's not 9).

And then you can show them how we can use multiplication to figure out how many pencils are actually in the boxes.

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    $\begingroup$ That's exactly the answer I wanted to write :) The number has some significance, and you must avoid operations that break that significance. If you're counting things, boxes and pencils add up. If you're counting pencils, adding the number of boxes makes the total meaningless - it's not a total of pencils anymore. In the end, the numbers must be useful for something - and that's the part that's often missing from maths education, where you just keep working with numbers that don't really mean anything. The kid has been trained to add numbers together, and now you pull the rug from under them $\endgroup$ – Luaan Aug 16 at 11:48
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    $\begingroup$ "In the end, the numbers must be useful for something - and that's the part that's often missing from maths education," - I totally disagree. You are confusing mathematics with elementary arithmetic. Teaching elementary arithmetic to kids is an excellent idea, but don't confuse it with "maths education" because whatever it says on your school timetable, they are completely different things. $\endgroup$ – alephzero Aug 16 at 22:21
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    $\begingroup$ This question brings up strong feelings. (In my. opinion) arithmetic and mathematics are not completely different things. There are differences, but they are parts of the same beast. $\endgroup$ – Sue VanHattum Aug 17 at 0:43
  • $\begingroup$ @Sue VanHattum: Regarding alephzero's but don't confuse it with "maths education", I don't see the point (unless snobbery is involved) because one could say this about every area of mathematics. For example --- teaching Lebesgue integration to graduate students in math is an excellent idea, but don't confuse it with "maths education"; teaching calculus to first year college students is an excellent idea, but don't confuse it with "maths education"; teaching Sylow's theorems to undergraduate students in math is an excellent idea, but don't confuse it with "maths education". $\endgroup$ – Dave L Renfro Aug 17 at 7:18
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I wonder if it would help to have them make up the problems, instead of you? Clearly, these students are already discounting the meaning of math.

There is lots of research on the efficacy of well-led classroom discussions about math topics. (Deborah Ball has written eloquently about this.)

One book I loved, set at this level, is Little Kids: Powerful Problem Solvers, by Angela Andrews.

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I see in one comment that you refer to 'him', so I now think you are working with one individual student. If he really doesn't see why you can only add like things, then he has already decided that math does not make sense.

If I were working with this student, I might focus on food. 1. You have 3 oranges on the counter and 2 oranges in the fridge. How many oranges do you have? 2. You have 3 oranges on the counter and 2 apples in the fridge. Can we put them together? How would we talk about them? (5 fruit.) After focusing on sensible problems, maybe you want to play with absurdity so you can laugh at silly answers together. You have 3 cars on the counter (they must be toys, not real, huh?) and 2 blueberries in the fridge. Can you put them together? (Maybe. As objects.)

Once he is clear on why adding only makes sense with like objects, then you can work on the multiplication issues.

Your student can make sense of this if you talk in language that works for him. You can find that through your connection with him better than we can find it through general principles.

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  • $\begingroup$ My original answer assumed a classroom group. I see you mentioned 'him', so I have edited my answer for work with one student. $\endgroup$ – Sue VanHattum Aug 16 at 15:22
  • $\begingroup$ (1) While the book you are suggesting may have some useful information, its authors have tarnished themselves by close links to NCTM; one chaired the K-4 team for the 1989 Standards, another participated in writing NCTM's 2000 Standards. NCTM's standards and the dozen of programs that were created in the early 1990s according to the Standards downplayed pencil-and-paper computational algorithms and understanding of abstract algebra, while pushing context-based, calculator-supported, proof-free curricula onto elementary, middle, and high school students. $\endgroup$ – Rusty Core Aug 16 at 22:51
  • $\begingroup$ (2) The Standards and the math programs were universally ridiculed, and many states created their own programs, for example California enacted its math standards in 1997. The failed programs retreated and for more than a decade stayed dormant, waiting for a good time to attack. NCTM participated in creation of Common Core standards. In their meetings NCTM execs did not shy from calling Common Core a continuation of NCTM Standards. The old math programs returned, aligned with Common Core they looked legit. $\endgroup$ – Rusty Core Aug 16 at 22:52
  • $\begingroup$ (3) Sadly, very few fight the new incarnations of these programs, the battles fought quarter century ago are almost forgotten, and their battlefields — that is, websites, forums and mailing lists — are razed and erased. For all the above reasons I am not inclined to take advice from NCTM onhangers, and I consider NCTM itself an enemy of the national education. (Some links: web.archive.org/web/20190309170039/http://…, csun.edu/~vcmth00m/riley.html, toomandre.com/my-articles/engeduc/Wars.pdf) $\endgroup$ – Rusty Core Aug 16 at 22:52
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    $\begingroup$ Your comments are partially incorrect on facts (eg "universally ridiculed" is incorrect) and quite opinionated. They have little to do with the question you asked. $\endgroup$ – Sue VanHattum Aug 16 at 23:19
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  1. When they are first learning multiplication, keep the numbers very small and allow them to do repeated addition. 2*4 and 4*2 are good ones to start with. Do 3*5 before 4*5. (Or a sequence of 1*5, 2*5, 3*5, 4*5, etc.)

  2. Try to keep things simpler. Not boxes of pencils. But groups of pencils. Boxes of pencils is a bit of a word problem and a conversion problem.

  3. Teach them the multiplication table via memorization and drill. Don't only approach multiplication in this manner...use concrete counting examples as well. But take a belt and suspenders approach. IOW, don't exclude learning of this kind. Having learned the table, kids are more ready to use it. Also, do not underestimate the joy in memorization and recall. Look how kids are with state capitals. Or how kids compete in games even simple drill.

  4. Just persist and prevail. Don't assume they are as smart as you or as experienced. Repeat, repeat, repeat. That is the path to instruction more than "killer explanation". But of course, as in 3, use explanations AS WELL. But don't expect concepts themselves to magically unlock a stuck door. Some people even need to just learn by imitation, practice, and correction. (See coaching in sports and music.)

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    $\begingroup$ You might have missed the question, which is: why adding pencils to pencils is ok, but adding pencils to pencil boxes is not? If someone asks you why you cannot add four boxes to five pencils, what do you answer? Do you say that there are no boxes, there are groups, which cannot be added to pencils? $\endgroup$ – Rusty Core Aug 16 at 4:22
  • $\begingroup$ Just say you can't do that because they're different. Don't belabor the explanation. Don't try to achieve victory with magic lightbulb blinking on. THEN shift to a different mode of instruction. You insist on thinking that the key to lock (explanation) is the path to training of new skills. And it isn't. $\endgroup$ – guest Aug 16 at 4:53
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    $\begingroup$ What does that actually teach the children, though? You told them "adding boxes and pencils is bad". That's a meaningless rule that only works in a very small context of the problem of adding pencils together. Counting how many fruits there are in a basket, adding apples to oranges is exactly what you do want to do. That's the distinction you need to teach, not a pseudo-general "you can't add different things". Heck, you're probably adding different things much more often than same things even if you consider e.g. all apples to be "same things". $\endgroup$ – Luaan Aug 16 at 11:58
  • $\begingroup$ Don't expect to teach solely by explication. Did Mr. Miyagi teach Ralph Machio to fight with the "paint the fence"? (yes) $\endgroup$ – guest Aug 16 at 23:09
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I am right now teaching Numbers to kids and I think it is the toughest part because Numbers are the true abstract thing in math. So I first taught them what 1(One) means and they got it pretty quickly. So now I am approaching bigger Numbers and I am trying to make them understand that bigger numbers are "COLLECTION" of smaller numbers. So the problem with "things" and representing them as "numbers" is that there is a transition between this real world, "things" and Mathematical Universe where those "things" are represented as numbers and this transition may not be as automatic and intuitive as that of a grown up. Addition can be used as an aid to explaining collection. There are additions where constituents are not visible after addition(7+2=9, or milk in milk) and add then there are additions where constituents are visible after addition(2 pencils + 4 pencils), these additions are collections of things. Also one of the things I try to make clear is there are various aspects of addition: 1. What is addition? 2. What is it used for? 3. What is the process of addition?

Difference between individual things, collections and parts of a thing are important and must be made clear to kids...

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