Xjyuk

"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?

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.

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?

You can add, though it's more tedious.

"Suppose you bought four boxes of pencils having five pencils in each, how many pencils do you have altogether?"

First box has five pencils. Second box has five pencils. Third box has five pencils. Fourth box has five pencils. Altogether, then, we have $underbrace5 text pencils_textbox 1 + underbrace5 text pencils_
textbox 2+underbrace5 text pencils_textbox 3 + underbrace5 text pencils_textbox 4 = 20$ pencils.

Or more conveniently, we can write $4 text box times dfrac5 text pencils textbox = 20$ pencils.

What you can't do is add one box of six apples, with 3 oranges, to get either 9 apples nor nine oranges, nor four apples, nor 4 oranges.

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.

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.)