The difficulty isn't with fractions. It's about understanding what "+" means.
Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie").
His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about the word "plus" to the symbol. People even use "+" (rather than "plus") to mean combining things. Maybe Addison saw "Nature's Path Pumpkin Seed + Flax Granola" at the grocery store. So why shouldn't he try using it that way?
Somebody needs to communicate to Addison that, in math class, "+" always means something specific. He doesn't know that yet, but he has been asked to use the symbol anyway.
Math uses a whole lot of lingo. If it's not covered well enough, stumbling over the terminology can be an impediment to learning. This includes both new words ("quotient", "integer") and words that are used in everyday language but differently in math ("where" meaning condition or definition instead of place, "of" meaning multiplication, "real" numbers).
That's my favourite explanation so far. That's how I would put it to the student:
Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples:
* At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table.
* Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables.
The first operation is what mathematicians call "+". They write "1/3 + 1/3 = 2/3"
The mathematicians do not have a good name for the second one, so let's invent one: "1/3 1/3 = 2/6"
What's better with this explanation, compared to the "ratios vs. fractions" thing, or the units thing, is that you do not have to introduce a separate category of numbers that sound very similar but act differently.
We explicitly encourage the "let's invent name for this similar but unnamed thing" with our students, and use whatever name they come up with for the thing for the rest of their time with us.
Sometimes those student created names become legendary. They really enjoy the idea that they can have "Jane's relation" be used by students after they graduate. (Of course, sometimes Jane's relation is really Euler's totient function, and we have to encourage them that even though they found something a very good mathematician also found, we're going to call it by the common name.)
I would say the issue with symbols is why I had trouble with linear algebra in college, specifically dot and cross products. The fact that they used the same symbols as scaler multiplication and are also called products confused my mind so much.
Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie").
His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about the word "plus" to the symbol. People even use "+" (rather than "plus") to mean combining things. Maybe Addison saw "Nature's Path Pumpkin Seed + Flax Granola" at the grocery store. So why shouldn't he try using it that way?
Somebody needs to communicate to Addison that, in math class, "+" always means something specific. He doesn't know that yet, but he has been asked to use the symbol anyway.
Math uses a whole lot of lingo. If it's not covered well enough, stumbling over the terminology can be an impediment to learning. This includes both new words ("quotient", "integer") and words that are used in everyday language but differently in math ("where" meaning condition or definition instead of place, "of" meaning multiplication, "real" numbers).