[jasode]

>For example, 3 bundles of 5 bananas is different from 5 bundles of 3 bananas although they total to the same number of bananas. Their structures are different.

I don't fully buy into this justification. The "5x3" problem on the test had "pure" numbers with no annotation of "objects". It's the blog writer that inserted an additional interpretation of "bananas" or "bundles".

Instead, the "5x3" can be interpreted as counting iterations of "rows" -- or -- "columns" of a rectangle. Whichever orientation the child picked in his head can yield 5+5+5 or 3+3+3+3+3. In fact, take a closer look at the photo and you'll see in Question #2 that the child had a "different rectangle orientation" than the teacher! The Q1 & Q2 should not have been marked as incorrect.

As for the other justification about possibly using a commutative law that's out of sequence with the learning curriculum, it still seems possible to interpret "5x3" using plain English as "take 5 and copy it out 3 times". No jumping ahead to Commutative Law required.

[tremendo]

When I learned English (second language) I remember thinking "wow, wonderful, the language of multiplication tells you exactly what to do!" which I read as, in this case 5 × 3 => "[five times] three" 3+3+3+3+3, as the teacher illustrated, but here the student apparently answered "five [three times]".

In my first language (Spanish) the multiplication is read as "five by three" which conjures up rectangles or lists, which can be vertical or horizontal oriented, and in either case, less clear and unambiguous than the English version.

Still I believe it's certainly teaching the wrong lesson to mark the answer as incorrect, especially when the red mark comes without explanation. Even if the problem states "Use the repeated addition strategy". The author mentions it's crucial to understand this but I don't believe important enough to discourage a young student this way. The explanation of what was requested and the method of arriving at it should be made explicit, and it may have happened in class, we just don't know.

[mattmcknight]

I feel the opposite way about the English reading "5 times 3". In English, the subject comes first, so I would expect the sentence to mean take 5, use "times" as a verb, and 3 as the adverb. Likewise, if you read it as "5 multiplied by 3", you would expect to take five, three times.

[signal11]

Exactly. And this is why we use math operators rather than English terms for math operations. If you want to specifically mean 5 times a group of 3, define a new operator for it, like 5 ○ 3. Don't 'overload' the x operator with your own arbitrary meaning.

[chris_wot]

Ummm... I think you just overloaded the x operator yourself!

[newjersey]

If you look at the next question, they go over the five by three in a rectangle approach.

Maybe we should do away with grading students based on exam performance altogether.

[chris_wot]

My wife is a teacher in the NSW education system (Australia) and I've seen her use the rectangle system. However, the rectangle system is used to also show that if you take the same rectangle with the items placed in it in a uniform distribution, the rotate the rectangle and its contents by 90 degrees the the number of items are the same, but the row and column numbers swap around.

If anything the rectangular system shows that multiplication is commutative, which I feel is its real value. Interestingly enough, that isn't ever explained to most teachers so I'm not surprised if it's being misapplied as a technique for learning!

[vixen99]

As against what happens out there in the tough world of work.

[dznkorxt76qb]

> "you'll see in Question #2 that the child had a "different rectangle orientation" than the teacher"

Vectors can be considered identical for the same reason... [1, 0] and [9, 0] are the same arrow if you move your head in the latter case. Here, the teacher is assuming the kid's head is at [0,0], when rotation (or translation) doesn't change the arrow any more than the rectangle. Neither is wrong. The lesson: memorize the teacher's poor use of language; ignore objective reality.

[jmilloy]

> The "5x3" problem on the test had "pure" numbers with no annotation of "objects"

It's not the "5x3" problem but the "repeated addition strategy" problem. I think that's part of the problem. Similarly, the bananas example isn't about the 5 and the 3 but about a difference between counting "x sets of y" and "y sets of x".

[jasode]

>a difference between counting "x sets of y" and "y sets of x".

You're making the same mistake as the blog writer by overlaying a difference between "x" and "y" that was not on the test.

The child did do the repeated addition strategy. It's just that the child's "shape" of the addition didn't exactly match the teacher's. If the point of the problem was the "repeated addition" instead of the final answer "15", the child still did it correctly. He/she showed his work of repeated addition!

The actual test problem was stated as "5 times 3" and not "5subscriptX times 3subscriptY" or "5subscriptBundles times 3subscriptBananas". You're arguing about a test the child didn't actually take.

[jmilloy]

The objective and obvious difference between the 5 and the 3 is that the 5 is first and the 3 is second. The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. (The bananas and bundles just illustrates an example for why, in another context, being first or second would be important. But on the test, 5 is still first.)

On the other hand, you are invoking a "repeated addition" that the student was never taught. Your repeated addition strategy is "add <one of numbers> together <the other number> times". The taught repeated addition strategy was "add <the second number> together <the first number> times".

[signal11]

> The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique.

Um, this is sophistry. The question asked for "5 x 3" using repeated addition. The x is a very well defined mathematical operator and "repeated addition" has a very well-defined meaning, and the child has demonstrated it by repeatedly adding 5 three times.

Yes, the child's cardinal sin is he Did Not Do As He Was Taught(tm), but seriously, that's more the teacher's and the school board's problem in my book, not the child's.

[chris_wot]

The way the child is being taught is because the teacher or course administrator has misunderstood the purpose of teaching arithmetic via repeated addition.

Repeated addition is relying on the fact that children see the world in a very concrete way and have not started to understand concepts in a more abstract fashion. Thus you use objects to explain concepts, like: every cat has one tail, I have 3 cats so how many tails are there in total?

You introduce notation in the class, but I can't see how it is valuable to use an abstract expression like 1x3 without a concrete description of the example of cats and tails. After all, you aren't really teaching repeated addition, you are just using it as scaffolding to provide an insight into multiplication!

The fact that the answer given can be shown as wrong has already demonstrated that the child (and parent!) was annoyed because it made little sense to mark it as wrong. It probably caused more harm than good, because now the child questions their understanding of the subject matter, yet ironically they do appear to have grasped the concept!

So at this point, the poor pedagogy of the teacher in misusing the counting technique means that the child starts to doubt themselves unnecessarily, they become locked in to a scaffolding technique that will later need to be discarded anyway. When they hit non-integer rational numbers - numbers with decimal points - they aren't going to be able to add these together, instead they will need to grasp that you can scale down numbers if you multiply any rational number between 0 and 1.

[jmilloy]

> After all, you aren't really teaching repeated addition, you are just using it as scaffolding to provide an insight into multiplication!

You may be right. This is the interesting part of the discussion, and you've framed it well. I think it can be scaffolding technique also for the application of definitions, the expansion of symbols to their definition. Perhaps there is a better way to say that (or other examples), but the point is that I don't think that the exclusive value in teaching the technique is soon-to-be discarded scaffolding for multiplying numbers.

> the child (and parent!) was annoyed because it made little sense to mark it as wrong

The impact that the -1 has on the child is also interesting. I think it scored 1 out of 2, so it wasn't marked "wrong" so mach as "partially correct". It should be clear to the student that they basically got it right but slightly misapplied the technique, due to the comment, shouldn't it? If it isn't, it's the result of too much focus on the grade and too little focus on the comment.

It seems to me more likely that it's parents and other adults who see this -1 so negatively, and impose that on the kids. I would have been upset as a kid, too, but the sooner someone could have gotten me to be okay with quantitative imperfection, the better.

[jmilloy]

It doesn't seem like a cardinal sin so much as a small quantitative note that the process was taught a different way that the teacher thinks is important.

Let's suppose one student can follow the procedure when asked but can't actually multiply in application, one student can't follow the procedure correctly but can multiply when needed, and a third can do both. Probably the first student will get questions on this quiz correct but will struggle on much of the rest of the unit, maybe get a low grade or hopefully get the help they need. The second (with the paper shown in the OP), will probably get an high grade because they got partial credit on a silly quiz. The third will get a higher high grade. What's so bad about that?

BTW, appealing to definitions won't work here, because the x does have a very well-defined mathematical meaning: a x b := b + ... + b.

[orless]

a x b = b + ... +b is not a strict definition, it's just a convention.

Check the English Wikipedia:

https://en.wikipedia.org/wiki/Multiplication

It's, as you say, 5x3 = 3 + 3 + 3 + 3 +3.

But now check Russian Wikipedia:

https://ru.wikipedia.org/wiki/%D0%A3%D0%BC%D0%BD%D0%BE%D0%B6...

There you'll see 5x3 = 5 + 5 + 5.

So much for the "very well-defined mathematical meaning".

[signal11]

> ... does have a very well-defined mathematical meaning: a x b := b + ... + b.

Please complete the definition. which is that a x b = b x a, so a x b can also be written as a x .. x a. There is nothing special about the order.

[jasode]

>The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique.

If you(royal-you) insist that the 5 being the first factor has a specific job and you teach such nonsense to a child, it means you're not teaching actual mathematics.

In _real_ math, the factors/mutiplicands have no notion of ordinal rank such as "first" or "second" or "specific jobs". Even if the child was not formerly taught The Commutative Law, it's not impossible for him to see multiplication tables[1]. (In fact, many are hung as big posters in elementary classrooms.) Any child with pattern recognition abilities beyond a chimpanzee would notice that the cells of XY have the same answer as YX. He/she would ask mom/dad/teacher "is xy always same as yx?".

In the world of _pseudo_ math that stresses bizarre hoop jumping, we overlay non-mathematical concepts such as "specific job" to factors. Maybe this skill is important and transferable to the enlisted man to make sure he makes his bed before cleaning his machine gun instead of the other way around so everyone in the squad doesn't get punished with 50 pushups. But don't pass it off as "teaching math."

[1]https://www.google.com/search?q=multiplication+table&es_sm=9...

[jmilloy]

I only have a bachelors degree in math, focusing on theory, but I think we have a different understanding of what actual mathematics is. For example, in some "real math", definitions, properties, and axioms are well-distinguished and mixing them up can get you in trouble.

More importantly, are we even trying to teach "real" math to elementary kids (I wish we did, but I don't think we do) or "computation"? Both are useful and interesting.

[chris_wot]

Yeah, but the problem is: repeated addition is attempting to take something very concrete like I give four children three marbles each, how many marbles does each child have?

You then use that addition technique to have them add up the number of marbles (in essence it's as if you are asking them to count on their hands, which is a valid technique at this level).

But that helps the child understand the concept of addition in a very literal and concrete fashion, because at the age of 4-5 years old (sometimes older), children don't think at a higher level of abstraction. And using symbols to represent multiplication IS a higher level of abstraction.

It seems to me, a non-educator, that the counting technique has value in word problems. But as soon as the child shows they understand the concept, then you introduce the notation (e.g. 5 x 3), explain the numbers can be added up either as five values added up three times, or three values added up five times.

That the test talked about a "strategy" is not really maths, and frankly it seems to be misapplying a solid teaching technique, leading to confusion, anger and a lack of confidence in the child. If that's happening, then I'd suggest the technique is not all that solid and teachers and other educators should seriously consider whether it is causing more harm than good.

P.S. If you have a Bachelors in Mathematics, then surely you can see that there is a fundamental problem if a child is taught that 5x3 is not the same as 3x5?

[3pt14159]

I was leaning that way, but when I tried to the plain English way, I defaulted to "Five threes" not the backwards "take five and copy it out three times".

Once he made the tie back to programming he was able to change my mind that this was important enough. Although it does seem unfair for the individual child.