Why 5 x 3 = 5 and 5 and 5 Was Marked Wrong (opens in new tab)

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.

When I took geometry I often was burned by my teacher when she'd ask me to come to the board and solve a problem. She'd say "You can't do that we haven't gotten to that part of the book yet, sit down already!" It all just seemed to logical to me. Maybe I didn't know what "it" was called but it was logical and easy to work out.

Most of my math teachers treated me this way, they'd be upset with me because I didn't write 100 copies of the problem on my homework but I'd pass all my tests. I was constantly in trouble with my teachers often being reported to the administration as a cheater. One teacher made me take my shirt off he was so convinced I must have written the answers on a sleeve or something. All because I didn't miss a single problem on his test but never turned in a single page of homework to his class.

The argument about bundles and bananas is besides the point. But even then it works because

3x(5 bananas) = (3x5) bananas = 5 x (3 bananas)

Of course, if I define my own special multiplication, then 3x5 != 5x3. If I put enough concepts on top of it an operate in obscure mathematical domains, sure, I'll need to be careful about things being accidentally equal but not equivalent. But I bet the student expected that multiplication behaves just like the multiplication an elementary student knows. Or do you expect them to define the operations they're working with?

When I grade the work of students, I will accept any answer that is in accordance with the question. Who cares how often they commute things that commute, or if they picked an entirely different approach that was never even discussed. If it says solve X using Y and they solved X using Y, they deserve the points.

If the teacher didn't make it clear enough in his question, then it's not the student's fault.

This is absolutely false. Becoming a 3rd grade teacher is not that hard (being a third grade teacher, on the other hand, surely is).

> It’s more important than ever for students to understand the difference between equal as a result and equivalence in meaning from a young age because it is a fundamental computer science concept.

It's not though, because you can learn these things later in life and still understand them just as well. What exactly is lost if you don't have this figured out on your 9th birthday?

People who don't understand a subject should not teach it. If they understand education but not math, then let them teach education.

Nonsense. By those definitions, equality is a special case of equivalence - one that simply neglects to strongly emphasize function (which could be taken as value; the latter still doesn't mean 'identical').

5 x 3 = 3 + 3 + 3 + 3 + 3 and 5 x 3 = 5 + 5 + 5

are both numerically equal and functionally equivalent. The student at least understands the commutative property of multiplication, unlike the teacher.

If the marker's motivation is to identify that difference, then this is horribly misguided. In my opinion the marker has just made an error.

Note the stated goal of the exercise: "I can use multiplication strategies to help me multiply".

Using commutativity is a multiplication strategy and it's an essential goal for students at this level to learn this as part of their work with number.

    x * 0 = 0
    x * S(y) = x + (x * y)

    5 * S(2) = 5 + 5 * 2
             = 5 + 5 + 5 * 1
             = 5 + 5 + 5 + 5 * 0
             = 5 + 5 + 5 + 0
             = 5 + 5 + 5

  Comment:
  This approach seems a great way to discourage smart young kids.Do we expect geometry students to grasp integrals? Want to convert kids to be math people? Laud the child for grasping the connection between multiplication and addition, use it as an opportunity to introduce the commutive property, and work on equivalence down the line….

  Brett Berry's Answer:
  I agree! Great opportunity for learning!!

Flagged.

There are algebraic structures where multiplication is not commutative, but I don't think this was the case here.

Those are definitely qualities you want in a programmer.

However, just becuase 53 is pronounced "5 times 3" doesn't mean that it's exactly the same thing. We pronounce it that way because it's convenient, "times" is a one syllable word that we can inject right where the multiplication sign goes, and have it be a meaningful, representative sentence. But that doesn't mean that "53" derives its semantics from "5 times 3". "53" is an abstract mathematical expression that has no concept of "grouping". It can't have a concept of grouping --- how would you extend grouping to something like matricies.

Edit: Also, "5"x3 or 5x"3"?

How many fingers do you see?

The worst teachers I've had graded like this this one did: demanding regurgitation of the precise idiosyncratic procedure they use, while refusing to recognize equivalent methods or equivalent terminology.

"To my mind, it makes no difference at all which is which. In fact, today it is more common to call them both "factors" and not make such a distinction. I wouldn't fight over this, on either side."

Does anyone have a more sane explanation of what the goal is? I can't think of any way this is going to be helpful to the student.

http://mathforum.org/library/drmath/view/57919.html

1. The teacher understands (and is expecting 9 year olds to learn) the pedantic difference between equivalence and equality. Keep in mind, this being elementary school, the teacher likely is a generalist and also teaches reading, science, and social studies.

OR

2. The teacher has a very rigid grading guide that specifies how much credit is given for any given answer/technique, and is simply blowing through 50+ tests at 1AM, applying this standard grading. Ironically, in this scenario, the teacher is just as pointlessly constrained to "following the rules" as the students. He/she may even agree that it's ridiculous to deduct a point.

Or at least that would make more sense than arguing the kid is wrong via Javascript and matrices.

Without access to this particular teacher's curriculum materials, it's not possible to know for sure what definition is being referenced by the "repeated addition strategy". I'm inclined to assume the teacher knows what they're doing and has graded the work appropriately.

There are many comments on this thread about multiplication being commutative by definition, but this is not quite correct. Following the same definition of multiplication I cited above, it is a theorem that $a \times b = b \times a$. When I teach Abstract Thinking (a sort of introduction to proof writing course for mathematics students), I have the students write proofs for this property of multiplication of natural numbers, and the other familiar properties (cancellation, distribution, etc.). If anyone is interested, I've broken the steps out into worksheets that I give to my students, and you can see them at the link below. [2] [pdf] (Multiplication of natural numbers is section 5.5.)

[1] https://en.wikipedia.org/wiki/Natural_number#Multiplication [2] [pdf] http://billkronholm.com/wp-content/uploads/2015/10/MATH280.p...

It’s more important than ever for students to understand the difference between equal as a result and equivalence in meaning from a young age because it is a fundamental computer science concept (...) Equivalent means not only are they equal, they are also of the same data type. In other words, they mean the same thing.

Except that this point is totally misguided because 5+5+5 and 3+3+3+3+3 are, in fact, the same thing. A member of the set of natural numbers, commonly known as 15, and that you can write as S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(0))))))))))))))) if you have the patience to do so.

In fact, the author uses the == and === operators in JavaScript to illustrate his point, but of course, 5+5+5 === 3+3+3+3+3 resolves to true in Javascript, and in any language under the sun that compares stuff by value.

And then, if you compare by reference, the result of == or similar operators doesn't depend at all on whether you made your integer by adding up three fives or five threes: in Java,

  Integer i1 = new Integer(5+5+5);
  Integer i2 = new Integer(3+3+3+3+3);
  Integer i3 = new Integer(3+3+3+3+3);
  System.out.println(i1==i2 + " and " + i2==i3);

So I don't see how this nonsense would teach kids anything useful about computer science. The only thing it can do is confuse them.

equivalent is defined as, “equal in value, amount, function, or meaning.” In the above problem 5 x 3 is equal to 5 + 5 + 5, but they’re not necessarily equivalent. Equivalence relates to meaning, so it depends on the meaning of multiplication, as the directions indicate.

First of all, the sign in the exam statement is an equal sign, not an equivalence sign. So if 15 is equal to 5x3, what the student wrote is perfectly fine. Also, "solving" a multiplication means finding out its value, which is what he did.

Secondly, you know what also is a fundamental computer science concept? The logical operation "or". The definition of "equivalent" in the blog is reported as “equal in value, amount, function, or meaning.” "Or", not "and". So the definition doesn't say anything about equality in meaning (however you define it) being a necessary condition for equivalence.

Maybe teachers are "experts on child education" but that doesn't exempt them from knowing something about maths if they are supposed to teach them.

Neither of these things have anything to do with the original problem. The best thing I can think of him doing, as a self-proclaimed "math evangelist", is to shut up about concepts that are obviously beyond his understanding.

Equal is defined as, “being the same in ...value.” Whereas equivalent is defined as, “equal in value...”

If you can't be precise in what you write yourself...

> The teacher obviously knows (I’m assuming) that 5 + 5 + 5 is the same as 3 + 3 + 3 + 3 + 3.

> So why would one method be preferred over the other?

> Because thinking of 5 x 3 as, literally, “five groups of three” will help them when they learn how to divide. (That’s what the Common Core standard here is getting at.)

from http://www.patheos.com/blogs/friendlyatheist/2015/10/21/why-...

Also notable is the explicit assumption of good faith:

> Let’s assume for a second that this teacher isn’t an idiot. (I know. I know. Bear with me for a minute.)

> What possible explanation could there be for deducting points from this poor child’s exam?

2) Javascript's equality operators can confuse even longtime practitioners and bringing them up muddles the point.

I think the author is trying to say that 5 x 3, 3 x 5, and 30 / 2 are equivalent but not equal because they represent different operations, which he is somehow equating to types in computer science? This is nonsense, all three expressions are obviously the same type, and they are referentially transparent mathematical expressions, so they would be equal not just equivalent.

3) Setting students up for matrix multiplication? Really?

Would probably depend on how how much the concept was presented in class and the books, etc.

5*(1 + 1 + 1) is equivalent. If the student had written (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1), should the teacher have marked it correct? It's essentially the approach in question 2.

I'd argue that in this example, it doesn't demonstrate an understanding of "repeated addition" and at the very least should warrant follow-up by the teacher. The commutative example is more subtle and context would be nice to understand, but if this was a no consequence homework assignment that led to a quick follow-up by the teacher then it seems like a good move.

The student got a grade of 4/6. There is no evidence from what can be seen on the photo if there are more than two questions or not - however it appears that after #2 there are instructions about showing work, etc. So it's entirely within reason to think that there were only two questions, yet many many people seem to accuse the teacher of being awful for not assigning partial credit. The evidence present on the page is not enough to conclude either way - there are solid arguments and inferences in each direction, yet everyone seems to argue from a "the answer was all right" or "the answer was all wrong" perspective.

Similarly - everyone is focusing just on the "learning to multiply whole numbers" aspect of this. But I presume this assignment was given in the course of a broader teaching curriculum. Perhaps the goal is to get the kids to arrange things in a certain way, because it provides a bit of foundational knowledge for next steps. Some next steps where pushing the "3+3+3+3+3" version of this is a "better" representation:

* algebraic concepts: 2x becomes x+x, etc. I don't know a way to write 2x in terms of 2+...

* fractions: (this is basically the same as above) 4 * 1/2 is 1/2 + 1/2 + 1/2 + 1/2, but again I don't know how to write it in terms of 4+ without requiring a bigger transform of first doing the multiplication then switching the sign.

* Matricies - here is a case of multiplication that isn't commutative

Point being - when teaching sometimes things are left out at first, for the sake of a simple consistent framework to built more concepts upon. Later - those concepts can help to understand additional properties or adjustments to the original facts. It's not required to teach the rules of commutative and associative and so on immediately.

Another funny thing about this is everyone just assumes that the paper as shown represents the totality of output from the teacher. It's entirely possible that after the quiz or assignment, the teacher gave a lesson on why this (perhaps common) 'mistake' is wrong.

I guess I'm rambling off my original point - but I don't really understand how this entire thing is causing so much vitriol and hate and unfounded speculation - other than a bunch of people projecting their own frustrations with some shitty teachers they had in the past.

Firstly, as we know, multiplication and addition are associative, which means if you ever teach a child that 5 x 3 is different to 3 x 5, you are imparting wrong information.

The issue is that the question asks the child to "use the repeated addition strategy to solve: 5x3". The reason this is a problem is because "repeated addition" is indeed a strategy to teach children the concept that if you take a multiple of some number the. It is like repeatedly adding that number of items, a number of times. It is used as a stepping stone towards fully understanding multiplication,after on, and takes into account that young children think I terms of what they see. So for example, they see that a dog has 4 legs, and if you have 3 dogs then you add the four legs together three times (one for each dog).

Notice that it's madness to teach this as a. "addition strategy", because at that age "strategy" is far too abstract a concept for most children to grasp. The irony is that teachers then attempt to teach using a technique that uses a low level of abstraction, but when they call it the "addition strategy" they have just attempted to teach this technique that is using a more concrete methodology via language that uses concepts that are arguably more abstract than the concept they are attempting to teach!

You can see that the whole point of that technique is missed completely on that exam because of the question being asked. In fact, to have a student demonstrate understanding the I fact the question should be "I have five jars of Jellybeans. Each jar has 3 Jellybeans in them. Show me how you would represent the number of jars times by the number of Jellybeans in each jar, using addition."

You see, the point of the strategy is entirely being missed here. The author protests that the child will get confused because if they rely on the law of association with subtraction and division they will get the answer wrong, and be confused. But that's not what is happening. The child has clearly understood that actually, 3+3+3+3+3 is the same as 5+5+5. In actual fact, the student has shown a clear understanding of multiplication via addition.

If you think that child will be confused, wait till they get to fractions and numbers with decimal places! Because at that point, you can't use addition to explain multiplication and then you need to explain multiplication in terms of scale. There's actually a case to answer that the entire technique of teaching multiplication via addition is fundamentally flawed and it's better to teach in terms of scale anyway. I don't subscribe to that view, but I can see why it might be held.

I have to also take issue with using the definition from what looks like the Cambridge Dictionary's noun definition is that this is NOT the same precise meaning as equivalence in mathematics. In fact, if you were to use first-order logic, then it would be:

iff 5x3=15 then 5+5+5=15

or,

(5x3=15) ≡ (5+5+5=15)

That satisfies the two expressions logical equivalence. So the statement that this is NOT logically equivalent is entirely wrong.

Furthermore, the author has not read the definition on Wikipedia carefully enough. It says:

The multiplication of two whole numbers is equivalent to adding as many copies of one of them, as the value of the other one

The assumption being made here is that Wikipedia is saying that the number that is to be added up multiple times is the leftmost number in the expression, but it does not in fact say this at all. It says to add "as many copies of one of them", which means it could be referring to the left or right hand value in the multiplication expression.

The common core and the techniques used to teach young children are solid. Unfortunately, it looks like the way they have been used and taught to educators is the problem here! The fact that you can see the framework leaking into a test question shows that there is a fundamental flaw in the pedagogy of whoever is teaching that class.

https://medium.com/@highsource/the-only-reason-for-this-answ...

The only reason for this answer to be marked as “wrong” is the teacher saying “You did not apply the strategy I’ve tought EXACTLY as I tought it. You have dared to understand the idea and acted on your understanding INSTEAD OF mechanically applying the actions I told you to.”

Most of your argument does not have a stand.

You’re quoting “the definition of multiplication” which says “adding as many copies of one of them as the value of another one” and use this as basis to argument that 3+3+3+3+3 would have been correct and 5+5+5 is wrong. But even this definition does not say “the first one” and “the second one”. It says “one of them” and “another one”. So 3+3+3+3+3 is just as correct as 5+5+5. Period.

You’re quiting definitions of equal “being the same in quantity, size, degree, or value” and “ equivalent” as “equal in value, amount, function, or meaning”. First point here: the task said nothing about “equivalence”. It just said “solve 5x3”, applying the repetitive addition strategy. So it absolutely does not matter if 5+5+5 is equivalent to 3+3+3+3+3 or not.

Next point, you say that 5+5+5 is equal to 3+3+3+3+3 but not equivalent. If you explicitly add some trivia like banana bundles then you can somehow argument that there is some difference in amount, function or meaning. But only if you explicitly add these details. In the original task, there are no such details so there is no way you can show difference in amount, function or meaning.

I agree that using a commutative property before it was itroduced would have been wrong. But the child here did not use the commutative property! Absolutely not. The child applied the repetitive addition strategy, just (obviously) not at the EXACT convention that the teacher taught. This has nothing to do with multiplication being commutative at this point.

The whole point of this answer being wrong is for the teacher to enforce application of the taught rules or strategies EXACTLY how they are taught. There is no sensible reason for the repetitive addition strategy to be applied as 5+5+5 instead of 3+3+3+3+3. Only the convention and “do as I said”.

Whether “do as I taught” is a good thing or a bad thing really depends. For some children it is really important that they follow the teacher mechanically, repeating exactly what they were told. This way they are at least guaranteed to manage the basic mechanical tasks. So the teacher is more or less guaranteed to have some borderline success with them.

But many children understand things on a much deeper level from the very beginning. They understand the sense and the reason and the logic of math much deeper than the basic mechanics. And once they understand the internals like the absolut truth of 5+5+5 and 3+3+3+3+3 giving the same result, it becomes illogical that one answer is right and the other one is wrong due to “you have not done this EXACTLY as I have tought you”. You see, math is the absolute truth, so if your conventions and enforcements contradict that, these conventions and enforcements are simply wrong. Yes, maybe you first have to do “wrong” for the better good later on, but don’t pretend you’re right.

Finally, you bring the point of “Respect the teachers” because they are “ they are qualified experts on child education”.

Oh, my, I don’t even know where to begin.

There are really different kinds of teachers, some doing great jobs and some, well, not-so-great. Of course you have to respect them as you would respect any other human being.

But this does not mean that teachers or teaching programs are infallible. Respect does not mean they are always right, because, you know, “they are qualified experts” and that they can’t be criticized.

I had around 12 or 15 different math classes in the university and really different kinds of professors. Most of them respected the thinking and understanding above all. They did not care if I did a proof exactly as they taught it— or came up with something original (which was, admittedly, mostly, because I skipped the lection). But there were also some which insisted on exactly the same proofs and even notations as they once wrote on the whiteboard. Reasoning: it was harder for them to check the correctness of the proof if it was not in their exact notation! Should I have respected this? I did not and I have brought a few cases to the higher university commissions and had all of the wrongful evaluations dismissed.

You point to the dangers of children later on not understanding matrix operations or “equals” vs. == vs. ===. For me much more dangerous is teaching mechanics and punishing for misunderstanding. I have never ever saw a student or a programmer who had troubles with matrix operations, vector multiplication, or === in JavaScript because of they’ve grasped the commutative property of multiplication for numbers too early.

But I have seen a lot of people thinking and working mechanically with once-learned mindsets which they are afraid or uncabale of leaving. I am afraid, this is exactly the mindset which is enforced by “5+5+5 is wrong because this is not how I taught it”.

Let me tell you this. If my kid would have brought this from school, I’d explain him that 3+3+3+3+3 is just as valid as 5+5+5. But I would have also point out that sometimes it’s not just math that you learn in the math lesson. That you also learn social skills — like that the teacher expects you to be conformant to his or her rules. You have to be able to recognize this in this person. You have to be not just clever enough to understand that 5+5+5 is the right answer. You have to be clever enough to see that there is the other correct answer, 3+3+3+3+3, and that the teacher probably expects that one instead.

If you were asked in the wood shop to make a drawer using dovetail joins and you instead make a drawer using a rabbet joins, well you shouldn't get full credit because the task wasn't about producing a drawer but about dovetail joins.

The learning objective may read "I can use control structures to solve tasks involving repetition", but if the question is "Use a for-loop to print the numbers from 1 to 10", and you used a while-loop, well again you would get partial credit but not full credit.

Many of us are computer programmers, and we know that many successful code solutions to the same problem are of equal quality. The worst code I work with is by people who never seem to care what the code looks like as long as it produced a correct result. Maybe they were only ever taught by teachers who would mark this 100% correct because they thought that answers matter and concepts don't.