
Why 5 x 3 = 5 and 5 and 5 Was Marked Wrong - nonameface
https://medium.com/i-math/why-5-x-3-5-5-5-was-marked-wrong-b34607a5b74c
======
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.

~~~
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!

------
kator
Funny when I read the image I thought the student was optimizing to less work.
His way was faster "I can use multiplication strategies to help me multiply."
Seems the student's strategy was to do less work by adding 5+5+5.

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.

~~~
ryandrake
Yep, the "erase your brain of that technique because we haven't gotten to that
part of the book yet" reasoning was the most frustrating part of math (and
science) classes in high school. It's a miracle I got through my teen years
still interested in STEM. Contrast with how they treat these situations at the
university level:

University: "Ahh, you seem to be pretty far ahead for MATH 140! You might as
well go test out of the class and enroll in 141 instead! Save some time."

High School: "Conform to the curriculum. Repeat this technique. Obey the rules
or fail."

------
tomrod
What a defense of nonsense. The marking punishes a student for achieving a
correct answer in an appropriate way, regardless of pedagogical justification.

~~~
signal11
I found this link interesting. Under this "Common Core" curriculum,
apparently, students are trained to read 5x3 as "five groups of three" which
is why 3+3+3+3+3 is right and 5+5+5 is wrong.
[http://www.businessinsider.com/why-55515-is-wrong-under-
the-...](http://www.businessinsider.com/why-55515-is-wrong-under-the-common-
core-2015-10?IR=T)

It's hilarious because I read 5x3 as "5, 3 times".

Anyhow, just goes to show Maths teachers have now been replaced by box tickers
who refuse to apply their brain. In my book, the kid demonstrated repeated
addition and should have got the mark.

~~~
pbreit
How do you not read it "5 times 3"? Why would you re-arrange where the "times"
is?

~~~
signal11
Because where I come from, we use the English equivalent of "into" rather than
"times". "5 into 3" roughly translates to "5, 3 times".

The meta-point here is that English (or any other language) is crap for math,
which is why we use mathematical notation. And this bullcrap syllabus is
trying to redefine the "x" operator, which gets my goat.

~~~
chris_wot
The syllabus does nothing of the sort. The addition technique is a way of
teaching very young children in a way they can grasp. However, it relies on
using concrete objects and so far as I can see, should be used as a technique
to aid understanding, and only then should the multiplication notation be
introduced.

------
QuantumRoar
There are just too many assumptions in this article. It says: Use the repeated
addition strategy to solve 5x3. That is clearly what the student did.

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.

~~~
jonsen

      3 bundle x 5 banana/bundle =
      5x3 bundlexbanana/bundle =
      5 bundle x 3 banana/bundle =
      15 banana

------
88e282102ae2e5b
_> They are qualified experts on child education._

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?

~~~
jmilloy
> What exactly is lost if you don't have this figured out on your 9th
> birthday?

Not much. But what exactly is lost if you get 1 out of 2 instead of 2 out of 2
on a quiz in 3rd grade?

If there is a problem, it's that we can't be told that we were partially
correct instead of fully correct on silly problems without it being a big deal
and a failing.

~~~
scrollaway
> But what exactly is lost if you get 1 out of 2 instead of 2 out of 2 on a
> quiz in 3rd grade?

Spoken like someone truly unaware of how children think! You should work in
education, there's plenty of people like that there.

The child _could_ in fact become horribly confused about multiplication
because of a bullshit technicality, and this could set them back months. Or
the child could be certain they're right and this breaks trust in authority --
non-obedient children are not inherently bad, but without careful handling
they can become extremely aggressive.

I certainly relate. In fact, you can fairly easily identify, in all these
comments, who has experienced similar BS and who is knee-deep stuck in theory
without understanding the human component behind it (looking at you, pohl).

The child doesn't see the -1 and think "Oh, I immediately understand why my
answer is wrong! Of course, I understood 3 groups of 5 instead of 5 groups of
3!". No, the child sees it, thinks "but you told me they're the same? ok...",
and is now more confused than ever about what's actually been taught in the
class. Most 9 year olds don't know how to introspect.

Urgh. The comments here are so infuriating _because_ this complete disconnect
is exactly the same as the one the people behind the design of the most
atrocious curriculums and methods have! Damn it, who here is actually taking
into account _their own age_ compared to the kid? (And fun trivia: It's the
same belittling, disconnected behaviour people have when they talk to 18-22
year olds about life experiences they can't reliably have had before the age
of 35... except it's a lot more flagrant here)

~~~
jmilloy
I think you would find we agree much more than we disagree. Though what I find
most infuriating is the blanket assumption (with similar level of disconnect)
that what is being taught is mindless or confusing with no value, often simply
because it's labeled as a "curriculum" or a "learning objective". You're not
automatically right because you "experienced similar BS"; instead you have to
realize that you, too, are coming into it with a bias and blindness.

What I see is a a bunch of people who can't stand seeing that red -1, maybe
because it has been ingrained in them that they have to be perfect. Or maybe
it's natural, and no one helped them git rid of that feeling.

It's so important for young students to feel like they understand and will
continue to understand, in order for them to then achieve new understanding. I
don't know how to write that without sounding like a theorist, but I sincerely
believe it to be true. You've got to get rid of that fear of red ink.

There are tons of poor ways to teach, and poor curricula. This teacher could
be doing a fine job with this student (and the parent's the ones that don't
get it), or could be seriously hindering the child. I certainly wouldn't teach
multiplication strategies this way. But it's not clear to me that marking this
particular answer as only partially correct is inherently and unquestionably
wrong.

~~~
scrollaway
You're still stuck in the theory, talking about how the people in this thread
feel when _none of them matter_. The child matters, that's it.

> _It 's so important for young students to feel like they understand and will
> continue to understand, in order for them to then achieve new understanding.
> I don't know how to write that without sounding like a theorist, but I
> sincerely believe it to be true. You've got to get rid of that fear of red
> ink._

 _None of what has been applied in the photo_ is pedagogical and will lead to
"getting rid of the fear of the red ink". Seriously man, take a step back,
punishing a child for being right will make it _worse_ if anything. Even if
what you were talking about was a thing (it's not - the closest thing that
comes to it is fear of failure and it's dealt with outside of tests), this
would NOT help it.

~~~
jmilloy
The child isn't being punished.

The child wasn't right.

That's okay.

They child might not know it's okay. In that case, they should receive
support.

Getting marked wrong doesn't help get rid of the fear of failure; we agree
about that. Not sure why you got the impression I thought otherwise.

Many people in the thread are reacting against the lesson and grading because
of how _they_ feel, not about how the child feels. That's why how they feel
matters, when discussing it in an ultimately irrelevant forum.

~~~
aninhumer
The purpose of marking a test is to communicate to the child whether they
understand the topic. Marking an almost correct answer wrong (without
elaboration) is bad feedback.

The problem that people are raising here isn't how the child feels, it's how
the child thinks. And one thing they might think as a result of this answer is
"Oh, I guess multiplication isn't the same both ways. I must have been
mistaken." and it might take some time for this misunderstanding to clear up.

------
mcnamaratw
What a Kafkaesque non-explanation. It is exactly as correct to say that the
second factor is the number of copies.

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

~~~
WalterSear
The person who marked that answer wrong doesn't understand either.

------
dznkorxt76qb
> Equal is defined as, “being the same in quantity, size, degree, or value.”
> Whereas equivalent is defined as, “equal in value, amount, function, or
> meaning.”

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.

------
learnstats2
This article defends multiplication marked incorrectly because of a semantic
difference between 5x3 and 3x5. I recognise there is semantic difference
(although I don't think the Wikipedia reference is correct about its nature).

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.

~~~
sophacles
Teaching necessarily forces a rigor not seen in most actual usage. This is
because there is a need to build concepts on top of one another. So while 3x5
and 5x3 are the same in practical usage, it this method helps in later steps
like algebra:

5x = x + x + x + x, i can't rearrange that into terms of 5+ without involving
even more concepts (like recursion etc)

~~~
aninhumer
>Teaching necessarily forces a rigor not seen in most actual usage. This is
because there is a need to build concepts on top of one another.

To be honest, what I take away from this is "It's easier for the teacher to
keep track of progress if everyone takes the same path to understanding". This
might be true, but is precise knowledge of progress more important than the
benefits of allowing different paths? I'm inclined to think it will stunt
their creativity and exploration in ways that slow them down overall.

------
arithma
The wikipedia definition includes commutativity. Multiplication is defined as
inherently commutative there. This is just wrong. It teaches children to hate
this type of math, where even the correct answer is wrong.

------
jgrahamc
I hope this kid goes to school and tells the teacher that they were just using
the Peano axioms of arithmetic.

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

So the original 5 * 3 would be

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

~~~
chris_wot
The smart alek teacher then tells the child that they are using second order
arithmetic :-)

------
hitekker
After reading this article, reading the comments here and then reading the
author's follow-up to a direct message:

    
    
      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!!
    

... I would say that this author has no idea what's he talking about. Saying
"I agree" to a refutation of your article is dangerously close to agreeing
that your article is more rhetoric than substance. Especially when considering
the self-righteous tone, this article seems little more than your garden-
variety mental gymnastics: dressed up in pretty rhetoric which barely obscures
the lesions of condescension, defensiveness, and disdain for others.

Flagged.

~~~
signal11
I think the meta-lesson for the child is that there'll always be people like
the teacher who marked "5+5+5" wrong and this guy who defends it with bullshit
reasoning, and part of learning to deal with the world involves learning that
sometimes one can be absolutely right and still be penalised/marked
wrong/disagreed with, and you have to deal with that, sometimes by just
answering the way they expect you to answer.

It's a pretty rough lesson to learn during a math test, though. :-\

------
Sharlin
Requiring students to blindly memorize and regurgitate arbitrary definitions
and punishing the use of intellect and common sense is the worst possible way
to teach math. Or anything else for that matter.

~~~
BarryReitman
I totally agree, Sharlin. But only because memorization techniques aren't
taught. It it were made simple — as it can be! — acquiring facts and data make
intellectual considerations much easier and more valuable.

Memory and intellect complement each other.

Barry Reitman Author: "Secrets, Tips, and Tricks of a Powerful Memory."
www.PowerfulMemorySecrets.com

------
howeman
Sure, Wikipedia defines it as 3+3+3+3+3, but the Oxford English dictionary
defines it the other way (as 5+5+5). They are both correct.

~~~
scintill76
> I did something today I’ve never done before, I looked up the definition of
> multiplication.

> And as I suspected in the definition of multiplication, the first factor is
> is the number of copies and the second is the number being repeated.

Yeah, something seems off if an educated adult has to look up something (on
the site that's always harped on for being untrustworthy in school) in order
to convince us that it's basic knowledge a grade-schooler should know. It
implies there's not an actual consensus (as indicated by my parent comment),
and/or that the fact in question does not really matter at all. I have no
memory of whether I was originally taught an order that's "right." It's
possible the student already learned it the opposite/"wrong" way -- what
purpose is served by forcing them to change? As illustrated by the more
complete photo here
([https://imgur.com/gallery/KtKNmXG](https://imgur.com/gallery/KtKNmXG)), the
student seems to understand the geometric difference between 7x4 and 4x7, when
it's more explicitly stated and the two are non-equivalent in the context.

~~~
JoeAltmaier
They are being multiplied, as in drawing a rectangle with one side equal to
each number. There's no 'first' and 'second'; the idea that one is a count and
the other being copied shows a fundamental misconstruction.

~~~
scintill76
I don't really support the pedantry that's going on in the grading, but I'm
not sure I agree with you. To the observer looking at a non-moving rectangle,
"length" and "width" are not interchangable. In the same way, there are indeed
a "first" and "second" by simple definition of the way English/math notation
work (in other words, "left" and "right".) The student was asked to use the
"repeated addition strategy" and an "array" \-- if the algorithm for doing
those was taught using a specific order of the operands, the student is
technically wrong to swap them. Whether or not it's fair or useful to deem
them wrong when they are giving an equal but non-equivalent answer, or whether
or not the algorithm should care about order when the underlying mathematical
operation is commutative, are other issues.

~~~
JoeAltmaier
Then we can only conclude, the syllabus is teaching nonsense. They are
structuring things that don't need (and shouldn't be) structured; they are
instilling fake rules in plastic young minds that will take enormous effort to
unlearn later.

Math is important. Teaching some witchcraft-inspired rote math is destructive
to real learning.

And rectangles exist regardless of how you view them. If I approach your desk
and see the rectangle from the side, its the same rectangle. Even from a
corner. Even in a mirror, its the same rectangle.

~~~
scintill76
Yes, the difference between length and width is only one of perception or
orientation. I emphasized it because I felt like you went too far in the other
direction, almost implying that which axis is which doesn't matter
geometrically. A 5x3 rectangle drawn in 2D space with labeled axes is not the
same as a 3x5 rectangle, even if they have equal perimeter and area. There
might be some value in trying to make sure the student understands that.

~~~
JoeAltmaier
The act of calculating the area surely does mean they are the same rectangle.
Because they can be written down in more than one way is a weakness of the
notation; the math is independent of that.

------
luckystarr
In the number-fields used outside university multiplication is commutative (it
even is for complex numbers!). So I can't imagine why this should not be
correct.

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

~~~
tomrod
On that point, matrix multiplication is not commutative (and oh that it
were!).

------
Comaleaf
I'm not sure why this argument tries to use programming as a justification but
then uses a different example from programming as the evidence. Yes, in JS, 4
=== "4" is false. But 3+3+3+3+3 === 5+5+5 is true. Because they are
mathematically identical. It's a universal truth that 3+3+3+3+3 will always be
equal to 5+5+5. If the idea is that the kid needs to understand the difference
between equality and identity then actually it seems important that the kid
does understand that 3+3+3+3+3 is identical to 5+5+5.

------
rockshassa
What nonsense. I'd write 5+5+5, every time, because it is less handwriting
than 3+3+3+3+3. To me, this shows that the kid was able to step back and
optimize his process, in addition to applying the rules of math.

Those are definitely qualities you want in a programmer.

------
allending
The leap from the implied definition on Wikipedia to the authoritative "by
definition" one sentence later is truly stunning, surpassed only by the sudden
parachuting of bundles and bananas into the unsuspecting 3s and 5s.

------
adrusi
5 _3 is pronounced "5 times 3", synonymous with "3, 5 times in sequence". So
if the problem given had been presented as "5 times 3" I would agree that that
means "3+3+3+3+3".

However, just becuase 5_3 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 "5 _3 " derives its semantics from "5
times 3". "5_3" 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.

~~~
jeremysmyth
It's not pronounced that universally either.

I say "5 _multiplied by_ 3", which informs my interpretation of which value is
repeated by which. Another commenter uses the phrase "5 into 3".

When you use an less formal (and inherently ambiguous) language (English) to
state a condition in a formal language (used to define multiplication), you're
gonna have problems of this nature.

------
grhino
Mathematics education should embrace discovery as well as memorizing
definitions. The commutative property for multiplication does not necessarily
need to be taught before a student discovers it. They may not be able to
rigorously prove it. Early on mathematics education should be about to
discover and curiosity and lighter on rigor.

------
_0ffh
I think the argument of the article is flawed. Reason being, the "=" sign
denotes equality and not equivalence. And the argument hangs on the
observation that while 5+5+5 and 3+3+3+3+3 may be equal, they're not
equivalent. Yeah, right, but that wasn't asked here, was it?

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

How many fingers do you see?

------
gherkin0
I like how all of the comments "Recommended by Brett Berry [the author]" are
the few that agree with him. What an ass.

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.

------
eyko
A similar discussion rescued from about 15 years ago:
[http://mathforum.org/library/drmath/view/58567.html](http://mathforum.org/library/drmath/view/58567.html)
from which I quote:

 _" 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."_

------
ricardobeat
"Common Core Math problem" \- yes, it does look like a special kind of math.
Marking that answer wrong is the equivalent of "you are not allowed to think
about this just yet". I sure hope that teacher at least sat him down to
explain how multiplication is commutative.

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.

------
natmaster
Multiplication is commutative. I don't care if you can rewrite the definition
on wikipedia, this is a fundamental truth of math. Far more important than
your semantic nonsense.

~~~
ja27
But the students aren't supposed to know that yet. That's literally two topics
later in the Common Core Standard, so they won't learn that for another day or
two. Smh.

[http://www.corestandards.org/Math/Content/3/OA/](http://www.corestandards.org/Math/Content/3/OA/)

~~~
Mindless2112
Punishing students who are ahead is an excellent way to make all students
equally disinterested. (Oh, but not equivalently disinterested.)

The question is whether the student knows about commutativity of
multiplication or if he/she didn't understand what was taught or made a
mistake.

Personally, I think the problem here is that math is taught as processes
rather than as concepts.

------
amadvance
This is like complaining with Gauss that he summed the first 100 numbers in
the wrong way ;)

[http://mathforum.org/library/drmath/view/57919.html](http://mathforum.org/library/drmath/view/57919.html)

------
lucio
A 5x3 rectangle is exactly the same as a 3x5 rectangle. It is not a bundle of
bananas, silly. It's just a rectangle, regardless of orientation.

~~~
ivan_ah
That's what I was about to write, but I'm a +1 you instead.

Inherently, one can understand addition as adding of lengths (think stick +
another stick = stick of combined length). Multiplication is about computing
areas where base-times-height vs. height-times-base obviously doesn't matter.

Brett Berry, you're an ass. You know how I know? Because you suggest 5+5+5 is
equally wrong as 30÷2, which has nothing to do with it.

------
ryandrake
This seems like a far-fetched justification. What's more likely?

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.

~~~
lucio
If "He/she may even agree that it's ridiculous to deduct a point", then don't
deduct it.

~~~
ryandrake
And get in trouble for not applying the government-mandated grading guide
fairly to all students?

------
droque
It's pretty obvious the kid is incorrect because 5*3 is using the sum monoid
and overloading notation.

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

------
wckronholm
If the author of this article instead references the definition of
multiplication of natural numbers on wikipedia [1], then the student is
correct since $a \times b = a + a + \dots + a$ with that definition.

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

~~~
pinealservo
The Peano Axioms are not about defining what 'addition' and 'multiplication'
are; they're about presenting a _model_ of the natural numbers along with the
operations of addition and multiplication _in first-order logic_. This makes
them great fodder for worksheets in proof-writing courses (and I did glance
through your worksheet; it looks like a great resource!), but doesn't
necessarily expose the standard mathematical notion of what 'addition' and
'multiplication' are! If you ask a random mathematician out of the blue what
the axioms of arithmetic are, my guess is that you won't often get the Peano
axioms as an answer, but rather the standard algebraic ring or field axioms.

Although it seems _very common_ to define multiplication as repeated addition
in dictionaries and materials for kids, it is in fact only a valid definition
for a rather narrow conception of numbers, i.e. the natural numbers. It
doesn't work without exceptions for the Integers, the Rationals, or the Reals.
Considering that we want students to eventually be able to deal with the Real
numbers, I think it would be better to avoid _defining_ multiplication to be
something that doesn't work outside of the Naturals! We would be in quite a
pickle trying to explain the calculation of the area of a circle in terms of
repeated addition...

By calling what they're teaching the 'repeated addition _strategy_ ' it seems
like they've thought about this; it's indeed a strategy for _calculating_ a
product of two natural numbers. But that makes the marking off of a point all
the more perplexing, because both repeated addition schemes are equally valid
strategies for computing the same product, by virtue of the commutative
property of multiplication! Which is indeed generally an axiom and not a
derived theorem in the more general case of multiplication, because
multiplication is not _generally_ defined in terms of repeated addition. In
general, the axioms only say that multiplication distributes over addition:
[https://en.wikipedia.org/wiki/Field_(mathematics)](https://en.wikipedia.org/wiki/Field_\(mathematics\))

My kids are actually going through this phase of their curriculum right now,
and I know that here, at least, they do teach the commutative property of
multiplication fairly quickly after multiplication is introduced. So I'm not
really sure what pedagogical point of the grading of this assignment would be,
but perhaps there is some point to it. Fortunately my kids have not run afoul
of this kind of thing.

------
emsy
I may have bought the explanation if the answer below hadn't been marked
incorrect as well, because the axis on an imaginary array are arbitrary.

------
Al-Khwarizmi
This post just made my BS-meter explode. How can this make the front page of
HN? I don't even know where to start.

 _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);
    

prints false and false, while using equals instead of == would print true and
true.

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.

------
boomlinde
This is a load of crap. "Equivalence" has a mathematical meaning and it isn't
the bullshit he tries to play it off as. After interpreting an uncited example
blurb on Wikipedia as "the definition of multiplication" he goes on to discuss
division, a non-commutative operator, how it's bad for kids to use the
commutative property of multiplication before the teacher has taught it, how
bundles of bananas of varying quantity (i.e. sets of different sizes) are not
equivalent, how the JavaScript type system is weird and how you can't simply
flip the dimensions of vectors and expect them to be equivalent.

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.

~~~
chris_wot
The definition he quotes is accurate. His understanding of what it says is
what is inaccurate! His error is that he believes the definition says that the
first number referred to in that definition has to be the left-most number,
when in fact the definition refers to either the LHA _or_ the RHS number.

------
lucio
I almost stop reading at the first paragraph: He makes a difference between
"Equals" and "Equivalency" and then he gives overlapping definitions:

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...

------
husam212
But the question is asking the student to find a solution not something
equivalent, I will agree with what this article says if the question is "Find
the equivalent repeated addition form of the following multiplication problem"
but it is not.

------
jules
This is how you ensure that somebody will never like math.

------
zck
Here's a view of why it might make sense for a teacher to emphasize one way
over the other that doesn't focus on "it's the definition and definitions are
important":

> 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-...](http://www.patheos.com/blogs/friendlyatheist/2015/10/21/why-
would-a-math-teacher-punish-a-child-for-saying-5-x-3-15/)

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?

------
ontoillogical
1) When I was learning what "5 times 3" was I don't think I could even
pronounce "commutative," let alone understand what it meant. I did know that 3
x 5 was the same as 5 x 3, but 5 / 3 was not 3 / 5 though!

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?

------
lips
I'd be fully ok with this if the kid in question was actually learning about
"Equals Versus Equivalency," but all of my school experience says that's
unlikely. I hope I'm wrong.

------
pbreit
The answers were wrong. But I would have given the student at least 1/2 point
if not a full point with an explanation.

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

~~~
chris_wot
The answer was NOT wrong.

------
phamilton
Potentially a straw man, but:

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.

------
sophacles
This whole thing is hilariously polarizing. People seem to project so much
stuff on to what's actually shown on the paper that the arguments don't even
make sense a lot of the time - either way. Some things I can get only looking
at the paper:

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.

------
drtz
The author forgets to mention that in mathematics = represents equivalency.

------
lupinglade
This is why saying 5 of 3 is better than 5 times 3.

------
JoeAltmaier
Its a trivial mistake; we don't know the context; the kid could (and maybe
was) tutored on the difference. All about nothing.

~~~
phamilton
Precisely. Is this a homework assignment? Did the teacher explicitly teach it
the way it was marked? Did the teacher follow up with the student to
understand why the student strayed from the path taught? Who knows, but if
they did then good job. Broken foundations have huge consequences, and if
being pedantic makes it easier to spot broken foundations, then it's the right
approach.

------
chris_wot
There are a number of issues with this explanation.

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.

~~~
chris_wot
Yow! I wrote associative when I meant commutative! Oops.

------
orless
Here's my response to this:

[https://medium.com/@highsource/the-only-reason-for-this-
answ...](https://medium.com/@highsource/the-only-reason-for-this-answer-to-be-
marked-as-wrong-is-the-teacher-saying-you-did-not-
apply-e92e64b4de92#.jns0a69vz)

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.

------
jmilloy
I am surprised to see exclusively negative reactions here. For one, the
question wasn't marked _wrong_ so much as partially incorrect (and partially
correct). Secondly, the question is _not_ "What is 5x3?" and the test is _not_
about multiplication. It's explicitly about a specific process, which the
teacher has presumably taught and which the student unequivocally got wrong.

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.

~~~
orless
The task was about repetitive multiplication strategy. Which was what the
child did. Just not by the EXACTLY taught convention.

So your analogy with dovetail joins would be not replacing them by rabbet
joins, but setting them just in the different order. And it would have been a
complete nonsense to ask for a specific order in a wood shop.

