8 ms·

Why 5 x 3 = 5 and 5 and 5 Was Marked Wrong

53 points by nonameface 11y ago

howeman 11y ago
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.
- deleted 11y ago
  [deleted]
- scintill76 11y ago
  > 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 11y ago
    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 11y ago
      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 11y ago
        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.
        signal11 11y ago
        Ironically, the person who posted the linked-to article has a another post called "Common Core is not the enemy" https://medium.com/i-math/common-core-math-is-not-the-enemy-c05b68f46b3e#.qrh3myw2s https://medium.com/i-math/common-core-math-is-not-the-enemy-... Reading the BS rationalizations in the linked-to artcle and I'm beginning maybe the problem with math education is learn-by-rote teachers who won't think for themselves.
        scintill76 11y ago
        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.
arithma 11y ago
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.
dznkorxt76qb 11y ago
> 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.
tomrod 11y ago
What a defense of nonsense. The marking punishes a student for achieving a correct answer in an appropriate way, regardless of pedagogical justification.
- tfigueroa 11y ago
  Agreed. Few children are going to get to the point where the commutative property is a concern, anyway. If we're going to be pedantic, use this as a learning opportunity. "Actually, 5x3 is slightly different than 3x5. Multiplication has this property..." Some of the kids won't care, but some will be intrigued. Teaching it this way is a gotcha.
- signal11 11y ago
  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-common-core-2015-10?IR=T http://www.businessinsider.com/why-55515-is-wrong-under-the-... 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 11y ago
    How do you not read it "5 times 3"? Why would you re-arrange where the "times" is?
    - TeMPOraL 11y ago
      You're joking, right? GP read it as: "5 x3", like he would read "copy x3", or "copy, three times". It's a natural way of reading "5x3", though I personally read it as "5x 3".
    - Retric 11y ago
      (5, times 3) or (5 times, 3) both work.
deleted 11y ago
[deleted]
mcnamaratw 11y ago
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 11y ago
  The person who marked that answer wrong doesn't understand either.
jasode 11y ago
>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 11y ago
  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 11y ago
    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.
natmaster 11y ago
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.
- SpinningCode 11y ago
  +1. In few years this kind of answers will be criminalized.
- ja27 11y ago
  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 11y ago
    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.
- orless 11y ago
  It actually even did not matter here if multiplication is commutative or not. The child had to apply the repetitive addition strategy and so he or she did. Just probably not with the taught convention.
luckystarr 11y ago
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 11y ago
  On that point, matrix multiplication is not commutative (and oh that it were!).
Sharlin 11y ago
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 11y ago
  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
88e282102ae2e5b 11y ago
> 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 11y ago
  > 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 11y ago
    > 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)
learnstats2 11y ago
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 11y ago
  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 11y ago
    >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.
grhino 11y ago
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.
pbreit 11y ago
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 11y ago
  The answer was NOT wrong.
allending 11y ago
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.
QuantumRoar 11y ago
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 11y ago
  3 bundle x 5 banana/bundle = 5x3 bundlexbanana/bundle = 5 bundle x 3 banana/bundle = 15 banana
eyko 11y ago
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."
_0ffh 11y ago
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?
rockshassa 11y ago
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.
zck 11y ago
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-would-a-math-teacher-punish-a-child-for-saying-5-x-3-15/ 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?
jgrahamc 11y ago
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
- jonsen 11y ago
  1 * n = n m * n = (m-1) * n + n
- chris_wot 11y ago
  The smart alek teacher then tells the child that they are using second order arithmetic :-)
kator 11y ago
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.
- ecspike 11y ago
  Yep. My dad taught me a bunch of math on his own and I would also get in trouble for "not showing my work" on tests.
- ryandrake 11y ago
  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."
amadvance 11y ago
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
jules 11y ago
This is how you ensure that somebody will never like math.
emsy 11y ago
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.
adrusi 11y ago
53 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 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.
- jeremysmyth 11y ago
  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.
ontoillogical 11y ago
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?
JoeAltmaier 11y ago
Its a trivial mistake; we don't know the context; the kid could (and maybe was) tutored on the difference. All about nothing.
- sparkie 11y ago
  It's the tutor who needs tutoring on the non-difference.
- phamilton 11y ago
  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.
Comaleaf 11y ago
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.
droque 11y ago
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.
lucio 11y ago
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...
lucio 11y ago
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 11y ago
  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.
jmilloy 11y ago
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 11y ago
  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.
gherkin0 11y ago
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.
deleted 11y ago
[deleted]
lips 11y ago
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.
hitekker 11y ago
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 11y ago
  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. :-\
drtz 11y ago
The author forgets to mention that in mathematics = represents equivalency.
husam212 11y ago
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.
phamilton 11y ago
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.
ricardobeat 11y ago
"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.
sophacles 11y ago
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.
Al-Khwarizmi 11y ago
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.
ryandrake 11y ago
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 11y ago
  If "He/she may even agree that it's ridiculous to deduct a point", then don't deduct it.
  - ryandrake 11y ago
    And get in trouble for not applying the government-mandated grading guide fairly to all students?
chris_wot 11y ago
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.
boomlinde 11y ago
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 11y ago
  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.
wckronholm 11y ago
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.pdf http://billkronholm.com/wp-content/uploads/2015/10/MATH280.p...
- pinealservo 11y ago
  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.
lupinglade 11y ago
This is why saying 5 of 3 is better than 5 times 3.
orless 11y ago
Here's my response to this: 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 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.

53 points by nonameface 11y ago

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.

deleted 11y ago
[deleted]
scintill76 11y ago
> 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 11y ago
  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 11y ago
    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 11y ago
      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.
      - signal11 11y ago
        Ironically, the person who posted the linked-to article has a another post called "Common Core is not the enemy" https://medium.com/i-math/common-core-math-is-not-the-enemy-c05b68f46b3e#.qrh3myw2s https://medium.com/i-math/common-core-math-is-not-the-enemy-... Reading the BS rationalizations in the linked-to artcle and I'm beginning maybe the problem with math education is learn-by-rote teachers who won't think for themselves.
      - scintill76 11y ago
        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.

arithma 11y ago

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.

dznkorxt76qb 11y ago

> 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.

tomrod 11y ago

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

tfigueroa 11y ago
Agreed. Few children are going to get to the point where the commutative property is a concern, anyway. If we're going to be pedantic, use this as a learning opportunity. "Actually, 5x3 is slightly different than 3x5. Multiplication has this property..." Some of the kids won't care, but some will be intrigued. Teaching it this way is a gotcha.
signal11 11y ago
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-common-core-2015-10?IR=T http://www.businessinsider.com/why-55515-is-wrong-under-the-... 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 11y ago
  How do you not read it "5 times 3"? Why would you re-arrange where the "times" is?
  - TeMPOraL 11y ago
    You're joking, right? GP read it as: "5 x3", like he would read "copy x3", or "copy, three times". It's a natural way of reading "5x3", though I personally read it as "5x 3".
  - Retric 11y ago
    (5, times 3) or (5 times, 3) both work.

deleted 11y ago

[deleted]

mcnamaratw 11y ago

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 11y ago
The person who marked that answer wrong doesn't understand either.

jasode 11y ago

>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 11y ago
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 11y ago
  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.

natmaster 11y ago

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.

SpinningCode 11y ago
+1. In few years this kind of answers will be criminalized.
ja27 11y ago
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 11y ago
  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.
orless 11y ago
It actually even did not matter here if multiplication is commutative or not. The child had to apply the repetitive addition strategy and so he or she did. Just probably not with the taught convention.

luckystarr 11y ago

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 11y ago
On that point, matrix multiplication is not commutative (and oh that it were!).

Sharlin 11y ago

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 11y ago
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

88e282102ae2e5b 11y ago

> 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 11y ago
> 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 11y ago
  > 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)

learnstats2 11y ago

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 11y ago
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 11y ago
  >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.

grhino 11y ago

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.

pbreit 11y ago

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 11y ago
The answer was NOT wrong.

allending 11y ago

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.

QuantumRoar 11y ago

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 11y ago
3 bundle x 5 banana/bundle = 5x3 bundlexbanana/bundle = 5 bundle x 3 banana/bundle = 15 banana

eyko 11y ago

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."

_0ffh 11y ago

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?

rockshassa 11y ago

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.

zck 11y ago

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-would-a-math-teacher-punish-a-child-for-saying-5-x-3-15/ 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?

jgrahamc 11y ago

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

jonsen 11y ago
1 * n = n m * n = (m-1) * n + n
chris_wot 11y ago
The smart alek teacher then tells the child that they are using second order arithmetic :-)

kator 11y ago

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.

ecspike 11y ago
Yep. My dad taught me a bunch of math on his own and I would also get in trouble for "not showing my work" on tests.
ryandrake 11y ago
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."

amadvance 11y ago

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

jules 11y ago

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

emsy 11y ago

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.

adrusi 11y ago

53 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 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.

jeremysmyth 11y ago
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.

ontoillogical 11y ago

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?

JoeAltmaier 11y ago

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

sparkie 11y ago
It's the tutor who needs tutoring on the non-difference.
phamilton 11y ago
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.

Comaleaf 11y ago

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.

droque 11y ago

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.

lucio 11y ago

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...

lucio 11y ago

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 11y ago
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.

jmilloy 11y ago

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 11y ago
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.

gherkin0 11y ago

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.

deleted 11y ago

[deleted]

lips 11y ago

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.

hitekker 11y ago

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 11y ago
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. :-\

drtz 11y ago

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

husam212 11y ago

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.

phamilton 11y ago

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.

ricardobeat 11y ago

"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.

sophacles 11y ago

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.

Al-Khwarizmi 11y ago

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.

ryandrake 11y ago

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 11y ago
If "He/she may even agree that it's ridiculous to deduct a point", then don't deduct it.
- ryandrake 11y ago
  And get in trouble for not applying the government-mandated grading guide fairly to all students?

chris_wot 11y ago

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.

boomlinde 11y ago

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 11y ago
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.

wckronholm 11y ago

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.pdf http://billkronholm.com/wp-content/uploads/2015/10/MATH280.p...

pinealservo 11y ago
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.

lupinglade 11y ago

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

orless 11y ago

Here's my response to this: 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 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.