Why 5 x 3 = 5 and 5 and 5 Was Marked Wrong • Hacker News Board

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

#121

Post by chris_wot » Sat, Oct 31, 2015, 5:19 PM UTC

jmilloy wrote: post #112

chris_wot wrote: post #105

Earlier quoted context omitted.

The way the child is being taught is because the teacher or course administrator has misunderstood the purpose of teaching arithmetic via repeated addition. Repeated addition is relying on the fact that children see the world in a very concrete way and have not started to understand concepts in a more abstract fashion. Thus you use objects to explain concepts, like: every cat has one tail, I have 3 cats so how many t…

> After all, you aren't really teaching repeated addition, you are just using it as scaffolding to provide an insight into multiplication! You may be right. This is the interesting part of the discussion, and you've framed it well. I think it can be scaffolding technique also for the application of definitions, the expansion of symbols to their definition. Perhaps there is a better way to say that (or other examples)…

Thanks, we are probably on the same page here :-)

The mark doesn't honestly seem to be the issue here though, at least so far as I can see, but rather that the teacher marked something as wrong when it was right.

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

#122

Post by wckronholm » Sat, Oct 31, 2015, 5:28 PM UTC

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

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

#123

Post by jhanschoo » Sat, Oct 31, 2015, 5:52 PM UTC

chris_wot wrote: post #120

jmilloy wrote: post #114

Earlier quoted context omitted.

That's not part of the definition . That's a separate property . I'm not sure we should care about that in elementary school, so the point is not to defend the teacher but only that you can't use the definition as an argument against the teacher.

No, I'm afraid you've not given a complete definition of multiplication. You need to also show that multiplication is commutative, which is indeed a property of multiplication but MUST be included in the definition. At the child's level (primary age child, NOT high-school) then it is unnecessary to introduce the distributive property. But you honestly have to make the associative property very, very clear of the chil…

I'm pretty sure that the homework was given as part of a course teaching multiplication. Perhaps what was desired was to first have children able to construct products from repeated addition, before teaching them the commutative property?

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

#124

Post by scrollaway » Sat, Oct 31, 2015, 5:57 PM UTC

jmilloy wrote: post #73

88e282102ae2e5b wrote: post #12

> 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…

> 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.

> 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)

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

#125

Post by lupinglade » Sat, Oct 31, 2015, 6:03 PM UTC

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

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

#126

Post by jmilloy » Sat, Oct 31, 2015, 6:21 PM UTC

chris_wot wrote: post #118

jmilloy wrote: post #116

Earlier quoted context omitted.

> P.S. If you have a Bachelors in Mathematics, then surely you can see that there is a fundamental problem if a child is taught that 5x3 is not the same as 3x5? It's not the same. I'm not sure when that should be taught to a student.

It is the same!

Right, for different definitions of "same". I think that the OP attempts to explain that there are different definitions of "same" that are each valid.

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

#127

Post by ivansavz » Sat, Oct 31, 2015, 6:36 PM UTC

lucio wrote: post #55

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.

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.

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

#128

Post by chris_wot » Sat, Oct 31, 2015, 6:37 PM UTC

chris_wot wrote: post #93

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 ch…

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

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

#129

Post by chris_wot » Sat, Oct 31, 2015, 6:39 PM UTC

jhanschoo wrote: post #123

chris_wot wrote: post #120

Earlier quoted context omitted.

No, I'm afraid you've not given a complete definition of multiplication. You need to also show that multiplication is commutative, which is indeed a property of multiplication but MUST be included in the definition. At the child's level (primary age child, NOT high-school) then it is unnecessary to introduce the distributive property. But you honestly have to make the associative property very, very clear of the chil…

I'm pretty sure that the homework was given as part of a course teaching multiplication. Perhaps what was desired was to first have children able to construct products from repeated addition, before teaching them the commutative property?

As I've said, that's a misuse of the repeated addition technique.

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

#130

Post by jmilloy » Sat, Oct 31, 2015, 6:41 PM UTC

scrollaway wrote: post #124

jmilloy wrote: post #73

Earlier quoted context omitted.

> 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.

> 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…

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

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

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

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