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

#111

Post by chris_wot » Sat, Oct 31, 2015, 4:52 PM UTC

newjersey wrote: post #37

tremendo wrote: post #30

Earlier quoted context omitted.

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, wh…

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.

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

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

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

#112

Post by jmilloy » Sat, Oct 31, 2015, 4:57 PM UTC

chris_wot wrote: post #105

signal11 wrote: post #95

Earlier quoted context omitted.

> The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. Um, this is sophistry. The question asked for "5 x 3" using repeated addition. The x is a very well defined mathematical operator and "repeated addition" has a very well-defined meaning, and the child has demonstrated it by repeatedly adding 5 three times. Yes, the child's cardinal sin is he Did…

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), but the point is that I don't think that the exclusive value in teaching the technique is soon-to-be discarded scaffolding for multiplying numbers.

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

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

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

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

#113

Post by chris_wot » Sat, Oct 31, 2015, 4:58 PM UTC

pbreit wrote: post #15

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.

The answer was NOT wrong.

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

#114

Post by jmilloy » Sat, Oct 31, 2015, 4:59 PM UTC

signal11 wrote: post #109

jmilloy wrote: post #103

Earlier quoted context omitted.

It doesn't seem like a cardinal sin so much as a small quantitative note that the process was taught a different way that the teacher thinks is important. Let's suppose one student can follow the procedure when asked but can't actually multiply in application, one student can't follow the procedure correctly but can multiply when needed, and a third can do both. Probably the first student will get questions on this q…

> ... does have a very well-defined mathematical meaning: a x b := b + ... + b. Please complete the definition. which is that a x b = b x a, so a x b can also be written as a x .. x a. There is nothing special about the order.

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.

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

#115

Post by boomlinde » Sat, Oct 31, 2015, 5:00 PM UTC

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.

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

#116

Post by jmilloy » Sat, Oct 31, 2015, 5:04 PM UTC

chris_wot wrote: post #110

jmilloy wrote: post #104

Earlier quoted context omitted.

I only have a bachelors degree in math, focusing on theory, but I think we have a different understanding of what actual mathematics is. For example, in some "real math", definitions, properties, and axioms are well-distinguished and mixing them up can get you in trouble. More importantly, are we even trying to teach "real" math to elementary kids (I wish we did, but I don't think we do) or "computation"? Both are us…

Yeah, but the problem is: repeated addition is attempting to take something very concrete like I give four children three marbles each, how many marbles does each child have? You then use that addition technique to have them add up the number of marbles (in essence it's as if you are asking them to count on their hands, which is a valid technique at this level). But that helps the child understand the concept of addi…

> 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.

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

#117

Post by chris_wot » Sat, Oct 31, 2015, 5:05 PM UTC

boomlinde wrote: post #115

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 quanti…

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.

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

#118

Post by chris_wot » Sat, Oct 31, 2015, 5:06 PM UTC

jmilloy wrote: post #116

chris_wot wrote: post #110

Earlier quoted context omitted.

Yeah, but the problem is: repeated addition is attempting to take something very concrete like I give four children three marbles each, how many marbles does each child have? You then use that addition technique to have them add up the number of marbles (in essence it's as if you are asking them to count on their hands, which is a valid technique at this level). But that helps the child understand the concept of addi…

> 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!

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

#119

Post by chris_wot » Sat, Oct 31, 2015, 5:12 PM UTC

signal11 wrote: post #91

pbreit wrote: post #23

Earlier quoted context omitted.

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

Because where I come from, we use the English equivalent of "into" rather than "times". "5 into 3" roughly translates to "5, 3 times". The meta-point here is that English (or any other language) is crap for math, which is why we use mathematical notation. And this bullcrap syllabus is trying to redefine the "x" operator, which gets my goat.

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

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

#120

Post by chris_wot » Sat, Oct 31, 2015, 5:16 PM UTC

jmilloy wrote: post #114

signal11 wrote: post #109

Earlier quoted context omitted.

> ... does have a very well-defined mathematical meaning: a x b := b + ... + b. Please complete the definition. which is that a x b = b x a, so a x b can also be written as a x .. x a. There is nothing special about the order.

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 child will potentially have problems down the track!

(Edit: brain fart - I said associative when I meant commutative. Oops!)