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

#101

Post by JoeAltmaier » Sat, Oct 31, 2015, 4:06 PM UTC

scintill76 wrote: post #68

JoeAltmaier wrote: post #50

Earlier quoted context omitted.

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.

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 a…

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.

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

#102

Post by chris_wot » Sat, Oct 31, 2015, 4:14 PM UTC

xworld21 wrote: post #33

signal11 wrote: post #19

Earlier quoted context omitted.

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-... 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 Anyhow, just goes to show Maths teachers have now been replaced by box tickers who refuse to apply their brain.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…

Funnily enough, the traditional formal definition of product is due to Peano and it would say 5x3=5+5+5: https://en.m.wikipedia.org/wiki/Peano_axioms#Multiplication

Paeno axioms do not define multiplication. Multiplication is considered to be part of second-order arithmetic, which includes Paeno arithmetic (which is a first order system) and augments it to form a stronger set of axioms, of which multiplication is one of them.

The definition of multiplication is indeed in the Wikipedia article, but if you reread it then you'll see it doesn't claim to be a Paeno axiom. A better article to read (after reading about Paeno axioms!) is here:

https://en.m.wikipedia.org/wiki/Second-order_arithmetic#Basi...

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

#103

Post by jmilloy » Sat, Oct 31, 2015, 4:18 PM UTC

signal11 wrote: post #95

jmilloy wrote: post #88

Earlier quoted context omitted.

The objective and obvious difference between the 5 and the 3 is that the 5 is first and the 3 is second. The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. (The bananas and bundles just illustrates an example for why, in another context, being first or second would be important. But on the test, 5 is still first.) On the other hand, you are invokin…

> 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…

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 quiz correct but will struggle on much of the rest of the unit, maybe get a low grade or hopefully get the help they need. The second (with the paper shown in the OP), will probably get an high grade because they got partial credit on a silly quiz. The third will get a higher high grade. What's so bad about that?

BTW, appealing to definitions won't work here, because the x does have a very well-defined mathematical meaning: a x b := b + ... + b.

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

#104

Post by jmilloy » Sat, Oct 31, 2015, 4:24 PM UTC

jasode wrote: post #100

jmilloy wrote: post #88

Earlier quoted context omitted.

The objective and obvious difference between the 5 and the 3 is that the 5 is first and the 3 is second. The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. (The bananas and bundles just illustrates an example for why, in another context, being first or second would be important. But on the test, 5 is still first.) On the other hand, you are invokin…

>The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. If you(royal-you) insist that the 5 being the first factor has a specific job and you teach such nonsense to a child, it means you're not teaching actual mathematics . In _real_ math, the factors/mutiplicands have no notion of ordinal rank such as "first" or "second" or "specific jobs". Even if th…

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 useful and interesting.

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

#105

Post by chris_wot » Sat, Oct 31, 2015, 4:26 PM UTC

signal11 wrote: post #95

jmilloy wrote: post #88

Earlier quoted context omitted.

The objective and obvious difference between the 5 and the 3 is that the 5 is first and the 3 is second. The point is that because the 5 is first, as everyone can see, it has a specific job in the repeated addition technique. (The bananas and bundles just illustrates an example for why, in another context, being first or second would be important. But on the test, 5 is still first.) On the other hand, you are invokin…

> 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 tails are there in total?

You introduce notation in the class, but I can't see how it is valuable to use an abstract expression like 1x3 without a concrete description of the example of cats and tails. After all, you aren't really teaching repeated addition, you are just using it as scaffolding to provide an insight into multiplication!

The fact that the answer given can be shown as wrong has already demonstrated that the child (and parent!) was annoyed because it made little sense to mark it as wrong. It probably caused more harm than good, because now the child questions their understanding of the subject matter, yet ironically they do appear to have grasped the concept!

So at this point, the poor pedagogy of the teacher in misusing the counting technique means that the child starts to doubt themselves unnecessarily, they become locked in to a scaffolding technique that will later need to be discarded anyway. When they hit non-integer rational numbers - numbers with decimal points - they aren't going to be able to add these together, instead they will need to grasp that you can scale down numbers if you multiply any rational number between 0 and 1.

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

#106

Post by signal11 » Sat, Oct 31, 2015, 4:30 PM UTC

JoeAltmaier wrote: post #101

scintill76 wrote: post #68

Earlier quoted context omitted.

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 a…

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 se…

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-...

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.

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

#107

Post by chris_wot » Sat, Oct 31, 2015, 4:33 PM UTC

jgrahamc wrote: post #26

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

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

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

#108

Post by chris_wot » Sat, Oct 31, 2015, 4:34 PM UTC

pohl wrote: post #44

Al-Khwarizmi wrote: post #41

Earlier quoted context omitted.

The student wrote 5+5+5, isn't that a repeated addition strategy?

It certainly is a repeated addition strategy, but is it the repeated addition strategy given to the students? The definition of the algorithm given to the student may involve language like "take the first number and..." The steps are the steps.

The steps are either being taught incorrectly, or they are wrong. Very worrying!

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

#109

Post by signal11 » Sat, Oct 31, 2015, 4:40 PM UTC

jmilloy wrote: post #103

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…

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.

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

#110

Post by chris_wot » Sat, Oct 31, 2015, 4:46 PM UTC

jmilloy wrote: post #104

jasode wrote: post #100

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. If you(royal-you) insist that the 5 being the first factor has a specific job and you teach such nonsense to a child, it means you're not teaching actual mathematics . In _real_ math, the factors/mutiplicands have no notion of ordinal rank such as "first" or "second" or "specific jobs". Even if th…

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 addition in a very literal and concrete fashion, because at the age of 4-5 years old (sometimes older), children don't think at a higher level of abstraction. And using symbols to represent multiplication IS a higher level of abstraction.

It seems to me, a non-educator, that the counting technique has value in word problems. But as soon as the child shows they understand the concept, then you introduce the notation (e.g. 5 x 3), explain the numbers can be added up either as five values added up three times, or three values added up five times.

That the test talked about a "strategy" is not really maths, and frankly it seems to be misapplying a solid teaching technique, leading to confusion, anger and a lack of confidence in the child. If that's happening, then I'd suggest the technique is not all that solid and teachers and other educators should seriously consider whether it is causing more harm than good.

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?