Why 5 x 3 = 5 and 5 and 5 Was Marked Wrong
medium.com>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.
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.
I feel the opposite way about the English reading "5 times 3". In English, the subject comes first, so I would expect the sentence to mean take 5, use "times" as a verb, and 3 as the adverb. Likewise, if you read it as "5 multiplied by 3", you would expect to take five, three times.
Exactly. And this is why we use math operators rather than English terms for math operations. If you want to specifically mean 5 times a group of 3, define a new operator for it, like 5 ○ 3. Don't 'overload' the x operator with your own arbitrary meaning.
Ummm... I think you just overloaded the x operator yourself!
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!
As against what happens out there in the tough world of work.
> "you'll see in Question #2 that the child had a "different rectangle orientation" than the teacher"
Vectors can be considered identical for the same reason... [1, 0] and [9, 0] are the same arrow if you move your head in the latter case. Here, the teacher is assuming the kid's head is at [0,0], when rotation (or translation) doesn't change the arrow any more than the rectangle. Neither is wrong. The lesson: memorize the teacher's poor use of language; ignore objective reality.
> The "5x3" problem on the test had "pure" numbers with no annotation of "objects"
It's not the "5x3" problem but the "repeated addition strategy" problem. I think that's part of the problem. Similarly, the bananas example isn't about the 5 and the 3 but about a difference between counting "x sets of y" and "y sets of x".
>a difference between counting "x sets of y" and "y sets of x".
You're making the same mistake as the blog writer by overlaying a difference between "x" and "y" that was not on the test.
The child did do the repeated addition strategy. It's just that the child's "shape" of the addition didn't exactly match the teacher's. If the point of the problem was the "repeated addition" instead of the final answer "15", the child still did it correctly. He/she showed his work of repeated addition!
The actual test problem was stated as "5 times 3" and not "5subscriptX times 3subscriptY" or "5subscriptBundles times 3subscriptBananas". You're arguing about a test the child didn't actually take.
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 invoking a "repeated addition" that the student was never taught. Your repeated addition strategy is "add <one of numbers> together <the other number> times". The taught repeated addition strategy was "add <the second number> together <the first number> times".
> 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 Not Do As He Was Taught(tm), but seriously, that's more the teacher's and the school board's problem in my book, not the child's.
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.
> 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.
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.
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.
a x b = b + ... +b is not a strict definition, it's just a convention.
Check the English Wikipedia:
https://en.wikipedia.org/wiki/Multiplication
It's, as you say, 5x3 = 3 + 3 + 3 + 3 +3.
But now check Russian Wikipedia:
https://ru.wikipedia.org/wiki/%D0%A3%D0%BC%D0%BD%D0%BE%D0%B6...
There you'll see 5x3 = 5 + 5 + 5.
So much for the "very well-defined mathematical meaning".
> ... 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!)
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.
But 5 + 5 + 5 WAS repeated addition!
Well, maybe you can find a source, but I can only find sources that define multiplication as I have and then mention that multiplication of, say, real numbers, is commutative.
>define $EQUIVALENCE [...] and then mention $PROPERTY
When you keep pointing back to "a x b = b+b...+b", as The Definition without including the properties, it means you're mixing up the orthography[0] of multiplication with the real underlying idea of multiplication.
A math definition includes that all properties must simultaneously be true. It's the limitations of writing (orthography[0]) that we state things one thing before the other. The phrase "and then" used as a sequential condition is not applicable. Instead, if all properties are true, you thus have the definition.
Here's another "definition"[1] that states the summation in reverse order: "In simple algebra, multiplication is the process of calculating the result when a number a is taken b times."
e.g. "when a number 5 is taken 3 times" ... which is the repeated addition the child carried out.
That wikipedia stated multiplication as "a x b = b+b...+b" while Wolfram MathWorld stated it as "a is taken b times." is a difference in orthography and not definition. Unfortunately, you're working backward from an arbitrary orthography and judging the child to be wrong.
[0]https://en.wikipedia.org/wiki/Orthography
[1]https://books.google.com/books?id=aFDWuZZslUUC&pg=PA1974&lpg...
The contents of the Weisstein book was also used in Wolfram MathWorld:
I'd just like to add a point to jasode's excellent point about getting hung up about orthography, which is that please don't define math using English. It's a terrible thing to do -- for example, the en-us 5x3 = 5 times 3 = 3+3+3+3+3 fails for Spanish speakers. For another, English itself is not very "standard" - some variants of British English would actually read 5x3 as "5, 3 times". Math exists outside of human language and teaching kids should adapt to this reality.
+1 data point: In my anglophone school system we say 5x3 as "5 multiplied by 3", not "5 times 3". These semantics lead children (and me) to think 5 + 5 + 5 instead of 3 + 3 + 3 + 3 + 3.
What are your sources?
>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 the child was not formerly taught The Commutative Law, it's not impossible for him to see multiplication tables[1]. (In fact, many are hung as big posters in elementary classrooms.) Any child with pattern recognition abilities beyond a chimpanzee would notice that the cells of XY have the same answer as YX. He/she would ask mom/dad/teacher "is xy always same as yx?".
In the world of _pseudo_ math that stresses bizarre hoop jumping, we overlay non-mathematical concepts such as "specific job" to factors. Maybe this skill is important and transferable to the enlisted man to make sure he makes his bed before cleaning his machine gun instead of the other way around so everyone in the squad doesn't get punished with 50 pushups. But don't pass it off as "teaching math."
[1]https://www.google.com/search?q=multiplication+table&es_sm=9...
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.
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?
> 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.
I can't see how!
I was leaning that way, but when I tried to the plain English way, I defaulted to "Five threes" not the backwards "take five and copy it out three times".
Once he made the tie back to programming he was able to change my mind that this was important enough. Although it does seem unfair for the individual child.
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.
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."
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.
What a defense of nonsense. The marking punishes a student for achieving a correct answer in an appropriate way, regardless of pedagogical justification.
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 my book, the kid demonstrated repeated addition and should have got the mark.
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...
Personally the way I learned to think of it was to read it in the order that organizes it into the fewest number of "groups". Afterall it is easier (and faster) to visualize three groups of five items, than it is to visualize five groups of three items.
For example what if it was 11 x 3. It would make no sense to try to think of it as eleven groups of three, when you can easily derive the answer far quicker as 11 + 11 = 22 + 11 = 33.
How do you not read it "5 times 3"? Why would you re-arrange where the "times" is?
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".
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.
(5, times 3) or (5 times, 3) both work.
I just consider "times" the name of the multiplication operator.
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.
"Use the repeated addition strategy..."
Critics of this are missing that the teacher is not asking the student to find the correct result. Instead, the teacher is asking the student to apply a specific algorithm.
If the teacher asks to apply Merge Sort to a list, but the student applies Insertion Sort, both strategies will result in the same sorted list.
But only one will demonstrate what the teacher asked the student to demonstrate.
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 the steps". Great advice if the purpose of school is to train people for rote factory work (we have robots for that). Not such a great way to prepare future leaders or creative problem solvers.
When are practising skills in school, sometimes we practice creativity and sometimes we practice techniques. Both are useful, and it's clear which is which.
Secondly, a student that knows the difference between different techniques and can call them up at will (such as the difference between 5 sets of 3 and 3 sets of 5) is better off than a student that only knows how to produce a particular answer for a particular question.
I seriously doubt this student "only knows how to produce a particular answer for a particular question". Are you claiming he could compute 3 times 5 but not 5 times 3?
On the contrary, I think this student may be showing that he knew the two techniques, and that he is smart enough to pick the easier computation.
But yes, if your goal is to kill any creativity in intelligent should punish kids that deviate from the lines hard.
If a teacher asked "compute 1000 x 1", no sane kid would do "1 plus 1 equals 2; 2 plus 1 equals 3;...; 999 plus one equals 1000".
This teacher would have failed Carl Friedrich Gauss, too, for computing sum(1,100) in seconds.
A student that sees a multiplication sign and reads it as "sets of" is not better off than one who reads it as multiplication.
Maybe, maybe not; I'm not sure.
But I am sure that a student who understands and can apply a specific process when required as well as produce a correct result is better off than one who can only produce a correct result.
Obviously I don't think that all of school should be rote application of techniques. But I've found that when you want to make sure your students have learned a specific technique, you sometimes have to create a somewhat artificial request. Have you taught enough to have found otherwise?
Nope, I'm not a teacher.
(I tutored math in college and did have the experience of helping people understand math...)
The steps ought to be a bridge to understanding the arithmetic as an abstraction.
Teaching and requiring that a commutative operation be ordered doesn't seem like it is going to contribute to that.
I agree with you. The learning objective is stated as "I can use multiplication strategies to help me multilpy", but it's important that the question asks about a specific multiplication strategy, and marks the question as partially correct because the specific multiplication strategy desired is used only partially correctly.
The steps are either being taught incorrectly, or they are wrong. Very worrying!
I don't think "merge sort vs. insertion sort" is a good analogy in this case. This seems more like the teacher is deducting points because of a wrong indentation or brace style.
It's not an analogy.
I was giving a different example of an algorithm than the one denoted by the assignment. "Use the repeated addition strategy..." is a clear call to demonstrate the application of a named procedure.
This is no accident, by the way. This is a deliberate design decision of those who created the common core standards. One of the goals is to teach algorithmic thinking.
It's pretty easy to google about if you can find your way through the reactionary hissy-fit memes.
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.
3 bundle x 5 banana/bundle = 5x3 bundlexbanana/bundle = 5 bundle x 3 banana/bundle = 15 banana
> 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?
> 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)
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.
You're still stuck in the theory, talking about how the people in this thread feel when none of them matter. The child matters, that's it.
> It's so important for young students to feel like they understand and will continue to understand, in order for them to then achieve new understanding. I don't know how to write that without sounding like a theorist, but I sincerely believe it to be true. You've got to get rid of that fear of red ink.
None of what has been applied in the photo is pedagogical and will lead to "getting rid of the fear of the red ink". Seriously man, take a step back, punishing a child for being right will make it worse if anything. Even if what you were talking about was a thing (it's not - the closest thing that comes to it is fear of failure and it's dealt with outside of tests), this would NOT help it.
The child isn't being punished.
The child wasn't right.
That's okay.
They child might not know it's okay. In that case, they should receive support.
Getting marked wrong doesn't help get rid of the fear of failure; we agree about that. Not sure why you got the impression I thought otherwise.
Many people in the thread are reacting against the lesson and grading because of how they feel, not about how the child feels. That's why how they feel matters, when discussing it in an ultimately irrelevant forum.
The purpose of marking a test is to communicate to the child whether they understand the topic. Marking an almost correct answer wrong (without elaboration) is bad feedback.
The problem that people are raising here isn't how the child feels, it's how the child thinks. And one thing they might think as a result of this answer is "Oh, I guess multiplication isn't the same both ways. I must have been mistaken." and it might take some time for this misunderstanding to clear up.
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.
The person who marked that answer wrong doesn't understand either.
> 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.
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.
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)
>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.
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.
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)
5 * S(2) = 5 + 5 * 2
= 5 + 5 + 5 * 1
= 5 + 5 + 5 + 5 * 0
= 5 + 5 + 5 + 0
= 5 + 5 + 5The smart alek teacher then tells the child that they are using second order arithmetic :-)
1 * n = n m * n = (m-1) * n + n
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!!
Flagged.
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. :-\
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.
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
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.
> 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), 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.
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 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.
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.
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.
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.
The act of calculating the area surely does mean they are the same rectangle. Because they can be written down in more than one way is a weakness of the notation; the math is independent of that.
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.
On that point, matrix multiplication is not commutative (and oh that it were!).
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.
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.
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.
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.
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.
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.
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?
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.
A similar discussion rescued from about 15 years ago: 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."
"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.
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.
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.
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.
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.
+1. In few years this kind of answers will be criminalized.
This is like complaining with Gauss that he summed the first 100 numbers in the wrong way ;)
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.
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.
If "He/she may even agree that it's ridiculous to deduct a point", then don't deduct it.
And get in trouble for not applying the government-mandated grading guide fairly to all students?
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.
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...
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)
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.
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.
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);
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.
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.
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.
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...
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.
This is how you ensure that somebody will never like math.
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-...
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?
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?
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.
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.
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.
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.
The author forgets to mention that in mathematics = represents equivalency.
This is why saying 5 of 3 is better than 5 times 3.
Its a trivial mistake; we don't know the context; the kid could (and maybe was) tutored on the difference. All about nothing.
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.
It's the tutor who needs tutoring on the non-difference.
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.
Yow! I wrote associative when I meant commutative! Oops.
Here's my response to this:
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.
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.
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.