> 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.
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.
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.
> 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.
What a defense of nonsense. The marking punishes a student for achieving a correct answer in an appropriate way, regardless of pedagogical justification.
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.
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.
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.
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:
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.
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 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.
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.
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.
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.
>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.
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!
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.
> "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.
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.
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?
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.
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.
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.
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.
>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."
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
"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."
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?
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.
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.)
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."
143 comments
[ 3.8 ms ] story [ 146 ms ] thread> 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.
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.
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.
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.
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.
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.
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".
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 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...
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.
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 definition of the algorithm given to the student may involve language like "take the first number and..."
The steps are the steps.
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.
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.
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.
People who don't understand a subject should not teach it. If they understand education but not math, then let them teach education.
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.
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.
Maybe we should do away with grading students based on exam performance altogether.
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!
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.
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".
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.
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".
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.
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.
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.
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.
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!)
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:
[2]http://mathworld.wolfram.com/Multiplication.html
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".
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.
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.
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.
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...
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.
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?
It's not the same. I'm not sure when that should be taught to a student.
http://www.corestandards.org/Math/Content/3/OA/
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.
There are algebraic structures where multiplication is not commutative, but I don't think this was the case here.
Memory and intellect complement each other.
Barry Reitman Author: "Secrets, Tips, and Tricks of a Powerful Memory." www.PowerfulMemorySecrets.com
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?
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.
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)
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.
> 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 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 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.
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.
5x = x + x + x + x, i can't rearrange that into terms of 5+ without involving even more concepts (like recursion etc)
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.
Would probably depend on how how much the concept was presented in class and the books, etc.
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.
"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."
Edit: Also, "5"x3 or 5x"3"?
How many fingers do you see?
Those are definitely qualities you want in a programmer.
> 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?
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.
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."
http://mathforum.org/library/drmath/view/57919.html