[HN Gopher] What's Wrong with "Multiplication Is Repeated Additi...
___________________________________________________________________
What's Wrong with "Multiplication Is Repeated Addition"? (2008)
Author : harperlee
Score : 130 points
Date : 2021-03-03 12:20 UTC (10 hours ago)
(HTM) web link (denisegaskins.com)
(TXT) w3m dump (denisegaskins.com)
| threatofrain wrote:
| The teacher mentions juggling units when adding and multiplying,
| but is 2 cm x 3 cm the same thing as 2 x 3?
| cturner wrote:
| Both examples use the same operator, but we resolve them to
| different functions. Mentally, we use the tuple of argument
| types to select the appropriate function. Addition does not
| provide equivalent functions for all multiplication cases.
| Hence, multiplication is not repeated addition.
| salawat wrote:
| You need to stop conflating the tyoe mumbo jumbo with
| multiplication.
|
| We don't define it based on what machines do with it, we
| define it based on what _WE_ do with it. In the context of
| teaching, dimensional analysis comes far later in the process
| of education.
|
| The pedants answer is that there is more than one
| multiplication operator taught. It's most basic incarnation
| is, for all intents and purposes, repeated addition.
|
| It's more advanced forms, which include extra auditing for
| dimensional analysis isn't merely repeated addition. Though
| I'd challenge that too.
|
| Dimensional analysis is an extra aspect stapled onto and
| seperate from the arithmetic operation. Therefore makes up an
| additional level of auditing seperately as a whole, and
| therefore inappropriate to define as an inherent property of
| the operator. You multiply numbers.
|
| People need to stop trying to teach unrelated things bundled
| by some programmer or language writer to kids/adults. It
| doesn't make one seem smart. It just confuses people by
| blurring orthogonal aspects of problem solving, which puts
| people off of it.
| d--b wrote:
| Arguably, what they're saying is that 2 x 3 should be
| understood as a unit-less "scaler" 2 that multiplies a number
| (of things) "3".
| caddemon wrote:
| Yeah I don't think they are the same. You can of course also
| have 2 cm * 3, which still fits with the repeated addition
| idea, as long as you frame it as 2 cm + 2 cm + 2 cm. I'm not an
| educator by any stretch, but that seems like an intuitive way
| to introduce units without causing confusion about previous
| number teachings.
| vishnugupta wrote:
| > but is 2 cm x 3 cm the same thing as 2 x 3?
|
| This made me think too, in particular the teacher's example.
| So; here's what I _think_ it should be calculated.
|
| 3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm).
|
| What you are doing above is adding two objects each of who are
| 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm"
| which then becomes "6cm".
|
| However 3 cm + 3 cm != 2 cm X 3 cm as the author states which
| is what leads them to wrong conclusion. In other words 2 cm X 3
| cm leads to an entirely new _thing_ in geometry which is
| "area". So while 3 cm + 3 cm is still length (and hence it
| should be translated to 2 X 3 cm i.e., one scalar unitless
| number and other number with unit) 2 cm x 3 cm is _not_ length.
| And the resultant area is correctly written as 6 cm^2; by
| multiplying units as well as the numbers. Now, how does one
| multiply numbers (2 X 3)? It 's indeed by repeated addition!
|
| The author, however is onto something. Which is when you
| multiply a something with unit (like 3Kg, 2Cm) with a unitless
| number that resultant product is still _something_ (that
| weight, length) that you began with. However when you multiply
| two things both of which have units the resultant product is
| something entirely new (i.e., area as opposed to length,
| pressure as opposed to weight etc.,)
| harperlee wrote:
| Another way to say it is that 2cm * 3cm is shorthand for
| (2)(1cm)(i) * (3)(1cm)(j). We are complecting 3 semantics:
| measure and unit (which compose a module, in vector terms)
| and "meaning" (dimension, in vector terms).
|
| This is why in high school physics you are taught to always
| write down the units of each number: typing them helps avoid
| a big amount of errors.
|
| Note that "typing" and "vector terms" are not essential, just
| helps me express it with the HN crowd easily.
| insert_coin wrote:
| The author is confusing math with physics.
|
| Math: 3x3 = 3^2 physics: cmxcm = cm^2
|
| 3cm x 2cm = (3x2)(cm x cm) = (math)(physics)
|
| The context gives the meaning.
|
| Or in computers, how many bits would the representation of a
| matrix of 2b x 2b occupy? 2b x 2b = 4b (not 4b^2 since memory
| is linear)
| EvanAnderson wrote:
| The HN crowd is way above me when it comes to math, but I'm going
| to chime in here w/ a question, come what may.
|
| I read about this a couple of years ago when I was doing basic
| arithmetic instruction at home to supplement my 5 y/o daughter's
| schooling. We were just doing basic numeracy activities at the
| time-- playing with physical representations of a number line,
| etc. I wanted to have some idea of how to approach multiplication
| when we got to that point (which we did about a year later).
|
| I was flummoxed by the whole "multiplication isn't repeated
| addition" debate because of my own arithmetic education nearly 40
| years before. The "light bulb" moment for me was reading an
| article that described multiplication of real numbers as scaling.
| I'm a embarrassed that, at >40 y/o, I had such an epiphany. (Then
| again, math has never been something I've had great intuition
| for.)
|
| I knew that her teacher was likely going to introduce
| multiplication as repeated addition and I felt like I wanted to
| have the first word on the subject with her.
|
| With that in mind when we got to multiplication in our home
| activities I pulled the physical number line analog out again (a
| board w/ some small nails in it) and used some elastic and rubber
| bands to illustrate "stretching" a number as multiplication of an
| integer by another integer (granted, for small numbers only-- I
| only had the patience to drive so many nails). I did a little bit
| of "shrinking" an integer by multiplying it by a fraction
| (because fractions are something we have covered extensively, if
| only as an excuse to have cake!) to show that multiplication is
| can scale a number to be both larger and smaller.
|
| It seems intuitive to me, but I'm a rank amateur. Was this a
| reasonable strategy? Did I do more harm than good?
| hinkley wrote:
| The new new math isn't just multiplication is addition, it's
| addition and substraction.
|
| 39 x 40 = 40 x 40 - 40
| EvanAnderson wrote:
| That seems like a reasonable way to do that computation, from
| a purely practical perspective. Simple "tricks" to arrive at
| the correct answer quickly don't at all seem like a bad idea.
| When we were doing addition of two digit integers and working
| with place value there was a lot of adding the ones
| separately from the tens, then adding those together. Later,
| when addition with carry / regrouping came up we used both
| strategies to solve the problems and demonstrated how they
| really were the same thing, just following a different
| "recipe" to get the result.
| matheist wrote:
| Math PhD here. You're a fantastic dedicated parent and you did
| no harm.
|
| First, as to the merits of your approach --- I think of
| multiplication as scaling myself. That's totally valid.
|
| Second, whether it's even possible to do harm here --- worst
| case scenario, the metaphor doesn't make sense to your kid and
| she doesn't use it in her own thinking about mathematics.
| Everyone has to develop their own intuitions --- like you had
| your epiphany --- and she'll develop hers even if you set her
| on a track that doesn't work, just as part of her learning and
| doing mathematics.
| EvanAnderson wrote:
| I appreciate the reply. I didn't feel too terribly off-base,
| but seeing the discourse here gave me cause to question.
| maxerickson wrote:
| I suspect that the problem isn't introducing a given model,
| rather the problem is then not moving on to additional models.
|
| Or putting it another way, I doubt that considering repeated
| addition gets in the way of understanding scaling.
| EvanAnderson wrote:
| I will say what we did work thru some multiplication of
| integers as repeated addition and scaling, just to show that
| the answers "come out the same". I used the scaling by a
| fraction to show that the repeated addition method doesn't
| work on all classes of numbers. (We've talked some about the
| difference between integers and real numbers.)
|
| The gist of my statement was something like: Adding
| repeatedly is a fine method to solve this kind of problem but
| it doesn't for all classes of numbers. We can't say, like we
| can w/ addition and subtraction, that multiplication implies
| a specific direction of movement (towards or away from zero)
| on the number line.
| maxerickson wrote:
| I wasn't really trying to analyze your approach, more
| speculating that getting in front of the teacher probably
| doesn't matter. It's the insight that multiplication isn't
| strictly equivalent to addition that matters, and
| illustrating that they are often equivalent probably
| doesn't block that insight.
| tarkin2 wrote:
| Why not just say:
|
| There is the symbol '*'. And when there's two integers it does
| one thing, and you can think about it as repeated addition. But
| when there's something different, it works differently. But now
| we're just going to concentrate on integers.
| jmchuster wrote:
| I like how 3blue1brown presents it in terms of animated number
| line translations
|
| https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=365
| rustybolt wrote:
| These articles bring back memories of teachers telling me I was
| wrong, because I didn't use the quadratic formula, but completed
| the square instead.
|
| You can definitely see multiplication as repeated addition. It's
| even a useful thing to do, how else would you define
| multiplication?
|
| This is the whole idea that made mathematics great. You define
| more complicated operations in terms of familiar operations.
| Then, if you gained enough intuition you are ready to use these
| more complicated operations as primitives as well. Then, you can
| generalize and apply your intuition to more abstract objects.
| baryphonic wrote:
| Funny, my teachers in middle and high school frowned on using
| the quadratic formula until we could demonstrate we knew how to
| complete the square.
| yakubin wrote:
| _> You can definitely see multiplication as repeated addition._
|
| Only for rational numbers. Doesn't work for real and complex
| numbers.
|
| _> It 's even a useful thing to do, how else would you define
| multiplication?_
|
| Axiomatically, not algorithmically.
| SAI_Peregrinus wrote:
| Multiplication of the Surreals is a recursive operation using
| sums (addition and subtraction on the left and right sets).
| Since the Reals are a strict subfield of the Surreals one can
| define multiplication of the reals using only the same
| recursive formula and restricting both operands to be Reals.
| eigenket wrote:
| Its probably worth emphasising that the recursion you need
| to "construct" the Surreals is infinite, in other words
| this does not give a reasonable algorithm to (for example)
| add two real numbers, you need S_omega in order to have
| even all rational numbers.
|
| The construction is rather involved but if we're only
| interested in the reals for now you can think of it as
| defining a real number by a set of rational numbers, in
| particular define a particular "real number" to be the set
| of all rational numbers less than it, for example sqrt(2)
| is defined to be the set of all rationals p/q such that
| p^2/q^2 < 2. We can "recursively" define addition of these
| real numbers in terms of addition of rational numbers
| because to add two reals you "just" have to add all the
| rationals in their respective sets.
|
| In general there is no sensible algorithm to do anything in
| the real numbers, since most real numbers aren't even
| computable (there is no way to represent an arbitrary real
| number on a Turing machine).
| marcosdumay wrote:
| Of course, once you decide to iterate over uncountable
| sets, infinity starts to appear all the time.
|
| This isn't the only case of infinite calculations that
| can't be computed in practice but that mathematics use
| all the time anyway; and it reflects quite well the fact
| that multiplying irrational numbers isn't something that
| one can do practice. There is no problem with it.
| alisonkisk wrote:
| Multiplying algebraics is trivial. A rectangle with sides
| sqrt(3) and sqrt(2) has area sqrt(6), which can be
| approximated by a decimal if needed.
|
| There are countable/computable/constructable subsets of
| the reals where multiplication has a finite algorithm and
| is it repeated addition.
|
| One example is the algebraics, as well as extensions the
| including a few special constants like pi. These are the
| subsets of the reals most commonly used for math and
| science. So in a wide range of problems areas,
| multiplication is not just repeated addition.
| eigenket wrote:
| I definitely wasn't trying to indicate that there was a
| problem with this, just pointing out that the path of
| using the Surreals (or anything else) to give an
| "algorithmic" description of real addition or
| multiplication is probably a bad idea.
| SAI_Peregrinus wrote:
| I wasn't even attempting to give an "algorithmic"
| description. Just a formulaic one, that depends on the
| axiom of infinity (and accepting transfinite induction as
| a valid process). Since even at least one of the usual
| process for constructing the Reals (Dedekind cuts) needs
| this I don't feel it's much of a stretch in reasoning.
| And the Surreals have a nicer recursive formula for
| multiplication that eventually turns into repeated sums,
| and they're a strict superset of the Reals, so it does
| apply there.
|
| If you want an algorithmic (no possible need an infinite
| number of steps) explicit construction of _anything_ on
| the Reals you 're going to be disappointed. They're an
| infinite set.
| eigenket wrote:
| You can get algorithmic explicit constructions of (for
| example) integer addition, rational addition and
| multiplication and even addition and multiplication of
| algebraic numbers on a Turing machine (all algebraic
| numbers are computable). All of these sets are infinite.
| The reals are particularly "badly behaved" even as far as
| infinite sets go.
| WallWextra wrote:
| If you define the reals axiomatically, you still need an
| existence proof. Which will involve addition and
| multiplication algorithms.
| qsort wrote:
| Well, no, not really. The standard definition of the reals
| is as the unique nontrivial totally-ordered, Dedekind-
| complete, Archimedean field up to isomorphism. So what you
| would really need is a uniqueness proof, with addition and
| multiplications "provided" by the hypothesis.
| howling wrote:
| And how do you prove that a totally-ordered, Dedekind-
| complete, Archimedean field does not lead to
| contradiction besides constructing it explicitly by
| bootstrapping from natural numbers?
| qsort wrote:
| Nothing in the construction requires you to show an
| algorithm that given x, y \in R allows you to compute x+y
| and xy. You would make the usual Dedekind construction
| and show it satisfies the axioms of such a field. (as a
| matter of fact, no such algorithm exists in full
| generality!)
|
| It's probably a tomato/tomato kind of thing, but I'm only
| objecting to the 'algorithm' part of parent's comment.
| yakubin wrote:
| Not algorithms. There will be infinite addition involved,
| and algorithms are finite.
|
| Thinking of multiplication as repeated addition also won't
| explain anything about it. It's a separate operation. Deal
| with it. For similar reasons, you can't calculate x-th
| power of a number, when x is irrational, by decomposing it
| into exponentiation and roots.
|
| This metaphor is just training wheels. At some point you
| should lose it.
|
| Together with the notion that "multiplication is repeated
| addition" comes the notion that numbers are quantities.
| Only some of them are, and this isn't really what makes
| them numbers. Now what exactly gets repeated, when you
| don't have quantities?
| prionassembly wrote:
| Multiplication on the _naturals_ is repeated addition.
|
| You might see the natural numbers as training wheels for
| higher mathematics, but number theorists might
| disagree...
| yakubin wrote:
| Number theorists work with integers, not naturals. And
| they too will agree with me, as they mostly work with
| polynomials, which show exactly that.
|
| I also didn't say natural numbers are training wheels,
| just this metaphor, which to use HN lingo: doesn't scale.
| alisonkisk wrote:
| Number theorists work with complex numbers and irrational
| numbers too
|
| https://www.britannica.com/science/Riemann-zeta-function
| titzer wrote:
| Algorithms can work on symbolic formulas, and symbols can
| represent anything; infinite objects, operations on
| infinite objects, infinite sets of operations on infinite
| objects, and so on.
| eru wrote:
| > There will be infinite addition involved, and
| algorithms are finite.
|
| People are just bit sloppy, and say algorithm when they
| mean something slightly different.
|
| See https://stackoverflow.com/questions/28841260/what-is-
| the-dif... and https://en.wikipedia.org/wiki/Corecursion
|
| Basically, you don't want an 'algorithm' here to produce
| the whole number.
|
| All you need is some scheme that will produce the next
| digit in finite time (and the next one and the next one
| etc).
| alisonkisk wrote:
| Which is a technical way of saying "in real life people
| use finite rational or algebraic approximations for
| reals, so uncountability of reals and infinite precision
| aren't a problem".
| bopbeepboop wrote:
| Algorithms aren't necessarily finite, eg "while True,
| print 1".
|
| Also, an irrational exponent is the product of component
| factors.
|
| b = Prod(0,inf) a^[x_i * 10^(-i)] = a^x
|
| So even with irrational numbers, operations can be
| decomposed - such as exponentiation into multiplication
| of integer exponents and roots.
|
| Which makes sense, because in the sequence definition of
| reals you need a way to generate the resulting sequence
| from the two original sequences.
|
| I think you're trying to claim more than is true.
| gugagore wrote:
| Algorithms are not finite. E.g. Newton's method.
| yakubin wrote:
| First sentence on Wikipedia[1]:
|
| _> In mathematics and computer science, an algorithm (
| /'aelg@rId@m/ (About this soundlisten)) is a finite
| sequence of well-defined, computer-implementable
| instructions, typically to solve a class of problems or
| to perform a computation._
|
| Newton's method is finite too. You perform finitely many
| iterations. It doesn't calculate roots. It calculates
| their approximations.
|
| [1]: <https://en.wikipedia.org/wiki/Algorithm>
| gugagore wrote:
| If you use a termination condition that has to do with
| convergence of iterates instead of a number of iterations
| (often the case), then you generally don't know
| beforehand the length of the finite sequences. Maybe you
| know a bound, but in general you might not even have
| that.
|
| In an important sense, it only becomes a finite
| algorithm. It isn't one. You cannot write the finite
| sequence of instructions down. It's got loops.
|
| To your point about approximations vs not, if you have an
| algorithm that, for any desired approximation accuracy
| can compute the square root to that accuracy in a finite
| number of steps, then that process is as much "the square
| root" as anything involving the real numbers.
| yakubin wrote:
| _> To your point about approximations vs not, if you have
| an algorithm that, for any desired approximation accuracy
| can compute the square root to that accuracy in a finite
| number of steps, then that process is as much "the
| square root" as anything involving the real numbers._
|
| Not really, since approximations, no matter how accurate,
| don't preserve algebraic properties. You only get to know
| what it's bigger/smaller than.
| gugagore wrote:
| I think I understand what you mean, so let me dial back
| "anything involving the real numbers".
|
| If you are representing or thinking of "sqrt(2)" as "the
| positive solution to x^2 = 2", then you preserve
| algebraic properties. But you generally (correct me if
| I'm wrong) _don 't_ get to know whether it's bigger or
| smaller than something else of the form "the
| _choose_uniquely_ solution to _some_equation_" unless you
| rely on an argument where you invoke approximations.
| xirbeosbwo1234 wrote:
| >Only for rational numbers. Doesn't work for real and complex
| numbers.
|
| Complex numbers aren't really relevant, in my opinion,
| because they are usually introduced as an extension of the
| rules for reals and polynomials. To multiply two complex
| numbers, you can totally forget that i is imaginary, do the
| multiplication as if it's just an ordinary variable, then
| substitute "i" back in. But that relies on being able to
| multiply polynomials, which would be difficult to define in
| terms of repeated multiplication.
|
| To some extent all of mathematics is a lie. We can do
| multiplication on the reals because we have decided that it's
| allowed. It is reasonable to define multiplication at first
| as repeated addition and then define a way to extend that to
| the reals that is consistent with the first definition.
| DanielMcLaury wrote:
| Multiplication of real and complex numbers is typically
| defined by starting from repeated addition, extending this
| notion to rationals, and then extending that notion to reals
| by taking limits.
|
| How exactly are you going to present multiplication of real
| numbers axiomatically without essentially including an axiom
| that bootstraps everything from repeated addition?
|
| I suppose you can try defining the reals as "the unique
| complete ordered field" or the complex numbers as "the unique
| algebraically closed field of characteristic zero with
| cardinality c," but I don't think either of those are
| pedagogically useful to someone who is still learning what
| multiplication is.
| alentist wrote:
| > You can definitely see multiplication as repeated addition.
|
| Doesn't work in every context where multiplication is defined.
|
| > It's even a useful thing to do, how else would you define
| multiplication?
|
| Perhaps: A multiplication is an endomorphism on an additive
| semigroup.
| scatters wrote:
| > how else would you define multiplication?
|
| In Church arithmetic, multiplication is defined as function
| composition; that is, the dot operator `.` in Haskell-like
| languages. Addition is considerably more complex; it is defined
| as `lift (.)`. Exponentiation is even simpler; it is function
| application, so the `id` function - except that we write
| exponentiation backwards, so actually `flip id`.
| shawnz wrote:
| Interesting, what about tetration then?
| scatters wrote:
| Good question: you can define m||n as: \m
| n -> n (exp m) 1'
|
| where `exp` is Church exponentiation, `flip id`, and `1'`
| is the Church numeral for 1, that is, `id`. According to
| pointfree.io this expands to: flip flip
| id . flip id . flip id
|
| Which isn't particularly illuminating. Pentation isn't much
| worse, though: flip flip id . flip id .
| flip flip id . flip id . flip id
| thrwn_frthr_awy wrote:
| > how else would you define multiplication?
|
| With my six year old I told them to say "groups of" instead of
| "times". So 3 X 2 isn't read as "three times two" it is "three
| groups of two" which I think has been helpful.
| Frost1x wrote:
| >These articles bring back memories of teachers telling me I
| was wrong, because I didn't use the quadratic formula, but
| completed the square instead.
|
| Education is probably a special case, at least from my
| educational experience. There's often multiple ways to
| abstractly represent a problem in mathematics and find the
| desired solution. The example you pointed out is such a case.
|
| When teaching mathematics, I think the goal is to introduce a
| lot of forms of mathematical thinking and approaches to solving
| a problem, to make you realize there are often multiple
| approaches and to take a peek at the insight of some of these
| approaches and how they often connect or think about different
| 'branches' of mathematics.
|
| Teaching math needs to be explicit with this though: solve this
| using method X. They should also explain this to kids as to why
| they're doing it. One of the biggest mistakes I see in
| mathematics teaching is the perception there's "only one right
| answer." Well, yes and no. Under certain condition a specific
| answer exists, sometimes it's a set of answers and sometimes
| you they're not really assessing so much that you can get the
| correct answer, but instead that you understand a specific
| method.
|
| The 'answer' is a means to an end to force you to step through,
| internalize a process, and hopefully at some point understand
| the deeper insight of that clever process you internalized and
| apply it to other problems you may encounter in the future. You
| may never use the exact process or insight behind it as-is,
| then again you may use facets of the reasoning you internalized
| later. If you don't work in professions that require this sort
| of abstract thinking I can see how the entire dance is quite
| silly but if you work in research, you often appreciate all
| these nuggets of deep insight and wisdom you've gained you can
| cobble together, remold, or lead to new insights. As a kid you
| probably have no idea where you'll work as an adult so maybe a
| lot of that effort is wasted.
| jordan_curve wrote:
| To make this more explicit, mathematics (even at the primary
| school level) is arguably more about "proving" things than it
| is about computation.
|
| Obviously you're not writing proofs in 3rd grade, but the
| emphasis should be on students being able to say why their
| answer is correct rather than just being able to produce the
| correct answer.
| swiley wrote:
| Anyone that teaches this way should find something else to do.
| Memorizing a formula is easy but if you want to do well later
| you _must_ be able to complete the square.
| daveFNbuck wrote:
| I never learned to complete the square, and I managed to do
| well enough later to get a degree in math.
| rafaelturk wrote:
| Yep, it is a clickbait
| parenthesis wrote:
| > These articles bring back memories of teachers telling me I
| was wrong, because I didn't use the quadratic formula, but
| completed the square instead.
|
| That's hilarious, since the quadratic formula is just what you
| get by solving ax^2 + bx + c = 0 by completing the square.
|
| In fact, I have a high school textbook that makes that
| derivation.
| sudosysgen wrote:
| We did that derivation in class, actually! I was so lucky to
| have a great math teacher in high school.
| IdiocyInAction wrote:
| Yeah, we got it as bonus homework, to derive it ourselves.
| eru wrote:
| That's not the norm?
| Jtsummers wrote:
| No, it's not. It depends a bit on your teachers but also
| on which "level" of math you're in (in the US). Lower
| level but still algebra/geometry classes tend to teach
| facts, not derivations from foundational concepts. Those
| are the classes aimed at non-Honors and maybe non-College
| Prep students (2 of the 3 typical "tracks" students end
| up in the US, names may vary by state and decade).
| shadowofneptune wrote:
| The way Geometry is taught in the US is awful. Instead of
| learning that you can use shapes to do useful
| calculations like square roots, you slog through
| postulates and theorums without any sense of why you have
| to do them. Rarely is what is learnt in geometry ever
| used in later high school courses, save for trigonometry.
| I hope it is different in other countries.
| loonster wrote:
| The best math class I ever had was a drafting class
| called "Descriptive Geometry". In that college class we
| used a drafting table to solve math problems.
|
| An easy example would be the length of line of the corner
| seam of a hip roof. Sadly I have forgotten what the more
| complex problems were. Fantastic class that isn't offered
| anymore.
| nathancahill wrote:
| Btw, the game Euclidea on iOS is fun for that. You
| "discover" and prove concepts and progress through
| higher-orders of geometry proofs.
| drdec wrote:
| The point of the Geometry course in that curriculum is to
| teach the student how to do proofs. The Geometry itself
| is secondary.
| jacobolus wrote:
| Unfortunately this is typically done in a
| counterproductive way.
|
| See http://jwilson.coe.uga.edu/TiMER/Schoenfeld%20(1988)%
| 20Good%...
| thaumasiotes wrote:
| > Instead of learning that you can use shapes to do
| useful calculations like square roots
|
| Huh?
|
| You can easily draw shapes that conceptually represent
| square roots, but how do you get from that to calculating
| the square root? You'd need an infinitely-graded ruler.
| sudosysgen wrote:
| All calculation is inherently imperfect. But if you want
| to have something of length sqrt(2) you can.
| thaumasiotes wrote:
| I'd say the opposite; the drawing is guaranteed to be
| imperfect, but the calculation won't be.
| throwaway894345 wrote:
| The US education system varies tremendously. I don't
| think you can credibly claim that this teaching method is
| "standard" in the US. At least my experience was much
| better than you described. Indeed it wasn't until
| university-level mathematics that I started to get into
| the "stop trying to understand how they work and just
| memorize these formulae". Fortunately my engineering
| classes provided a through-line for understanding how
| those formulae worked, and I was much stronger for it
| than my math-major contemporaries.
| meepmorp wrote:
| Oh god, no. Not at all, especially in the US.
| jeff-davis wrote:
| Are you sure? My teacher did that.
|
| It doesn't take very long, and I can definitely see
| someone forgetting all about it.
| setr wrote:
| I distinctly remember "accidentally" deriving it when I
| forgot the formula on a test; perhaps my proudest math
| moment (though really I was just scrambling apply any
| rule I could think of that got me closer to the vague
| form I remembered)
|
| And then I got really annoyed that no one ever told me to
| do that before and started discounting teachers for years
| onwards.. probably to my own detriment.
| the_only_law wrote:
| I can tell you how great my math education was. I don't
| even recall the quadratic formula or completing the
| square is. I'm almost certain they were part of the
| curriculum at some point.
|
| I recall in later math course, notably calculus, the
| teacher assumed we we're familiar with some concept
| because we we're supposedly taught it the prior year, yet
| not a single student in the class could recall it having
| been taught before.
| redshirtrob wrote:
| I don't think so. I was on the Calculus track in high
| school so we derived it in...Pre-Calculus.
|
| Prior to that the quadratic formula was something that
| seemed to be handed down from on high. We used it in
| Algebra II and maybe even before that, but I had no idea
| where it came from.
|
| It was a mind-opening experience when we derived it in
| class one day. Our teacher didn't ruin the surprise. She
| just said, let's complete the square on a general
| quadratic equation. And there it was. The quadratic
| formula!
| thaumasiotes wrote:
| > Our teacher didn't ruin the surprise. She just said,
| let's complete the square on a general quadratic
| equation.
|
| How is this not ruining the surprise? The only possible
| outcomes of doing that are that (1) you make a mistake;
| or (2) you get a formula for solving quadratic equations.
| Quadratic equations have the same solutions regardless of
| your methodology, so there's only one formula you can
| get.
| [deleted]
| betterunix2 wrote:
| "That's hilarious, since the quadratic formula is just what
| you get by solving ax^2 + bx + c = 0 by completing the
| square."
|
| That is one way to derive it, but there are others, including
| my personal favorite, the resolvents method (which also works
| for deriving cubic and quartic formulas).
| pdonis wrote:
| _> You can definitely see multiplication as repeated addition_
|
| Only if you restrict to whole numbers. But multiplication does
| not just apply to whole numbers. How would you see
| multiplication by pi as repeated addition?
|
| _> how else would you define multiplication?_
|
| As a separate operation with its own properties. The subject
| article gives several examples of properties of multiplication
| that are simply different from properties of addition.
| dash2 wrote:
| Multiplication by rational fractions p/q is just
| multiplication by whole p, followed by multiplication by 1/q.
| 1/q is just the number that when multiplied by q gives 1. So
| you can cash that all out as repeated addition.
|
| Multiplication by irrationals is just an infinite sum of
| multiplication by rationals, no? x times pi = x times 3, plus
| x times 1 / 10, plus....
|
| I don't see there's any conceptual issue here. Mathematicians
| feel free to correct me.
| pdonis wrote:
| _> multiplication by 1 /q_
|
| Is not repeated addition. You can't add a number to itself
| 1/q times. At least, not unless you're willing to adopt
| increasingly perverse interpretations of "repeated
| addition" as you try to cover more and more numbers. See my
| response to wruza upthread.
| dash2 wrote:
| Hmm, so just go about it differently: x * p/q with p,q
| integers is (x*p) / q. The first bit is repeated
| addition. The second bit means, find the number r such
| that r * q = x * p. Even if r is not integer, q is
| integer, so we can try different numbers, add them to
| themselves q times, and close in on the answer.
|
| So I still think conceptually it's fine to think of it as
| repeated addition? It might be algorithmically a bad way
| to do it ("increasingly perverse", taking limits when the
| number is irrational etc.). I don't know how computers
| actually implement multiplication - though wikipedia
| (https://en.wikipedia.org/wiki/Binary_multiplier) says it
| works via shifts and adds.
|
| Or maybe, it's bad to teach it to kids this way? But then
| I think we need evidence from educationalists.
| pdonis wrote:
| _> The first bit is repeated addition_
|
| So now your definition of "repeated addition" is
| "repeated addition, plus a version of multiplication".
| Division is the inverse of multiplication, so your
| definition is circular: you're "defining" multiplication
| in terms of repeated addition _and multiplication_.
|
| Similar objections apply to another poster's contention
| upthread that the "repeated addition" definition is
| justified because of the distributive law. The
| distributive law defines A times (b plus c) in terms of A
| times b plus A times c. So it's useless as a _definition_
| of multiplication in terms of addition.
| im3w1l wrote:
| You can do it like this:
|
| First define a*b for integers by addition.
|
| Define the the rational numbers as ordered pairs of
| integers where the last number may not be 0. Consider the
| rational number (a,b) to be equivalent to (c, d) if a*d =
| b*c
|
| (Exercise for the reader: Prove that this is an
| equivalence relation)
|
| Now define (a, b) + (c, d) = (ad + bc, bd), (a, b) * (c,
| d) = (ac, bd)
|
| (Exercise for the reader: Prove that the subset of
| rationals (n, 1) behaves just like the integers)
| wruza wrote:
| _How would you see multiplication by pi as repeated
| addition?_
|
| By taking 3 times as usual and then another .14
| approximately. E.g. for 2, it is 3+3+28/100, roughly 6.28.
| (edit: if you're confused by /100, read it as "two places
| next to a point")
|
| If you're asking how to do that _exactly_ , first tell how do
| you even add pi to some number exactly? Let's start with one,
| 1 + pi:
| pdonis wrote:
| _> By taking 3 times as usual and then another .14
| approximately_
|
| First, "taking 3 .14 times" doesn't make sense, at least
| not in the elementary school student's understanding of
| "repeated addition".
|
| Second, pi is irrational, as you evidently realize since
| you say "approximately". There is no way to add 3 pi times.
|
| _> If you're asking how to do that exactly, first tell how
| do you even add pi to some number exactly? Let's start with
| one, 1 + pi:_
|
| Add 1 to the 3 and keep the part to the right of the
| decimal point the same. Addition is commutative.
|
| Basically, your argument is that multiplication is repeated
| addition as long as we're willing to adopt increasingly
| perverse interpretations of "repeated addition" as we
| expand the scope of the numbers we use. Wouldn't it be
| better to just admit up front that multiplication is _not_
| repeated addition, although the two are similar for some
| types of numbers?
| RandallBrown wrote:
| > not in the elementary school student's understanding of
| "repeated addition".
|
| Pi isn't particularly well understood by elementary
| school students either.
| wruza wrote:
| But what _is_ that number? You only described an
| algorithm that relies on a vague definition of "part to
| the right". See, you can't even write it down (unlike 1 +
| 3.14 = 4.14), because pi is not really a number, in a
| sense. It is an infinite calculation that happens to
| converge between 3.14 and 3.15. You can never get rid of
| "pi" in your calculations unless it cancels out
| naturally, so for practical reasons it's 3.14 and for
| theoretical reasons it's "pi".
|
| Let's make it clear: you cannot add pi or take pi times
| at all. You only can add to factors to the left of it
| (same as a+bi). That's why you cannot multiply by pi by
| repeated addition, not because the numeric method is
| wrong.
|
| _Wouldn 't it be better to just admit up front that
| multiplication is not repeated addition, although the two
| are similar for some types of numbers?_
|
| For generalized reasoning, yes. For teaching, not sure.
| Which is better, a student who knows that mul is rep-add,
| or a student who gave up and doesn't know even that? When
| they learn, they ask themselves why and it's a rabbit
| hole. You should stop somewhere before turning into maths
| professor, if that's not your goal.
| pdonis wrote:
| _> what is that number?_
|
| Pi minus 3. If you are saying you can't actually perform
| that operation, great! That means you are agreeing with
| me. See below.
|
| _> you cannot multiply by pi by repeated addition_
|
| Thank you for agreeing with my main point. See below.
|
| _> not because the numeric method is wrong._
|
| If someone is going to claim that multiplication _is_
| repeated addition, then their definition of
| multiplication in terms of repeated addition must cover
| all cases. You are saying it doesn 't cover pi, which
| means it doesn't cover all cases. So the definition is
| wrong.
|
| _> For generalized reasoning, yes. For teaching, not
| sure._
|
| Why not? What's wrong with telling kids, multiplication
| in general is a distinct primitive operation, we are
| teaching you how to multiply whole numbers using repeated
| addition, but be aware that this method will not
| generalize to all cases?
|
| _> Which is better, a student who knows that mul is rep-
| add, or a student who gave up and doesn't know even
| that?_
|
| A student who has been told what I just described above
| is not in either of these positions, so you are arguing
| against a straw man.
|
| _> When they learn, they ask themselves why and it's a
| rabbit hole._
|
| No, it's natural human curiosity which should be
| encouraged, not stomped on. At some point they'll either
| realize that they're up against material they're not yet
| ready for, and put it aside for later, or they'll end up
| being a math prodigy. Both outcomes are better ones than
| them just being told "this is it, don't ask any
| questions".
| whatgoodisaroad wrote:
| Exactly. Probably every discovery in math starts out with
| something that's only close to the truth (like multiplication-
| is-repeated-addition) but generalizing it to open up whole new
| areas.
|
| This part of math is so exciting and compelling, I feel like it
| should be more of a focus in schools.
| johndoe42377 wrote:
| Literally nothing. It is even in the language - multiple times.
| Multiply n times. And even square makes perfect sense - whatever
| side you choice, which is commutativity.
|
| Disconnected from reality concepts is the root of all evil.
| jeff-davis wrote:
| Is addition a shortcut for counting, or is that wrong, too?
| tromp wrote:
| In some settings, "repeated addition" is the only sensible way to
| think of multiplication. Such as multiplying an elliptic curve
| point with a scalar.
| sumtechguy wrote:
| Multiplication is a shorthand for 'repeated addition'. Much
| like power is short hand for 'repeated multiplication'. Even
| with imaginary numbers. You can create an in infinite series of
| additions that looks a lot like the imaginary number.
|
| The argument they are making is kind of strange. But not
| 'wrong' per se. If you mix 2 types of number systems you get a
| different number system out of it. It is a similar argument I
| make about fractions. It is basically physically impossible to
| have a fraction of some things. If I have a car and divide it
| in half. I no longer have a car. I have two of something else.
| They are clearly not a car because they can not act like a car
| anymore.
| WalterBright wrote:
| I know how addition is done in hardware. I've programmed
| multiplication and division on computers that only had shift and
| add instructions. I've implemented IEEE 754 floating point
| packages, too.
|
| https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL...
|
| Multiplication is repeated addition.
|
| Yes, I know about roundoff errors, overflows, etc.
| OlleTO wrote:
| Proof: The derivative of x^2 is x.
|
| x^2 = x * x = x + x ... + x (x times)
|
| Therefore,
|
| d/dx(x^2) =
|
| d/dx(x * x) =
|
| d/dx (x + x + ... + x (x times)) =
|
| d/dx(x) + d/dx(x) + ... + d/dx(x) (x times) =
|
| 1 + 1 + ... + 1 (x times) =
|
| x
| michaelwu wrote:
| It's actually 2x...
| OlleTO wrote:
| Yes, because multiplication isn't actually repeated addition
| :)
| t0r0nat0r wrote:
| I don't think the aruments made here are compelling.
| Multiplication can be defined in this way. Saying that
| multiplication is repeated addition is not the same as saying
| that multiplication IS addition. Addition relates to
| multiplication as multiplication relates to powers.
| Agingcoder wrote:
| I don't understand.
|
| The author is mixing up physics (dimensional analysis) and maths,
| and trying to give the multiplicand a special role (I didn't even
| know there was a distinction between the two terms - to me they
| are both factors). This might be true in the physical world, but
| in the world of numbers, I think the distinction is irrelevant.
|
| Furthermore, being able to compute/define multiplication through
| repeated addition doesn't prevent you from looking at the special
| properties of this new operator.
| drewcoo wrote:
| Agreed. Multiplier and multiplicand are different words but the
| commutative property says their values can swap equivalently,
| so . . . what was the author's point again?
| tomxor wrote:
| This was exactly my thought, I don't understand why those two
| word exist since it's commutative. Perhaps all this confusion
| is purely semantic.
| ThePadawan wrote:
| They exist to distinguish the element being operated on
| (the LHS) and what it is operated on by (the RHS).
|
| Total technicality, but I could see myself using the term
| multiplicand/multiplier in my code if I had to implement
| e.g. a stack-based parser for arithmetic expressions.
| tomxor wrote:
| > to distinguish the element being operated on (the LHS)
|
| I'm not sure what you mean, they are both operated on, it
| is a binary operator and commutative, there is literally
| no difference.
| simias wrote:
| I agree that it's a generally pointless distinctions,
| although it might be useful in some cases such as number
| systems where the multiplication _isn 't_ commutative, or
| for a particular implementation where the distinction
| matters.
|
| After all if you were to code a multiplication that was
| implemented naively as a series of additions it'd be
| generally much faster todo 2x1000 than 1000x2.
|
| To return to TFA I think the author is talking from a
| pedagogical standpoint, that teaching that multiplication
| is just a bunch of additions under a trench coat is not
| the best way to go. I'm not sure that I agree personally.
|
| In particular this bit regarding multiplier/multiplicand
| makes zero sense to me:
|
| >Different names indicate a difference in function. The
| multiplier and the multiplicand are not conceptually
| interchangeable. It is true that multiplication is
| commutative, but (2 rows x 3 chairs/row) is not the same
| as (3 rows x 2 chairs/row), even though both sets contain
| 6 chairs.
|
| Of course 3 rows and 2 rows aren't the same, but what
| does it have to do with the order of the multiplication?
| Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2
| rows? It's a bizarre argument.
| Double_Cast wrote:
| Suppose your friend Alice arranges your wedding. You ask
| her to arrange the lawn chairs in an arrangement of 2
| rows and 3 columns. But she misinterprets your request as
| 3 rows and 2 columns. Oops. Now a particular family can't
| all sit in a single row without rearranging the chairs.
|
| If all you care about is the total number of chairs, the
| order of operands is irrelevant. but if you care about
| the structure, "2 x 3" may encode information that "6"
| does not.
| bosswipe wrote:
| It distinguishes these two rectangles.
| *** ** *** ** **
| anbende wrote:
| If we are just looking at numbers with no units attached
| there is no difference. As soon as there are units, a
| difference arises.
|
| I will collect 12 apples 5 times. The 12 apples are the
| multiplicand and the 5 times is the multiplier.
|
| The answer retains the units from the multiplicand.
| np_tedious wrote:
| Still don't see the difference.
|
| Density times volume is mass. Volume times density is
| mass. Distance times force is work/energy...
|
| "5 times I will connect 12 apples"
| KMag wrote:
| You're correct. The GP in incorrect about their
| dimensional analysis. 5 apples 12 times yields apples
| because it's apples times a dimensionless scalar (count).
| Newtons times meters is always Newton-meters, never
| Newtons or meters. Units are never magically dropped in
| dimensional analysis.
| SAI_Peregrinus wrote:
| Multiplication over the reals is commutative. Matrix
| multiplication of non-square matrices isn't. Multiplication
| in a Ring isn't necessarily commutative. Other algebraic
| structures also have non-commutative multiplication.
|
| One could argue that these things aren't "multiplication"
| even if they are "products" since they don't satisfy all
| the properties of multiplication over the reals. But it is
| common to call the use of the product operation
| "multiplication", at least in cases where there's only one
| product operation to use. EG Geometric Algebra has Inner,
| Outer, and Geometric products, so calling them
| "multiplication" seems less common IME.
| qsort wrote:
| I really don't get the point you're making. If we're
| going to pull out random examples, monoids aren't
| guaranteed to be abelian; strings and concatenation form
| a monoid that's not abelian.
|
| If you have enough mathematical sophistication to
| conceptualize a non-commutative ring, you're well past
| the point where naming conventions are even remotely an
| issue.
|
| The original article was contrasting addition and
| multiplication on the basis that addends are called the
| same while factors are supposed to be called differently,
| which not only makes no sense (it's just a naming
| convention), but it also breaks down when you have more
| than two factors: what is the "c" in a x b x c called? Or
| we're talking about _non-associative_ operations now?
| SAI_Peregrinus wrote:
| My point is only that the commutative property is not
| inherent to all multiplication operations, so there can
| be a distinction between the operands. It's not
| necessarily a _useful_ distinction, and in the usual use
| of multiplication it 's utterly useless and only adds
| confusion.
|
| But matrix multiplication is taught in high school (and
| usually promptly forgotten), it's not particularly
| advanced math.
|
| Personally I'm of the opinion that the terminology is
| muddled. There's no need to distinguish the operands of a
| multiplication over any of the usual domains (reals,
| rationals, integers, etc). And when you reach the point
| where it does become important there's generally more
| than one product operation and we should stop calling it
| multiplication. "Matrix multiplication" is a bad term.
| You also typically wouldn't name the operands, since as
| you note there can be more than two!
| qsort wrote:
| Sure, my counterpoint was just that the same reasoning
| technically applies to addition as well, but as you note
| "matrix multiplication" is taught in high school, while
| my example wouldn't come up.
|
| I agree not calling it multiplication would probably
| help, perhaps something like "linear transformation
| composition" might encourage students to keep it
| separated from real multiplication, I jsut found the
| argument in the original article kind of ridiculous to be
| honest.
| Retric wrote:
| Yea, math tends to reuse terms and notation across
| disciplines in ways that adds confusion rather than
| clarity. It's much better to think of infinity for
| example as multiple independent concepts than assume it's
| all the same idea.
| karmakaze wrote:
| Not all multiplication is commutative, e.g. matrices.
| gugagore wrote:
| If, for a moment, you conceptualize of multiplication on non-
| negative whole numbers as repeated addition, then this is the
| algorithm: procedure product(multiplier,
| multiplicand) acc := 0 for i = 1 to
| multiplier acc := acc + multiplicand
| end return acc end
|
| Swapping the arguments is a different computation. But after
| thinking about it, you realize that you get the same answer
| all the same. That's the point of
|
| > (2 rows x 3 chairs/row) is not the same as (3 rows x 2
| chairs/row), even though both sets contain 6 chairs.
|
| The point with bringing up dimensional analysis is that the
| above algorithm doesn't work because what does it mean to do
| `for i = 1 to 3 chairs/row`? You might think of it like
| procedure product(multiplier, multiplicand) acc :=
| "0" # an "absolute" zero that cooperates with any dimension
| each single_multiplier in multiplier acc := acc
| + (multiplicand * single_multiplier) end
| return acc end
|
| But then what is `(multiplicand * single_multiplier)` ?
| midjji wrote:
| Not defending the article, but how would you compute: \pi*\pi
| using repeated addition?
| thrower123 wrote:
| 3.14159 + 3.14159 + 3.14159 + (3.14159*0.14159)
| bitlax wrote:
| red pen time
| thrower123 wrote:
| It's fine if you're going by significant figures
| horsawlarway wrote:
| pi + pi + pi + (.141592... of pi) ~= 9.8696
|
| You need the concept of a ratio, so arguably I'm using
| multiplication to define multiplication, but you're sort of
| cheating by asking about a fractional number.
| masswerk wrote:
| However, pi is already a ratio...
| horsawlarway wrote:
| Right, that's why it's cheating. He's asking about a
| number set that requires the concept of ratios, and
| ratios are multiplication.
| bidirectional wrote:
| Pi is definitely not a fractional number... I don't think
| it's cheating at all, multiplication on the naturals is
| repeated addition, that's not the case for the reals.
| spacedcowboy wrote:
| Interesting how computers (which only understand '1' and
| '0' can do multiplication of reals, then. Unless you're
| intel, of course... (no, I will never let it go :)
| Cerium wrote:
| Can we not consider that the algorithm taught for
| multiplication of real numbers is repeated multiplication
| of natural numbers which can be seen as repeated addition
| of natural numbers so we could define an addition only
| algorithm for multiplication of real numbers.
| andrewprock wrote:
| This is an interesting statement.
|
| While it's true, it obscures the fact that pi is defined
| as a fraction: Circumference/Diameter.
|
| That this fraction cannot be represented as a numeric
| fraction is one of the great insights of early
| mathematics.
| DanielMcLaury wrote:
| I wouldn't call the fact that pi is irrational an insight
| of "early mathematics." We knew that sqrt(2) was
| irrational around 500 BC if not earlier. We didn't know
| pi was irrational until around the time of the American
| revolution.
| horsawlarway wrote:
| Sure it is, as long as you're willing to repeat in
| increments of real numbers.
|
| And that's my point, basically - By the time we're
| discussing real numbers, we need multiplication as an
| operator, because we're discussing ratios already.
| scarmig wrote:
| How would you compute \pi+\pi using addition?
|
| Irrationals break all our intuitions.
| eigenket wrote:
| You first define multiplication of natural numbers to be
| repeated addition, then define multiplication of rationals in
| terms of multiplication and addition of naturals, then define
| multiplication of reals in terms of multiplication of Cauchy
| sequences of rationals ;)
| codethief wrote:
| Exactly. I've scrolled through hundreds of comments here
| now and it really is beyond me how the question of whether
| you can define x*y in terms of addition for x and y being
| arbitrary reals is even a matter of debate.
| spacedcowboy wrote:
| I have a PhD in physics and more maths qualifications than I
| can shake a stick at; to me, multiplication is repeated
| addition.
|
| I'm not sure what the teacher is trying to do here, but I do
| think the outcome of what they're trying to do is far more
| complicated than the simple "multiplication is repeated
| addition".
|
| I also happen to have an 8-year-old going through third grade
| right now, and when we were talking through his homework, it
| was quite clear that using simple concepts he already knew
| (addition & subtraction) to explain slightly more complex
| things that he was learning (multiplication and division) was
| _really_ useful to him. As I recall it being to me.
|
| [aside] I think the maths schedule is more advanced now than it
| was in my day anyway - he only did multiplication and division
| this year, but he also did algebra and simultaneous linear
| equations now, as in:
|
| a + b + 8 = 24
|
| a - b = 4
|
| "Solve for a and b"
|
| Pretty sure I only did that in senior school (11 and up), not
| at age 8. No powers as yet (presumably they'll come after the
| multiplication/division stuff), so no quadratic formula, but
| still...
|
| [/aside]
| agumonkey wrote:
| Since you're a physician, do you think that helps for
| multiplicative relationships in real world laws ? It took me
| decades .. sadly, to be comfy with handling U = RI formulas.
| On the algebraic level it's stupid simple, but for real world
| physics the meaning is more bidirectional coupling of ratios
| and amplitudes and taps into a different part of my brain.
| spacedcowboy wrote:
| Physicist, not physician, but really - I haven't used my
| physics knowledge directly in a few decades now... I've
| been a software engineer for most of my life :)
|
| As for V=IR (I had to google U=RI, maybe U is the more
| modern version, but it was always V=IR when I were a lad),
| I don't really have a problem with ratios. When I was
| learning equations, the simple rule is "do unto one side
| whatever you do to the other", so ...
|
| V = IR, divide by R -> V/R = I
|
| I was happy with either representation, and I didn't think
| of it as multiplying, dividing, adding or subtracting, it's
| just "do the same thing" on each side. The problems I had
| were more "when do you apply Kirchoff's laws to figure
| something out, and when do you apply Ohm's law; that sort
| of thing you just get by experience, I think.
| munchbunny wrote:
| I think the useful distinction is that, when you teach
| multiplication as a mechanical computation (arithmetic) it's
| useful to talk about it as repeated addition.
|
| As you get to negative numbers, rational/irrational numbers,
| complex numbers, matrices, etc. it becomes more useful to
| think about multiplication in more abstract ways, among which
| repeated addition is still often a useful way to look at it.
|
| I also think it's not particularly useful to talk about those
| other ways to think about multiplication until you actually
| need to.
|
| It's too easy once you've mastered the concepts to forget how
| beginners look at them and struggle to understand them. I
| think the author isn't remembering what it's like to try to
| understand multiplication as a new concept - I certainly
| can't remember.
| agumonkey wrote:
| so the iterative pattern doesn't really hold when you go
| into abstract algebra ?
| spacedcowboy wrote:
| Your last two paragraphs are, I believe, the crux of my own
| argument. Sure, matrices aren't even commutative, and
| complex numbers have their own quirks because of _i_ ^2 ==
| -1, but these concepts _build_ on the earlier axioms the
| kid has learnt. Our entire education system is built on
| "lies-to-children", and as you progress they point out that
| what you comfortably believed was a gross simplification.
| This is no different.
| eternalban wrote:
| Speaking of "lies to children", and since you are a
| physicist, it was only in my last year studying EE that
| Feynman's QED came my way and just imagine my surprise on
| finding that photons do not travel in a straight line.
|
| (I do agree with your pedagogical position.)
| jschwartzi wrote:
| I think the heart of the issue is whether it's more useful
| to teach children how to multiply two abstract numbers
| together as a kind of "mathematical procedure" that they
| need to memorize, or whether it's more useful to teach
| children that if they measure two sides of a square with a
| measuring tape, they can "multiply" the measurement and
| that the result is now in "square inches" rather than
| regular inches. And the schism is that some people believe
| that procedural memorization is useful because after 20+
| years of education they've gotten through the good part,
| and other people believe that the procedural memorization
| does kids a disservice by divorcing mathematical thinking
| from the concrete world entirely.
| munchbunny wrote:
| _And the schism is that some people believe that
| procedural memorization is useful because after 20+ years
| of education they 've gotten through the good part, and
| other people believe that the procedural memorization
| does kids a disservice by divorcing mathematical thinking
| from the concrete world entirely._
|
| In my very limited teaching experience, I think the
| answer differs based on the student. At the individual
| level I don't think there's much controversy. Just align
| with the student's learning style. At scale, I have no
| idea and do not have the data/experience to have a well-
| formed opinion.
|
| I was perfectly happy to focus on mechanical mastery of
| the multiplication rituals well before I had any concrete
| reasons to use them. I know plenty of others didn't work
| that way.
| sidlls wrote:
| I have got almost exactly the same situation: PhD, physics,
| forgotten more math than most folks ever learn, etc., and
| with an 8 year old learning the same level of mathematics as
| you describe.
|
| I think the only difference might be I used "iterated" rather
| than "repeated" when helping him. Anyone who is just learning
| multiplication likely lacks the depth of experience necessary
| to make use of the "correct" jargon and abstract concepts as
| a starting point. "Repeated addition" is a useful aid in
| learning the operation to build that experience.
| cobaltoxide wrote:
| Another physics PhD chiming in here. I have never before
| noted the difference between "multiplier" and
| "multiplicand". The whole article has me rolling my eyes.
|
| In fact, I would argue that multiplication being
| associative shows that this distinction is meaningless.
| musicale wrote:
| Calculator multiplication can be performed by what is essentially
| repeated addition.
|
| Probably the author would argue that what calculators do is
| neither addition nor multiplication because of precision loss. In
| practice it works pretty well though.
| bmc7505 wrote:
| Does anyone have an intuition for why Presburger arithmetic is
| decidable, but once you introduce x, it is not? If + can be
| rewritten as repeated succ, (which is decidable) and x can be
| rewritten as repeated +, then it seems logical that x should be
| decidable, but as any undergraduate CS student knows, Peano
| arithmetic is undecidable. What is so special about x exactly?
| wizzwizz4 wrote:
| It's a variable number of addition operations.
| bmc7505 wrote:
| Isn't + a variable number of succ's? Why does x break
| decidability, but + does not? It seems to me you should be
| able to define a second order Presburger system with +
| replacing succ, and x replacing + to restore decidability. Or
| is there some issue with countability or assumption that is
| broken?
| eigenket wrote:
| Its probably worth pointing out that the system with only
| multiplication and no addition is also decidable (Skolem
| arithmetic). Things get interesting when you have both
| multiplication and addition defined and the reason for that is
| basically the fundamental theorem of arithmetic which gives you
| unique prime factorisation and directly the Godel encoding
| (writing out statements about integers as integers).
|
| The fundamental theorem of arithmetic says for any non-zero
| natural number x there is a unique finite sequence of integers
| a_i such that x = 2^a_0 . 3^a_1 . 5^a_3 ... p_n^a_n and that
| given x the sequence a_i is computable (similarly given a_i you
| can easily construct their x).
|
| This basically gives you a (bijective) mapping between integers
| and tuples of integers x <-> (a_0, a_1, ... a_n) which lets you
| build up more complicated structures and eventually construct
| the sentence you need for an incompleteness theorem (if you're
| as smart as Godel).
| bmc7505 wrote:
| That's a great explanation, thank you!
| IAmLiterallyAB wrote:
| > but as any undergraduate CS student knows, Peano arithmetic
| is undecidable
|
| We went to very different universities it seems. Neither Peano
| arithmetic nor decidability were ever mentioned
| jmull wrote:
| To me the "So What's the Problem?" section has a _lot_ of
| irrelevant stuff.
|
| The problem with "multiplication is repeated addition" is that
| the concept breaks down once you move on from integers.
|
| 2 X 3: "add 2 together, three times" - works well
|
| 1/2 X 1/2: oof. You can sorta do it, like "add 1/2 a half time",
| but the concept is an impediment that isn't helping anymore.
|
| I can see why teachers who aren't that comfortable with math
| might be getting confused about this though. Repeated addition is
| a reasonable algorithm when you've got one operand that is a
| relatively small integer.
|
| If you spend a long time teaching small integer multiplication
| you may forget that "You can solve these multiplication problems
| with repeated addition" does not mean "Multiplication _is_
| repeated addition. "
| bawolff wrote:
| > 1/2 X 1/2:
|
| Or just re-arrange the terms to be (1*1)/(2*2). No need to add
| half, half a time, just understand that multiplication
| distributes over fractions.
|
| That said i think the concept would make sense to most school
| children as a way of extending the intuitive concept of
| multiplication of natural numbers.
| hc-taway wrote:
| > 1/2 X 1/2: oof. You can sorta do it, like "add 1/2 a half
| time", but the concept is an impediment that isn't helping
| anymore.
|
| On the contrary, continuing to conceptualize fractional
| multiplication exactly as your "1/2 X 1/2" as addition example
| served me well. My wife--with a higher measured IQ than mine,
| FWIW--evidently _can 't_ see it that way. She basically stopped
| learning math when multiplication of fractions was introduced
| (it's interesting that one can continue to earn decent math
| grades, regardless) as all the results felt arbitrary and
| magical to her, and still do in middle-age. This is actually a
| very common point for students to stop following WTF is going
| on in their math classes and never really get back on track
| (yes, all the way down in, what, 2nd or 3rd grade) from what
| I've seen; factoring is another, later on.
|
| "1/2, added 1/2 times" is precisely how I think of it. If I
| can't just pattern-match or rule-follow my way to a solution
| (because I've forgotten the rules, say) that's _still_ my line
| of reasoning to figure out what to do to get the solution.
|
| I do exactly the same sort of thing to come back to my senses
| if I get lost or forget exactly what is happening in fractional
| division. "How many times does 1/2 fit in 1/4? 1/2. How many
| times does 1/4 fit in 1/2? 2." Even if I have to manipulate
| some things to figure out the result with uglier fractions,
| that's absolutely how I _think_ about what I 'm doing as I do
| it, and it keeps me focused on the ultimate purpose of the
| calculation. I imagine that's also wrong, according to the
| author.
|
| In short I can vouch that yes, that exact thing was _very_
| helpful to this particular person, including for multiplication
| of fractions. I truly don 't know how else I might have
| understood those problems, to avoid joining the ranks of the
| mathematically-lost as early as lower elementary school.
| amelius wrote:
| By the way, why doesn't exponentiation have units in physics?
|
| E.g. why don't we ever have something like kg^s?
| rovolo wrote:
| This is purely a guess, but I think it's because all the
| exponents are scalar. You get exponents when the derivative of
| a function is proportional to itself: d/dx
| (f) = k*f
|
| So exponentials will be scalars because the exponential part is
| scaling the original value.
|
| Example: population growth is exponential.
| d/dt(p) = (ln(2)/r)*p p(t) = p0 * 2^(t/r) p :
| population p0: population t : days r :
| days (time it takes for the population to double) t/r :
| scalar
| kmill wrote:
| I think it would have to play well with Taylor series -- for
| example e^s would be 1 + s + s^2/2 + s^3/6 + s^4/24 + ... --
| and that would be mixing units. That's not to say it's
| meaningless, but rather you'd have to accept measurements like
| 4s + 5s^2, with incompatible units hanging around. (There's
| something similar in geometric algebra, I think, where
| expressions contain terms of different dimensions added
| together.)
| cryptica wrote:
| As someone with a pragmatic, visual mind, I rely entirely on this
| kind of reasoning in order to understand and apply math. For
| example:
|
| - Multiplication is repeated addition.
|
| - Power is repeated multiplication.
|
| - The square of a number can be visualized as a geometric square
| when you duplicate a row of x items x number of times.
|
| - The cube of a number can be visualized as a geometric cube when
| you duplicate a row of x items x number of times and then you
| take the resulting geometric square and duplicate it x times
| along the orthogonal axis.
|
| - n to the power of p can be visualized as a tree with height p
| such that each branch splits up into n branches at each level.
|
| - The logarithm base x of y is the height 'number of levels' of
| the tree when each branch splits into x branches at each level
| until the number of tips is equal to y.
|
| - A factorial can be visualized as a tree whose branches split up
| in such a way that there is one fewer branch at each level until
| the branches cannot be split anymore.
| raverbashing wrote:
| I'll tell you what's wrong: the mentality of the author of this
| text
|
| Multiplication has started as repeated addition. That's where the
| idea came from.
|
| > It is as if there were two types of addition: regular, random,
| "wild" addition and the specially-bred variety of addition to
| which we give the name multiplication.
|
| Nobody. Literally nobody said that
|
| Of course, you'll need to forget a bit the idea of repeated
| multiplication when you get into the rationals/reals/complex
| numbers, but even there it kinda makes sense
|
| So no, I think this is the kind of teacher that makes the
| students even more confused and prone to hating math
| contravariant wrote:
| Now that's an interesting point, _did_ it start as repeated
| addition?
|
| It is just as conceivable that people were faced with a problem
| like 'each person needs 2 apples, we have 5 people, so we need
| 10 apples'?
|
| In this case repeated addition is a perfectly fine algorithm to
| calculate the product but the product itself is not _defined_
| as repeated addition, it 's the solution to a particular type
| of problem.
| cyborgx7 wrote:
| Your problem just reinforces the notion that multiplication
| is repeated addition. "So we need 2 for him and 2 for her and
| 2 for him and 2 for him and 2 for her. 2+2+2+2+2 = 5 x 2 =
| 10"
| contravariant wrote:
| I disagree, translating from "2 for (him + her + him + him
| + her)" to "2 for him + 2 for her + 2 for him + 2 for him +
| 2 for her" is _using_ the distributive property of
| multiplication.
|
| The sentence "2 for him and 2 for her and 2 for him and 2
| for him and 2 for her" is unnatural and not the way people
| generally think (unless they're just tallying up but then
| we're not even talking about multiplication anymore, a
| tally rarely consists of all equal numbers)
| cryptica wrote:
| It doesn't make sense to say:
|
| 2 apples + 3 apples = 5 apples
|
| This confuses a lot of people when they later learn algebra and
| believe that it is equivalent to:
|
| 2a + 3a = 5a
|
| Both of these equations look correct at face value but the
| critical difference is that an apple is not a variable. Do we
| mean the weight of the apple? Or the price of the apple? Or its
| energy content? Apple is too concrete to be used as a variable
| and it's not quantifiable. The word 'apples' invites people to
| think of the variable as a concrete object instead of a
| placeholder for a quantifiable property of an object.
| whatgoodisaroad wrote:
| I like this article, but I feel like it misses the value of
| teaching analysis to students. It's true that it's not-quite-
| right to say multiplication _is_ repeated addition, but it 's
| also a really nice demonstration of the analytical/generalization
| approach anyone could use to "invent" multiplication on their
| own.
|
| This same pattern can be seen driving discoveries in math. -
| Integrals are generalizations of Riemann sums. - Fractional
| exponents generalize taking square roots. - The gamma function
| sorta generalizes factorials. It's okay to start with something
| not-quite-right and explore how it generalizes.
| mxfh wrote:
| This is more confusing then it helps.
|
| Why mix concepts of Linear Algebra/Physics with basic Arithmetics
| for no good reason?
|
| The multiplication, that is just repeated addition, is very
| possible in dimensional analysis and is called _scalar
| multiplication_ :
|
| The outright worst part is this weird complication:
|
| _2 baskets x 3 apples per basket = 6 apples_
|
| of 2 x 3 apples = 6 apples.
|
| Just why? This is just a scalar operation.
|
| Staying in strictly mathematical concepts, the operation
| described in OP are closer to the _cross product_ of orthogonal
| vectors, then multiplication in my opinion.
|
| [4 0 0] x [0 3 0] = [0 0 12]
|
| https://en.wikipedia.org/wiki/Scalar_multiplication
|
| https://en.wikipedia.org/wiki/Cross_product#Geometric_meanin...
| draw_down wrote:
| Damn, this is so crazy. Repeated addition is absolutely how I
| intuitively understand multiplication. But then again I'm no
| galaxy-brain mathematician.
| mensetmanusman wrote:
| It may be wrong, but it is good enough to run all of our
| computation at the transistor level :)
|
| The fascinating thing is that now these transistors are helping
| make new math proofs beyond human comprehension.... how does that
| come out of addition?
| flohofwoe wrote:
| AFAIK hardware multipliers are not quite as dumb and work more
| like this though:
|
| 5*11 = 5*(8+2+1) = 5*8 + 5*2 + 5*1 = (5<<3) + (5<<1) + (5<<0)
|
| ...and then doing all this in parallel.
|
| Instead of:
|
| 5*11 = 5+5+5+5+5+5+5+5+5+5+5
|
| ;)
| devinplatt wrote:
| (Unrelated to pedagogy, but related to when multiplication is not
| repeated addition.)
|
| When I was an early college student I started to become curious
| about the prime numbers. Why are some numbers in particular
| prime, and others aren't? I started with a thought experiment:
|
| What if it were that 2*2 = 5 ?
|
| Pretty quickly I realized that the relationship between addition
| and multiplication could not remain the same, because the
| distributive property guarantees that 2*2=4 2*2 =
| (1+1) * (1+1) = 1+1+1+1 = 4
|
| So I sought to define some form of multiplication that was as
| close to regular multiplication as possible, but without the
| distributive property. I ended up defining this generalized form
| of multiplication as a function m:NxN -> N (N = the natural
| numbers 1,2,3,4,...) with the following properties
|
| 1. Associative
|
| 2. Commutative
|
| 3. Multiplicative Identity
|
| 4. Increasing ( If i,j > 1, then m(i,j) > i,j )
|
| 5. Bigger number, bigger product ( If j > i, then m(j,k) > m(i,k)
| )
|
| 6. Uniqueness of Prime Factorization
|
| An easy example is the odd numbers. 1,2,3,4,5,
| ... n 1,3,5,7,9, ... odd(n) odd(n) = 2n-1
| inverse_odd(x) = (x+1)/2 m(i,j) = inverse_odd( odd(i) *
| odd(j) ) = 2ij - i - j + 1
|
| Indeed, for the odd numbers m(2,2) = 5! The second odd number (3)
| multiplied with the second odd number (3) equals the fifth odd
| number (9).
|
| It turns out that this definition is really a (nice) subset of
| something known as the Beurling Integers. The Beurling Integers
| are neat, because you can basically choose whatever distribution
| of prime factorizations you want (following the rules 4, 5, 6
| above) and find a sequence of real numbers that satisfies that
| distribution. The catch is that we had to sever the ties of
| addition and multiplication.
| fchu wrote:
| Learning isn't possible without unlearning whatever temporary
| construct we used as a crutch, but it doesn't mean using that
| temporary construct is wrong even if it's not technically
| correct.
|
| More specifically in mathematics, the interplay between formalism
| and intuition, like a mental danse or gymnastics, is a powerful
| process in furthering our understanding of mathematical truths:
|
| From a formal perspective, mathematical objects can be created in
| so many ways, some constructions being more intuitive and
| beautiful than others (axioms, groups, rings, fields). The
| formalism itself let us see what intuition can't.
|
| From an intuitive perspective, it's useful to latch on whatever
| concept one have to learn the next level of abstraction, while
| acknowledging that the intuition might not be 100% correct. Like
| using addition to intuitively understand multiplication, or
| addition and multiplication to intuitively understand fields. The
| intuition let us familiarize with otherwise novel ideas.
|
| Ironically, this article wants to be very normative about which
| mathematical intuition is better (which there isn't, I'm sure
| many don't think of "multiplicand" as something special), while
| disregarding any cues from any formalism.
| klodolph wrote:
| The author is speaking from personal experiences teaching
| children K-10 (ages 5-16). It sounds like she is documenting
| her experience _teaching_ these various temporary constructs,
| and that her experience is that teaching multiplication as
| repeated addition is a less effective way of teaching it.
|
| > Ironically, this article wants to be very normative about
| which mathematical intuition is better...
|
| No, the article is not being normative about "which intuition
| is better," this is an incorrect reading of the article.
|
| The article is giving advice about how to teach multiplication.
| Advice that is apparently based on _years of experience_
| teaching multiplication.
|
| It's not uncommon for people to experience teachers who
| prescribe specific intuitions about math rather than
| accommodate different intuitions--but the author is not doing
| that. What she's doing here is outlining the various ways in
| which one particular intuition may fail you.
| bitwize wrote:
| In mathematics, sometimes a wrong approach can be used as a
| teaching tool until you get the concept, at which the correct
| definition may be disclosed. Example: derivative calculus is
| easier to teach with infinitesimals, even though the derivative
| is correctly defined in terms of limits.
| WalterBright wrote:
| I distinctly remember when I realized that the times in "4 times
| 3" is 3 added 4 times. That was the end of trying to memorize the
| times tables. If I forgot what "6 times 9" was I just thought "6
| times 10 minus 6" and there it was.
| scythe wrote:
| The nicest formal definition of multiplication is "the operation
| that distributes over addition". When defining operations on
| abelian groups that might be called "multiplication", it's the
| distributive property that makes the term useful, and brings us
| into the categories of ring, semiring, [weird math prefix
| here]-ring, etc.
|
| But the thing is: repeated addition _does_ distribute over
| addition! So there is really a very natural extension from
| "multiplication is repeated addition" to "multiplication is any
| operation that preserves the nicest property of repeated
| addition, which is distribution".
|
| I hasten to add that as a GTA I have had untold numbers of
| students who apparently did not learn the distributive property
| correctly -- likewise, the biggest difficulty my students seem to
| have with dimensional analysis _in practice_ is that they have
| trouble dividing fractions symbolically. Also, substitution
| (replacing an expression with a letter) continues to trip
| students up: e.g. when pointing out that, say, newtons per
| coulomb is the same as volts per meter. (Students are no doubt
| tired of hearing me yak about how math expressions are a form of
| communication...)
|
| Sometimes, I do think we need to teach students to "manipulate
| expressions" rather than just "solve problems", but then again,
| don't we do that already?
| harperlee wrote:
| I just found this interesting article. If you are interested, it
| is a review of the following "conversation of articles" (which
| are reachable from the OP but I guess it's useful enough to
| list): -
| https://www.maa.org/external_archive/devlin/devlin_09_07.html
| - https://www.maa.org/external_archive/devlin/devlin_06_08.html
| - https://denisegaskins.com/2008/07/01/if-it-aint-repeated-
| addition/ - (the link I posted that creates this thread)
| Jtsummers wrote:
| Clickable:
|
| -
| https://www.maa.org/external_archive/devlin/devlin_09_07.htm...
|
| -
| https://www.maa.org/external_archive/devlin/devlin_06_08.htm...
|
| - https://denisegaskins.com/2008/07/01/if-it-aint-repeated-
| add...
|
| (Sorry, it just bothers me when people don't use clickable
| links here.)
| alberto_ol wrote:
| I wanted to post the same links, you can find the same articles
| and others about that subject by googling "Keith Devlin
| addition multiplication"
| andrewprock wrote:
| tl;dr: "Nothing"
| thysultan wrote:
| Philosophical mathematicians trying to confuse the plebs.
| thothamon wrote:
| I think the author is trying to say that _from a teaching point
| of view_, thinking of multiplication as a kind of addition is
| suboptimal.
|
| Because you can indeed multiply through repeated addition (which
| she acknowledges), repeated addition is one valid way to look at
| multiplication. But it might not be the maximally helpful
| framework for students.
| yarg wrote:
| It's not circular reasoning - it's bootstrapping the rational
| implementation atop the integer implementation.
| bikenaga wrote:
| I think it's okay to say: "It's useful to _visualize_ (or _think
| of_ ) multiplication as repeated addition, in contexts where that
| makes sense." Properly qualified the statement seems true and
| unbojectionable, and I think it's a useful way to help students
| understand multiplication. (It's not the only way.)
|
| The reservation I'd have with the statement "Multiplication _is_
| repeated addition " is that the use of the word "is" implies an
| _identification_ , or a _necessary_ derivation of one from the
| other. The operations aren 't identical, and one is not
| _necessarily_ derived from the other. As operations, they 're
| distinct.
|
| (The issue also has nothing per se to do with units. When you
| introduce units of length or area, you're _using_ numbers in a
| _particular applied context_. Outside of that context, numbers
| don 't have units.)
|
| In _everyday usage_ "addition" and "multiplication" are
| conventional designations. Conventions always have ambiguities
| and edge cases.
|
| The most common _formal structure_ in which you have both
| addition and multiplication is in a _ring_. A ring is a set with
| two operations, which we _call_ "addition" and "multiplication"
| --- but those are just names. We could just as well have used
| "foo" and "bar". We tend to use the standard addition and
| multiplication symbols for those two operations, but that is also
| just a convention/convenience. (Note also that those operations
| are not unique.)
|
| Rings include the integers, rationals, reals, complex numbers -
| but also (for instance) polynomials with real coefficients, 42 x
| 42 matrices with real entries, quaternions, or finite fields.
|
| Now consider what happens as you list the axioms for a ring. You
| say that addition is associative, addition has an identity
| element (conventionally denoted "0"), every element has an
| additive inverse, and addition is commutative. That's addition.
|
| Then you say multiplication is associative (and nowadays, since
| people find it convenient to assume this) and multiplication has
| an identity element (conventionally denoted "1").
|
| At the moment, you have two independently defined operations,
| which have nothing to do with one another. "Multiplication" is
| therefore _not identical to_ (or _defined as_ ) repeated
| addition. But it's not too useful to have two independent
| unrelated operations. You _connect_ the two operations by
| introducting the distributive axiom: For all a, b, c in your
| ring, a * (b + c) = a * b + a * c and (a + b)
| * c = a * c + b * c.
|
| (scythe pointed out the importance of the distributive law in
| another reply.) The distributve axiom is huge! Consider
| a * (b + c) = a * b + a * c.
|
| From left to right, it says that "you can multiply out"; from
| right to left, it says that "you can take out a common factor",
| which are both standard operations in algebra.
|
| _Once the distributive property is available to connect the
| operations_ , it explains why you can "think of" multiplication
| as repeated addition: For instance, 3 * 2 = 3 *
| (1 + 1) = 3 * 1 + 3 * 1 = 3 + 3.
|
| So you say "3 times 2" _can be thought of_ as "3 added to itself
| 2 times". It _does not_ say that multiplication _is_ repeated
| addition, if "is" means "defined as" or "derived from",
|
| Or consider one of the standard illustrations given to kids in
| grade school: 3 * 2 is $ $ $ which is [ $ $
| $ ] $ $ $ [ $ $ $ ]
|
| This is 2 groups of 3, i.e. 3 + 3. So 3 * 2 = 3 + 3.
|
| But what happened here? We relied on our physical intuition to
| "know" that putting the original 6 dollars into 2 bags in the
| second step didn't change the number of dollars. But formally, it
| is 3 * 2 = 3 * (1 + 1) = 3 * 1 + 3 * 1,
|
| which is the distributive law again.
|
| (BTW regrouping the dollars again into 3 groups of 2 is a
| standard way of motivating commutativity of multiplication, since
| 3 groups of 2 is visibly "the same as" 2 groups of 3.)
|
| (Someone might suggest from the dollars example that
| multiplication might have arisen historically as a shorthand for
| repeated addition. I don't know math history well enough to say,
| but historical derivation doesn't imply identity or logical
| derivation.)
|
| So I think saying "Multiplication _is_ repeated addition " is a
| little sloppy in the use of the word "is", but we can agree to
| disagree about how much sloppiness is okay. The statement is fine
| as a way of giving students one (of many) ways to think about
| multiplication. (Pictures are important, too!)
|
| Here's something to think about. (I don't have an opinion
| myself.) Suppose we have a complex multiplication: (7 i)(3 + 4
| i). Following the interpretation above, I describe this as "7 i
| added to itself (3 + 4 i) times". If you don't like the sound of
| that ... why? You might say "You can't have '3 + 4 i' things."
| Well, in the real world discrete "things" come in nonnegative
| integer quantities ... or do they? For instance, we could make an
| agreement that "-3 things" means "I'm missing 3 things" or "you
| owe me 3 things". Nothing stops us from making agreements about
| the use of words. So maybe we could "agree" that "(3 + 4 i)
| times" means exactly an occurrence of the expression "3 + 4 i" in
| a context like this one. Is there any harm in that? :-)
| bawolff wrote:
| I feel like people here have a weird definition of "truth".
| Platonism aside, there is no outside ground truth on what
| "addition" or "multiplication" is. It is an abstract symbol
| defined solely by what it does to numbers. As long as your
| definition describes its behaviour, your definition is no more or
| less right than any other definition.
|
| If you want to define multiplication as repeated addition (and
| addition as repeated invocation of the succesor function), by all
| means do so. Its not wrong.
|
| Sure you could say it doesn't work for complex numbers, but
| usually when you introduce complex numbers you say something
| like, multiplication works as normal except ixi=-1. I hardly see
| that as a problem with the original definition of multiplication.
| Its more about the definition of i than the definition of
| multiplication.
| Garlef wrote:
| My favourite way of thinking about these operations comes from
| category theory:
|
| * addition: the cardinality of the disjoint union of two sets
|
| * multiplication: the cardinality of the direct product of two
| sets
|
| * exponentiation: a^b is the cardinality of the set of maps from
| B to A where B is a set of cardinality b and A is a set of
| cardinality a
| t0r0nat0r wrote:
| This seems a bit restricted. How do you go about extending it
| to real and complex numbers?
| Garlef wrote:
| From what I know: This is still an open question.
|
| Here's a list of some approaches.
|
| https://mathoverflow.net/questions/310004/categorifications-.
| ..
| johndoe42377 wrote:
| Yes, very clever social signaling of assumed smartness.
|
| However, both set theory and category theory are canonical
| examples of synthetic purely abstract theories, which include
| paradoxes, while repeated addition (which is just merging
| together) is deeply grounded in reality.
|
| Putting an abstract theories first is the same flaw as with
| Froidian psychology, Marxism, and all the other abstract
| bullshitting, from Hegel onwards.
| AzzieElbab wrote:
| That is exactly how I explained multiplication to my 5 years
| old, right after getting him into programming by playing with
| profunctor optics
| Garlef wrote:
| I learned adding from counting disjoint unions of dots.
|
| I learned multiplication from counting squares of dots.
|
| But agreed: I learned about exponentiation first via repeated
| multiplication.
| AzzieElbab wrote:
| That is very cool. I prefer geometric/visual approaches.
| Like move that many steps along the line, figure out area
| of a rectangle and so on
| YeGoblynQueenne wrote:
| You waited 5 years to teach programming to your 5 year old?
| [deleted]
| Rerarom wrote:
| This is just set theory, not category theory.
| Garlef wrote:
| Neither is more foundational than the other. So I think the
| use of "just" is not justified here.
|
| https://mathoverflow.net/questions/360578/category-theory-
| an...
| [deleted]
| nicky0 wrote:
| But talking about category theory makes you sound smarter.
| Garlef wrote:
| I'm personally more scared of set theory.
| eigenket wrote:
| Yeah but your comment didn't use the concept of a
| "category" at all, whereas you used "set" three times. Its
| definitely more natural to say these definitions come from
| set theory.
| Garlef wrote:
| To me, "disjoint union", "direct product" and "set of
| maps" (as a special case of "internal hom") are the
| central keywords in what I wrote: Concepts from category
| theory.
| eigenket wrote:
| Those keywords all appeared in the context of set theory
| long before category theory was thought of.
|
| Even ignoring that as a historical accident (we invented
| set theory before category theory), those ideas are all
| natural, meaningful and interesting to examine in the
| context of sets even if one has never heard of a
| category.
| spekcular wrote:
| Yes, there's no need to invoke any category theory
| concepts here. Everything is happening in the category of
| sets.
| spekcular wrote:
| First, that's not what your link says. (See, e.g., the
| comments to the first answer.)
|
| Second, I find that claim highly dubious. To do
| metamathematics and talk about the relative strength of
| various axiomatic systems, you need to talk about large
| cardinals. So that, at least to me, seems like a good
| argument that set theory is foundational to mathematics in
| a way that category theory is not, since the latter has no
| non-eliminable place in the (study of) contemporary
| foundations of mathematics.
| threatofrain wrote:
| Can this perspective accommodate fractional exponents, like
| 2^(0.5)?
| Garlef wrote:
| No (At least not yet: open research question). But neither
| can "repeated multiplication". How would you multiply 2 with
| itself "a half" times?
| prionassembly wrote:
| Operationally, unsure. But if there was an answer, it would
| be something that multiplied with itself twice would have
| be the same as 2 multiplied with itself once.
|
| I think it was Gerolamo Cardano who in a book of worked
| exercises came up with a problem like "obtain two numbers
| that add to 10 and multiply to 40" and arrived at the
| solution had to involve square roots of -1. Then he goes
| "this makes no sense, but let's calculate nevertheless..."
| Garlef wrote:
| > it would be something that multiplied with itself twice
| would have be the same as 2 multiplied with itself once
|
| from what i recall that's the way to go:
|
| You can use this to extend the definition of exponentials
| as "repeated addition" from allowing only natural numbers
| expoonents to allowing positive rational numbers.
| Afterwards you can "complete" this to allow real
| exponents.
|
| But this requires some hard maths to do rigorously.
| prionassembly wrote:
| You're absolutely right about rigor: I'm not claiming
| that Cardano discovered complex analysis or anything...
| bawolff wrote:
| > Is that really how we want our students to think?
| Multiplication is not a mere sub-species of addition.
| Multiplication is its own animal, an independent operation.
|
| Wtf, why not? Why wouldn't you want people to draw connections
| between different parts of math. The whole point of math is to
| find the patterns and interconnections.
| maweki wrote:
| I think what the author is missing is the word "scalar". I do
| understand the sentiment that in general multiplication is not
| repeated addition. But multiplication with a scalar is.
|
| In the usual curriculum you do the unit analysis as a separate
| step, as it's usually in the physics lessons where units are
| used. In math it's unitless.
|
| Of course, we should start teaching immediately that you can't
| add apples and pears. Maybe we should just do this explicitly
| there, that 5 times 3 apples is fine, as 5 is scalar. Explain
| that immediately and then add the unit analysis in physics.
| pdonis wrote:
| _> I do understand the sentiment that in general multiplication
| is not repeated addition. But multiplication with a scalar is._
|
| "Scalar" is too broad; irrational numbers are scalars, but
| multiplication by an irrational number is not repeated
| addition.
| y7 wrote:
| I think the author's argument becomes more clear when you
| consider multiplication in rings other than the integers, for
| example (square) matrices. The product of two matrices A*B does
| not correspond to repeated addition, and it is not commutative
| (A*B does not equal B*A in general).
|
| I can see that having an engrained belief that multiplication is
| defined via addition becomes problematic at some point when
| learning about mathematics. However, this is true for a lot of
| basic properties that hold over the integers and not in other
| domains, so I'm not really convinced that it's actually wrong to
| teach kids about multiplication this way.
| prionassembly wrote:
| The better line of thought here might be "A*B is not
| multiplication, it's function composition".
|
| It literally looks nothing like multiplication when carried out
| manually.
| eigenket wrote:
| It looks exactly like scalar multiplication when your
| matrices are diagonal.
| prionassembly wrote:
| Then: all matrices look like the identity matrix if applied
| to a suitably rescaled eigenvector.
| eigenket wrote:
| We're talking about matrix/matrix multiplication here,
| not matrix/vector, right?
| prionassembly wrote:
| I was talking about matrix/matrix _composition_ (rather
| than multiplication) at first. Then I talked about matrix
| _application_.
|
| A matrix is a function. Not all functions can be
| represented by matrices (although all smooth functions
| can be represented by a Taylor series that sums over
| matrices (and "the sum C=A+B" really means "the function
| C such that Cx = Ax+Bx for all x in the range of both A
| and B"))
| eigenket wrote:
| I rather dislike this way of saying things. A matrix is a
| matrix, its a table of numbers, and one can usefully
| define operations like addition and multiplication on
| them.
|
| A linear map is a linear map, it maps between vector
| spaces and it obeys some nice axioms. You can define
| addition and composition as operations on them.
|
| It is a quite interesting and non-trivial theorem that if
| you fix a particular choice of basis then you get "for
| free" a bijection between linear maps and matrices. The
| bijection between matrices and linear maps is completely
| dependent on the basis you choose, however, and there
| certainly isn't a canonical way to choose the basis.
|
| Often it is natural to change basis to make it easier
| solve some particular problem, and then the matrix that
| represents a particular linear map will change, but the
| properties of the linear map won't change (for example
| its rank, kernel, eigenvectors/values etc).
| nimish wrote:
| Not the zero matrix, or any non-invertible matrix over a
| field with 0 characteristic.
|
| The Jordan-Chevalley decomposition makes the difference
| precise -- the nilpotent part (off diagonal 1's in the
| jordan normal form)
| mantap wrote:
| The mathematical term is "product". It's a product.
| jmull wrote:
| It's just that matrix multiplication is not the same thing as
| scaler multiplication. I think you need to understand that no
| matter how you define or think of scaler multiplication.
| ducaale wrote:
| https://www.youtube.com/watch?v=F_0yfvm0UoU
| GuB-42 wrote:
| > Is there really a difference between multiplication and
| repeated addition, or am I tilting at windmills here?
|
| You are tilting at windmills
|
| > Is it even necessary for teachers to define multiplication?
|
| No, at least not in formal terms
|
| > Or is the teacher's job to provide plenty of examples of
| multiplication in action?
|
| I think so
|
| > Should we let the students intuit their own definition(s)?
|
| Yes
|
| > Will it help students if we change our focus from "how to get
| the answer" and teach them to identify the multiplicand, the
| "this per that" unit? Or will that introduce new difficulties I
| haven't considered?
|
| It will introduce new difficulties
|
| > Or do we already teach this way, only in different words?
|
| We don't, thankfully
|
| > If you are an elementary teacher, how do you teach
| multiplication to your students?
|
| I will let elementary teacher answer
|
| > Are some students clueless because, no matter how we explain
| it, they just don't pay attention?
|
| While we can't always blame teachers when students don't pay
| attention, if it is systematic, maybe there is a problem with
| your teaching.
|
| > Have you tried using bar diagrams to model elementary
| arithmetic situations? And if so, how did your students respond?
|
| I will let teachers respond
|
| Ok, I hope you are still there and didn't just click the
| "downvote" button.
|
| I think the author raises good points for at least high-school
| level students in STEM fields. But in reality, the important
| thing in elementary school is that we want people to know how to
| count. 5 apples at $0.50 each is $2.50, this kind of thing.
|
| Dimensional analysis is very important in engineering, and I
| definitely think is should be taught, as well as everything in
| the article, but at college level, not in elementary schools.
| Teaching such abstract concepts too soon is a recipe for
| disaster, I know because it has been tried. The expectation was
| that it will make better engineers in the future, the reality is
| that while it may have helped a tiny minority get college
| degrees, it mostly produced kids who couldn't do simple
| arithmetic.
| forinti wrote:
| The more methods one studies, the better one understands. There
| is no one true method.
| mansoor_ wrote:
| He makes a good case for multiplication being a special subset of
| (repeated) addition, whilst semantically trying to argue
| otherwise :D
| scelerat wrote:
| One of the best concise guides I've encountered to the
| relationship between the various algebraic operations -- and how
| they circle back to one another -- is chapter 22 "Algebra" of
| volume I of the Feynman Lectures on Physics. For some reason this
| tied everything together in my head in a way that made so much
| sense, when the pedagogy I head leading up to it simply did not
| supply me the same vantage point.
|
| https://www.feynmanlectures.caltech.edu/I_22.html
|
| From there I guess I've always thought as the various algebraic
| operations as functions which build upon one another. I think the
| article's point about not overlooking multiplication's
| Dimensionality as a key component is a good one, but neither
| should that focus avoid talking about the functional
| relationships of the various operations.
| LJD_E wrote:
| I think the author mixes variables with numbers. If you add units
| in to the mix then you have to think like it variables so for
| example that's taken "2 baskets x 3 apples per basket = 6 apples"
| would correspond to 2x _3y /x=6y removing the numbers and the
| equation becomes clear xy/x=y or we the numbers for them selves
| 2_3=6, That is because numbers are like their own variable but
| with connection to other numbers(variables). So if we would
| define 1 as x then 2 would be 1+x or we could call it y but y
| would still be 1+x in relation to 1.
|
| And also"3 cm + 3 cm = 6 cm" as 3x+3x=6x and removing the numbers
| it become 3x/3+3x/3=6x/3 => x+x=2x it makes more sense. And"2 cm
| x 3 cm = 6 cm^2'' can be then seen as 2x*3x=6x^2.
|
| Multiplication doesn't change anything. We must see what we add,
| in this case the unit and calculate accordingly both for addition
| as for multiplication.
| cjfd wrote:
| Ah yes, there is Keith Devlin involved.... I am sure he enjoys it
| a lot to be in a position where he can write pedantic falsehoods
| while at the same time being taken seriously.
|
| You know, most, if not all, mathematical objects of significance
| can be defined in many, many different ways. In fact, this is the
| hallmark of an mathematical object of significance: it keeps
| popping up in many context and can therefore be defined in any or
| all of these contexts if one enjoys doing so. There are far
| ranging context to define an object and less far ranging. I am
| quite sure that in grade school it is quite unhelpful to look for
| the furthest ranging context in which one could define
| multiplication. E.g, let us start in grade 1 with the definition
| of an algebra and derive everything from that.
| jschwartzi wrote:
| Except this isn't a pedantic falsehood. Multiplication has
| nothing to do with addition and at some point we have to stop
| teaching students that it's related. Understanding ratio is key
| to understanding a lot of the physical sciences and you can
| build a really good intuitive understanding of a lot of simple
| physical concepts if you can just do dimensional analysis. But
| if you think of multiplication and division as a kind of
| addition then it all breaks down and you can't reason about
| much at all with those tools. In fact you can't even get over
| the hurdle of doing dimensional analysis in chemistry or
| physics in the first place. It doesn't make any sense. And
| you'll have been set up to fail by your teachers.
|
| They're not saying "we should be teaching kindergarteners using
| Euclid" they're suggesting that focusing on the process of
| doing multiplication hides a lot of very useful insights about
| why someone would do multiplication in the first place.
|
| The biggest problem with math as it was taught to me in grade
| school is that it was focused on rote memorization of a
| procedure. Which is fine if you need to calculate how long to
| cut 2x4s or stair stringers but it doesn't help you at all if
| you need to understand why that works the way it does. And god
| help you if nobody ever shows you the exact method of
| calculation for those two things in the first place. You're set
| up to fail if you only focus on process.
| cannabis_sam wrote:
| > Multiplication has nothing to do with addition and at some
| point we have to stop teaching students that it's related.
|
| My apologies if this is a stupid question, but when does the
| intuitive (layman) understanding of multiplication as
| repeated addition break down (mathematically)?
| betterunix2 wrote:
| When you multiply two negative numbers.
| 613style wrote:
| We can't really talk about what multiplication "is" or
| "isn't" independently of context. It's an operation on two
| objects and the context its used in is necessary for defining
| the operation.
|
| Though it can be helpful to think of multiplication as
| scaling or rotation in certain contexts, or as repeated
| addition in others, none of those are a universal truth.
|
| To say it's not related to addition at all is also too broad
| to be true. They're related in many useful ways that other
| commenters have pointed out, in addition to the obvious way
| that in some situations you can define one in terms of the
| other. Even at the cutting edge, our inability to prove the
| Goldbach Conjecture might have something to do with not fully
| understanding the deepest relationships between the
| operations.
| aaplok wrote:
| Multiplication is _literally_ repeated addition, as in
| "taking a number _multiple_ times ". It is in the name.
|
| The fact that the operation is so useful that it has been
| generalised to the point where that original meaning is
| eventually lost through more and more abstractions doesn't
| invalidate that, because with all the generalisation and
| abstraction, _multiplication as repeated addition still
| works,_ and any generalisation is expected to leave that
| property intact.
|
| This is not unique to multiplication. Modern mathematics is
| all about extracting fundamental properties that make
| something true, to generalize results while keeping the
| original simpler statement true. Starting from the
| generalisations without explaining the intellectual process
| it took us to get there means just giving facts without
| explaining them.
| alentist wrote:
| Define p times e in terms of repeated addition.
|
| I wouldn't say multiplication is _literally_ repeated
| addition. I 'd say it _reduces to_ repeated addition when
| the multiplier is a natural number.
| codethief wrote:
| Maybe it is not _literally_ addition but it certainly
| reduces to addition in _all_ cases:
| https://news.ycombinator.com/item?id=26329855
| alentist wrote:
| Nope. I'll reply there.
| betterunix2 wrote:
| "Multiplication is literally repeated addition"
|
| How do you repeat something a negative number of times?
| What does it mean to multiply two negatives, if
| multiplication is repeated addition? This is not an overly
| generalized concept, we teach negative numbers to
| elementary school students.
| dragontamer wrote:
| > Multiplication has nothing to do with addition and at some
| point we have to stop teaching students that it's related.
|
| I just finished studying Galois Fields, which LITERALLY
| redefines multiplication and addition operators to study new
| forms of math (IE fields in particular)
|
| In all fields and rings, multiplication and addition are
| related by the distributive property.
|
| A(b + c) is equal to Ab + Ac, in literally every form of math
| you can every think of. A and b could be matrices, vectors,
| polynomials, prime numbers, integers, rational, complex,
| real, or Galois polynomials over a weird modulus ring thingy
| inside of a vector inside of a matrix. Doesn't matter,
| addition and multiplication are and always defined in that
| manner.
| pdonis wrote:
| _> In all fields and rings, multiplication and addition are
| related by the distributive property._
|
| That's a necessary condition for "repeated addition" to
| work for multiplication, but not a sufficient one. Try
| plugging in A = pi to the distributive formula and see how
| well "repeated addition" works.
| dragontamer wrote:
| pi(3 + 5) == 3pi + 5pi == (Pi + Pi + Pi) + (Pi + Pi + Pi
| + Pi + Pi) == 8-pi.
|
| Am I missing something here?
|
| --------
|
| This "multiplication is repeated addition through the
| distributed property" thing works on freaking
| __matricies__. They don't even have to be numbers or even
| related. Pi * ([ 1 0 ; + [ 1 0 ;
| 0 1 ] 2 1 ]) Pi * [ 1 0 ; + Pi
| * [ 1 0 ; 0 1 ] 2 1 ]
| [ Pi 0 ; + [ Pi 0 ; 0 Pi ] 2Pi Pi ]
| [ 2Pi 0 ; 2Pi 2Pi ]
|
| You can even "break up" the two matricies into its
| component parts: Pi * ( 1 0 + 0 0 + 1
| 0 + 0 0 + 0 0 ) 0 0 0 1 0 0 2 0
| 0 1
|
| And you'll still get the correct answer: multiplication
| is repeated addition. Even with matricies. Even with
| vectors. Even in Galois Fields. Even with rational
| numbers. Even with real numbers. Even with Complex
| numbers. This property holds through all forms of math
| that I'm aware of.
| iio8999 wrote:
| I think the issue is more about:
|
| 3 * pi = pi + pi + pi
|
| but how do you represent the other distribution, where 3
| is added together pi times?
| dragontamer wrote:
| Break-up pi in whatever means you think is reasonable.
|
| 3 * (3 + 0.1 + 0.04 + 0.001 + 0.0005...)
|
| Aka: 9.4245...
|
| You know, how we've been multiplying 3 * pi for our whole
| lives. We split pi up into an infinite sum of component
| numbers (3, 1, 4, 1, 5, 9, 2, 6...) and then combine them
| together by individually multiplying the parts (3 * 3 + 3
| * 0.1 + 3 * 0.04...)
| pdonis wrote:
| _> Break-up pi in whatever means you think is
| reasonable._
|
| You can't. Pi is irrational.
| dragontamer wrote:
| > You can't. Pi is irrational.
|
| You just need to break it up infinitely times. We usually
| call the sequence 31415926...
| pdonis wrote:
| _> You just need to break it up infinitely times._
|
| I've responded to this elsewhere in the thread: I don't
| think adopting increasingly perverse interpretations of
| "repeated addition" as you try to include more and more
| numbers is a useful way to teach multiplication.
| beaconstudios wrote:
| what they described is identical to long multiplication.
| That's not perverse, it's how most people multiply
| numbers.
| jimhefferon wrote:
| You extend to the rationals in the natural way, and then
| use continuity to define what happens for irrationals.
| (Of course, in a discrete context that doesn't apply. But
| the article's author is a grade school teacher.)
| pdonis wrote:
| _> Am I missing something here?_
|
| Sorry, I should have said substitute b = pi or c = pi. Or
| see the sibling post by iio8999.
| Gibbon1 wrote:
| pi isn't a counting number like you teach to children.
| mnl wrote:
| Yes, they are two binary operations and depending on the
| sets you consider and which properties you impose for those
| operations you have different algebraic structures. (This
| used to be taught at school before "modern mathematics"
| were considered harmful, maybe they were but at least they
| were correct).
|
| The thing is that as you can write m (let it be a positive
| integer) as m=1+...+1 (m-times), you can write
| n*m=n*(1+...+1), invoke the distributive property for * wrt
| + and express it as: n*m=n+...+n (m-times), so it looks
| like "repeated addition" for integers in this case. But
| it's not a good idea to let ourselves get carried away, we
| still have two binary operations going on. At any rate we
| have to impose that n*0=0, which can't be writen cleverly
| as "repeated addition" and worked up backwards.
| thaumasiotes wrote:
| > At any rate we have to impose that n*0=0, which can't
| be writen cleverly as "repeated addition" and worked up
| backwards.
|
| Are you kidding? This is the exact opposite of the truth;
| the nature of multiplication as repeated addition is _the
| entire reason_ why multiplying by 0 gives the additive
| identity. It 's exactly the same as how exponentiating by
| 0 gives the multiplicative identity, since exponentiation
| is just repeated multiplication. And this is so
| fundamental that 1 is frequently referred to by this
| property, as "the empty product".
| dragontamer wrote:
| > At any rate we have to impose that n*0=0, which can't
| be writen cleverly as "repeated addition" and worked up
| backwards.
|
| n * 0 = n * (1 + (-1)) = n + (-n) = 0.
|
| -----
|
| The 0-element in a Galois Field works identically btw. In
| GF(5), the 0 element is 5 (5 mod 5 == 0).
|
| n * (5) == n * 0 == n * (1 + (-1)) == n * (1 + 4) == 0
| mod5.
|
| For example, if we take "n" == 2, 2 * 5 == 10 mod 5 == 0.
|
| 2 * (1 + 4) == 2 + 8 == 2 + 3 (mod 5) == 5 mod 5 == 0.
| Etc. etc.
|
| -------
|
| This property literally holds in all fields and rings
| (but not groups).
| thaumasiotes wrote:
| > n * 0 = n * (1 + (-1)) = n + (-n) = 0
|
| You don't think it's easier to say "n*5 = n*(5+0) = n*5 +
| n*0"?
| dragontamer wrote:
| The (1 + (-1)) thing was very precisely written.
|
| All rings have 0 and 1 as elements. 0 is the additive
| identity. 1 is the multiplicative identity. 0 and 1 are
| NOT necessarily numbers. In Linear Algebra of 2x2
| matricies, 0 is: [ 0 0 0 0 ]
|
| And 1 is: [ 1 0 0 1 ]
|
| Because A * 1 == A, A * 0 == [0 0; 0 0]... etc. etc. In
| general, you cannot really assume much more than "0
| exists" and "1 exists" when working with Rings (at least,
| if you want your proof to extend out to all possible ring
| algebras). 0 and 1 may look like something you wildly
| don't expect... they're abstract labels that are kind of
| undefined aside from being additive-identity or
| multiplicative-identity, respectively.
|
| --------------
|
| As such, the concept of "5" does not necessarily exist in
| all possible Ring-systems. "5" exists in GF(5) for
| example, but not really in GF(3). Case in point, what
| does "5" mean in 2x2 Matrix Linear Algebra over GF(2)?
|
| --------
|
| "-A" is called the additive inverse of A, which also
| exists in all rings. A - A = 0.
|
| In GF(5), -1 is 4 for example. In 2x2 Linear Algebra, -1
| is [-1 0; 0 -1]. In Real Numbers, -1 is... well... -1.
|
| ----------
|
| Anyway, the A * 0 == A * (1 + (-1)) == A - A == 0 thing
| is built up from fundamental portions of Ring theory. As
| such, the proof I constructed at first applies to all
| rings. (And then later, I did an example in the GF(5)
| system as a specific example).
| thaumasiotes wrote:
| > As such, the concept of "5" does not necessarily exist
| in all possible Ring-systems.
|
| Don't think of 5 as a quantity; think of it as a variable
| name. The proof only depends on the concept of addition
| and an additive identity (and distribution of
| multiplication over addition, which you're using anyway);
| no property of 5 appeared.
|
| > the A * 0 == A * (1 + (-1)) == A - A == 0 thing is
| built up from fundamental portions of Ring theory. As
| such, the proof I constructed at first applies to all
| rings.
|
| To repeat myself: n*5 = n*(5+0)
| [definition of 0] n*(5+0) = n*5 + n*0
| [multiplication is distributive over addition] n*0
| = 0 [definition of 0]
|
| Every step in that proof is a direct application of one
| of the ring axioms; it doesn't matter what 5 is.
| [deleted]
| [deleted]
| dragontamer wrote:
| I see what you're saying now. So to answer your original
| question...
|
| > You don't think it's easier to say...
|
| No. I disagree, your way of thinking is harder for me to
| think. :-)
|
| You're correct, but my mind didn't work like yours. But
| that's the beautiful thing about mathematics: we both are
| correct. We just had different viewpoints about how
| things work. Ultimately, it seems like we're both saying
| the same thing, although we tweaked the formulas to look
| like the simplest ways for our own brains.
| thaumasiotes wrote:
| Fair enough.
| mnl wrote:
| Repeated addition that ain't...
|
| It boggles my mind that you've been studying Galois
| theory yet somehow try to reduce the algebraic structures
| associated to two binary operations to playing with one
| of them.
| dragontamer wrote:
| The algebraic structures multiplication brings is
| different from the algebraic structures from addition.
| That's the point of rings (as opposed to groups).
|
| Nonetheless, in a ring (and all fields are rings),
| multiplication must and always is related to addition,
| through the distributed property (which I argue, the
| distributed property IS the mathematical term for
| "repeated addition").
|
| Without the distributed property, you have no ring. You
| at best only have a group. Therefore, all multiplication
| operators ever defined (or more precisely, all rings)
| must have multiplication related to addition: (A * (B+C)
| == AB+AC)
| mnl wrote:
| Without the distributive property you have two magmas in
| general for the same set that don't see each other. I'm
| not sure what your point is. When you have two binary
| operations you need some sort of distributive property to
| build a structure.
|
| All this is trivial, if you consider a ring, you get
| a*0=0 as a property, if your starting point is the Peano
| axioms for the arithmetic of natural numbers that's one
| of them, for the latter seeing it as "repeated addition"
| makes no sense, for the former, well you have a ring, you
| have two binary operations, not one, and of course you
| have some form of distributive property or else you'd be
| studying this set with just one binary operation at a
| time.
|
| I'd like to see how "repeated addition" works in
| polynomial rings.
| dragontamer wrote:
| > I'd like to see how "repeated addition" works in
| polynomial rings.
|
| Consider the following polynomial: x0 * b^0 + x1 * b^1 +
| x2 * b^2 ... xn * b^n, where "n" goes to both positive
| infinity and negative infinity.
|
| When "b = 10" and when "x" can be numbers from [0-9], we
| have the so called base-10 set of real numbers, do we
| not? IIRC, if b = sqrt(-1) * 10, we then have the set of
| complex numbers (a non-intuitive result. I may have made
| a mistake somewhere, but I assure you there's a
| surprising property along those lines).
|
| That's the funny thing about real numbers and complex-
| numbers. Real numbers and even complex-numbers ARE
| polynomials, and therefore a polynomial ring. 3.1415926
| == 3 * 10^0 + 1 * 10 ^-1 + 4 * 10 ^-2 ...
|
| ------------
|
| I'm using a lot of words here. But all I'm saying is once
| again: Pi * 3 == 3.14... * 3 == 3 * 3 + 0.1 * 3 + 0.04 *
| 3 + ... == 9.42...
|
| We can evaluate 3 * Pi by splitting Pi up into a set of
| additions (3 + 0.1 + 0.04 + 0.001...), even if that set
| of additions is infinite. Then evaluate 3*(each
| component). This is possible because Pi is easily
| represented as a polynomial X0 * 10^0 + X1 * 10^-1 + ...
| Xn * 10^-n.
|
| There's a reason why polynomial multiplication is usually
| called "Carry-free multiplication". Because Real-numbers
| are just polynomials where you have that annoying "carry
| the one" property to keep track of. Remove the "carry the
| one" property (in say: Galois extension fields), and all
| the math still works.
|
| --------
|
| Hmmmm... I probably could have said all that in fewer
| words. TL;DR: "Real numbers ARE a polynomial ring". (And
| complex numbers are probably a polynomial ring, I just
| forgot how to prove that factoid)
|
| > I'd like to see how "repeated addition" works in
| polynomial rings.
|
| EDIT: Just thought of a cute and simple retort. You ever
| do a CRC32 check? There ya go.
| mnl wrote:
| A polynomial is a polynomial, a decimal representation of
| a real number is a decimal representation of a real
| number, and your representation of complex numbers has
| funny properties once you begin exponentiating that.
|
| I'd like to see how you'd show kids that:
| (1+x+x^2)*(1-x^3) is a "repeated addition", both belong
| to the ring Z[x].
| dragontamer wrote:
| > I'd like to see how you'd show kids that:
| (1+x+x^2)*(1-x^3) is a "repeated addition", both belong
| to the ring Z[x].
|
| Is that not just (1 * (1-x^3) + x * (1-x^3) + x^2 *
| (1-x^3)) ??
|
| The polynomial itself gives us the means at which we
| logically split up the multiplication into component
| parts.
|
| Just as 3.14 * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3, when we
| move onto polynomials, we do the same exact thing. EDIT:
| remember, ALL REAL NUMBERS ARE POLYNOMIALS with a base of
| 10.
|
| Or to put it another way: when x == 10, your polynomial
| of (1 + x + x^2) * (1-x^3) == 111 * (-999). That is to
| say: real numbers are simply polynomials where "x" has
| been defined to be a particular number, instead of an
| abstract entity. We call that number the radix-base.
|
| If you instead defined the base to be x = 16 (hexadecimal
| numbers), you'd get 111 * (-FFF), which you'll find will
| satisfy similar properties. Now leave x-undefined (since
| it could be 10 or 16), and what do you get?
|
| Polynomial math. Or so called "Carry-less multiplication"
| (https://en.wikipedia.org/wiki/Carry-less_product). We
| don't have a ring yet though: we still need to perform a
| modulus on all those polynomials to return to a proper
| ring (and if the modulus is irreducable, we have a Galois
| field). But we can already see how polynomials and the
| Reals are so closely related.
| mnl wrote:
| Precisely, you're not repeating p(x) q(x)-times, you've
| used that p(x) is a linear combination of monomials and
| then the distributive property of Z[x].
|
| Now, you could argue that this is exactly a way to "add
| repeatedly", but at some point pushing analogies stops
| being helpful to your students.
| dragontamer wrote:
| > Now, you could argue that this is exactly a way to "add
| repeatedly", but at some point pushing analogies stops
| being helpful to your students.
|
| That's not what this blogpost is arguing about. This
| blogpost is arguing that "Multiplication is Repeated
| Addition" is unhelpful at the elementary school level and
| stops being true at some point.
|
| -------
|
| My argument is otherwise. "Multiplication is Repeated
| Addition" is clearly helpful in grade school. Almost
| everybody I know has learned Multiplication through that
| method.
|
| Secondly: I cannot think of a single instance where its
| not true. Yes, I've had to use linear-combinations to
| extend it out to polynomials, but clearly the property
| holds even in polynomial-land.
|
| Its not useful to teach multiplication of polynomials
| with "Repeated Addition". But the advice is "not wrong",
| in fact, polynomial multiplication continues to see many
| similarities with Real and Complex multiplication.
| Especially if we consider a "Basis" to be analogs to the
| thing that's repeatedly-added.
| cannabis_sam wrote:
| > But it's not a good idea to let ourselves get carried
| away, we still have two binary operations going on. At
| any rate we have to impose that n*0=0, which can't be
| writen cleverly as "repeated addition" and worked up
| backwards.
|
| Why is this not a good idea?
|
| Can't we just accept/postulate that the additive identity
| is different from the multiplicative identity?
|
| And still define a relationship between the addition and
| multiplication?
|
| Maybe I misunderstand the issue..
|
| (I'm not trying to be pedantic, but my math background
| has some holes :)
| macspoofing wrote:
| >I am sure he enjoys it a lot to be in a position where he can
| write pedantic falsehoods
|
| Are you sure it's pedantic? As a matter of practically, neither
| people, nor computers actually compute multiplication in such a
| way. Even children, though they may be exposed to the
| 'multiplication is repeated addition' concept as an
| introduction to multiplication, are quickly ushered past this
| and it is never brought up again - because it isn't helpful as
| you incorporate fractions, and negative numbers.
| dragonwriter wrote:
| > As a matter of practically, neither people, nor computers
| actually compute multiplication in such a way. Even children,
| though they may be exposed to the 'multiplication is repeated
| addition' concept as an introduction to multiplication, are
| quickly ushered past this and it is never brought up again.
|
| The standard manual method of multiplication of large numbers
| relies on leveraging heavily:
|
| (1) the fact that my multiplication is equivalent to repeated
| addition for nonnegative integers,
|
| (2) the fact that shifting digit positions are equivalent to
| multiplication by the base (usually 10) or it's
| multiplicative inverse, depending on direction, and
|
| (3) memorization of multiplication tables for single digits
| in the base.
|
| So, no, I don't think the "multiplication is repeated
| addition" thing is something people are exposed to and then
| never use.
| pdonis wrote:
| _> there is Keith Devlin involved.... I am sure he enjoys it a
| lot to be in a position where he can write pedantic falsehoods
| while at the same time being taken seriously_
|
| I don't think his argument (or the similar argument being made
| in the subject article of this thread) for not teaching
| students that multiplication is repeated addition is a
| "pedantic falsehood". I think he has a valid point: that there
| are downsides to teaching students one thing, and then later
| coming back and saying "well, that thing you were taught before
| isn't actually correct...", and that a different approach is
| possible. One can argue about the pros and cons of each
| approach, but I don't think it is helpful to just dismiss one
| side of the argument as "pedantic falsehoods".
| lhorie wrote:
| > teaching students one thing, and then later coming back and
| saying "well, that thing you were taught before isn't
| actually correct..."
|
| My kids are learning multiplication. I honestly don't feel
| like they're being lied to, or confused about what they are
| being taught. Yes, they start with addition and subtraction
| as crutches (e.g. `9 x N` is initially taught as `10 x N -
| N`). It's certainly not how _I_ learned (I had to do rote
| memorization of the tables), but hey they can recite tables
| now too.
|
| Easing into multiplication via addition doesn't feel wrong
| IMHO. In fact, at later grades, I'd have to learn about
| associative/commutative properties anyways.
|
| My daughter in particular has a somewhat peculiar story: she
| is in kinder and learning multiplication from a game[0] that
| her older brother started playing, and the way they
| introduced multiplication is by telling her that to calculate
| the area of a rectangle, she needs to literally count the
| number of blocks that compose it.
|
| Is it "wrong"? I guess. Blocks aren't really proper units,
| but she gets that a rectangle made of 4x5 blocks has an area
| of 20 blocks, first by stumbling n' counting, then adding
| rows then eventually just memorizing the multiplication fact,
| so mission accomplished? Things build up from there, first
| without the grid, then into area of more complex polygons,
| spin off into word problems, etc.
|
| IMHO what helps kids learn is just repeatedly seeing where
| the memorized mechanisms can be applied and where they can't.
| Eventually multiplication becomes second nature and the
| addition/subtraction crutches come off.
|
| [0] https://www.prodigygame.com/
| com2kid wrote:
| > (e.g. `9 x N` is initially taught as `10 x N - N`)
|
| That isn't a crutch, that is how large numbers can be
| multiplied in one's head!
|
| 90 * 21 is (90 * 20) + 90
|
| Right after removing that extra 90, now you have 9 * 2 * 10
| * 10 + 90, which is easy to do mentally.
|
| Heck if you ask me what 9 * 14 is, I'm going to do 10 * 14
| - 14.
|
| Did I have to memorize 0x0 through 12x12? Yup. But now days
| I have a few key mid points memorized and I'll add my way
| from there. I don't remember 7 * 6 but I know its 7 * 7 -
| 7.
| FPGAhacker wrote:
| > Heck if you ask me what 9 * 14 is, I'm going to do 10 *
| 14 - 14.
|
| Yeah, that's interesting how we do things in our head
| like that.
|
| I would frequently break that down as 9*10 + 9*4 in my
| head.
|
| But I might also do it as 10*14 - 10 - 4.
|
| I'll try different ways in my head until I can solve one
| of them easily.
| pdonis wrote:
| _> I honestly don 't feel like they're being lied to, or
| confused about what they are being taught._
|
| They might not be. If they're just being taught how to
| multiply in particular cases (whole numbers) using repeated
| addition, then they're not being lied to. If they're being
| told straight up that, while multiplication is not
| identical to repeated addition, they can ease into
| multiplication by learning repeated addition, even better.
|
| They're only being lied to if they're being told,
| authoritatively, that multiplication _is_ repeated
| addition, no ifs, ands, or buts. And yes, I have had
| teachers like that, and I suspect many others have too.
| jacobolus wrote:
| > _Yes, they start with addition and subtraction as
| crutches (e.g. `9 x N` is initially taught as `10 x N -
| N`)._
|
| This is not a "crutch", beyond the extent to which a
| decimal place-value system is a crutch. I would instead
| call it a broader and more fluent view of the number
| system.
|
| Describing e.g. 18 = 2*10 - 2*1 instead of 1*10 + 8*1 is a
| perfectly valid alternative representation which happens to
| often be more convenient when multiplying.
|
| Either way when we multiply we break each multiplicand into
| a sum, multiply the components from each combinatorially,
| and then add the results together.
|
| 18*6 = (1*10 + 8)6 = 10(1*6) + 8*6 = 60 + 48 = 108
|
| vs.
|
| 18*6 = (2*10 - 2)6 = 10(2*6) - 2*6 = 120 - 12 = 108
|
| Regularly discussing the alternative ways to represent a
| number and choosing the most convenient for the current
| goal builds what is called "number sense": fluency with the
| place-value system, basic properties of integers,
| relationships between numbers, base ten, and in a broader
| way facility with manipulating data structures.
| hyperpallium2 wrote:
| Pre-university mathematics is mostly like this.
|
| But there is a way to teach mathematics from axioms that is
| intuitive: geometry. Euclid's _Elements_ was the key
| mathematics textbook for about 2000 years.
|
| Of course, you simply can't cover as much ground if you must
| derive everything.
|
| But I do wonder if the collapse in public discourse is partly
| because of faith-based mathematics. i.e. taught as: it's true
| because we tell you it's true, not because you can see for
| yourself that it's true.
| JosephRedfern wrote:
| I never got over being lied to about electron shells and the
| 2-8-8 "rule".
| pdonis wrote:
| How were you lied to about that?
| JosephRedfern wrote:
| In GCSE Chemistry (taught to 15-16ish year olds), we were
| taught that the first shell contained two electrons, and
| subsequent shells contained 8 (up to the final, which may
| contain fewer).
|
| Then, after GCSEs, during A-levels, we were told (by the
| same teachers, in the same classrooms) to forget this
| model, and that the situation was actually more
| complicated, with s, p, d, f orbitals etc.
|
| I realise that this is was a simplification and not
| necessarily an outright lie, and can understand why they
| did it this way. But it was the first time I realised I'd
| been deliberately taught something that was incomplete or
| inaccurate.
| Sharlin wrote:
| To be fair, it once was the best model we had, before we
| figured out quantum mechanics. That brings up an
| interesting point, though: the way many of these things are
| taught closely mirrors the history of how we discovered
| them in the first place. This is definitely not a
| coincidence, but I'm also not sure whether it is the most
| effective way to teach.
| zamadatix wrote:
| The general concept is known as
| https://en.wikipedia.org/wiki/Lie-to-children and I think
| you'd be hard pressed to teach/learn/apply anything of
| significance without this "well actually it's more
| complicated than that" approach and recognizing how far you
| need to go with it to aptly do some task.
| pdonis wrote:
| There's no need to lie to children. Telling children
| "multiplication is a separate operation on numbers, but it
| works like repeated addition for the counting numbers
| you're familiar with" is not a lie.
| CodeMage wrote:
| Let's not take the word "lie" out of its context and then
| get hung up on it. We're not _actually_ lying to children
| by simplifying the explanation down to what they can
| comprehend with the tools they have at that stage.
| pdonis wrote:
| _> Let 's not take the word "lie" out of its context_
|
| I'm not the one that used that word. If people are going
| to reference a concept that's been published in a book
| and uses that word, they should own it and be prepared to
| justify it. If they can't do that, they should not use
| the word in the first place.
|
| To be clear, I'm not saying _you_ are one of those
| people. But the poster I was responding to is.
| [deleted]
| majormajor wrote:
| The continuing series of qualifications "it's kinda like
| this, for what you're talking about" gets pretty rough
| there, even when explaining, say, Kubernetes to adults!
| It can be a confidence crusher.
|
| There's also the flip side response which is always
| asking a lot of questions about "well then what are the
| other sorts of numbers" and eventually getting shut down
| "we're not talking about that now" which comes back to
| the "who decides what you're smart enough to hear about
| now" question in its own way. Or "oh the teacher doesn't
| actually know what the difference is, or why this isn't
| 'true' 100%."
|
| Despite the cutesy name, I don't think omission of detail
| is the same as lying. It's often impossible to tell 100%
| the truth. You probably don't even know it yourself!
| smolder wrote:
| Refusing to answer curious students questions isn't
| helpful. Giving them a hint and telling them you'll get
| back to it works fine. They're smart enough to understand
| they aren't having to wait because they're incapable of
| understanding, but because other things need teaching
| first. I know I did. The good teachers were encouraging
| of the curiosity while back-burnering something, others
| responded less patiently with things like "we're not
| there yet!", which made them seem like bad teachers to
| me, even back to elementary.
| pdonis wrote:
| _> The continuing series of qualifications "it's kinda
| like this, for what you're talking about" gets pretty
| rough there_
|
| The qualifications are there; that's just a fact. Being
| told about them, or at least about their existence if not
| every detail of them, up front seems better to me than
| finding out about them later on when your mental model is
| solidified around the simplified version that you then
| find out doesn't always work.
|
| _> I don 't think omission of detail is the same as
| lying._
|
| Saying "multiplication is repeated addition", without
| qualification and without any caveats, is not "omission
| of detail". It's a false, categorical statement, i.e.,
| lying.
|
| As for where the line is where you stop giving details,
| obviously that will depend on the circumstances. A
| teacher who says "we don't have time to talk about that
| during class today, but yes, there is much more detail
| here that you can look into on your own" is not lying and
| is not saying the child is "not smart enough" to take in
| all the detail now. (Bonus points if the teacher says
| "see me after class and I'll give you some pointers on
| where to go for more information".) A parent who says
| something similar because they have to get dinner ready
| and the child needs to do the rest of their homework
| before bed is also not lying and not saying the child is
| "not smart enough". Limitations of time are a fact of
| life, and children need to deal with it just like the
| rest of us.
|
| A teacher who just says "we're not talking about that",
| or who doesn't even know about the qualifications, or who
| gets snippy when a child asks a natural question, is
| obviously not doing the child any good; but that is
| _because of_ the teacher fixating on a simplified model
| and treating it as "the Truth", so doing more of that
| won't fix it.
| roenxi wrote:
| > It can be a confidence crusher.
|
| If confidence gets priority over truth in explanations,
| then society will churn out people who are confidently
| wrong. This is a bad idea even if everybody does that and
| even if it is the traditional approach.
|
| If people were honest that they don't know something then
| the world at large would be a lot nicer to live in.
| ahepp wrote:
| I think this is backwards. The public at large are pretty
| honest that they don't know math. The world would be a
| much better place if they were able to do practical
| algebra, even if the higher truths of abstract
| mathematics never enter their minds.
|
| I really don't see people being confidently wrong about
| abstract mathematics as an issue. I certainly don't know
| the rigorous definitions of an integral, but I can apply
| the concepts of calculus to everyday life perfectly fine.
| People who care about rigorous math can do rigorous math,
| and I'm glad if they teach me an intuitive understanding
| that lets me live a happy, productive life.
|
| Personally, I find layers of abstraction necessary for
| learning. Maybe there are people who don't, I suspect
| they would have to be prodigies though. Tell me how to
| add fractions practically, then teach me the principles
| when I need to know them. Framing that as a lie seems
| wrong to me, I'd call it "bounded knowledge".
| kortilla wrote:
| The confidence is in the ability to learn a topic, not
| confidence about the knowledge.
| pdonis wrote:
| It seems to me that the way to build people's confidence
| in their ability to learn is to allow them to learn. That
| means _not_ making oversimplified statements because you
| don 't think they're "ready" for more details. It means
| _giving_ them the details, and letting _them_ decide when
| they 've had enough for now. (Of course one's time, say
| in a classroom, will be limited, so at some point one has
| to say "we don't have time to go into this further in
| class now"--but that's still letting them know that there
| _are_ more details, and they can dig into them further on
| their own.)
| peterwoerner wrote:
| But you can define multiplication based on abstracting
| repeated addition. That how we did it my analysis class,
| although I forget some of the details. So if you say
| multiplication is repeated addition, it really isn't a
| lie.
| threatofrain wrote:
| But did your analysis class define addition on the
| naturals? Then that's an operator for the naturals. And
| then for integers, rationals, etc.
| aidenn0 wrote:
| If you don't lie to children you confuse and discourage
| them. I know because my own internal desire for precision
| often is to the detriment of my pedagogy.
| SilasX wrote:
| Well, that would be true if all children were as smart
| and analytically adept as you.
| hrktb wrote:
| It looks like splitting hairs, but I also think it makes
| a difference to state upfront that it's an
| oversimplification.
|
| Kids actually understand the point, and they can decide
| if they care enough to ask more questions or if it's good
| enough for them.
|
| We faced that when teaching divisions. Saying upfront
| we'd explain falsehoods for the sake of simplicity helped
| set aside the more difficult questions (infinity, etc.)
| that came right after. We just say it's the complicated
| parts and move on.
|
| The best aspect is they are more receptive to have their
| mental model broken afterwards, instead of clinging to
| what you explained as fully true.
| pdonis wrote:
| I think this viewpoint is pernicious. A child doesn't
| have to be "smart" in order to deserve being told the
| truth.
|
| Or I could turn your remark around: what makes you, the
| adult, think you are so much smarter than the child that
| you can correctly judge what lies are OK to tell them?
| Are all _adults_ really that smart? (Are _any_ of us?)
| [deleted]
| allturtles wrote:
| An abstraction/simplification/shorthand is not a lie.
| People don't say "multiplication is repeated addition"
| because they are trying to hide the truth for some
| selfish reason. It's a pedagogical strategy to help
| people learn a new abstraction by analogy to an old one.
| These kind of crutches are a necessity, you can't
| introduce all the complexity of the world to someone all
| at once. This applies to every subject - science,
| history, writing. Simple notions and shorthands are
| introduced first, and complexity and nuance added on
| later.
|
| Is it a "lie" to teach kids just learning chess that
| queens are worth more than any other piece and you should
| always protect your queen, even though there are advanced
| situations when it makes sense to sacrifice your queen
| for no immediate material gain?
|
| In this specific case, introducing the ideas of
| "operations" and "counting numbers" into the picture
| muddies the waters, most kids who are just learning
| multiplication won't have any idea what you mean by those
| concepts.
| pdonis wrote:
| _> An abstraction /simplification/shorthand is not a
| lie._
|
| "Multiplication is repeated addition" is not "an
| abstraction/simplification/shorthand". Doing that for
| multiplication would be saying something like
| "multiplication is a distinct primitive operation, but it
| works like repeated addition for whole numbers, so that's
| what we'll be learning how to do now." Is that really so
| hard?
|
| _> Is it a "lie" to teach kids just learning chess that
| queens are worth more than any other piece and you should
| always protect your queen, even though there are advanced
| situations when it makes sense to sacrifice your queen
| for no immediate material gain?_
|
| If you tell them everything you just said, no, you're not
| lying. But if you just tell them "always protect your
| queen", without explaining anything about why and without
| saying that there are some advanced situations where you
| might break this rule, yes, you're lying. It only takes a
| couple of sentences to add that extra information. Again,
| is that really so hard?
|
| _> In this specific case, introducing the ideas of
| "operations" and "counting numbers" into the picture
| muddies the waters, most kids who are just learning
| multiplication won't have any idea what you mean by those
| concepts._
|
| Um, what? We're assuming they already know about addition
| of whole numbers. So it is simple to tell them "this
| addition thing that you learned, that's an example of an
| operation", and "those whole number thingies that you
| learned how to add, those are numbers". Once more, is
| that really so hard?
| perl4ever wrote:
| > but it works like repeated addition for whole numbers,
| so that's what we'll be learning how to do now
|
| I think it's an arbitrary perspective, whether you treat
| the whole number case as primary or the generalization as
| primary.
|
| People may prefer to consider the extended definition
| more "real", but I think the argument for going the other
| way is that usually the original limited form of
| something is more likely agreed upon by most people,
| whereas the generalization can be done in multiple ways
| which may owe something to history and culture, or
| context.
|
| I feel like math is fundamentally different than physics,
| where the more advanced theory is objectively closer to
| correct. With math, it's more of an arbitrary aesthetic
| or social judgment. Nothing ever stops you from
| generalizing anything even more than anyone did yet,
| right?
| pdonis wrote:
| _> I think it 's an arbitrary perspective, whether you
| treat the whole number case as primary or the
| generalization as primary._
|
| Axiomatically, I think it can go either way. But one
| still has to recognize, as you do, that there are more
| cases than just the whole number case, and that what
| works for the whole number case might not work for other
| cases.
|
| _> the generalization can be done in multiple ways_
|
| There are certainly cases of this, but I don't think the
| case under discussion is one of them. There is only one
| generalization of the whole numbers under discussion
| here, the one from whole numbers to rationals to reals
| (and on to complex numbers if you want to take it that
| far, and still further on to matrices for some people in
| this discussion). There aren't multiple ways to do that:
| the rationals, reals, and complex numbers are all unique
| sets.
| SilasX wrote:
| Along with what allturtles said, I don't think it's right
| to call it a lie, which, to me, implies moving someone's
| model _away_ from the truth (on the basis of them
| trusting you to convey the correct one). An
| oversimplified, "wrong" model doesn't do that; it moves
| them from ignorance toward the correct model (i.e.
| increases their prediction accuracy).
|
| And yes, _even when I 'm in the learner's shoes_, I
| prefer that a teacher start with an approximate model,
| and then refine it as they go further. Starting with the
| full thing is barely comprehensible.
| pdonis wrote:
| _> I don 't think it's right to call it a lie_
|
| It's not a lie to tell them "this is a simplified model
| that doesn't include everything, but you'll be able to
| add more complexities to it later". But that's not what
| "multiplication is repeated addition" says. You would say
| something like: "repeated addition is a simplified model
| of multiplication that works for whole numbers, but
| doesn't work well in more complicated cases that you'll
| learn about later".
| SilasX wrote:
| Yeah, and that's still too much inferential distance to
| feed to an elementary schooler.
| pdonis wrote:
| You appear to have a much more impoverished view of kids
| and their ability to learn than I do. My experience (not
| to mention my memory of how I was myself as a kid) is
| that kids grasp the fact that there can be more to a
| subject than adults are able to teach them at a
| particular time and place, so they're ok with adults
| honestly admitting that. But they do not like adults
| telling them categorical statements that later turn out
| to be wrong.
| kortilla wrote:
| You're view of children seems to be more based on your
| memory of high school. That's a lot different than your
| behavior as a 6 year old when these simple primitives are
| being taught.
|
| Little kids don't care about nuance when they're still
| having difficulties with carries in addition. Your ideal
| world where we first explain children that base10 isn't
| the only way to represent numbers and whatever other
| caveats simply doesn't exist.
|
| They don't have the knowledge required yet to even
| understand the scenarios when "exceptions to the rule"
| apply.
| SilasX wrote:
| There's a difference between prefacing a course with "oh
| hey these are simplifications that you'll improve upon in
| higher grades" vs loading down literally every claim with
| that long chain of caveats.
| pdonis wrote:
| _> loading down literally every claim with that long
| chain of caveats._
|
| I have never proposed doing the latter, so you are
| attacking a straw man. Once it's understood that you're
| teaching a simplified, approximate model, you don't have
| to repeat in every sentence that you're teaching a
| simplified, approximate model. You just have to _not_ say
| it 's "the Truth", without approximation and without
| qualification.
| SilasX wrote:
| I'm relying on these examples you gave of how to do it:
|
| >"multiplication is a separate operation on numbers, but
| it works like repeated addition for the counting numbers
| you're familiar with"
|
| >"repeated addition is a simplified model of
| multiplication that works for whole numbers, but doesn't
| work well in more complicated cases that you'll learn
| about later"
|
| If you disagree that that's "long" or would feel that way
| in having to do it in every sentence, we can have a great
| discussion about that, but it is not a strawman -- you
| seem to reject the idea of giving the one caveat at the
| beginning of the course, and instead want to make each
| sentence rigorous.
|
| If you recognize that your complicated sentences are
| probably not ideal for teaching math to second graders,
| then I think we're in agreement.
| pdonis wrote:
| _> If you disagree that that 's "long"_
|
| I do. Some of the words might be changed, depending on
| what words have been used to describe the operation of
| addition and the set of counting numbers. But, as I think
| I've pointed out elsewhere in this thread, the very fact
| that the children know about addition and the counting
| numbers means they know what an operation is ("a thingie
| like addition") and what a set of numbers is ("a thingie
| like the counting numbers").
|
| _> you seem to reject the idea of giving the one caveat
| at the beginning of the course_
|
| I don't know where you're getting that from. I have
| already said the contrary--once you've said it, you don't
| need to repeat in every sentence.
| [deleted]
| smolder wrote:
| They don't need this information when learning that 3
| groups of 5 fruit are 15 fruits. It's not in the natural
| order of understanding principles.
| pdonis wrote:
| _> They don 't need this information when learning that 3
| groups of 5 fruit are 15 fruits._
|
| They also don't need the "information" that
| multiplication is repeated addition, period, full stop.
| But that's what others in this discussion appear to be
| trying to argue for.
| wcarey wrote:
| How would you teach physics without lying to children?
| pdonis wrote:
| Um, by not lying to them?
|
| Do you have a specific example of where you think you
| need to lie to teach physics?
| JackFr wrote:
| We lie when when we teach calculus by relying on hand-
| wavy 'limit' proofs, without grounding students in the
| foundational aspects of point set topology.
|
| It's much more productive pedagogically to get an
| intuition for slope and area than it is to get an
| intuition for compactness and the infinite intersection
| of open sets, but slope and area are A LIE.
| djur wrote:
| Explaining the structure of an atom without having to
| explain atomic orbitals and standing waves. Explaining
| classical mechanics without including a bunch of caveats
| about relativistic speeds.
| roenxi wrote:
| "When they performed [this experiment] at [experimental
| accuracy] they observed [that result] which makes them
| think [atomic orbitals and standing waves]."
|
| Someone who thinks they understand physics without
| considering experimental accuracy _doesn 't_ understand
| physics.
| pdonis wrote:
| _> Explaining the structure of an atom without having to
| explain atomic orbitals and standing waves._
|
| My high school chemistry teacher had no problem
| explaining this to me, when teaching the periodic table
| of the elements, without telling any lies and without
| going into the details of the quantum mechanics involved.
| The Pauli exclusion principle and a general statement
| that the details of the quantum mechanics were out of
| scope for that class was enough.
|
| _> Explaining classical mechanics without including a
| bunch of caveats about relativistic speeds._
|
| My high school physics teacher had no problem explaining
| classical mechanics including the caveats. The caveats
| took only a few minutes early in the semester. What's the
| problem?
| pessimizer wrote:
| Why would you explain classical mechanics at all? It's an
| inaccurate simplification.
| kortilla wrote:
| Classical mechanics covers the vast majority of the
| interactions in day to day life people will have. There
| is a reason it's "classical mechanics".
| pdonis wrote:
| It's much simpler to compute answers with classical
| mechanics, and the answers are accurate enough for many
| practical purposes. As the saying goes, all models are
| wrong but some are useful.
| WalterBright wrote:
| Teaching Newtonian Mechanics to students is also a lie. But I
| don't believe starting with General Relativity is going to do
| anything but confuse students.
|
| I was never confused by "3 baskets of 6 apples, how many
| apples" questions or thought I'd been lied to about addition.
| pdonis wrote:
| _> Teaching Newtonian Mechanics to students is also a lie._
|
| Not if you tell the students that it's an approximate model
| that works well in the domain they're currently studying,
| but doesn't work well in a more expanded domain.
|
| Of course if you insist on acting like an authority and
| telling students that Newtonian Mechanics is "the Truth",
| then yes, you are lying to them. But you don't have to tell
| them that to teach them Newtonian Mechanics.
| tvanantwerp wrote:
| I still encounter this as a grown adult. I'm working through
| a course on a programming framework I'm unfamiliar with right
| now, and several times have gone through a half-hour setup to
| encounter "Now that I've shown you the awful way to do this,
| here's a built-in way that's far simpler and less error-
| prone!" Extremely irritating.
| curiousllama wrote:
| Idk discovering all the lies I believe is one of the most fun
| parts of learning for me.
| pdonis wrote:
| _> discovering all the lies I believe_
|
| I have no problem with discovering lies I believe because I
| told them to myself (and I think most of the lies anyone
| believes are lies they've told themselves). What I have a
| problem with is discovering that someone else deliberately
| lied to me. But probably not everyone feels the same way I
| do about such things.
| D13Fd wrote:
| I've been teaching my 5 year old multiplication and division for
| the last couple of weeks. I'm at a loss how you would teach it
| without explaining that it is repeated addition.
|
| For example, the other day I asked her how many fingers and toes
| the three kids at the table had, and she came up with "20 fingers
| and toes each times 3 kids means there are 60 fingers and toes."
| I think the units are intuitive in most cases and all of the more
| advanced concepts the author (and the related articles) discuss
| can be taught later.
| pdonis wrote:
| _> I 'm at a loss how you would teach it without explaining
| that it is repeated addition._
|
| How much will they later have to unlearn when they find out
| about other kinds of numbers for which multiplication is not
| repeated addition?
|
| Quite possibly the same approach won't work for all children;
| some will be better served by telling them multiplication is
| repeated addition at first, and then later explaining that it
| isn't always, it only is for whole numbers. Others might be
| better served by telling them up front that multiplication is a
| separate operation, but can be "emulated" for whole numbers by
| repeated addition. (As a child I was in the latter group; I
| hated it when teachers told me one thing in one grade, and then
| said "well, what you were taught before isn't actually
| correct..." in a later grade. How many more times would they
| change the rules?)
| kmill wrote:
| Maybe I've been doing too much computer-assisted proofs
| recently, but I'm not understanding what all the fuss is. At
| some point you need a `Nat.mul` that's concretely defined,
| and probably the only reasonable definition is recursively
| using repeated addition (and if your natural numbers are in
| unary, which is theoretically the simplest, then even
| `Nat.add` is going to be defined recursively by repeated
| incrementing! like taking a bead from a pile one at a time
| and putting it into another).
|
| Then you can define an interface with abstract operations
| like addition and subtraction, then any given number system
| can try to implement this interface. Naturals, integers,
| rationals, reals, polynomials, rational functions, complex
| numbers, and many others can implement at least the semiring
| interface. But what's important, I think, is that there's no
| universal multiplication operation -- it's just a word that
| fills in for whatever is the right multiplication operation
| at the moment. It's some ad-hoc polymorphism.
|
| Unless you're accepting the real line as being axiomatic
| (maybe from geometry, where you can define all the basic
| operations using ruler-and-compass constructions), all the
| definitions of multiplication for all the above number
| systems are going to be, eventually, based on the one for the
| naturals.
| pdonis wrote:
| _> At some point you need a `Nat.mul` that 's concretely
| defined_
|
| And since your computer can only perform finite operations
| on numbers with a finite bit size, which means operations
| on a set of numbers isomorphic to the integers, then yes,
| repeated addition works fine for your concrete
| implementation.
|
| But that's not at all the same as saying that
| multiplication _is_ repeated addition, without any
| qualification whatsoever. The latter claim is the claim I
| 'm arguing against.
|
| _> Unless you 're accepting the real line as being
| axiomatic_
|
| Your computer can't manipulate real numbers. It can only
| implement finite operations on numbers with a finite bit
| size, which, as above, is a set of numbers isomorphic to
| the integers. So if you want to work with the real numbers
| themselves, abstractly, then you can't expect what the
| computer does to carry over.
| kmill wrote:
| My computer can manipulate and prove things about real
| numbers just fine. I think it can do at least everything
| I can do normally, though much more laboriously.
|
| What I was mostly responding to in your previous comment
| is the idea of "emulating" multiplication for whole
| numbers by repeated addition. What I got from that is
| that you were thinking of the whole numbers as being
| inside the real numbers. My point was that to get the
| real numbers you usually have to start with the whole
| numbers and build up to the reals (say by Dedekind cuts
| or Cauchy sequences), and then you can embed the whole
| numbers inside and pretend that that was where they were
| the whole time.
|
| I do get that you were giving different pedagogical
| approaches, and this was just one of them, but I think
| this one is at least as bad of a misrepresentation as
| saying "multiplication is repeated addition" without
| specifying "for whole numbers." That is, unless you're
| like the Greek geometers and accept a continuum from the
| start.
|
| (There are other number systems that the integers sit
| inside, though, that don't themselves sit inside the real
| or complex numbers, which is a reason I would hesitate to
| think about whole numbers this way in general.)
| pdonis wrote:
| _> My computer can manipulate and prove things about real
| numbers just fine._
|
| If you mean it can, with appropriate software, do
| symbolic manipulations of general formulas that are valid
| for real numbers, yes, of course. But that's not the same
| as doing specific concrete computations with them.
|
| _> What I was mostly responding to in your previous
| comment is the idea of "emulating" multiplication for
| whole numbers by repeated addition. What I got from that
| is that you were thinking of the whole numbers as being
| inside the real numbers. My point was that to get the
| real numbers you usually have to start with the whole
| numbers and build up to the reals (say by Dedekind cuts
| or Cauchy sequences)_
|
| Ah, I see. As far as I know the axiomatic reasoning
| involved can go either way. But in any case, that wasn't
| what I was trying to get at with the term "emulating";
| I'm sorry if my use of that term caused confusion, and I
| agree with you that telling a child something like "the
| whole numbers are inside the real numbers" without
| qualification would also be a misrepresentation. My point
| was simply that teaching a child a specific computational
| procedure, repeated addition, in order to get an answer
| to particular multiplication problems does not require
| telling the child that multiplication _is_ repeated
| addition, without qualification. The two things are
| distinct, and I am fine with the former; I only object to
| the latter.
| prionassembly wrote:
| You can teach (some intuition for) multiplication without
| _numbers_ by multiplying lengths to get areas. Then show e.g.
| how the number of seats in a theater is the number of seats in
| a row * the number of rows.
|
| Let the school teacher press the algorithms on him, teach him
| your hacker's sense of wonder.
| magicalhippo wrote:
| > Then show e.g. how the number of seats in a theater is the
| number of seats in a row * the number of rows.
|
| How do you do that without adding up the number of seats in
| the rows or columns (ie repeated addition)?
| prionassembly wrote:
| Using a calculator.
|
| Not a troll: the ability to compute products and the idea
| that (horizontal length) times (vertical length) = (area)
| are different things.
| magicalhippo wrote:
| So teaching multiplication ends up as "whatever the
| calculator does when I press this button"?
| whizzter wrote:
| Having been teaching programming I can see an immediate problem
| with it, whilst it works great as a starter tying it too hard
| in the mind will require more un-doing when it comes to more
| advanced concepts where it isn't an appropriate level.
|
| On the other hand knowing the relation is a good hint at how to
| explain exponentiation.
|
| In the end teachers should be aware of what will be needed at
| higher grades to use tools such as it as a starting point but
| not ingrain it too much and try to push students onto thinking
| of multiplication as it's own functional primitive once they
| get the hang of basic tables and start moving on to long
| multiplication.
| Double_Cast wrote:
| I tend to think of multiplication as a transformation.
| "repeated additional" almost makes it sounds like we're merely
| telescoping an interval within a single dimension. Whereas
| "transformation" (to me), evokes more of a Cartesian Product or
| Quadrature sort of mental image.
| fifilura wrote:
| Lego is a great tool for this.
|
| I also taught prime numbers by handing out (round) 1x1 pieces
| and told him to figure out which ones could be made into
| squares.
|
| The teacher was surprised when - in the middle of a more
| convoluted explanation for primes - he just said "primes are
| numbers that can't be arranged as rectangles".
| eigenket wrote:
| *rectangles (hopefully)
| [deleted]
| tgv wrote:
| You start by teaching rings, duh.
|
| But seriously, her first objection is
|
| > To define multiplication as repeated addition is to make
| multiplication a sub-species of addition.
|
| To me, that means the author is wrong from the get-go. Not all
| cases of multiplication might be easily seen as repeated
| addition, but to get hung up on a deliberate linguistic
| misunderstanding? And to add insult to injury, she finishes by
| proposing to see multiplication as the answer to "how many or
| how much OF the unit", and shows a diagram with _repeating_
| units.
|
| Oh, never mind, the answer is at the end: she's got books to
| sell, and probably tries to improve her sales by appealing to
| parents with children that can't multiply 83 by 17 when they're
| 8.
| [deleted]
| sandworm101 wrote:
| >> Multiplication requires different units. >> 2 baskets x 3
| apples per basket = 6 apples
|
| No it doesn't. In that example the "baskets" and "per baskets"
| (ie x/baskets) are cancelling each other. 2 baskets x 3
| apples/baskets = 2 x 3 apples = 6 apples. Once you remove the
| unnecessary terms the actual multiplication is still basic
| repeated addition.
|
| 2 hours x 3 miles per hour = 2 x 3 miles = 3miles+3miles = 6
| miles.
| cuspycode wrote:
| Yes, that was the most glaring mistake in the article, in my
| eyes. The runner up strange argument was the one about
| multiplication and addition having different identities and
| inverses. But why would the operations "Addition" and "Repeated
| Addition" have exactly the same properties? The operations are
| not the same.
|
| I think the real issue is about how to interpret the adjective
| "Repeated" when we turn to more general domains than the
| natural numbers. But that's not a huge issue, it's mostly a
| matter of terminology.
| glitchc wrote:
| A bit meandering and muddled in some respects, but agree with the
| gist: Multiplication is not just repeated addition.
| rafaelturk wrote:
| As software developers we can defenetly point the many flaws in
| this article:
|
| 1. IS A CLICKBAIT
|
| 2. Course you can decopose multiplications in a series of
| repeated additions.
|
| 3. Extra points for the flawed argument of adding different, well
| you can't because they are different.
| EGreg wrote:
| A few years ago, I was teaching a class and decided to record a
| short series on the fundamentals of mathematics, for complete
| "beginners" - whether they be kids or adults.
|
| This is the first video in the series Thinking Mathematically
| (after the introductory video, "Why think Mathematically?") which
| I put on YouTube under a channel of the same name. It proceeds
| through the sets of numbers, N -> Z -> Q -> R -> C and yes it's
| for beginners. Would love some feedback:
|
| https://www.youtube.com/watch?v=Wd1lzqR3xW0
|
| The other videos you'll find on that channel are:
| 1. Why think mathematically? 2. Numbers and Algebra
| 3. Sets and Infinity 4. Logic and Probability
|
| Here is the channel with all the videos. They might be useful to
| share with people who you want to understand these concepts from
| the ground up:
|
| https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA
|
| (PS: I am generalizing this approach in our upcoming app,
| https://teaching.app)
| fullshark wrote:
| Imagine explaining the content of this post to a five year old
| instead of that multiplication is repeated addition.
| chrisandchips wrote:
| I strongly believe that helping younger students gain strong
| intuition for these operators pays dividends towards their later
| success in maths.
|
| I've always run into the following problem: I try to motivate
| multiplication as repeated addition, which does help with
| intuition, but then things totally fall apart when we move on
| from integers into fractional values.
|
| 1/2 * 1/2 -> 1/4.
|
| Sure you can teach someone to simply multiple the numerator and
| denominator, but it doesn't necessarily help them make clear
| sense of what's going on.
| kmill wrote:
| I think it's sort of an illusion (though certainly a useful
| one) that numbers are all part of the same system.
| Multiplication of natural numbers is defined to be repeated
| addition, and I'm not really sure how you could define natural
| number multiplication in any other way. From the natural
| numbers you can go on to define the integers, rationals, then
| reals, and each has its own definition of multiplication,
| though they each depend on the definition of multiplication
| from the previous system.
|
| There's a standard way of lifting each type of number to the
| next type, and this lift is compatible with all the basic
| operations (the lift is a "homomorphism"), so it's easy to
| pretend that the real numbers (or complex numbers if you want)
| are the universal system.
|
| So with your example of going to fractional values, you're
| right, repeated addition falls apart -- but I'd say that's
| because it's not the definition for multiplication of rational
| numbers! Multiplying numerators and denominators is the usual
| definition, but that gives about as much intuition as does the
| definition for multiplying natural numbers. Sort of "the point"
| of multiplication of naturals, I think, is that it represents
| how many things you have if you arrange them in an n by m grid.
| Rational numbers show up in geometry with similar shapes
| (scaling), and for a few reasons you'd want multiplication to
| represent by how much something scales after a composition of
| scalings; maybe "the point" of rational number multiplication
| (at least algebraically) is that you can defer dividing until
| later, i.e. (a/b) * (c/d) is (a*c)/b / d.
| NineStarPoint wrote:
| Fractional numbers make sense to me as an extension, but it
| also requires an intuition of division on the same lines.
|
| Take number n and multiply it by number x/y. To do this, you
| have to split number n into y parts and take x number of them.
| So to multiply 8 by 3/4, you split 8 into 4 parts (2 + 2 + 2 +
| 2) and then take 3 of those parts. This ends up being 2 + 2 + 2
| = 6.
|
| For multiplying two fractions, you have to extend it to n/m *
| x/y. Since you can multiply the top and bottom of a fraction by
| the same number, you can write n/m as ny/my. Then you can have
| n/my be your "equal part", and take x of them. So 1/2 * 1/2,
| you take 2/4 and split it into 1/4 + 1/4 and 1 of them, so the
| answer is 1/4.
|
| To me at least, this makes sense as an extension of
| multiplication is repeated addition. It's when you get to
| irrationals that it starts to fall apart, and even then the
| intuitions the above way of thinking led me to have served me
| well.
| tsian2 wrote:
| As someone with an almost solely intuitive understanding of
| mathematics, I still find the "repeated addition" idea useful
| when it comes to fractions. If 4 * 4 is 4 repeated 4 times (4 +
| 4 + 4 + 4). Then 1/2 * 1/2 is a half repeated half times, or a
| half of a half, which is a quarter. The numbers get hard to
| work with but the intuitive idea is still there.
| ubasu wrote:
| To put it in terms of what the audience here can relate to:
|
| multiplication is an overloaded operation, or in more modern
| terms, it is doing multiple dispatch, depending on whether the
| input is a whole number or integer or rational or real or complex
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