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What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

• harperlee 6y ago
  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)
  • alberto_ol 6y ago
    I wanted to post the same links, you can find the same articles and others about that subject by googling "Keith Devlin addition multiplication"
  • Jtsummers 6y ago
    Clickable: - https://www.maa.org/external_archive/devlin/devlin_09_07.html https://www.maa.org/external_archive/devlin/devlin_09_07.htm... - https://www.maa.org/external_archive/devlin/devlin_06_08.html https://www.maa.org/external_archive/devlin/devlin_06_08.htm... - https://denisegaskins.com/2008/07/01/if-it-aint-repeated-addition/ https://denisegaskins.com/2008/07/01/if-it-aint-repeated-add... (Sorry, it just bothers me when people don't use clickable links here.)
• threatofrain 6y ago
  The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?
  • cturner 6y ago
    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 6y ago
      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 6y ago
    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 6y ago
    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 6y ago
    > but is 2 cm × 3 cm the same thing as 2 × 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 6y ago
      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 6y ago
      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)
• jmchuster 6y ago
  I like how 3blue1brown presents it in terms of animated number line translations https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=365 https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=365
• tromp 6y ago
  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 6y ago
    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.
• t0r0nat0r 6y ago
  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 6y ago
  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 6y ago
    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 6y ago
      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 6y ago
        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 6y ago
          > 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.
          • anbende 6y ago
            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.
• bmc7505 6y ago
  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 6y ago
    It's a variable number of addition operations.
    • bmc7505 6y ago
      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 6y ago
    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 Gödel 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 Gödel).
    • bmc7505 6y ago
      That’s a great explanation, thank you!
• mensetmanusman 6y ago
  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 6y ago
    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 ;)
• Garlef 6y ago
  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
  • Rerarom 6y ago
    This is just set theory, not category theory.
    • Garlef 6y ago
      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-and-set-theory-just-a-different-language-or-different-foundati https://mathoverflow.net/questions/360578/category-theory-an...
      • deleted 6y ago
        [deleted]
      • nicky0 6y ago
        But talking about category theory makes you sound smarter.
        • Garlef 6y ago
          I'm personally more scared of set theory.
      • eigenket 6y ago
        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.
        • 6y ago
• forinti 6y ago
  The more methods one studies, the better one understands. There is no one true method.
• rustybolt 6y ago
  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.
  • yakubin 6y ago
    > 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.
    • WallWextra 6y ago
      If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.
      • yakubin 6y ago
        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 6y ago
• raverbashing 6y ago
  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 6y ago
    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 6y ago
      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 6y ago
        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)
• y7 6y ago
  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 6y ago
    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 6y ago
      It looks exactly like scalar multiplication when your matrices are diagonal.
      • prionassembly 6y ago
        Then: all matrices look like the identity matrix if applied to a suitably rescaled eigenvector.
        • eigenket 6y ago
          We're talking about matrix/matrix multiplication here, not matrix/vector, right?
          • prionassembly 6y ago
            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"))
• D13Fd 6y ago
  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.
  • prionassembly 6y ago
    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 6y ago
      > 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 6y ago
        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 6y ago
          So teaching multiplication ends up as "whatever the calculator does when I press this button"?
  • whizzter 6y ago
    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.
• rafaelturk 6y ago
  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 6y ago
  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 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 https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA (PS: I am generalizing this approach in our upcoming app, https://teaching.app https://teaching.app)
• fullshark 6y ago
  Imagine explaining the content of this post to a five year old instead of that multiplication is repeated addition.
• thothamon 6y ago
  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.
• GuB-42 6y ago
  > 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.
• bikenaga 6y ago
  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? :-)
• glitchc 6y ago
  A bit meandering and muddled in some respects, but agree with the gist: Multiplication is not just repeated addition.
• scythe 6y ago
  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?
• cjfd 6y ago
  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 6y ago
    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.
    • dragontamer 6y ago
      > 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.
• mansoor_ 6y ago
  He makes a good case for multiplication being a special subset of (repeated) addition, whilst semantically trying to argue otherwise :D
• jeff-davis 6y ago
  Is addition a shortcut for counting, or is that wrong, too?
• andrewprock 6y ago
  tl;dr: "Nothing"
• scelerat 6y ago
  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 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.
• johndoe42377 6y ago
  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.
• LJD_E 6y ago
  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 × 3 apples per basket = 6 apples” would correspond to 2x3y/x=6y removing the numbers and the equation becomes clear xy/x=y or we the numbers for them selves 23=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 × 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.
• chrisandchips 6y ago
  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 6y ago
    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.
  • tsian2 6y ago
    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.
• OlleTO 6y ago
  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 6y ago
    It's actually 2x...
    • OlleTO 6y ago
      Yes, because multiplication isn't actually repeated addition :)
• maweki 6y ago
  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 6y ago
    > 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.
• whatgoodisaroad 6y ago
  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.
• sandworm101 6y ago
  >> Multiplication requires different units. >> 2 baskets × 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 6y ago
    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.
• jmull 6y ago
  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 ½ X ½: oof. You can sorta do it, like "add ½ 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 6y ago
    > ½ X ½: 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 6y ago
    > ½ X ½: oof. You can sorta do it, like "add ½ 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.
• cryptica 6y ago
  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.
• tarkin2 6y ago
  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.
• fchu 6y ago
  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 6y ago
    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.
• EvanAnderson 6y ago
  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 6y ago
    The new new math isn't just multiplication is addition, it's addition and substraction. 39 x 40 = 40 x 40 - 40
    • EvanAnderson 6y ago
      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.
• cryptica 6y ago
  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.
• devinplatt 6y ago
  (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.
• ubasu 6y ago
  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
• mxfh 6y ago
  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 × 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] × [0 3 0] = [0 0 12] https://en.wikipedia.org/wiki/Scalar_multiplication https://en.wikipedia.org/wiki/Scalar_multiplication https://en.wikipedia.org/wiki/Cross_product#Geometric_meaning https://en.wikipedia.org/wiki/Cross_product#Geometric_meanin...
• amelius 6y ago
  By the way, why doesn't exponentiation have units in physics? E.g. why don't we ever have something like kg^s?
  • kmill 6y ago
    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.)
    • amelius 6y ago
      Interesting point. But I think that the Taylor series is just a mathematical tool, and as such it hides the units that exist in these expressions. From a physics point of view, every term in the Taylor series needs to be augmented with a unit that follows from the definition of the Taylor series.
  • rovolo 6y ago
    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
• thysultan 6y ago
  Philosophical mathematicians trying to confuse the plebs.
• WalterBright 6y ago
  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.
• WalterBright 6y ago
  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/FLOAT.ASM https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL... Multiplication is repeated addition. Yes, I know about roundoff errors, overflows, etc.
• bitwize 6y ago
  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.
• bawolff 6y ago
  > 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.
• ducaale 6y ago
  https://www.youtube.com/watch?v=F_0yfvm0UoU https://www.youtube.com/watch?v=F_0yfvm0UoU
• bawolff 6y ago
  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 i×i=-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.
• yarg 6y ago
  It's not circular reasoning - it's bootstrapping the rational implementation atop the integer implementation.
  • yarg 6y ago
    The euclidean algorithm can compute the GCD of the factors of the naive rational form, repeated subtraction (down to zero) of the GCD from the denominator and numerator can be used to find them in their reduced forms.
• draw_down 6y ago
  Damn, this is so crazy. Repeated addition is absolutely how I intuitively understand multiplication. But then again I'm no galaxy-brain mathematician.
• musicale 6y ago
  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.
• ac42 6y ago
  Many commenters have already said it: the article is probably not helpful when teaching kids. Or is it? I recall having asked myself these questions at school: if lines are infinitely thin, how is it repeated addition that defines surface areas? What is "inside" the variables we use as units? Measure theory (for your volume) and the structure of dimensional quantities [1] are non-trivial topics, and while probably not giving good answers, I think the article is certainly asking good questions. [1] https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis https://terrytao.wordpress.com/2012/12/29/a-mathematical-for...
• Koshkin 6y ago
  In 2п, п can be though of as added to itself; on the other hand, it is not immediately obvious how to think about 2 being added to itself п times.