The article you linked argues for a change in the way multiplication is explained to children, not the way it is defined.
> Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.
I disagree. Understanding multiplication as repeated addition has always been an invaluable intuition, especially in the beginning, where explicit calculations are important. The biggest hurdle when introducing multiplication is getting them to understand the multiplication table. The fact that it is defined as a separate operation in the definition of ring/field is almost irrelevant in the pedagogical context, just as we don't start teaching real numbers with Dedekind cuts.
That feels really intuitive now, but less so for a student working their way up to it.
Even more basic is the 2.7 * 3.1 = 27 * 31 and what do I do with the decimal place question. Kids first intuition is often 83.7 because it was one from the right in the numbers they started with.
In that context pi * e exposes several different challenges to peoples mental models of multiplication. Granted most people are just going to plug it into a calculator and trust the answer without much thought, but such is life.
It's clear the multiplication as repeated addition holds for the natural numbers, which form a closed ring anyway. Pi and E are sufficiently advanced that by the time you get there you must understand that the operation being described by multiplication is not the same operation at all (depending on how one constructs the real numbers, multiplication of reals is multiplication of sets or functions)
Could not disagree more with this take. Multiplication of fractions is simply the division of two whole number multiplications. Which, if you are teaching fractions, division has already been taught. Seems like a contrarian take for the sake of being contrarian rather than based on pedagogy. Glad he wasn’t my teacher as he would have confused me.
What about irrational numbers? There's no neat way to view multiplication of two irrational numbers as repeated addition. And even if there were a way I don't think it's a useful way to think or teach after the first couple years because it makes obvious things like √2×√2 = 2 seem weird and mysterious.
Others have handled the appropriate construction of irrational numbers. The claim here is not that every student needs to know irrational numbers are a limiting sequence but this is exactly how mathematicians think of it. So it is strange to hear another mathematician claim that repeated addition is somehow lying. At some point, people kind of just accept that it is just something you punch into your calculator and don't even think about it anymore.
Your example is quite bad because sqrt(2)*sqrt(2) = sqrt(2*2) = sqrt(4) = 2. So repeated addition works fine. Let's focus instead on pi*pi. The way calculators do this is precisely as some type of limiting sequence depending on how much precision you want. Because, one cannot "calculate" pi*pi exactly because it is irrational. So, you have 3*3, then 3.1*3.1, then 3.14*3.14, etc. which are all repeated additions with some division (e.g. 314*314/(100*100)). In reality, when multiplying two irrational numbers, we just use enough decimal points for floating point precision and then chop off any potentially erroneous digits after the multiplication.*
Irrational numbers are limits of sequences of rational numbers.
Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones.
That's a pretty far departure from the original "multiplication is just repeated addition". Regardless, I don't think any student would find it helpful to hear "Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones". In my country irrational numbers are introduced two or three years before limits so you couldn't teach it in schools effectively either.
Multiplication outside of positive integers is not "repeated addition".
It took us thousands of years to properly define real numbers. High school students can live without a perfect explanation, or we can just teach limits before college since they are the fundamental concept if calculus.
Multiplication is repeated additions is the informal way of stating the distributive property of multiplication and addition.
Probably you were taught how to multiply irrational by the property of powers (a^b * c^b = (a*c)^b etc.).
You were not taught a grand unifying theory of multiplication, you were taught how to manipulate operations to turn them into more useful operations.
Teaching these laws also prepares you for when a and b are just symbolic reals with no structure and those laws are the only thing you can use to manipulate them.
You don't need a rigorous notion of limits to informally notice that irrationals have arbitrarily close rational approximations, e.g. by adding successive digits.
It's a kind of generalization of repeated addition. Once you've been taught about π, you know that 2π lies between 23 and 24, without being told about limits. It may not be rigorous, but it's a good start.
You can't teach "the truth" (whatever you hold that to be). It would set back education instead of advancing it. In this case too, perfect is the enemy of good.
I agree with your disagreement. By this standard we shouldn't teach F=ma in introductory physics, and we should require kindergartners to understand the ZFC axioms before we can tell them what "3" is.
That's actually the exact opposite of what I'm saying and the exact approach the author is saying. It is unclear what they are proposing, but it smells awfully similar to jumping to modern understanding mathematics in one shot to avoid "repeatedly lying" to students.
As others have pointed out, repeated addition as multiplication readily extends to rational numbers, then to irrational numbers as limits of rational sequences. This is exactly the progression that is taught in Rudin's analysis book and the way to construct the real numbers. At no point in time do you need to backtrack on repeated addition but you need to introduce new concepts division and limits. This is exactly teaching F=ma and then introducing relativity and quantum as the students gain more depth and break past the classical setting.
I mean, this is true for every operation once you extend it to a new domain.
What is exponentiation? is 5^6 multiplying 5 for 6 times? sure, but how about 5^(-6)? what's up with that? and 5^(1/2)? and don't get me started on 5^(2/3)
Those follow. 5^(1/2) is multiplying by 5 one-halfth of a time. It is half the operation of multiplying by 5. Applying that operation twice results in 5.
5^(2/3) is two-thirds of the operation of multiplying by 5. Applying that operation three times results in multiplying by 5 for six-thirds times, or twice, and the result is 25.
5^-6 is multiplying by 5 negative-six times. What is multiplying a negative number of times? Dividing. You divide by 5 six times.
https://www.maa.org/external_archive/devlin/devlin_06_08.htm... https://www.maa.org/external_archive/devlin/devlin_0708_08.h...