54

This does not directly concern the $\infty+1=\infty$ issue and I am not certain that I understand what you mean by his previous understanding of mathematics, but I wanted to give the following suggestion: Ask your child to name the biggest number he knows (besides $\infty$). (Let's say he answers $1000$); Tell him to add $1$ to it; Ask him again what is the ...


27

I think that actually trying to get students at this age to contemplate infinities in a rigorous way is probably ill advised. I do think that exploring counting from both an "ordinal" and a "cardinal" point of view is probably a good idea. Example for a 5 year old: Something you could do is have 20 stuffed animals, only 18 of them wearing hats. You can ...


25

First of all, regardless of age, people need to understand that "infinity" is not a number, and not a placeholder for a number, but an attribute of them (i.e. the fact that you can increase numbers without ever getting to an end). For my children, the concept somehow came into their mind all alone due to the book "Guess How Much I Love You" by Sam McBratney....


18

I'm not sure why the two basic things adults seem to say about infinity are "infinity is not a number" and "∞+1=∞", both of which are at best misleading. (Infinity doesn't name a number, but it does refer to a property some numbers can have. ∞+1 is nonsense, $\aleph_0+1=\aleph_0$, and $\omega+1\neq\omega$.) The problem with talking about infinity with ...


18

On a piece of paper, he started with writing 10, then 100, then 1000, .... and he stopped after writing 40 zeros with 1. Then he came to me and said, "I understand infinity now; infinity is a number with infinite zeros." The main point is that as most of you suggested, he has now registered infinity in his brain as a concept rather than a number, which is ...


15

Speaking as someone who was that kid, you might be able to explain $\infty + 1 = \infty$ via the Hilbert hotel. Imagine a hotel that has an infinite number of rooms, one for every number. Imagine the hotel's full, and another guest shows up. You can make room for that guest by having the guest in room 1 move to room 2, the guest in room 2 move to room 3, ...


12

You can easily make them draw $\aleph$s; however the rest is much more demanding. There is a nice analysis from a researcher group taking a constructivist perspective. They distinguish potential infinity from actual infinity. The first one is covered by the idea of "counting on forever" the second one needs infinitely many things to exist in your thinking at ...


10

Lillian R. Lieber Author of Infinity: Beyond the Beyond the Beyond


9

Consider a Sumerian person, living around 2500BC, who owns a flock. She hires shepherds to take the flock to pasture. Not being a scribe, she does not know Sumerian numbers- instead, for each sheep that passes her, she places a stone in an urn. When the shepherd returns, for each sheep that comes back, she removes a stone from the urn. If the last sheep ...


8

My son, also 6 yo, regularly talks about millions and billions and infinity. Obviously, large numbers have some attraction to children of this age. I try to explain that infinity is not a number. Instead, infinity is an order of magnitude which has its own algebraic rules. Plus, minus, divison and multiplication do not work the way children learn in ...


7

This has certainly been tried before. See for example, H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. On-line Edition. This has also been published in print by Dover. Kathleen Sullivan. The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. The American Mathematical Monthly Vol. 83, No. 5 (May, 1976), pp. 370-...


6

This is what I came up with thinking about your question. I would start by exploring what it means to say that a line is 'made up of' points, because I think that is a really important thing that doesn't get taught in schools. Students know 'the equation of a line', but usually have no real concept of what that means. Admittedly, that will be somewhat ...


6

I am going to answer your question by suggesting a couple of books which might be fun to read with your son: The Phantom Tollbooth by Norton Juster. The book is a rather surreal adventure trip through a Wonderland-style setting populated by mad grammarians and mathemagical wizards (among others). There is a section somewhere in the middle where the ...


5

My understanding of this question is that it proposes the idea of a "monolingual" freshman calc course in which students mostly learn the language of NSA, and limits are largely or completely neglected. This makes it different from this question by Mikhail Katz, which asks whether it's a good idea for students to be "bilingual." I have some experience ...


5

Speaking as a former student, though an engineering one . . . It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system as well. And I don't remember ever needing the rigorous definition of a limit, standard or nonstandard. The most I needed even in complex calculus was an ...


5

Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the $x$-axis at $\frac{p}{q}$ with radius $\frac{1}{2q^2}$ where we always assume $p$ and $q$ are relatively prime. This gives an intriguing family of circles with special tangency ...


4

I would expect children can understand the idea that there are as many even numbers as there are natural numbers, as long as it's presented in a lively style, asking them questions and drawing pictures. How many numbers are there? Infinity. (Write the first few naturals) Can you tell me what an even number is? What are the first few? (Write them ...


4

There is a well-known Christian hymn, Amazing Grace, whose last lyric captures the idea of (countable) infinity quite well, and may be more effective to a five year old because it includes a context in which the notion of infinity can be applied. The lyrics goes: When we’ve been there ten thousand years, Bright shining as the sun, We’ve no less days to ...


3

I suppose one problem is that your son looks at $\infty$ the same way he looks at $10$. But infinity is not a natural or real number, even though it has a symbol and can be used in "equations" like $\infty + 1 = \infty$. These equations do not have the same meaning and do not follow the same rules as with "normal" numbers — the reason being that at ...


3

My children both learned about infinity at around four to five years old (now 5 and 7). For both of them it was fairly straightforward; it came about with my eldest when he was talking to other kids at school about the biggest number. We talked about trillion, quadrillion, etc.; as they were at a Montessori, it was easy to understand these. Then we talked ...


3

I agree with the premise, lines are (in some way) made out of points, and points have no length. If, restricting ourselves to a straight line, we can consider these as subsets of the real numbers $\Bbb R$. A line is a subset of this with the property, if $a, b \in \Bbb L$ then any real number between $a, b$ is in $\Bbb L$. Calling elements of a ...


3

It's not really that relevant since the bulk of a normal calculus course (e.g. AP BC, Thomas Finney, Stewart) just does a small amount of epsilon-delta (so student is exposed to it) and then moves to "x+h". The bulk of the course is about learning derivatives, antiderivatives, methods of integration, classic applied problems, a bit if polar coordinates, bit ...


3

Los Alamos National Laboratory wrote a nice lesson plan in the mid-90s around an embellished version of the Hilbert Hotel story, including the alignment to the 1989 NCTM standards: Hotel Infinity (you'll need to click around, as the interface is very 90s...) http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html If you prefer a youtube video (...


3

Power sets, maybe? I was just pondering this because of course preschoolers don't understand decimals.


3

I would start by saying something along the following lines... "You're asking some very grown-up questions for someone that's only 5. Are you ready to do some really, really, grown-up thinking about the answers?" He will of course, answer, 'yes'. I would respond... 'Ok, but this is serious stuff. You need to be ready to take this thinking very ...


3

Is the distinction between cardinal numbers and ordinal numbers taught as part of mathematics (as opposed to part of learning the language distinction between "one" and "first") in pre-college or early-college math instruction outside of set theory? If so, where is it used? Where is it needed? I'm not aware of any early-college courses in which the ...


3

I don't know if this is always in a set theory course or not, but it makes a difference when you define the natural numbers using the Peano axioms (which is ordinal) or the Frege construction (which is cardinal).


2

One can give them some intuition. Draw a line segment of length 'one'. Ask them how long it is. Then draw a single point. Ask them how much room it takes up (the answer is zero). Draw two points; how much room do they take up? (Zero) Keep drawing poiints one by one. Say, 'No matter how far I go, even if I go forever, to normal infinity, they take up no ...


2

To get playing with one-to-one correspondence, The Cat in Numberland, by Ivar Ekeland, is marvelous. Of course that only helps with ℵ0, but it's a step in the right direction. I wonder if there's a way to tell a story about ℵ1...


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