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 ...


35

I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful. The issue here (whether the category "rectangles" includes or excludes the category "squares") is one aspect of a larger question having to do with whether the classification of quadrilaterals ...


33

I don't think there's anything wrong with the wording; it's clear what is being asked. Your example with the three dollars is also not always the way we speak in everyday language. If you ask someone with three children if they have two children, they're unlikely to say "yes" and leave it at that. Getting more silly, a bicycle isn't a unicycle ...


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....


20

I'm nearly sure I did this with my child when she was young. First, establish that she understands that a number, like three, is equal to $1+1+1$. Hold three fingers up and ask her "how many is this"? Then spread them out and ask the same question. Are we adding $1+1+1$? Try holding up eight fingers (keep your thumbs down, for example) and ask her to ...


20

When we describe counts in natural language, there's almost always an implicit "exactly" when phrasing like this. We use phrases like "at least 4" when we want a more general description. Most children who have reached a development level where this quiz would be reasonable will probably already have learned this. In fact, this is why ...


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 ...


17

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 ...


16

Perhaps "shows" instead of "has". If you asked me to show you 4 apples, I can't think of a logical argument in favor of me grabbing 5 apples and smiling smugly.


15

Kindergartners are generally at an early stage of geometric development, in which shapes are recognized by how well they resemble prototypical images, rather than by whether or not they conform to a definition. Thus, for example, the shape on the left below is likely to be recognized as a "triangle" (despite the fact that it has four sides), the shape in ...


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

My immediate response is 'wait a few years'. I've spent a fair amount of time with 3 year olds, and most of them are busy learning how to be a person in their own right, how to have a conversation, what the difference is between real and make-believe, and (often) how to tell when they need the toilet. I've read that they can't understand metaphors by that ...


11

Nearly every test like this includes instructions to choose the "best answer" to cover exactly this scenario. This looks like it's part of a test of basic counting skills, and in that context, the best answer is B. While one could make an argument for either C or D, I can't imagine an argument for either of those being the best answer when B is ...


10

To supplement the "games not problems" answer above (I love that answer): Certain games provide interesting opportunities for discovery...even if you don't know anything about the rules! An example is Tantrix. The tiles feel good in one's hands, and most preschoolers will spontaneously start tiling if given some of these. If the tiling continues, ...


10

Edit (May 2016): From The Atlantic is: Boaler, J. & Chen, L. "Why Kids Should Use Their Fingers in Math Class." Apr 2016. Link. "Evidence from brain science suggests that far from being “babyish,” the technique is essential for mathematical achievement." (See the link for more!) Counting is one of the most important activities to engage in with ...


9

Addition/subtraction by measurement I'm not sure whether this is at all what you had in mind, but here comes anyway. When I was something like 2-3 years old my Dad taught me to add and subtract single digit numbers by sawing (and sanding for safety) pieces of wood of dimensions about half an inch x $2$ inches x $2n$ inches, where $n$ is the integer to be ...


9

There is a model of how people progress towards abstract reasoning through the subject of geometry called the Van Hiele model. The model describes five levels: visualization, analysis, abstraction, deduction, rigor. It dates to the 1950's, and continues to influence curricula. In the analysis level, children do not allow overlaps in categories, and will ...


8

Not formally an answer (so if you have one, do not let this prevent you to post it), but I think challenging the implicit assumption that it would be good to have your kid use any such software deserves more than a comment. First, I remember that several studies failed to prove any improvement on teaching math using computer technologies. This lets me ...


8

A good math curriculum to check out for Pre-K to K is Big Math for Little Kids. You can find some information about it in this interview with co-developer Herb Ginsburg. Ginsburg is also working on software that will include problems for the range you have specified. In particular, he and others (including doctoral students at Teachers College Columbia ...


8

Another great game is Rush hour. It requires the important but in my opinion underemphasized skills of nonverbal problem solving, working backwards and trying all possible options.


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 ...


8

I showed this question to my three-year old son. His response - because he counted the apples one by one in each picture, passing "4" each time - was B, C and D. Hence, we need to take into account how children arrive at their conclusion, since they do not apply formal logic. The thought process is very different from the abstract approach a ...


7

I think Set is a great commercial game. There's a daily instance of the game that can be played at the same website I just linked to.


6

(Migrated from the comments.) Here are three specific suggestions: Teachley (company page) Tiggly (company page) Bedtime Math (wikipage) I mentioned the first two in an earlier answer to MESE 750; I became aware of these two companies because the founders (all three at #1; one at #2) worked with my graduate advisor (Herb Ginsburg, whose expertise is on ...


6

Symmetry Suppose the blue line is a mirror, how the figure would look like in it (i.e. put the blocks at the right side to recreate the left side)? The purpose of this problem is to train pattern recognition. There are multiple follow-up challenges (point-symmetry, scaling, etc.).


6

Odd one out There is a bird, a plane, a square with a hole and a cloud, which one does not fit? The purpose of this problem is to highlight that explanations are as important as answers. In fact, any object can be singled out: the bird, because it's the only animal; the plane, because it's the only mechanical object; the square, because it's ...


6

I was working with a student (high school level, algebra) and she blushed when she realized that she was counting on her fingers in front of me. She quickly told me her friends made fun of her. "35 years ago, my classmates poked fun that I counted on my fingers. Then we took our SATs, and I aced the math portion, an 800. They stopped laughing." I'm a bit ...


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 ...


6

Blocks work well for thinking about addition. Have her count out 8 blocks, and then ask her about all the addition problems that have 8 blocks as the answer. A lovely children's book which looks at all the sum pairs for 7 is Quack and Count, by Keith Baker. (You can buy it used here.) It has luscious pictures, a driving rhythm, and a lovely storyline. (“...


Only top voted, non community-wiki answers of a minimum length are eligible