Dear Mr. George Stoica and Mr. Sudev Naduvath I am very thankful for your answers.

I started examining some of these properties in the following way. The curriculum for the first grade in Berlin school is for N. till 20, for the second grade till 100, third grade till 1000. I got an impression that some kids in the first grade think that 20 is the largest number of all, and similarly think the second graders for the 100.  Does curriculum maybe "prevent" their intuition development? Similarly, one five-grader denied the existence of negative integers, although she had information of such, just because they were not treated in school. She said her grandma has explained that, but "we haven't studied that in school" (part of an interview). On the other way, she knew that there exist infinitely many numbers. So my question is: How can we research pupils' intuition of the properties (the Peano's axioms), without their explicit statements, without precise axiomatic definition?

I started this as a case study, but I'm not sure if I'm on the right way. I would be very thankful to hear your opinion.

Your observation is interesting.

Maybe it is unwise to stop the learned number sets at 10,20,100 etc..

If we stopped instead at 11, 21, 101 etc.. learning facilitators would not misguide their students into believing that there is a highest or even just special number..

"Stopping" at e.g. 21 may tempt them to ask questions or create expectations and their own theories how numbers may work.

I suspect the (false) concept of a highest number may well be acquired by misleading educational methods.

The naturally highest number seems to be "many" rather than 20.

1,2, many... just like in many tribal languages or "kids language".

Dear Mr. Stoica, that's exactly what I'm talking about. What is a "good time"? Why schould some pupils wait for the next school year to come in order to inspect something that their intuition sasuggests. This "good time" may be too late (or too soon in many cases) according to curent ways of framing math educ in the curricula.

Mature people would not certainly deny the existence of negative numbers, I have no doubts in that.

Dear Mr. Petritsch, your idea seems interesting, especially "Stopping" at e.g. 21 may tempt them to ask questions or create expectations and their own theories how numbers may work." That's what we want for our children, to increase their couriosity and interest for maths. But, artifitial fraiming mathematical knowledge and presenting it to children when we think it is the most apropriate, may "harm" it. I think that even teachers sometimes forget such important issues. They often don't have time for investing more in conceptual knowledge, rather in pure arithmetic.

Do you know if this idea has already been tested or researched. Do you maybe have some recourses?

I am not aware of any formal studies of this idea. All I know is that it worked with my three children from what I can tell.

Whenever we used numbers I tried not to stop at the "classics" 10,20,100 etc...

My seven year old today told me that there is no highest number. He just entered first grade here in Finland.

My five year old could counted pretty well past101.. already many months ago. I think she is also aware that there is no end.

I think we grown-ups introduce the concept of numbers to them in the first place. Depending how carefully we do this, they will develop their understanding accordingly.

By the way, if a child (or grown up) claims negative numbers do not exist, I would say this can be a sign of rather high level thinking ability. It depends on their arguments though.

That is great to hear. Parents' influence is of a tremendous importance.

My daughter is 7 and a half and already starting third grade. I once heard her competing with her classmates on who can tell a larger number and it attracted my attention. Some were saying unexciting "numbers"-their own creations of number words they have heard somewhere. When she said, there is no largest numbers, they unexpectedly changed the game and ended up in drawing. I regret I didn't record the conversation. Maybe it could have been a good resource for further investigations.

Regarding negative numbers: I agree with you, but in the previously mentioned case study, that was not the case.

Thank you very much Mr. Petritsch for sharing.

My first position after graduating college was being a teacher with Project SEED.  We taught advanced mathematics to underprivileged children.  Our goal was to raise 5th to 7th grade students' academic self-confidence by teaching them math differently.  We taught them a simplified and visual style of proof and had them learn to invent negative numbers, fractions, and eventually imaginary numbers with a mathematical approach much like Peano Arithmetic.  I taught in an inner city project of Dallas and my students got it!  I agree with the prior responses that most school children do not learn to think in the deeper mathematical way that you're asking about.  How unfortunate.  Most college professors (outside mathematics) don't appreciate axiomatic systems either.  I have adapted Project SEED's style of proof to illustrate mathematical axiomatic systems to even graduate students and professors (covering years of Project SEED curriculum in an hour).

My first teaching experience still shapes the way I educate, no matter what subject or what level I'm teaching.  Why don't students understand the deeper level?  It's not that students lack the ability.  It's that *we*, the teachers, have lacked the ability and creativity to figure out how to start from where our students are, and take them to the deeper concepts.

http://projectseed.org/

Hi Kevin.

I took a look at the "sample curriculum topics by grade level" in the frame of the project SEED which you have suggested. It seems to me that the topics are defined from a higher point of view, for example, "Group properties of integers and rational numbers" in grade 4. Though it may look too abstract for non-mathematicians, I agree that we should not forget mathematics with all of its complexity, along the way of searching new teaching approaches.

Thank you very much for sharing your experience!

There is an ongoing international study on that topic lead by ICMI. You might find something interesting in the proceedings of the conference already done:

http://www.umac.mo/fed/ICMI23/proceedings.html

There is going to be a book published by Springer on that topic but it will take some time before it comes out.

Dear Mr. Silva,

I am planning on writing a manuscript on this topic exactly for the ICME13 in Hamburg 2016.

Thank you very much for sharing the conference proceedings. I think that they are very valuable recourse for me at the moment. I am immediately starting reading :)

Hope to meet you in Hamburg next year.

Yes, ICME-13 is a huge forum for mathematics education, and I hope to be there also. There is one Topic Study Group around the topic that interests you:

TSG 8 Teaching and learning of arithmetic and number systems (focus on primary education)

http://www.icme13.org/topic_study_groups

Yes, that is the one I have on my mind!

Although I was also thinking about TGS 32 Maths education in a multilingual and multicultural environment (because of my experience, for ex. https://www.researchgate.net/publication/281399396_Verbal_Counting_in_Bilingual_Contexts ).

Maybe you would like to read my writing before the submission and provide some feedback!?

Thank you!

Article Verbal Counting in Bilingual Contexts

A Philosophy of Learning Arithmetic Towards an Evaluation of Teaching Methods

http://dare.uva.nl/cgi/arno/show.cgi?fid=527166

Dear colleagues and friends participating in the discussion,

I warmly invite you to the oral presentation of the paper IS THERE A LARGEST NUMBER OF ALL? PUPILS’ REFLECTIONS ON NATURAL NUMBERS, at the ICME 13, Topic study Group 8. We may continue our discussion in Hamburg :)