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Does anybody know what are the logical basic concepts that a baby could understand? For example regarding numbers, I think until the age of 1 they cannot distinguish between the concepts of one and many. And until the age of two they cannot distinguish between two and three.

Furthermore, what about the logical connectives, like and, or, not and more... With my daughter it seems (20 months) that there is no problem with and, but she still seem to not understand the or.

I know that at that age every single baby follows his own way. But, I know also that they learn according the external incentives.

EDIT: This edit is made after the answers of balanced mama and Valkyrie and Pip

Thank you for you answers. First of all let me apologize if my question was unclear (some trouble with English that isn't my native language, so please edit all the mistakes). If think that, regarding logical and mathematical concepts, there are three different levels of understanding. The first level is knowing the difference between number, the second is the ability of managing numbers (at different levels depending on the age). The third level is to be fascinate from the beauty of math (and in particular of numbers). Just in trying to explain the differences between this three levels, let me give one example.

If my family (me, my wife and my daughter) goes to an art museum, I would be the one who would enjoy it the least. Also if technically I'm better than my daughter (translating I'm better in level one and two), I'm not at all able to understand (especially with abstract art) the essence of a paint (I'm lower in level three).

Resuming three different levels of understanding means three different types of external incentives.

It seem to me (but I didn't already see the links) that the good answers of balanced mama and Valkyrie cover, almost completely, the first two types. But what about the last level?

Some years ago I read a book of Daniel Tammet. He is an autistic savant with particular talent in remembering numbers. What is his strategy? He associates to numbers surfaces, colours and objects. This is quite common among mathematicians: a demonstration could become a story, where the logical passages became landscapes and objects became characters.

In general I think that, starting with numbers, our brain makes, for a certain point of view,completely irrational connections. Moreover I think that our brain starts to make this connections really early.

I would like that my daughter will appreciate mathematics from this point of view. The question now is: what kind of external stimuli accomplish to this task?

Pip give me an answer. Does somebody know more about that?

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I found a couple of links that might be useful based on your update; they're in my answer. –  Valkyrie Nov 26 '13 at 18:55

4 Answers 4

up vote 8 down vote accepted

Because you care about mathematical concepts and your daughter learning them, most likely she will learn them. You will point them out and talk about them. "see honey, you had one slice of banana, now, you have more slices of banana." You will be drawn to stories that contain math concepts (yes, they are out there) and games that teach mathematical concepts. When your daughter picks up a new concept well enough to apply it, you will notice and celebrate her accomplishment with her. All of these things are huge factors in helping her learn, but they happen more naturally than you might realize when exposure is occuring.

A baby's learning will be very concrete in orientation, as opposed to abstract as she is unlikely to be developmentally ready for abstraction yet. "and" is much more concrete than "or" so she is following standard patterns of growth in understanding well so far.

To help your baby learn the multitude of things she can start learning now, I suggest exposure without pressure Think of it this way, learning takes place because a child is exposed to stimuli, recognizes predictable patterns in that stimuli and responds accordingly. Babies are doing a lot of experimenting. "Last time I dropped this cup, mommy came over and picked it up. Will that happen again?" Now, it isn't quite that conscious in their little brains, but that is essentially how it works and why the do so many things over and over and over again. The key is to expose and then be okay with just letting the experience/idea/concept marinate in her brain and wait it out. Exposure allows her to use the concept and/or vocabulary as soon as she is developmentally ready for it, rather than being ready for it and then having to gain the exposure.

Ways you can support your child's learning in your home include having things to count, talking about "a little" and "a lot/many" choosing story books to read that contain mathematical concepts, noticing sequences and patterns and pointing those out, comparing sizes (play games like lining up cups of assorted sizes in order from biggest to smallest) doing a lot of "sorting" (by shape, color, or size) and working with cause and effect - "Oh did you hear the loud noise the plate made when you dropped it? Will the same thing happen if you drop the plate again?" and generally encouraging experimentation and manipulation of objects.

I read Born on a Blue Day by the way! In regard to how your daughter's brain percieves numbers and whether or not she develops a passion for it and sees beauty in those numbers the way Daniel Tammet describes - I think that is one of those parts of the equation that is more nature than nurture or at least, unknown.

Having said that, the brain is elastic and mold-able - especially during these younger days (where your daughter is). Studies have shown that the structure of the brain of a child who learns music before five, is quite different than that of a child who does not (I'm sorry, I don't remember the reference to cite). This info is important to you because kids that learn to read and play music, also tend to do better in math and languages. Likewise, kids with tons of early exposure to written language and tons of stories at an early age, seem to be better wired for academic learning in general and tend to live more enriched lives down the road (partially because they have higher income - causality is difficult here).

I taught kids like Daniel Tammet would have been during his younger years if he had been diagnosed more accurately sooner. The school where I taught did a lot of in-house training and brought in researchers to do presentations etc. In my experience, we simply don't know a lot about the brain yet - let alone how learning takes place. What we have in knowledge in these two areas is increasing in leaps and bounds in the last twenty/thirty years or so, but the science is in its infancy so anything that one could do is still really in earliest theory stages (meaning they are little more than hypotheses at this point.

My suggestion

  • Continue to love and be passionate about math and logic yourself - passions of the parents often rub off on the kids even when the parent tries hard not to push. That first paragraph in this incredibly long answer still applies here.
  • Help your daughter explore everything you can allow her to explore - include music and storytelling as a component of every single day. Mathematic concepts show themselves in the most un-expected places sometimes so if she is getting exposure to a wide variety of things, she has more opportunities to make more connections logical or not between mathematics and the tangible world.
  • Talk about how you see numbers and their relationships with her - some of it will rub off on her.
  • Don't be disappointed if your daughter doesn't see math and its language the way you do, she has a loving father and she will find beauty and passion - even if it isn't in numbers. Isn't the most important thing that she feel fulfilled?

http://ponce.inter.edu/cai/tesis/lmrivera/cap2.htm goes over learning stages of cognitive development as represented by Piaget's theory (called constuctivism) and the psychological elements used within the process of learning.

http://www.assessingmathconcepts.com/criticallearningphases.html has a downloadable page that lists a lot of the skills teachers are looking at and measuring in Kindergarten upon a child's arrival at school - while I understand this is not the right "age" for this question, it will offer insight into what level is generally aimed for to be successful once it is time to start attending school.

Finally, http://www.education.com/magazine/article/preschool-math/ gives ideas on continuing to support the growth of mathematcal learning and conceptualization in your home as your baby moves from infancy into toddler-hood.

List of books for early elementary ages that present mathematical concepts and themes.

Hope this helps.

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I will take a look at your links. If you are interested I also added an edit to the answer. –  amorvincomni Nov 26 '13 at 14:00
    
Thanks you for the suggestions. Me and my wife are fair to be stressing (and stressed) parents. As many parents does, I would like to teach my passions to my daughter. But I also know that I have to not be oppressive and in general I always try to be enthusiastic of her little and at the same time great new discovers! I will take in exam what you had suggested to me! –  amorvincomni Nov 26 '13 at 15:14

This may seem counterintuitive, but babies and small children will do the best at math if they are told stories and learn to tell stories themselves. O'Neille et al. (below) found that storytelling is an essential precursor for the development of logical thinking. This makes sense when you think about the fact that storytelling is ingrained in centuries of human neuro-development and oral tradition. Exposure to storytelling helps young kids' brains learn to abstract information and think sequentially, which is an essential basis for counting and logical operations. In general, many behaviors which look like play (such as pretending and making up stories) are actually part of laying essential ground-blocks for academic skills.

See O'Neill, D., M. Pearce, and J. Pick. (2004). "Predictive Relations Between Aspects of Preschool Children's Narratives and Performance on the Peabody Individualized Achievement Test-Revised: Evidence of a Relation Between Early Narrative and Later Mathematical Ability." First Language 24 (June): 149-183.

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This is what I'm looking for. See the self answer below –  amorvincomni Nov 26 '13 at 8:41
    
Sorry, I turn the self-answer in a edit. –  amorvincomni Nov 26 '13 at 13:57

I'm not convinced there's an age at which kids CANNOT learn something. You certainly need the basic building blocks before you can start constructing a tower, but at any age they seem to have the ability to learn all sorts of mathematical concepts. Verbalizing them is something entirely different, of course.

First, to directly address your question about concepts and ages, some links to some related research:

National Academy of Sciences

LiveScience: Infant Math Ability

Education.com: What Mathematical Concepts Do Infants and Toddlers Learn?

HeadStart: Supporting Early Math Learning for Infants and Toddlers

All I can offer, besides some links to research, is anecdata. My daughter (who admittedly has a Klieg light in her head) was able to count to 70 at 17 months, and understood the differences between 1 and many several months before that, and understood the concept of 0 at the age of 31 months. At 5.5 she's able to discuss the differences between nulls and zero with me as she watches me work (I'm a data guru and spend a lot of time architecting and implementing designs, so math is integral to my profession), and has groked negative numbers and has the basics of multiplication down pat (memorizing the multiplication tables, not so much). In the car last Friday, we discussed the difference between 1-to-n relationships and n-to-n relationships, in the context of tornadoes (tornadoes are always spawned by a storm like a thunderstorm or a hurricane, but not all thunderstorms and hurricanes spawn tornadoes).

My son is following a similar trajectory, except for the counting to 70 (he skipped that because he is not nearly as verbally advanced as his sister, although he's demonstrated a clear understanding of the other concepts). The hunger to learn these concepts is almost entirely within them, too; we have made a point to demonstrate how math is everywhere in our lives, but never pushed them to chase down topics and consume them with the vigor they've displayed.

Update:

Based on the OP's update, I've found some links to research about how to encourage mathematical visualization in small children, and the data behind these theories:

Mathematics Learning In Early Childhood

Number Sense Series (a site for STEM (Science, Technology, Engineering and Math) teachers

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I think teachers and some homeschoolers are now starting to teach math earlier and earlier because when the kids are younger, they have a tremendous and almost instinctual ability to pick up concepts. My kids started showing signs of understanding rather early and I helped them (without overwhelming them with too much information) by allowing them to download math apps for kids or letting them play cool math games (son calls them cool). There are tonnes of resources online that you would find helpful. Since you know your daughter best, you will be best able to gauge what is best for her depending on her learning style.

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