What is the best way to teach the standard arithmetic operations ($+ - \times \div$) to a 3 year old child?
Also: How we do know if the child has really understood it?
can a kid be 3 years old prodigy?
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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 age, and my guess is that abstraction is a fairly similar ability.
Maria Montessori pioneered the study of childhood education, and the materials she developed for mathematics remain state-of-the-art. In this article the author encourages taking an actual lesson from a Montessori-trained teacher, which I would also encourage. I participated in an event called "The Journey" at a local Montessori school where parents were invited to witness how children are taught the materials from ages 3-14. Some parents literally wept as they (1) understood certain mathematical concepts, such as the binomial cube, for the first time, and (2) understood that their children were developing an understanding of math that they never had.
The original text of the whole method provides a context for understanding the specifics of the teaching of numeration, which consists of 11 pages. I realize that the rules of a place like math.stackexchange.com frown on citing by reference. I'm not sure that math educators are greatly served by too much cutting and pasting of original literature, nor by the casual attempt to rephrase such in the interest of brevity. But here goes...
Maria Montessori observed that ”all children pass through 'sensitive periods' that stimulate them to carry out certain activities and to acquire certain experiences.” Crucially, she observed and understood that when children followed these inner directives, which characterize sensitive periods, the child is able to structure their development in relation their environment, which is the essence of learning. She understood that efforts of a teacher to either pour knowledge into "an absorbent mind" (or worse, to force it) works not only against nature, but also against the child's own sense of self. Montessori's genius was in reporting that it is through the child’s own activity that development transpires. In the literature, sensitive periods have four basic characteristics:
The construction of a Montessori environment is all about optimizing development according to sensitive periods. To wit: the environment must contain all the necessary elements that pertain to a given sensitive period, and the child must have access to the elements as well as the agency to engage the elements in the environment. Montessori wrote, “If the child lacks suitable external means, he will never be able to make use of the great energies with which nature has endowed him. He experiences the instinctive urge toward activities worthy of engaging all his energy, because this is the way nature incites him to perfect the acquisitions of his faculties. But if there is nothing there to satisfy the impulse, what can the child do but dissipate his energies aimlessly, with disorderly unruliness?”
The proper construction of a Montessori environment is the subject of many books, but suffice it to say that the materials for teaching mathematics are specifically designed to engage the sensitive periods related to numeracy, not as "this is first grade, this is second grade, this is third grade" but rather as holistic elements which can be understood with increasing levels of sophistication, starting at age 3 and continuing into middle school. Moreover, the environment is child-centered, meaning that it affords the child a choice in the materials they engage, and a liberally allotted work cycle in which to explore. The child is not constantly interrupted by a classroom bell as is so common in conventional classrooms.
The first step is to introduce the concepts of numbers as tangible things that can be counted (including zero), typically done with "golden beads". The second step is to engage the child in simple activities that have mathematically observable consequences, such as making change. This gets you to + and -. The third step is to use didactic materials, such as sets of rods with lengths 1 to 10 (and zero if you use your imagination), one set being red and one set being blue. These rods can confirm the understanding of addition and subtraction as the child plays with all possible ways to add numbers 1 to 9 to get 10. Then the child can try to construct a series of numbers by using two rods to do so. The child may discover that 1+1 = 2, 2+2 = 4, 3+3 = 6, 4+4 = 8, 5+5 = 10, and may thus learn the concept of "two things of one kind equal one thing of another kind". This unlocks * and /.
Most importantly, it is the child who pushes forward and asks the questions, which can happen when they enjoy an environment that is tuned to their sensitive periods.
For my own child, who was counting past 20 at the age of 2, I have found that visual learning and using games has worked well. The ideas of addition and subtraction are simple to introduce by using familiar objects that can be counted (e.g. marbles, toys, fruit, etc). At the age of three, we would practice questions like "how many will you have left if I take away this many?" or "how many will you have if we put these two piles together?". When first learning these concepts, children will need to re-count the piles of objects to determine the new amount. This is not really adding/subtracting in a typical sense, but it helps a new learner understand the concepts and verify the answers. Eventually, with practice, a 3 year old can visualize the answer in their head without having to move pieces around. Though, we stick with numbers below 20.
Multiplication and division are not so simple. In order to develop this skill, we have been counting pairs/matches when we do memory games. The concept of division makes sense to our (now) 4 year old (e.g. she knows that if she has 10 cards, when she splits them into pairs, she has 5 piles). However, multiplication has not interested her, and she does not yet understand multiplication even in basic form. We have been working with laying out the cards to play the memory matching game in different grids (e.g. 4 x 6, 5 x 3), but she sees no benefit to thinking about it any other way than just "counting". Me feeling is that in order for multiplication to make much sense to her, she will need to be able to count by 2s, 3s, etc., which she can do a little. But, in general, I feel the real complication is finding an activity that is both interesting and educational. In general, if it helps her win the game (or determine if she has won), she will try to learn the concept. So, I would like to find a children's game that could put multiplication to use. I've read here: Mathematical problems for preschoolers about the use of legos (which she loves) so we will explore that idea.
Assuming the kid can count, and also understands that the sequence of utterances 'one', 'two', ... corresponds to amounts of stuff, rather than being rote, then I'd say there are lots of ways to talk about these things age appropriately.
If I take three of these apples, how many will be left?
The cow has four legs! How about two cows? Three cows?
You and Mom and I are going to share these cookies. How many do we each get? ... But that's not fair! Mom has more than me! Make it fair!