# Teaching arithmetic operations ($+ - \times \div$) to a 3 year old

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?

• Can the kid count at all? Because if he / she doesn't know what "nine" would mean in terms of, say, a pile of toys, then trying to teach addition would be difficult, to say the least.
– Arthur
Nov 21 '15 at 10:49
• yes its difficult to know when kid really started to understand that he has one sister or one brother. Dont know how we start realizing this counting concept. i think this counting is hardwired in our brains Nov 21 '15 at 10:51
• Here's my story in a related thread. As the migrating moderator of Math.SE I apologize if this is not an appropriate question at ME.SE. I am not active enough here to be able to judge. Pointers to relevant discussion in ME.SE are, of course, welcome. Nov 25 '15 at 7:49
• What information can you provide about the child's current mathematical thinking? Two concrete example questions: What is the biggest number the child can count to? Biggest the child can count by (e.g., 7s: 7, 14, 21, ...)? Here is a post about counting (MESE 5866). How does the child do on the five principles excerpted there? And so forth... Nov 25 '15 at 8:43

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.

• How does this answer the question? Nov 28 '15 at 7:46
• @farrenthorpe The question was how best to teach a 3-year-old arithmetic. My answer is 'don't teach it at all'. A small number of children will be ready (answering the newly added question: yes a child can be a prodigy at 3, but very few are), and they will probably pick up things without formal instruction. The vast majority of children will not be ready at that age, and trying to teach them something they don't need to know ignores the more important things they need to learn about. Nov 28 '15 at 9:30
• @farrenthorpe I see from your answer that you've had one of the exception kids. Nov 28 '15 at 9:33
• @farrenthorpe I guess you speak a different version of English to me. Where I come from, 'don't' is a valid answer to 'how do I...', unless you're being deliberately antagonistic. Nov 29 '15 at 7:56
• Nov 29 '15 at 12:37

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:

1. They provide the intrinsic motivation for a specific aspect of human development. These human acquisitions relate to order, sensory perception, language, and movement--particularly of the hand and walking upright. The result is human intelligence.
2. Internal mental growth precedes external manifestation of activity. Organizational brain activity takes place prior any visible expression of the sensitive period.
3. They are unconscious; the child is completely unaware of them. Not only is the child unaware of why he wants to do a particular activity; adults cannot directly influence the onset of sensitive periods.
4. They are temporary windows for optimal development. After the sensitive period has passed, the particular acquisition is much more difficult, if it is possible at all.

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.

• this answer seems highly biased. I know of many students who have come to me from montessori school's who have below average mathematical understanding Nov 25 '15 at 13:04
• Your observation could be highly biased. Namely, children who come to you from Montessori schools do so because parents know that the school is failing the child. Alas, it is difficult to judge whether that's a valid statistical strike against Maria Montessori's methods as described in her 1912 paper, or the school's actual ability to practice the methods as described in the paper. During the seven years in which I was on the board of a local Montessori school, the children consistently achieved 2.5 standard deviations above statewide averages for both language and math skills. Nov 25 '15 at 14:47
• I have made some edits to make the answer more StackExchange-y. Nov 25 '15 at 22:25
• I guess that the concern of @celeriko is that the "article" you link to is entitled Everyone Should Learn Math the Montessori Way from mariamontessori[dot]com's Montessori Blog. This does not invalidate the content, but it is not an ideal source. Nov 25 '15 at 22:59

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!