My kids Cindy (5 1/2, in Kindergarden) and Jamie (4, in pre-K) are taking some math enrichment classes. I have been telling and teaching them to count items and do single-digit arithmetic in their head, hoping to train their brains rather than working the problems mechanically.

Below is the verbatim evaluation I received from their tutors.

Although I fully believe that people (not just kids!) should be able to do arithmetic in their head, I am wondering if I should wait a few more years.

"Cindy is very good at math, especially being able to visualize an answer. However, I'd like to get her to a point where she can calculate the answer before guessing so that she has the absolute correct answer. We went over simple addition after this set and I had her practice counting on her fingers. She did extremely well with this and got the hang of it right away."

"James was required to add more objects to a given amount. He did very well but it didn't seem as if he had any strategy to solve each problem. When I asked him why he chose a particular answer he replied, "because you said that." We pertained using our finger to help him find the answer. It was hard to determine whether Jamie was doing the work on his head and couldn't verbalize it or if his guessed were" lucky guesses.

• i see nothing wrong with doing math on your fingers. I am a high school math teacher and I still will occasionally use my fingers to keep track of numbers. It gives you immediate visualization of the problem you are trying and it also makes the problem tactile, which is especially helpful for younger learners. While mental math can be useful in a pinch, nothing beats physically displaying numbers (whether with fingers, manipulatives, or just writing it down on a piece of paper). Let your kids use their fingers and I guarantee they will develop the skills needed to do math in their head – celeriko Nov 17 '14 at 19:23
• Was James consistently getting the answers correct? If so, it is possible that he could add or multiply, but had not memorized the names of most of the numbers between hrair and the answer. – Jasper Nov 22 '14 at 19:05

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 anti-calculator, for multiple reasons, but as far as fingers go, I think it's a great way to avoid simple errors, and after adding 5 to 7, counting 8,9,10,11,12 on your hand, it eventually sinks in. It's when student use the calculator to perform even simpler math, that I worry. The finger math has a time honored tradition going back to the abacus.

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."

Counting is one of the most important activities to engage in with young children; it is also one of the key precursors for developing number sense later on. If you are looking to help young children with their mathematical development, then I would say that the single best choice is to count with them (another important activity is to practice using words meaningfully, e.g., identifying different shapes). If you are wondering about how to help them count, the standard approach is to begin with the number chant (it is pretty tough till around 13, and then gets a bit easier in the later teens, and then goes rapidly from 20 on); next, move on to skip counting (by 2s, by 5s, by 10s); and counting in reverse order.

As to the question here, there is a lot written about the subject. For example, Gelman writes here:

The real reason to post such an excerpt is to indicate how complex a task this is!

You can find some good work on counting by searching for AJ Baroody and HP Ginsburg.

If you are not so keen to read through the literature yourself and would just like a quick answer, here is the best that I can do (though, I reiterate, counting is not so simple!): The ultimate goal is for children to be able to count without needing to use their fingers; however, in building up to that goal, it is totally fine for them to count using their fingers. Keep practicing the chanting, the skip counting, and then try counting groups of objects! It is fine for children to tap each object when they are counting them; later on, they might just point; later still, they may be able to count without needing their fingers; ultimately, the goal is to count without even having the objects in front of themselves. But the time that this takes can vary.

(Note about multiplication and finger counting: Later on, when students move on from basic facts about addition to "basic facts" about multiplication, i.e., multiplying pairs of whole numbers from 1 to 10, they can still use finger counting to help them: especially with the skip counting numbers of 2, 5, and 10. For example, one way to compute 2x8 is to hold up 8 fingers, and then count on them by 2s (2, 4, 6, 8, 10, 12, 14, 16), putting down a finger each time it has been used. Again, the ultimate goal is to move towards abstraction and not needing a concrete representation, but using fingers to count - for addition or for multiplication, which can be thought of at that age as repeated addition - can be very helpful.)

Perhaps one of the stressful factors is not knowing what the norms are for a given age (or stage). The language that Jamie is using looks totally age-appropriate; my bigger concern would be that his tutors are not quite sure how to elicit his reasoning. For example, the response of because you said that looks very familiar if you have spoken with children about mathematics. One good way to try to understand their thinking is by using clinical interviews; a single particular technique is to say something like, what if a little kid asked you to explain your thinking? Rather than delve deeply into techniques for using the clinical interview, I will give the same recommendation I included in earlier answers here and here (see the very end of each):

Ginsburg, H. P. (2009). Early mathematics education and how to do it. Handbook of child development and early education: Research to practice, 423-428.

Finally, if you would like to know a bit more about norms for kids learning mathematics at an early age, I recommend the following book (or its 1989 reprinting):

Ginsburg, H. (1977). Children's arithmetic: The learning process. D. van Nostrand.

Here is an excerpt from Chapter 2, Learning to Count, to give you an idea around norms:

(Too long for a comment...)

Your question suggests a distinction between 'using ones brain' to do mathematics and 'performing a mechanical process' to do mathematics. This is an idea I always love to jump on.

The truth is that much of mathematics is mechanical (this is why computers can do it so well), and the machine involved in performing these mechanical processes is your children's brains.

Human brains obtain skills via long repetition, be it a motor skill (how many hours do violinists spend drilling the fingering for scales x y and z?) or an intellectual skill (in this case counting and the internalizing the concept of addition/subtraction). My own inclination would be to let them count until they abandon it of their own accord after it has become boring for them and/or their accumulated practice leaves them able to instantly know that five and three are eight.

• With regard to the initial idea you are talking about: You might look up APOS Theory for a formal framework to think about moving from the mechanical process to the using of one's brain (to speak colloquially). As for the "my own inclination" piece: You may wish to read up on things like count on, count all, subitizing... – Benjamin Dickman Nov 19 '14 at 1:36