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The reason finger counting is discouraged is because it slows down the whole calculation process. Fast recall of of arithmetic facts are essential for questions from "There are 8 eggs in a basket and 3 are taken out. How many are left?" through to "Solve 7x + 3 = 52" and beyond.
Yes there are calculation methods which use the fingers to work out calculations at super fast speed, which is great if you're going to train your child to do that but leaving your child to finger count while her classmates move ahead because they have memorised the arithmetic facts is just not fair. Try starting from the basics, memorising +1's first, then +2's , then +3's. I talk in more detail about this on my blog 
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Many people confuse arithmetic with mathematics. Remember that most of what is called "math" at this level is really just manipulating numbers, which is a different beast all together. I'm a great mathematician. My arithmetic is soso. You don't hit what I call mathematics until algebra or so, unless you're really lucky and get a lot of logic mathematics earlier. It saddens me that so many people get turned off the number crunching so early they never see that math is just a series of logical statements.

I also think this need society seems to have for "fast math" is silly. Doing things faster increases the chance of a mistake. There are only a few situations outside of a test where math is required to be fast and no everyone is going to want to be a nurse, or work for shuttle launches or something like that where fast math can be life or death. Most people will only use their math for basic everyday situations where it doesn't matter if it takes 1 minute or 30 seconds or .5 seconds to get the answer.

Actually that was my point  you can do multivariate calc and still count on your fingers. They're different skills.

I also think this need society seems to have for "fast math" is silly. Doing things faster increases the chance of a mistake.

I mostly agree with you. I see a difference between memorization and being fast, though. I see a lot of value in generally memorizing the multiplication table because it's really useful when inverting it all (dividing) or working with the general properties of mathematics (algebra). Having this memorized makes things more efficient and more painless when doing a series of calculations. 
It seems more likely to me that you have a feel for the pattern of the 6s and 8s, than that you've simply rote memorized them. The pattern of the 6s and 8s give them a rhythm that the 7 lack. If you were going solely by rote memory, the less rhythmic 7s would be just as easy.

However, I dealt with a student every week in my office hours last quarter who could neither figure out when to add or multiply (mathematics) or do 201 / 3 using long division (arithmetic). I'm pretty sure that the process of long division would have been significantly easier if he'd actually known 3x6=18, which he did not. Using ones fingers would have made it just as painful and tedious. The OP should know that you don't hit this kind of problem until 3rd or 4th grade.

However, I dealt with a student every week in my office hours last quarter who could neither figure out when to add or multiply (mathematics) or do 201 / 3 using long division (arithmetic). I'm pretty sure that the process of long division would have been significantly easier if he'd actually known 3x6=18, which he did not. Using ones fingers would have made it just as painful and tedious. If there's another learning method you can tell me to help students like this, I'm all ears. It was a painful quarter for both of us. 
The 6x table doesn't even need to be memorized to do multiplication of 6.

In that situation I would have taken that student right back to the basics. Start with 1 + 1 or 1 x 1 , memorise the 1's first, then the 2,s, then review the 1's and 2'sbefor moving on to the 3's, then review the 1's,2's and 3's before moving onto the 4's. At each stage make sure there is fluency before moving to the next.
It's not a quick fix, but even if you had limited time it would be better for that student to know some of the basics solidly which would likely increase his confidence as well. 
However, I dealt with a student every week in my office hours last quarter who could neither figure out when to add or multiply (mathematics) or do 201 / 3 using long division (arithmetic). I'm pretty sure that the process of long division would have been significantly easier if he'd actually known 3x6=18, which he did not. Using ones fingers would have made it just as painful and tedious. The OP should know that you don't hit this kind of problem until 3rd or 4th grade.

I get quite a few students every semester like this. These are university students taking a science course. They are not students who aspire to technical careers. They want to teach ELEMENTARY SCHOOL.
The high school physics and math teachers that I talk to tell me the roots of the problem are in elementary school, so I feel overly sensitive about that. 
I don't see fingercounting as a crutch at all. In fact, I kind of see it as a reflection of a true understanding of math and using all tools available.
My DD1 is 6 and uses her fingers to count. She touches the finger to her lips to count each one and I think it's adorable! 
Musiciandad, your arguments are both reasoned and persuasive, but if we return to the OP's question, her child only recently started finger counting and was previously able to recall number facts to 10. What I want to draw attention to is the unspoken idea that not only is it ok not to be able to recall number facts with ease but that as a parent, the OP shouldn't try to help her child memorise these facts.
Yes, I'm sure her child could still become a great mathematician even if she continued finger counting, but let me support the OP in her attempt to help her child. 
Musiciandad, your arguments are both reasoned and persuasive, but if we return to the OP's question, her child only recently started finger counting and was previously able to recall number facts to 10. What I want to draw attention to is the unspoken idea that not only is it ok not to be able to recall number facts with ease but that as a parent, the OP shouldn't try to help her child memorise these facts.
Yes, I'm sure her child could still become a great mathematician even if she continued finger counting, but let me support the OP in her attempt to help her child. 
2. Finger counting does not discourage memorization. Working through a problem in a way that is effective for the student encourages memorization. Again, there are people, the more times they count on their fingers to figure out the equation puts the equation into the "I did it myself, so I have an easier time remembering next time" category. I have seen this happen too, it's how I memorized the equations I have memorized. 
I don't think there is any importance to memorizing the multiplication tables at all. I think it's a product of the "fast math" mind set that we force kids to memorize a series of numbers instead of teaching them how to figure out how to get those numbers and it sets up a fairly intelligent portion of the population for stress in math class because they can't memorize a series of random equations.
What makes properties of math painless and more efficient is teaching students the way they learn best. For some, rote memorization may be the way to go, but for others it is nothing but a painful inefficient means of making them feel inadequate. 
In my opinion this is precisely the problem we have with our secondary education. Every few years or so, the schools here in U.S. follow certain educational fads that talk a lot about how all students learn differently and how teachers must create and utilize various teaching styles to fit each student's abilities (imagine that in the classroom of 3040 students!!!), and how knowledge of the subject matter is no longer important, and that students must learn the "concepts" rather than actual subject matter, and on and on.... The reality is that the students must learn the substance before they can apply the â€śconceptsâ€ť . They need to memorize the multiplication table because this will make their life MUCH easier down the road. Poor memory is just an excuse for not doing the "boring" memorization. Memory must be developed and improved through sets of simple exercises. For example, a child should read poetry as often as he/she can; this is the best and easiest way to improve oneâ€™s memory. This is what we all did in elementary school, by the way; we read poetry A LOT and we had to recite poems on the weekly basis. Also, the multiplication table hung on the wall in my bedroom and I looked at it every time when I went to bed or when I got up. As I recall, no one in my class had problems with memorizing multiplication tables; and there were 20 students in my class in elementary school. Hell, wake me up at night, and I can give you the whole table within five minutes. And I am eternally grateful to my teachers that they made me do that. Memorization is a part of learning process and while it is could be boring, it is absolutely necessary. My students often complain to me that they have poor memories and this makes their learning experience much more difficult that it shouldâ€™ve been. They tell me that they never were asked to memorize and recite poetry, to remember important historical dates, and yes, they were not required to memorize the multiplication table as well. And I see them failing my and other professorâ€™s classes, getting frustrated and dropping out.


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