At the end of my previous post, I hinted at another reason why fluency is important for students and how they can “go even faster…”.
Fluency is not only the ability to recall effortlessly and accurately, but also the ability to recognize patterns and to apply shortcuts and ‘tricks’ correctly. For example, in teaching data analysis units, some students have problems with finding means– either forgetting to divide, or more usually, making a mistake in the addition of the set of numbers and getting an unending decimal when they try to divide. For example:
Find the mean of the following numbers: 5 + 9 + 4 + 3 + 2 + 8 + 6 + 7 + 1 + 10= ? (this equals 55!)To a student who is not fluent in addition, this looks hard! The student thinks, “Too many numbers to add up. Then I have to divide. And I’ll probably get a decimal that doesn’t end…” The student has no confidence in themselves, and will not want to apply the effort to add all the numbers.
However, you may have seen it quickly because you are fluent in adding 1-digit numbers:
8 + 2 = 10 7 + 3 = 10 6 + 4 = 10 9 + 1 = 10
What’s left? 5 and a 10. Obviously, much easier to add 10 + 10 + 10 + 10 + 10 + 5 = 55.
I understand that this is a specific, made up example to illustrate my point, but the fact the fluency allows us to instantly see ‘groups’ of 10’s makes the addition go much faster and easier.
What about multiplication? Teachers, have you seen this? (I did in 7th grade):
413
x 200
000
0000
82600
as opposed to this:
413
x 200
82600
or even this: “I know that I can just add two zeros at the end since I am multiplying by 200. And since 4×2=8, 1×2=2, 3×2=6, the answer is 82,600”
The problem with this multiplication problem is writing down way too much information (then consider how bad some handwriting can be, especially boys!)– which can introduce mistakes – not related to actual multiplication– but to organization. Forgetting to add a place holder 0, or lining up the columns correctly, OR actually multiplying incorrectly. And this example does not have any carry values.
The second example is a lot less writing and computing, so there is a lot less chance for error. The last example there is only single-digit multiplication.
I am all for students to learn ‘tricks’ and shortcuts…ONLY after I know they have mastered basic addition and multiplication. One of the problems that occurs when students learn a trick before mastery and understanding is that they will apply the shortcut incorrectly. Teaching tricks and shortcuts over process can lead to confusion on the student’s part.
So, the importance of being fluent is that it allows students to go faster, make fewer mistakes, and be more accurate. Under the pressure to perform well on standardized tests, I firmly believe students who are fluent have a huge advantage of those who still have to compute 5 + 9 + 4 + 3 + 2 + 8 + 6 + 7 + 1 + 10.
It is not so much fluency that allows children to do what you talk about above but rather it is the result of having number sense. Children don’t gain number sense through “tricks” or memorizing but through thoughtfully working with numbers and applying strategies they discover or learn. Fluency need not be automatic but rather a smooth application of a strategy to work toward a solution. Thus in adding the string of numbers above the children I work with learn to look for friendly combinations that can help them work efficiently toward a solution; in your example the strategy is obviously making tens. In the multiplication example you give if kids understand that if they have one hundred 413’s they have 41,300 so 200 413 would be 82,600, or double the amount. There are no tricks involved here just understanding what the numbers mean and knowing when to apply a process. I want my students to look and think about what they know about the numbers presented first and then think bout what strategy they might choose to use to reach a solution. They are not limited by a finite number of strategies and they are thinking always about efficiency.
Agreed Paula. I consider that number sense and fluency go hand-in-hand…And fluency is much more that memorization. I think that fluency is a necessary step to number sense and pickup up patterns in numbers/operations which allows them to understand ‘tricks’