Arithmetic Fluency: Some Ideas for Achieving It
By Michael Maloney
Printed in Practical Homeschooling #59, 2004.
Aside from teaching reading, the most frequent curriculum concern for many homeschoolers lies in teaching math. Like other skills, competent math skills require a solid foundation. The foundation for fluent math skills begins with counting skills.
We have all seen children who struggle with math problems. Some parents explain it as "not having an aptitude for math." Others consider that it has something to do with left-brain versus right-brain functioning. I think it has mostly to do with not teaching the right skills in the right order to a high enough standard so that the child has really mastered it.
Many times we are just not sure what those standards should look like or how we could easily measure them. Using the fluencies that have been developed with hundreds of thousands of students over 30 years, provides us a new approach
The major variable we see in how soon and well children reach various math fluencies is the amount of instruction and practice each requires to reach the standard. Some do it quickly and effortlessly; others achieve it only after great effort. Very few cannot reach fluent levels of performance across a wide range of arithmetic skills if we are prepared to do the teaching and practice.
Counting Skills
There are many kinds of counting and children need to learn all of them to succeed at math. Most parents begin by teaching children to count from 1 to 10, then 1 to 20, and so on up to 100. Sometimes we neglect to teach them to count fluently from numbers other than 1. Then when they begin addition, they have problems because they have never counted from a number other than 1. So they start at 1 and count to the first number and then continue to count for the second number. To be fluent counters, we must also teach them to count from a number to a number.
The ability to count backwards from 100 or from a number to a number becomes important when you teach subtraction. Children who are not fluent at counting backward will get mixed up when they count from 80 to 79. They cannot determine what number comes just before 80. They will hesitate and often guess.
Using recursive counting patterns for each number is the easiest way to teach skip counting. It has a great impact on learning multiplication and division where counting groups is the objective of the exercise. The recursive pattern for 3's is given below:
| 3 |
6 |
9 |
12 |
15 |
| 18 |
21 |
24 |
27 |
30 |
| 33 |
36 |
39 |
42 |
45 |
| 48 |
51 |
54 |
57 |
60 |
| 63 |
66 |
69 |
72 |
75 |
| 78 |
81 |
84 |
87 |
90 |
Seeing a recurring pattern helps children to remember how to count by threes or any other number more easily. Each number has its own recursive pattern.
Children can easily count at 300 counts per minute if they are fluent. Typically counting is done for 15 seconds or 30 seconds and the results multiplied to get a count/minute score. Some typical counting tasks and their standards are listed below.
See and say numbers forward from 1-10, 1-20, 1-50, or 1-100 at 300 counts per minute with 0-2 errors. (Numbers are on a page in front of the student as a visual aid.)
Think and say numbers from 1-10, 1-20, 1-50, or 1-100 at 300 counts per minute with 0-2 errors. (Student is working from memory, no visual aids)
See and say numbers (forward or backwards) from one number to another number at 300 counts per minute with 0-2 errors. (E.g. count from 7 to 14, or from 23 to 12, etc., with use of a visual aid.)
Think and say numbers (forward or backwards) from one number to another number at 300 counts per minute with 0-2 errors. (e.g. count from 7 to 14, or from 23-12, etc., from memory). For free scripted lessons of counting tasks, visit www.teachyourchildrenwell.ca/freebies
Basic Arithmetic Operations
Children who are fluent at using arithmetic operations can add, subtract, multiply, or divide simple math facts at 60 to 80 facts per minute with no more than 2 errors. Children who count on their fingers can never reach these fluent levels. Children can either say or write the answers to demonstrate fluency. Limiting errors to two per 60-80 problems ensures a high level of accuracy as well as high pace of responding. We do not want to create fast, sloppy arithmeticians.
Input and Output Channels
Arithmetic tasks can be done many different ways. Children can count using a visual aid ("See and say numbers") or from memory ("Think and say numbers"). They can say the answers to math facts from flashcards or math sheets ("See/Say addition facts") or they can write the answers ("See/Write addition facts"). Sometimes children have poor number writing skills that would never allow them to answer 60-80 problems per minute. In such a situation, we would change the response channel from "see and write" and have the child "see and say" the answers. As we change the input and output channels, we are changing the task itself and the student's scores will change as a direct result. Hearing and saying the answers to math facts is quite different from seeing and writing the answers to the same math facts. Comparing the results of the same task done two different ways will clearly demonstrate the discrepancies. For children with certain conditions, being able to change the input and output channels is a critical variable to success.
Number Writing
To write 60-80 simple arithmetic facts per minute, children need to be able to write 120-160 numerals per minute. If each multiplication fact requires a 2-digit answer, the student needs to be able to write 120 to 160 digits to be able to complete the task to a fluent level. To determine the student's current skill level in digit writing, a simple measure is available. Have the child write the numbers from 0-9 as many times as possible for a minute to see if they have fluent number writing skills before expecting them to deliver on fluent fact writing. On the first line of a sheet of paper, the student writes:
1 2 3 4 5 6 7 8 9
Then the student writes that line of numerals as many times as possible within the 30-second or one-minute period. Any numeral that cannot be easily deciphered is marked as an error. The student practices this task daily for a couple of minutes, records the scores and continues developing number writing skills until he or she can write 120-160 numerals per minute with no more than 2 errors.
Fluency and Advanced Calculations
Fluency allows the child to concentrate on the problem, not on the computation. Dysfluent students often bog down in the calculations and lose their place in the process of completing a multi-step problem. When they have to stop in the middle of the problem to try to figure out what six times seven is, they often then make a simple error which causes the entire problem to be incorrect. They put the correct number in the wrong column, or forget to write down the number that they are carrying, or such like. This is particularly true when word problems are involved.
Fluency and Rules
Many aspects of arithmetic are governed by rules. In addition, you always start in the ones column. When multiplying simple fractions, you always "multiply the top by the top and the bottom by the bottom." Children have to learn the rules and the procedures of arithmetic to fluent levels. They should be able to recite the rules without hesitation. Students who cannot articulate the rule or describe the procedure quickly and accurately do not really understand it and will soon forget it.
Practice Sheets and Flash Cards
Most commercially available math materials will not have 60-80 arithmetic facts on a single page for the student to solve. You may have to make up worksheets with at least 60 facts for each operation. Flash cards work well for math facts, except that children can only flip about 50 cards per minute, which limits their learning.
The Effect of Developing Fluent Performances
Children who develop fluent levels of performance can do arithmetic quickly and easily. They are ready for the next more complex concept or operation. They do not lose their skills over periods without practice. They relearn effortlessly. Children who are not fluent look and act entirely differently when presented with a mathematical challenge. The difference in all but a very few cases is found in the instruction and practice given to reach fluent levels.
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