Help students develop fluent performance
Not long ago I worked with a very intelligent and charming high school student who needed help in his algebra course. I remember very vividly watching him work through one problem that required him to calculate 9 x 5, and he proceded to use his fingers to skip count by 5’s nine times. He was fairly dextrous in extending his fingers while doing this calculation, and he got the correct value for 9 x 5. Now, using one's fingers to do calculations is a very useful step for students learning beginning arithmetic. But it's use by a high school student in an algebra course put on clear display the primary source of his difficulties, namely, he had never advanced from counting on his fingers to memorizing the multiplication facts.
In the first blog posts of this series on unlocking your child’s math abilities, I emphasized the importance of grounding lessons in student exploration of counting both physical things and counting using drawn models. Such exploration, I believe, is necessary to develop genuine understanding of basic mathematical patterns and relationships, and this understanding is the basis for mastering rapid calculation procedures, applying math skills to solve problems, and enjoying and appreciating math.
While exploration of counting both using things and models ought to be the foundation of math lessons, it is, however, in and of itself neither sufficient to develop a student’s proficiency with basic calculations nor to develop the student’s skills in such a way that she can learn new concepts as quickly and efficiently as possible. If we want students to develop advanced skills as well as to enjoy math, we need to make sure as well that they develop fluent performance of basic math skills, and this means that students have to be able to recall basic relationships and procedures rapidly and with little effort.
First of all, rapid recall of basic number relationships helps students do many calculations easily. Even though we now have at our disposal electronic calculators, there are still many calculations, such as adding 16 and 5, or multiplying 16 by 5, that a person can do much more quickly in one’s head than by punching the numbers into a calculator, or even asking a digital assistant such as Alexa or Siri.
A more important reason, though, is that rapid recall of basic number relationships helps deepen a student’s understanding of math. It does so in two ways. First of all, the ability to see patterns requires seeing and investigating lots and lots of examples. If students have to do problems by counting on their fingers, they will not be able to do as many problems in the same amount of time as a student who can recall basic facts readily. Furthermore, doing a calculation on one’s fingers is much more demanding and tiring than retrieving a known fact from memory, and so a student’s ability to attend to what he is doing is diminished.
A related benefit of automatic recall of basic number relationships is that the student is able to recognize new number patterns more quickly. For example, when a student is learning about multiplication, the ability to rapidly add a one-digit number to a two-digit number will help the student become familiar with skip-counting patterns more readily, and this familiarity deepens the students understanding of just what multiplication is, and accelerates her acquisition of the multiplication facts.
Likewise, a student who knows the multiplication facts can do some fraction calculations using drawn models and discover that 1/2 of 2/3 = 2/6 and 3/4 of 2/5 = 6/20, and then readily see the relationships between the numerators and denominators on each side of the equal sign. Without knowing the multiplication facts, it takes much longer to see this relationship.
So there are clear cognitive benefits to help students develop fluent, automatic recall of basic number relationships:
- It helps them do calculations rapidly
- it increases the number of math experiences they can have a given time
- it improves their ability to identify new patterns more readily.
In addition to these cognitive benefits, there is as well a very important affective or emotional benefit to helping students acquire rapid recall of basic number facts. Without this ability, doing any sort of math investigations will remain unnecessarily laborious. In general, people like to do things that are relatively familiar and easy, and would rather avoid things that are difficult, especially if the difficulty persists for months or years. So if a student, even a student who is otherwise quite successful in school, is still counting on her fingers to do a basic multiplication problem into 5th or beyond, she will be taking much longer to do the work than her fellow students who have memorized these patterns. Such struggles often lead students to conclude that they are “just not that good at math,” or don’t have “math brains” or some such thing. And they certainly contribute to the unnecessary scene I sketched at the beginning of today’s blog, the situation in which an otherwise successful and motivated 14-year old was struggling in his algebra class.
Now, few would disagree that automatic recall of basic number relationships is a good thing. So, why are there so many students who never develop this automatic recall? One culprit is essentially administrative: it is much easier to evaluate or grade the performance of students after a certain period of time than it is to make sure that a sufficient amount of time is provided for virtually all of them to acquire fluent performance.
Another important reason has to do with the inefficiency of most of the practice students receive. All students in schools get some kind of practice to memorize basic math facts, but much of this practice is extremely inefficient. So not only is it difficult to provide sufficient time for all students to develop fluent skills, the practice given usually does not use the readily available time very well.
So, what are the characteristics of efficient practice. That will be the topic in my next post.