Approximately 5-10 % of children and adults have difficulty in at least one domain of mathematics. Since remediation programs need to be practical, effective and up to date, researchers are trying to learn more about the underlying mechanisms of mathematical cognition. My research is primarily concerned with the role of motor skills in the development of basic numerical skills and mathematical school achievement. I have also looked at the relationship between motor skills and executive functions, and at their different roles in the development of mathematical skills.

###### How children learn about mathematics

The development of mathematical skills begins long before children enter school. Children are born with an informal awareness of numbers and magnitudes, or what is often referred to as number sense. At the age of only four and a half months, babies are already able to distinguish between groups of small numbers of objects. This and similar non-symbolic skills are believed to be intuition-based and relatively imprecise, and are therefore also referred to as the approximate number system (ANS).

Symbolic skills are acquired when children are able to name the numbers or use Arabic numerals. Symbolic skills are exact by nature; it is possible to determine precisely how many objects are contained in a collection, not just whether there are more or fewer than in another collection. Once children have mastered the counting principles ­– such as the one-to-one-principle, which states that there is only one name for one quantity of objects ­– they are able to solve simple arithmetic problems by counting up addends.

Usually they do this by counting on their fingers, using one hand for each addend. Take, for example, the following problem: “If I have three toy cars and my sister gives me one, how many do I have all together?” The child might show three fingers on his right hand and one on the left, and then count the fingers on both hands to arrive at the answer: four. With time and experience, a child’s strategies for solving similar arithmetic problems become more sophisticated and complex.

Basic numerical skills and an understanding of magnitudes are of special interest to researchers, since they help to prepare children for what they will learn in school. With more experience, children develop an understanding of a spatially arranged representation of magnitudes, expressed by the mental number line. In Western cultures, smaller numbers are mentally represented on the left and larger numbers on the right.

Several studies have investigated the accuracy of children’s ability to compare magnitudes and the linearity of their number line estimations, because these skills are predictive of later mathematics achievement. To tailor remediation programs to the needs of children who have difficulties in this area, it is crucial to know what mechanisms underlie the most basic skills and the other skills that support them during the development process.

###### Is there a relationship between motor skills and mathematical cognition?

Two theories suggest that cognition is based on early sensorimotor sensation and experience. Psychologist Jean Piaget believed that the first phase of life is determined by sensorimotor representation; that is, infants learn new information through movements and sensations.

The second theory, called embodied cognition, posits that cognition, for the most part mathematical cognition, is grounded in bodily processes beyond the brain itself. According to these theories, there should be a strong relationship between children’s numerical or mathematical skills and their motor skills.

As pointed out above, at first children count and perform simple arithmetic using their fingers. It seems quite clear that there is some relationship between fine motor skills (finger movements) and mathematical skills (in this case counting and simple arithmetic). Some studies have looked at this relationship. They have often found correlations and have observed the activation of certain brain regions, which they have attributed to experiences early in childhood. It is important to determine whether there is a similar relationship between fine motor skills and mathematical skills in childhood as well.

Once again, several studies have investigated this relationship, but most have also included other “supporting skills,” such as executive functions. “Executive function” is an umbrella term for the ability to adapt, regulate, monitor, and control one’s own behavior or information processes.

Executive functions are particularly important for skills or tasks that require a certain amount of higher-order processing, such as shifting the focus of one’s attention or excluding irrelevant information, as well as for retaining relevant information in short-term or working memory. They are commonly the dominant predictor of mathematics achievement in school and closely correlated with basic numerical skills.

For my PhD project, I conducted a longitudinal study with two objectives: The first was to find out whether executive functions and motor skills are differently correlated with symbolic and non-symbolic skills; and the second was to determine if and how executive functions, motor skills and basic numerical skills predict later mathematical skills.

The study included over 150 children in kindergarten and second grade, who were asked to compare magnitudes and estimate number lines in a symbolic and non-symbolic version. They also completed a motor-skill battery and were asked to solve computer-based tasks assessing their executive functions. The second graders also completed a standard mathematics test with equations, sequences and simple subtraction and addition problems.

###### Motor skills are related to non-symbolic skills

The results showed that motor skills and non-symbolic skills were correlated, as were executive function and symbolic skills. This pattern supports Jean Piaget’s theory, showing that the non-symbolic skills that are acquired earlier in a child’s development depend on motor skills, while the symbolic skills that develop later are supported by executive functions. Overall, the results point in the same direction: Basic numerical skills, which are predicted by executive functions and motor skills (as mentioned above), are the dominant predictor of mathematics achievement in second grade.

I also found that executive functions are an indirect predictor of later mathematics achievement (through basic numerical skills) and that factors of executive functions are differently associated to various basic numerical skills. However, I was unable to find evidence that motor skills predict mathematics achievement.

“Motor skills play an important role in the transition from non-symbolic to symbolic skills, through finger counting, and are thus indirectly associated with second-grade mathematics achievement.”

Another interesting finding may offer a possible explanation for these results. Motor skills and executive functions are highly correlated. This strong relationship may provide another indirect pathway for motor skills to affect mathematics achievement. If better motor skills lead to better executive function (and vice versa), and better executive function (and motor skills) lead to better basic numerical skills, which, in turn, lead to better mathematics achievement, it can be argued that motor skills indirectly predict mathematics achievement.

The results suggest that motor skills are important for the non-symbolic basic numerical skills that develop early in life. In addition, however, motor skills play an important role in the transition from non-symbolic to symbolic skills, through finger counting, and are thus indirectly associated with second-grade mathematics achievement.

## One comment

1. Lyelle Palmer says:

As a special education professor teaching for more than 30 years with techniques for learning disabled and brain stimulation techniques for super-advanced school readiness (S.M.A.R.T. program at actg.org in Minneapolis) I can tell you what cripples children for life in math and what boosts them to automatic precision. Instead of counting, please introduce children to quantity recognition. We do this as early as age 2 by matching “friends” using common black and white dominos and demonstrating by naming: “Here are “two dots” and I’m finding another “two dots,” and they can go together like this.” The quantities/values are mentioned informally and the child learns quickly to match successfully and feel satisfaction through accomplishment and playing this matching/naming “game” together.

The saddest thing is to see children at fourth grade (and well beyond) slowly counting on their fingers (and miscounting) in a discovery approach to quantities. The futile strategies observed for counting by ones even in middle school (making tally marks on paper, then mis-adding, mis-multiplying, etc) makes us want to weep because of all the work and the resulting failure because of miscounting complex lists of tally marks. Young children must begin with quantity recognition, not counting (objects or on fingers). The naming of quantities must be modeled.

Another mistake is constantly quiz children, a common practice of adults that creates a context of inquisition with children. Questions are not the only way to interact and should not be confused with teaching (movies typically use testing as the example of teachers in action). Many alternatives to questions are available, so learn to use them. Young children feel the pressure and hate being tested. Automatic recognition of sets, subsets and combinations and divisions is fun (at older ages/grades put number ladders into hop-scotch so that the counting is not by ones). Counting by ones will occur naturally when listening to others count and generally does not have to require much repetition.

Teachers are generally not math adept and are not trained at a high level in understanding math. In fact, teachers at primary and elementary levels are regarded as fearing math and some teachers pass along to students messages of inadequacy expectations and sympathy. It is sad to have to make these statements that appear so obvious. Young children love math when taught rather than tested, and find satisfaction in handling the various languages representing quantities. Of course we teach cardinal and ordinal names of numbers, but quick calculation relies on perception of quantities. Most people count every day, but counting by ones should not occupy much time. One of the first uses of counting is to recognize a quantity and the count on with the added quantity. With dominos however, young children can master the dot quantities of combinations to ten and later to twenty, and then combinations of two quantities to 100, etc. The basic quantity concepts are not counting by ones because one is counting individual trees while not recognizing the forest (quantity). Parents and teachers are amazed to see how young children quickly master concepts when presented the way the brain learns.

One more thing: young children learn number sequences vertically rather than the lateral number line. Much labor is avoided when the number line is placed vertically reading from bottom to top for number sequences at different points in mastery, such as 1-10, by 2s 2-20, 3s 3-30, 10s 10 to 100, 5s 5-100, 11s and 12s, 15s and 25s for older students. Make this curriculum fun with adult enthusiasm. Please do not make children think that math is mastered by completing 1000 pages of math problems since that approach is not teach, but just laborious testing.

I hope that you will share these tips for developing confidence with early math. Remember that the early years curriculum is different from all others and is essential for basics. Just ask a music teacher or other performance coach about basic foundations that prevent plateaus and future progress. What is taught first often determines outcomes years later.

Mr. Lyelle Palmer, Ph.D., Professor Emeritus in Special Education, Winona State University, Minnesota

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