Comments on The Science of Early Learning
Deans for Impact encourages robust debate on the science of learning. Here, Dr. Karen Fuson of Northwestern University offers commentary on some of the principles raised in the numeracy section of our recent The Science of Early Learning report. We are grateful to Dr. Fuson for sharing her thoughts, which we publish without endorsement.
Because understanding numeracy is so important for young children, I want to comment upon the research discussed in The Science of Early Learning report to emphasize crucial aspects of our knowledge about children’s thinking that can support effective teaching.
How do young children learn to count?
Counting is a complex activity in which children must match each counting word said in time to an object being counted, which is usually in some spatial location. Some indicating act such as pointing connects the word said in time to the object in space. Many different errors can occur, but with effort, help and feedback, and practice, children can move from counting small sets to counting large sets accurately. Children enjoy counting, and many will do so spontaneously. Counting even when they can do so accurately helps to internalize the number word list.
Children also need to learn to give a number word to small sets without counting. This is called “subitizing.” This helps them understand cardinality (how many in a set) and addition and subtraction as they begin to subitize numbers inside other numbers, such as seven is five and two.
How do young children develop abstract knowledge of mathematical concepts?
Meanings of mathematical words and symbols are created in children’s minds by relating actions on concrete objects and pictures/drawings to abstract math symbols like spoken words and written numerals. Jerome Bruner’s three modes of representation—enactive, iconic, and symbolic—are often misunderstood to mean separate stages that follow each other, but it is the relating that creates the synergistic meaning-making power of using actions, visual models, and abstract math symbols together.
As symbols and abstract representations for numbers take on meaning for each child, the child will gradually become able to use the written symbols and abstract representations instead of concrete and visual representations. These symbols and the counting words eventually become mental representational tools for more advanced thinking (see levels 2 and 3 in the next section about arithmetic).
The research about board games has two aspects that need to be emphasized.
First, board games for young children involve number paths in which numbers one, two, three, and so on are written on circles, squares, or other shapes. They are not number lines that are length models such as rulers, with numbers below little vertical marks on a line. All National Research Council Reports since 2000 and the Common Core State Standards emphasize that young children make errors using number lines, and that number lines should only be introduced beginning in second grade when measurement of length and rulers are introduced.
Second, Siegler’s research had children count along the number path saying the numbers on the path. So if a child’s token was on five and a 1 was spun, the children moved the token to six and said, “six.” If a 2 was spun, the child said, “six, seven.” This is in contrast to a usual method of playing board games where you count “one” or “one, two” or “one, two, three” and thus just practice words you already know well.
How do young children learn arithmetic?
Educators need to support students to progress through the learning path for adding and subtracting by connecting representations and discussing methods along this path. This support should begin before kindergarten and may take until second grade to be complete for most children. These methods are:
- Level 1 counting all: Find 5 + 3 by counting out or drawing 5 objects, then 3 objects, and then counting all of the objects.
- Level 2 counting on: Start counting on from 5 and count on 3 more objects: 5, 6, 7, 8.
- Level 3 derived facts (recomposing methods): Make-a-ten by decomposing the larger number into the amount to make ten and the rest: 8 + 6: 8 + 2 + 4 = 10 + 4 = 14. Children can learn the three prerequisites for make-a-ten in kindergarten: the amount to make ten, breaking apart a number in various ways, and 10 + n is what teen number. Many children also recompose to make doubles plus or minus 1 or 2: 7 + 6 = 6 + 1 + 6 = 12 + 1 = 13.
These addition methods have related subtraction methods at each level. Methods become abbreviated and internalized at each level, and the counting word sequence becomes a mental tool for adding and subtracting.
Knowing additions and subtractions quickly is valuable because it frees up working memory for more complex problems. However, any emphasis on speed should be limited to problems that children can solve readily. Practice with larger numbers for which children are still using strategies is also useful but with an emphasis on accuracy and understanding and being able to explain and not just on speed. Estimation requires knowledge of numbers and place value and so belongs in Grade 2 where many children have built such knowledge. At present a lot of learning time is wasted and children get bored and discouraged by too much practice on facts they already know. Practice for speed or for accuracy should be in the learning zone of individual children.
Karen C. Fuson is Professor Emerita atNorthwestern University in the School of Education and Department of Psychology.
Cross, C. T., Woods, T. A., & Schweingruber, H. (Eds.). (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: National Academies Press. doi:10.17226/12519
Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.
National Council of Teachers of Mathematics. 2009. Focus in Grade 1: Teaching with the Curriculum Focal Points. Reston, VA: NCTM.
National Council of Teachers of Mathematics. 2010. Focus in Kindergarten: Teaching with the Curriculum Focal Points. Reston, VA: NCTM.
National Council of Teachers of Mathematics. 2010. Focus in Prekindergarten: Teaching with the Curriculum Focal Points. Reston, VA: NCTM.