Let us take care of the children 
For they have a long way to go.
    ~ Nelson Mandela



Research into Practice: Mathematical Thinking in the Early Years

Research Into Practice Volume 1

Rethinking Math

by Iris Berger

When we think about math we often think about numbers. Yet, mathematical thinking for young children has a completely different meaning. Children naturally use an array of concepts and skills that enable them to see, as well as to process and organize relationships and connections among different elements in the world (McWhorter —­ Colvin, 1996). Mathematical concepts, such as comparing, measuring, and patterning help children understand phenomenon, solve real life problems, appreciate design, and make predictions about the future. In this context math should definitely be something that is enjoyed by young children.

Development of Mathematical Thinking

Early on children learn about math intuitively by using their senses and body to make sense of their social and physical environment. For example, they experience small and big when they cuddle in their mother's lap (Leeb-Lundberg, 1985). Such experiences prepare them for the understanding and naming of math vocabulary —­ small, medium, big and mathematical concepts such as sequencing. From ages 3 through 6 children move from an intuitive to a more organized/formal mathematical thinking. During this time "children need many experiences that call on them to relate their knowledge to the vocabulary and conceptual framework of mathematics — in other words to 'mathematize' what they intuitively grasp" (Joint position statement of NAEYC and NCTM, 2002).

Mathematics is About Thinking, Not Only Doing

Learning math in the early years involves more than providing children with manipulatives, such as pattern blocks or peg boards to explore, because mathematics is about thinking, not just doing something with objects. In order for children to mathematize what they intuitively grasp, teachers and parents intervene through providing the appropriate vocabulary ("When you cut it in the middle you divide it in half") and a dialogue that promotes reflection and further thinking: ("Look and see if the pieces are the same, what will happen if you cut the smaller pieces in half?").

Mathawareness: Math is part of daily life

When we embrace a broad definition of math we become aware of the many opportunities in our daily lives for learning and teaching mathematical concepts. Children, however, are unaware that during play and daily activities they often explore mathematical ideas and processes ("He has more than I do!", "It won't fit cause it's too big."). Whether it is while cooking, reading a book, shopping, cleaning up, buying new shoes, or setting up the table, be aware of opportunities where you can effectively facilitate the 'mathematization' of children's experience by engaging them in thinking about related mathematical ideas: "How should we sort the groceries?", "How can we divide the cookies so each one gets the same?", "What will happen if I cut the apple once, twice, three, or four times?"

Mathematical concepts and activities for early childhood

As children come to understand mathematical relationships in a quantitative manner, which involves numbers and formal operations (such as adding, subtracting, multiplying, and dividing), they need many opportunities to think and make qualitative judgment about the world around them using the following concepts:

Classifying: sorting into groups by discovering likeness and differences (from strong contrast to subtle differences).

Take advantage of real life opportunities to sort objects; e.g. clean up time, sorting laundry. Create sorting games: sorting stones collections by color, sorting cars by size. "How else can we sort these?" Involve children in decisions about classification: "Are these farm or wild animals?" "Are these reptiles or mammals?" "Is a whale a fish?"

Ordering: arranging in a sequence by size, amount, or time (from a few objects to many).

Ask questions such as: "Which one comes first, second, third?" "Why" "Which one is the longest, shortest, heaviest?" "How can you tell?"

Patterning: a form of ordering that contains an element of repetition (from becoming aware of patterns to extending and then creating patterns).

Focus children's attention on patterns in daily routines, songs, rhymes, artwork, nature (night and day, seasons, snail spiral, zebra's stripes). Encourage children to notice and describe patterns in the world, extend dialogue about patterns, translate and identify relationships among objects and phenomena. Is there repetition? Does this geometrical pattern symmerical? Extend children's thinking by asking to predict how the pattern will evolve (will it change?). Ask children to make up a pattern.

Shape and space: understanding positions, distances, boundaries, and recognition of form (from experimenting with space and shape to planning a structure).

Allow children to explore the environment and use vocabulary that relates to child's or object's position in the space: up, down, above, under, behind, in front. Provide puzzles with varied degrees of difficulty. Help identify, name, and compare shapes through games, books, and objects in the environment. Explore properties of regular (circle, triangle, etc.) and irregular shapes. "Which shape has pointy edges?" "How many sides does it have?" Round things roll, squares don't — "Why?". Provide building materials and ask about balance, strength, and design of structures.

Non-standard Measurement: compare length, weight, volume (from direct comparison side by side to using non-standard tools and estimation).

Start with simple comparisons of less than and more than, as big as my hand, as tall as the teacher by direct comparison. Provide non-standard measuring tools to extend children's interest in measuring. They can measure their height by using blocks, a string, or tape. Involve children in making decision about measurement: "How long do you think it will last?" "How much should I add?" "Is this bowl big enough?"

Counting and naming numbers: counting by rote, counting objects, recognizing numerals.

Play rhyming games with numbers. Play age appropriate board games. Ask questions or make suggestions that make children think numerically and make qualitative judgment: "Did you get as many as I did?" "Count again and see if you get the same number!" Encourage children to count in a meaningful context. "Do we have enough cups for everyone?" "Count how many trucks you take outside so that you will know how many to bring back." Counting can help solve a conflict if the dispute is about who has more. Ask children to figure out how many children are missing, how many sit around the table, etc.

A Word About Numbers

Number identification and counting are important features of early math development; however, it is important to note that the ability to say the number name does not mean that the child knows the quantitative value of a number, or understands the relationships between numbers (for example young children are not aware that number 2 includes number 1, that number 3 includes numbers 1 and 2, or that number 4 includes 2 sets of 2, etc.). Comprehension of the quantitative value of number develops gradually from early childhood to the primary years.

Math Enhancing Attitudes

Establish an atmosphere that fosters attitudes of curiosity, risk-taking, trial and error. Let children experience autonomous problem - solving. This promotes thinking, reasoning, verbalization & connection of ideas, and reflection.

Math — Assessment

The best way to find out what the child knows about math is to figure out how the child thinks. This can be done in two ways; observation and intervention. While observing children look for mathematical thinking beyond counting, identifying shapes, and simple sorting. Look for comparison, estimation, patterns, symmetry, and understanding of spatial relationships (Kyoung-Hye Seo, 2003).

Observation can lead to intervention through planning of future math activities, and/or to immediate intervention. Intervene when mathematical thinking is stalled, or when children are unaware that they are making a mathematical discovery; e.g. when children build a symmetrical structure with blocks an adult can name the concept of symmetry and suggest looking for other places where it may be found. When intervening ask questions that reveal the child's thinking process: "How do you know?" "Can you show me?" Through reflecting on their own thinking children construct their mathematical understanding. It is important to be aware that children's mathematical ideas are often different from those of adults (For example, children will say that many pennies are more money than 1 loonie. That is, they focus on the number of objects and not the value of the currency as an adult might). Therefore, look for understanding as opposed to correctness.

As teachers and parents our goal is to lay the conceptual groundwork that will serve children in the long journey of mathematical development and learning (NAEYC & NCTM, 2002).


K.-H., Seo. 2003. "What children's play tells us about teaching mathematics", Young Children, January.

Leeb-Lundberg, K. 1985. "Mathematics is more than numbers." Olney, MD. ACEI

National Association for the Education of Young Children (NAEYC) & National Council of Teachers of Mathematics (NCTM). 2002. Early childhood mathematics: Promoting good beginnings.

McWhorter-Colvin, S. 1996. "Math Milestones: Abilities in children of different ages", In More Than Numbers: Mathematical thinking in the early years ed. D. Palmer Wolf & B. Neugebauer. Child Care information Exchange.

Mathematical Discourse in storybook reading

Anderson, A., Anderson, J. & Shapiro, J.
Journal for research in mathematics education. January 2004

Storybook reading is promoted as a means through which to teach mathematical concepts to young children. However, except for a small number of case studies (e.g. Anderson & Anderson, 1995) there has been little research documenting the mathematical learning that occurs when parents and children read storybooks. In the present study, 21 parents (17 mothers, 4 fathers) were audio taped as they read One Snowy Night to their child at home, usually at bedtime. Four dyads who accounted for the most mathematics related interactions were selected and their discourse or talk about mathematical concepts was analyzed. Results showed that parents and children conversed about several mathematical concepts as they made sense of the text, including subitizing — the ability to identify the number of small groups of objects (e.g. 2, 3, or 4) without counting each object separately – counting, comparing size, and some addition/subtraction problem solving. While parents initiated most of the talk about mathematics, some of the children also initiated such conversations. The participants were well educated, middle class families, who spoke English as their first language. Still, there was considerable variety in the manner in which they shared storybooks and, in turn, supported mathematical learning. This diversity is consistent with other research and leads us to question the assumption that there is “best” or “correct” way to share storybooks that we believe is implied in some of the professional literature.