The Development of Mathematical Thinking

DreamBox Learning
March 22, 2010

Development is multi-fold in nature: it proceeds in patterns that are predictable enough that psychologists, doctors, and educators can point to broad social, physical, and mental behaviors that are typical of any age group. In addition, as the child grows each of these areas develop and impact each other; none develop in isolation. Development is also unpredictable in that it is very individual, and patterns of growth can be unruly: at times aligning with the child's chronological age, or at other times, not. Parents and teachers know this from experience.

Young children already possess a rich assortment of mathematical cognitive abilities when they enter school. Through play with their toys and everyday family activities, they have spontaneously compared, sorted, arranged, and counted objects, explaining what they did and challenging others' explanations. (When their senses falter, they can always fall back on their fail-safe question, why?). Young children are intensely curious about their environment and interact directly with it. Much of what they know is filtered through their perceptions, which are particular to them and can be very unreliable. Children at this intuitive stage will believe that a quantity changes when the arrangement is changed, even if they have counted several times. What's more, they don't question their belief.

Although we describe each age separately as a way to organize the information, we will describe the cognitive and mathematical development of children ages four, five, and six in broad strokes. There are no sharp divisions from one stage to another, but rather there are overlaps that indicate the transitional nature of growth.

As children's minds continue to develop, their brains go through successive stages of growth, and children become less dependent on perception. The quality of children's thinking changes; they think differently and are puzzled by their earlier belief that the quantity changed when the arrangement changed. Their more logical thinking tells them that no matter the arrangement, the quantity will remain the same.

At Four Years Old

  • A four year old may easily compare sets and know which has more, but may not know how many he has.
  • He may struggle to count each object only once, and to name the number for each object.
  • Watch your child play to understand her mathematical knowledge. When she counts, does she touch each object once? Is her voice in sync with her tag?
  • When walking, collect objects she likes. At home count and sort this collection in different ways.

At Five Years Old

  • Five year olds can determine 1-to-1 correspondence - 5 kids need 5 pencils.
  • Once she can count on, she may know which set is more and may sequence sets from smallest to largest.
  • To think about the permanence of a set, put 6 pennies in a row, then change the arrangement. Ask, "Did the quantity change?"
  • Five year olds love repetition. Clapping patterns help him discover sequences and predict what comes next.

 At Six Years Old

  • A six year old knows that 6 can be 5 and 1, or 3 and 3. And he knows that all sets of 6, no matter what objects, are equivalent.
  • He understands that "nothing" is represented by 0, and that any number can be written with the digits 0-9.
  • Dice, cards, and board games can help your child learn addition combinations.
  • Count 8 pennies, then hide 4. Ask "How many are hidden?"

For even more helpful information about your child's mathematical development, visit DreamBox Learning. There you can find individualized online math games that your children can play and even more in-depth research about your child's mathematical development.

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