Importance of Informal Mathematics Knowledge

Seo and Ginsburg conducted an interesting study of the types of informal mathematical activities in which four- and five-year-old children were engaged in natural settings (2004). The researchers classified observable activities by their mathematics characteristics:

• Classification activities involved sorting, grouping, or categorizing objects.
• Magnitude activities were statements made about global magnitude of objects, direct or side-by-side comparisons, or judgments without quantification.
• Enumeration activities involved saying number words, counting, subitizing, and even reading and writing numbers.
• Dynamics involved putting things together, taking them apart, or making other transformations such as turning and flipping.
• Pattern and shape activities included identifying or creating patterns or shapes and exploring the properties and relationships of shapes. (Seo & Ginsburg, 2004, pp. 93–94)

After coding 15-minute videotaped segments of ninety children, the researchers concluded that most children (88%) engaged in mathematical activities naturally and that about forty-three percent of the time observed was spent in math-like activities. Very significant in their findings was the conclusion that there were no income level or gender differences in these activity levels. In general, children engaged in pattern and shape activities the most, and classification the least, and were capable of achieving quite complex levels of performance. For example, some children demonstrated estimating the number in a set without counting and transforming a rhombus shape into a trapezoid.

Both quadrilaterals because of their four straight sides, a trapezoid is a quadrilateral with two parallel sides; a rhombus is a quadrilateral with both pairs of opposite sides parallel (also called a rhomb or diamond).

Informal mathematics knowledge is also important for older students. It may be harder to extract informal from formal learning and some misconceptions may be more rigidly held by older students, but teachers need to assess prior knowledge in whatever form for better connections to new mathematics learning. For example, Jamie is a student in Mrs. Banks’ third-grade class and was taught in previous grades that applying multiplication to whole numbers would result in larger numbers. This concept was confirmed with Jamie’s experiences grocery shopping with her mother. If one apple costs 20¢, then four apples cost 80¢. Buying a six-pack of soda for $2.50 was cheaper than buying six sodas at the individual price of 50¢. Mrs. Banks is attempting to explain why would result in a smaller number. She could begin with Jamie’s understanding that six sodas 50¢ would result in a $3.00 purchase. Shown another way, that would be of a dollar, or 3 dollars.

Misconceptions of older students are often caused by inadequate concept development through a wide enough range of examples during formal instruction or by limited informal experiences. For example, if a child had only cube-shaped blocks to play with, he could not compare other three-dimensional shapes and their properties. Viewing a diagram of a pyramid would be confusing for this child.