Perhaps the subtitle of "Math Wise!" ($29.95 in paperback from Jossey-Bass), by Jim Overholt and Laurie Kincheloe, sums it up: "Over 100 Hands-On Activities that Promote Real Math Understanding" for Grades K-8. Now in its second edition, the book is aimed at helping students understand concepts that range from counting to data analysis.
Overholt is an education professor at Chico State University; Kincheloe teaches mathematics at Butte College. Both have extensive experience working with K-12 students and their parents, and especially with elementary school teachers.
A grade level is provided for each activity, which include those that are "concrete/manipulative" (such as illustrating division using paper clips); "visual/pictorial" (3 x 5 can be shown by three horizontal lines crossed by five vertical ones); and "abstract procedures" (in Post-It Mental Math a student tries to guess the numerals stuck to his or her back using clues provided by others in the group). "Math Wise!" encourages teachers not only to "instruct students in regard to mathematical mechanics but also enable them to gain a true understanding of the concepts involved."
Each activity explains what the exercise is designed to accomplish (such as practicing computation, getting friendly with fractions, or learning about probability). There's a list of materials needed and detailed steps and examples that give ample guidance for teachers. There are also "extensions" that provide additional ways of using the activity. Many of the activities in the book are for entire classes, others are for small groups, and still others work best as independent projects.
My own mathematical knowledge is, uh, a fraction of what it should be, and I found myself delighted at the creative approaches--and sophistication--that "Math Wise!" embraces. Take "Palindromic Addition." (A palindromic number is reversible.) The activity uses pencil and paper. Pick a non-palindromic number less than 1000, then add its reverse, and continue until a palindrome is produced.
The example given is the number 158. When 851 is added to it, the sum is 1009, but it's not a palindrome. So add 9001 to 1009 and you get 10010--still not a palindrome. Ah, but add 01001 to 10010 and you get 11011. Our palindrome at last! Great fun for kids in school--and for adults in boring meetings.