Moomaw_1598571196_ch02_019-044.qxd
2/1/11
8:54 AM
Page 24
Excerpted from Teaching Mathematics in Early Childhood
by Sally Moomaw, Ed.D.
Teaching Mathematics in Early Childhood
24
then begin skipping some objects as their pointing finger moves faster than they say the
number words. Conversely, children may appear to forget which objects they have already
counted and count them more than once. In terms of stable order, many children know the
order of the counting words up to a certain point, but then say numbers in an incorrect order
or repeat numbers that they have already used. It is interesting, though, that even after they
have used their memorized string of counting words, children usually fill in with number
words rather than with other labels, such as letter names. Finally, some children can count
objects but do not understand that the last object they counted represents the total for the
entire group. They have not yet constructed cardinality. The assessment section of this chapter focuses on how to analyze children’s counting in a variety of situations and discusses
how this information guides teachers’ instructional practice.
Application of the order-irrelevant principle is rarely apparent in preschool, but may
begin to emerge during kindergarten. The following is an example of one child’s “ah-ha”
moment when she suddenly made the connection between the order-irrelevant principle
and cardinality.
EXAMPLE 2.4
The students in a summer kindergarten class were interested in how many days they would
meet over the summer. Each day, during the beginning circle time, the teacher added one
teddy bear shape to the bulletin board to represent that day. Together, the teacher and children then counted all of the bears to determine how many days they had already met. They
always counted the row of bears from left to right, in typical reading fashion.
One day the teacher decided to try an experiment. When it was time to count the bears,
she counted from right to left rather than from left to right. The children protested vociferously. “Now we have no idea how many days of summer school we’ve had so far,” they
informed her. Surprised, the teacher asked, “Do you want me to count in the other direction?” The reply was a definitive and unanimous “Yes!” The class then counted the bears
from left to right. None of the children seemed to notice that the total was the same as the
total the teacher had previously counted.
Intrigued by their response, the teacher decided to repeat the procedure on the following day. Once again the children protested. The bears were re-counted in the “correct” direction, and the children seemed oblivious to the fact that the totals from both directions were
the same. The same scenario unfolded each day for two weeks. On the 10th day of the
teacher’s experiment, something different happened. Once again, the teacher had counted
from right to left, the children had protested, and the teacher had asked whether she should
count from left to right. Suddenly, Angela began bouncing up and down on her mat in
excitement. “Wait, wait, wait!” she exclaimed. “It doesn’t matter. Whichever way you
count, you still get the same answer.” This was a big moment for Angela and a revealing
one for the teacher. Angela had just figured out the principle of order irrelevance and accurately explained it to her classmates. Her teacher was excited to have witnessed this event.
She realized that her “off the cuff” experiment had provided the experiences that Angela
(and, shortly thereafter, many of her classmates) had needed to understand this big idea. The
teacher integrated this knowledge into her pedagogy and began counting sets of objects in
various orders during many kindergarten math experiences.
What about the abstraction principle? Would teachers ever see an example of children
wrestling with that idea during preschool or kindergarten? Actually, an example earlier in
this chapter described a group of children who took turns, each adding a cookie to a jar. The
children were confident that they could find out how many cookies were in the jar by simply counting the cookies; however, they were stumped when their teacher asked for another
Brookes Publishing | www.brookespublishing.com | 1-800-638-3775
© 2011 | All rights reserved
Order Form
BROOKES PUBLISHING • P.O. Box 10624, Baltimore, MD 21285-0624
ABOUT YOU (write in your specialty and check one field that best applies)
Specialty
Birth to Five
K–12
Clinical/Medical Personnel
4-year College/Grad.
Comm. College/Vocational
Association/Foundation
Comm. Services
Name
Address
residential
commercial
City
State
ZIP
Country
Your list code or savings code (if applicable):
E-mail
Yes! I want to receive e-mail about new titles and special offers.
(Your e-mail address will not be shared with any other party.)
We auto-confirm all orders by mail; please provide an email address to receive confirmation of order and shipping.
Qty
Stock #
Title
Price
Subtotal
PAYMENT METHOD
State sales tax (MD and WA*)
Check enclosed (payable to Brookes Publishing Co.)
(GST) Business tax (CAN only)
Purchase Order attached (bill my institution)
*Add 2% to product total for P.O. handling fee
P.O. customers: 2% of subtotal
American Express (15 digits)
Shipping (see chart below)
Discover (16 digits)
Total (in U.S. dollars)
MasterCard (16 digits)
Visa (13 or 16 digits)
*WA residents: Please add applicable sales tax by region.
Sales tax should be calculated based on the total order
(including shipping) in U.S. dollars. If sales tax is calculated
incorrectly the customer service department will correct it
prior to processing your order and the adjusted total will
appear on the invoice included with your shipment.
Credit card account number __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __
Security code (3 or 4 digit code on back of card): __ __ __ __
Expiration date __ __ /__ __ Signature
Convenient ways to order:
CALL toll-free 1-800-638-3775 M-F, 9 a.m. to 5 p.m. ET.; FAX 410-337-8539;
MAIL order form to: Brookes Publishing Co., P.O. Box 10624, Baltimore, MD 21285-0624:
ON-LINE www.brookespublishing.com
Money-back guarantee! Ordering with Brookes is risk-free. If you are not completely satisfied, you may return books and videotapes within
30 days for a full credit of the purchase price (unless otherwise indicated). Refunds will be issued for prepaid orders. Items must be returned in resalable
condition. All prices in U.S.A. dollars. Policies and prices subject to change without notice. Prices may be higher outside the U.S.A.
STANDARD GROUND SHIPPING & HANDLING
(For other shipping options and rates, call 1-800-638-3775, in the U.S.A. and Canada, and 410-337-9580, worldwide.)
Continental U.S.A. territories & protectorates ; AK, HI & PR
Add*
For subtotal of
Canada
For subtotal of
Add*
US$55.00 and under
$6.49
US$67.00 and under
$9.99
US$55.01 and over
12%
US$67.01 and over
15%
**
†
‡
Continental U.S.A. orders ship via UPS Ground Delivery.
†
U.S.A. territories & protectorates orders ship via USPS.
‡
AK, HI, and PR please add an additional US$12.00. Orders ship via UPS Air.
Please call or email for expedited shipping options and rates.
**
11/11
03/10
Orders for Canada are consolidated for shipping twice each month.
For minimum shipping time, please place your orders by the 9th or
24th of each month.
*calculate percentage on subtotal
printed from www.brookespublishing.com