Using a Journal Article as a Professional Development Experience
Number and Operations
Recommended Grade Level: Teachers of Grades K-2
Title: Counting Collections
Authors: Schwerdtfeger, Kern Julie and Angela Chan
Journal: Teaching Children Mathematics
Issue: March 2007, Volume 13, Issue 7, pp. 356-361
Title: Counting Skills - A Foundation for Early Mathematics
Author: Frank, Alan R.
Journal: Arithmetic Teacher
Issue: September 1989, Issue 1, pp. 14-17
Rational/Suggestions for Use
Although counting is one of the best ways to help children develop number sense and other important mathematical ideas, we do not do enough of it in elementary schools. Children need many and varied experiences with counting to learn which numbers come next, how this number sequence is related to the objects in front of them, how to keep track of which ones have been counted and which still need to be counted (Fuson 1988). Experience with counting provides a solid foundation for future experience with addition, subtraction, multiplication, and division (National Research Council, 2001). This article provides a rationale for why and a guide for how to have children count objects all school year long.
- Ask participants, “What does ‘counting’ mean?” Allow a minute of alone time for thinking, then pair with some one else in the room to share for two minutes and finally have a representative of each group share what they discussed. Facilitator records the different types of counting shared in a way that all participants can observe the recording.
- At this time the facilitator should refer to or have participants read the article by Frank, Alan R., “Counting Skills - A Foundation for Early Mathematics" Arithmetic Teacher (September 1989).
This article offers the following types of counting:
Next have the participants compare their listing with the types in the Frank article.
Have all the participants read the article by Kern Schwerdtfeger and Julie and Angela Chan, “Counting Collections”, Teaching Children Mathematics (March 2007).
As a group write a sequence for counting based on the two articles. Again, record so all can see.
Divide the class into smaller groups and write assessable goals for all the steps in the sequence.
- Rote counting: Saying numbers in sequence
- Point counting: Point to individual objects more than once
- Rational counting: This is an extension of point counting. Students learn to apply the cardinality rule: The last number said after counting tells how many are in a set
- Counting on: Rote or rational counting beginning with a number other than one
- Skip counting: Counting every nth number in a series
As a closing, ask the participants:
- Were any of the ways of counting new to them?
- Are there other ways to count that are not included?
- Have they done all these types of counting with their class?
- In addition to partner sharing in a classroom, what other ways are there for children to be taught counting?
Extensions (Session One)
Have participants read the second paragraph on page 359 and discuss it in relation to experiences their students have with keeping track while counting. Have participants discuss the differences between counting skills and organizational skills which students use to keep track.
Give the assignment and ask for questions. Be sure to explain that the assignment will be the basis for the second session.
- Write a summative assessment for each of the assessable goals written in class.
- Design a 15-minute lesson that teaches different organizing methods for counting and implement the lesson with your class.
- Be prepared to share you lesson and results in Session Two.
- Begin by reviewing the different forms of counting. Ask if any one has any additions or comments to make about counting.
- Have participants share their assessments for the goals. Allow discussions and questions as participants share. The instructor may want to devise a system for sharing these assessments with everyone in the class.
- Have participants share the basic design for their 15-minute lesson on organization for counting. Allow discussions and questions as participants share. Again the instructor may want to devise a system for sharing these lessons with everyone in the class.
- To wrap-up, have participants respond to the statement, “Although counting is one of the best ways to help children develop number sense and other important mathematical ideas, we do not do nearly enough of it in elementary schools”.
Connections to Other NCTM Publications
- Barker, L. (2009, February). Ten is the magic number!Teaching Children Mathematics, 15, 336-346.
- Christy, D., Lambe, K., Payson, C., Carnevale, P., & Scarpelli, D. (2008, April). Alice in numberland: Through the standards in wonderland. Teaching Children Mathematics, 14, 436-446.
- Frank, A. R. (1989, September). A foundation for early mathematics. Arithmetic Teacher, 37, 14-17.
- Goodrow, A. M., & Kidd, K. (2008, September). “We all have something that has to do with tens”: Counting school days, decomposing number, and determining place value. Teaching Children Mathematics, 15, 74-79.
- Kamii, C., & Rummelsburg, J. (2008, March). Arithmetic for first graders lacking number concepts. Teaching Children Mathematics, 14, 389-394.
- Kastberg, S. E., & Walker, V. (2008, May). Insights into our understandings of large numbers. Teaching Children Mathematics, 14, 530-536.
- National Council of Teachers of Mathematics (2000). Principles and Standards of School Mathematics. Reston, VA: Author.
- Whitin, D. J. (2008, April). Learning our way to one million. Teaching Children Mathematics, 14, 448-453.