COURSE TITLE:
USING CHILDREN'S MATH THINKING TO GUIDE MATH INSTRUCTION
NO OF CREDITS:
5 QUARTER CREDITS
[semester equivalent = 3.33 credits]
INSTRUCTOR:
CHRISTINE WILLIAMS, Master of Teaching
206-286-0771
chris@alcuinschool.com
WA CLOCK HRS:
OREGON PDUs:
50
50
COURSE DESCRIPTION:
Cognitively guided instruction (CGI) is a research-based method that allows teachers to guide
math instruction based on the childrenʼs own understanding of math, rather than using a predetermined
sequence provided by textbook guides.
Children enter school with a great deal of intuitive knowledge about math. Rather than teaching
teachers how to teach, this course will provide teachers with a body of knowledge about how children
think about math and provide guidance in how to use that information to guide math instruction.
This course is appropriate for teachers of prekindergarten through fifth grade. You will need the
book Childrenʼs Mathematics: Cognitively Guided Instruction by Thomas P. Carpenter, et al (CD included)
and a study packet from the instructor.
LEARNING OUTCOMES:
Upon completion of this course, participants will have:
1. Learned how to understand childrenʼs thinking and use that information to inform math instruction.
2. Learned 11 types of addition and subtraction problems, and observed the 6 types of direct modeling
strategies and 5 types of counting strategies used by children to solve those problems.
3. Observed and learned the 3 types of grouping and partitioning problems and corresponding types of
direct modeling strategies used by children to solve them, as well as the 3 types of related problems.
4. Learned how to use cognitively guided multiplication and division problem solving to solve area, array
and combination problems.
5. Learned how to use word problems and manipulatives to provide problem-solving scenarios that elicit
grouping by ten and promote an understanding of the base-ten concept.
6. Learned to recognize the 3 direct modeling and counting strategies children use to solve multi-digit
problems.
7. Learned the 3 types of invented algorithms and how to promote the use of them in order to avoid
some of the misconceptions children develop when using traditional algorithms.
8. Learned the relationship between problem solving and the development of basic math skills (e.g.,
counting, math facts) and how to integrate them for maximum benefit.
9. How to introduce cognitively guided instruction into math instruction.
10. Learned how to create cognitively guided math lesson plans that are effective, efficient and engaging.
COURSE REQUIREMENTS:
Completion of all specified assignments is required for issuance of hours or credit. The Heritage Institute
does not award partial credit.
HOURS EARNED:
Completing the basic assignments (Section A. Information Acquisition) for this course automatically earns
participantʼs their choice of 50 Washington State Clock Hours or 50 Oregon PDUs. The Heritage Institute
is an approved provider of Washington State Clock Hours and Oregon PDUs.
Using Childrenʼs Math Thinking
1
Approved 4/18/2013
UNIVERSITY QUARTER CREDIT INFORMATION
REQUIREMENTS FOR UNIVERSITY QUARTER CREDIT
Continuing Education Quarter credits are awarded by Antioch University Seattle (AUS). AUS requires
75% or better for credit at the 400 level (Upper Division) and 85% or better to issue credit at the 500 level
(Post-Baccalaureate). These criteria refer both to the amount and quality of work submitted.
1. Completion of Information Acquisition assignments 30%
2. Completion of Learning Application assignments 40%
3. Completion of Integration Paper assignment30%
CREDIT/NO CREDIT (No Letter Grades or Numeric Equivalents on Transcripts)
Antioch University Seattle (AUS) Continuing Education (CE) Quarter credit is offered on a Credit/No
Credit basis; neither letter grades nor numeric equivalents are on a transcript. 400 level credit is equal to
a “C” or better, 500 level credit is equal to a “B” or better. This information is on the back of the transcript.
AUS CE quarter credits may or may not be accepted into degree programs. Prior to registering determine
with your district personnel, department head or state education office the acceptability of these credits for
your purpose.
ADDITIONAL COURSE INFORMATION
NOTES:
•
You may work collaboratively with other teachers and submit joint assignments on all but the final
Integration Paper, which must be individually authored and submitted.
•
Alternatives to written assignments (video or audio tape, photo collage, a collection of products,
letters to editor, brochure and Web pages) may be submitted as substitute assignments with the
instructorʼs prior approval.
•
To maintain privacy, please do not refer to students in your papers by their actual names, but
rather use an alias or designation such as “Student A.”
REQUIRED TEXT:
• Childrenʼs Mathematics: Cognitively Guided Instruction by Thomas P. Carpenter, Elizabeth Fennema,
Megan Loef Franke and Linda Levi. This book comes with a CD.
• Study packet, ordered from the instructor.
MATERIALS FEE:
• Childrenʼs Mathematics is available from Amazon for about $15.
• To request the study packet and copy of Childrenʼs Mathematics: Cognitively Guided Instruction, email
the instructor. There is no fee.
HEADING REQUIRED FOR ALL ASSIGNMENTS:
A heading is required; please use the following format:
Your Name:
Instructor Name:
Course Number:
Course Name:
Date:
Level: Clock/ PDU/Credit (400 or 500)
Assignment #:
Using Childrenʼs Math Thinking
2
Approved 4/18/2013
ASSIGNMENTS REQUIRED FOR HOURS OR UNIVERSITY QUARTER CREDIT
A. INFORMATION ACQUISITION
Assignment #1:
Introduce yourself with a 1-2 page background statement that includes the following: a) Your current
professional situation? b) Why you decided to take this course?
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #1ʼ.
Assignment #2:
Read Chapter 1 in Childrenʼs Mathematics and write a 2-4 page response highlighting the following: (a)
How do the strategies that adults and children use to solve simple problems reflect the conceptual
understanding they have of different kinds of subtraction problems? (b) How do Joseʼs self-taught
counting strategies reflect a higher level of abstract mathematical understanding than Tanyaʼs? (c) What
kind of direct modeling, counting or derived math fact strategies have you seen children use in your
classes?
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #2ʼ.
Assignment #3:
Request the study packet from the instructor. Read Chapters 2 through 8, watching the video lessons
provided. Respond to the questions in the Study Question Guide provided in the study packet.
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #3ʼ.
Assignment #4:
Fill in the CGI Chart, providing the type of problem, strategy or story problems.
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #4ʼ.
Assignment #5:
Create two addition story problems and two subtraction story problems that promote the development of
invented strategies, referring to Chapter 15, p. 80.
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #5ʼ.
Assignment #6:
Review Chapters 7 and 8 and write a 2-3 page Classroom Plan that reflects the changes your classroom
organization and teaching that would be necessary to implement CGI. Divide the paper into four sections
and answer: 1) How will the problems be set (e.g., determined by students? by teacher?) How will using
CGI affect your lesson planning practices? What kind of changes do you foresee? 2) How will you
organize your classroom for problem solving (e.g., grouping)? For discussion? 3) How will the children
share their strategies (e.g., with partners? with class?)? What role will you play during these discussions?
4) How will you track individual progress? How will you determine over time, for example, if individual
students are progressing from basic modeling to more sophisticated strategies like counting or using
derived facts? How will you assess individual studentsʼ conceptual understanding?
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #6ʼ.
This completes the assignments required for Hours.
Continue to the next section for additional assignments required for University Quarter Credit.
Using Childrenʼs Math Thinking
3
Approved 4/18/2013
ADDITIONAL ASSIGNMENTS REQUIRED FOR UNIVERSITY QUARTER CREDIT
B. LEARNING APPLICATION
In this section you will apply your learning to your professional situation. This course assumes that most
participants are classroom teachers who have access to students. If you are not teaching in a classroom,
please contact the instructor for course modifications. If you are a classroom teacher and start or need to
complete this course during the summer, please try to apply your ideas when possible with youth from
your neighborhood, at a local public library or parks department facility (they will often be glad to sponsor
community-based learning) or with students in another teacherʼs summer classroom in session.
Assignment #7: (Required for 400 and 500 Level)
Write one age-appropriate lesson plan for your class, for addition or subtraction, using the Lesson
Planning Guide included in the packet sent by the instructor.
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #7ʼ.
Assignment #8: (Required for 400 and 500 Level)
Write two age-appropriate lesson plans for your class, for multiplication or division, using the Lesson
Planning Guide.
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #8ʼ.
Assignment #9: (Required for 400 and 500 Level)
Write two age-appropriate lesson plans for your class, for multi-digit numbers using the Lesson Planning
Guide.
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #9ʼ.
Assignment #10: (Required for 400 and 500 Level)
Video the delivery of one of your lesson plans (delivered by yourself or by a colleague) and post it on
YouTube as Unlisted, then email the instructor the link.
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #10ʼ.
Assignment #11: (Required for 400 and 500 Level)
Write a 2-3 page critique of your delivery in assignment 10, dividing the paper into four sections,
answering the following questions and providing specific examples for each. 1) What are the strengths
and weaknesses of the lesson? 2) Was the lesson effective? Did it meet all the childrenʼs needs? How did
you determine that? How did the instructor help struggling children? 3) Was the lesson efficient? Did it
use the instructorʼs and childrenʼs time and effort in an efficient manner? 4) Was the lesson engaging?
Did the lesson gain and maintain the childrenʼs attention and focus? How did you determine that?
Send critique to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #11ʼ.
Assignment #12: (Required for 400 and 500 Level)
Describe in 2-3 pages the benefits and limitations of implementing CGI in math instruction. What kinds of
problems did you experience in the delivery of the lesson that might suggest areas of weakness in this
instruction method that you will need to address? Will you need to use any other kind of math instruction
to supplement CGI?
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #12ʼ.
Using Childrenʼs Math Thinking
4
Approved 4/18/2013
500 LEVEL ASSIGNMENT
Assignment #13: (500 Level only)
In addition to the 400 level assignments, complete one of the following and send to instructor:
Option A)
Request from the instructor and read Should There Be a Three-Strikes Rule Against Pure Discovery
Learning? The Case for Guided Methods of Instruction by Richard E. Mayer. Write a 3-4 page paper,
dividing it into sections to answer these questions: 1) Summarize the research that indicates that
unguided discovery methods are not effective instructional methods. 2) Explain how these criticisms
reflect and/or do not reflect the CGI approach to math instruction. What might be potential weaknesses of
CGI, given this research? 3) What role will corrective feedback, modeling, focused goals, hints and other
guided discovery methods play in your CGI math instruction?
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #13Aʼ.
OR
Option B)
Another assignment of your own design with the instructorʼs prior approval.
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #13Bʼ.
C. INTEGRATION PAPER
(Required for 400 and 500 Level)
Assignment #14:
Write a 2-3 page Integration Paper responding to these questions:
1. What did you learn vs. what you expected to learn from this course?
2. What aspects of the course were most helpful and why?
3. What further knowledge and skills in this general area do you feel you need?
4. How, when and where will you use what you have learned?
5. How and with what other school or community members might you share what you learned?
Send to instructor: chris@alcuinschool.com. Subject line to read ʻCGI #14ʼ.
INSTRUCTOR COMMENTS ON YOUR WORK:
Please indicate by email to the instructor that you would like to receive comments on your assignments.
QUALIFICATIONS FOR TEACHING THIS COURSE:
Christine Williams, Master of Teaching, received an Association Montessori Internationale (AMI)
diploma from the Centro de Estudios in Mexico City in 1979, her BA from the University of Texas in 1986,
and a Master of Teaching with an Early Childhood Endorsement from Seattle University in 2002. She is
trained in early dyslexia intervention, Linguistic Remedies and Handwriting Without Tears. She has
worked with ages 3 to 8 since 1981, specializing in meeting the needs of early learners, with a focus on
early literacy and math. She has guest lectured on early math instruction at the Antioch University teacher
training program and has taught courses at the Heritage Institute since 2006.
Using Childrenʼs Math Thinking
5
Approved 4/18/2013
USING CHILDREN'S MATH THINKING TO GUIDE MATH INSTRUCTION
BIBLIOGRAPHY
Carpenter, Thomas P., Franke, M. L., and Levi, L. Thinking Mathematically: Integrating Arithmetic and
Algebra in Elementary School, Heinemann, 2003, paperback, 160 pages, ISBN 978-0-325-00565-2. This
volume helps children deepen their understanding of arithmetic and provides a foundation for learning
algebra. CD included. As with the volume on fractions and decimals, teachers can use these texts to
deepen their own understanding of arithmetic and algebra.
Empson, Susan B., and Levi, L. Extending Children's Mathematics: Fractions & Decimals: Innovations
In Cognitively Guided Instruction, Heinemann, 2011, paperback, 272 pages, ISBN-10: 0325030537. For
those of you working with older students, this follow-up to Children's Mathematics: Cognitively Guided
Instruction, covers fractions and decimals.
Hieber, James, Carpenter, T. P. Fennema, E., Fuson, K. C., Wearne, D., Murray, H. Olivier, A., and
Human, P. Making Sense: Teaching and Learning Mathematics with Understanding, Heinemann, 1997,
978-2854186055. Making the leap from traditional instruction to cognitively guided instruction is not easy.
Making Sense focuses on issues, such as the role of the teacher, the culture of the classroom and
accessibility, rather than focusing the nuts and bolts of ʻhow toʼ teach math. It is, therefore, a helpful tool
for teachers who would like to become more aware of the culture of teaching that surrounds them. Its
chapters on A Day in the Life are particularly helpful for helping teachers understand the difference
between cognitively guided instruction and traditional instruction.
Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental
Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series),
Routledge, 2nd Edition, 2010, 978-0415873840. This book was a bestseller when it first came out, with
good reason. A favorite of mine, Lipingʼs book focuses not on student understanding of math, but on
teacher understanding. She demonstrates that it is not a teacherʼs education background that determines
the depth of mathematical understanding, but to their approach to math instruction. (Chinese teachers
have less formal education than American teachers, but demonstrate deeper conceptual understanding.)
Contrasting the conceptual knowledge of American teachers with that of Chinese teachers, she
demonstrates that Chinese teachersʼ approach to mathematical instruction helps deepen their own
conceptual understanding, while American instruction is limited to an understanding based on algorithms
and other procedural approaches. As with the previous book in this bibliography, this book will help you
understand the culture of teaching that influences your instruction.
Using Childrenʼs Math Thinking
6
Approved 4/18/2013
ADDITIONAL RESOURCES
For reviews of research about CGi by researchers other than the authors, read...
Borko, H, and R. Putnam. 1996. “Learning to Teach.” In Handbook of Educational Psychology, edited by
D. Berliner & R. C. Calfee. New York: Macmillan.
Decorte, E., Greer, B., and L Verschaffel. 1996. “Mathematics Teaching and Learning.” In Handbook of
Educational Psychology, edited by D. Berliner & R. C. Calfee. New York: Macmillan.
Ginsburg, H. P., Klein, A, and P. Starkey. 1998. “The Development of Childrenʼs Mathematical Thinking:
Connecting Research with Practice.” In Handbook of Child Psychology, edited by I. E. Sigel and K. A.
Renninger. New York: Wiley.
Wilson, S. M. and J. Berne. 1999. “Teacher Learning and the Acquisition of Professional Knowledge: An
Examination of Research on Contemporary Professional Development.” In Review of Research in
Education (Vol. 24), edited by A. Iran-Nejad and P. D. Pearson. Washington, DC: AERA.
Using Childrenʼs Math Thinking
7
Approved 4/18/2013