CI 512: Teaching and Learning
Tuesday, 7/26: Week 2
Conceptual vs. Procedural Understanding
Class Outline
Conceptual and Procedural Understanding
Note Taker: Derek Baily
Observer: Karen Czarnecki
 Logistics (9:00-9:05)
 Theory vs. Method (9:05-9:10)
 Conceptual and Procedural Understanding
 Small Group Discussions (9:10-9:40)
 Whole Class Discussion (9:40-10:10)
 Break (10:10-10:20)
 TIMSS Video Study
 Video Viewing (10:20-11:20)
 Whole Class Discussion (11:20-11:40)
 Observer Observations (11:40-11:45)
 Conclusions and Exit Cards (11:45-11:50)
Logistics: Readings for Thursday
Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.).
(1999). How People Learn: Brain, Mind, Experience,
and School. Washington D.C.: National Academy Press.
Lobato, J. (2003). How design research can inform a
rethinking of transfer and vice versa. Educational
Researcher, 32(1), 17-20.
Carraher, T., Carraher, D., & Schliemann, A. (1985).
Mathematics in the streets and in schools. British
Journal of Developmental Psychology, 3, 21-29
 Optional Reading:
Robertson, W. (1990). Detection of cognitive structure with
protocol data: Prediction performance on physics
transfer problems. Cognitive Science 14, 253-280.
Reading Responses
Keep it brief and to the point
No longer a 350 word limit
Exit Card Reflections
 Readings and discussion don’t focus on my specific
teaching context
 I/we need to do a better job at bringing conversation back
to how these examples relate to our specific foci
 Request for more reflection time
 Small Group Discussions
 Some small groups had successful discussions
 Other small groups were dominated by a single speaker
 Written questions for the small group time
Theory vs. Method
 Behaviorism as a theory of learning
 Learning is exhibited only through external
behaviors
 Classical Conditioning: Subjects can be
conditioned to have involuntary responses to
artificial stimulus
 Operant Conditioning: Positive (or negative)
reinforcement strengths the bonds between
stimulus and response
 Drill and practice as a method for
instruction
What should we learn?
 Conceptual and procedural understandings
do not form a theory for learning: instead
they seek to answer the question “What
should one learn?”
 Orientations toward conceptual and
procedural understandings have
implications for teaching practices
Small Groups
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
*Mike P.
*Westie
*Mike T.
*Karen
*Kyle
*Greg
Sean
Derek
Michael
Casey
Teale
Colin
Martin
Nick
Arielle
Chad
Mike M.
Chai
Laura
Carlos
Iman
* Group Moderator:
• Help encourage all group members to participate in the
discussion
• Ensure that all of the discussion questions are covered
Small Group Questions
 What is procedural/instrumental understanding? Why is
it important?
 What is conceptual/relational understanding? Why is it
important?
 Give a description of both conceptual and procedural
understanding for (one or more) of the following topics:
 The vertex of a parabola
 The human circulatory system
 Gravity
 A topic of your group’s choosing
 For one of the above topics, design an activity/questions
that help to assess or develop:
(a) conceptual understanding
(b) procedural understanding
Strands of Mathematical Proficiency
 Conceptual understanding—comprehension of
mathematical concepts, operations, and relations
 Procedural fluency—skill in carrying out procedures
flexibly, accurately, efficiently, and appropriately
 Strategic competence—ability to formulate, represent,
and solve mathematical problems
 Adaptive reasoning—capacity for logical thought,
reflection, explanation, and justification
 Productive disposition—habitual inclination to see
mathematics as sensible, useful, and worthwhile,
coupled with a belief in diligence and one’s own efficacy.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding It
Up: Helping Children Learn Mathematics.
download.nap.edu/catalog.php?record_id=9822
10 minute Break
TIMSS Study
 Trends in International Math and Science Study
 Collected data on 4th and 8th grade students in more
than 60 countries over a 12 year period (1995, 1999,
2003, 2007)
 Video data was collected in three countries: America,
Japan and Germany
 More information at nces.ed.gov/timss/or google TIMSS
TIMSS Methodology
 A subset of schools in each nation were randomly
selected, and a random class period from each school
was selected to be filmed. No substitutions were
allowed.
 100 classes in Germany, 50 classes in Japan, and 81
classes in the U.S.
 Teachers were asked to teach a typical lesson, and selfreported data on the previous lesson was collected
 Although the presence of video cameras might affect
classroom behavior, it is unlikely to change the types of
follow-up questions a teacher would ask students
TIMSS Video Questions
 What types of (a) procedural (b) conceptual
understandings are evidenced in this video?
 What do you think the teachers’ goals are in
terms of conceptual and procedural
understanding?
 How does the teacher encourage (a) procedural
and (b) conceptual understanding in this video?
 How would you characterize the teachers’ beliefs
with respect to conceptual and procedural
understanding?
Analysis of TIMSS
Stiegler & Heibert (1999) The
Teaching Gap
International group of teachers and
teacher educators reviewed the
videos
Complex analytic process of
repeated video viewing
Analysis of TIMSS
Mottos for Teaching Mathematics
 Japanese: “Structured problem solving”
 German: “Developing advance
procedures”
 American: “Learning terms and practicing
procedures”
Analysis of TIMMS
(Stiegler & Hiebert)
“In the Japanese lessons, there is
the mathematics on the one hand,
and the students on the other. The
students engage with the
mathematics, and the teacher
mediates the relationship between
the two.”
Analysis of TIMMS
(Stiegler & Hiebert)
“In Germany, there is the
mathematics as well, but the
teacher owns the mathematics and
parcels it out to students as he sees
fit, giving facts and information at
just the right time.”
Analysis of TIMMS
(Stiegler & Hiebert)
“In U.S. lessons, there are the
students and there is the teacher. I
have trouble finding the
mathematics. I just see interactions
between students and teachers.”
Observer Observations
Exit Cards
 Rate your level of participation for today
(0-3)
 What was one thing you learned during
today’s class?
 What is one thing you would like to know
more about from today’s class?