Teaching for Understanding: Urban School Teachers

1
Teaching for Understanding Urban School
Teachers Implementation of a Rich Mathematics
Problem
Spencer, J. LessonLab Research
Institute University of San Diego

• Trends
• Teachers provided warm-up problems that
  mirrored or were very similar to On An Elevator
• Example There are 7 pieces of fruit in a
  basket. The basket cannot hold more than 50
  pounds. The watermelons weigh 7 pounds each and
  the cantaloupes weigh 3 pounds each. How many
  watermelons and how many cantaloupes might there
  me in the basket?
• 2. Teachers gave away a great deal of the work
  and direction necessary for solving the problem
  at the beginning of the lesson.
• Example Students were directed to create a chart
  conceived of and partially or completely filled
  in by the teacher
• 3. Computational work- not conceptualizing,
  conjecturing or proving- dominated the public
  work of the classroom.
• Example The majority of the private
  teacher-student talk dealt with student
  multiplication and addition errors resulting from
  different groupings of teachers and students

• Background
• TIMSS 1999 Video Study Relevant Findings
• Teachers in high achieving countries assign
  mathematically rich problems to their students at
  similar rates to US teachers
• High achieving countries engage students in
  conceptual thinking, maintaining the richness of
  the problems they assign.
• U.S. teachers reduce the complexity of rich
  problems by providing students with step by step
  procedures.

Institute of Education Sciences 2007 Research
Conference
On An Elevator There are 7 people on an elevator.
The elevator can hold no more than one thousand
pounds. The average teacher weighs 150 pounds
and the average student weighs 90 pounds. How
many teachers and how many students might there
be on the elevator?

• What do U.S. Teachers Need in order to Maintain
  the Complexity of Rich Problems?
• Deep understanding of mathematics and of
  students difficulties with specific concepts.
• Models of what it looks like to implement
  problems in rich ways

• Research Design
• On day-2 of each module, teachers analyzed a
  video-taped lesson featuring the teaching of a
  rich problem
• Participating teachers taught the analyzed
  lesson to their students and subsequently engaged
  in discussion of their lessons with fellow
  teachers
• At the culmination of all modules, both ET and
  DT teachers prepared and taught a lesson based on
  a novel rich problem, On an Elevator. (For this
  culminating task, teachers did not watch a
  video-taped lesson based on the problem)
• All culminating lessons were video-taped
• A sample of video-taped lessons were coded and
  analyzed based on features of teaching
  mathematics for understanding
• Videos were further analyzed for trends

• Research Questions
• How did ALFA teachers implement the rich
  problem,On an Elevator?
• What Opportunities did the lessons provide for
  students to develop mathematical understanding?

150t 90s 1000

• Trends, continued
• 4. Gross misunderstanding of variable and the
  role of variables in the problem.
• Example
• Teachers used variables to represent the constant
  value of (weight) as in t 150 and t90 vs. t
  number of teachers and s number of students
• Resulting solutions were a set of equations 6s
  1t 1000 5s2t 1000 4s 3t 1000, etc. vs.
  150t 90s 1000
• Teachers provided common definition of the term
  variable as, A variable is a letter that
  represents a number
• Most teachers did represent the problem
  symbolically as an inequality
• One teacher provided in-depth and accurate
  discussion of variable
• 5. Reformy practices such as group work and
  statements such as, there may be more than one
  answer, occurred, but were not exploited to the
  benefit of deepening mathematical understanding.
• Example
• Computational complexity of student solutions
  varied yet was not interrogated
• Distinction not made between number of solution
  methods (i.e.addition vs. multiplication) and the
  number of solutions (i.e. 5 students and 2
  teachers, 4 teachers and 3 students, etc.)
• Group work was used mainly for the purpose of
  students checking their computation with one
  another

• PD Structure and Content
• 40 hours (6 pull-out days 3 1-hour meetings)
• Paced according to district instructional guide
• Technology-enhanced modules focus on three
  6th-grade core concepts fractions, ratio and
  proportion, and expressions and equations.
• Two pull-out days plus a 1-hour sharing meeting
  per module.
• Day 1 Content
• Exploration
• Teachers deepen their
• understanding of key
• math content through the
• analysis of other teachers
• videotaped math discussions
• and through problem solving tasks.
• Day 2 Lesson

Sample Twelve video-taped lessons of On an
Elevator
Measuring Teaching Mathematics for
Understanding Each of the 12 videos was scored
using a Teaching Mathematics for Understanding
Rubric Cognitive Development Extent to which
the series of lessons promotes command of the
central concepts or big ideas of the
discipline, and instruction generalizes from
specific instances to larger concepts or
relationships. Extent to which teacher listens
to students and responds in ways that scaffold
student understanding toward larger
understanding. Explanation Justification Extent
to which students are expected to explain and
justify their reasoning and how they arrived at
solutions in problems (both orally and in written
assignments). The extent to which students
mathematical explanations and justifications
incorporate conceptual, as well as computational
and procedural arguments. Mathematical
Discourse Extent to which the classroom social
norms foster a sense of community in which
students feel free to express their mathematical
ideas honestly and openly. Extent to which the
teacher and students talk mathematics, and
students are expected to communicate their
mathematical thinking clearly to their peers and
teacher using the language of mathematics.
Average Scores in each dimension were between 1
and 2 on the 4-point rubric Adapted from,
Mathematics Observation Rating Guide, (2003),
CRESST Artifact Project, Battey, D., Borko, H.,
Gilbert, M., Kuffner, K., Spencer, J. Stecher,
B.

• Next Steps/Implications
• Video-taped lessons demonstrate a need for
  greater understanding and focus on variable in PD
• Year 2 videos will be analyzed to study the
  effects of second year of PD on teacher practice
• Teachers must be assisted in moving beyond
  surface aspects of reform based mathematics
  instruction

Contact information Joi Spencerjoi.spencer_at_sandie
go.edu
This study is funded by IES through the Teacher
Quality Program Grant R305M030154