Transcript The Mathematics Problem Solving Model

The Mathematics
Problem Solving
Model
A Professional Development
Program
Presented by Sarah Enoch
September 10, 2011
The Mathematics Problem Solving
Model



Designed by Education Northwest, an
educational research lab in Portland, OR
Research-based framework
Central themes of the model:


Teaching mathematics through problem-solving
Formative assessment as a tool to enhance
instruction
Assumptions about Student Learning


Two important influences on how children
learn mathematics are the tasks and
problems they engage in and the
interactions they have about them.
Teachers’ ability to understand students’
mathematical development is enhanced by
their ability to notice and describe what
students say and do.
Two important influences on how children learn mathematics
are the tasks and problems they engage in…



Effective instructional practice revolves around the
use of problem solving tasks and student-centered
discussions about these tasks (See, for example,
Boaler & Humphreys, 2005; Hiebert, Carpenter,
Fennema, Fuson, Wearne, Murray, …& Human,
1997; Kilpatrick, Swaffor, & Findell, 2001).
it was found that higher learning gains were
achieved when teachers implemented tasks that
were cognitively demanding (Stein and Lane, 1996)
Students that engage in cognitively demanding
problem solving tasks have opportunities to learn
problem solving, reasoning skills, and higher order
thinking (Wood and Turner-Vorbeck, 2001) .
…and the interactions they have about them


It is necessary to create opportunities for students to
clarify their solutions, providing their reasoning and
justification for the approaches they take and
connecting it to relevant mathematical content
(Hiebert et al., 1997; Kazemi & Stipek, 2001)
Also, students should have opportunities to look for
patterns and make generalizations around problem
solving tasks, attempting to defend and justify their
conjectures, with reasoning playing a central role in
the process (Yackel & Hanna, 2003).
Teachers’ ability to understand students’ mathematical
development is enhanced by their ability to notice and describe
what students say and do



Analysis of student work is a valuable tool for
creating effective mathematical instruction.
An awareness of students’ understanding and
thinking around mathematical concepts and tasks
makes it possible to appropriately choose tasks that
are of high cognitive demand, but still within the
students’ reach (Stein, Grover, et al., 1996)
Knowledge of student work also supports
appropriate teacher questions useful for building
upon student thinking to develop and make
connections between mathematical ideas (Grouws,
2003).
Teacher’s knowledge about students’ thinking can be
improved in the following ways:



Posing problems specifically designed to access students’
thinking (Lesh, Hoover, Hole, Kelly, & Post, 2000)
Asking good questions during instruction that access students’
thinking (Wiliam, 2007) and listening attentively to their students’
thinking (Schifter, 1998)
Examining student work to make a connection between their
idiosyncratic ways of thinking and more conventional
mathematics (Mewborn, 2003)
Mathematical Learning
Objectives



Teachers’ instructional move that lead towards
building conceptual understanding should begin with
and be subsequently guided by the identification of
an instructional learning goal (Sherin, 2002)
the implementation of a problem solving task should
take place concurrently with the mathematical
concepts and procedures in the classroom’s
curriculum (NCTM 2000; Lesh & Zawojewski, 2007)
Implementing a problem-solving tasks is NOT
“Problem-Solving Friday”.
Mathematical Learning
Objectives
Teachers are supported in this element of the
MPSM by:
 Examining the mathematical content and
skills required for implementation of the
MPSM tasks
 Looking at their own curriculum to see where
these tasks would best fit.
Mathematical Tasks

Problem-solving tasks are open-ended




have multiple solution methods
lead to multiple possible solutions
the focus is not on finding the answer, but on the
processes that students use to arrive at their
solutions.
These problems should be non-routine and
challenging, but not inaccessible (Becker &
Shimada, 1997).
Mathematical Tasks: Cognitive Demand

Low-level tasks




require little to no cognitive demand to complete
are focused on producing correct answers
have no connections to related concepts
require little explanations.
Memorization tasks are straight-forward tasks that
require students to recall previously learned
information to solve and do not afford opportunities
to use procedures.
Procedures without connections tasks are algorithmic
in nature, requiring the use of previously learned
procedures.
Mathematical Tasks: Cognitive Demand
High-level tasks:
 require students to explore mathematical relationships,
processes, and concepts
 demand self-monitoring of one’s thinking
 require students to access relevant knowledge from past
experiences
Procedures with connections tasks focus students on the use of
broad procedural pathways for the purpose of deepening
understanding of the concepts underlying the procedures,
requiring students to engage with conceptual ideas in order to
successfully complete the task.
Doing mathematics tasks require complex, non-algorithmic thinking,
and require considerable cognitive effort.

Mathematical Tasks: Cognitive Demand
Teachers are supported in this element of the MPSM
by:
 Being supplied with tasks which were identified by
the professional developers as doing mathematics
tasks.
 analyzing problem solving tasks for the purpose of
categorizing tasks by their level of cognitive demand
 taking low-level tasks from their own curricula and
adapting them to raise the cognitive demand of the
tasks
Implementing Tasks
It was found that several factors contributed to the
decline of cognitive demand during
implementation:
1)
2)
3)
4)
5)
6)
tasks became non-problematic as the teacher either
reduced the requirements of the task or completed the
challenging steps for the students;
the task was inappropriate for the students, indicating a
lack of knowledge about students on the part of the
teacher;
the focus shifted to finding the correct answer;
too much or too little time was allotted to complete the
task;
a lack of accountability on the part of the students;
classroom management issues.
Implementing Tasks
Several factors contribute to the maintenance of
cognitive demand:
1)
2)
3)
4)
5)
6)
7)
the task built on students’ prior knowledge;
an appropriate amount of time was allotted for students
to work on the task;
a high level of performance was modeled;
there was sustained pressure for explanation and
meaning;
scaffolding was provided without taking away from the
complexity of the task;
students were encouraged to self-monitor their work;
the teacher helped the students to draw conceptual
connections
Implementing Tasks: The Use
of Teacher Questioning
Maintaining the cognitive demand of a problem-solving task is
particularly challenging for teachers when they see their students
are struggling with a task and they want to relieve that anxiety for
their students (Henningsen & Stein, 1997).
Questions that support students in moving their thinking forward
should encourage students to provide mathematical
argumentation and make mathematical connections (Kazemi &
Stipeck, 2001).
Implementing Tasks

Teachers were supported in this element of
the MPSM through


Discussion of questioning frameworks to
distinguish different types of questions (Boaler &
Humphrys, 2005; NCTM, 2000)
Observation of video recordings followed by
discussion of the questioning practices of
teachers that successfully maintained the
cognitive demand of problem solving tasks in their
classrooms.
Feedback Guide as Lens

Five problem solving traits :






Conceptual Understanding
Strategies & Reasoning
Communication
Computation & Execution
Insights
These traits are distinct from, but are useful
for supporting, the Standards for
Mathematical Practice (see handout)
Analyzing Student Work

The analysis of student work allows for
teachers to gain insight into how their
students think about and understand
mathematics, allowing teachers to…


better understand how they can support their
students in moving forward their thinking and
shape subsequent instruction (Sowder, 2007;
Franke et al., 2007).
Plan Next Step

Two possible avenues:


Classroom Group “Feedback” (discourse around
the problem solving task)
Follow up on written feedback
Written Feedback



The primary purpose of formative feedback is to
increase the knowledge, skills, or understanding of a
student within a certain content area (Shute, 2008).
An important characteristic of good feedback is that
the nature of feedback provided should be guided
by the teacher’s instructional goals and the
teacher’s knowledge of the student (or students) in
question (Narciss & Huth, 2004).
Research shows that the most learning gains are
associated with feedback in which the student is
given information about correct results, some
explanation, and suggestions for specific activities to
undertake in order to improve (Nyquist, 2003) .
Formative Feedback Guide




Organized by the five traits of problems solving:
Conceptual Understanding, Strategies & Reasoning,
Communication, Computation & Execution, and
Insights.
Designed for teachers to give more detailed
information when providing written feedback
suggests possible avenues for improving the issues
evident in the student’s work
Example of suggested feedback from the guide:

I’m not sure you used __(mathematical term)__ correctly.
Check the definition and see if there is a better word to
use.
Classroom Use of Feedback


Good feedback is not useful when students do not
reflect upon it.
Ways that feedback can be put to use in the
classroom:




Revise and resubmit
peer feedback
student reflection on written feedback
Particularly these last two processes encourage
student autonomy through metacognitive behavior
(Shute, 2008)
Classroom Group “Feedback”: Discourse
around Problem Solving Tasks



Students sharing their problem-solving strategies is a valuable
way for students to build new mathematical thinking (Hiebert,
2003)
Students randomly volunteering to share can lead to limited
opportunities for students to engage in mathematical thinking
around a problem solving task (Chazan & Ball, 2001; Nathan &
Knuth, 2003; Stein, Engle, Smith, & Hughes, 2008)
The MPSM proposes planning for discourse as follows:



Identify which students’ work will be discussed
Determine the order in which the work will be
presented
plan appropriate questions to make the mathematics
salient (Stein et al, 2008; Stein et al, 2009).
Identification of Student Work

Identification of student work for classroom
group “feedback” may be based upon





the mathematical ideas present in the solution
representations used
misconceptions that are evident in the work
sharing a particularly well-written solution
making sure that a student who hasn’t presented
in a while gets an opportunity to share
Sequencing of Student Work

Strategies for sequencing student work include, but
are not limited to




starting with the least sophisticated solutions and
progressively building a sophisticated solution (Groves &
Doig, 2004).
sharing incorrect solutions up front to eliminate
misconceptions
sharing two conceptually similar solutions together to
create opportunities for students to make connections
between the strategies
sequencing in such a way to allow a mathematical lesson
to emerge (Stein et al., 2008).
Planning Questions for Discourse

Questions a teacher might ask include




requests for students to provide justification for the
strategies they used (Hiebert et al., 1997; Kazemi & Stipek,
2001)
questions that lead students to make sense of the
mathematical ideas used to solve the task (Boaler &
Humphreys, 2005; Sherin, 2002)
questions that prompt students to make connections
between strategies (Hiebert & Wearne, 1993; Kazemi &
Stipek, 2001)
questions that encourage students to formulate and prove
conjectures and generalizations around the mathematics in
the task (Fraivillig, Murphy, & Fuson., 1999; Hiebert, &
Wearne, 2003; Yackel & Hanna, 2003).
Questions?
Please feel free to email me any requests for further information (including
a reference list) at [email protected]