Mathematical Tasks and Higher Level Mathematics

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Transcript Mathematical Tasks and Higher Level Mathematics

Mathematical Tasks, Cognitive
Demand and Higher Level
Essential questions:
1. What do we mean by higher level
mathematics in middle school?
2. How can teachers ensure that middle school
English Language Learners have maximum
opportunities to learn high level
Quasar Project
University of Pittsburgh
• Conducted five years of research in urban middle
• Key finding: mathematically-rich tasks,
implemented appropriately in classrooms is the
key to success for urban students
• Developed a teaching approach to ensure that all
students are engaged in problem solving,
mathematical thinking, mathematical
“… students who performed best … were
in classrooms in which tasks were … set
up and implemented at high levels of
cognitive demand … for these students,
having the opportunity to work on
challenging tasks in a supportive
classroom environment translated into
substantial learning gains on an
instrument designed to measure student
thinking, reasoning, problem solving and
Consider two tasks with the same
underlying math content:
Task 1. Find the surface area and volume of a rectangular
prism that measures 2” x 4” x 24”.
Task 2. The EZ-Play Toy Company makes a set of 24 alphabet
blocks. Each block is a 2” cube. EZ-Play plans to ship the
blocks in a box that uses the smallest amount of cardboard.
What are the dimensions of the box that EZ-Play should use?
Organize your work so that anyone who reads it will understand
what you have done.
Cognitive Demands of Task 1
• Students know how to compute volume and
surface area of a rectangular prism.
• Teacher will help students remember which
formula to use.
• Task provides practice with arithmetic
(addition and multiplication)
• Task at low level of cognitive demand:
procedures without connections
Cognitive demands of Task 2
• Students decide what is needed to solve the problem and
make their own plans for solving it.
• Students determine volume of 24 blocks and generate
several different rectangular prisms with that volume.
• Students compute and compare surface areas of several
rectangular prisms, finding the prism with the smallest
surface area.
• Students explain their thinking clearly.
• Students have opportunity to practice arithmetic
• Task is at a high level of cognitive demand: procedures
with connections.
Levels of Cognitive demand
Lowest to Highest
Memorization tasks
Procedures without Connections
Procedures with connections
Doing Mathematics
Memorization tasks require:
• Memorizing definitions, facts and formulas
• Using memorized definitions, facts and
formulas, exactly as learned
• Unambiguous correct answers
• No connection to underlying concepts
Procedures Without Connection
tasks require:
• Using known algorithms to produce correct
• Following steps in a given order
• Little doubt or confusion about what is to be
• No explanations other than describing the
• No connection to underlying concepts
Procedures with Connections
tasks require:
• Using procedures to help illustrate or understand
underlying concepts
• Making connections among multiple
representations (diagrams manipulatives, symbols,
formulas, problem statements)
• Drawing on conceptual understanding needed to
solve problem
• Following broad procedures rather than narrow
Doing mathematics tasks require:
• Analyzing a task and constraints
• Making a plan without knowing all steps or
procedures ahead of time
• Accessing relevant knowledge and exploring to
deepen understanding of mathematical concepts
• Self-monitoring of cognitive processes
• Using multiple representations to solve problems
and explain thinking