Mathematical Tasks, Cognitive Demand and Higher Level Mathematics

Essential questions: 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 mathematics?

Quasar Project University of Pittsburgh Conducted five years of research in urban middle schools 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 communication

“… 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 communication.”

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 answers Following steps in a given order Little doubt or confusion about what is to be done No explanations other than describing the procedure 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 algorithms

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