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Four Fours and Operations Problem • Use four 4s and some symbols +, x, -, ÷,and ( ) to give expressions for the whole numbers from 0 through 9: for example 5 = (4 x 4 + 4) ÷ 4. • Solve the problem. • Sharing solutions
Analyzing the Problem • What mathematics did you use to solve the problem? • When would you use this type of problem? • What related problems are usually found in textbooks? • What makes this task a worthwhile task?
The teacher of mathematics should pose tasks that are based on --- • Sound and significant mathematics; • Knowledge of students’ understandings, interests, and experiences; • Knowledge of the range of ways that diverse students learn mathematics; • And that engage students’ intellect; • Develop students’ mathematical understandings and skills; • Stimulate students to make connections and develop a coherent framework for mathematical ideas;
Cont’d • Call for problem formulation, problem solving, and mathematical reasoning; • Promote communication about mathematics; • Represent mathematics as an ongoing human activity; • Display sensitivity to, and draw on, students’ diverse background experiences and dispositions; • Promote the development of all students’ dispositions to do mathematics. Reference: National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM
How to Determine Worthwhile Tasks • In selecting, adapting, or generating mathematical tasks, teachers must base their decision on three areas of concern: mathematical content, the students, and the ways in which students learn mathematics.
Mathematical Content • Teachers should consider how appropriately the task represents the concepts and procedures entailed. • Teachers must also use a curricular perspective, considering the potential of the task to help students progress in their cumulative understanding of a particular domain and to make connections among ideas they have studied in the past and those they will encounter in the future. • Teachers must also assess what the task conveys about what is entailed in doing mathematics. • Teachers must also consider how well a task helps in the development of appropriate skill and automaticity.
Students • Teachers must consider what they know about students in deciding on the appropriateness of a given task. • Teachers must consider what they know about students from psychological, cultural, sociological, and political perspectives. • When selecting tasks, teachers must think about what their students already know and can do, what they need to work on, and how much they seem ready to stretch intellectually. • Teachers must know their students interests, dispositions, and experiences.
Knowledge About Ways In Which Students Learn Mathematics • The mode of activity, the kind of thinking required, and the way in which students are led to explore the particular content all contribute to the kind of learning opportunity afforded by a task. • Teachers must be aware of common misconceptions about mathematical concepts. • Teachers should deliberately select tasks that provide them with windows into students’ thinking. Reference: National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM
Problem Solving Situation based on 1 Hunter by Pat Hutchins A hunter walks past two elephants, three giraffes, four ostriches, …, and ten parrots. How many animals were there including the hunter?
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 3 7 11 15 19 10 10 10 26 10 55 animals 36 10 10+10+10+10+10+5 = 55 animals 54 animals (student miscounted) Resourceful and Flexible Problem Solving 12
Pockets in Our Clothing Keith Lynda Mark Nicki Octavio Paula Quinton Robert Sam Wendy Victor Yolanda Anthony Barbara Christine Donald Eleanor Fred Gertrude Hannah Ian 1 2 3 4 5 6 7 8 9 10 Number of Pockets Kindergarten Students– Number of Pockets
X X X Number of Students X X X X X X X X X X X X X X X X X X X X X X X X X 1 2 3 4 5 6 7 8 9 10 How Many Pockets Number of Pockets
Find several ways to determine the number of dots on the boundary of the square and then represent your solutions as equations.
4 x 8 + 4 = 36 4 x 10 – 4 = 36 10 + 8 + 10 + 8 = 36 Equivalence 16
B A Geometric Reasoning Starting with two identical rectangular regions, cut each of the two rectangles in half as shown. Is region A equal in size to region B. Explain your answer. Adapted from Tierney and Berle-Carman (1998, p.10)
Difference Frequency Flexible and Resourceful Problem Solvers Roll two number cubes and subtract the smaller number from the larger number. Is one particular difference more likely than any other differences? If you do this twenty times, what will be the results?
Tasks that foster skill development even as students engage in problem solving and reasoning • Rolling pairs of dice as part of an investigation of probability can simultaneously provide students with practice in addition. • Trying to figure out how many ways 36 desks can be arranged in equal-sized groups--and whether there are more or fewer possible groupings with 36, 37, 38, 39, or 40 desks--presses students to produce each number’s factors quickly.
Where can we find worthwhile tasks? • Good ideas can be found in articles in the Arithmetic Teacher, Mathematics Teaching in the Middle School,Mathematics Teacher, Teaching Children Mathematics in the PSSM, the Addenda Series, and Navigations. • Change from products to explanations • Use models • Collect and use a variety of contexts