Moving Beyond Algorithm Through Problem Solving - Ncs
MOVING BEYOND ALGORITHMTHROUGH PROBLEM SOLVINGWang Dichen, DaisyThe University of Hong Kong
OBJECTIVES Recognise problem solving as a tool for learningmaths and as a goal of learning in itself Caring diversity in problem solving activities Analyse problem-solving activities
THREE PERSPECTIVES Problem solving as a goal: Learn about how to problemsolve. Problem solving as a process: Extend and learn mathconcepts through solving selected problems. Problem solving as a tool for applications and modelling:Apply math to real-world or word problems, and usemathematics to model the situations in these problems.
THE PROBLEMS Contextual problems offering opportunities forstudents to develop informal solution strategies, andare used to support mathematical concept building The context may even be rather unrealistic or withinmathematics, if concept development requires it The contextual problem must be experienced as areal problem by the students(Doorman, Michiel, Drijvers. et al, 2007)
Good problem solving activities provide an entrypoint that allows all students to be working on thesame problem.
Suppose 39 students want to share 5 candy bars fairly.How much can each student get?Leo: That’s 5 divided by 39, and we decided last year that you can’t dividea bigger number into a smaller number.Anthony: I think that 39 5 will be 7 remainder 4, but I think that 5 39will make a decimal number.Jackson: I think that you will end up with a fraction of a number because,well, because 5 and 39—you can’t divide 5 by 39 equally. I think it’s goingto be a number below 0.After some further discussion about which notation (39 5 or 5 39)actually represents the situation in this problem and what sorts ofnumbers might be possible answers (e.g., fractions, decimals, remainders,“smaller numbers”)This scenario is adapted from Benefits of Teaching through Problem Solving (Diana V. Lambdin, 2003, pp.3-5)
Suppose 39 students want to share 5 candy bars fairly.How much can each student get?Mitchell: So if each kid was going to get equal shares, they would have tocut the five candy bars into little equal pieces.Teacher (MaryAnn): Can you name those equal pieces?Mitchell: They might be candy bars.Teacher: Can you name the fraction that they might be?Teacher: How many people think that you can do the problem 5 39?How many think no, you can’t?The results are yes, 13; no, 15.This scenario is adapted from Benefits of Teaching through Problem Solving (Diana V. Lambdin, 2003, pp.3-5)
After a pause, Leo says that he wants to change his no to a yes.Cynthia quickly responds that Leo’s representation cannot be correctbecause it does not yield equal shares. “That’s a problem,” she says.Laila: If I cut each of the five candy bars into thirty-nine pieces and thengive each kid one piece from each candy bar, you could have each kidhave five-thirty-ninths of a candy bar.After further discussion, most of the class seems convinced that Laila hasproposed a valid solution to the problemThis scenario is adapted from Benefits of Teaching through Problem Solving (Diana V. Lambdin, 2003, pp.3-5)
BENEFITS OF TEACHING THROUGHPROBLEM SOLVING Opportunities for exploring, discussing, experimenting with, andattempting to make sense of mathematical ideas Confident feeling that ideas make sense Promotes understanding Helps memory Enhances Transfer Become autonomous learners(Diana V. Lambdin, 2003)
STORY: AREA OF POLYGON A classroom with NCS students Promoting problem solving activities
LEARNING TRAJECTORY:AREA OF POLYGON
AREA OF TRIANGLE(First version)
AREA OFTRIANGLE(Second version)
AREA OFTRIANGLE(Second version)
AREA OFPARALLELOGRAM
AREA OF PARALLELOGRAM
AREA OFTRAPEZIUM
AREA OF TRAPEZIUM
Hints card
Opportunities for low-achievers
REDUCE UNKNOWN TOKNOWNGrade 3: Multiplication
EXHAUSTIVE LISTING OFFACTORSGrade 4: Factors and Multiples
VOLUME OF CUBOID
CAKE DISSECTION
DISCOVERING VOLUMEFORMULA
How to find all the possiblenets of cubes?
情況 ：位於同側：情況 ：位於兩側：
BRAINSTORM Think about your lesson plans for next month. Pick thelesson you see as important and design a problembased task for your students. Any task or activity which students have no prescribedrules or memorized procedures that they can use tosolve it Need not be complex or elaborate
Wang Dichen, Daisydcwang@hku.hk
REFERENCE Doorman, Michiel, Drijvers, Paul, Dekker, Truus, Heuvel-Panhuizen, Marja, Lange, Jan, &Wijers, Monica. (2007). Problem solving as a challenge for mathematics education in TheNetherlands. ZDM, 39(5-6), 405-418. Lester, F., & National Council of Teachers of Mathematics. (2003). Teaching mathematicsthrough problem solving : Prekindergarten-grade 6. Reston, Va.: National Council ofTeachers of Mathematics. ��嗎？《數學教育》22期，25－30。 96）。香港大學教育學院。
THREE PERSPECTIVES Problem solving as a goal: Learn about how to problem solve. Problem solving as a process: Extend and learn math concepts through solving selected problems. Problem solving as a tool for applications and modelling: Apply math to real-world or word problems, and use mathematics to model the situations in these problems.
Dual moving averages are moving averages of moving averages, and according to symbols are written as MA (k k), which means moving averages as much as k periods of moving averages as much as k periods [10]. The steps used in calculating a double moving average are as follows: 1. Calculates the first moving average Mt Yt Yt-1 Yt-2 n (1) 2.
Algorithms and Data Structures Marcin Sydow Desired Properties of a Good Algorithm Any good algorithm should satisfy 2 obvious conditions: 1 compute correct (desired) output (for the given problem) 2 be e ective ( fast ) ad. 1) correctness of algorithm ad. 2)complexity of algorithm Complexity of algorithm measures how fast is the algorithm
Algorithm 19 - Timer Functions 125 Algorithm 20 - Totalization 129 Algorithm 21 - Comparator 133 Algorithm 22 - Sequencer 136 Algorithm 23 - Four Channel Line Segment (Version 1.1 or Later) 152 Algorithm 24 - Eight Channel Calculator (Version 1.1 or Lat
table of contents 1.0 introduction 3 2.0 overview 7 3.0 algorithm description 8 3.1 theoretical description 8 3.1.1 physical basis of the cloud top pressure/temperature/height algorithm 9 3.1.2 physical basis of infrared cloud phase algorithm 14 3.1.3 mathematical application of cloud top pressure/temperature/height algorithm 17 3.1.4 mathematical application of the cloud phase algorithm 24
Customize new algorithm (Check Script Algorithm) To program your own algorithm using MBP script, select Script Algorithm. The new algorithm will be independent of MBP default Java Algorithm. For more details, refer to the MBP Script Programming Manual and contact Agilent support team (mbp_pdl-eesof@agilent.com).
Lecture 18 Algorithms Solving the Problem Dijkstra’s algorithm Solves only the problems with nonnegative costs, i.e., c ij 0 for all (i,j) E Bellman-Ford algorithm Applicable to problems with arbitrary costs Floyd-Warshall algorithm Applicable to problems with arbitrary costs Solves a more gene
learn to solve problem through hands-on learning experience [16]. Problem solving is also defined as learning that uses the problems of daily activities and problem situations that are simulated as a context for learning mathematics [17, 18]. The problem in problem solving can also be non-routine problem. Non routine-problem is problem where
Introduction 1 Part I Ancient Greek Criticism 7 Classical Literary Criticism: Intellectual and Political Backgrounds 9 1 Plato (428–ca. 347 bc)19 2 Aristotle (384–322 bc)41 Part II The Traditions of Rhetoric 63 3 Greek Rhetoric 65 Protagoras, Gorgias, Antiphon, Lysias, Isocrates, Plato, Aristotle 4 The Hellenistic Period and Roman Rhetoric 80 Rhetorica, Cicero, Quintilian Part III Greek .
